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Polyanin-Manshirov Handbook_of_Integral_Equations 2e

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Published reference book by Andrei D. Polyanin and Alexander V. Manzhirov (Chapman & Hall/CRC, 2008), kept in the archive as a downloaded PDF of someone else's work. Part I catalogs exact solutions of linear integral equations of the first and second kind with variable limit of integration, organized by kernel type: power-law, exponential, hyperbolic, logarithmic, trigonometric and special functions. The contents shown suggest later parts on methods, but only the front matter and early table of contents were seen.

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Andrei D.Polyanin Alexander V.Manzhirov ivs rlANDBOOK OF INTEGRAL rc "I (acyEQUATIONS SECOND EDITION odSosemen&Haven HANDBOOK OF SECOND EDITIONINTEGRAL EQUATIONS Handbooks of Mathematical Equations Handbook of Linear Partial Differential Equations for Engineers and Scientists A. D. Polyanin, 2002 Handbook of First Order Partial Differential Equations A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, 2002 Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition A. D. Polyanin and V. F. Zaitsev, 2003 Handbook of Nonlinear Partial Differential Equations A. D. Polyanin and V. F. Zaitsev, 2004 Handbook of Integral Equations, 2nd Edition A. D. Polyanin and A. V. Manzhirov, 2008 See also: Handbook of Mathematics for Engineers and Scientists A. D. Polyanin and A. V. Manzhirov, 2007 HANDBOOK OF SECOND EDITIONINTEGRAL EQUATIONS Andrei D. Polyanin Alexander V. Manzhirov Chapman & Hall/CRC Taylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-507-8 (Hardcover)This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy-ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For orga-nizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Polianin, A. D. (Andrei Dmitrievich) Handbook of integral equations / Andrei D. Polyanin and Alexander V. Manzhirov. -- 2nd ed. p. cm. Includes bibliographical references and index.ISBN-13: 978-1-58488-507-8 (hardcover : alk. paper)ISBN-10: 1-58488-507-6 (hardcover : alk. paper)1. Integral equations--Handbooks, manuals, etc. I. Manzhirov, A. V. (Aleksandr Vladimirovich) II. Title. QA431.P65 2008 515’.45--dc22 2007035725 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com CONTENTS Authors ................................................................... xxix Preface .................................................................... xxxi Some Remarks and Notation ................................................. xxxiii Part I. Exact Solutions of Integral Equations 1. Linear Equations of the First Kind with Variable Limit of Integration ............ 3 1.1. Equations Whose Kernels Contain Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1-1. Kernels Linear in the Arguments xandt................................ 4 1.1-2. Kernels Quadratic in the Arguments xandt............................. 4 1.1-3. Kernels Cubic in the Arguments xandt................................ 5 1.1-4. Kernels Containing Higher-Order Polynomials in xandt.................. 6 1 . 1 - 5 . K e r n e l sC o n t a i n i n gR a t i o n a lF u n c t i o n s ................................. 7 1.1-6. Kernels Containing Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 . 1 - 7 . K e r n e l sC o n t a i n i n gA r b i t r a r yP o w e r s .................................. 1 2 1.1-8. Two-Dimensional Equation of the Abel Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2. Equations Whose Kernels Contain Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2-1. Kernels Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2-2. Kernels Containing Power-Law and Exponential Functions . . . . . . . . . . . . . . . . . 19 1.3. Equations Whose Kernels Contain Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 22 1 . 3 - 1 . K e r n e l sC o n t a i n i n gH y p e r b o l i cC o s i n e ................................. 2 2 1 . 3 - 2 . K e r n e l sC o n t a i n i n gH y p e r b o l i cS i n e ................................... 2 8 1.3-3. Kernels Containing Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.3-4. Kernels Containing Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.3-5. Kernels Containing Combinations of Hyperbolic Functions . . . . . . . . . . . . . . . . . 39 1.4. Equations Whose Kernels Contain Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 42 1.4-1. Kernels Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.4-2. Kernels Containing Power-Law and Logarithmic Functions . . . . . . . . . . . . . . . . . 45 1.5. Equations Whose Kernels Contain Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 46 1 . 5 - 1 . K e r n e l sC o n t a i n i n gC o s i n e .......................................... 4 6 1 . 5 - 2 . K e r n e l sC o n t a i n i n gS i n e ............................................ 5 2 1 . 5 - 3 . K e r n e l sC o n t a i n i n gT a n g e n t .......................................... 6 0 1.5-4. Kernels Containing Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621.5-5. Kernels Containing Combinations of Trigonometric Functions . . . . . . . . . . . . . . 63 1.6. Equations Whose Kernels Contain Inverse Trigonometric Functions . . . . . . . . . . . . . . . . 66 1 . 6 - 1 . K e r n e l sC o n t a i n i n gA r c c o s i n e ........................................ 6 6 1 . 6 - 2 . K e r n e l sC o n t a i n i n gA r c s i n e .......................................... 6 8 1 . 6 - 3 . K e r n e l sC o n t a i n i n gA r c t a n g e n t ....................................... 7 0 1 . 6 - 4 . K e r n e l sC o n t a i n i n gA r c c o t a n g e n t ..................................... 7 1 v vi CONTENTS 1.7. Equations Whose Kernels Contain Combinations of Elementary Functions . . . . . . . . . . 73 1.7-1. Kernels Containing Exponential and Hyperbolic Functions . . . . . . . . . . . . . . . . . 73 1.7-2. Kernels Containing Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 771.7-3. Kernels Containing Exponential and Trigonometric Functions . . . . . . . . . . . . . . . 781.7-4. Kernels Containing Hyperbolic and Logarithmic Functions . . . . . . . . . . . . . . . . . 83 1.7-5. Kernels Containing Hyperbolic and Trigonometric Functions . . . . . . . . . . . . . . . 84 1.7-6. Kernels Containing Logarithmic and Trigonometric Functions . . . . . . . . . . . . . . 85 1.8. Equations Whose Kernels Contain Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 1.8-1. Kernels Containing Error Function or Exponential Integral . . . . . . . . . . . . . . . . . 861 . 8 - 2 . K e r n e l sC o n t a i n i n gS i n ea n dC o s i n eI n t e g r a l s ........................... 8 71 . 8 - 3 . K e r n e l sC o n t a i n i n gF r e s n e lI n t e g r a l s................................... 8 7 1.8-4. Kernels Containing Incomplete Gamma Functions . . . . . . . . . . . . . . . . . . . . . . . . 88 1.8-5. Kernels Containing Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881.8-6. Kernels Containing Modi ﬁe dB e s s e lF u n c t i o n s .......................... 9 7 1.8-7. Kernels Containing Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 1.8-8. Kernels Containing Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . 107 1.8-9. Kernels Containing Con ﬂu e n tH y p e r g e o m e t r i cF u n c t i o n s .................. 1 0 7 1.8-10. Kernels Containing Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081.8-11. Kernels Containing Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 1.8-12. Kernels Containing Laguerre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 1.8-13. Kernels Containing Jacobi Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101.8-14. Kernels Containing Other Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 1.9. Equations Whose Kernels Contain Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 111 1.9-1. Equations with Degenerate Kernel: K(x,t)=g 1(x)h1(t)+g2(x)h2(t) ......... 1 1 1 1.9-2. Equations with Difference Kernel: K(x,t)=K(x–t) ..................... 1 1 4 1 . 9 - 3 . O t h e rE q u a t i o n s ................................................... 1 2 2 1 . 1 0 . S o m eF o r m u l a sa n dT r a n s f o r m a t i o n s ....................................... 1 2 4 2. Linear Equations of the Second Kind with Variable Limit of Integration .......... 127 2.1. Equations Whose Kernels Contain Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . 127 2.1-1. Kernels Linear in the Arguments xandt................................ 1 2 7 2.1-2. Kernels Quadratic in the Arguments xandt............................. 1 2 9 2.1-3. Kernels Cubic in the Arguments xandt................................ 1 3 2 2.1-4. Kernels Containing Higher-Order Polynomials in xandt.................. 1 3 3 2 . 1 - 5 . K e r n e l sC o n t a i n i n gR a t i o n a lF u n c t i o n s ................................. 1 3 62.1-6. Kernels Containing Square Roots and Fractional Powers . . . . . . . . . . . . . . . . . . . 138 2 . 1 - 7 . K e r n e l sC o n t a i n i n gA r b i t r a r yP o w e r s .................................. 1 3 9 2.2. Equations Whose Kernels Contain Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 144 2.2-1. Kernels Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.2-2. Kernels Containing Power-Law and Exponential Functions . . . . . . . . . . . . . . . . . 151 2.3. Equations Whose Kernels Contain Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 154 2 . 3 - 1 . K e r n e l sC o n t a i n i n gH y p e r b o l i cC o s i n e ................................. 1 5 42 . 3 - 2 . K e r n e l sC o n t a i n i n gH y p e r b o l i cS i n e ................................... 1 5 62.3-3. Kernels Containing Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 2.3-4. Kernels Containing Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 2.3-5. Kernels Containing Combinations of Hyperbolic Functions . . . . . . . . . . . . . . . . . 164 2.4. Equations Whose Kernels Contain Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 164 2.4-1. Kernels Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 2.4-2. Kernels Containing Power-Law and Logarithmic Functions . . . . . . . . . . . . . . . . . 165 CONTENTS vii 2.5. Equations Whose Kernels Contain Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 166 2 . 5 - 1 . K e r n e l sC o n t a i n i n gC o s i n e .......................................... 1 6 6 2 . 5 - 2 . K e r n e l sC o n t a i n i n gS i n e ............................................ 1 6 9 2 . 5 - 3 . K e r n e l sC o n t a i n i n gT a n g e n t .......................................... 1 7 4 2.5-4. Kernels Containing Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1752.5-5. Kernels Containing Combinations of Trigonometric Functions . . . . . . . . . . . . . . 176 2.6. Equations Whose Kernels Contain Inverse Trigonometric Functions . . . . . . . . . . . . . . . . 176 2 . 6 - 1 . K e r n e l sC o n t a i n i n gA r c c o s i n e ........................................ 1 7 6 2 . 6 - 2 . K e r n e l sC o n t a i n i n gA r c s i n e .......................................... 1 7 7 2 . 6 - 3 . K e r n e l sC o n t a i n i n gA r c t a n g e n t ....................................... 1 7 8 2 . 6 - 4 . K e r n e l sC o n t a i n i n gA r c c o t a n g e n t ..................................... 1 7 8 2.7. Equations Whose Kernels Contain Combinations of Elementary Functions . . . . . . . . . . 179 2.7-1. Kernels Containing Exponential and Hyperbolic Functions . . . . . . . . . . . . . . . . . 179 2.7-2. Kernels Containing Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 180 2.7-3. Kernels Containing Exponential and Trigonometric Functions . . . . . . . . . . . . . . . 181 2.7-4. Kernels Containing Hyperbolic and Logarithmic Functions . . . . . . . . . . . . . . . . . 185 2.7-5. Kernels Containing Hyperbolic and Trigonometric Functions . . . . . . . . . . . . . . . 186 2.7-6. Kernels Containing Logarithmic and Trigonometric Functions . . . . . . . . . . . . . . 187 2.8. Equations Whose Kernels Contain Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 2.8-1. Kernels Containing Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 2.8-2. Kernels Containing Modi ﬁe dB e s s e lF u n c t i o n s .......................... 1 8 9 2.9. Equations Whose Kernels Contain Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2.9-1. Equations with Degenerate Kernel: K(x,t)=g 1(x)h1(t)+···+gn(x)hn(t) . . . . 191 2.9-2. Equations with Difference Kernel: K(x,t)=K(x–t) ..................... 2 0 3 2 . 9 - 3 . O t h e rE q u a t i o n s ................................................... 2 1 2 2 . 1 0 . S o m eF o r m u l a sa n dT r a n s f o r m a t i o n s ....................................... 2 1 5 3. Linear Equations of the First Kind with Constant Limits of Integration ........... 217 3.1. Equations Whose Kernels Contain Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . 217 3.1-1. Kernels Linear in the Arguments xandt................................ 2 1 7 3.1-2. Kernels Quadratic in the Arguments xandt............................. 2 1 9 3.1-3. Kernels Containing Integer Powers of xandtor Rational Functions . . . . . . . . . . 220 3.1-4. Kernels Containing Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 3 . 1 - 5 . K e r n e l sC o n t a i n i n gA r b i t r a r yP o w e r s .................................. 2 2 3 3.1-6. Equations Containing the Unknown Function of a Complicated Argument . . . . . 227 3.1-7. Singular Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 3.2. Equations Whose Kernels Contain Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 231 3.2-1. Kernels Containing Exponential Functions of the Form eλ|x–t|............... 2 3 1 3.2-2. Kernels Containing Exponential Functions of the Forms eλxandeµt......... 2 3 4 3.2-3. Kernels Containing Exponential Functions of the Form eλxt................ 2 3 4 3.2-4. Kernels Containing Power-Law and Exponential Functions . . . . . . . . . . . . . . . . . 236 3.2-5. Kernels Containing Exponential Functions of the Form eλ(x±t)2............. 2 3 6 3 . 2 - 6 . O t h e rK e r n e l s ..................................................... 2 3 7 3.3. Equations Whose Kernels Contain Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 238 3 . 3 - 1 . K e r n e l sC o n t a i n i n gH y p e r b o l i cC o s i n e ................................. 2 3 83 . 3 - 2 . K e r n e l sC o n t a i n i n gH y p e r b o l i cS i n e ................................... 2 3 8 3.3-3. Kernels Containing Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 3.3-4. Kernels Containing Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 viii CONTENTS 3.4. Equations Whose Kernels Contain Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 242 3.4-1. Kernels Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2423.4-2. Kernels Containing Power-Law and Logarithmic Functions . . . . . . . . . . . . . . . . . 244 3.4-3. Equation Containing the Unknown Function of a Complicated Argument . . . . . . 246 3.5. Equations Whose Kernels Contain Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 246 3 . 5 - 1 . K e r n e l sC o n t a i n i n gC o s i n e .......................................... 2 4 6 3 . 5 - 2 . K e r n e l sC o n t a i n i n gS i n e ............................................ 2 4 7 3 . 5 - 3 . K e r n e l sC o n t a i n i n gT a n g e n t .......................................... 2 5 13.5-4. Kernels Containing Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 3.5-5. Kernels Containing a Combination of Trigonometric Functions . . . . . . . . . . . . . . 252 3.5-6. Equations Containing the Unknown Function of a Complicated Argument . . . . . 2543.5-7. Singular Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 3.6. Equations Whose Kernels Contain Combinations of Elementary Functions . . . . . . . . . . 255 3.6-1. Kernels Containing Hyperbolic and Logarithmic Functions . . . . . . . . . . . . . . . . . 2553.6-2. Kernels Containing Logarithmic and Trigonometric Functions . . . . . . . . . . . . . . 256 3.6-3. Kernels Containing Combinations of Exponential and Other Elementary F u n c t i o n s ........................................................ 2 5 7 3.7. Equations Whose Kernels Contain Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 3.7-1. Kernels Containing Error Function, Exponential Integral or Logarithmic Integral 258 3.7-2. Kernels Containing Sine Integrals, Cosine Integrals, or Fresnel Integrals . . . . . . 2583 . 7 - 3 . K e r n e l sC o n t a i n i n gG a m m aF u n c t i o n s ................................. 2 6 0 3.7-4. Kernels Containing Incomplete Gamma Functions . . . . . . . . . . . . . . . . . . . . . . . . 260 3.7-5. Kernels Containing Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . 2613.7-6. Kernels Containing Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . 264 3.7-7. Kernels Containing Combinations of the Bessel Functions . . . . . . . . . . . . . . . . . 265 3.7-8. Kernels Containing Modi ﬁed Bessel Functions of the First Kind . . . . . . . . . . . . . 266 3.7-9. Kernels Containing Modi ﬁed Bessel Functions of the Second Kind . . . . . . . . . . 266 3.7-10. Kernels Containing a Combination of Bessel and Modi ﬁed Bessel Functions . . 269 3 . 7 - 1 1 . K e r n e l sC o n t a i n i n gL e g e n d r eF u n c t i o n s ............................... 2 7 03.7-12. Kernels Containing Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . 271 3.7-13. Kernels Containing Kummer Conﬂ uent Hypergeometric Functions . . . . . . . . . . 272 3.7-14. Kernels Containing Tricomi Con ﬂuent Hypergeometric Functions . . . . . . . . . . 274 3.7-15. Kernels Containing Whittaker Con ﬂuent Hypergeometric Functions . . . . . . . . . 274 3.7-16. Kernels Containing Gauss Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . 276 3.7-17. Kernels Containing Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . 2763.7-18. Kernels Containing Other Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 3.8. Equations Whose Kernels Contain Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 278 3 . 8 - 1 . E q u a t i o n sw i t hD e g e n e r a t eK e r n e l .................................... 2 7 83.8-2. Equations Containing Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 3.8-3. Equations with Difference Kernel: K(x,t)=K(x–t) ..................... 2 8 4 3.8-4. Other Equations of the Form⎝integraltext b aK(x,t)y(t)dt=F(x) ..................... 2 8 5 3.8-5. Equations of the Form⎝integraltextb aK(x,t)y(···)dt=F(x) ........................ 2 8 9 3 . 9 . D u a lI n t e g r a lE q u a t i o n so ft h eF i r s tK i n d ..................................... 2 9 5 3.9-1. Kernels Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 3.9-2. Kernels Containing Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . 2973.9-3. Kernels Containing Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . 299 3.9-4. Kernels Containing Legendre Spherical Functions of the First Kind, i 2= –1 . . . 299 CONTENTS ix 4. Linear Equations of the Second Kind with Constant Limits of Integration ......... 301 4.1. Equations Whose Kernels Contain Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . 301 4.1-1. Kernels Linear in the Arguments xandt................................ 3 0 1 4.1-2. Kernels Quadratic in the Arguments xandt............................. 3 0 4 4.1-3. Kernels Cubic in the Arguments xandt................................ 3 0 7 4.1-4. Kernels Containing Higher-Order Polynomials in xandt.................. 3 1 1 4 . 1 - 5 . K e r n e l sC o n t a i n i n gR a t i o n a lF u n c t i o n s ................................. 3 1 4 4 . 1 - 6 . K e r n e l sC o n t a i n i n gA r b i t r a r yP o w e r s .................................. 3 1 7 4.1-7. Singular Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 4.2. Equations Whose Kernels Contain Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 320 4.2-1. Kernels Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3204.2-2. Kernels Containing Power-Law and Exponential Functions . . . . . . . . . . . . . . . . . 326 4.3. Equations Whose Kernels Contain Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 327 4 . 3 - 1 . K e r n e l sC o n t a i n i n gH y p e r b o l i cC o s i n e ................................. 3 2 7 4 . 3 - 2 . K e r n e l sC o n t a i n i n gH y p e r b o l i cS i n e ................................... 3 2 9 4.3-3. Kernels Containing Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3324.3-4. Kernels Containing Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 4.3-5. Kernels Containing Combination of Hyperbolic Functions . . . . . . . . . . . . . . . . . 334 4.4. Equations Whose Kernels Contain Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 334 4.4-1. Kernels Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 4.4-2. Kernels Containing Power-Law and Logarithmic Functions . . . . . . . . . . . . . . . . . 335 4.5. Equations Whose Kernels Contain Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 335 4 . 5 - 1 . K e r n e l sC o n t a i n i n gC o s i n e .......................................... 3 3 5 4 . 5 - 2 . K e r n e l sC o n t a i n i n gS i n e ............................................ 3 3 7 4 . 5 - 3 . K e r n e l sC o n t a i n i n gT a n g e n t .......................................... 3 4 24.5-4. Kernels Containing Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 4.5-5. Kernels Containing Combinations of Trigonometric Functions . . . . . . . . . . . . . . 344 4.5-6. Singular Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 4.6. Equations Whose Kernels Contain Inverse Trigonometric Functions . . . . . . . . . . . . . . . . 344 4 . 6 - 1 . K e r n e l sC o n t a i n i n gA r c c o s i n e ........................................ 3 4 44 . 6 - 2 . K e r n e l sC o n t a i n i n gA r c s i n e .......................................... 3 4 5 4 . 6 - 3 . K e r n e l sC o n t a i n i n gA r c t a n g e n t ....................................... 3 4 6 4 . 6 - 4 . K e r n e l sC o n t a i n i n gA r c c o t a n g e n t ..................................... 3 4 7 4.7. Equations Whose Kernels Contain Combinations of Elementary Functions . . . . . . . . . . 348 4.7-1. Kernels Containing Exponential and Hyperbolic Functions . . . . . . . . . . . . . . . . . 3484.7-2. Kernels Containing Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 349 4.7-3. Kernels Containing Exponential and Trigonometric Functions . . . . . . . . . . . . . . . 349 4.7-4. Kernels Containing Hyperbolic and Logarithmic Functions . . . . . . . . . . . . . . . . . 351 4.7-5. Kernels Containing Hyperbolic and Trigonometric Functions . . . . . . . . . . . . . . . 352 4.7-6. Kernels Containing Logarithmic and Trigonometric Functions . . . . . . . . . . . . . . 353 4.8. Equations Whose Kernels Contain Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 4.8-1. Kernels Containing Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3534.8-2. Kernels Containing Modi ﬁe dB e s s e lF u n c t i o n s .......................... 3 5 5 4.9. Equations Whose Kernels Contain Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 357 4.9-1. Equations with Degenerate Kernel: K(x,t)=g 1(x)h1(t)+···+gn(x)hn(t) . . . . 357 4.9-2. Equations with Difference Kernel: K(x,t)=K(x–t) ..................... 3 7 2 4.9-3. Other Equations of the Form y(x)+⎝integraltextb aK(x,t)y(t)dt=F(x) ............... 3 7 4 4.9-4. Equations of the Form y(x)+⎝integraltextb aK(x,t)y(···)dt=F(x) ................... 3 8 1 4 . 1 0 . S o m eF o r m u l a sa n dT r a n s f o r m a t i o n s ....................................... 3 9 0 x CONTENTS 5. Nonlinear Equations of the First Kind with Variable Limit of Integration ......... 393 5.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters . . . . . . . . . . . 393 5.1-1. Equations of the Form⎝integraltextx 0y(t)y(x–t)dt=f(x) .......................... 3 9 3 5.1-2. Equations of the Form⎝integraltextx 0K(x,t)y(t)y(x–t)dt=f(x) .................... 3 9 5 5.1-3. Equations of the Form⎝integraltextx 0y(t)y(···)dt=f(x) ........................... 3 9 6 5.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions . . . . . . . . . . . . 397 5.2-1. Equations of the Form⎝integraltextx aK(x,t)[Ay (t)+By2(t)]dt=f(x) ................ 3 9 7 5.2-2. Equations of the Form⎝integraltextx aK(x,t)y(t)y(ax+bt)dt=f(x) .................. 3 9 8 5 . 3 . E q u a t i o n sw i t hN o n l i n e a r i t yo fG e n e r a lF o r m.................................. 3 9 9 5.3-1. Equations of the Form⎝integraltextx aK(x,t)f(t,y(t))dt=g(x) ...................... 3 9 9 5 . 3 - 2 . O t h e rE q u a t i o n s ................................................... 4 0 1 6. Nonlinear Equations of the Second Kind with Variable Limit of Integration ....... 403 6.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters . . . . . . . . . . . 403 6.1-1. Equations of the Form y(x)+⎝integraltextx aK(x,t)y2(t)dt=F(x) ................... 4 0 3 6.1-2. Equations of the Form y(x)+⎝integraltextx aK(x,t)y(t)y(x–t)dt=F(x) .............. 4 0 6 6.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions . . . . . . . . . . . . 406 6.2-1. Equations of the Form y(x)+⎝integraltextx aK(x,t)y2(t)dt=F(x) ................... 4 0 6 6 . 2 - 2 . O t h e rE q u a t i o n s ................................................... 4 0 7 6 . 3 . E q u a t i o n sw i t hP o w e r - L a wN o n l i n e a r i t y ...................................... 4 0 8 6.3-1. Equations Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 6.3-2. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 6.4. Equations with Exponential Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 6.4-1. Equations Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.4-2. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 6.5. Equations with Hyperbolic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 6.5-1. Integrands with Nonlinearity of the Form cosh[ βy(t) ] ..................... 4 1 4 6.5-2. Integrands with Nonlinearity of the Form sinh[ βy(t) ] ..................... 4 1 5 6.5-3. Integrands with Nonlinearity of the Form tanh[ βy(t) ] ..................... 4 1 6 6.5-4. Integrands with Nonlinearity of the Form coth[ βy(t) ] ..................... 4 1 8 6 . 6 . E q u a t i o n sw i t hL o g a r i t h m i cN o n l i n e a r i t y ..................................... 4 1 9 6.6-1. Integrands Containing Power-Law Functions of xandt.................... 4 1 9 6.6-2. Integrands Containing Exponential Functions of xandt................... 4 1 9 6 . 6 - 3 . O t h e rI n t e g r a n d s................................................... 4 2 0 6.7. Equations with Trigonometric Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 6.7-1. Integrands with Nonlinearity of the Form cos[ βy(t) ] ...................... 4 2 0 6.7-2. Integrands with Nonlinearity of the Form sin[ βy(t) ] ...................... 4 2 2 6.7-3. Integrands with Nonlinearity of the Form tan[ βy(t) ] ...................... 4 2 3 6.7-4. Integrands with Nonlinearity of the Form cot[ βy(t) ] ...................... 4 2 4 6 . 8 . E q u a t i o n sw i t hN o n l i n e a r i t yo fG e n e r a lF o r m.................................. 4 2 5 6.8-1. Equations of the Form y(x)+⎝integraltext x aK(x,t)G⎝parenleftbig y(t)⎝parenrightbig dt=F(x) ................. 4 2 5 6.8-2. Equations of the Form y(x)+⎝integraltextx aK(x–t)G⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x) .............. 4 2 8 6 . 8 - 3 . O t h e rE q u a t i o n s ................................................... 4 3 1 7. Nonlinear Equations of the First Kind with Constant Limits of Integration ........ 433 7.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters . . . . . . . . . . . 433 7.1-1. Equations of the Form⎝integraltextb aK(t)y(x)y(t)dt=F(x) ........................ 4 3 3 7.1-2. Equations of the Form⎝integraltextb aK(t)y(t)y(xt)dt=F(x) ....................... 4 3 5 7 . 1 - 3 . O t h e rE q u a t i o n s ................................................... 4 3 6 CONTENTS xi 7.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions . . . . . . . . . . . . 437 7.2-1. Equations of the Form⎝integraltextb aK(t)y(t)y(···)dt=F(x) ....................... 4 3 7 7.2-2. Equations of the Form⎝integraltextb a[K(x,t)y(t)+M(x,t)y2(t)]dt=F(x) ............. 4 4 3 7.3. Equations with Power-Law Nonlinearity That Contain Arbitrary Functions . . . . . . . . . . 444 7.3-1. Equations of the Form⎝integraltextb aK(t)yµ(x)yγ(t)dt=F(x) ...................... 4 4 4 7.3-2. Equations of the Form⎝integraltextb aK(t)yγ(t)y(xt)dt=F(x) ...................... 4 4 4 7.3-3. Equations of the Form⎝integraltextb aK(t)yγ(t)y(x+βt)dt=F(x) ................... 4 4 5 7.3-4. Equations of the Form⎝integraltextb a[K(x,t)y(t)+M(x,t)yγ(t)]dt=f(x) ............. 4 4 6 7 . 3 - 5 . O t h e rE q u a t i o n s ................................................... 4 4 6 7 . 4 . E q u a t i o n sw i t hN o n l i n e a r i t yo fG e n e r a lF o r m.................................. 4 4 7 7.4-1. Equations of the Form⎝integraltextb aϕ⎝parenleftbig y(x)⎝parenrightbig K⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x) .................... 4 4 7 7.4-2. Equations of the Form⎝integraltextb ay(xt)K⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x) ...................... 4 4 7 7.4-3. Equations of the Form⎝integraltextb ay(x+βt)K⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x) ................... 4 4 9 7.4-4. Equations of the Form⎝integraltextb a[K(x,t)y(t)+ϕ(x)Ψ(t,y(t))]dt=F(x) ........... 4 5 0 7 . 4 - 5 . O t h e rE q u a t i o n s ................................................... 4 5 1 8. Nonlinear Equations of the Second Kind with Constant Limits of Integration ...... 453 8.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters . . . . . . . . . . . 453 8.1-1. Equations of the Form y(x)+⎝integraltextb aK(x,t)y2(t)dt=F(x) ................... 4 5 3 8.1-2. Equations of the Form y(x)+⎝integraltextb aK(x,t)y(x)y(t)dt=F(x) ................. 4 5 4 8.1-3. Equations of the Form y(x)+⎝integraltextb aK(t)y(t)y(···)dt=F(x) ................. 4 5 5 8.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions . . . . . . . . . . . . 456 8.2-1. Equations of the Form y(x)+⎝integraltextb aK(x,t)y2(t)dt=F(x) ................... 4 5 6 8.2-2. Equations of the Form y(x)+⎝integraltextb a⎝summationtextKnm(x,t)yn(x)ym(t)dt=F(x),n+m≤2 457 8.2-3. Equations of the Form y(x)+⎝integraltextb aK(t)y(t)y(···)dt=F(x) ................. 4 6 0 8 . 3 . E q u a t i o n sw i t hP o w e r - L a wN o n l i n e a r i t y ...................................... 4 6 4 8.3-1. Equations of the Form y(x)+⎝integraltextb aK(x,t)yβ(t)dt=F(x) ................... 4 6 4 8 . 3 - 2 . O t h e rE q u a t i o n s ................................................... 4 6 5 8.4. Equations with Exponential Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 8.4-1. Integrands with Nonlinearity of the Form exp[ βy(t) ] ...................... 4 6 7 8 . 4 - 2 . O t h e rI n t e g r a n d s................................................... 4 6 8 8.5. Equations with Hyperbolic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 8.5-1. Integrands with Nonlinearity of the Form cosh[ βy(t) ] ..................... 4 6 8 8.5-2. Integrands with Nonlinearity of the Form sinh[ βy(t) ] ..................... 4 6 9 8.5-3. Integrands with Nonlinearity of the Form tanh[ βy(t) ] ..................... 4 6 9 8.5-4. Integrands with Nonlinearity of the Form coth[ βy(t) ] ..................... 4 7 0 8 . 5 - 5 . O t h e rI n t e g r a n d s................................................... 4 7 1 8 . 6 . E q u a t i o n sw i t hL o g a r i t h m i cN o n l i n e a r i t y ..................................... 4 7 2 8.6-1. Integrands with Nonlinearity of the Form ln[ βy(t) ] ....................... 4 7 2 8 . 6 - 2 . O t h e rI n t e g r a n d s................................................... 4 7 3 8.7. Equations with Trigonometric Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 8.7-1. Integrands with Nonlinearity of the Form cos[ βy(t) ] ...................... 4 7 3 8.7-2. Integrands with Nonlinearity of the Form sin[ βy(t) ] ...................... 4 7 4 8.7-3. Integrands with Nonlinearity of the Form tan[ βy(t) ] ...................... 4 7 5 8.7-4. Integrands with Nonlinearity of the Form cot[ βy(t) ] ...................... 4 7 5 8 . 7 - 5 . O t h e rI n t e g r a n d s................................................... 4 7 6 xii CONTENTS 8 . 8 . E q u a t i o n sw i t hN o n l i n e a r i t yo fG e n e r a lF o r m.................................. 4 7 7 8.8-1. Equations of the Form y(x)+⎝integraltextb aK(|x–t|)G⎝parenleftbig y(t)⎝parenrightbig dt=F(x) ............... 4 7 7 8.8-2. Equations of the Form y(x)+⎝integraltextb aK(x,t)G⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x) ............... 4 7 9 8.8-3. Equations of the Form y(x)+⎝integraltextb aG⎝parenleftbig x,t,y(t)⎝parenrightbig dt=F(x) ................... 4 8 3 8.8-4. Equations of the Form y(x)+⎝integraltextb ay(xt)G⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x) ................. 4 8 5 8.8-5. Equations of the Form y(x)+⎝integraltextb ay(x+βt)G⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x) .............. 4 8 7 8 . 8 - 6 . O t h e rE q u a t i o n s ................................................... 4 9 4 Part II. Methods for Solving Integral Equations 9. Main De ﬁnitions and Formulas. Integral Transforms .......................... 501 9.1. Some De ﬁn i t i o n s ,R e m a r k s ,a n dF o r m u l a s .................................... 5 0 1 9.1-1. Some De ﬁn i t i o n s .................................................. 5 0 1 9.1-2. Structure of Solutions to Linear Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 502 9 . 1 - 3 . I n t e g r a lT r a n s f o r m s ................................................ 5 0 3 9.1-4. Residues. Calculation Formulas. Cauchy’s Residue Theorem . . . . . . . . . . . . . . . 504 9 . 1 - 5 . J o r d a nL e m m a .................................................... 5 0 5 9 . 2 . L a p l a c eT r a n s f o r m ....................................................... 5 0 5 9.2-1. De ﬁn i t i o n .I n v e r s i o nF o r m u l a ........................................ 5 0 5 9 . 2 - 2 . I n v e r s eT r a n s f o r m so fR a t i o n a lF u n c t i o n s ............................... 5 0 69.2-3. Inversion of Functions with Finitely Many Singular Points . . . . . . . . . . . . . . . . . 507 9.2-4. Convolution Theorem. Main Properties of the Laplace Transform . . . . . . . . . . . . 507 9 . 2 - 5 . L i m i tT h e o r e m s ................................................... 5 0 79.2-6. Representation of Inverse Transforms as Convergent Series . . . . . . . . . . . . . . . . . 509 9.2-7. Representation of Inverse Transforms as Asymptotic Expansions as x→∞ . . . 509 9 . 2 - 8 . P o s t – W i d d e rF o r m u l a .............................................. 5 1 0 9.3. Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 9.3-1. De ﬁn i t i o n .I n v e r s i o nF o r m u l a ........................................ 5 1 0 9.3-2. Main Properties of the Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5119.3-3. Relation Among the Mellin, Laplace, and Fourier Transforms . . . . . . . . . . . . . . . 511 9 . 4 . F o u r i e rT r a n s f o r m ....................................................... 5 1 2 9.4-1. De ﬁn i t i o n .I n v e r s i o nF o r m u l a ........................................ 5 1 2 9 . 4 - 2 . A s y m m e t r i cF o r mo ft h eT r a n s f o r m ................................... 5 1 2 9 . 4 - 3 . A l t e r n a t i v eF o u r i e rT r a n s f o r m ........................................ 5 1 2 9.4-4. Convolution Theorem. Main Properties of the Fourier Transforms . . . . . . . . . . . 513 9 . 5 . F o u r i e rC o s i n ea n dS i n eT r a n s f o r m s ......................................... 5 1 4 9 . 5 - 1 . F o u r i e rC o s i n eT r a n s f o r m ........................................... 5 1 4 9 . 5 - 2 . F o u r i e rS i n eT r a n s f o r m ............................................. 5 1 4 9 . 6 . O t h e rI n t e g r a lT r a n s f o r m s ................................................. 5 1 5 9 . 6 - 1 . H a n k e lT r a n s f o r m ................................................. 5 1 5 9 . 6 - 2 . M e i j e rT r a n s f o r m .................................................. 5 1 6 9 . 6 - 3 . K o n t o r o v i c h – L e b e d e vT r a n s f o r m ..................................... 5 1 6 9.6-4. Y- t r a n s f o r m ...................................................... 5 1 6 9 . 6 - 5 . S u m m a r yT a b l eo fI n t e g r a lT r a n s f o r m s ................................. 5 1 7 10. Methods for Solving Linear Equations of the Form⎝integraltext⎝integraltext x aK(x,t)y(t)dt=f(x)..... 519 1 0 . 1 . V o l t e r r aE q u a t i o n so ft h eF i r s tK i n d ........................................ 5 1 9 10.1-1. Equations of the First Kind. Function and Kernel Classes . . . . . . . . . . . . . . . . 5191 0 . 1 - 2 . E x i s t e n c ea n dU n i q u e n e s so faS o l u t i o n .............................. 5 2 0 10.1-3. Some Problems Leading to V olterra Integral Equations of the First Kind . . . . 520 CONTENTS xiii 10.2. Equations with Degenerate Kernel: K(x,t)=g1(x)h1(t)+···+gn(x)hn(t) ......... 5 2 2 10.2-1. Equations with Kernel of the Form K(x,t)=g1(x)h1(t)+g2(x)h2(t) ....... 5 2 2 1 0 . 2 - 2 . E q u a t i o n sw i t hG e n e r a lD e g e n e r a t eK e r n e l ............................ 5 2 3 10.3. Reduction of V olterra Equations of the First Kind to V olterra Equations of the Second K i n d ................................................................. 5 2 41 0 . 3 - 1 . F i r s tM e t h o d .................................................... 5 2 41 0 . 3 - 2 . S e c o n dM e t h o d ................................................. 5 2 4 10.4. Equations with Difference Kernel: K(x,t)=K(x–t) .......................... 5 2 4 10.4-1. Solution Method Based on the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . 524 10.4-2. Case in Which the Transform of the Solution is a Rational Function . . . . . . . . 5251 0 . 4 - 3 . C o n v o l u t i o nR e p r e s e n t a t i o no faS o l u t i o n ............................. 5 2 6 10.4-4. Application of an Auxiliary Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 10.4-5. Reduction to Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 52710.4-6. Reduction of a V olterra Equation to a Wiener–Hopf Equation . . . . . . . . . . . . . 528 1 0 . 5 . M e t h o do fF r a c t i o n a lD i f f e r e n t i a t i o n ........................................ 5 2 9 10.5-1. De ﬁnition of Fractional Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 10.5-2. De ﬁnition of Fractional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 1 0 . 5 - 3 . M a i nP r o p e r t i e s ................................................. 5 3 0 1 0 . 5 - 4 . S o l u t i o no ft h eG e n e r a l i z e dA b e lE q u a t i o n ............................ 5 3 1 10.5-5. Erd ´e l y i – K o b e rO p e r a t o r s .......................................... 5 3 2 10.6. Equations with Weakly Singular Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 1 0 . 6 - 1 . M e t h o do fT r a n s f o r m a t i o no ft h eK e r n e l .............................. 5 3 2 10.6-2. Kernel with Logarithmic Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 1 0 . 7 . M e t h o do fQ u a d r a t u r e s .................................................. 5 3 4 1 0 . 7 - 1 . Q u a d r a t u r eF o r m u l a s ............................................. 5 3 4 1 0 . 7 - 2 . G e n e r a lS c h e m eo ft h eM e t h o d ..................................... 5 3 5 1 0 . 7 - 3 . A l g o r i t h mB a s e do nt h eT r a p e z o i d a lR u l e ............................. 5 3 610.7-4. Algorithm for an Equation with Degenerate Kernel . . . . . . . . . . . . . . . . . . . . . 536 10.8. Equations with Inﬁ n i t eI n t e g r a t i o nL i m i t .................................... 5 3 7 10.8-1. Equation of the First Kind with Variable Lower Limit of Integration . . . . . . . . 537 10.8-2. Reduction to a Wiener–Hopf Equation of the First Kind . . . . . . . . . . . . . . . . . 538 11. Methods for Solving Linear Equations of the Form y(x)–⎝integraltext⎝integraltext x aK(x,t)y(t)dt=f(x) 539 1 1 . 1 . V o l t e r r aI n t e g r a lE q u a t i o n so ft h eS e c o n dK i n d ............................... 5 3 9 11.1-1. Preliminary Remarks. Equations for the Resolvent . . . . . . . . . . . . . . . . . . . . . 53911.1-2. Relationship Between Solutions of Some Integral Equations . . . . . . . . . . . . . . 540 11.2. Equations with Degenerate Kernel: K(x,t)=g 1(x)h1(t)+···+gn(x)hn(t) ......... 5 4 0 11.2-1. Equations with Kernel of the Form K(x,t)=ϕ(x)+ψ(x)(x–t) ........... 5 4 0 11.2-2. Equations with Kernel of the Form K(x,t)=ϕ(t)+ψ(t)(t–x) ............ 5 4 1 11.2-3. Equations with Kernel of the Form K(x,t)=⎝summationtextn m=1ϕm(x)(x–t)m–1....... 5 4 2 11.2-4. Equations with Kernel of the Form K(x,t)=⎝summationtextn m=1ϕm(t)(t–x)m–1....... 5 4 3 11.2-5. Equations with Degenerate Kernel of the General Form . . . . . . . . . . . . . . . . . . 543 11.3. Equations with Difference Kernel: K(x,t)=K(x–t) .......................... 5 4 4 11.3-1. Solution Method Based on the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . 544 11.3-2. Method Based on the Solution of an Auxiliary Equation . . . . . . . . . . . . . . . . . 546 11.3-3. Reduction to Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 54711.3-4. Reduction to a Wiener–Hopf Equation of the Second Kind . . . . . . . . . . . . . . . 547 11.3-5. Method of Fractional Integration for the Generalized Abel Equation . . . . . . . . 548 1 1 . 3 - 6 . S y s t e m so fV o l t e r r aI n t e g r a lE q u a t i o n s ............................... 5 4 9 xiv CONTENTS 11.4. Operator Methods for Solving Linear Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 549 11.4-1. Application of a Solution of a “Truncated” Equation of the First Kind . . . . . . 54911.4-2. Application of the Auxiliary Equation of the Second Kind . . . . . . . . . . . . . . . . 551 11.4-3. Method for Solving “Quadratic” Ope r a t o rE q u a t i o n s .................... 5 5 2 11.4-4. Solution of Operator Equations of Polynomial Form . . . . . . . . . . . . . . . . . . . . 553 1 1 . 4 - 5 . S o m eG e n e r a l i z a t i o n s ............................................ 5 5 4 11.5. Construction of Solutions of Integral Equations with Special Right-Hand Side . . . . . . . 555 1 1 . 5 - 1 . G e n e r a lS c h e m e ................................................. 5 5 511.5-2. Generating Function of Exponential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 1 1 . 5 - 3 . P o w e r - L a wG e n e r a t i n gF u n c t i o n .................................... 5 5 7 11.5-4. Generating Function Containing Sines and Cosines . . . . . . . . . . . . . . . . . . . . . 558 1 1 . 6 . M e t h o do fM o d e lS o l u t i o n s ............................................... 5 5 9 1 1 . 6 - 1 . P r e l i m i n a r yR e m a r k s ............................................. 5 5 9 1 1 . 6 - 2 . D e s c r i p t i o no ft h eM e t h o d ......................................... 5 6 0 11.6-3. Model Solution in the Case of an Exponential Right-Hand Side . . . . . . . . . . . 561 11.6-4. Model Solution in the Case of a Power-Law Right-Hand Side . . . . . . . . . . . . . 562 11.6-5. Model Solution in the Case of a Sine-Shaped Right-Hand Side . . . . . . . . . . . . 56211.6-6. Model Solution in the Case of a Cosine-Shaped Right-Hand Side . . . . . . . . . . 563 1 1 . 6 - 7 . S o m eG e n e r a l i z a t i o n s ............................................ 5 6 3 11.7. Method of Differentiation for Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 11.7-1. Equations with Kernel Containing a Sum of Exponential Functions . . . . . . . . 56411.7-2. Equations with Kernel Containing a Sum of Hyperbolic Functions . . . . . . . . . 564 11.7-3. Equations with Kernel Containing a Sum of Trigonometric Functions . . . . . . . 564 11.7-4. Equations Whose Kernels Contain Combinations of Various Functions . . . . . . 565 11.8. Reduction of V olterra Equations of the Second Kind to V olterra Equations of the First K i n d ................................................................. 5 6 5 1 1 . 8 - 1 . F i r s tM e t h o d .................................................... 5 6 5 1 1 . 8 - 2 . S e c o n dM e t h o d ................................................. 5 6 6 1 1 . 9 . S u c c e s s i v eA p p r o x i m a t i o nM e t h o d ......................................... 5 6 6 1 1 . 9 - 1 . G e n e r a lS c h e m e ................................................. 5 6 6 1 1 . 9 - 2 . F o r m u l af o rt h eR e s o l v e n t ......................................... 5 6 7 1 1 . 1 0 . M e t h o do fQ u a d r a t u r e s ................................................. 5 6 8 1 1 . 1 0 - 1 . G e n e r a lS c h e m eo ft h eM e t h o d ................................... 5 6 8 1 1 . 1 0 - 2 . A p p l i c a t i o no ft h eT r a p e z o i d a lR u l e ............................... 5 6 8 1 1 . 1 0 - 3 . C a s eo faD e g e n e r a t eK e r n e l ..................................... 5 6 9 11.11. Equations with In ﬁn i t eI n t e g r a t i o nL i m i t ................................... 5 6 9 11.11-1. Equation of the Second Kind with Variable Lower Integration Limit . . . . . . 570 11.11-2. Reduction to a Wiener–Hopf Equation of the Second Kind . . . . . . . . . . . . . 571 12. Methods for Solving Linear Equations of the Form⎝integraltext⎝integraltext b aK(x,t)y(t)dt=f(x)..... 573 12.1. Some De ﬁnition and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 12.1-1. Fredholm Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . 573 12.1-2. Integral Equations of the First Kind with Weak Singularity . . . . . . . . . . . . . . . 574 12.1-3. Integral Equations of Convolution Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5741 2 . 1 - 4 . D u a lI n t e g r a lE q u a t i o n so ft h eF i r s tK i n d ............................. 5 7 5 12.1-5. Some Problems Leading to Integral Equations of the First Kind . . . . . . . . . . . 575 1 2 . 2 . I n t e g r a lE q u a t i o n so ft h eF i r s tK i n dw i t hS y m m e t r i cK e r n e l ..................... 5 7 7 12.2-1. Solution of an Integral Equation in Terms of Series in Eigenfunctions of Its K e r n e l ......................................................... 5 7 7 1 2 . 2 - 2 . M e t h o do fS u c c e s s i v eA p p r o x i m a t i o n s ............................... 5 7 9 CONTENTS xv 12.3. Integral Equations of the First Kind with Nonsymmetric Kernel . . . . . . . . . . . . . . . . . . 580 12.3-1. Representation of a Solution in the Form of Series. General Description . . . . 580 12.3-2. Special Case of a Kernel That is a Generating Function . . . . . . . . . . . . . . . . . . 58012.3-3. Special Case of the Right-Hand Side Represented in Terms of Orthogonal F u n c t i o n s ...................................................... 5 8 2 1 2 . 3 - 4 . G e n e r a lC a s e .G a l e r k i n ’ sM e t h o d ................................... 5 8 2 12.3-5. Utilization of the Schmidt Kernels for the Construction of Solutions of E q u a t i o n s ...................................................... 5 8 2 12.4. Method of Differentiation for Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 12.4-1. Equations with Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 12.4-2. Other Equations. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 1 2 . 5 . M e t h o do fI n t e g r a lT r a n s f o r m s ............................................ 5 8 6 1 2 . 5 - 1 . E q u a t i o nw i t hD i f f e r e n c eK e r n e lo nt h eE n t i r eA x i s ..................... 5 8 6 12.5-2. Equations with Kernel K(x,t)=K(x/t)o nt h eS e m i a x i s ................ 5 8 7 12.5-3. Equation with Kernel K(x,t)=K(xt)a n dS o m eG e n e r a l i z a t i o n s .......... 5 8 7 12.6. Krein’s Method and Some Other Exact Methods for Integral Equations of Special Types 588 12.6-1. Krein’s Method for an Equation with Difference Kernel with a Weak Singularity 588 12.6-2. Kernel is the Sum of a Nondegenerate Kernel and an Arbitrary Degenerate K e r n e l ......................................................... 5 8 9 12.6-3. Reduction of Integral Equations of the First Kind to Equations of the Second K i n d .......................................................... 5 9 1 1 2 . 7 . R i e m a n nP r o b l e mf o rt h eR e a lA x i s ........................................ 5 9 2 12.7-1. Relationships Between the Fourier Integral and the Cauchy Type Integral . . . . 592 1 2 . 7 - 2 . O n e - S i d e dF o u r i e rI n t e g r a l s ........................................ 5 9 3 12.7-3. Analytic Continuation Theorem and the Generalized Liouville Theorem . . . . 59512.7-4. Riemann Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 12.7-5. Problems with Rational Coef ﬁc i e n t s ................................. 6 0 1 12.7-6. Exceptional Cases. The Homogeneous Problem . . . . . . . . . . . . . . . . . . . . . . . . 602 12.7-7. Exceptional Cases. The Nonhomogeneous Problem . . . . . . . . . . . . . . . . . . . . . 604 12.8. Carleman Method for Equations of the Convolution Type of the First Kind . . . . . . . . . 606 1 2 . 8 - 1 . W i e n e r – H o p fE q u a t i o no ft h eF i r s tK i n d .............................. 6 0 612.8-2. Integral Equations of the First Kind with Two Kernels . . . . . . . . . . . . . . . . . . . 607 12.9. Dual Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 12.9-1. Carleman Method for Equations with Difference Kernels . . . . . . . . . . . . . . . . 610 12.9-2. General Scheme of Finding Solutions of Dual Integral Equations . . . . . . . . . . 611 12.9-3. Exact Solutions of Some Dual Equations of the First Kind . . . . . . . . . . . . . . . . 613 12.9-4. Reduction of Dual Equations to a Fredholm Equation . . . . . . . . . . . . . . . . . . . 615 12.10. Asymptotic Methods for Solving Equations with Logarithmic Singularity . . . . . . . . . 618 1 2 . 1 0 - 1 . P r e l i m i n a r yR e m a r k s ........................................... 6 1 8 12.10-2. Solution for Large λ............................................ 6 1 9 12.10-3. Solution for Small λ............................................ 6 2 0 1 2 . 1 0 - 4 . I n t e g r a lE q u a t i o no fE l a s t i c i t y .................................... 6 2 1 12.11. Regularization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 1 2 . 1 1 - 1 . L a v r e n t i e vR e g u l a r i z a t i o nM e t h o d ................................ 6 2 112.11-2. Tikhonov Regularization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 12.12. Fredholm Integral Equation of the First Kind as an Ill-Posed Problem . . . . . . . . . . . . 623 12.12-1. General Notions of Well-Posed and Ill-Posed Problems . . . . . . . . . . . . . . . . 623 12.12-2. Integral Equation of the First Kind is an Ill-Posed Problem . . . . . . . . . . . . . 624 xvi CONTENTS 13. Methods for Solving Linear Equations of the Form y(x)–⎝integraltext⎝integraltextb aK(x,t)y(t)dt=f(x) 625 13.1. Some De ﬁnition and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 13.1-1. Fredholm Equations and Equations with Weak Singularity of the Second Kind 625 1 3 . 1 - 2 . S t r u c t u r eo ft h eS o l u t i o n .......................................... 6 2 613.1-3. Integral Equations of Convolution Type of the Second Kind . . . . . . . . . . . . . . 6261 3 . 1 - 4 . D u a lI n t e g r a lE q u a t i o n so ft h eS e c o n dK i n d ........................... 6 2 7 13.2. Fredholm Equations of the Second Kind with De generate Kernel. Some Generalizations 627 1 3 . 2 - 1 . S i m p l e s tD e g e n e r a t eK e r n e l ........................................ 6 2 71 3 . 2 - 2 . D e g e n e r a t eK e r n e li nt h eG e n e r a lC a s e ............................... 6 2 813.2-3. Kernel is the Sum of a Nondegenerate Kernel and an Arbitrary Degenerate K e r n e l ......................................................... 6 3 1 13.3. Solution as a Power Series in the Parameter. Method of Successive Approximations . . 632 1 3 . 3 - 1 . I t e r a t e dK e r n e l s ................................................. 6 3 21 3 . 3 - 2 . M e t h o do fS u c c e s s i v eA p p r o x i m a t i o n s ............................... 6 3 3 1 3 . 3 - 3 . C o n s t r u c t i o no ft h eR e s o l v e n t ...................................... 6 3 3 13.3-4. Orthogonal Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 13.4. Method of Fredholm Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 1 3 . 4 - 1 . F o r m u l af o rt h eR e s o l v e n t ......................................... 6 3 5 1 3 . 4 - 2 . R e c u r r e n tR e l a t i o n s .............................................. 6 3 6 13.5. Fredholm Theorems and the Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 13.5-1. Fredholm Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 13.5-2. Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 13.6. Fredholm Integral Equations of the Second Kind with Symmetric Kernel . . . . . . . . . . . 639 1 3 . 6 - 1 . C h a r a c t e r i s t i cV a l u e sa n dE i g e n f u n c t i o n s ............................. 6 3 913.6-2. Bilinear Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 1 3 . 6 - 3 . H i l b e r t – S c h m i d tT h e o r e m ......................................... 6 4 1 13.6-4. Bilinear Series of Iterated Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64213.6-5. Solution of the Nonhomogeneous Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 64213.6-6. Fredholm Alternative for Symmetric Equations . . . . . . . . . . . . . . . . . . . . . . . . 643 1 3 . 6 - 7 . R e s o l v e n to faS y m m e t r i cK e r n e l ................................... 6 4 4 13.6-8. Extremal Properties of Characteristic Values and Eigenfunctions . . . . . . . . . . 64413.6-9. Kellog’s Method for Finding Characteristic Values in the Case of Symmetric K e r n e l ......................................................... 6 4 5 13.6-10. Trace Method for the Approximation of Characteristic Values . . . . . . . . . . . . 646 13.6-11. Integral Equations Reducible to Symmetric Equations . . . . . . . . . . . . . . . . . . 64713.6-12. Skew-Symmetric Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 13.6-13. Remark on Nonsymmetric Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 13.7. Integral Equations with Nonnegative Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 13.7-1. Positive Principal Eigenvalues. Gen e r a l i z e dJ e n t z c hT h e o r e m ............. 6 4 8 13.7-2. Positive Solutions of a Nonhomogeneous Integral Equation . . . . . . . . . . . . . . . 649 1 3 . 7 - 3 . E s t i m a t e sf o rt h eS p e c t r a lR a d i u s ................................... 6 4 9 13.7-4. Basic De ﬁnition and Theorems for Oscillating Kernels . . . . . . . . . . . . . . . . . . 651 1 3 . 7 - 5 . S t o c h a s t i cK e r n e l s ............................................... 6 5 4 13.8. Operator Method for Solving Integral Equations of the Second Kind . . . . . . . . . . . . . . 655 1 3 . 8 - 1 . S i m p l e s tS c h e m e ................................................ 6 5 5 13.8-2. Solution of Equations of the Second Kind on the Semiaxis . . . . . . . . . . . . . . . 655 CONTENTS xvii 13.9. Methods of Integral Transforms and Model Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 656 1 3 . 9 - 1 . E q u a t i o nw i t hD i f f e r e n c eK e r n e lo nt h eE n t i r eA x i s ..................... 6 5 613.9-2. Equation with the Kernel K(x,t)=t –1Q(x/t)o nt h eS e m i a x i s ............ 6 5 7 13.9-3. Equation with the Kernel K(x,t)=tβQ(xt)o nt h eS e m i a x i s ............. 6 5 8 13.9-4. Method of Model Solutions for Equations on the Entire Axis . . . . . . . . . . . . . 659 13.10. Carleman Method for Integral Equations of Convolution Type of the Second Kind . . 660 13.10-1. Wiener–Hopf Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . 66013.10-2. Integral Equation of the Second Kind with Two Kernels . . . . . . . . . . . . . . . 66413.10-3. Equations of Convolution Type with Variable Integration Limit . . . . . . . . . . 668 13.10-4. Dual Equation of Convolution Type of the Second Kind . . . . . . . . . . . . . . . 670 1 3 . 1 1 . W i e n e r – H o p fM e t h o d .................................................. 6 7 1 1 3 . 1 1 - 1 . S o m eR e m a r k s ................................................ 6 7 113.11-2. Homogeneous Wiener–Hopf Equation of the Second Kind . . . . . . . . . . . . . 67313.11-3. General Scheme of the Method. The Factorization Problem . . . . . . . . . . . . 676 13.11-4. Nonhomogeneous Wiener–Hopf Equation of the Second Kind . . . . . . . . . . 677 13.11-5. Exceptional Case of a Wiener–Hopf Equation of the Second Kind . . . . . . . 678 1 3 . 1 2 . K r e i n ’ sM e t h o df o rW i e n e r – H o p fE q u a t i o n s ................................. 6 7 9 1 3 . 1 2 - 1 . S o m eR e m a r k s .T h eF a c t o r i z a t i o nP r o b l e m ......................... 6 7 913.12-2. Solution of the Wiener–Hopf Equations of the Second Kind . . . . . . . . . . . . 6811 3 . 1 2 - 3 . H o p f – F o c kF o r m u l a ............................................ 6 8 3 13.13. Methods for Solving Equations with Difference Kernels on a Finite Interval . . . . . . . 683 1 3 . 1 3 - 1 . K r e i n ’ sM e t h o d ............................................... 6 8 31 3 . 1 3 - 2 . K e r n e l sw i t hR a t i o n a lF o u r i e rT r a n s f o r m s .......................... 6 8 513.13-3. Reduction to Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 686 13.14. Method of Approximating a Kernel by a Degenerate One . . . . . . . . . . . . . . . . . . . . . . 687 1 3 . 1 4 - 1 . A p p r o x i m a t i o no ft h eK e r n e l ..................................... 6 8 71 3 . 1 4 - 2 . A p p r o x i m a t eS o l u t i o n .......................................... 6 8 8 1 3 . 1 5 . B a t e m a nM e t h o d ...................................................... 6 8 9 1 3 . 1 5 - 1 . G e n e r a lS c h e m eo ft h eM e t h o d ................................... 6 8 9 1 3 . 1 5 - 2 . S o m eS p e c i a lC a s e s ............................................ 6 9 0 1 3 . 1 6 . C o l l o c a t i o nM e t h o d .................................................... 6 9 2 1 3 . 1 6 - 1 . G e n e r a lR e m a r k s .............................................. 6 9 21 3 . 1 6 - 2 . A p p r o x i m a t eS o l u t i o n .......................................... 6 9 3 13.16-3. Eigenfunctions of the Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 1 3 . 1 7 . M e t h o do fL e a s tS q u a r e s ................................................ 6 9 5 1 3 . 1 7 - 1 . D e s c r i p t i o no ft h eM e t h o d ....................................... 6 9 51 3 . 1 7 - 2 . C o n s t r u c t i o no fE i g e n f u n c t i o n s ................................... 6 9 6 13.18. Bubnov–Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 1 3 . 1 8 - 1 . D e s c r i p t i o no ft h eM e t h o d ....................................... 6 9 71 3 . 1 8 - 2 . C h a r a c t e r i s t i cV a l u e s ........................................... 6 9 7 1 3 . 1 9 . Q u a d r a t u r eM e t h o d .................................................... 6 9 8 13.19-1. General Scheme for Fredholm Equations of the Second Kind . . . . . . . . . . . 69813.19-2. Construction of the Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 13.19-3. Speci ﬁc Features of the Application of Quadrature Formulas . . . . . . . . . . . . 700 13.20. Systems of Fredholm Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . 701 1 3 . 2 0 - 1 . S o m eR e m a r k s ................................................ 7 0 113.20-2. Method of Reducing a System of Equations to a Single Equation . . . . . . . . 701 xviii CONTENTS 13.21. Regularization Method for Equations with In ﬁnite Limits of Integration . . . . . . . . . . . 702 13.21-1. Basic Equation and Fredholm Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7021 3 . 2 1 - 2 . R e g u l a r i z i n gO p e r a t o r s ......................................... 7 0 31 3 . 2 1 - 3 . R e g u l a r i z a t i o nM e t h o d .......................................... 7 0 4 14. Methods for Solving Singular Integral Equations of the First Kind .............. 707 14.1. Some De ﬁn i t i o n sa n dR e m a r k s ............................................ 7 0 7 14.1-1. Integral Equations of the First Kind with Cauchy Kernel . . . . . . . . . . . . . . . . . 707 14.1-2. Integral Equations of the First Kind with Hilbert Kernel . . . . . . . . . . . . . . . . . 707 1 4 . 2 . C a u c h yT y p eI n t e g r a l .................................................... 7 0 8 14.2-1. De ﬁnition of the Cauchy Type Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 14.2-2. H ¨older Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 14.2-3. Principal Value of a Singular Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7091 4 . 2 - 4 . M u l t i v a l u e dF u n c t i o n s ............................................ 7 1 1 14.2-5. Principal Value of a Singular Curvilinear Integral . . . . . . . . . . . . . . . . . . . . . . . 712 14.2-6. Poincar ´e – B e r t r a n dF o r m u l a ........................................ 7 1 4 14.3. Riemann Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 14.3-1. Principle of Argument. The Generalized Liouville Theorem . . . . . . . . . . . . . . 71414.3-2. Hermite Interpolation Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7161 4 . 3 - 3 . N o t i o no ft h eI n d e x .............................................. 7 1 6 1 4 . 3 - 4 . S t a t e m e n to ft h eR i e m a n nP r o b l e m .................................. 7 1 8 1 4 . 3 - 5 . S o l u t i o no ft h eH o m o g e n e o u sP r o b l e m ............................... 7 2 014.3-6. Solution of the Nonhomogeneous Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72114.3-7. Riemann Problem with Rational Coef ﬁc i e n t s .......................... 7 2 3 1 4 . 3 - 8 . R i e m a n nP r o b l e mf o raH a l f - P l a n e .................................. 7 2 5 14.3-9. Exceptional Cases of the Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 14.3-10. Riemann Problem for a Multiply Connected Domain . . . . . . . . . . . . . . . . . . . 73114.3-11. Riemann Problem for Open Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73414.3-12. Riemann Problem with a Discontinuous Coef ﬁc i e n t .................... 7 3 9 1 4 . 3 - 1 3 . R i e m a n nP r o b l e mi nt h eG e n e r a lC a s e .............................. 7 4 114.3-14. Hilbert Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 14.4. Singular Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 1 4 . 4 - 1 . S i m p l e s tE q u a t i o nw i t hC a u c h yK e r n e l ............................... 7 4 314.4-2. Equation with Cauchy Kernel on the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . 7431 4 . 4 - 3 . E q u a t i o no ft h eF i r s tK i n do naF i n i t eI n t e r v a l ......................... 7 4 4 14.4-4. General Equation of the First Kind with Cauchy Kernel . . . . . . . . . . . . . . . . . . 74514.4-5. Equations of the First Kind with Hilbert Kernel . . . . . . . . . . . . . . . . . . . . . . . . 746 14.5. Multhopp–Kalandiya Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 14.5-1. Solution That is Unbounded at the Endpoints of the Interval . . . . . . . . . . . . . . 74714.5-2. Solution Bounded at One Endpoint of the Interval . . . . . . . . . . . . . . . . . . . . . . 74914.5-3. Solution Bounded at Both Endpoints of the Interval . . . . . . . . . . . . . . . . . . . . . 750 14.6. Hypersingular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 14.6-1. Hypersingular Integral Equations with Cauchy- and Hilbert-Type Kernels . . . 751 14.6-2. De ﬁnition of Hypersingular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 14.6-3. Exact Solution of the Simplest Hypersingular Equation with Cauchy-Type K e r n e l ......................................................... 7 5 3 14.6-4. Exact Solution of the Simplest Hypersingular Equation with Hilbert-Type K e r n e l ......................................................... 7 5 4 14.6-5. Numerical Methods for Hypersingular Equations . . . . . . . . . . . . . . . . . . . . . . . 754 CONTENTS xix 15. Methods for Solving Complete Singular Integral Equations .................... 757 15.1. Some De ﬁn i t i o n sa n dR e m a r k s ............................................ 7 5 7 1 5 . 1 - 1 . I n t e g r a lE q u a t i o n sw i t hC a u c h yK e r n e l ............................... 7 5 7 15.1-2. Integral Equations with Hilbert Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 15.1-3. Fredholm Equations of the Second Kind on a Contour . . . . . . . . . . . . . . . . . . . 759 1 5 . 2 . C a r l e m a nM e t h o df o rC h a r a c t e r i s t i cE q u a t i o n s................................ 7 6 1 1 5 . 2 - 1 . C h a r a c t e r i s t i cE q u a t i o nw i t hC a u c h yK e r n e l ........................... 7 6 1 1 5 . 2 - 2 . T r a n s p o s e dE q u a t i o no faC h a r a c t e r i s t i cE q u a t i o n ...................... 7 6 4 1 5 . 2 - 3 . C h a r a c t e r i s t i cE q u a t i o no nt h eR e a lA x i s ............................. 7 6 5 1 5 . 2 - 4 . E x c e p t i o n a lC a s eo faC h a r a c t e r i s t i cE q u a t i o n ......................... 7 6 7 1 5 . 2 - 5 . C h a r a c t e r i s t i cE q u a t i o nw i t hH i l b e r tK e r n e l ........................... 7 6 9 1 5 . 2 - 6 . T r i c o m iE q u a t i o n ................................................ 7 6 9 15.3. Complete Singular Integral Equations Solvable in a Closed Form . . . . . . . . . . . . . . . . . 770 15.3-1. Closed-Form Solutions in the Case of Constant Coef ﬁc i e n t s .............. 7 7 0 1 5 . 3 - 2 . C l o s e d - F o r mS o l u t i o n si nt h eG e n e r a lC a s e ........................... 7 7 1 15.4. Regularization Method for Complete Singular Integral Equations . . . . . . . . . . . . . . . . . 772 15.4-1. Certain Properties of Singular Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 1 5 . 4 - 2 . R e g u l a r i z e r ..................................................... 7 7 4 15.4-3. Methods of Left and Right Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775 1 5 . 4 - 4 . P r o b l e mo fE q u i v a l e n tR e g u l a r i z a t i o n ................................ 7 7 615.4-5. Fredholm Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 15.4-6. Carleman–Vekua Approach to the Regularization . . . . . . . . . . . . . . . . . . . . . . . 778 1 5 . 4 - 7 . R e g u l a r i z a t i o ni nE x c e p t i o n a lC a s e s ................................. 7 7 9 1 5 . 4 - 8 . C o m p l e t eE q u a t i o nw i t hH i l b e r tK e r n e l .............................. 7 8 0 15.5. Analysis of Solutions Singularities for Co mplete Integral Equations with Generalized C a u c h yK e r n e l s ........................................................ 7 8 3 1 5 . 5 - 1 . S t a t e m e n to ft h eP r o b l e ma n dP r e l i m i n a r yR e m a r k s ..................... 7 8 3 15.5-2. Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 15.5-3. Equations for the Exponents of Singularity of a Solution . . . . . . . . . . . . . . . . . 787 15.5-4. Analysis of Equations for Singularity Exponents . . . . . . . . . . . . . . . . . . . . . . . 789 15.5-5. Application to an Equation Arising in Fracture Mechanics . . . . . . . . . . . . . . . . 791 15.6. Direct Numerical Solution of Singular Integral Equations with Generalized Kernels . . 792 1 5 . 6 - 1 . P r e l i m i n a r yR e m a r k s ............................................. 7 9 2 15.6-2. Quadrature Formulas for Integrals with the Jacobi Weight Function . . . . . . . . 793 15.6-3. Approximation of Solutions in Terms of a System of Orthogonal Polynomials 795 15.6-4. Some Special Functions and Their Calculations . . . . . . . . . . . . . . . . . . . . . . . . 797 15.6-5. Numerical Solution of Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . 79915.6-6. Numerical Solutions of Singular Integral Equations of Bueckner Type . . . . . . 801 16. Methods for Solving Nonlinear Integral Equations ............................ 805 16.1. Some De ﬁn i t i o n sa n dR e m a r k s ............................................ 8 0 5 16.1-1. Nonlinear Equations with Variable Limit of Integration (V olterra Equations) . 805 16.1-2. Nonlinear Equations with Constant Integration Limits (Urysohn Equations) . . 806 16.1-3. Some Special Features of Nonlinear Integral Equations . . . . . . . . . . . . . . . . . . 807 16.2. Exact Methods for Nonlinear Equations with Variable Limit of Integration . . . . . . . . . . 809 1 6 . 2 - 1 . M e t h o do fI n t e g r a lT r a n s f o r m s ..................................... 8 0 9 16.2-2. Method of Differentiation for Nonlinear Equations with Degenerate Kernel . . 810 xx CONTENTS 16.3. Approximate and Numerical Methods for Nonlinear Equations with Variable Limit of I n t e g r a t i o n ............................................................ 8 1 1 1 6 . 3 - 1 . S u c c e s s i v eA p p r o x i m a t i o nM e t h o d .................................. 8 1 1 16.3-2. Newton–Kantorovich Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 1 6 . 3 - 3 . C o l l o c a t i o nM e t h o d .............................................. 8 1 51 6 . 3 - 4 . Q u a d r a t u r eM e t h o d .............................................. 8 1 6 16.4. Exact Methods for Nonlinear Equations with Constant Integration Limits . . . . . . . . . . 817 16.4-1. Nonlinear Equations with Degenerate Kernels . . . . . . . . . . . . . . . . . . . . . . . . . 817 1 6 . 4 - 2 . M e t h o do fI n t e g r a lT r a n s f o r m s ..................................... 8 1 916.4-3. Method of Differentiating for Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 82016.4-4. Method for Special Urysohn Equations of the First Kind . . . . . . . . . . . . . . . . . 82116.4-5. Method for Special Urysohn Equations of the Second Kind . . . . . . . . . . . . . . . 822 1 6 . 4 - 6 . S o m eG e n e r a l i z a t i o n s ............................................ 8 2 4 16.5. Approximate and Numerical Methods for Nonlinear Equations with Constant Integration L i m i t s ................................................................ 8 2 6 1 6 . 5 - 1 . S u c c e s s i v eA p p r o x i m a t i o nM e t h o d .................................. 8 2 6 16.5-2. Newton–Kantorovich Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8271 6 . 5 - 3 . Q u a d r a t u r eM e t h o d .............................................. 8 2 916.5-4. Tikhonov Regularization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 16.6 Existence and Uniqueness Theorems for Nonlinear Equations . . . . . . . . . . . . . . . . . . . . 830 1 6 . 6 - 1 . H a m m e r s t e i nE q u a t i o n s ........................................... 8 3 0 1 6 . 6 - 2 . U r y s o h nE q u a t i o n s ............................................... 8 3 2 16.7. Nonlinear Equations with a Parameter: Eigenfunctions, Eigenvalues, Bifurcation Points 834 16.7-1. Eigenfunctions and Eigenvalues of Nonlinear Integral Equations . . . . . . . . . . . 834 16.7-2. Local Solutions of a Nonlinear Integral Equation with a Parameter . . . . . . . . . 835 16.7-3. Bifurcation Points of Nonlinear Integral Equations . . . . . . . . . . . . . . . . . . . . . . 835 17. Methods for Solving Multidimensional Mixed Integral Equations ............... 839 17.1. Some De ﬁnition and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 1 7 . 1 - 1 . B a s i cC l a s s e so fF u n c t i o n s......................................... 8 3 9 1 7 . 1 - 2 . M i x e dE q u a t i o n so naF i n i t eI n t e r v a l ................................. 8 4 0 1 7 . 1 - 3 . M i x e dE q u a t i o no naR i n g - S h a p e d( C i r c u l a r )D o m a i n ................... 8 4 117.1-4. Mixed Equations on a Closed Bounded Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842 17.2. Methods of Solution of Mixed Integral Equations on a Finite Interval . . . . . . . . . . . . . . 843 17.2-1. Equation with a Hilbert–Schmidt Kernel and a Given Right-Hand Side . . . . . . 843 17.2-2. Equation with Hilbert–Schmidt Ker nel and Auxiliary Conditions . . . . . . . . . . 845 17.2-3. Equation with a Schmidt Kernel and a Given Right-Hand Side on an Interval . 84817.2-4. Equation with a Schmidt Kernel and Auxiliary Conditions . . . . . . . . . . . . . . . 851 17.3. Methods of Solving Mixed Integral Equations on a Ring-Shaped Domain . . . . . . . . . . 855 17.3-1. Equation with a Hilbert–Schmidt Kernel and a Given Right-Hand Side . . . . . . 85517.3-2. Equation with a Hilbert–Schmidt Ker nel and Auxiliary Conditions . . . . . . . . . 856 17.3-3. Equation with a Schmidt Kernel and a Given Right-Hand Side . . . . . . . . . . . . 859 17.3-4. Equation with a Schmidt Kernel and Auxiliary Conditions on Ring-Shaped D o m a i n ........................................................ 8 6 2 17.4. Projection Method for Solving Mixed Equations on a Bounded Set . . . . . . . . . . . . . . . . 866 17.4-1. Mixed Operator Equation with a Given Right-Hand Side . . . . . . . . . . . . . . . . . 866 17.4-2. Mixed Operator Equations with Auxiliary Conditions . . . . . . . . . . . . . . . . . . . 869 1 7 . 4 - 3 . G e n e r a lP r o j e c t i o nP r o b l e mf o rO p e r a t o rE q u a t i o n...................... 8 7 3 CONTENTS xxi 18. Application of Integral Equations for the Investigation of Differential Equations ..875 18.1. Reduction of the Cauchy Problem for ODEs to Integral Equations . . . . . . . . . . . . . . . . 875 18.1-1. Cauchy Problem for First-Order ODEs. Uniqueness and Existence Theorems 87518.1-2. Cauchy Problem for First-Order ODEs. Method of Successive Approximations 87618.1-3. Cauchy Problem for Second-Order ODEs. Method of Successive A p p r o x i m a t i o n s ................................................. 8 7 6 18.1-4. Cauchy Problem for a Special n- O r d e rL i n e a rO D E ..................... 8 7 6 18.2. Reduction of Boundary Value Problems for ODEs to V olterra Integral Equations. C a l c u l a t i o no fE i g e n v a l u e s ............................................... 8 7 7 18.2-1. Reduction of Differential Equations to V olterra Integral Equations . . . . . . . . . 87718.2-2. Application of V olterra Equations to the Calculation of Eigenvalues . . . . . . . . 879 18.3. Reduction of Boundary Value Problems for ODEs to Fredholm Integral Equations with t h eH e l po ft h eG r e e n ’ sF u n c t i o n ........................................... 8 8 1 18.3-1. Linear Ordinary Differential Equations. Fundamental Solutions . . . . . . . . . . . 881 18.3-2. Boundary Value Problems for nth Order Differential Equations. Green’s F u n c t i o n ....................................................... 8 8 2 18.3-3. Boundary Value Problems for Second- Order Differential Equations. Green’s F u n c t i o n ....................................................... 8 8 3 18.3-4. Nonlinear Problem of Nonisothermal Flow in Plane Channel . . . . . . . . . . . . . 884 18.4. Reduction of PDEs with Boundary Conditions of the Third Kind to Integral Equations 887 18.4-1. Usage of Particular Solutions of PDEs for the Construction of Other Solutions 88718.4-2. Mass Transfer to a Particle in Fluid Flow Complicated by a Surface Reaction 88818.4-3. Integral Equations for Surface Concentration and Diffusion Flux . . . . . . . . . . 890 18.4-4. Method of Numerical Integration of the Equation for Surface Concentration . 891 18.5. Representation of Linear Boundary Value Problems in Terms of Potentials . . . . . . . . . . 892 18.5-1. Basic Types of Potentials for the Laplace Equation and Their Properties . . . . . 89218.5-2. Integral Identities. Green’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89518.5-3. Reduction of Interior Dirichlet and Neumann Problems to Integral Equations . 89518.5-4. Reduction of Exterior Dirichlet and Neumann Problems to Integral Equations 896 18.6. Representation of Solutions of Nonlinear PDEs in Terms of Solutions of Linear Integral E q u a t i o n s( I n v e r s eS c a t t e r i n g ) ............................................. 8 9 8 18.6-1. Description of the Zakharov–Shabat Method . . . . . . . . . . . . . . . . . . . . . . . . . . 89818.6-2. Korteweg–de Vries Equation and Other Nonlinear Equations . . . . . . . . . . . . . 899 Supplements Supplement 1. Elementary Functions and Their Properties ....................... 905 1.1. Power, Exponential, and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 1 . 1 - 1 . P r o p e r t i e so ft h eP o w e rF u n c t i o n ...................................... 9 0 5 1.1-2. Properties of the Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9051.1-3. Properties of the Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906 1.2. Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907 1 . 2 - 1 . S i m p l e s tR e l a t i o n s ................................................. 9 0 71 . 2 - 2 . R e d u c t i o nF o r m u l a s ................................................ 9 0 71.2-3. Relations Between Trigonometric Functions of Single Argument . . . . . . . . . . . . 9081.2-4. Addition and Subtraction of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 908 1.2-5. Products of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908 1.2-6. Powers of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908 1.2-7. Addition Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909 xxii CONTENTS 1.2-8. Trigonometric Functions of Multiple Arguments . . . . . . . . . . . . . . . . . . . . . . . . . 909 1.2-9. Trigonometric Functions of Half Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9091 . 2 - 1 0 . D i f f e r e n t i a t i o nF o r m u l a s ........................................... 9 1 0 1 . 2 - 1 1 . I n t e g r a t i o nF o r m u l a s .............................................. 9 1 0 1 . 2 - 1 2 . E x p a n s i o ni nP o w e rS e r i e s .......................................... 9 1 01.2-13. Representation in the Form of In ﬁnite Products . . . . . . . . . . . . . . . . . . . . . . . . . 910 1.2-14. Euler and de Moivre Formulas. Relationship with Hyperbolic Functions . . . . . 911 1.3. Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911 1.3-1. De ﬁnitions of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 911 1 . 3 - 2 . S i m p l e s tF o r m u l a s ................................................. 9 1 21 . 3 - 3 . S o m eP r o p e r t i e s ................................................... 9 1 2 1.3-4. Relations Between Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 912 1.3-5. Addition and Subtraction of Inverse Trigonometric Functions . . . . . . . . . . . . . . . 912 1 . 3 - 6 . D i f f e r e n t i a t i o nF o r m u l a s ............................................ 9 1 3 1 . 3 - 7 . I n t e g r a t i o nF o r m u l a s ............................................... 9 1 3 1 . 3 - 8 . E x p a n s i o ni nP o w e rS e r i e s ........................................... 9 1 3 1 . 4 . H y p e r b o l i cF u n c t i o n s ..................................................... 9 1 3 1.4-1. De ﬁn i t i o n so fH y p e r b o l i cF u n c t i o n s ................................... 9 1 3 1 . 4 - 2 . S i m p l e s tR e l a t i o n s ................................................. 9 1 3 1.4-3. Relations Between Hyperbolic Functions of Single Argument ( x≥0 ) ........ 9 1 4 1.4-4. Addition and Subtraction of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . 9141.4-5. Products of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914 1.4-6. Powers of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914 1.4-7. Addition Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9151.4-8. Hyperbolic Functions of Multiple Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9151 . 4 - 9 . H y p e r b o l i cF u n c t i o n so fH a l fA r g u m e n t ................................ 9 1 5 1 . 4 - 1 0 . D i f f e r e n t i a t i o nF o r m u l a s ........................................... 9 1 6 1 . 4 - 1 1 . I n t e g r a t i o nF o r m u l a s .............................................. 9 1 61 . 4 - 1 2 . E x p a n s i o ni nP o w e rS e r i e s .......................................... 9 1 61.4-13. Representation in the Form of In ﬁnite Products . . . . . . . . . . . . . . . . . . . . . . . . . 916 1.4-14. Relationship with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916 1 . 5 . I n v e r s eH y p e r b o l i cF u n c t i o n s .............................................. 9 1 7 1.5-1. De ﬁn i t i o n so fI n v e r s eH y p e r b o l i cF u n c t i o n s ............................. 9 1 7 1 . 5 - 2 . S i m p l e s tR e l a t i o n s ................................................. 9 1 7 1 . 5 - 3 . R e l a t i o n sB e t w e e nI n v e r s eH y p e r b o l i cF u n c t i o n s ......................... 9 1 7 1.5-4. Addition and Subtraction of Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . 9171 . 5 - 5 . D i f f e r e n t i a t i o nF o r m u l a s ............................................ 9 1 71 . 5 - 6 . I n t e g r a t i o nF o r m u l a s ............................................... 9 1 8 1 . 5 - 7 . E x p a n s i o ni nP o w e rS e r i e s ........................................... 9 1 8 Supplement 2. Finite Sums and In ﬁnite Series ................................... 919 2 . 1 . F i n i t eN u m e r i c a lS u m s ................................................... 9 1 9 2 . 1 - 1 . P r o g r e s s i o n s ...................................................... 9 1 9 2.1-2. Sums of Powers of Natural Numbers Having the Form⎝summationtextk m............... 9 1 9 2.1-3. Alternating Sums of Powers of Natural Numbers,⎝summationtext(–1)kkm............... 9 2 0 2.1-4. Other Sums Containing Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 920 2.1-5. Sums Containing Binomial Coef ﬁc i e n t s ................................ 9 2 0 2 . 1 - 6 . O t h e rN u m e r i c a lS u m s .............................................. 9 2 1 CONTENTS xxiii 2 . 2 . F i n i t eF u n c t i o n a lS u m s ................................................... 9 2 2 2.2-1. Sums Involving Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.2-2. Sums Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922 2.3. Inﬁ n i t eN u m e r i c a lS e r i e s .................................................. 9 2 4 2 . 3 - 1 . P r o g r e s s i o n s ...................................................... 9 2 4 2 . 3 - 2 . O t h e rN u m e r i c a lS e r i e s ............................................. 9 2 4 2.4. In ﬁn i t eF u n c t i o n a lS e r i e s .................................................. 9 2 5 2 . 4 - 1 . P o w e rS e r i e s ...................................................... 9 2 5 2.4-2. Trigonometric Series in One Variable Involving Sine . . . . . . . . . . . . . . . . . . . . . . 9272.4-3. Trigonometric Series in One Variable Involving Cosine . . . . . . . . . . . . . . . . . . . . 928 2.4-4. Trigonometric Series in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930 Supplement 3. Tables of Inde ﬁnite Integrals .................................... 933 3 . 1 . I n t e g r a l sI n v o l v i n gR a t i o n a lF u n c t i o n s ....................................... 9 3 3 3.1-1. Integrals Involving a+bx........................................... 9 3 3 3.1-2. Integrals Involving a+xandb+x.................................... 9 3 3 3.1-3. Integrals Involving a 2+x2.......................................... 9 3 4 3.1-4. Integrals Involving a2–x2.......................................... 9 3 5 3.1-5. Integrals Involving a3+x3.......................................... 9 3 6 3.1-6. Integrals Involving a3–x3.......................................... 9 3 6 3.1-7. Integrals Involving a4±x4.......................................... 9 3 7 3.2. Integrals Involving Irrati onal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937 3.2-1. Integrals Involving x1/2............................................. 9 3 7 3.2-2. Integrals Involving ( a+bx)p/2....................................... 9 3 8 3.2-3. Integrals Involving ( x2+a2)1/2....................................... 9 3 8 3.2-4. Integrals Involving ( x2–a2)1/2....................................... 9 3 8 3.2-5. Integrals Involving ( a2–x2)1/2....................................... 9 3 9 3.2-6. Integrals Involving Arbitrary Powers. Reduction Formulas . . . . . . . . . . . . . . . . . 939 3.3. Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9403 . 4 . I n t e g r a l sI n v o l v i n gH y p e r b o l i cF u n c t i o n s ..................................... 9 4 0 3.4-1. Integrals Involving cosh x........................................... 9 4 0 3.4-2. Integrals Involving sinh x............................................ 9 4 1 3.4-3. Integrals Involving tanh xor coth x................................... 9 4 2 3.5. Integrals Involving Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9433.6. Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944 3.6-1. Integrals Involving cos x(n=1 ,2 , ...) ................................ 9 4 4 3.6-2. Integrals Involving sin x(n=1 ,2 , ...) ................................ 9 4 5 3.6-3. Integrals Involving sin xand cos x..................................... 9 4 7 3 . 6 - 4 . R e d u c t i o nF o r m u l a s ................................................ 9 4 7 3.6-5. Integrals Involving tan xand cot x..................................... 9 4 7 3.7. Integrals Involving Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948 Supplement 4. Tables of De ﬁnite Integrals ...................................... 951 4 . 1 . I n t e g r a l sI n v o l v i n gP o w e r - L a wF u n c t i o n s ..................................... 9 5 1 4 . 1 - 1 . I n t e g r a l sO v e raF i n i t eI n t e r v a l ....................................... 9 5 1 4.1-2. Integrals Over an In ﬁn i t eI n t e r v a l ..................................... 9 5 2 4.2. Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9544 . 3 . I n t e g r a l sI n v o l v i n gH y p e r b o l i cF u n c t i o n s ..................................... 9 5 54.4. Integrals Involving Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955 xxiv CONTENTS 4.5. Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956 4 . 5 - 1 . I n t e g r a l sO v e raF i n i t eI n t e r v a l ....................................... 9 5 6 4.5-2. Integrals Over an In ﬁn i t eI n t e r v a l ..................................... 9 5 7 4 . 6 . I n t e g r a l sI n v o l v i n gB e s s e lF u n c t i o n s ......................................... 9 5 8 4.6-1. Integrals Over an In ﬁn i t eI n t e r v a l ..................................... 9 5 8 4 . 6 - 2 . O t h e rI n t e g r a l s .................................................... 9 5 9 Supplement 5. Tables of Laplace Transforms ................................... 961 5 . 1 . G e n e r a lF o r m u l a s ........................................................ 9 6 15 . 2 . E x p r e s s i o n sw i t hP o w e r - L a wF u n c t i o n s ...................................... 9 6 35.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963 5.4. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964 5.5. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965 5.6. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9665 . 7 . E x p r e s s i o n sw i t hS p e c i a lF u n c t i o n s.......................................... 9 6 7 Supplement 6. Tables of Inverse Laplace Transforms ............................. 969 6 . 1 . G e n e r a lF o r m u l a s ........................................................ 9 6 96 . 2 . E x p r e s s i o n sw i t hR a t i o n a lF u n c t i o n s ......................................... 9 7 16 . 3 . E x p r e s s i o n sw i t hS q u a r eR o o t s ............................................. 9 7 5 6 . 4 . E x p r e s s i o n sw i t hA r b i t r a r yP o w e r s .......................................... 9 7 7 6.5. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9786.6. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9796.7. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9806.8. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9816 . 9 . E x p r e s s i o n sw i t hS p e c i a lF u n c t i o n s.......................................... 9 8 1 Supplement 7. Tables of Fourier Cosine Transforms ............................. 983 7 . 1 . G e n e r a lF o r m u l a s ........................................................ 9 8 3 7 . 2 . E x p r e s s i o n sw i t hP o w e r - L a wF u n c t i o n s ...................................... 9 8 3 7.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9847.4. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9857.5. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9857.6. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9867 . 7 . E x p r e s s i o n sw i t hS p e c i a lF u n c t i o n s.......................................... 9 8 7 Supplement 8. Tables of Fourier Sine Transforms ................................ 989 8 . 1 . G e n e r a lF o r m u l a s ........................................................ 9 8 98 . 2 . E x p r e s s i o n sw i t hP o w e r - L a wF u n c t i o n s ...................................... 9 8 9 8.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990 8.4. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9918.5. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.6. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928 . 7 . E x p r e s s i o n sw i t hS p e c i a lF u n c t i o n s.......................................... 9 9 3 CONTENTS xxv Supplement 9. Tables of Mellin Transforms ..................................... 997 9 . 1 . G e n e r a lF o r m u l a s ........................................................ 9 9 7 9 . 2 . E x p r e s s i o n sw i t hP o w e r - L a wF u n c t i o n s ...................................... 9 9 8 9.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998 9.4. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 9.5. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9999.6. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 Supplement 10. Tables of Inverse Mellin Transforms ............................. 1001 10.1. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 10.2. Expressions with Exponential and Logar ithmic Functions . . . . . . . . . . . . . . . . . . . . . . . 1002 10.3. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003 10.4. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004 Supplement 11. Special Functions and Their Properties .......................... 1007 11.1. Some Coef ﬁcients, Symbols, and Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 11.1-1. Binomial Coef ﬁcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 11.1-2. Pochhammer Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 11.1-3. Bernoulli Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100811.1-4. Euler Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008 11.2. Error Functions. Exponential and Logarithmic Integrals . . . . . . . . . . . . . . . . . . . . . . . . 1009 11.2-1. Error Function and Complementary Error Function . . . . . . . . . . . . . . . . . . . . . 100911.2-2. Exponential Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 11.2-3. Logarithmic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 11.3. Sine Integral and Cosine Integral. Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 11.3-1. Sine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 11.3-2. Cosine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101111.3-3. Fresnel Integrals and Generalized Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . 1012 11.4. Gamma Function, Psi Function, and Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 11.4-1. Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 11.4-2. Psi Function (Digamma Function) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013 11.4-3. Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 11.5. Incomplete Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 11.5-1. Incomplete Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 11.5-2. Incomplete Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015 11.6. Bessel Functions (Cylindrical Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 11.6-1. De ﬁnitions and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 11.6-2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 1017 11.6-3. Zeros of Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019 11.6-4. Orthogonality Properties of Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 101911.6-5. Hankel Functions (Bessel Functions of the Third Kind) . . . . . . . . . . . . . . . . . . 1020 11.7. Modi ﬁed Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 11.7-1. De ﬁnitions. Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 11.7-2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 1022 11.8. Airy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 11.8-1. De ﬁnition and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 11.8-2. Power Series and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 xxvi CONTENTS 11.9. Conﬂ uent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 11.9-1. Kummer and Tricomi Con ﬂuent Hypergeometric Functions . . . . . . . . . . . . . . 1024 11.9-2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 102711.9-3. Whittaker Con ﬂuent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 1027 11.10. Gauss Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028 11.10-1. Various Representations of the Gauss Hypergeometric Function . . . . . . . . . 102811.10-2. Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028 11.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions . . . 1030 11.11-1. Legendre Polynomials and Legendre Functions . . . . . . . . . . . . . . . . . . . . . . 103011.11-2. Associated Legendre Functions with Integer Indices and Real Argument . . 103111.11-3. Associated Legendre Functions. General Case . . . . . . . . . . . . . . . . . . . . . . . 1032 11.12. Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034 11.12-1. De ﬁnitions. Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034 11.12-2. Integral Representations, Asymptotic Expansions, and Linear Relations . . . 1035 11.13. Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 11.13-1. Complete Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103511.13-2. Incomplete Elliptic In tegrals (Elliptic Integrals) . . . . . . . . . . . . . . . . . . . . . . 1037 11.14. Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038 11.14-1. Jacobi Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103911.14-2. Weierstrass Elliptic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042 11.15. Jacobi Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 11.15-1. Series Representation of the Jacobi Theta Functions. Simplest Properties . . 104311.15-2. Various Relations and Formulas. C onnection with Jacobi Elliptic Functions 1044 11.16. Mathieu Functions and Modi ﬁed Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 11.16-1. Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104511.16-2. Modi ﬁed Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 11.17. Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047 11.17-1. Laguerre Polynomials and Generalized Laguerre Polynomials . . . . . . . . . . . 104711.17-2. Chebyshev Polynomials and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104811.17-3. Hermite Polynomials and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 11.17-4. Jacobi Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 11.17-5. Gegenbauer Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 11.18. Nonorthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 11.18-1. Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105211.18-2. Euler Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053 Supplement 12. Some Notions of Functional Analysis ............................ 1055 12.1. Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 12.1-1. De ﬁnition of a Function of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . 1055 12.1-2. Classes of Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 12.1-3. Properties of Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 12.1-4. Criteria for Functions to Have Bounded Variation . . . . . . . . . . . . . . . . . . . . . . 105712.1-5. Properties of Continuous Functions of Bounded Variation . . . . . . . . . . . . . . . . 1057 12.2. Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 12.2-1. Basic De ﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 12.2-2. Properties of the Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105812.2-3. Existence Theorems for the Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 1058 CONTENTS xxvii 12.3. Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059 12.3-1. Riemann Integral and the Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 105912.3-2. Sets of Zero Measure. Notion of “Almost Everywhere” . . . . . . . . . . . . . . . . . . 1060 12.3-3. Step Functions and Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 12.3-4. De ﬁnition and Properties of the Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . 1061 12.3-5. Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062 12.3-6. Integration Over Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106312.3-7. Case of an In ﬁnite Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063 12.3-8. Case of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064 12.3-9. Spaces L p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064 12.4. Linear Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 12.4-1. Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106512.4-2. Linear Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 12.4-3. Space of Continuous Functions C(a,b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 12.4-4. Lebesgue Space L p(a,b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 12.4-5. H ¨older Space Cα(0, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 12.4-6. Space of Functions of Bounded Variation V(0, 1) . . . . . . . . . . . . . . . . . . . . . . . 1066 12.5. Euclidean and Hilbert Sp aces. Linear Operators in Hilbert Spaces . . . . . . . . . . . . . . . . 1067 12.5-1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 12.5-2. Euclidean and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106712.5-3. Linear Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 References ................................................................. 1071 Index ..................................................................... 1081 AUTHORS Andrei D. Polyanin, D.Sc., Ph.D., is a well-known scientist of broad interests and is active in various areas of mathematics, me- chanics, and chemical engineering sciences. He is one of the most prominent authors in the ﬁeld of reference literature on mathemat- ics and physics. Professor Polyanin graduated with honors from the Depart- ment of Mechanics and Mathematics of Moscow State University in 1974. He received his Ph.D. degree in 1981 and D.Sc. degree in1986 at the Institute for Problems in Mechanics of the Russian (for- mer USSR) Academy of Sciences. Since 1975, Professor Polyanin has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences; he is also Professor of Mathematics at Bauman Moscow State Technical University. He is a member ofthe Russian National Committee on Theoretical and Applied Me- chanics and of the Mathematics and Mechanics Expert Council of the Higher Certiﬁ cation Committee of the Russian Federation. Professor Polyanin has made important contributions to exact and approximate analytical meth- ods in the theory of differential equations, mathematical physics, integral equations, engineering mathematics, theory of heat and mass transfer, and chemical hydrodynamics. He obtained exact solutions for several thousand ordinary differential, partial differential, and integral equations. Professor Polyanin is an author of more than 30 books in English, Russian, German, and Bulgar- ian as well as over 120 research papers and thr ee patents. He has written a number of fundamental handbooks, including A. D. Polyanin and V . F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations , CRC Press, 1995 and 2003; A. D. Polyanin and A. V . Manzhirov, Handbook of Integral Equations , CRC Press, 1998; A. D. Polyanin, Handbook of Linear Partial Differen- tial Equations for Engineers and Scientists , Chapman & Hall/CRC Press, 2002; A. D. Polyanin, V . F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations ,T a y l o r & Francis, 2002; A. D. Polyanin and V . F. Zaitsev, Handbook of Nonlinear Partial Differential Equations , Chapman & Hall/CRC Press, 2004, and A. D. Polyanin and A. V . Manzhirov, Handbook of Mathematics for Engineers and Scientists , Chapman & Hall/CRC Press, 2007. Professor Polyanin is editor of the book series Differential and Integral Equations and Their Applications , Chapman & Hall/CRC Press, London/Boca Raton, and Physical and Mathematical Reference Literature , Fizmatlit, Moscow. He is also Editor-in-Chief of the international scienti ﬁc- educational Website EqWorld—The World of Mathematical Equations (http://eqworld.ipmnet.ru), which is visited by over 1700 users a day worldwide. Professor Polyanin is a member of the Editorial Board of the journal Theoretical Foundations of Chemical Engineering. In 1991, Professor Polyanin was awarded a Chaplygin Prize of the Russian Academy of Sciences for his research in mechanics. In 2001, he received an award from the Ministry of Education of theRussian Federation. Address: Institute for Problems in Mechanics, Vernadsky Ave. 101 Bldg 1, 119526 Moscow, Russia Home page: http://eqworld.ipmnet.ru/polyanin-ew.htm xxix xxx AUTHORS Alexander V . Manzhirov, D.Sc., Ph.D., is a noted scientist in the ﬁelds of mechanics and applied mathematics, integral equations, and their applications. After graduating with honors from the Department of Mechan- ics and Mathematics of Rostov State University in 1979, Alexander Manzhirov attended postgraduate courses at Moscow Institute ofCivil Engineering. He received his Ph.D. degree in 1983 at Moscow Institute of Electronic Engineering Industry and D.Sc. degree in 1993 at the Institute for Problems in Mechanics of the Russian(former USSR) Academy of Sciences. Since 1983, Alexander Manzhirov has been working at the Institute for Problems in Me- chanics of the Russian Academy o f Sciences. Currently, he is head of the Laboratory for Modeling in Solid Mechanics at the same institute. Professor Manzhirov is also head of a branch of the Department of Applied Mathematics at Bauman Moscow State Technical University, profe ssor of mathematics at Moscow State University of Engineering and Computer Science,vice-chairman of Mathematics and Mechanics Expert Councilof the Higher Certi ﬁcation Committee of the Russian Federation, executive secretary of Solid Mechanics Scienti ﬁc Council of the Russian Academy of Sciences, and expert in mathematics, mechanics, and computer science of the Russian Foundation for Basic Research. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and the European Mechanics Society (EUROMECH), and member of the editorial board of the journal Mechanics of Solids and the international scienti ﬁc-educational Website EqWorld—The World of Mathematical Equations (http://eqworld.ipmnet.ru). Professor Manzhirov has made important contributions to new mathematical methods for solving problems in the ﬁelds of integral equations and their applications, mechanics of growing solids, contact mechanics, tribology, viscoelasticity, and creep theory. He is an author of more than ten books (including Contact Problems in Mechanics of Growing Solids [in Russian], Nauka, Moscow, 1991; Handbook of Integral Equations , CRC Press, Boca Raton, 1998; Handbuch der Integralgleichungen: Exacte L ¨osungen , Spektrum Akad. Verlag, Heidelberg, 1999; Contact Problems in the Theory of Creep [in Russian], National Academy of Sciences of Armenia, Erevan, 1999; A. D. Polyanin and A. V . Manzhirov, Handbook of Mathematics for Engineers and Scientists , Chapman & Hall/CRC Press, Boca Raton, 2007), more than 70 research papers, and two patents. Professor Manzhirov is a winner of the First Competition of the Science Support Foundation 2001, Moscow. Address: Institute for Problems in Mechanics, Vernadsky Ave. 101 Bldg 1, 119526 Moscow, Russia. Home page: http://eqworld.ipmnet.ru/en/board/manzhirov.htm. PREFACE TO THE NEW EDITION Handbook of Integral Equations , Second Edition, a unique reference for engineers and scientists, contains over 2,500 integral equations with solutions, as well as analytical and numerical methods for solving linear and nonlinear equations. It considers V olterra, Fredholm, Wiener–Hopf, Hammerstein, Urysohn, and other equations, which arise in mathematics, physics, engineering sciences, economics,etc. In total, the number of equations described is an order of magnitude greater than in any other book available. The second edition has been substantially update d, revised, and exte nded. It includes new chapters on mixed multidimensional equations, m ethods of integral equations for ODEs and PDEs, and about 400 new equations with exact solutions. It presents a considerable amount of newmaterial on V olterra, Fredholm, singular, hypersingular, dual, and nonlinear integral equations, integral transforms, and special functions. Many examples were added for illustrative purposes. The new edition has been increased by a total of over 300 pages. Note that the ﬁrst part of the book can be used as a database of test problems for numerical and approximate methods for solving linear and nonlinear integral equations. We would like to express our deep gratitude to Alexei Zhurov and Vasilii Silvestrov for fruitful discussions. We also appreciate the help of Grigory Yosi ﬁan in translating new sections of this book and valuable remarks. The authors hope that the handbook will prove helpful for a wide audience of researchers, college and university teachers, engineers, and students in various ﬁelds of applied mathematics, mechanics, physics, chemistry, biology, economics, and engineering sciences. A. D. Polyanin A. V . Manzhirov PREFACE TO THE FIRST EDITION Integral equations are encountered in various ﬁelds of science and numerous applications (in elasticity, plasticity, heat and mass transfer, oscillation theory, ﬂuid dynamics, ﬁltration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical en-gineering, economics, medicine, etc.). Exact (closed-form) solutions of integral equations play an important role in the proper un- derstanding of qualitative features of many phenomena and processes in various areas of naturalscience. Lots of equations of physics, chemistry, and biology contain functions or parameters which are obtained from experiments and hence are not strictly ﬁxed. Therefore, it is expedient to choose the structure of these functions so that it would be easier to analyze and solve the equation. As a possible selection criterion, one may adopt the re quirement that the model integral equation admits a solution in a closed form. Exact solutions can be used to verify the consistency and estimate errorsof various numerical, asymptotic, and approximate methods. More than 2,100 integral equations and their solutions are given in the ﬁrst part of the book (Chapters 1–6). A lot of new exact solutions to linear and nonlinear equations are included. Specialattention is paid to equations of general form, which depend on arbitrary functions. The other equations contain one or more free parameters (the book actually deals with families of integral xxxi xxxii PREFACE equations); it is the reader’s option to ﬁx these parameters. In total, the number of equations described in this handbook is an order of magnitude greater than in any other book currentlyavailable. The second part of the book (Chapters 7–14) presents exact, approximate analytical, and numer- ical methods for solving linear and nonlinear integral equations. Apart from the classical methods, some new methods are also described. When selecting the material, the authors have given a pronounced preference to practical aspects of the matter; that is, to methods that allow effectively“constructing” the solution. For the reader’s be tter understanding of the methods, each section is supplied with examples of speci ﬁc equations. Some sections may be used by lecturers of colleges and universities as a basis for courses on integra l equations and mathematical physics equations for graduate and postgraduate students. For the convenience of a wide audience with different mathematical backgrounds, the authors tried to do their best,wherever possible, to avoid special terminology. Therefore, some of the methods are outlined in a schematic and somewhat simpli ﬁed manner, with necessary references made to books where these methods are considered in more detail. For some nonlinear equations, only solutions of the simplest form are given. The book does not cover two-, three-, and multidimensional integral equations. The handbook consists of chapters, sections, and subsections. Equations and formulas are numbered separately in each section. The equati ons within a section are arranged in increasing order of complexity. The extensive table of conten ts provides rapid access to the desired equations. For the reader’s convenience, the main material is followed by a number of supplements, where some properties of elementary and special functions are described, tables of inde ﬁnite and de ﬁnite integrals are given, as well as tables of Laplace, Mellin, and other transforms, which are used in thebook. Theﬁrst and second parts of the book, just as many s ections, were written so that they could be read independently from each other. This allows th e reader to quickly get to the heart of the matter. We would like to express our deep gratitude to Rolf Sulanke and Alexei Zhurov for fruitful discussions and valuable remarks. We also appr eciate the help of Vlad imir Nazaikinskii and Alexander Shtern in translating the second part of this book, and are thankful to Inna Shingareva for her assistance in preparing the camera-ready copy of the book. The authors hope that the handbook will prove helpful for a wide audience of researchers, college and university teachers, e ngineers, and students in various ﬁelds of mathematics, mechanics, physics, chemistry, biology, economics, and engineering sciences. A. D. Polyanin A. V . Manzhirov SOME REMARKS AND NOTATION 1.In Chapters 1–11, 14, and 18 in the original integral equations, the independent variable is denoted by x, the integration variable by t, and the unknown function by y=y(x). 2.For a function of one variable f=f(x), we use the following notation for the derivatives: f/prime x=df dx,f/prime/prime xx=d2f dx2,f/prime/prime/prime xxx=d3f dx3,f/prime/prime/prime/prime xxxx =d4f dx4,a n d f(n) x=dnf dxnforn≥5. Occasionally, we use the similar notation for par tial derivatives of a function of two variables, for example, K/prime x(x,t)=∂ ∂xK(x,t). 3.In some cases, we use the operator notation⎝bracketleftBig f(x)d dx⎝bracketrightBign g(x), which is de ﬁned recursively by ⎝bracketleftbigg f(x)d dx⎝bracketrightbiggn g(x)=f(x)d dx⎝braceleftbigg⎝bracketleftbigg f(x)d dx⎝bracketrightbiggn–1 g(x)⎝bracerightbigg . 4.It is indicated in the beginning of Chapters 1–8 that f=f(x),g=g(x),K=K(x), etc. are arbitrary functions, and A,B, etc. are free parameters. This means that: (a)f=f(x),g=g(x),K=K(x), etc. are assumed to be continuous real-valued functions of real arguments;* (b) if the solution contains derivatives of these functions, then the functions are assumed to be sufﬁciently differentiable;** (c) if the solution contains integrals with these functions (in combination with other functions), then the integrals are supposed to converge; (d) the free parameters A,B, etc. may assume any real values for which the expressions occurring in the equation and the solution make sense (for example, if a solution contains a factorA 1–A, then it is implied that A≠1; as a rule, this is not speci ﬁed in the text). 5.The notations Re zand Im zstand, respectively, for the real and the imaginary part of a complex quantity z. 6.In the ﬁrst part of the book (Chapters 1–8) when refe rencing a particular equation, we use a notation like 2.3.15, which implies equation 15 from Section 2.3. 7.To highlight portions of the text, the following symbols are used in the book: /trianglerightsldindicates important information pertaining to a group of equations (Chapters 1–8); indicates the literature used in the preparation of the text in speci ﬁc equations (Chapters 1–8) or sections (Chapters 9–18). * Less severe restrictions on these functions are presented in the second part of the book. ** Restrictions (b) and (c) imposed on f=f(x),g=g(x),K=K(x), etc. are not mentioned in the text. xxxiii Part I Exact Solutions of Integral Equations Chapter 1 Linear Equations of the First Kind with Variable Limit of Integration /trianglerightsld Notation: f=f(x),g=g(x),h=h(x),K=K(x), andM=M(x)are arbitrary functions (these may be composite functions of the argument depending on two variables xandt);A,B,C,D,E, a,b,c,α,β,γ,λ, andµare free parameters; and mandnare nonnegative integers. /trianglerightsldPreliminary remarks. For equations of the form ⎝integraldisplayx aK(x,t)y(t)dt=f(x), a≤x≤b, where the functions K(x,t)a n d f(x) are continuous, the right-hand side must satisfy the following conditions: 1◦.I fK(a,a)≠0, then we must have f(a) = 0 (for example, the right-hand sides of equations 1.1.1 and 1.2.1 must satisfy this condition). 2◦.I fK(a,a)=K/prime x(a,a)=···=K(n–1) x(a,a)=0 , 0<⎝vextendsingle⎝vextendsingleK(n) x(a,a)⎝vextendsingle⎝vextendsingle<∞, then the right-hand side of the equation must s atisfy the conditions f(a)=f/prime x(a)=···=f(n) x(a)=0 . For example, with n= 1, these are constraints for the right-hand side of equation 1.1.2. 3◦.I fK(a,a)=K/prime x(a,a)=···=K(n–1) x(a,a)=0 , K(n) x(a,a)=∞, then the right-hand side of the equation must satisfy the conditions f(a)=f/prime x(a)=···=f(n–1) x(a)=0 . For example, with n= 1, this is a constraint for the right-hand side of equation 1.1.30. 4◦. For unbounded K(x,t) with integrable power-law or logarithmic singularity at x=tand continuous f(x), no additional conditions are imposed on th e right-hand side of the integral equation (e.g., see Abel’s equation 1.1.36). In the case of a difference kernel, K(x,t)=K(x–t), that can be represented as x→tin the form K(x–t)=A(x–t)λ+o⎝parenleftbig (x–t)λ⎝parenrightbig (0 < |A|<∞), the right-hand side of the integral equation, for λ≥0, must satisf y the conditions f(a)=f/prime x(a)=···=f([λ]) x(a)=0 , where [ λ] is the integer part of λ. For –1 < λ< 0, there are no additional conditions imposed on the function f(x). In Chapter 1, conditions 1◦–3◦are as a rule not speciﬁed. 3 4 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 1.1. Equations Whose Kernels Contain Power-Law Functions 1.1-1. Kernels Linear in the Arguments xandt. 1.⎝integraldisplay ⎝integraldisplayx ay(t)dt=f(x). Solution: y(x)=f/prime x(x). 2.⎝integraldisplay ⎝integraldisplayx a(x–t)y(t)dt=f(x). Solution: y(x)=f/prime/prime xx(x). 3.⎝integraldisplay ⎝integraldisplayx a(Ax +Bt +C)y(t)dt=f(x). This is a special case of equation 1.9.5 with g(x)=x. 1◦. Solution with B≠–A: y(x)=d dx⎝braceleftbigg⎝bracketleftbig (A+B)x+C⎝bracketrightbig–A A+B⎝integraldisplayx a⎝bracketleftbig (A+B)t+C⎝bracketrightbig–B A+Bf/prime t(t)dt⎝bracerightbigg . 2◦. Solution with B=–A: y(x)=1 Cd dx⎝bracketleftbigg exp⎝parenleftBig –A Cx⎝parenrightBig⎝integraldisplayx aexp⎝parenleftBigA Ct⎝parenrightBig f/prime t(t)dt⎝bracketrightbigg . 1.1-2. Kernels Quadratic in the Arguments xandt. 4.⎝integraldisplay ⎝integraldisplayx a(x–t)2y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Solution: y(x)=1 2f/prime/prime/prime xxx(x). 5.⎝integraldisplay ⎝integraldisplayx a(x2–t2)y(t)dt=f(x), f(a)=f/prime x(a)=0 . This is a special case of equation 1.9.2 with g(x)=x2. Solution: y(x)=1 2x2⎝bracketleftbig xf/prime/prime xx(x)–f/prime x(x)⎝bracketrightbig . 6.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ax2+Bt2⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=x2.F o rB=–A, see equation 1.1.5. Solution: y(x)=1 A+Bd dx⎝bracketleftbigg x–2A A+B⎝integraldisplayx at–2B A+Bf/prime t(t)dt⎝bracketrightbigg . 7.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ax2+Bt2+C⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.5 with g(x)=x2. Solution: y(x)=s i g n ϕ(x)d dx⎝braceleftbigg |ϕ(x)|–A A+B⎝integraldisplayx a|ϕ(t)|–B A+Bf/prime t(t)dt⎝bracerightbigg ,ϕ(x)=(A+B)x2+C. 1.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 5 8.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Ax2+(B–A)xt–Bt2⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Differentiating with respect to xyields an equation of the form 1.1.3: ⎝integraldisplayx a[2Ax+(B–A)t]y(t)dt=f/prime x(x). Solution: y(x)=1 A+Bd dx⎝bracketleftbigg x–2A A+B⎝integraldisplayx atA–B A+Bf/prime/prime tt(t)dt⎝bracketrightbigg . 9.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ax2+Bt2+Cx +Dt +E⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Ax2+Cxandh(t)=Bt2+Dt+E. 10.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axt +Bt2+Cx +Dt +E⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=x,h1(t)=At+C,g2(x)=1 ,a n d h2(t)=Bt2+Dt+E. 11.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ax2+Bxt +Cx +Dt +E⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Bx+D,h1(t)=t,g2(x)=Ax2+Cx+E, andh2(t)=1 . 1.1-3. Kernels Cubic in the Arguments xandt. 12.⎝integraldisplay ⎝integraldisplayx a(x–t)3y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Solution: y(x)=1 6f/prime/prime/prime/prime xxxx(x). 13.⎝integraldisplay ⎝integraldisplayx a(x3–t3)y(t)dt=f(x), f(a)=f/prime x(a)=0 . This is a special case of equation 1.9.2 with g(x)=x3. Solution: y(x)=1 3x3⎝bracketleftbig xf/prime/prime/prime xxx(x)–2f/prime x(x)⎝bracketrightbig . 14.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ax3+Bt3⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=x3.F o rB=–A, see equation 1.1.13. Solution with 0 ≤a≤x:y(x)=1 A+Bd dx⎝bracketleftbigg x–3A A+B⎝integraldisplayx at–3B A+Bf/prime t(t)dt⎝bracketrightbigg . 15.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ax3+Bt3+C⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.5 with g(x)=x3. 6 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 16.⎝integraldisplay ⎝integraldisplayx a(x2t–xt2)y(t)dt=f(x), f(a)=f/prime x(a)=0 . This is a special case of equation 1.9.11 with g(x)=x2andh(x)=x. Solution: y(x)=1 xd2 dx2⎝bracketleftbigg1 xf(x)⎝bracketrightbigg . 17.⎝integraldisplay ⎝integraldisplayx a(Ax2t+Bxt2)y(t)dt=f(x). This is a special case of equation 1.9.12 with g(x)=x2andh(x)=x.F o r B=–A,s e e equation 1.1.16. Solution: y(x)=1 (A+B)xd dx⎝braceleftbigg x–A A+B⎝integraldisplayx at–B A+Bd dt⎝bracketleftbigg1 tf(t)⎝bracketrightbigg dt⎝bracerightbigg . 18.⎝integraldisplay ⎝integraldisplayx a(Ax3+Bxt2)y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Ax3,h1(t)=1 ,g 2(x)=Bx,a n dh2(t)=t2. 19.⎝integraldisplay ⎝integraldisplayx a(Ax3+Bx2t)y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Ax3,h1(t)=1 , g2(x)=Bx2,a n d h2(t)=t. 20.⎝integraldisplay ⎝integraldisplayx a(Ax2t+Bt3)y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Ax2,h1(t)=t,g2(x)=B,a n dh2(t)=t3. 21.⎝integraldisplay ⎝integraldisplayx a(Axt2+Bt3)y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Ax,h1(t)=t2,g2(x)=B,a n dh2(t)=t3. 22.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig A3x3+B3t3+A2x2+B2t2+A1x+B1t+C⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=A3x3+A2x2+A1x+Candh(t)= B3t3+B2t2+B1t. 1.1-4. Kernels Containing Higher-Order Polynomials in xandt. 23.⎝integraldisplay ⎝integraldisplayx a(x–t)ny(t)dt=f(x), n=1 , 2 , ... It is assumed that the right-hand of the equation satisﬁes the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=1 n!f(n+1) x(x). Example. Forf(x)=Axm,w h e r e mis a positive integer, m>n, the solution has the form y(x)=Am! n!(m–n–1 ) !xm–n–1. 1.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 7 24.⎝integraldisplay ⎝integraldisplayx a(xn–tn)y(t)dt=f(x), f(a)=f/prime x(a)=0 , n=1 , 2 , ... Solution: y(x)=1 nd dx⎝bracketleftbiggf/prime x(x) xn–1⎝bracketrightbigg . 25.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig tnxn+1–xntn+1⎝parenrightbig⎝parenrightbig y(t)dt=f(x), n=2 , 3 , ... This is a special case of equation 1.9.11 with g(x)=xn+1andh(x)=xn. Solution: y(x)=1 xnd2 dx2⎝bracketleftbiggf(x) xn⎝bracketrightbigg . 1.1-5. Kernels Containing Rational Functions. 26.⎝integraldisplay ⎝integraldisplayx 0y(t)dt x+t=f(x). 1◦. For a polynomial right-hand side, f(x)=N⎝summationtext n=0Anxn, the solution has the form y(x)=N⎝summationdisplay n=0An Bnxn,Bn= (–1)n⎝bracketleftbigg ln 2 +n⎝summationdisplay k=1(–1)k k⎝bracketrightbigg . 2◦.F o rf(x)=xλN⎝summationtext n=0Anxn,w h e r e λis an arbitrary number ( λ> –1), the solution has the form y(x)=xλN⎝summationdisplay n=0An Bnxn,Bn=⎝integraldisplay1 0tλ+ndt 1+t. 3◦.F o rf(x)=l nx⎝parenleftBigN⎝summationtext n=0Anxn⎝parenrightBig , the solution has the form y(x)=l nxN⎝summationdisplay n=0An Bnxn+N⎝summationdisplay n=0AnIn B2nxn, Bn= (–1)n⎝bracketleftbigg ln 2 +n⎝summationdisplay k=1(–1)k k⎝bracketrightbigg ,In= (–1)n⎝bracketleftbiggπ2 12+n⎝summationdisplay k=1(–1)k k2⎝bracketrightbigg . 4◦.F o rf(x)=N⎝summationtext n=0An⎝parenleftbig lnx)n, the solution of the equation has the form y(x)=N⎝summationdisplay n=0AnYn(x), where the functions Yn=Yn(x)a r eg i v e nb y Yn(x)=⎝braceleftbiggdn dλn⎝bracketleftbiggxλ I(λ)⎝bracketrightbigg⎝bracerightbigg λ=0,I(λ)=⎝integraldisplay1 0zλdz 1+z. 8 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 5◦.F o rf(x)=N⎝summationtext n=1Ancos(λ nlnx)+N⎝summationtext n=1Bnsin(λnlnx), the solution of the equation has the form y(x)=N⎝summationdisplay n=1Cncos(λ nlnx)+N⎝summationdisplay n=1Dnsin(λnlnx), where the constants CnandDnare found by the method of undetermined coefﬁcients. 6◦. For arbitrary f(x), the transformation x=1 2e2z,t=1 2e2τ,y(t)=e–τw(τ),f(x)=e–zg(z) leads to an integral equation with difference kernel of the form 1.9.27: ⎝integraldisplayz –∞w(τ)dτ cosh(z –τ)=g(z). 27.⎝integraldisplay ⎝integraldisplayx 0y(t)dt ax +bt=f(x), a>0 , a+b>0 . 1◦. For a polynomial right-hand side, f(x)=N⎝summationtext n=0Anxn, the solution has the form y(x)=N⎝summationdisplay n=0An Bnxn,Bn=⎝integraldisplay1 0tndt a+bt. 2◦.F o rf(x)=xλN⎝summationtext n=0Anxn,w h e r e λis an arbitrary number ( λ> –1), the solution has the form y(x)=xλN⎝summationdisplay n=0An Bnxn,Bn=⎝integraldisplay1 0tλ+ndt a+bt. 3◦.F o rf(x)=l nx⎝parenleftBigN⎝summationtext n=0Anxn⎝parenrightBig , the solution has the form y(x)=l nxN⎝summationdisplay n=0An Bnxn–N⎝summationdisplay n=0AnCn B2nxn,Bn=⎝integraldisplay1 0tndt a+bt,Cn=⎝integraldisplay1 0tnlnt a+btdt. 4◦. For some other special forms of the right-hand side (see items 4 and 5, equation 1.1.26), the solution may be found by the method of undetermined coefﬁcients. 28.⎝integraldisplay ⎝integraldisplayx 0y(t)dt ax2+bt2=f(x), a>0 , a+b>0 . 1◦. For a polynomial right-hand side, f(x)=N⎝summationtext n=0Anxn, the solution has the form y(x)=N⎝summationdisplay n=0An Bnxn+1,Bn=⎝integraldisplay1 0tn+1dt a+bt2. 1.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 9 Example. Fora=b=1a n d f(x)=Ax2+Bx+C, the solution of the integral equation is: y(x)=2A 1–l n2x3+4B 4–πx2+2C ln 2x. 2◦.F o rf(x)=xλN⎝summationtext n=0Anxn,w h e r e λis an arbitrary number ( λ> –1), the solution has the form y(x)=xλN⎝summationdisplay n=0An Bnxn+1,Bn=⎝integraldisplay1 0tλ+n+1dt a+bt2. 3◦.F o rf(x)=l nx⎝parenleftBigN⎝summationtext n=0Anxn⎝parenrightBig , the solution has the form y(x)=l nxN⎝summationdisplay n=0An Bnxn+1–N⎝summationdisplay n=0AnCn B2nxn+1,Bn=⎝integraldisplay1 0tn+1dt a+bt2,Cn=⎝integraldisplay1 0tn+1lnt a+bt2dt. 29.⎝integraldisplay ⎝integraldisplayx 0y(t)dt axm+btm=f(x), a>0 , a+b>0 , m=1 , 2 , ... 1◦. For a polynomial right-hand side, f(x)=N⎝summationtext n=0Anxn, the solution has the form y(x)=N⎝summationdisplay n=0An Bnxm+n–1,Bn=⎝integraldisplay1 0tm+n–1dt a+btm. 2◦.F o rf(x)=xλN⎝summationtext n=0Anxn,w h e r e λis an arbitrary number ( λ> –1), the solution has the form y(x)=xλN⎝summationdisplay n=0An Bnxm+n–1,Bn=⎝integraldisplay1 0tλ+m+n–1dt a+btm. 3◦.F o rf(x)=l nx⎝parenleftBigN⎝summationtext n=0Anxn⎝parenrightBig , the solution has the form y(x)=l nxN⎝summationdisplay n=0An Bnxm+n–1–N⎝summationdisplay n=0AnCn B2nxm+n–1, Bn=⎝integraldisplay1 0tm+n–1dt a+btm,Cn=⎝integraldisplay1 0tm+n–1lnt a+btmdt. 1.1-6. Kernels Containing Square Roots. 30.⎝integraldisplay ⎝integraldisplayx a√ x–ty(t)dt=f(x). Differentiating with respect to x, we arrive at Abel’s equation 1.1.36: ⎝integraldisplayx ay(t)dt √ x–t=2f/prime x(x). Solution: y(x)=2 πd2 dx2⎝integraldisplayx af(t)dt √ x–t. 10 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 31.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig√ x–√ t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.1.45 with µ=1 2. Solution: y(x)=2d dx⎝bracketleftbig√ xf/prime x(x)⎝bracketrightbig . 32.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig A√ x+B√ t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.1.46 with µ=1 2. 33.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig 1+b√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Differentiating with respect to x, we arrive at Abel’s equation of the second kind 2.1.46: y(x)+b 2⎝integraldisplayx ay(t)dt √ x–t=f/prime x(x). 34.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig t√ x–x√ t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.11 with g(x)=√ xandh(x)=x. 35.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig At√ x+Bx√ t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.12 with g(x)=√ xandh(t)=t. 36.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ x–t=f(x). Abel’s equation. Solution: y(x)=1 πd dx⎝integraldisplayx af(t)dt √ x–t=f(a) π√ x–a+1 π⎝integraldisplayx af/prime t(t)dt √ x–t. Reference: E. T. Whittaker and G. N. Watson (1958). 37.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbigg ⎝parenleftbigg b+1 √ x–t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). Let us rewrite the equation in the form ⎝integraldisplayx ay(t)dt √ x–t=f(x)–b⎝integraldisplayx ay(t)dt. Assuming the right-hand side to be known, we s olve this equation as Abel’s equation 1.1.36. After some manipulations, we arrive at Abel’s equation of the second kind 2.1.46: y(x)+b π⎝integraldisplayx ay(t)dt √ x–t=F(x), where F(x)=1 πd dx⎝integraldisplayx af(t)dt √ x–t. 1.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 11 38.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbigg ⎝parenleftbigg1 √ x–1 √ t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 1.1.45 with µ=–1 2. Solution: y(x)=– 2⎝bracketleftbig x3/2f/prime x(x)⎝bracketrightbig/prime x,a>0 . 39.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbigg ⎝parenleftbiggA √ x+B √ t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 1.1.46 with µ=–1 2. 40.⎝integraldisplay ⎝integraldisplayx –x⎝radicalbigg x–t x+ty(t)dt=f(x). Solution: y(x)=signx 2π⎝bracketleftbiggd dx⎝integraldisplay|x| 0f(t)–f(–t) √ x2–t2dt–1 xd dx⎝integraldisplay|x| 0t[f(t)–f(–t)] √ x2–t2dt⎝bracketrightbigg . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992). 41.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ x2–t2=f(x). Solution: y=2 πd dx⎝integraldisplayx atf(t)dt √ x2–t2. Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975). 42.⎝integraldisplay ⎝integraldisplayx 0y(t)dt √ ax2+bt2=f(x), a>0 , a+b>0 . 1◦. For a polynomial right-hand side, f(x)=N⎝summationtext n=0Anxn, the solution has the form y(x)=N⎝summationdisplay n=0An Bnxn,Bn=⎝integraldisplay1 0tndt √ a+bt2. 2◦.F o rf(x)=xλN⎝summationtext n=0Anxn,w h e r e λis an arbitrary number ( λ> –1), the solution has the form y(x)=xλN⎝summationdisplay n=0An Bnxn,Bn=⎝integraldisplay1 0tλ+ndt √ a+bt2. 3◦.F o rf(x)=l nx⎝parenleftBigN⎝summationtext n=0Anxn⎝parenrightBig , the solution has the form y(x)=l nxN⎝summationdisplay n=0An Bnxn–N⎝summationdisplay n=0AnCn B2nxn,Bn=⎝integraldisplay1 0tndt √ a+bt2,Cn=⎝integraldisplay1 0tnlnt √ a+bt2dt. 4◦.F o rf(x)=N⎝summationtext n=0An⎝parenleftbig lnx)n, the solution of the equation has the form y(x)=N⎝summationdisplay n=0AnYn(x), 12 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION where the functions Yn=Yn(x)a r eg i v e nb y Yn(x)=⎝braceleftbiggdn dλn⎝bracketleftbiggxλ I(λ)⎝bracketrightbigg⎝bracerightbigg λ=0,I(λ)=⎝integraldisplay1 0zλdz √ a+bz2. 5◦.F o rf(x)=N⎝summationtext n=1Ancos(λ nlnx)+N⎝summationtext n=1Bnsin(λnlnx), the solution of the equation has the form y(x)=N⎝summationdisplay n=1Cncos(λ nlnx)+N⎝summationdisplay n=1Dnsin(λnlnx), where the constants CnandDnare found by the method of undetermined coefﬁcients. 1.1-7. Kernels Containing Arbitrary Powers. 43.⎝integraldisplay ⎝integraldisplayx a(x–t)λy(t)dt=f(x), f(a)=0 , 0< λ<1 . Differentiating with respect to x, we arrive at the generalized Abel equation 1.1.47: ⎝integraldisplayx ay(t)dt (x–t)1–λ=1 λf/prime x(x). Solution: y(x)=kd2 dx2⎝integraldisplayx af(t)dt (x–t)λ,k=sin(πλ) πλ. Reference: F. D. Gakhov (1977). 44.⎝integraldisplay ⎝integraldisplayx a(x–t)µy(t)dt=f(x). Forµ= 0 ,1 ,2 , ..., see equations 1.1.1, 1.1.2, 1.1.4, 1.1.12, and 1.1.23. For 0 < µ<1 ,s e e equation 1.1.43. Setµ=n–λ,w h e r e n=1 ,2 , ...and 0 ≤λ<1 ,a n d f(a)=f/prime x(a)=···=f(n–1) x(a)=0 . On differentiating the equation ntimes, we arrive at an equation of the form 1.1.47: ⎝integraldisplayx ay(t)dτ (x–t)λ=Γ(µ–n+1 ) Γ(µ+1 )f(n) x(x), whereΓ(µ) is the gamma function. Example. Setf(x)=Axβ,w h e r e β≥0, and let µ>– 1a n d µ–β≠0, 1, 2, ...In this case, the solution has the form y(x)=AΓ(β+1 ) Γ(µ+1 )Γ(β–µ)xβ–µ–1. Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971). 45.⎝integraldisplay ⎝integraldisplayx a(xµ–tµ)y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=xµ. Solution: y(x)=1 µ⎝bracketleftbig x1–µf/prime x(x)⎝bracketrightbig/prime x. 1.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 13 46.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axµ+Btµ⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=xµ.F o rB=–A, see equation 1.1.44. Solution: y(x)=1 A+Bd dx⎝bracketleftbigg x–Aµ A+B⎝integraldisplayx at–Bµ A+Bf/prime t(t)dt⎝bracketrightbigg . 47.⎝integraldisplay ⎝integraldisplayx ay(t)dt (x–t)λ=f(x), 0 < λ<1 . The generalized Abel equation. Solution: y(x)=sin(πλ) πd dx⎝integraldisplayx af(t)dt (x–t)1–λ=sin(πλ) π⎝bracketleftbiggf(a) (x–a)1–λ+⎝integraldisplayx af/prime t(t)dt (x–t)1–λ⎝bracketrightbigg . Reference: E. T. Whittaker and G. N. Watson (1958). 48.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbigg ⎝bracketleftbigg b+1 (x–t)λ⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), 0 < λ<1 . Rewrite the equation in the form ⎝integraldisplayx ay(t)dt (x–t)λ=f(x)–b⎝integraldisplayx ay(t)dt. Assuming the right-hand side to be known, w e solve this equation as the generalized Abel equation 1.1.47. After some manipulations, we arrive at Abel’s equation of the second kind 2.1.60: y(x)+bsin(πλ) π⎝integraldisplayx ay(t)dt (x–t)1–λ=F(x), where F(x)=sin(πλ) πd dx⎝integraldisplayx af(t)dt (x–t)1–λ. 49.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig√ x–√ t⎝parenrightbig⎝parenrightbigλy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=k √ x⎝parenleftbigg√ xd dx⎝parenrightbigg2⎝integraldisplayx af(t)dt √ t⎝parenleftbig√ x–√ t⎝parenrightbigλ,k=sin(πλ) πλ. 50.⎝integraldisplay ⎝integraldisplayx ay(t)dt ⎝parenleftbig⎝parenleftbig√ x–√ t⎝parenrightbig⎝parenrightbigλ=f(x), 0 < λ<1 . Solution: y(x)=sin(πλ) 2πd dx⎝integraldisplayx af(t)dt √ t⎝parenleftbig√ x–√ t⎝parenrightbig1–λ. 51.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλ+Btµ⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Axλandh(t)=Btµ. 14 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 52.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig 1+A(xλtµ–xλ+µ)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.13 with g(x)=Axµandh(x)=xλ. Solution: y(x)=d dx⎝braceleftbiggxλ Φ(x)⎝integraldisplayx a⎝bracketleftbig t–λf(t)⎝bracketrightbig/prime tΦ(t)dt⎝bracerightbigg ,Φ(x)=e x p⎝parenleftBig –Aµ µ+λxµ+λ⎝parenrightBig . 53.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axβtγ+Bxδtλ⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axβ,h1(t)=tγ,g2(x)=Bxδ,a n d h2(t)=tλ. 54.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axλ(tµ–xµ)+Bxβ(tγ–xγ)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.47 with g1(x)=Axλ,h1(x)=xµ,g2(x)=Bxβ,a n d h2(x)=xγ. 55.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axλtµ+Bxλ+βtµ–β–(A+B)xλ+γtµ–γ⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.49 with g(x)=x. 56.⎝integraldisplay ⎝integraldisplayx atσ(xµ–tµ)λy(t)dt=f(x), σ> –1, µ>0 , λ> –1. The transformation τ=tµ,z=xµ,w(τ)=tσ–µ+1y(t) leads to an equation of the form 1.1.43: ⎝integraldisplayz A(z–τ)λw(τ)dτ=F(z), where A=aµandF(z)=µf(z1/µ). Solution with –1 < λ<0 : y(x)=–µsin(πλ) πxσd dx⎝bracketleftbigg⎝integraldisplayx atµ–1(xµ–tµ)–1–λf(t)dt⎝bracketrightbigg . 57.⎝integraldisplay ⎝integraldisplayx 0y(t)dt (x+t)µ=f(x). This is a special case of equation 1.1.58 with λ=1a n d a=b=1 . The transformation x=1 2e2z,t=1 2e2τ,y(t)=e(µ–2)τw(τ),f(x)=e–µzg(z) leads to an equation with difference kernel of the form 1.9.27: ⎝integraldisplayz –∞w(τ)dτ coshµ(z–τ)=g(z). 1.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 15 58.⎝integraldisplay ⎝integraldisplayx 0y(t)dt (axλ+btλ)µ=f(x), a>0 , a+b>0 . 1◦. The substitution t=xzleads to a special case of equation 3.8.45: ⎝integraldisplay1 0y(xz)dz (a+bzλ)µ=xλµ–1f(x). (1) 2◦. For a polynomial right-hand side, f(x)=n⎝summationtext m=0Amxm, the solution has the form y(x)=xλµ–1n⎝summationdisplay m=0Am Imxm,Im=⎝integraldisplay1 0zm+λµ–1dz (a+bzλ)µ. The integrals Imare supposed to be convergent. 3◦. The solution structure for some other right-hand sides of the integral equation may be obtained using (1) and the results presented for the more general equation 3.8.53 (see also equations 3.8.34–3.8.40). 4◦.F o ra=b, the equation can be reduced, just as equation 1.1.57, to an integral equation with difference kernel of the form 1.9.27. 59.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig√ x+√ x–t⎝parenrightbig⎝parenrightbig2λ+⎝parenleftbig⎝parenleftbig√ x–√ x–t⎝parenrightbig⎝parenrightbig2λ 2tλ√ x–ty(t)dt=f(x). The equation can be rewritten in terms of the Gaussian hypergeometric functions in the form ⎝integraldisplayx a(x–t)γ–1F⎝parenleftBig λ,–λ,γ;1–x t⎝parenrightBig y(t)dt=f(x), where γ=1 2. See 1.8.135 for the solution of this equation. 1.1-8. Two-Dimensional Equation of the Abel Type. 60.⎝integraldisplay⎝integraldisplay⎝integraldisplay⎝integraldisplay ∆u(x,y)dx dy ⎝radicalbig (y0–y)2–(x0–x)2=f(x0,y0). Here∆is an isosceles right triangle with apex at the point ( x0,y0) and base on the x-axis. Solution: u(x0,y0)=1 2π2⎝parenleftbigg∂2g ∂x2 0–∂2g ∂y2 0⎝parenrightbigg ,g(x0,y0)=⎝integraldisplay⎝integraldisplay ∆f(x,y)dx dy ⎝radicalbig (y0–y)2–(x0–x)2. Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975). 1.2. Equations Whose Kernels Contain Exponential Functions 1.2-1. Kernels Containing Exponential Functions. 1.⎝integraldisplay ⎝integraldisplayx aeλ(x–t)y(t)dt=f(x). Solution: y(x)=f/prime x(x)–λf(x). Example. In the special case a=0a n d f(x)=Ax, the solution has the form y(x)=A(1 –λx). 16 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 2.⎝integraldisplay ⎝integraldisplayx aeλx+βty(t)dt=f(x). Solution: y(x)=e–(λ+β)x⎝bracketleftbig f/prime x(x)–λf(x)⎝bracketrightbig . Example. In the special case a=0a n d f(x)=Asin(γx), the solution has the form y(x)=Ae–(λ+β)x× [γcos(γx)–λsin(γx)]. 3.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig eλ(x–t)–1⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=1 λf/prime/prime xx(x)–f/prime x(x). 4.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig eλ(x–t)+b⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Forb= –1, see equation 1.2.3. Differentiating with respect to xyields an equation of the form 2.2.1: y(x)+λ b+1⎝integraldisplayx aeλ(x–t)y(t)dt=f/prime x(x) b+1. Solution: y(x)=f/prime x(x) b+1–λ (b+1 )2⎝integraldisplayx aexp⎝bracketleftbiggλb b+1(x–t)⎝bracketrightbigg f/prime t(t)dt. 5.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig eλx+βt+b⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=eλx,h1(t)=eβt,g2(x)=1 ,a n d h2(t)=b. Forβ=–λ, see equation 1.2.4. 6.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig eλx–eλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . This is a special case of equation 1.9.2 with g(x)=eλx. Solution: y(x)=e–λx⎝bracketleftbigg1 λf/prime/prime xx(x)–f/prime x(x)⎝bracketrightbigg . 7.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig eλx–eλt+b⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.3 with g(x)=eλx.F o rb= 0, see equation 1.2.6. Solution: y(x)=1 bf/prime x(x)–λ b2eλx⎝integraldisplayx aexp⎝parenleftbiggeλt–eλx b⎝parenrightbigg f/prime t(t)dt. 8.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Aeλx+Beλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=eλx.F o rB=–A, see equation 1.2.6. Solution: y(x)=1 A+Bd dx⎝bracketleftbigg exp⎝parenleftBig –Aλ A+Bx⎝parenrightBig⎝integraldisplayx aexp⎝parenleftBig –Bλ A+Bt⎝parenrightBig f/prime t(t)dt⎝bracketrightbigg . 9.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Aeλx+Beλt+C⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.5 with g(x)=eλx. 1.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 17 10.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Aeλx+Beµt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aeλxandh(t)=Beµt.F o rλ=µ,s e e equation 1.2.8. 11.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig eλ(x–t)–eµ(x–t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=1 λ–µ⎝bracketleftbig f/prime/prime xx–(λ+µ)f/prime x+λµf⎝bracketrightbig ,f=f(x). 12.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeλ(x–t)+Beµ(x–t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeλx,h1(t)=e–λt,g2(x)=Beµx,a n d h2(t)=e–µt.F o rB=–A, see equation 1.2.11. Solution: y(x)=eλx A+Bd dx⎝braceleftbigg e(µ–λ)xΦ(x)⎝integraldisplayx a⎝bracketleftbiggf(t) eµt⎝bracketrightbigg/prime tdt Φ(t)⎝bracerightbigg ,Φ(x)=e x p⎝bracketleftbiggB(λ–µ) A+Bx⎝bracketrightbigg . 13.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeλ(x–t)+Beµ(x–t)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.2.14 with β=0 . 14.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeλ(x–t)+Beµ(x–t)+Ceβ(x–t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Differentiating the equation with respect to xyields (A+B+C)y(x)+⎝integraldisplayx a⎝bracketleftbig Aλeλ(x–t)+Bµeµ(x–t)+Cβeβ(x–t)⎝bracketrightbig y(t)dt=f/prime x(x). Eliminating the term with eβ(x–t)with the aid of the original equation, we arrive at an equation of the form 2.2.10: (A+B+C)y(x)+⎝integraldisplayx a⎝bracketleftbig A(λ–β)eλ(x–t)+B(µ–β)eµ(x–t)⎝bracketrightbig y(t)dt=f/prime x(x)–βf(x). In the special case A+B+C= 0, this is an equation of the form 1.2.12. 15.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeλ(x–t)+Beµ(x–t)+Ceβ(x–t)–A–B–C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Differentiating with respect to x, we arrive at an equation of the form 1.2.14: ⎝integraldisplayx a⎝bracketleftbig Aλeλ(x–t)+Bµeµ(x–t)+Cβeβ(x–t)⎝bracketrightbig y(t)dt=f/prime x(x). 16.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig eλx+µt–eµx+λt⎝parenrightbig⎝parenrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . This is a special case of equation 1.9.11 with g(x)=eλxandh(t)=eµt. Solution: y(x)=f/prime/prime xx–(λ+µ)f/prime x(x)+λµf(x) (λ–µ)e x p [ ( λ+µ)x]. 18 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 17.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Aeλx+µt+Beµx+λt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.12 with g(x)=eλxandh(t)=eµt.F o rB=–A,s e e equation 1.2.16. Solution: y(x)=1 (A+B)eµxd dx⎝braceleftbigg ΦA(x)⎝integraldisplayx aΦB(t)d dt⎝bracketleftbiggf(t) eµt⎝bracketrightbigg dt⎝bracerightbigg ,Φ(x)=e x p⎝parenleftBigµ–λ A+Bx⎝parenrightBig . 18.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Aeλx+µt+Beβx+γt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeλx,h1(t)=eµt,g2(x)=Beβx,a n d h2(t)=eγt. 19.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ae2λx+Be2βt+Ceλx+Deβt+E⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Ae2λx+Ceλxandh(t)=Be2βt+Deβt+E. 20.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Aeλx+βt+Be2βt+Ceλx+Deβt+E⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=eλx,h1(t)=Aeβt+C,a n dg2(x)=1 , h2(t)=Be2βt+Deβt+E. 21.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ae2λx+Beλx+βt+Ceλx+Deβt+E⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Beλx+D,h1(t)=eβt,a n dg2(x)= Ae2λx+Ceλx+E,h2(t)=1 . 22.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig 1+Aeλx(eµt–eµx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.13 with g(x)=eµxandh(x)=Aeλx. Solution: y(x)=d dx⎝braceleftbigg eλxΦ(x)⎝integraldisplayx a⎝bracketleftbiggf(t) eλt⎝bracketrightbigg/prime tdt Φ(t)⎝bracerightbigg ,Φ(x)=e x p⎝bracketleftbiggAµ λ+µe(λ+µ)x⎝bracketrightbigg . 23.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeλx(eµx–eµt)+Beβx(eγx–eγt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.47 with g1(x)=Aeλx,h1(t)=–eµt,g2(x)=Beβx,a n d h2(t)=–eγt. 24.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig Aexp(λx +µt)+Bexp[(λ +β)x+(µ–β)t] –(A+B)e x p [ ( λ+γ)x+(µ–γ)t]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 1.9.49 with g1(x)=ex. 1.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 19 25.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig eλx–eλt⎝parenrightbig⎝parenrightbigny(t)dt=f(x), n=1 , 2 , ... Solution: y(x)=1 λnn!eλx⎝parenleftBig1 eλxd dx⎝parenrightBign+1 f(x). 26.⎝integraldisplay ⎝integraldisplayx a√ eλx–eλty(t)dt=f(x), λ>0 . Solution: y(x)=2 πeλx⎝parenleftBig e–λxd dx⎝parenrightBig2⎝integraldisplayx aeλtf(t)dt √ eλx–eλt. 27.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ eλx–eλt=f(x), λ>0 . Solution: y(x)=λ πd dx⎝integraldisplayx aeλtf(t)dt √ eλx–eλt. 28.⎝integraldisplay ⎝integraldisplayx a(eλx–eλt)µy(t)dt=f(x), λ>0 , 0< µ<1 . Solution: y(x)=keλx⎝parenleftBig e–λxd dx⎝parenrightBig2⎝integraldisplayx aeλtf(t)dt (eλx–eλt)µ,k=sin(πµ) πµ. 29.⎝integraldisplay ⎝integraldisplayx ay(t)dt (eλx–eλt)µ=f(x), λ>0 , 0< µ<1 . Solution: y(x)=λsin(πµ) πd dx⎝integraldisplayx aeλtf(t)dt (eλx–eλt)1–µ. 1.2-2. Kernels Containing Power-Law and Exponential Functions. 30.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig A(x–t)+Beλ(x–t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Differentiating with respect to x, we arrive at an equation of the form 2.2.4: By(x)+⎝integraldisplayx a⎝bracketleftbig A+Bλeλ(x–t)⎝bracketrightbig y(t)dt=f/prime x(x). 31.⎝integraldisplay ⎝integraldisplayx a(x–t)eλ(x–t)y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=f/prime/prime xx(x)–2λf/prime x(x)+λ2f(x). 32.⎝integraldisplay ⎝integraldisplayx a(Ax +Bt +C)eλ(x–t)y(t)dt=f(x). The substitution u(x)=e–λxy(x) leads to an equation of the form 1.1.3: ⎝integraldisplayx a(Ax+Bt+C)u(t)dt=e–λxf(x). 20 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 33.⎝integraldisplay ⎝integraldisplayx a(Axeλt+Bteµx)y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Ax,h1(t)=eλt,a n dg2(x)=Beµx, h2(t)=t. 34.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axeλ(x–t)+Bteµ(x–t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axeλx,h1(t)=e–λt,g2(x)=Beµx,a n d h2(t)=te–µt. 35.⎝integraldisplay ⎝integraldisplayx a(x–t)2eλ(x–t)y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Solution: y(x)=1 2⎝bracketleftbig f/prime/prime/prime xxx(x)–3λf/prime/prime xx(x)+3λ2f/prime x(x)–λ3f(x)⎝bracketrightbig . 36.⎝integraldisplay ⎝integraldisplayx a(x–t)neλ(x–t)y(t)dt=f(x), n=1 , 2 , ... It is assumed that f(a)=f/prime x(a)=···=f(n) x(a)=0 . Solution: y(x)=1 n!eλxdn+1 dxn+1⎝bracketleftbig e–λxf(x)⎝bracketrightbig . 37.⎝integraldisplay ⎝integraldisplayx a(Axβ+Beλt)y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Axβandh(t)=Beλt. 38.⎝integraldisplay ⎝integraldisplayx a(Aeλx+Btβ)y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aeλxandh(t)=Btβ. 39.⎝integraldisplay ⎝integraldisplayx a(Axβeλt+Btγeµx)y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axβ,h1(t)=eλt,g2(x)=Beµx,a n d h2(t)=tγ. 40.⎝integraldisplay ⎝integraldisplayx aeλ(x–t)√ x–ty(t)dt=f(x). Solution: y(x)=2 πeλxd2 dx2⎝integraldisplayx ae–λtf(t)dt √ x–t. 41.⎝integraldisplay ⎝integraldisplayx aeλ(x–t) √ x–ty(t)dt=f(x). Solution: y(x)=1 πeλxd dx⎝integraldisplayx ae–λtf(t)dt √ x–t. 1.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 21 42.⎝integraldisplay ⎝integraldisplayx a(x–t)λeµ(x–t)y(t)dt=f(x), 0 < λ<1 . Solution: y(x)=keµxd2 dx2⎝integraldisplayx ae–µtf(t)dt (x–t)λ,k=sin(πλ) πλ. 43.⎝integraldisplay ⎝integraldisplayx aeλ(x–t) (x–t)µy(t)dt=f(x), 0 < µ<1 . Solution: y(x)=sin(πµ) πeλxd dx⎝integraldisplayx ae–λtf(t) (x–t)1–µdt. 44.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig√ x–√ t⎝parenrightbig⎝parenrightbigλeµ(x–t)y(t)dt=f(x), 0 < λ<1 . The substitution u(x)=e–µxy(x) leads to an equation of the form 1.1.49: ⎝integraldisplayx a⎝parenleftbig√ x–√ t⎝parenrightbigλu(t)dt=e–µxf(x). 45.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)y(t)dt ⎝parenleftbig⎝parenleftbig√ x–√ t⎝parenrightbig⎝parenrightbigλ=f(x), 0 < λ<1 . The substitution u(x)=e–µxy(x) leads to an equation of the form 1.1.50: ⎝integraldisplayx au(t)dt (√ x–√ t)λ=e–µxf(x). 46.⎝integraldisplay ⎝integraldisplayx aeλ(x–t) √ x2–t2y(t)dt=f(x). Solution: y=2 πeλxd dx⎝integraldisplayx ate–λt √ x2–t2f(t)dt. 47.⎝integraldisplay ⎝integraldisplayx aexp[λ(x2–t2)]y(t)dt=f(x). Solution: y(x)=f/prime x(x)–2λxf(x). 48.⎝integraldisplay ⎝integraldisplayx a[exp(λx2)–e x p ( λt2)]y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=e x p ( λx2). Solution: y(x)=1 2λd dx⎝bracketleftbiggf/prime x(x) xexp(λx2)⎝bracketrightbigg . 49.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aexp(λx2)+Bexp(λt2)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.5 with g(x)=e x p ( λx2). 50.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aexp(λx2)+Bexp(µt2)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aexp(λx2)a n dh(t)=Bexp(µt2). 22 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 51.⎝integraldisplay ⎝integraldisplayx a√ x–texp[λ(x2–t2)]y(t)dt=f(x). Solution: y(x)=2 πexp(λx2)d2 dx2⎝integraldisplayx aexp(–λt2) √ x–tf(t)dt. 52.⎝integraldisplay ⎝integraldisplayx aexp[λ(x2–t2)] √ x–ty(t)dt=f(x). Solution: y(x)=1 πexp(λx2)d dx⎝integraldisplayx aexp(–λt2) √ x–tf(t)dt. 53.⎝integraldisplay ⎝integraldisplayx a(x–t)λexp[µ(x2–t2)]y(t)dt=f(x), 0 < λ<1 . Solution: y(x)=kexp(µx2)d2 dx2⎝integraldisplayx aexp(–µt2) (x–t)λf(t)dt,k=sin(πλ) πλ. 54.⎝integraldisplay ⎝integraldisplayx aexp[λ(xβ–tβ)]y(t)dt=f(x). Solution: y(x)=f/prime x(x)–λβxβ–1f(x). 55.⎝integraldisplay ⎝integraldisplayx 0(–1)[(x–t)/b]y(t)dt=f(x), f(0) =f/prime x(0) = 0. Hereb= const and [ A] stands for the integer part of the number A. Solution: y(x)=1 2⎝integraldisplayx 0⎝parenleftbigg 2⎝bracketleftbiggx–t b⎝bracketrightbigg +1⎝parenrightbigg f/prime/prime tt(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 434). 1.3. Equations Whose Kernels Contain Hyperbolic Functions 1.3-1. Kernels Containing Hyperbolic Cosine. 1.⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)]y(t)dt=f(x). Solution: y(x)=f/prime x(x)–λ2⎝integraldisplayx af(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 435). 2.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig cosh[λ (x–t)] – 1⎝bracerightbig ⎝bracerightbig y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(x)=0 . Solution: y(x)=1 λ2f/prime/prime/prime xxx(x)–f/prime x(x). 1.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 23 3.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig cosh[λ (x–t)] +b⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). Forb= 0, see equation 1.3.1. For b= –1, see equation 1.3.2. For λ= 0, see equation 1.1.1. Differentiating the equation with respect to x, we arrive at an equation of the form 2.3.16: y(x)+λ b+1⎝integraldisplayx asinh[λ(x–t)]y(t)dt=f/prime x(x) b+1. 1◦. Solution with b(b+1 )<0 : y(x)=f/prime x(x) b+1–λ2 k(b+1 )2⎝integraldisplayx asin[k(x–t)]f/prime t(t)dt,w h e r e k=λ⎝radicalbigg –b b+1. 2◦. Solution with b(b+1 )>0 : y(x)=f/prime x(x) b+1–λ2 k(b+1 )2⎝integraldisplayx asinh[k(x–t)]f/prime t(t)dt,w h e r e k=λ⎝radicalbigg b b+1. 4.⎝integraldisplay ⎝integraldisplayx acosh(λx +βt)y(t)dt=f(x). Forβ=–λ, see equation 1.3.1. Differentiating the equation with respect to xtwice, we obtain cosh[(λ +β)x]y(x)+λ⎝integraldisplayx asinh(λx+βt)y(t)dt=f/prime x(x), (1) ⎝braceleftbig cosh[(λ +β)x]y(x)⎝bracerightbig/prime x+λsinh[(λ+β)x]y(x)+λ2⎝integraldisplayx acosh(λx+βt)y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the ﬁrst-order linear ordinary differential equation w/prime x+λtanh[(λ+β)x]w=f/prime/prime xx(x)–λ2f(x), w= cosh[( λ+β)x]y(x). (3) Setting x=ain (1) yields the initial condition w(a)=f/prime x(a). On solving equation (3) with this condition, after some manipulati ons we obtain the solution of t he original integral equation in the form y(x)=1 cosh[(λ +β)x]f/prime x(x)–λsinh[(λ+β)x] cosh2[(λ+β)x]f(x) +λβ coshk+1[(λ+β)x]⎝integraldisplayx af(t)c o s hk–2[(λ+β)t]dt,k=λ λ+β. 5.⎝integraldisplay ⎝integraldisplayx a[cosh( λx)–c o s h ( λt)]y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=c o s h ( λx). Solution: y(x)=1 λd dx⎝bracketleftbiggf/prime x(x) sinh(λx)⎝bracketrightbigg . 6.⎝integraldisplay ⎝integraldisplayx a[Acosh(λx )+Bcosh(λt )]y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x) = cosh( λx). For B=–A, see equation 1.3.5. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig cosh(λx)⎝bracketrightbig–A A+B⎝integraldisplayx a⎝bracketleftbig cosh(λt )⎝bracketrightbig–B A+Bf/prime t(t)dt⎝bracerightbigg . 24 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 7.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosh(λx )+Bcosh(µt )+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acosh(λx)a n d h(t)=Bcosh(µt)+ C. 8.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A1cosh[λ 1(x–t)] +A2cosh[λ 2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). The equation is equivalent to the equation⎝integraldisplayx a⎝braceleftbig B1sinh[λ1(x–t)] +B2sinh[λ2(x–t)]⎝bracerightbig y(t)dt=F(x), B1=A1 λ1,B2=A2 λ2,F(x)=⎝integraldisplayx af(t)dt, of the form 1.3.49. (Differen tiating this equation yields the original equation.) 9.⎝integraldisplay ⎝integraldisplayx acosh2[λ(x–t)]y(t)dt=f(x). Differentiation yields an equation of the form 2.3.16: y(x)+λ⎝integraldisplayx asinh[2λ (x–t)]y(t)dt=f/prime x(x). Solution: y(x)=f/prime x(x)–2λ2 k⎝integraldisplayx asinh[k(x–t)]f/prime t(t)dt,w h e r e k=λ√ 2. 10.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cosh2(λx)–c o s h2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=1 λd dx⎝bracketleftbiggf/prime x(x) sinh(2 λx)⎝bracketrightbigg . 11.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosh2(λx)+Bcosh2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o s h2(λx). For B=–A, see equation 1.3.10. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig cosh(λx)⎝bracketrightbig–2A A+B⎝integraldisplayx a⎝bracketleftbig cosh(λt )⎝bracketrightbig–2B A+Bf/prime t(t)dt⎝bracerightbigg . 12.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosh2(λx)+Bcosh2(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acosh2(λx), and h(t)=Bcosh2(µt)+C. 13.⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)] cosh[λ (x+t)]y(t)dt=f(x). Using the formula cosh(α –β)c o s h ( α+β)=1 2[cosh(2α )+c o s h ( 2 β)],α=λx, β=λt, we transform the original equation to an equation of the form 1.3.6 with A=B=1 :⎝integraldisplayx a[cosh(2λx)+c o s h ( 2 λt)]y(t)dt=2f(x). Solution: y(x)=d dx⎝bracketleftbigg1 √ cosh(2 λx)⎝integraldisplayx af/prime t(t)dt √ cosh(2 λt)⎝bracketrightbigg . 1.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 25 14.⎝integraldisplay ⎝integraldisplayx a[cosh( λx)c o s h ( µt)+c o s h ( βx)c o s h ( γt)]y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=c o s h ( λx),h1(t)=c o s h ( µt),g2(x)= cosh(βx), and h2(t)=c o s h ( γt). 15.⎝integraldisplay ⎝integraldisplayx acosh3[λ(x–t)]y(t)dt=f(x). Using the formula cosh3β=1 4cosh 3 β+3 4coshβ, we arrive at an equation of the form 1.3.8: ⎝integraldisplayx a⎝braceleftbig1 4cosh[3 λ(x–t)] +3 4cosh[λ (x–t)]⎝bracerightbig y(t)dt=f(x). 16.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cosh3(λx)–c o s h3(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=1 3λd dx⎝bracketleftbiggf/prime x(x) sinh(λx)c o s h2(λx)⎝bracketrightbigg . 17.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosh3(λx)+Bcosh3(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o s h3(λx). For B=–A, see equation 1.3.16. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig cosh(λx)⎝bracketrightbig–3A A+B⎝integraldisplayx a⎝bracketleftbig cosh(λt )⎝bracketrightbig–3B A+Bf/prime t(t)dt⎝bracerightbigg . 18.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosh2(λx)c o s h ( µt)+Bcosh(βx )c o s h2(γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Acosh2(λx),h1(t)=c o s h ( µt),g2(x)= Bcosh(βx), and h2(t)=c o s h2(γt). 19.⎝integraldisplay ⎝integraldisplayx acosh4[λ(x–t)]y(t)dt=f(x). Let us transform the kernel of the integral equation using the formula cosh4β=1 8cosh 4 β+1 2cosh 2 β+3 8,w h e r e β=λ(x–t), and differentiate the resulting equation with respect to x. Then we obtain an equation of the form 2.3.18: y(x)+λ⎝integraldisplayx a⎝braceleftbig1 2sinh[4λ (x–t)] + sinh[2λ (x–t)]⎝bracerightbig y(t)dt=f/prime x(x). 20.⎝integraldisplay ⎝integraldisplayx a[cosh( λx)–c o s h ( λt)]ny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=sinh(λx) λnn!⎝bracketleftbigg1 sinh(λx)d dx⎝bracketrightbiggn+1 f(x). 26 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 21.⎝integraldisplay ⎝integraldisplayx a√ coshx–c o s h ty(t)dt=f(x). Solution: y(x)=2 πsinhx⎝parenleftBig1 sinhxd dx⎝parenrightBig2⎝integraldisplayx asinhtf(t)dt √ coshx–c o s h t. 22.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ coshx–c o s h t=f(x). Solution: y(x)=1 πd dx⎝integraldisplayx asinhtf(t)dt √ coshx–c o s h t. 23.⎝integraldisplay ⎝integraldisplayx a(coshx–c o s h t)λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=ksinhx⎝parenleftBig1 sinhxd dx⎝parenrightBig2⎝integraldisplayx asinhtf(t)dt (coshx–c o s h t)λ,k=sin(πλ) πλ. 24.⎝integraldisplay ⎝integraldisplayx a(coshµx–c o s hµt)y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=c o s hµx. Solution: y(x)=1 µd dx⎝bracketleftbiggf/prime x(x) sinhxcoshµ–1x⎝bracketrightbigg . 25.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Acoshµx+Bcoshµt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o s hµx.F o rB=–A, see equation 1.3.24. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig cosh(λx)⎝bracketrightbig–Aµ A+B⎝integraldisplayx a⎝bracketleftbig cosh(λt )⎝bracketrightbig–Bµ A+Bf/prime t(t)dt⎝bracerightbigg . 26.⎝integraldisplay ⎝integraldisplayx ay(t)dt (coshx–c o s h t)λ=f(x), 0 < λ<1 . Solution: y(x)=sin(πλ) πd dx⎝integraldisplayx asinhtf(t)dt (coshx–c o s h t)1–λ. 27.⎝integraldisplay ⎝integraldisplayx a(x–t)c o s h [ λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . Differentiating the equation twice yields y(x)+2λ⎝integraldisplayx asinh[λ(x–t)]y(t)dt+λ2⎝integraldisplayx a(x–t)c o s h [ λ(x–t)]y(t)dt=f/prime/prime xx(x). Eliminating the third term on the right-hand side with the aid of the original equation, we arrive at an equation of the form 2.3.16: y(x)+2λ⎝integraldisplayx asinh[λ(x–t)]y(t)dt=f/prime/prime xx(x)–λ2f(x). 1.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 27 28.⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)] √ x–ty(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=2 πλ⎝integraldisplayx acosh[λ(x–t)] √ x–t[f/prime/prime tt(t)–λ2f(t)]dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 436). 29.⎝integraldisplay ⎝integraldisplayx a√ x–tcosh⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=1 π⎝integraldisplayx acos⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf/prime t(t)dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 437), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 30.⎝integraldisplay ⎝integraldisplayx acosh⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig √ x–ty(t)dt=f(x). Solution: y(x)=1 πd dx⎝integraldisplayx acos⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 437), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 31.⎝integraldisplay ⎝integraldisplay∞ xcosh⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig √ t–xy(t)dt=f(x). Solution: y(x)=–1 πd dx⎝integraldisplay∞ xcos⎝parenleftbig λ√ t–x⎝parenrightbig √ t–xf(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 439), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 32.⎝integraldisplay ⎝integraldisplayx 0cosh⎝parenleftbig⎝parenleftbig λ√ x2–t2⎝parenrightbig⎝parenrightbig √ x2–t2y(t)dt=f(x). Solution: y(x)=2 πd dx⎝integraldisplayx 0tcos⎝parenleftbig λ√ x2–t2⎝parenrightbig √ x2–t2f(t)dt. 33.⎝integraldisplay ⎝integraldisplay∞ xcosh⎝parenleftbig⎝parenleftbig λ√ t2–x2⎝parenrightbig⎝parenrightbig √ t2–x2y(t)dt=f(x). Solution: y(x)=–2 πd dx⎝integraldisplay∞ xtcos⎝parenleftbig λ√ t2–x2⎝parenrightbig √ t2–x2f(t)dt. 28 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 34.⎝integraldisplay ⎝integraldisplayx 0cosh⎝parenleftbig⎝parenleftbig λ√ xt–t2⎝parenrightbig⎝parenrightbig √ x–ty(t)dt=f(x). Solution: y(x)=1 πx⎝integraldisplayx 0cos⎝parenleftbig λ√ x2–xt⎝parenrightbig √ x–t[f(t)/2+tf/prime t(t)]dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 438), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 35.⎝integraldisplay ⎝integraldisplayx 0cosh⎝parenleftbig⎝parenleftbig λ√ x2–xt⎝parenrightbig⎝parenrightbig √ x–ty(t)dt=f(x). Solution: y(x)=√ x πd dx⎝bracketleftBig√ x⎝integraldisplayx 0cos⎝parenleftbig λ√ xt–t2⎝parenrightbig √ x–tf(t)dt⎝bracketrightBig . References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 438), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 36.⎝integraldisplay ⎝integraldisplayx acosh⎝bracketleftbig⎝bracketleftbig λ√ (x–t)(x –t+γ)⎝bracketrightbig⎝bracketrightbig √ x–ty(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Solution: y(x)=2 πλ2⎝integraldisplayx asinh⎝bracketleftbig λ√ (x–t)(x–t–γ)⎝bracketrightbig √ x–t–γ⎝integraldisplayt asinh[λ(t–s)]⎝parenleftBigd2 ds2–λ2⎝parenrightBig2 f(s)dsdt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 438), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 37.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axβ+Bcoshγ(λt)+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Axβandh(t)=Bcoshγ(λt)+C. 38.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoshγ(λx)+Btβ+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acoshγ(λx)a n d h(t)=Btβ+C. 39.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλcoshµt+Btβcoshγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=c o s hµt,g2(x)=Bcoshγx, andh2(t)=tβ. 1.3-2. Kernels Containing Hyperbolic Sine. 40.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=1 λf/prime/prime xx(x)–λf(x). Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 435). 1.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 29 41.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] √ x–ty(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=2 πλ⎝integraldisplayx asinh[λ(x–t)] √ x–t[f/prime/prime tt(t)–λ2f(t)]dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 436). 42.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] (x–t)3/2y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=2 πλ⎝integraldisplayx asinh[λ(x–t)] √ x–t⎝bracketleftbigg f/prime/prime tt(t)–λ2f(t)+f/prime(t) x–t⎝bracketrightbigg dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 437). 43.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig sinh[λ(x–t)] +b⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). Forb= 0, see equation 1.3.40. Assume that b≠0. Differentiating the equation with respect to x, we arrive at an equation of the form 2.3.3: y(x)+λ b⎝integraldisplayx acosh[λ (x–t)]y(t)dt=1 bf/prime x(x). Solution: y(x)=1 bf/prime x(x)+⎝integraldisplayx aR(x–t)f/prime t(t)dt, R(x)=λ b2exp⎝parenleftbigg –λx 2b⎝parenrightbigg⎝bracketleftbiggλ 2bksinh(kx)–c o s h ( kx)⎝bracketrightbigg ,k=λ√ 1+4b2 2b. 44.⎝integraldisplay ⎝integraldisplayx asinh(λx +βt)y(t)dt=f(x). Forβ=–λ, see equation 1.3.40. Assume that β≠–λ. Differentiating the equation with respect to xtwice yields sinh[(λ+β)x]y(x)+λ⎝integraldisplayx acosh(λx +βt)y(t)dt=f/prime x(x), (1) ⎝braceleftbig sinh[(λ+β)x]y(x)⎝bracerightbig/prime x+λcosh[( λ+β)x]y(x)+λ2⎝integraldisplayx asinh(λx+βt)y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the ﬁrst-order linear ordinary differential equation w/prime x+λcoth[(λ +β)x]w=f/prime/prime xx(x)–λ2f(x), w= sinh[( λ+β)x]y(x). (3) Setting x=ain (1) yields the initial condition w(a)=f/prime x(a). On solving equation (3) with this condition, after some manipulati ons we obtain the solution of t he original integral equation in the form y(x)=1 sinh[(λ+β)x]f/prime x(x)–λcosh[(λ +β)x] sinh2[(λ+β)x]f(x) –λβ sinhk+1[(λ+β)x]⎝integraldisplayx af(t)s i n hk–2[(λ+β)t]dt,k=λ λ+β. 30 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 45.⎝integraldisplay ⎝integraldisplayx a[sinh(λx)–s i n h ( λt)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . This is a special case of equation 1.9.2 with g(x) = sinh(λx). Solution: y(x)=1 λd dx⎝bracketleftbiggf/prime x(x) cosh(λx)⎝bracketrightbigg . 46.⎝integraldisplay ⎝integraldisplayx a[Asinh(λx)+Bsinh(λt)]y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=s i n h ( λx). For B=–A, see equation 1.3.45. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig sinh(λx)⎝bracketrightbig–A A+B⎝integraldisplayx a⎝bracketleftbig sinh(λt)⎝bracketrightbig–B A+Bf/prime t(t)dt⎝bracerightbigg . 47.⎝integraldisplay ⎝integraldisplayx a[Asinh(λx)+Bsinh(µt)]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asinh(λx)a n d h(t)=Bsinh(µt). 48.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig µsinh[λ(x–t)] –λsinh[µ(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Solution: y(x)=f/prime/prime/prime/prime xxxx –(λ2+µ2)f/prime/prime xx+λ2µ2f µλ3–λµ3,f=f(x). 49.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A1sinh[λ1(x–t)] +A2sinh[λ2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x),f(a)=f/prime x(a)=0 . 1◦. Introduce the notation I1=⎝integraldisplayx asinh[λ1(x–t)]y(t)dt,I2=⎝integraldisplayx asinh[λ2(x–t)]y(t)dt, J1=⎝integraldisplayx acosh[λ 1(x–t)]y(t)dt,J2=⎝integraldisplayx acosh[λ 2(x–t)]y(t)dt. Let us successively differentiate the integral equation four times. As a result, we have (the ﬁrst line is the original equation): A1I1+A2I2=f,f=f(x), (1) A1λ1J1+A2λ2J2=f/prime x,( 2) (A1λ1+A2λ2)y+A1λ2 1I1+A2λ2 2I2=f/prime/prime xx,( 3) (A1λ1+A2λ2)y/prime x+A1λ3 1J1+A2λ3 2J2=f/prime/prime/prime xxx,( 4) (A1λ1+A2λ2)y/prime/prime xx+(A1λ3 1+A2λ3 2)y+A1λ4 1I1+A2λ4 2I2=f/prime/prime/prime/prime xxxx.( 5 ) Eliminating I1andI2from (1), (3), and (5), we arrive at the following second-order linear ordinary differential equation with constant coefﬁcients: (A1λ1+A2λ2)y/prime/prime xx–λ1λ2(A1λ2+A2λ1)y=f/prime/prime/prime/prime xxxx –(λ2 1+λ2 2)f/prime/prime xx+λ2 1λ22f.( 6 ) The initial conditions can be obtained by substituting x=ainto (3) and (4): (A1λ1+A2λ2)y(a)=f/prime/prime xx(a), (A 1λ1+A2λ2)y/prime x(a)=f/prime/prime/prime xxx(a). (7) Solving the differential equation (6) under conditions (7) allows us to ﬁnd the solution of the integral equation. 1.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 31 2◦. Denote ∆=λ1λ2A1λ2+A2λ1 A1λ1+A2λ2. 2.1. Solution for ∆>0 : (A1λ1+A2λ2)y(x)=f/prime/prime xx(x)+Bf(x)+C⎝integraldisplayx asinh[k(x–t)]f(t)dt, k=√ ∆,B=∆–λ2 1–λ2 2,C=1 √ ∆⎝bracketleftbig ∆2–(λ2 1+λ2 2)∆+λ2 1λ22⎝bracketrightbig . 2.2. Solution for ∆<0 : (A1λ1+A2λ2)y(x)=f/prime/prime xx(x)+Bf(x)+C⎝integraldisplayx asin[k(x–t)]f(t)dt, k=√ –∆,B=∆–λ2 1–λ2 2,C=1 √ –∆⎝bracketleftbig ∆2–(λ2 1+λ2 2)∆+λ2 1λ22⎝bracketrightbig . 2.3. Solution for ∆=0 : (A1λ1+A2λ2)y(x)=f/prime/prime xx(x)–(λ2 1+λ2 2)f(x)+λ2 1λ22⎝integraldisplayx a(x–t)f(t)dt. 2.4. Solution for ∆=∞: y(x)=f/prime/prime/prime/prime xxxx –(λ2 1+λ2 2)f/prime/prime xx+λ2 1λ22f A1λ31+A2λ3 2,f=f(x). In the last case, the relation A1λ1+A2λ2= 0 is valid, and the right-hand side of the integral equation is assumed to satisfy the conditions f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . 50.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig Asinh[λ(x–t)] +Bsinh[µ(x–t)] +Csinh[β(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). It assumed that f(a)=f/prime x(a) = 0. Differentiating the integral equation twice yields (Aλ+Bµ+Cβ)y(x)+⎝integraldisplayx a⎝braceleftbig Aλ2sinh[λ(x–t)] +Bµ2sinh[µ(x–t)]⎝bracerightbig y(t)dt +Cβ2⎝integraldisplayx asinh[β(x–t)]y(t)dt=f/prime/prime xx(x). Eliminating the last integral with the aid of the original equation, we arrive at an equation of the form 2.3.18: (Aλ+Bµ+Cβ)y(x) +⎝integraldisplayx a⎝braceleftbig A(λ2–β2)s i n h [λ(x–t)] +B(µ2–β2)s i n h [ µ(x–t)]⎝bracerightbig y(t)dt=f/prime/prime xx(x)–β2f(x). In the special case Aλ+Bµ+Cβ= 0, this is an equation of the form 1.3.49. 51.⎝integraldisplay ⎝integraldisplayx asinh2[λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Differentiating yields an equation of the form 1.3.40: ⎝integraldisplayx asinh[2λ (x–t)]y(t)dt=1 λf/prime x(x). Solution: y(x)=1 2λ–2f/prime/prime/prime xxx(x)–2f/prime x(x). 32 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 52.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig sinh2(λx)–s i n h2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=1 λd dx⎝bracketleftbiggf/prime x(x) sinh(2 λx)⎝bracketrightbigg . 53.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinh2(λx)+Bsinh2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=s i n h2(λx). For B=–A, see equation 1.3.52. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig sinh(λx)⎝bracketrightbig–2A A+B⎝integraldisplayx a⎝bracketleftbig sinh(λt)⎝bracketrightbig–2B A+Bf/prime t(t)dt⎝bracerightbigg . 54.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinh2(λx)+Bsinh2(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asinh2(λx)a n d h(t)=Bsinh2(µt). 55.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] sinh[ λ(x+t)]y(t)dt=f(x). Using the formula sinh(α–β) sinh(α +β)=1 2[cosh(2α )–c o s h ( 2 β)],α=λx, β=λt, we reduce the original equation to an equation of the form 1.3.5: ⎝integraldisplayx a[cosh(2λx)–c o s h ( 2 λt)]y(t)dt=2f(x). Solution: y(x)=1 λd dx⎝bracketleftbiggf/prime x(x) sinh(2 λx)⎝bracketrightbigg . 56.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinh(λx)s i n h ( µt)+Bsinh(βx)s i n h ( γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Asinh(λx),h1(t)=s i n h ( µt),g2(x)= Bsinh(βx), and h2(t) = sinh(γt ). 57.⎝integraldisplay ⎝integraldisplayx asinh3[λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Using the formula sinh3β=1 4sinh 3β–3 4sinhβ, we arrive at an equation of the form 1.3.49: ⎝integraldisplayx a⎝braceleftbig1 4sinh[3 λ(x–t)] –3 4sinh[λ(x–t)]⎝bracerightbig y(t)dt=f(x). 58.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig sinh3(λx)–s i n h3(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . This is a special case of equation 1.9.2 with g(x)=s i n h3(λx). 1.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 33 59.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinh3(λx)+Bsinh3(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=s i n h3(λx). Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig sinh(λx)⎝bracketrightbig–3A A+B⎝integraldisplayx a⎝bracketleftbig sinh(λt)⎝bracketrightbig–3B A+Bf/prime t(t)dt⎝bracerightbigg . 60.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinh2(λx) sinh( µt)+Bsinh(βx)s i n h2(γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Asinh2(λx),h1(t)=s i n h ( µt),g2(x)= Bsinh(βx), and h2(t)=s i n h2(γt). 61.⎝integraldisplay ⎝integraldisplayx asinh4[λ(x–t)]y(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=···=f/prime/prime/prime/prime xxxx(a)=0 . Let us transform the kernel of the integral equation using the formula sinh4β=1 8cosh 4 β–1 2cosh 2 β+3 8,w h e r e β=λ(x–t), and differentiate the resulting equation with respect to x. Then we arrive at an equation of the form 1.3.49: λ⎝integraldisplayx a⎝braceleftbig1 2sinh[4 λ(x–t)] – sinh[2 λ(x–t)]⎝bracerightbig y(t)dt=f/prime x(x). 62.⎝integraldisplay ⎝integraldisplayx asinhn[λ(x–t)]y(t)dt=f(x), n=2 , 3 , ... It is assumed that f(a)=f/prime x(a)=···=f(n) x(a)=0 . 1◦. Let us differentiate the equation with respect to xtwice and transform the kernel of the resulting integral equation using the formula cosh2β=1+s i n h2β,w h e r e β=λ(x–t). Then we have λ2n2⎝integraldisplayx asinhn[λ(x–t)]y(t)dt+λ2n(n–1 )⎝integraldisplayx asinhn–2[λ(x–t)]y(t)dt=f/prime/prime xx(x). Eliminating the ﬁrst term on the left-hand side with the aid of the original equation, we obtain ⎝integraldisplayx asinhn–2[λ(x–t)]y(t)dt=1 λ2n(n–1 )⎝bracketleftbig f/prime/prime xx(x)–λ2n2f(x)⎝bracketrightbig . This equation has the same form as the original equation, but the exponent of the kernel has been reduced by two. By applying this technique sufﬁciently many times, we ﬁnally arrive at simple integral equations of the form 1.1.1 (for even n)o r1 . 3 . 4 0( f o ro d dn ). 2◦. Solution: y(x)=1 λnn!⎝parenleftbiggd dx+nλ⎝parenrightbigg⎝parenleftbiggd dx+(n–2 )λ⎝parenrightbigg ...⎝parenleftbiggd dx–nλ⎝parenrightbigg f(x). Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 436). 34 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 63.⎝integraldisplay ⎝integraldisplayx asinh⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=2 πλd2 dx2⎝integraldisplayx acos⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 437), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 64.⎝integraldisplay ⎝integraldisplay∞ xsinh⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=2 πλd2 dx2⎝integraldisplay∞ xcos⎝parenleftbig λ√ t–x⎝parenrightbig √ t–xf(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 439), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 65.⎝integraldisplay ⎝integraldisplayx a√ sinhx–s i n h ty(t)dt=f(x). Solution: y(x)=2 πcoshx⎝parenleftBig1 coshxd dx⎝parenrightBig2⎝integraldisplayx acoshtf(t)dt √ sinhx–s i n h t. 66.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ sinhx–s i n h t=f(x). Solution: y(x)=1 πd dx⎝integraldisplayx acoshtf(t)dt √ sinhx–s i n h t. 67.⎝integraldisplay ⎝integraldisplayx a(sinhx–s i n h t)λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=kcoshx⎝parenleftBig1 coshxd dx⎝parenrightBig2⎝integraldisplayx acoshtf(t)dt (sinhx–s i n h t)λ,k=sin(πλ) πλ. 68.⎝integraldisplay ⎝integraldisplayx a(sinhµx–s i n hµt)y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=s i n hµx. Solution: y(x)=1 µd dx⎝bracketleftBigf/prime x(x) coshxsinhµ–1x⎝bracketrightBig . 69.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinhµ(λx)+Bsinhµ(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=s i n hµ(λx). Solution with B≠–A: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig sinh(λx)⎝bracketrightbig–Aµ A+B⎝integraldisplayx a⎝bracketleftbig sinh(λt)⎝bracketrightbig–Bµ A+Bf/prime t(t)dt⎝bracerightbigg . 1.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 35 70.⎝integraldisplay ⎝integraldisplayx ay(t)dt (sinhx–s i n h t)λ=f(x), 0 < λ<1 . Solution: y(x)=sin(πλ) πd dx⎝integraldisplayx acoshtf(t)dt (sinhx–s i n h t)1–λ. 71.⎝integraldisplay ⎝integraldisplayx a(x–t)s i n h [ λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Double differentiation yields 2λ⎝integraldisplayx acosh[λ (x–t)]y(t)dt+λ2⎝integraldisplayx a(x–t)s i n h [ λ(x–t)]y(t)dt=f/prime/prime xx(x). Eliminating the second term on the left-hand side with the aid of the original equation, we arrive at an equation of the form 1.3.1: ⎝integraldisplayx acosh[λ (x–t)]y(t)dt=1 2λ⎝bracketleftbig f/prime/prime xx(x)–λ2f(x)⎝bracketrightbig . Solution: y(x)=1 2λf/prime/prime/prime xxx(x)–λf/prime x(x)+1 2λ3⎝integraldisplayx af(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 436). 72.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] √ x–ty(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=2 πλ⎝integraldisplayx asinh[λ(x–t)] √ x–t[f/prime/prime tt(t)–λ2f(t)]dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 436). 73.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] (x–t)3/2y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=2 πλ⎝integraldisplayx asinh[λ(x–t)] √ x–t⎝bracketleftBig f/prime/prime tt(t)–λ2f(t)+f/prime(t) x–t⎝bracketrightBig dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 437). 74.⎝integraldisplay ⎝integraldisplayx asinh⎝bracketleftbig⎝bracketleftbig λ√ (x–t)(x –t+γ)⎝bracketrightbig⎝bracketrightbig √ x–t+γy(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Solution: y(x)=2 πλ2⎝integraldisplayx acosh⎝bracketleftbig λ√ (x–t)(x–t–γ)⎝bracketrightbig √ x–t⎝integraldisplayt asinh[λ(t–s)]⎝parenleftBigd2 ds2+λ2⎝parenrightBig2 f(s)dsdt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 438), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 36 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 75.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axβ+Bsinhγ(λt)+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Axβandh(t)=Bsinhγ(λt)+C. 76.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinhγ(λx)+Btβ+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asinhγ(λx)a n d h(t)=Btβ+C. 77.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλsinhµt+Btβsinhγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=s i n hµt,g2(x)=Bsinhγx, andh2(t)=tβ. 1.3-3. Kernels Containing Hyperbolic Tangent. 78.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tanh(λx) – tanh( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=t a n h ( λx). Solution: y(x)=1 λ⎝bracketleftbig cosh2(λx)f/prime x(x)⎝bracketrightbig/prime x. 79.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanh(λx)+Btanh(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=t a n h ( λx). For B=–A, see equation 1.3.78. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig tanh(λx)⎝bracketrightbig–A A+B⎝integraldisplayx a⎝bracketleftbig tanh(λt)⎝bracketrightbig–B A+Bf/prime t(t)dt⎝bracerightbigg . 80.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanh(λx)+Btanh(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanh(λx)a n d h(t)=Btanh(µt)+C. 81.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tanh2(λx)–t a n h2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=t a n h2(λx). Solution: y(x)=d dx⎝bracketleftbiggcosh3(λx)f/prime x(x) 2λsinh(λx)⎝bracketrightbigg . 82.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanh2(λx)+Btanh2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=t a n h2(λx). For B=–A, see equation 1.3.81. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig tanh(λx)⎝bracketrightbig–2A A+B⎝integraldisplayx a⎝bracketleftbig tanh(λt)⎝bracketrightbig–2B A+Bf/prime t(t)dt⎝bracerightbigg . 83.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanh2(λx)+Btanh2(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanh2(λx)a n d h(t)=Btanh2(µt)+C. 1.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 37 84.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tanh(λx) – tanh( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=1 λnn!c o s h2(λx)⎝bracketleftbigg cosh2(λx)d dx⎝bracketrightbiggn+1 f(x). 85.⎝integraldisplay ⎝integraldisplayx a√ tanhx–t a n h ty(t)dt=f(x). Solution: y(x)=2 πcosh2x⎝parenleftBig cosh2xd dx⎝parenrightBig2⎝integraldisplayx af(t)dt cosh2t√ tanhx–t a n h t. 86.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ tanhx–t a n h t=f(x). Solution: y(x)=1 πd dx⎝integraldisplayx af(t)dt cosh2t√ tanhx–t a n h t. 87.⎝integraldisplay ⎝integraldisplayx a(tanhx–t a n h t)λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=sin(πλ) πλcosh2x⎝parenleftBig cosh2xd dx⎝parenrightBig2⎝integraldisplayx af(t)dt cosh2t(tanhx–t a n h t)λ. 88.⎝integraldisplay ⎝integraldisplayx a(tanhµx–t a n hµt)y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=t a n hµx. Solution: y(x)=1 µd dx⎝bracketleftbiggcoshµ+1xf/prime x(x) sinhµ–1x⎝bracketrightbigg . 89.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Atanhµx+Btanhµt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=t a n hµx.F o rB=–A, see equation 1.3.88. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig tanh(λx)⎝bracketrightbig–Aµ A+B⎝integraldisplayx a⎝bracketleftbig tanh(λt)⎝bracketrightbig–Bµ A+Bf/prime t(t)dt⎝bracerightbigg . 90.⎝integraldisplay ⎝integraldisplayx ay(t)dt [tanh(λx )–t a n h ( λt)]µ=f(x), 0 < µ<1 . This is a special case of equation 1.9.44 with g(x)=t a n h ( λx)a n d h(x)≡1. Solution: y(x)=λsin(πµ) πd dx⎝integraldisplayx af(t)dt cosh2(λt)[tanh( λx)–t a n h ( λt)]1–µ. 38 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 91.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axβ+Btanhγ(λt)+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Axβandh(t)=Btanhγ(λt)+C. 92.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanhγ(λx)+Btβ+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanhγ(λx)a n d h(t)=Btβ+C. 93.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλtanhµt+Btβtanhγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=t a n hµt,g2(x)=Btanhγx, andh2(t)=tβ. 1.3-4. Kernels Containing Hyperbolic Cotangent. 94.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig coth(λx )–c o t h ( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=c o t h ( λx). Solution: y(x)=–1 λd dx⎝bracketleftbig sinh2(λx)f/prime x(x)⎝bracketrightbig . 95.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoth(λx )+Bcoth(λt )⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o t h ( λx). For B=–A, see equation 1.3.94. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig tanh(λx)⎝bracketrightbigA A+B⎝integraldisplayx a⎝bracketleftbig tanh(λt)⎝bracketrightbigB A+Bf/prime t(t)dt⎝bracerightbigg . 96.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoth(λx )+Bcoth(µt )+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acoth(λx)a n d h(t)=Bcoth(µt)+C. 97.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig coth2(λx)–c o t h2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=c o t h2(λx). Solution: y(x)=–d dx⎝bracketleftbiggsinh3(λx)f/prime x(x) 2λcosh(λx)⎝bracketrightbigg . 98.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoth2(λx)+Bcoth2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o t h2(λx). For B=–A, see equation 1.3.97. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig tanh(λx)⎝bracketrightbig2A A+B⎝integraldisplayx a⎝bracketleftbig tanh(λt)⎝bracketrightbig2B A+Bf/prime t(t)dt⎝bracerightbigg . 99.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoth2(λx)+Bcoth2(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acoth2(λx)a n d h(t)=Bcoth2(µt)+C. 1.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 39 100.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig coth(λx )–c o t h ( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=(–1)n λnn!s i n h2(λx)⎝bracketleftbigg sinh2(λx)d dx⎝bracketrightbiggn+1 f(x). 101.⎝integraldisplay ⎝integraldisplayx a(cothµx–c o t hµt)y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=c o t hµx. Solution: y(x)=–1 µd dx⎝bracketleftbiggsinhµ+1xf/prime x(x) coshµ–1x⎝bracketrightbigg . 102.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Acothµx+Bcothµt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o t hµx.F o rB=–A, see equation 1.3.101. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingletanhx⎝vextendsingle⎝vextendsingleAµ A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingletanht⎝vextendsingle⎝vextendsingleBµ A+Bf/prime t(t)dt⎝bracerightbigg . 103.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axβ+Bcothγ(λt)+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Axβandh(t)=Bcothγ(λt)+C. 104.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acothγ(λx)+Btβ+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acothγ(λx)a n d h(t)=Btβ+C. 105.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλcothµt+Btβcothγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=c o t hµt,g2(x)=Bcothγx, andh2(t)=tβ. 1.3-5. Kernels Containing Combinations of Hyperbolic Functions. 106.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig cosh[λ (x–t)] +Asinh[µ(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). Let us differentiate the equation with respect to xand then eliminate the integral with the hyperbolic cosine. As a result, we arrive at an equation of the form 2.3.16: y(x)+(λ–A2µ)⎝integraldisplayx asinh[µ(x–t)]y(t)dt=f/prime x(x)–Aµf (x). 107.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosh(λx )+Bsinh(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acosh(λx)a n d h(t)=Bsinh(µt)+C. 40 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 108.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosh2(λx)+Bsinh2(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acosh2(λx)a n d h(t)=Bsinh2(µt)+C. 109.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cosh[λ (x+t)]y(t)dt=f(x). Using the formula sinh(α–β)c o s h ( α+β)=1 2⎝bracketleftbig sinh(2α ) – sinh(2β )⎝bracketrightbig ,α=λx, β=λt, we reduce the original equation to an equation of the form 1.3.45: ⎝integraldisplayx a⎝bracketleftbig sinh(2 λx)–s i n h ( 2 λt)⎝bracketrightbig y(t)dt=2f(x). Solution: y(x)=1 λd dx⎝bracketleftbiggf/prime x(x) cosh(2 λx)⎝bracketrightbigg . 110.⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)] sinh[ λ(x+t)]y(t)dt=f(x). Using the formula cosh(α –β)s i n h ( α+β)=1 2⎝bracketleftbig sinh(2 α)+s i n h ( 2 β)⎝bracketrightbig ,α=λx, β=λt, we reduce the original equation to an equation of the form 1.3.46 with A=B=1 : ⎝integraldisplayx a⎝bracketleftbig sinh(2 λx)+s i n h ( 2 λt)⎝bracketrightbig y(t)dt=2f(x). 111.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosh(λx ) sinh( µt)+Bcosh(βx ) sinh( γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Acosh(λx), h1(t)=s i n h ( µt),g2(x)= Bcosh(βx), and h2(t)=s i n h ( γt). 112.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig sinh(λx)c o s h ( µt)+s i n h ( βx)c o s h ( γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x) = sinh(λx), h1(t)=c o s h ( µt),g2(x)= sinh(βx), and h2(t)=c o s h ( γt). 113.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cosh(λx )c o s h ( µt)+s i n h ( βx) sinh( γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=c o s h ( λx),h1(t)=c o s h ( µt),g2(x)= sinh(βx), and h2(t) = sinh(γt ). 114.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoshβ(λx)+Bsinhγ(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acoshβ(λx)a n d h(t)=Bsinhγ(µt). 1.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 41 115.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinhβ(λx)+Bcoshγ(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asinhβ(λx)a n d h(t)=Bcoshγ(µt). 116.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλcoshµt+Btβsinhγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=c o s hµt,g2(x)=Bsinhγx, andh2(t)=tβ. 117.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig (x–t)s i n h [ λ(x–t)] –λ(x–t)2cosh[λ (x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). Solution: y(x)=⎝integraldisplayx ag(t)dt, where g(t)=⎝radicalbigg π 2λ1 64λ5⎝parenleftbiggd2 dt2–λ2⎝parenrightbigg6⎝integraldisplayt a(t–τ)5 2I5 2[λ(t–τ)]f(τ)dτ. 118.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbigg ⎝braceleftbiggsinh[λ(x–t)] x–t–λcosh[λ (x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x). Solution: y(x)=1 2λ4⎝parenleftbiggd2 dx2–λ2⎝parenrightbigg3⎝integraldisplayx asinh[λ(x–t)]f(t)dt. 119.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig sinh⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig –λ√ x–tcosh⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x),f(a)=f/prime x(a)=0 . Solution: y(x)=–4 πλ3d3 dx3⎝integraldisplayx acos⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. 120.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλsinhµt+Btβcoshγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=s i n hµt,g2(x)=Bcoshγx, andh2(t)=tβ. 121.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanh(λx)+Bcoth(µt )+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanh(λx)a n d h(t)=Bcoth(µt)+C. 122.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanh2(λx)+Bcoth2(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanh2(λx)a n d h(t)=Bcoth2(µt). 123.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tanh(λx)c o t h ( µt)+t a n h ( βx)c o t h ( γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=t a n h ( λx),h1(t)=c o t h ( µt),g2(x)= tanh(βx), and h2(t)=c o t h ( γt). 42 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 124.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig coth(λx )t a n h ( µt)+c o t h ( βx) tanh( γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=c o t h ( λx),h1(t)=t a n h ( µt),g2(x)= coth(βx), and h2(t)=t a n h ( γt). 125.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tanh(λx) tanh( µt)+c o t h ( βx)c o t h ( γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=t a n h ( λx),h1(t)=t a n h ( µt),g2(x)= coth(βx), and h2(t)=c o t h ( γt). 126.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanhβ(λx)+Bcothγ(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanhβ(λx)a n d h(t)=Bcothγ(µt). 127.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acothβ(λx)+Btanhγ(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acothβ(λx)a n d h(t)=Btanhγ(µt). 128.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλtanhµt+Btβcothγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=t a n hµt,g2(x)=Bcothγx, andh2(t)=tβ. 129.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλcothµt+Btβtanhγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=c o t hµt,g2(x)=Btanhγx, andh2(t)=tβ. 1.4. Equations Whose Kernels Contain Logarithmic Functions 1.4-1. Kernels Containing Logarithmic Functions. 1.⎝integraldisplay ⎝integraldisplayx a(lnx–l nt)y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=l nx. Solution: y(x)=xf/prime/prime xx(x)+f/prime x(x). 2.⎝integraldisplay ⎝integraldisplayx 0ln(x–t)y(t)dt=f(x). Solution: y(x)=–⎝integraldisplayx 0f/prime/prime tt(t)dt⎝integraldisplay∞ 0(x–t)ze–Cz Γ(z+1 )dz–f/prime x(0)⎝integraldisplay∞ 0xze–Cz Γ(z+1 )dz, where C= lim k→∞⎝parenleftBig 1+1 2+···+1 k+1–l nk⎝parenrightBig = 0.5772 ...is the Euler constant and Γ(z)i s the gamma function. References: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971), A. G. Butkovskii (1979). 1.4. E QUATIONS WHOSE KERNELS CONTAIN LOGARITHMIC FUNCTIONS 43 3.⎝integraldisplay ⎝integraldisplayx a[ln(x –t)+A]y(t)dt=f(x). Solution: y(x)=–d dx⎝integraldisplayx aνA(x–t)f(t)dt,νA(x)=d dx⎝integraldisplay∞ 0xze(A–C)z Γ(z+1 )dz, whereC= 0.5772 ...is the Euler constant and Γ(z) is the gamma function. Fora= 0, the solution can be written in the form y(x)=–⎝integraldisplayx 0f/prime/prime tt(t)dt⎝integraldisplay∞ 0(x–t)ze(A–C)z Γ(z+1 )dz–f/prime x(0)⎝integraldisplay∞ 0xze(A–C)z Γ(z+1 )dz. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 4.⎝integraldisplay ⎝integraldisplayx a(Alnx+Blnt)y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=l nx.F o rB=–A, see equation 1.4.1. Solution: y(x)=sign(ln x) A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsinglelnx⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsinglelnt⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . 5.⎝integraldisplay ⎝integraldisplayx a(Alnx+Blnt+C)y(t)dt=f(x). This is a special case of equation 1.9.5 with g(x)=x. 6.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig ln2(λx)–l n2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=d dx⎝bracketleftbiggxf/prime x(x) 2l n (λx)⎝bracketrightbigg . 7.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aln2(λx)+Bln2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=l n2(λx). For B=–A, see equation 1.4.6. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingleln(λx)⎝vextendsingle⎝vextendsingle–2A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingleln(λt)⎝vextendsingle⎝vextendsingle–2B A+Bf/prime t(t)dt⎝bracerightbigg . 8.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aln2(λx)+Bln2(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aln2(λx)a n d h(t)=Bln2(µt)+C. 9.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig ln(x/t )⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=1 n!x⎝parenleftbigg xd dx⎝parenrightbiggn+1 f(x). 44 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 10.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig ln2x–l n2t⎝parenrightbig⎝parenrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=lnx 2nn!x⎝parenleftbiggx lnxd dx⎝parenrightbiggn+1 f(x). 11.⎝integraldisplay ⎝integraldisplayx aln⎝parenleftbigg ⎝parenleftbiggx+b t+b⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=l n ( x+b). Solution: y(x)=(x+b)f/prime/prime xx(x)+f/prime x(x). 12.⎝integraldisplay ⎝integraldisplayx a⎝radicalbig ln(x/t )y(t)dt=f(x). Solution: y(x)=2 πx⎝parenleftbigg xd dx⎝parenrightbigg2⎝integraldisplayx af(t)dt t⎝radicalbig ln(x/t). 13.⎝integraldisplay ⎝integraldisplayx ay(t)dt ⎝radicalbig ln(x/t )=f(x). Solution: y(x)=1 πd dx⎝integraldisplayx af(t)dt t⎝radicalbig ln(x/t). 14.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig lnµ(λx)–l nµ(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=l nµ(λx). Solution: y(x)=1 µd dx⎝bracketleftbig xln1–µ(λx)f/prime x(x)⎝bracketrightbig . 15.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Alnβ(λx)+Blnγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Alnβ(λx)a n d h(t)=Blnγ(µt)+C. 16.⎝integraldisplay ⎝integraldisplayx a[ln(x/t )]λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=k x⎝parenleftbigg xd dx⎝parenrightbigg2⎝integraldisplayx af(t)dt t[ln(x/t )]λ,k=sin(πλ) πλ. 17.⎝integraldisplay ⎝integraldisplayx ay(t)dt [ln(x/t )]λ=f(x), 0 < λ<1 . This is a special case of equation 1.9.44 with g(x)=l nxandh(x)≡1. Solution: y(x)=sin(πλ) πd dx⎝integraldisplayx af(t)dt t[ln(x/t )]1–λ. 1.4. E QUATIONS WHOSE KERNELS CONTAIN LOGARITHMIC FUNCTIONS 45 18.⎝integraldisplay ⎝integraldisplayx 0ln√ x+√ x–t √ x–√ x–ty(t)dt=f(x). Solution: y(x)=1 πd dx⎝integraldisplay ⎝integraldisplayx 0√ t √ x–td dtf(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 451). 19.⎝integraldisplay ⎝integraldisplay∞ xln√ t+√ t–x √ t–√ t–xy(t)dt=f(x). Solution: y(x)=1 π1 √ xd dx⎝integraldisplay ⎝integraldisplay∞ xt √ t–xd dtf(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 452). 1.4-2. Kernels Containing Power-Law and Logarithmic Functions. 20.⎝integraldisplay ⎝integraldisplayx a(x–t)⎝bracketleftbig⎝bracketleftbig ln(x–t)+A⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=–d2 dx2⎝integraldisplayx aνA(x–t)f(t)dt,νA(x)=d dx⎝integraldisplay∞ 0xze(A–C)z Γ(z+1 )dz, whereC= 0.5772 ...is the Euler constant and Γ(z) is the gamma function. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 21.⎝integraldisplay ⎝integraldisplayx aln(x–t)+A (x–t)λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=–sin(πλ) πd dx⎝integraldisplayx aF(t)dt (x–t)1–λ,F(x)=⎝integraldisplayx aνh(x–t)f(t)dt, νh(x)=d dx⎝integraldisplay∞ 0xzehz Γ(z+1 )dz,h=A+ψ(1 –λ), where Γ(z) is the gamma function and ψ(z)=⎝bracketleftbig Γ(z)⎝bracketrightbig/prime zis the logarithmic derivative of the gamma function. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 22.⎝integraldisplay ⎝integraldisplayx a(x–t)α–1 Γ(α)[ln(x –t)+A]y(t)dt=f(x), α>0 . Solution: y(x)=–1 Γ([α]–α+1 )⎝parenleftbiggd dx⎝parenrightbigg[α]+1⎝integraldisplay ⎝integraldisplayx aF(t)dt (x–t)α–[α],F(x)=⎝integraldisplayx aνh(x–t)f(t)dt, νh(x)=d dx⎝integraldisplay∞ 0xzehz Γ(z+1 )dz,h=A+ψ(α), where Γ(z) is the gamma function and ψ(z)=⎝bracketleftbig Γ(z)⎝bracketrightbig/prime zis the logarithmic derivative of the gamma function. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993, p. 483). 46 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 23.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig tβlnλx–xβlnλt)y(t)dt=f(x). This is a special case of equation 1.9.11 with g(x)=l nλxandh(t)=tβ. 24.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Atβlnλx+Bxµlnγt)y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Alnλx,h1(t)=tβ,g2(x)=Bxµ,a n d h2(t)=l nγt. 25.⎝integraldisplay ⎝integraldisplayx aln⎝parenleftbigg ⎝parenleftbiggxµ+b ctλ+s⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=l n ( xµ+b)a n dh(t)=–l n ( ctλ+s). 1.5. Equations Whose Kernels Contain Trigonometric Functions 1.5-1. Kernels Containing Cosine. 1.⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)]y(t)dt=f(x). Solution: y(x)=f/prime x(x)+λ2⎝integraldisplayx af(x)dx. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 442). 2.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig cos[λ (x–t)] – 1⎝bracerightbig ⎝bracerightbig y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Solution: y(x)=–1 λ2f/prime/prime/prime xxx(x)–f/prime x(x). 3.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig cos[λ (x–t)] +b⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). Forb= 0, see equation 1.5.1. For b= –1, see equation 1.5.2. For λ= 0, see equation 1.1.1. Differentiating the equation with respect to x, we arrive at an equation of the form 2.5.16: y(x)–λ b+1⎝integraldisplayx asin[λ(x–t)]y(t)dt=f/prime x(x) b+1. 1◦. Solution with b(b+1 )>0 : y(x)=f/prime x(x) b+1+λ2 k(b+1 )2⎝integraldisplayx asin[k(x–t)]f/prime t(t)dt,w h e r e k=λ⎝radicalbigg b b+1. 2◦. Solution with b(b+1 )<0 : y(x)=f/prime x(x) b+1+λ2 k(b+1 )2⎝integraldisplayx asinh[k(x–t)]f/prime t(t)dt,w h e r e k=λ⎝radicalbigg –b b+1. 1.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 47 4.⎝integraldisplay ⎝integraldisplayx acos(λx +βt)y(t)dt=f(x). Differentiating the equation with respect to xtwice yields cos[(λ +β)x]y(x)–λ⎝integraldisplayx asin(λx+βt)y(t)dt=f/prime x(x), (1) ⎝braceleftbig cos[(λ +β)x]y(x)⎝bracerightbig/prime x–λsin[(λ+β)x]y(x)–λ2⎝integraldisplayx acos(λx +βt)y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the ﬁrst-order linear ordinary differential equation w/prime x–λtan[(λ+β)x]w=f/prime/prime xx(x)+λ2f(x), w=c o s [ ( λ+β)x]y(x). (3) Setting x=ain (1) yields the initial condition w(a)=f/prime x(a). On solving equation (3) under this condition, after some transformati ons we obtain the solution of t he original integral equation in the form y(x)=1 cos[(λ +β)x]f/prime x(x)+λsin[(λ+β)x] cos2[(λ+β)x]f(x) –λβ cosk+1[(λ+β)x]⎝integraldisplayx af(t)c o sk–2[(λ+β)t]dt,k=λ λ+β. 5.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cos(λx )–c o s ( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=c o s ( λx). Solution: y(x)=–1 λd dx⎝bracketleftbiggf/prime x(x) sin(λx)⎝bracketrightbigg . 6.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos(λx )+Bcos(λt )⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o s ( λx). For B=–A, see equation 1.5.5. Solution with B≠–A: y(x)=sign cos( λx) A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsinglecos(λx)⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsinglecos(λt )⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . 7.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos(λx )+Bcos(µt )+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acos(λx)a n d h(t)=Bcos(µt)+ C. 8.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A1cos[λ 1(x–t)] +A2cos[λ 2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). The equation is equivalent to the equation ⎝integraldisplayx a⎝braceleftbig B1sin[λ1(x–t)] +B2sin[λ2(x–t)]⎝bracerightbig y(t)dt=F(x), B1=A1 λ1,B2=A2 λ2,F(x)=⎝integraldisplayx af(t)dt, which has the form 1.5.41. (Differentiation of this equation yields the original integral equation.) 48 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 9.⎝integraldisplay ⎝integraldisplayx acos2[λ(x–t)]y(t)dt=f(x). Differentiating yields an equation of the form 2.5.16: y(x)–λ⎝integraldisplayx asin[2λ(x–t)]y(t)dt=f/prime x(x). Solution: y(x)=f/prime x(x)+2λ2 k⎝integraldisplayx asin[k(x–t)]f/prime t(t)dt,w h e r e k=λ√ 2. 10.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cos2(λx)–c o s2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=–1 λd dx⎝bracketleftbiggf/prime x(x) sin(2λx)⎝bracketrightbigg . 11.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos2(λx)+Bcos2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o s2(λx). For B=–A, see equation 1.5.10. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig cos(λx)⎝bracketrightbig–2A A+B⎝integraldisplayx a⎝bracketleftbig cos(λt )⎝bracketrightbig–2B A+Bf/prime t(t)dt⎝bracerightbigg . 12.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos2(λx)+Bcos2(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acos2(λx)a n d h(t)=Bcos2(µt)+C. 13.⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)] cos[λ (x+t)]y(t)dt=f(x). Using the trigonometric formula cos(α –β)c o s (α+β)=1 2⎝bracketleftbig cos(2α)+c o s ( 2 β)⎝bracketrightbig ,α=λx, β=λt, we reduce the original equation to an equation of the form 1.5.6 with A=B=1 : ⎝integraldisplayx a⎝bracketleftbig cos(2λx)+c o s ( 2 λt)⎝bracketrightbig y(t)dt=2f(x). Solution with cos(2 λx)>0 : y(x)=d dx⎝bracketleftbigg1 √ cos(2λx)⎝integraldisplayx af/prime t(t)dt √ cos(2λt)⎝bracketrightbigg . 14.⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)] cos[µ (x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=1 ⎝radicalbig λ2–µ2⎝bracketleftbiggd2 dx2+(λ+µ)2⎝bracketrightbigg⎝bracketleftbiggd2 dx2+(λ–µ)2⎝bracketrightbigg⎝integraldisplayx a⎝integraldisplayt asin⎝bracketleftbig⎝radicalbig λ2+µ2(t–s)⎝bracketrightbig f(s)dsdt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 444). 1.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 49 15.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos(λx )c o s (µt)+Bcos(βx )c o s (γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Acos(λx), h1(t)=c o s ( µt),g2(x)= Bcos(βx ), and h2(t)=c o s ( γt). 16.⎝integraldisplay ⎝integraldisplayx acos3[λ(x–t)]y(t)dt=f(x). Using the formula cos3β=1 4cos 3β+3 4cosβ, we arrive at an equation of the form 1.5.8: ⎝integraldisplayx a⎝braceleftbig1 4cos[3λ(x–t)] +3 4cos[λ (x–t)]⎝bracerightbig y(t)dt=f(x). 17.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cos3(λx)–c o s3(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=–1 3λd dx⎝bracketleftbiggf/prime x(x) sin(λx)c o s2(λx)⎝bracketrightbigg . 18.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos3(λx)+Bcos3(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o s3(λx). For B=–A, see equation 1.5.17. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig cos(λx)⎝bracketrightbig–3A A+B⎝integraldisplayx a⎝bracketleftbig cos(λt )⎝bracketrightbig–3B A+Bf/prime t(t)dt⎝bracerightbigg . 19.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cos2(λx)c o s (µt)+c o s ( βx)c o s2(γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=c o s2(λx),h1(t)=c o s ( µt),g2(x)=c o s ( βx), andh2(t)=c o s2(γt). 20.⎝integraldisplay ⎝integraldisplayx acos4[λ(x–t)]y(t)dt=f(x). Let us transform the kernel of the integral equation using the trigonometric formula cos4β= 1 8cos 4β+1 2cos 2β+3 8,w h e r e β=λ(x–t), and differentiate the resulting equation with respect to x. Then we arrive at an equation of the form 2.5.18: y(x)–λ⎝integraldisplayx a⎝braceleftbig1 2sin[4λ(x–t)] + sin[2λ (x–t)]⎝bracerightbig y(t)dt=f/prime x(x). 21.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cos(λx )–c o s ( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=(–1)n λnn!sin(λx)⎝bracketleftbigg1 sin(λx)d dx⎝bracketrightbiggn+1 f(x). 50 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 22.⎝integraldisplay ⎝integraldisplayx a√ cost–c o sxy(t)dt=f(x). This is a special case of equation 1.9.40 with g(x)=1–c o s x. Solution: y(x)=2 πsinx⎝parenleftBig1 sinxd dx⎝parenrightBig2⎝integraldisplayx asintf(t)dt √ cost–c o sx. 23.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ cost–c o sx=f(x). Solution: y(x)=1 πd dx⎝integraldisplayx asintf(t)dt √ cost–c o sx. 24.⎝integraldisplay ⎝integraldisplayx a(cost–c o sx)λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=ksinx⎝parenleftBig1 sinxd dx⎝parenrightBig2⎝integraldisplayx asintf(t)dt (cost–c o sx)λ,k=sin(πλ) πλ. 25.⎝integraldisplay ⎝integraldisplayx a(cosµx–c o sµt)y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=c o sµx. Solution: y(x)=–1 µd dx⎝bracketleftbiggf/prime x(x) sinxcosµ–1x⎝bracketrightbigg . 26.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Acosµx+Bcosµt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o sµx.F o rB=–A, see equation 1.5.25. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsinglecosx⎝vextendsingle⎝vextendsingle–Aµ A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsinglecost⎝vextendsingle⎝vextendsingle–Bµ A+Bf/prime t(t)dt⎝bracerightbigg . 27.⎝integraldisplay ⎝integraldisplayx ay(t)dt (cost–c o sx)λ=f(x), 0 < λ<1 . Solution: y(x)=sin(πλ) πd dx⎝integraldisplayx asintf(t)dt (cost–c o sx)1–λ. 28.⎝integraldisplay ⎝integraldisplayx a(x–t)c o s [λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . Differentiating the equation twice yields y(x)–2λ⎝integraldisplayx asin[λ(x–t)]y(t)dt–λ2⎝integraldisplayx a(x–t)c o s [ λ(x–t)]y(t)dt=f/prime/prime xx(x). Eliminating the third term on the left-hand side with the aid of the original equation, we arrive at an equation of the form 2.5.16: y(x)–2λ⎝integraldisplayx asin[λ(x–t)]y(t)dt=f/prime/prime xx(x)+λ2f(x). 1.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 51 29.⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)] √ x–ty(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=2 πλ⎝integraldisplayx asin[λ(x–t)] √ x–t[f/prime/prime tt(t)+λ2f(t)]dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 445). 30.⎝integraldisplay ⎝integraldisplayx a√ x–tcos⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=1 π⎝integraldisplayx acosh⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf/prime t(t)dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 445–446), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 31.⎝integraldisplay ⎝integraldisplayx acos⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig √ x–ty(t)dt=f(x). Solution: y(x)=1 πd dx⎝integraldisplayx acosh⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 446), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 32.⎝integraldisplay ⎝integraldisplay∞ xcos⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig √ t–xy(t)dt=f(x). Solution: y(x)=–1 πd dx⎝integraldisplay∞ xcosh⎝parenleftbig λ√ t–x⎝parenrightbig √ t–xf(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 448), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 33.⎝integraldisplay ⎝integraldisplayx 0cos⎝parenleftbig⎝parenleftbig λ√ x2–t2⎝parenrightbig⎝parenrightbig √ x2–t2y(t)dt=f(x). Solution: y(x)=2 πd dx⎝integraldisplayx 0tcosh⎝parenleftbig λ√ x2–t2⎝parenrightbig √ x2–t2f(t)dt. 34.⎝integraldisplay ⎝integraldisplay∞ xcos⎝parenleftbig⎝parenleftbig λ√ t2–x2⎝parenrightbig⎝parenrightbig √ t2–x2y(t)dt=f(x). Solution: y(x)=–2 πd dx⎝integraldisplay∞ xtcosh⎝parenleftbig λ√ t2–x2⎝parenrightbig √ t2–x2f(t)dt. 35.⎝integraldisplay ⎝integraldisplayx 0cos⎝parenleftbig⎝parenleftbig λ√ xt–t2⎝parenrightbig⎝parenrightbig √ x–ty(t)dt=f(x). Solution: y(x)=1 πx⎝integraldisplayx 0cosh⎝parenleftbig λ√ x2–xt⎝parenrightbig √ x–t[f(t)/2+tf/prime t(t)]dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 446). 52 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 36.⎝integraldisplay ⎝integraldisplayx 0cos⎝parenleftbig⎝parenleftbig λ√ x2–xt⎝parenrightbig⎝parenrightbig √ x–ty(t)dt=f(x). Solution: y(x)=√ x πd dx⎝bracketleftbigg√ x⎝integraldisplayx 0cosh⎝parenleftbig λ√ xt–t2⎝parenrightbig √ x–tf(t)dt⎝bracketrightbigg . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 446). 37.⎝integraldisplay ⎝integraldisplayx acos⎝bracketleftbig⎝bracketleftbig λ√ (x–t)(x –t+γ)⎝bracketrightbig⎝bracketrightbig √ x–ty(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Solution: y(x)=2 πλ2⎝integraldisplayx asin⎝bracketleftbig λ√ (x–t)(x–t–γ)⎝bracketrightbig √ x–t–γ⎝integraldisplayt asin[λ(t–s)]⎝parenleftbiggd2 ds2+λ2⎝parenrightbigg2 f(s)dsdt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 447). 38.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axβ+Bcosγ(λt)+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Axβandh(t)=Bcosγ(λt)+C. 39.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosγ(λx)+Btβ+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acosγ(λx)a n d h(t)=Btβ+C. 40.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλcosµt+Btβcosγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=c o sµt,g2(x)=Bcosγx, andh2(t)=tβ. 1.5-2. Kernels Containing Sine. 41.⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=1 λf/prime/prime xx(x)+λf(x). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 442). 42.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig sin[λ(x–t)] +b⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). Forb= 0, see equation 1.5.41. Assume that b≠0. Differentiating the equation with respect to xyields an equation of the form 2.5.3: y(x)+λ b⎝integraldisplayx acos[λ (x–t)]y(t)dt=1 bf/prime x(x). 1.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 53 43.⎝integraldisplay ⎝integraldisplayx asin(λx +βt)y(t)dt=f(x). Forβ=–λ, see equation 1.5.41. Assume that β≠–λ. Differentiating the equation with respect to xtwice yields sin[(λ+β)x]y(x)+λ⎝integraldisplayx acos(λx +βt)y(t)dt=f/prime x(x), (1) ⎝braceleftbig sin[(λ+β)x]y(x)⎝bracerightbig/prime x+λcos[(λ +β)x]y(x)–λ2⎝integraldisplayx asin(λx+βt)y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the ﬁrst-order linear ordinary differential equation w/prime x+λcot[(λ+β)x]w=f/prime/prime xx(x)+λ2f(x), w=s i n [ ( λ+β)x]y(x). (3) Setting x=ain (1) yields the initial condition w(a)=f/prime x(a). On solving equation (3) under this condition, after some transformation we obtain the solution of the original integral equation in the form y(x)=1 sin[(λ+β)x]f/prime x(x)–λcos[(λ +β)x] sin2[(λ+β)x]f(x) –λβ sink+1[(λ+β)x]⎝integraldisplayx af(t)s i nk–2[(λ+β)t]dt,k=λ λ+β. 44.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig sin(λx)–s i n ( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=s i n ( λx). Solution: y(x)=1 λd dx⎝bracketleftbiggf/prime x(x) cos(λx)⎝bracketrightbigg . 45.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asin(λx)+Bsin(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=s i n ( λx). For B=–A, see equation 1.5.44. Solution with B≠–A: y(x)=sign sin( λx) A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsinglesin(λx)⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsinglesin(λt)⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . 46.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asin(λx)+Bsin(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asin(λx)a n d h(t)=Bsin(µt)+C. 47.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig µsin[λ(x–t)] –λsin[µ(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Solution: y(x)=f/prime/prime/prime/prime xxxx +(λ2+µ2)f/prime/prime xx+λ2µ2f λµ3–λ3µ,f=f(x). 54 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 48.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A1sin[λ1(x–t)] +A2sin[λ2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x),f(a)=f/prime x(a)=0 . This equation can be solved in the same manner as equation 1.3.49, i.e., by reducing it to a second-order linear ordinary differential equation with constant coefﬁcients. Let ∆=–λ1λ2A1λ2+A2λ1 A1λ1+A2λ2. 1◦. Solution for ∆>0 : (A1λ1+A2λ2)y(x)=f/prime/prime xx(x)+Bf(x)+C⎝integraldisplayx asinh[k(x–t)]f(t)dt, k=√ ∆,B=∆+λ2 1+λ2 2,C=1 √ ∆⎝bracketleftbig ∆2+(λ2 1+λ2 2)∆+λ2 1λ22⎝bracketrightbig . 2◦. Solution for ∆<0 : (A1λ1+A2λ2)y(x)=f/prime/prime xx(x)+Bf(x)+C⎝integraldisplayx asin[k(x–t)]f(t)dt, k=√ –∆,B=∆+λ2 1+λ2 2,C=1 √ –∆⎝bracketleftbig ∆2+(λ2 1+λ2 2)∆+λ2 1λ22⎝bracketrightbig . 3◦. Solution for ∆=0 : (A1λ1+A2λ2)y(x)=f/prime/prime xx(x)+(λ2 1+λ2 2)f(x)+λ2 1λ22⎝integraldisplayx a(x–t)f(t)dt. 4◦. Solution for ∆=∞: y(x)=–f/prime/prime/prime/prime xxxx +(λ2 1+λ2 2)f/prime/prime xx+λ2 1λ22f A1λ31+A2λ3 2,f=f(x). In the last case, the relation A1λ1+A2λ2= 0 holds and the right-hand side of the integral equation is assumed to satisfy the conditions f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Remark. The solution can be obtained from the solution of equation 1.3.49 in which the change of variables λk→iλk,Ak→–iAk,i2=– 1 ( k= 1, 2), should be made. 49.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig Asin[λ(x–t)] +Bsin[µ(x–t)] +Csin[β(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). It is assumed that f(a)=f/prime x(a) = 0. Differentiating the integral equation twice yields (Aλ+Bµ+Cβ)y(x)–⎝integraldisplayx a⎝braceleftbig Aλ2sin[λ(x–t)] +Bµ2sin[µ(x–t)]⎝bracerightbig y(t)dt –Cβ2⎝integraldisplayx asin[β(x–t)]y(t)dt=f/prime/prime xx(x). Eliminating the last integral with the aid of the original equation, we arrive at an equation of the form 2.5.18: (Aλ+Bµ+Cβ)y(x)+⎝integraldisplayx a⎝braceleftbig A(β2–λ2)s i n [λ(x–t)] +B(β2–µ2)s i n [µ(x–t)]⎝bracerightbig y(t)dt=f/prime/prime xx(x)+β2f(x). In the special case Aλ+Bµ+Cβ= 0, this is an equation of the form 1.5.41. 1.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 55 50.⎝integraldisplay ⎝integraldisplayx asin2[λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Differentiation yields an equation of the form 1.5.41: ⎝integraldisplayx asin[2λ(x–t)]y(t)dt=1 λf/prime x(x). Solution: y(x)=1 2λ–2f/prime/prime/prime xxx(x)+2f/prime x(x). 51.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig sin2(λx)–s i n2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=1 λd dx⎝bracketleftbiggf/prime x(x) sin(2λx)⎝bracketrightbigg . 52.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asin2(λx)+Bsin2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=s i n2(λx). For B=–A, see equation 1.5.51. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsinglesin(λx)⎝vextendsingle⎝vextendsingle–2A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsinglesin(λt)⎝vextendsingle⎝vextendsingle–2B A+Bf/prime t(t)dt⎝bracerightbigg . 53.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asin2(λx)+Bsin2(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asin2(λx)a n d h(t)=Bsin2(µt)+C. 54.⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] sin[λ(x+t)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . Using the trigonometric formula sin(α–β)s i n (α+β)=1 2⎝bracketleftbig cos(2β)–c o s ( 2 α)⎝bracketrightbig ,α=λx, β=λt, we reduce the original equation to an equation of the form 1.5.5: ⎝integraldisplayx a⎝bracketleftbig cos(2λx)–c o s ( 2 λt)⎝bracketrightbig y(t)dt=– 2f(x). Solution: y(x)=1 λd dx⎝bracketleftbiggf/prime x(x) sin(2λx)⎝bracketrightbigg . 55.⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] sin[µ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Solution: y(x)=⎝bracketleftbiggd2 dx2+(λ+µ)2⎝bracketrightbigg⎝bracketleftbiggd2 dx2+(λ–µ)2⎝bracketrightbigg1 2λµ⎝integraldisplayx af(t)dt. Reference A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 443). 56 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 56.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig sin(λx)s i n (µt)+s i n ( βx)s i n (γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=s i n ( λx),h1(t)=s i n ( µt),g2(x)=s i n ( βx), andh2(t)=s i n ( γt). 57.⎝integraldisplay ⎝integraldisplayx asin3[λ(x–t)]y(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Using the formula sin3β=–1 4sin 3β+3 4sinβ, we arrive at an equation of the form 1.5.48: ⎝integraldisplayx a⎝braceleftbig –1 4sin[3λ(x–t)] +3 4sin[λ(x–t)]⎝bracerightbig y(t)dt=f(x). 58.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig sin3(λx)–s i n3(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), f(a)=f/prime x(a)=0 . This is a special case of equation 1.9.2 with g(x)=s i n3(λx). 59.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asin3(λx)+Bsin3(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=s i n3(λx). For B=–A, see equation 1.5.58. Solution: y(x)=sign sin( λx) A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsinglesin(λx)⎝vextendsingle⎝vextendsingle–3A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsinglesin(λt)⎝vextendsingle⎝vextendsingle–3B A+Bf/prime t(t)dt⎝bracerightbigg . 60.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig sin2(λx)s i n (µt)+s i n ( βx)s i n2(γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=s i n2(λx),h1(t)=s i n ( µt),g2(x)=s i n ( βx), andh2(t)=s i n2(γt). 61.⎝integraldisplay ⎝integraldisplayx asin4[λ(x–t)]y(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=···=f/prime/prime/prime/prime xxxx(a)=0 . Let us transform the kernel of the integral equation using the trigonometric formula sin4β=1 8cos 4β–1 2cos 2β+3 8,w h e r e β=λ(x–t), and differentiate the resulting equation with respect to x. Then we obtain an equation of the form 1.5.48: λ⎝integraldisplayx a⎝braceleftbig –1 2sin[4λ(x–t)] + sin[2λ (x–t)]⎝bracerightbig y(t)dt=f/prime x(x). 62.⎝integraldisplay ⎝integraldisplayx asinn[λ(x–t)]y(t)dt=f(x), n=2 , 3 , ... It is assumed that f(a)=f/prime x(a)=···=f(n) x(a)=0 . 1◦. Let us differentiate the equation with respect to xtwice and transform the kernel of the resulting integral equation using the formula cos2β=1–s i n2β,w h e r e β=λ(x–t). We have –λ2n2⎝integraldisplayx asinn[λ(x–t)]y(t)dt+λ2n(n–1 )⎝integraldisplayx asinn–2[λ(x–t)]y(t)dt=f/prime/prime xx(x). 1.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 57 Eliminating the ﬁrst term on the left-hand side with the aid of the original equation, we obtain ⎝integraldisplayx asinn–2[λ(x–t)]y(t)dt=1 λ2n(n–1 )⎝bracketleftbig f/prime/prime xx(x)+λ2n2f(x)⎝bracketrightbig . This equation has the same form as the original equation, but the degree characterizing the kernel has been reduced by two. By applying this technique sufﬁciently many times, we ﬁnally arrive at simple integral equations of the form 1.1.1 (for even n)o r1 . 5 . 4 1( f o ro d dn ). 2◦. Solution: y(x)=1 λnn!⎝parenleftbiggd dx⎝parenrightbigg1–α β⎝productdisplay k=1⎝bracketleftBigd2 dx2+( 2k+α)λ2⎝bracketrightBig f(x), where α=n–2 [n/2],β=[ (n+1 )/2], [A] denotes the integer part of number A. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 443). 63.⎝integraldisplay ⎝integraldisplayx a(x–t)s i n [λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Solution: y(x)=1 2λ⎝parenleftbiggd2 dx2+λ2⎝parenrightbigg2⎝integraldisplayx af(t)dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 444), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 64.⎝integraldisplay ⎝integraldisplayx asin⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=2 πλd2 dx2⎝integraldisplayx acosh⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. See also Example 2 in Section 10.4. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 445), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 65.⎝integraldisplay ⎝integraldisplay∞ xsin(λ√ t–x)y(t)dt=f(x). Solution: y(x)=2 πλd2 dx2⎝integraldisplay∞ xcos(λ√ t–x) √ t–xf(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 447). 66.⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] √ x–ty(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=2 πλ⎝integraldisplayx acos[λ (x–t)] √ x–t[f/prime/prime tt(t)+λ2f(t)]dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 445), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 58 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 67.⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] (x–t)3/2y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=2 πλ2⎝integraldisplayx asin[λ(x–t)] √ x–t⎝bracketleftbigg f/prime/prime tt(t)+λ2f(t)+f/prime(t) x–t⎝bracketrightbigg dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 445), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 68.⎝integraldisplay ⎝integraldisplayx asin⎝bracketleftbig⎝bracketleftbig λ√ (x–t)(x –t+γ)⎝bracketrightbig⎝bracketrightbig √ x–t+γy(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Solution: y(x)=2 πλ2⎝integraldisplayx acos⎝bracketleftbig λ√ (x–t)(x–t–γ)⎝bracketrightbig √ x–t⎝integraldisplayt asin[λ(t–s)]⎝parenleftbiggd2 ds2+λ2⎝parenrightbigg2 f(s)dsdt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 447). 69.⎝integraldisplay ⎝integraldisplayx a√ sinx–s i nty(t)dt=f(x). Solution: y(x)=2 πcosx⎝parenleftbigg1 cosxd dx⎝parenrightbigg2⎝integraldisplayx acostf(t)dt √ sinx–s i nt. 70.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ sinx–s i nt=f(x). Solution: y(x)=1 πd dx⎝integraldisplayx acostf(t)dt √ sinx–s i nt. 71.⎝integraldisplay ⎝integraldisplayx a(sinx–s i nt)λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=kcosx⎝parenleftBig1 cosxd dx⎝parenrightBig2⎝integraldisplayx acostf(t)dt (sinx–s i nt)λ,k=sin(πλ) πλ. 72.⎝integraldisplay ⎝integraldisplayx a(sinµx–s i nµt)y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=s i nµx. Solution: y(x)=1 µd dx⎝bracketleftbiggf/prime x(x) cosxsinµ–1x⎝bracketrightbigg . 73.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A|sin(λx)|µ+B|sin(λt)|µ⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=|sin(λx)|µ. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsinglesin(λx)⎝vextendsingle⎝vextendsingle–Aµ A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsinglesin(λt)⎝vextendsingle⎝vextendsingle–Bµ A+Bf/prime t(t)dt⎝bracerightbigg . 1.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 59 74.⎝integraldisplay ⎝integraldisplayx ay(t)dt [sin(λx )–s i n ( λt)]µ=f(x), 0 < µ<1 . This is a special case of equation 1.9.44 with g(x)=s i n ( λx)a n d h(x)≡1. Solution: y(x)=λsin(πµ) πd dx⎝integraldisplayx acos(λt )f(t)dt [sin(λx)–s i n ( λt)]1–µ. 75.⎝integraldisplay ⎝integraldisplayx a(x–t)s i n [λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Double differentiation yields 2λ⎝integraldisplayx acos[λ (x–t)]y(t)dt–λ2⎝integraldisplayx a(x–t)s i n [λ(x–t)]y(t)dt=f/prime/prime xx(x). Eliminating the second integral on the left-hand side of this equation with the aid of the original equation, we arrive at an equation of the form 1.5.1: ⎝integraldisplayx acos[λ (x–t)]y(t)dt=1 2λ⎝bracketleftbig f/prime/prime xx(x)+λ2f(x)⎝bracketrightbig . Solution: y(x)=1 2λf/prime/prime/prime xxx(x)+λf/prime x(x)+1 2λ3⎝integraldisplayx af(t)dt. 76.⎝integraldisplay ⎝integraldisplayx a|sin(λ(x–t))|y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Solution: y(x)=1 λ⎝integraldisplayx a(–1)[λ(x–t)/π]⎝parenleftbig f/prime/prime/prime ttt(t)+λ2f/prime t(t)⎝parenrightbig dt, where [ A] denotes the integer part of number A. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 443). 77.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axβ+Bsinγ(λt)+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Axβandh(t)=Bsinγ(λt)+C. 78.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinγ(λx)+Btβ+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asinγ(λx)a n d h(t)=Btβ+C. 79.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλsinµt+Btβsinγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=s i nµt,g2(x)=Bsinγx, andh2(t)=tβ. 60 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 1.5-3. Kernels Containing Tangent. 80.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tan(λx)–t a n ( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=t a n ( λx). Solution: y(x)=1 λd dx⎝bracketleftbig cos2(λx)f/prime x(x)⎝bracketrightbig . 81.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atan(λx)+Btan(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=t a n ( λx). For B=–A, see equation 1.5.80. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig tan(λx)⎝bracketrightbig–A A+B⎝integraldisplayx a⎝bracketleftbig tan(λt)⎝bracketrightbig–B A+Bf/prime t(t)dt⎝bracerightbigg . 82.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atan(λx)+Btan(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atan(λx)a n d h(t)=Btan(µt)+C. 83.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tan2(λx)–t a n2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=t a n2(λx). Solution: y(x)=d dx⎝bracketleftbiggcos3(λx)f/prime x(x) 2λsin(λx)⎝bracketrightbigg . 84.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atan2(λx)+Btan2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=t a n2(λx). For B=–A, see equation 1.5.83. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingletan(λx)⎝vextendsingle⎝vextendsingle–2A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingletan(λt)⎝vextendsingle⎝vextendsingle–2B A+Bf/prime t(t)dt⎝bracerightbigg . 85.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atan2(λx)+Btan2(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atan2(λx)a n d h(t)=Btan2(µt)+C. 86.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tan(λx)–t a n ( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=1 λnn!c o s2(λx)⎝bracketleftbigg cos2(λx)d dx⎝bracketrightbiggn+1 f(x). 87.⎝integraldisplay ⎝integraldisplayx a√ tanx–t a nty(t)dt=f(x). Solution: y(x)=2 πcos2x⎝parenleftBig cos2xd dx⎝parenrightBig2⎝integraldisplayx af(t)dt cos2t√ tanx–t a nt. 1.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 61 88.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ tanx–t a nt=f(x). Solution: y(x)=1 πd dx⎝integraldisplayx af(t)dt cos2t√ tanx–t a nt. 89.⎝integraldisplay ⎝integraldisplayx a(tanx–t a nt)λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=sin(πλ) πλcos2x⎝parenleftBig cos2xd dx⎝parenrightBig2⎝integraldisplayx af(t)dt cos2t(tanx–t a nt)λ. 90.⎝integraldisplay ⎝integraldisplayx a(tanµx–t a nµt)y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=t a nµx. Solution: y(x)=1 µd dx⎝bracketleftbiggcosµ+1xf/prime x(x) sinµ–1x⎝bracketrightbigg . 91.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Atanµx+Btanµt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=t a nµx.F o rB=–A, see equation 1.5.90. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig tan(λx)⎝bracketrightbig–Aµ A+B⎝integraldisplayx a⎝bracketleftbig tan(λt)⎝bracketrightbig–Bµ A+Bf/prime t(t)dt⎝bracerightbigg . 92.⎝integraldisplay ⎝integraldisplayx ay(t)dt [tan(λx)–t a n ( λt)]µ=f(x), 0 < µ<1 . This is a special case of equation 1.9.44 with g(x)=t a n ( λx)a n d h(x)≡1. Solution: y(x)=λsin(πµ) πd dx⎝integraldisplayx af(t)dt cos2(λt)[tan(λx)–t a n ( λt)]1–µ. 93.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axβ+Btanγ(λt)+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Axβandh(t)=Btanγ(λt)+C. 94.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanγ(λx)+Btβ+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanγ(λx)a n d h(t)=Btβ+C. 95.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλtanµt+Btβtanγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=t a nµt,g2(x)=Btanγx, andh2(t)=tβ. 62 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 1.5-4. Kernels Containing Cotangent. 96.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cot(λx)–c o t ( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=c o t ( λx). Solution: y(x)=–1 λd dx⎝bracketleftbig sin2(λx)f/prime x(x)⎝bracketrightbig . 97.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acot(λx)+Bcot(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o t ( λx). For B=–A, see equation 1.5.96. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig tan(λx)⎝bracketrightbigA A+B⎝integraldisplayx a⎝bracketleftbig tan(λt)⎝bracketrightbigB A+Bf/prime t(t)dt⎝bracerightbigg . 98.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acot(λx)+Bcot(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acot(λx)a n d h(t)=Bcot(µt)+C. 99.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cot2(λx)–c o t2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=c o t2(λx). Solution: y(x)=–d dx⎝bracketleftbiggsin3(λx)f/prime x(x) 2λcos(λx)⎝bracketrightbigg . 100.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acot2(λx)+Bcot2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o t2(λx). For B=–A, see equation 1.5.99. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingletan(λx)⎝vextendsingle⎝vextendsingle2A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingletan(λt)⎝vextendsingle⎝vextendsingle2B A+Bf/prime t(t)dt⎝bracerightbigg . 101.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acot2(λx)+Bcot2(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acot2(λx)a n d h(t)=Bcot2(µt)+C. 102.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cot(λx)–c o t ( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=(–1)n λnn!s i n2(λx)⎝bracketleftbigg sin2(λx)d dx⎝bracketrightbiggn+1 f(x). 103.⎝integraldisplay ⎝integraldisplayx a(cotµx–c o tµt)y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=c o tµx. Solution: y(x)=–1 µd dx⎝bracketleftbiggsinµ+1xf/prime x(x) cosµ–1x⎝bracketrightbigg . 1.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 63 104.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Acotµx+Bcotµt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=c o tµx.F o rB=–A, see equation 1.5.103. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingletanx⎝vextendsingle⎝vextendsingleAµ A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingletant⎝vextendsingle⎝vextendsingleBµ A+Bf/prime t(t)dt⎝bracerightbigg . 105.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Axβ+Bcotγ(λt)+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Axβandh(t)=Bcotγ(λt)+C. 106.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acotγ(λx)+Btβ+C]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acotγ(λx)a n d h(t)=Btβ+C. 107.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλcotµt+Btβcotγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=c o tµt,g2(x)=Bcotγx, andh2(t)=tβ. 1.5-5. Kernels Containing Combinations of Trigonometric Functions. 108.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig cos[λ (x–t)] +Asin[µ(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). Differentiating the equation with respect to xfollowed by eliminating the integral with the cosine yields an equation of the form 2.3.16: y(x)–(λ+A2µ)⎝integraldisplayx asin[µ(x–t)]y(t)dt=f/prime x(x)–Aµf (x). 109.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos(λx )+Bsin(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acos(λx)a n d h(t)=Bsin(µt)+C. 110.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asin(λx)+Bcos(µt )+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asin(λx)a n d h(t)=Bcos(µt)+ C. 111.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos2(λx)+Bsin2(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acos2(λx)a n d h(t)=Bsin2(µt). 64 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 112.⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] cos[λ (x+t)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . Using the trigonometric formula sin(α–β)c o s (α+β)=1 2⎝bracketleftbig sin(2α)–s i n ( 2 β)⎝bracketrightbig ,α=λx, β=λt, we reduce the original equation to an equation of the form 1.5.44: ⎝integraldisplayx a⎝bracketleftbig sin(2λx)–s i n ( 2 λt)⎝bracketrightbig y(t)dt=2f(x). Solution: y(x)=1 λd dx⎝bracketleftbiggf/prime x(x) cos(2λx)⎝bracketrightbigg . 113.⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)] sin[λ(x+t)]y(t)dt=f(x). Using the trigonometric formula cos(α –β)s i n (α+β)=1 2⎝bracketleftbig sin(2α)+s i n ( 2 β)⎝bracketrightbig ,α=λx, β=λt, we reduce the original equation to an equation of the form 1.5.45 with A=B=1 : ⎝integraldisplayx a⎝bracketleftbig sin(2λx)+s i n ( 2 λt)⎝bracketrightbig y(t)dt=2f(x). Solution with sin(2 λx)>0 : y(x)=d dx⎝bracketleftbigg1 √ sin(2λx)⎝integraldisplayx af/prime t(t)dt √ sin(2λt)⎝bracketrightbigg . 114.⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] cos[µ (x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Solution with µ<λ: y(x)=1 λ⎝radicalbig λ2–µ2⎝bracketleftbiggd2 dx2+(λ+µ)2⎝bracketrightbigg⎝bracketleftbiggd2 dx2+(λ–µ)2⎝bracketrightbigg⎝integraldisplayx asin⎝bracketleftbig⎝radicalbig λ2–µ2(x–t)⎝bracketrightbig f(t)dt. Solution with µ>λ: y(x)=1 λ⎝radicalbig λ2–µ2⎝bracketleftbiggd2 dx2+(λ+µ)2⎝bracketrightbigg⎝bracketleftbiggd2 dx2+(λ–µ)2⎝bracketrightbigg⎝integraldisplayx asinh⎝bracketleftbig⎝radicalbig µ2–λ2(x–t)⎝bracketrightbig f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 444). 115.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos(λx )s i n (µt)+Bcos(βx )s i n (γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Acos(λx), h1(t)=s i n ( µt),g2(x)= Bcos(βx ), and h2(t)=s i n ( γt). 1.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 65 116.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asin(λx)c o s (µt)+Bsin(βx)c o s (γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Asin(λx),h1(t)=c o s ( µt),g2(x)= Bsin(βx), and h2(t)=c o s ( γt). 117.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos(λx )c o s (µt)+Bsin(βx)s i n (γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Acos(λx), h1(t)=c o s ( µt),g2(x)= Bsin(βx), and h2(t)=s i n ( γt). 118.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosβ(λx)+Bsinγ(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acosβ(λx)a n d h(t)=Bsinγ(µt). 119.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinβ(λx)+Bcosγ(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asinβ(λx)a n d h(t)=Bcosγ(µt). 120.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλcosµt+Btβsinγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=c o sµt,g2(x)=Bsinγx, andh2(t)=tβ. 121.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλsinµt+Btβcosγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=s i nµt,g2(x)=Bcosγx, andh2(t)=tβ. 122.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig (x–t)s i n [λ(x–t)] –λ(x–t)2cos[λ (x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). Solution: y(x)=⎝integraldisplayx ag(t)dt, where g(t)=⎝radicalbigg π 2λ1 64λ5⎝parenleftbiggd2 dt2+λ2⎝parenrightbigg6⎝integraldisplayt a(t–τ)5/2J5/2[λ(t–τ)]f(τ)dτ. 123.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbigg ⎝braceleftbiggsin[λ(x–t)] x–t–λcos[λ (x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x). Solution: y(x)=1 2λ4⎝parenleftbiggd2 dx2+λ2⎝parenrightbigg3⎝integraldisplayx asin[λ(x–t)]f(t)dt. 124.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig sin⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig –λ√ x–tcos⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x),f(a)=f/prime x(a)=0 . Solution: y(x)=4 πλ3d3 dx3⎝integraldisplayx acosh⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. 66 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 125.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atan(λx)+Bcot(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atan(λx)a n d h(t)=Bcot(µt)+C. 126.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atan2(λx)+Bcot2(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atan2(λx)a n d h(t)=Bcot2(µt). 127.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tan(λx)c o t (µt)+t a n ( βx)c o t (γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=t a n ( λx),h1(t)=c o t ( µt),g2(x)=t a n ( βx), andh2(t)=c o t ( γt). 128.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig cot(λx)t a n (µt)+c o t ( βx)t a n (γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=c o t ( λx),h1(t)=t a n ( µt),g2(x)=c o t ( βx), andh2(t)=t a n ( γt). 129.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tan(λx)t a n (µt)+c o t ( βx)c o t (γt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=t a n ( λx),h1(t)=t a n ( µt),g2(x)=c o t ( βx), andh2(t)=c o t ( γt). 130.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanβ(λx)+Bcotγ(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanβ(λx)a n d h(t)=Bcotγ(µt). 131.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acotβ(λx)+Btanγ(µt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acotβ(λx)a n d h(t)=Btanγ(µt). 132.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλtanµt+Btβcotγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=t a nµt,g2(x)=Bcotγx, andh2(t)=tβ. 133.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλcotµt+Btβtanγx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=c o tµt,g2(x)=Btanγx, andh2(t)=tβ. 1.6. Equations Whose Kernels Contain Inverse Trigonometric Functions 1.6-1. Kernels Containing Arccosine. 1.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arccos( λx) – arccos( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x) = arccos( λx). Solution: y(x)=–1 λd dx⎝bracketleftBig√ 1–λ2x2f/prime x(x)⎝bracketrightBig . 1.6. E QUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS 67 2.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarccos( λx)+Barccos( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x) = arccos(λx). For B=–A, see equation 1.6.1. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig arccos( λx)⎝bracketrightbig–A A+B⎝integraldisplayx a⎝bracketleftbig arccos( λt)⎝bracketrightbig–B A+Bf/prime t(t)dt⎝bracerightbigg . 3.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarccos( λx)+Barccos( µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aarccos( λx)a n d h(t)=Barccos( µt)+C. 4.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arccos( λx) – arccos( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=(–1)n λnn!√ 1–λ2x2⎝parenleftbigg√ 1–λ2x2d dx⎝parenrightbiggn+1 f(x). 5.⎝integraldisplay ⎝integraldisplayx a⎝radicalbig arccos( λt) – arccos(λx )y(t)dt=f(x). This is a special case of equation 1.9.40 with g(x)=1–a r c c o s ( λx). Solution: y(x)=2 πϕ(x)⎝parenleftbigg1 ϕ(x)d dx⎝parenrightbigg2⎝integraldisplayx aϕ(t)f(t)dt √ arccos( λt) – arccos( λx),ϕ(x)=1 √ 1–λ2x2. 6.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ arccos( λt) – arccos(λx )=f(x). Solution: y(x)=λ πd dx⎝integraldisplayx aϕ(t)f(t)dt √ arccos( λt) – arccos( λx),ϕ(x)=1 √ 1–λ2x2. 7.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arccos( λt) – arccos(λx )⎝bracketrightbig⎝bracketrightbigµy(t)dt=f(x), 0 < µ<1 . Solution: y(x)=kϕ(x)⎝parenleftbigg1 ϕ(x)d dx⎝parenrightbigg2⎝integraldisplayx aϕ(t)f(t)dt [arccos( λt) – arccos( λx)]µ, ϕ(x)=1 √ 1–λ2x2,k=sin(πµ) πµ. 8.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arccosµ(λx) – arccosµ(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x) = arccosµ(λx). Solution: y(x)=–1 λµd dx⎝bracketleftbiggf/prime x(x)√ 1–λ2x2 arccosµ–1(λx)⎝bracketrightbigg . 68 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 9.⎝integraldisplay ⎝integraldisplayx ay(t)dt ⎝bracketleftbig⎝bracketleftbig arccos( λt) – arccos( λx)⎝bracketrightbig⎝bracketrightbigµ=f(x), 0 < µ<1 . Solution: y(x)=λsin(πµ) πd dx⎝integraldisplayx aϕ(t)f(t)dt [arccos( λt) – arccos( λx)]1–µ,ϕ(x)=1 √ 1–λ2x2. 10.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarccosβ(λx)+Barccosγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aarccosβ(λx)a n d h(t)=Barccosγ(µt)+C. 1.6-2. Kernels Containing Arcsine. 11.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arcsin( λx)–a r c s i n ( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x) = arcsin( λx). Solution: y(x)=1 λd dx⎝bracketleftBig√ 1–λ2x2f/prime x(x)⎝bracketrightBig . 12.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarcsin( λx)+Barcsin( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=a r c s i n ( λx). For B=–A, see equation 1.6.11. Solution: y(x)=signx A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsinglearcsin( λx)⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsinglearcsin( λt)⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . 13.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarcsin( λx)+Barcsin( µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aarcsin( λx)a n d h(t)=Barcsin( µt)+C. 14.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arcsin( λx)–a r c s i n ( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=1 λnn!√ 1–λ2x2⎝parenleftbigg√ 1–λ2x2d dx⎝parenrightbiggn+1 f(x). 15.⎝integraldisplay ⎝integraldisplayx a⎝radicalbig arcsin( λx)–a r c s i n ( λt)y(t)dt=f(x). Solution: y(x)=2 πϕ(x)⎝parenleftbigg1 ϕ(x)d dx⎝parenrightbigg2⎝integraldisplayx aϕ(t)f(t)dt √ arcsin( λx) – arcsin( λt),ϕ(x)=1 √ 1–λ2x2. 1.6. E QUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS 69 16.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ arcsin( λx)–a r c s i n ( λt)=f(x). Solution: y(x)=λ πd dx⎝integraldisplayx aϕ(t)f(t)dt √ arcsin( λx) – arcsin( λt),ϕ(x)=1 √ 1–λ2x2. 17.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arcsin( λx)–a r c s i n ( λt)⎝bracketrightbig⎝bracketrightbigµy(t)dt=f(x), 0 < µ<1 . Solution: y(x)=kϕ(x)⎝parenleftbigg1 ϕ(x)d dx⎝parenrightbigg2⎝integraldisplayx aϕ(t)f(t)dt [arcsin( λx) – arcsin( λt)]µ, ϕ(x)=1 √ 1–λ2x2,k=sin(πµ) πµ. 18.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arcsinµ(λx)–a r c s i nµ(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x) = arcsinµ(λx). Solution: y(x)=1 λµd dx⎝bracketleftbiggf/prime x(x)√ 1–λ2x2 arcsinµ–1(λx)⎝bracketrightbigg . 19.⎝integraldisplay ⎝integraldisplayx ay(t)dt ⎝bracketleftbig⎝bracketleftbig arcsin( λx)–a r c s i n ( λt)⎝bracketrightbig⎝bracketrightbigµ=f(x), 0 < µ<1 . Solution: y(x)=λsin(πµ) πd dx⎝integraldisplayx aϕ(t)f(t)dt [arcsin( λx) – arcsin( λt)]1–µ,ϕ(x)=1 √ 1–λ2x2. 20.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarcsinβ(λx)+Barcsinγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aarcsinβ(λx)a n d h(t)=Barcsinγ(µt)+C. 21.⎝integraldisplay ⎝integraldisplayx 0arcsin⎝radicalbigg 1–t xy(t)dt=f(x). Solution: y(x)=2 π1 √ xd dx⎝integraldisplay ⎝integraldisplayx 0t √ x–td dtf(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 452). 22.⎝integraldisplay ⎝integraldisplay∞ xarcsin⎝radicalbigg 1–x ty(t)dt=f(x). Solution: y(x)=2 πd dx⎝integraldisplay ⎝integraldisplay∞ x√ t √ t–xd dtf(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 453). 70 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 1.6-3. Kernels Containing Arctangent. 23.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arctan( λx) – arctan( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x) = arctan( λx). Solution: y(x)=1 λd dx⎝bracketleftbig (1 +λ2x2)f/prime x(x)⎝bracketrightbig . 24.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarctan( λx)+Barctan( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x)=a r c t a n ( λx). For B=–A, see equation 1.6.21. Solution: y(x)=signx A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsinglearctan( λx)⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsinglearctan( λt)⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . 25.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarctan( λx)+Barctan( µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aarctan( λx)a n d h(t)=Barctan( µt)+C. 26.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arctan( λx) – arctan( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=1 λnn!( 1+λ2x2)⎝parenleftbigg (1 +λ2x2)d dx⎝parenrightbiggn+1 f(x). 27.⎝integraldisplay ⎝integraldisplayx a⎝radicalbig arctan( λx) – arctan( λt)y(t)dt=f(x). Solution: y(x)=2 πϕ(x)⎝parenleftbigg1 ϕ(x)d dx⎝parenrightbigg2⎝integraldisplayx aϕ(t)f(t)dt √ arctan( λx) – arctan( λt),ϕ(x)=1 1+λ2x2. 28.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ arctan( λx)–a r c t a n ( λt)=f(x). Solution: y(x)=λ πd dx⎝integraldisplayx aϕ(t)f(t)dt √ arctan( λx) – arctan( λt),ϕ(x)=1 1+λ2x2. 29.⎝integraldisplay ⎝integraldisplayx a√ tarctan⎝parenleftbigg ⎝parenleftbigg⎝radicalbigg x–t t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). The equation can be rewritten in terms of the Gaussian hypergeometric function in the form ⎝integraldisplayx a(x–t)γ–1F⎝parenleftBig α,β,γ;1–x t⎝parenrightBig y(t)dt=f(x), where α=1 2,β=1 , γ=3 2. See 1.8.135 for the solution of this equation. 1.6. E QUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS 71 30.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arctan( λx) – arctan( λt)⎝bracketrightbig⎝bracketrightbigµy(t)dt=f(x), 0 < µ<1 . Solution: y(x)=kϕ(x)⎝parenleftbigg1 ϕ(x)d dx⎝parenrightbigg2⎝integraldisplayx aϕ(t)f(t)dt [arctan( λx) – arctan( λt)]µ, ϕ(x)=1 1+λ2x2,k=sin(πµ) πµ. 31.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arctanµ(λx) – arctanµ(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x) = arctanµ(λx). Solution: y(x)=1 λµd dx⎝bracketleftbigg(1 +λ2x2)f/prime x(x) arctanµ–1(λx)⎝bracketrightbigg . 32.⎝integraldisplay ⎝integraldisplayx ay(t)dt ⎝bracketleftbig⎝bracketleftbig arctan( λx) – arctan( λt)⎝bracketrightbig⎝bracketrightbigµ=f(x), 0 < µ<1 . Solution: y(x)=λsin(πµ) πd dx⎝integraldisplayx aϕ(t)f(t)dt [arctan( λx) – arctan( λt)]1–µ,ϕ(x)=1 1+λ2x2. 33.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarctanβ(λx)+Barctanγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aarctanβ(λx)a n d h(t)=Barctanγ(µt)+C. 1.6-4. Kernels Containing Arccotangent. 34.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arccot( λx) – arccot( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x) = arccot( λx). Solution: y(x)=–1 λd dx⎝bracketleftbig (1 +λ2x2)f/prime x(x)⎝bracketrightbig . 35.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarccot( λx)+Barccot( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.4 with g(x) = arccot( λx). For B=–A, see equation 1.6.34. Solution: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig arccot(λx)⎝bracketrightbig–A A+B⎝integraldisplayx a⎝bracketleftbig arccot(λt )⎝bracketrightbig–B A+Bf/prime t(t)dt⎝bracerightbigg . 36.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarccot( λx)+Barccot( µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aarccot(λx)a n d h(t)=Barccot(µt)+ C. 72 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 37.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arccot( λx) – arccot( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=(–1)n λnn!( 1+λ2x2)⎝parenleftbigg (1 +λ2x2)d dx⎝parenrightbiggn+1 f(x). 38.⎝integraldisplay ⎝integraldisplayx a⎝radicalbig arccot( λt) – arccot( λx)y(t)dt=f(x). Solution: y(x)=2 πϕ(x)⎝parenleftbigg1 ϕ(x)d dx⎝parenrightbigg2⎝integraldisplayx aϕ(t)f(t)dt √ arccot(λt ) – arccot(λx),ϕ(x)=1 1+λ2x2. 39.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ arccot( λt) – arccot( λx)=f(x). Solution: y(x)=λ πd dx⎝integraldisplayx aϕ(t)f(t)dt √ arccot(λt ) – arccot(λx),ϕ(x)=1 1+λ2x2. 40.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arccot( λt) – arccot( λx)⎝bracketrightbig⎝bracketrightbigµy(t)dt=f(x), 0 < µ<1 . Solution: y(x)=kϕ(x)⎝parenleftbigg1 ϕ(x)d dx⎝parenrightbigg2⎝integraldisplayx aϕ(t)f(t)dt [arccot(λt ) – arccot(λx)]µ, ϕ(x)=1 1+λ2x2,k=sin(πµ) πµ. 41.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig arccotµ(λx) – arccotµ(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x) = arccotµ(λx). Solution: y(x)=–1 λµd dx⎝bracketleftbigg(1 +λ2x2)f/prime x(x) arccotµ–1(λx)⎝bracketrightbigg . 42.⎝integraldisplay ⎝integraldisplayx ay(t)dt ⎝bracketleftbig⎝bracketleftbig arccot( λt) – arccot( λx)⎝bracketrightbig⎝bracketrightbigµ=f(x), 0 < µ<1 . Solution: y(x)=λsin(πµ) πd dx⎝integraldisplayx aϕ(t)f(t)dt [arccot(λt ) – arccot(λx)]1–µ,ϕ(x)=1 1+λ2x2. 43.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarccotβ(λx)+Barccotγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Aarccotβ(λx)a n d h(t)=Barccotγ(µt)+C. 1.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 73 1.7. Equations Whose Kernels Contain Combinations of Elementary Functions 1.7-1. Kernels Containing Exponential and Hyperbolic Functions. 1.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝braceleftbig ⎝braceleftbig A1cosh[λ 1(x–t)] +A2cosh[λ 2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.8: ⎝integraldisplayx a⎝braceleftbig A1cosh[λ 1(x–t)] +A2cosh[λ 2(x–t)]⎝bracerightbig w(t)dt=e–µxf(x). 2.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)cosh2[λ(x–t)]y(t)dt=f(x). Solution: y(x)=ϕ(x)–2λ2 k⎝integraldisplayx aeµ(x–t)sinh[k(x–t)]ϕ(x)dt,k=λ√ 2,ϕ(x)=f/prime x(x)–µf(x). 3.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)cosh3[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.15: ⎝integraldisplayx acosh3[λ(x–t)]w(t)dt=e–µxf(x). 4.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)cosh4[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.19: ⎝integraldisplayx acosh4[λ(x–t)]w(t)dt=e–µxf(x). 5.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝bracketleftbig⎝bracketleftbig cosh(λx )–c o s h ( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... Solution: y(x)=1 λnn!eµxsinh(λx)⎝bracketleftbigg1 sinh(λx)d dx⎝bracketrightbiggn+1 Fµ(x), Fµ(x)=e–µxf(x). 6.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)√ coshx–c o s h ty(t)dt=f(x), f(a)=0 . Solution: y(x)=2 πeµxsinhx⎝parenleftBig1 sinhxd dx⎝parenrightBig2⎝integraldisplayx ae–µtsinhtf(t)dt √ coshx–c o s h t. 7.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)y(t)dt √ coshx–c o s h t=f(x). Solution: y(x)=1 πeµxd dx⎝integraldisplayx ae–µtsinhtf(t)dt √ coshx–c o s h t. 74 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 8.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)(coshx–c o s h t)λy(t)dt=f(x), 0 < λ<1 . The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.23: ⎝integraldisplayx a(coshx–c o s h t)λw(t)dt=e–µxf(x). 9.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bcoshλx⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=Bcoshλx, andh2(t)=1 . 10.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bcoshλt⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=B,a n d h2(t)=c o s hλt. 11.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)(coshλx–c o s hλt)y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.24: ⎝integraldisplayx a(coshλx–c o s hλt)w(t)dt=e–µxf(x). 12.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Acoshλx+Bcoshλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.25: ⎝integraldisplayx a⎝parenleftbig Acoshλx+Bcoshλt⎝parenrightbig w(t)dt=e–µxf(x). 13.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)y(t)dt (coshx–c o s h t)λ=f(x), 0 < λ<1 . Solution: y(x)=sin(πλ) πeµxd dx⎝integraldisplayx ae–µtsinhtf(t)dt (coshx–c o s h t)1–λ. 14.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝braceleftbig ⎝braceleftbig A1sinh[λ1(x–t)] +A2sinh[λ2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.49: ⎝integraldisplayx a⎝braceleftbig A1sinh[λ1(x–t)] +A2sinh[λ2(x–t)]⎝bracerightbig w(t)dt=e–µxf(x). 15.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sinh2[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.51: ⎝integraldisplayx asinh2[λ(x–t)]w(t)dt=e–µxf(x). 1.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 75 16.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sinh3[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.57: ⎝integraldisplayx asinh3[λ(x–t)]w(t)dt=e–µxf(x). 17.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sinhn[λ(x–t)]y(t)dt=f(x), n=2 , 3 , ... The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.62: ⎝integraldisplayx asinhn[λ(x–t)]w(t)dt=e–µxf(x). 18.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sinh⎝parenleftbig⎝parenleftbig k√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=2 πkeµxd2 dx2⎝integraldisplayx ae–µtcos⎝parenleftbig k√ x–t⎝parenrightbig √ x–tf(t)dt. 19.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)√ sinhx–s i n h ty(t)dt=f(x). Solution: y(x)=2 πeµxcoshx⎝parenleftBig1 coshxd dx⎝parenrightBig2⎝integraldisplayx ae–µtcoshtf(t)dt √ sinhx–s i n h t. 20.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)y(t)dt √ sinhx–s i n h t=f(x). Solution: y(x)=1 πeµxd dx⎝integraldisplayx ae–µtcoshtf(t)dt √ sinhx–s i n h t. 21.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)(sinhx–s i n h t)λy(t)dt=f(x), 0 < λ<1 . The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.67: ⎝integraldisplayx a(sinhx–s i n h t)λw(t)dt=e–µxf(x). 22.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)(sinhλx–s i n hλt)y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.68: ⎝integraldisplayx a(sinhλx–s i n hλt)w(t)dt=e–µxf(x). 23.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Asinhλx+Bsinhλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.69: ⎝integraldisplayx a⎝parenleftbig Asinhλx+Bsinhλt⎝parenrightbig w(t)dt=e–µxf(x). 76 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 24.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bsinhλx⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=Bsinhλx, andh2(t)=1 . 25.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bsinhλt⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=B,a n d h2(t)=s i n hλt. 26.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)y(t)dt (sinhx–s i n h t)λ=f(x), 0 < λ<1 . Solution: y(x)=sin(πλ) πeµxd dx⎝integraldisplayx ae–µtcoshtf(t)dt (sinhx–s i n h t)1–λ. 27.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Atanhλx+Btanhλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.89: ⎝integraldisplayx a⎝parenleftbig Atanhλx+Btanhλt⎝parenrightbig w(t)dt=e–µxf(x). 28.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Atanhλx+Btanhβt+C⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.9.6 with g(x)=Atanhλx, h(t)=Btanhβt+C: ⎝integraldisplayx a⎝parenleftbig Atanhλx+Btanhβt+C⎝parenrightbig w(t)dt=e–µxf(x). 29.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Btanhλx⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=Btanhλx, andh2(t)=1 . 30.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Btanhλt⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=B,a n d h2(t)=t a n hλt. 31.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Acothλx+Bcothλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.3.102: ⎝integraldisplayx a⎝parenleftbig Acothλx+Bcothλt⎝parenrightbig w(t)dt=e–µxf(x). 1.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 77 32.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Acothλx+Bcothβt+C⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.9.6 with g(x)=Acothλx, h(t)=Bcothβt+C: ⎝integraldisplayx a⎝parenleftbig Acothλx+Bcothβt+C⎝parenrightbig w(t)dt=e–µxf(x). 33.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bcothλx⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=Bcothλx, andh2(t)=1 . 34.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bcothλt⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=B,a n d h2(t)=c o t hλt. 1.7-2. Kernels Containing Exponential and Logarithmic Functions. 35.⎝integraldisplay ⎝integraldisplayx aeλ(x–t)(lnx–l nt)y(t)dt=f(x). Solution: y(x)=eλx⎝bracketleftbig xϕ/prime/prime xx(x)+ϕ/prime x(x)⎝bracketrightbig ,ϕ(x)=e–λxf(x). 36.⎝integraldisplay ⎝integraldisplayx 0eλ(x–t)ln(x–t)y(t)dt=f(x). The substitution w(x)=e–λxy(x) leads to an equation of the form 1.4.2: ⎝integraldisplayx 0ln(x–t)w(t)dt=e–λxf(x). 37.⎝integraldisplay ⎝integraldisplayx aeλ(x–t)(Alnx+Blnt)y(t)dt=f(x). The substitution w(x)=e–λxy(x) leads to an equation of the form 1.4.4: ⎝integraldisplayx a(Alnx+Blnt)w(t)dt=e–λxf(x). 38.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝bracketleftbig⎝bracketleftbig Aln2(λx)+Bln2(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). The substitution w(x)=e–λxy(x) leads to an equation of the form 1.4.7: ⎝integraldisplayx a⎝bracketleftbig Aln2(λx)+Bln2(λt)⎝bracketrightbig w(t)dt=e–λxf(x). 78 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 39.⎝integraldisplay ⎝integraldisplayx aeλ(x–t)⎝bracketleftbig⎝bracketleftbig ln(x/t )⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... Solution: y(x)=1 n!xeλx⎝parenleftbigg xd dx⎝parenrightbiggn+1 Fλ(x), Fλ(x)=e–λxf(x). 40.⎝integraldisplay ⎝integraldisplayx aeλ(x–t)⎝radicalbig ln(x/t )y(t)dt=f(x). Solution: y(x)=2eλx πx⎝parenleftbigg xd dx⎝parenrightbigg2⎝integraldisplayx ae–λtf(t)dt t⎝radicalbig ln(x/t). 41.⎝integraldisplay ⎝integraldisplayx aeλ(x–t) ⎝radicalbig ln(x/t )y(t)dt=f(x). Solution: y(x)=1 πeλxd dx⎝integraldisplayx ae–λtf(t)dt t⎝radicalbig ln(x/t). 42.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Blnν(λx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=Blnν(λx), andh2(t)=1 . 43.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Blnν(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=B,a n d h2(t)=l nν(λt). 44.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)[ln(x/t )]λy(t)dt=f(x), 0 < λ<1 . The substitution w(x)=e–µxy(x) leads to an equation of the form 1.4.16: ⎝integraldisplayx a[ln(x/t)]λw(t)dt=e–µxf(x). 45.⎝integraldisplay ⎝integraldisplayx aeµ(x–t) [ln(x/t )]λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=sin(πλ) πeµxd dx⎝integraldisplayx af(t)dt teµt[ln(x/t)]1–λ. 1.7-3. Kernels Containing Exponential and Trigonometric Functions. 46.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)cos[λ (x–t)]y(t)dt=f(x). Solution: y(x)=f/prime x(x)–µf(x)+λ2⎝integraldisplayx aeµ(x–t)f(t)dt. 1.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 79 47.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝braceleftbig ⎝braceleftbig A1cos[λ 1(x–t)] +A2cos[λ 2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.8: ⎝integraldisplayx a⎝braceleftbig A1cos[λ 1(x–t)] +A2cos[λ 2(x–t)]⎝bracerightbig w(t)dt=e–µxf(x). 48.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)cos2[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.9. Solution: y(x)=ϕ(x)+2λ2 k⎝integraldisplayx aeµ(x–t)sin[k(x–t)]ϕ(t)dt,k=λ√ 2,ϕ(x)=f/prime x(x)–µf(x). 49.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)cos3[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.16: ⎝integraldisplayx acos3[λ(x–t)]w(t)dt=e–µxf(x). 50.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)cos4[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.20: ⎝integraldisplayx acos4[λ(x–t)]w(t)dt=e–µxf(x). 51.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝bracketleftbig⎝bracketleftbig cos(λx )–c o s ( λt)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=(–1)n λnn!eµxsin(λx)⎝bracketleftbigg1 sin(λx)d dx⎝bracketrightbiggn+1 Fµ(x), Fµ(x)=e–µxf(x). 52.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)√ cost–c o sxy(t)dt=f(x). Solution: y(x)=2 πeµxsinx⎝parenleftBig1 sinxd dx⎝parenrightBig2⎝integraldisplayx ae–µtsintf(t)dt √ cost–c o sx. 53.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)y(t)dt √ cost–c o sx=f(x). Solution: y(x)=1 πeµxd dx⎝integraldisplayx ae–µtsintf(t)dt √ cost–c o sx. 80 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 54.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)(cost–c o sx)λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=keµxsinx⎝parenleftBig1 sinxd dx⎝parenrightBig2⎝integraldisplayx ae–µtsintf(t)dt (cost–c o sx)λ,k=sin(πλ) πλ. 55.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)(cosλx–c o sλt)y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.25: ⎝integraldisplayx a(cosλx–c o sλt)w(t)dt=e–µxf(x). 56.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Acosλx+Bcosλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.26: ⎝integraldisplayx a⎝parenleftbig Acosλx+Bcosλt⎝parenrightbig w(t)dt=e–µxf(x). 57.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)y(t)dt (cost–c o sx)λ=f(x), 0 < λ<1 . The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.27: ⎝integraldisplayx aw(t)dt (cost–c o sx)λ=e–µxf(x). 58.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bcosν(λx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=Bcosν(λx), andh2(t)=1 . 59.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bcosν(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=B,a n d h2(t)=c o sν(λt). 60.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sin[λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=1 λ⎝bracketleftbig f/prime/prime xx(x)–2µf/prime x(x)+(λ2+µ2)f(x)⎝bracketrightbig . 61.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝braceleftbig ⎝braceleftbig A1sin[λ1(x–t)] +A2sin[λ2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.48: ⎝integraldisplayx a⎝braceleftbig A1sin[λ1(x–t)] +A2sin[λ2(x–t)]⎝bracerightbig w(t)dt=e–µxf(x). 1.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 81 62.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sin2[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.50: ⎝integraldisplayx asin2[λ(x–t)]w(t)dt=e–µxf(x). 63.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sin3[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.57: ⎝integraldisplayx asin3[λ(x–t)]w(t)dt=e–µxf(x). 64.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sinn[λ(x–t)]y(t)dt=f(x), n=2 , 3 , ... The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.62: ⎝integraldisplayx asinn[λ(x–t)]w(t)dt=e–µxf(x). 65.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sin⎝parenleftbig⎝parenleftbig k√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=2 πkeµxd2 dx2⎝integraldisplayx ae–µtcosh⎝parenleftbig k√ x–t⎝parenrightbig √ x–tf(t)dt. 66.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)√ sinx–s i nty(t)dt=f(x). Solution: y(x)=2 πeµxcosx⎝parenleftBig1 cosxd dx⎝parenrightBig2⎝integraldisplayx ae–µtcostf(t)dt √ sinx–s i nt. 67.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)y(t)dt √ sinx–s i nt=f(x). Solution: y(x)=1 πeµxd dx⎝integraldisplayx ae–µtcostf(t)dt √ sinx–s i nt. 68.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)(sinx–s i nt)λy(t)dt=f(x), 0 < λ<1 . Solution: y(x)=keµxcosx⎝parenleftBig1 cosxd dx⎝parenrightBig2⎝integraldisplayx ae–µtcostf(t)dt (sinx–s i nt)λ,k=sin(πλ) πλ. 69.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)(sinλx–s i nλt)y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.72: ⎝integraldisplayx a(sinλx–s i nλt)w(t)dt=e–µxf(x). 82 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 70.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Asinλx+Bsinλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.9.4 with g(x)=s i nλx: ⎝integraldisplayx a⎝parenleftbig Asinλx+Bsinλt⎝parenrightbig w(t)dt=e–µxf(x). 71.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)y(t)dt (sinx–s i nt)λ=f(x), 0 < λ<1 . The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.74: ⎝integraldisplayx aw(t)dt (sinx–s i nt)λ=e–µxf(x). 72.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bsinν(λx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=Bsinν(λx), andh2(t)=1 . 73.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bsinν(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=B,a n d h2(t)=s i nν(λt). 74.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Atanλx+Btanλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.91: ⎝integraldisplayx a⎝parenleftbig Atanλx+Btanλt⎝parenrightbig w(t)dt=e–µxf(x). 75.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Atanλx+Btanβt+C⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.9.6: ⎝integraldisplayx a⎝parenleftbig Atanλx+Btanβt+C⎝parenrightbig w(t)dt=e–µxf(x). 76.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Btanν(λx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=Btanν(λx), andh2(t)=1 . 77.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Btanν(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=B,a n d h2(t)=t a nν(λt). 1.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 83 78.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Acotλx+Bcotλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.5.104: ⎝integraldisplayx a⎝parenleftbig Acotλx+Bcotλt⎝parenrightbig w(t)dt=e–µxf(x). 79.⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝parenleftbig⎝parenleftbig Acotλx+Bcotβt+C⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 1.9.6: ⎝integraldisplayx a⎝parenleftbig Acotλx+Bcotβt+C⎝parenrightbig w(t)dt=e–µxf(x). 80.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bcotν(λx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=Bcotν(λx), andh2(t)=1 . 81.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeµ(x–t)+Bcotν(λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Aeµx,h1(t)=e–µt,g2(x)=B,a n d h2(t)=c o tν(λt). 1.7-4. Kernels Containing Hyperbolic and Logarithmic Functions. 82.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoshβ(λx)+Blnγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acoshβ(λx)a n d h(t)=Blnγ(µt)+C. 83.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoshβ(λt)+Blnγ(µx)+ C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Blnγ(µx)+Candh(t)=Acoshβ(λt). 84.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinhβ(λx)+Blnγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asinhβ(λx)a n d h(t)=Blnγ(µt)+C. 85.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinhβ(λt)+Blnγ(µx)+ C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Blnγ(µx)a n dh(t)=Asinhβ(λt)+C. 86.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanhβ(λx)+Blnγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanhβ(λx)a n d h(t)=Blnγ(µt)+C. 84 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 87.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanhβ(λt)+Blnγ(µx)+ C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Blnγ(µx)a n dh(t)=Atanhβ(λt)+C. 88.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acothβ(λx)+Blnγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acothβ(λx)a n d h(t)=Blnγ(µt)+C. 89.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acothβ(λt)+Blnγ(µx)+ C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Blnγ(µx)a n dh(t)=Acothβ(λt)+C. 1.7-5. Kernels Containing Hyperbolic and Trigonometric Functions. 90.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoshβ(λx)+Bcosγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acoshβ(λx)a n d h(t)=Bcosγ(µt)+C. 91.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoshβ(λt)+Bsinγ(µx)+ C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Bsinγ(µx)+Candh(t)=Acoshβ(λt). 92.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoshβ(λx)+Btanγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acoshβ(λx)a n d h(t)=Btanγ(µt)+C. 93.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinhβ(λx)+Bcosγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asinhβ(λx)a n d h(t)=Bcosγ(µt)+C. 94.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinhβ(λt)+Bsinγ(µx)+ C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Bsinγ(µx)a n dh(t)=Asinhβ(λt)+C. 95.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinhβ(λx)+Btanγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asinhβ(λx)a n d h(t)=Btanγ(µt)+C. 96.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanhβ(λx)+Bcosγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanhβ(λx)a n d h(t)=Bcosγ(µt)+C. 97.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanhβ(λx)+Bsinγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Atanhβ(λx)a n d h(t)=Bsinγ(µt)+C. 1.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 85 98.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] – sin[λ (x–t)]y(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Solution: y(x)=1 2λ3⎝parenleftbiggd4 dx4–λ4⎝parenrightbigg f(x). Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 449). 99.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] sin[λ(x–t)]y(t)dt=f(x), f(a)=f/prime x(a)=f/prime/prime xx(a)=0 . Solution: y(x)=1 2λ2⎝parenleftbiggd4 dx4+4λ4⎝parenrightbigg⎝integraldisplayx af(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 449). 100.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cos[λ (x–t)]y(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Solution: y(x)=1 √ 2λ2⎝parenleftbiggd4 dx4+4λ4⎝parenrightbigg⎝integraldisplayx asinh[√ 2λ(x–t)]f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 449). 101.⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)] sin[λ(x–t)]y(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 . Solution: y(x)=1 √ 2λ2⎝parenleftbiggd4 dx4+4λ4⎝parenrightbigg⎝integraldisplayx asin[√ 2λ(x–t)]f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 450). 102.⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)] cos[λ (x–t)]y(t)dt=f(x), f(a)=0 . Solution: y(x)=1 2⎝parenleftbiggd4 dx4+4λ4⎝parenrightbigg⎝integraldisplayx a(x–t)2f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 450). 1.7-6. Kernels Containing Logarithmic and Trigonometric Functions. 103.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosβ(λx)+Blnγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Acosβ(λx)a n d h(t)=Blnγ(µt)+C. 86 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 104.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosβ(λt)+Blnγ(µx)+ C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Blnγ(µx)+Candh(t)=Acosβ(λt). 105.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinβ(λx)+Blnγ(µt)+C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Asinβ(λx)a n d h(t)=Blnγ(µt)+C. 106.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinβ(λt)+Blnγ(µx)+ C⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=Blnγ(µx)a n dh(t)=Asinβ(λt)+C. 1.8. Equations Whose Kernels Contain Special Functions∗ 1.8-1. Kernels Containing Error Function or Exponential Integral. 1.⎝integraldisplay ⎝integraldisplayx 0erf(λ√ x–t)y(t)dt=f(x), f(0) =f/prime x(0) = 0. Here erf zis the error function (see Supplement 11.2-1). Solution: y(x)=1 √ πλe–λ2xd dx⎝integraldisplayx 0eλ2t √ x–tf/prime t(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 458). 2.⎝integraldisplay ⎝integraldisplay∞ xerf(λ√ t–x)y(t)dt=f(x). Solution: y(x)=1 √ πλeλ2xd dx⎝integraldisplay∞ x1 eλ2t√ t–xf/prime t(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 459). 3.⎝integraldisplay ⎝integraldisplayx 0Ei(λ(t–x))y(t)dt=f(x), f(0) =f/prime x(0) = 0. Here Ei(z ) is the exponential integral (see Supplement 11.2-2). Solution: y(x)=–1 λ⎝integraldisplayx 0eλ(t–x)ν(λ(x–t))⎝parenleftbiggd2 dt2+λd dt⎝parenrightbigg f(t)dt, where ν(z)=⎝integraldisplay∞ 0zξdξ Γ(ξ+1 ). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 455). * For notation and properties of special functions, see Supplement 11. 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 87 1.8-2. Kernels Containing Sine and Cosine Integrals. 4.⎝integraldisplay ⎝integraldisplayx 0[sin(x –t)S i (x–t)–c o s ( x–t)c i (x–t)]y(t)dt=f(x), f(0) =f/prime x(0) = 0. Here Si(z ) is the sine integral and ci( z) is the cosine integral (see Supplements 11.3-1 and 11.3-2). Solution: y(x)=⎝integraldisplayx 0ν(x–t)⎝parenleftbiggd2 dt2+1⎝parenrightbigg f(t)dt, where ν(z)=⎝integraldisplay∞ 0zξdξ Γ(ξ+1 ). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 458). 5.⎝integraldisplay ⎝integraldisplayx 0[cos(x–t)S i (x–t)–s i n ( x–t)c i (x–t)]y(t)dt=f(x), f(0) =f/prime x(0) =f/prime/prime xx(0) = 0. Solution: y(x)=⎝integraldisplayx 0ν(x–t)⎝parenleftbiggd3 dt3+d dt⎝parenrightbigg f(t)dt, where ν(z)=⎝integraldisplay∞ 0zξdξ Γ(ξ+1 ). Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 458). 1.8-3. Kernels Containing Fresnel Integrals. 6.⎝integraldisplay ⎝integraldisplayx 0S(x–t)y(t)dt=f(x), f(0) =f/prime x(0) =f/prime/prime xx(0) =f/prime/prime/prime xxx(0) = 0. HereS(z) is the Fresnel sine integral (see Supplement 11.3-3). Solution: y(x)=4⎝integraldisplayx 0C(x–t)⎝parenleftbiggd4 dt4+d2 dt2⎝parenrightbigg y(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 460). 7.⎝integraldisplay ⎝integraldisplayx 0C(x–t)y(t)dt=f(x), f(0) =f/prime x(0) =f/prime/prime xx(0) =f/prime/prime/prime xxx(0) = 0. HereC(z) is the Fresnel cosine integral (see Supplement 11.3-3). Solution: y(x)=4⎝integraldisplayx 0S(x–t)⎝parenleftbiggd4 dt4+d2 dt2⎝parenrightbigg y(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 460). 88 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 1.8-4. Kernels Containing Incomplete Gamma Functions. 8.⎝integraldisplay ⎝integraldisplayx 0γ(ν,λ(x–t))y(t)dt=f(x). Hereγ(ν,z) is the incomplete gamma function (see Supplement 11.5-1). 1◦.L e t R e ν>0 ,m=[ R eν] + 1, where [Re ν] denotes the integer part of the number Re ν, andf(0) =f/prime x(0) =···=f(m) x(0) = 0. Then the solution is y(x)=λ–ν Γ(ν)Γ(m–ν)e–λx⎝parenleftbiggd dx⎝parenrightbiggm⎝integraldisplayx 0eλt (x–t)ν–m+1f/prime t(t)dt. 2◦.L e tν=n/2, where nis a positive integer, and f(0) =f/prime(0) =···=f(n+1)(0) = 0. Then the solution is y(x)=λ–n Γ2(n/2)⎝integraldisplayx 0γ⎝parenleftBign 2,λ(x–t)⎝parenrightBigd2 dt2⎝parenleftbiggd dt+λ⎝parenrightbiggn f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 461). 9.⎝integraldisplay ⎝integraldisplay∞ xγ(ν,λ(t–x))y(t)dt=f(x). Solution: y(x)=–λ–ν Γ(ν)Γ(m–ν)eλx⎝parenleftbigg –d dx⎝parenrightbiggm⎝integraldisplay∞ xe–λt (t–x)ν–m+1f/prime t(t)dt, where Re ν>0 ,m=[ R eν]+1 ,a n d[ R e ν] denotes the integer part of the number Re ν. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 462). 10.⎝integraldisplay ⎝integraldisplayx 0Γ(ν,λ(x–t))y(t)dt=f(x). Solution: y(x)=e–λx Γ(ν)⎝integraldisplayx 0Eν⎝parenleftbig [λ(x–t)]ν⎝parenrightbig⎝parenleftbiggd2 dt2–λd dt⎝parenrightbigg⎝parenleftbig eλtf(t)⎝parenrightbig dt, where Re ν>0a n d Eν(z) are the Weber function, Eν(z)=1 π⎝integraldisplayπ 0sin(νt–zsint)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 462). 1.8-5. Kernels Containing Bessel Functions. 11.⎝integraldisplay ⎝integraldisplayx aJ0(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.17 with n=0a n d J0(z) is the Bessel function (see Supplement 11.6-1). If f(a)=f/prime x(a) = 0 then the solution is y(x)=⎝integraldisplayx aJ0(λ(x–t))⎝parenleftbiggd2 dt2+λ2⎝parenrightbigg f(t)dt. Example. In the special case λ=1a n d f(x)=Asinx, the solution has the form y(x)=AJ0(x). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 470). * For notation and properties of special functions, see Supplement 11. 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 89 12.⎝integraldisplay ⎝integraldisplayx a[J0(λx)–J0(λt)]y(t)dt=f(x). Solution: y(x)=–d dx⎝bracketleftbiggf/prime x(x) λJ1(λx)⎝bracketrightbigg . 13.⎝integraldisplay ⎝integraldisplayx a[AJ 0(λx)+BJ 0(λt)]y(t)dt=f(x). ForB=–A, see equation 1.8.12. We consider the interval [ a,x]i nw h i c h J0(λx) does not change its sign. Solution with B≠–A: y(x)=±1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingleJ0(λx)⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingleJ0(λt)⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . Here the sign of J0(λx) should be taken. 14.⎝integraldisplay ⎝integraldisplayx a(x–t)J0(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.18 with n=0 . I f f(a)=f/prime x(a)=f/prime/prime xx(a)=0t h e nt h e solution is y(x)=⎝integraldisplayx aJ0(λ(x–t))⎝parenleftbiggd2 dt2+λ2⎝parenrightbigg2 F(t)dt,F(t)=⎝integraldisplayt af(s)ds. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472). 15.⎝integraldisplay ⎝integraldisplayx a(x–t)J1(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.17 with n=1 .I f f(a)=f/prime x(a)=0t h e nt h es o l u t i o ni s y(x)=f/prime x(x) λ+1 λ⎝integraldisplayx aJ0(λ(x–t))⎝parenleftbiggd2 dt2+λ2⎝parenrightbigg f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 471). 16.⎝integraldisplay ⎝integraldisplayx a(x–t)2J1(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.18 with n=1 .I f f(a)=f/prime x(a)=···=f/prime/prime/prime/prime xxxx(a)=0t h e n the solution is y(x)=1 3λ⎝integraldisplayx aJ0(λ(x–t))⎝parenleftbiggd2 dt2+λ2⎝parenrightbigg3 F(t)dt,F(t)=⎝integraldisplayt af(s)ds. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472). 17.⎝integraldisplay ⎝integraldisplayx a(x–t)nJn(λ(x–t))y(t)dt=f(x), n=0 ,1 ,2 , ... Iff(a)=f/prime x(a)=···=f(2n+1) x (a)=0t h e nt h es o l u t i o ni s y(x)=2nn! (2n)!λn⎝integraldisplayx aJ0(λ(x–t))⎝parenleftbiggd2 dt2+λ2⎝parenrightbiggn+1 f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 471–472). 90 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 18.⎝integraldisplay ⎝integraldisplayx a(x–t)n+1Jn(λ(x–t))y(t)dt=f(x), n=0 ,1 ,2 , ... Iff(a)=f/prime x(a)=···=f(2n+2) x (a)=0t h e nt h es o l u t i o ni s y(x)=2n+1(n+1 ) ! (2n+2 ) !λn⎝integraldisplayx aJ0(λ(x–t))⎝parenleftbiggd2 dt2+λ2⎝parenrightbiggn+2 F(t)dt,F(t)=⎝integraldisplayt af(s)ds. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472). 19.⎝integraldisplay ⎝integraldisplayx a(x–t)1/2J1/2(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.23 with n=1 .I f f(a)=f/prime x(a)=0t h e nt h es o l u t i o ni s y(x)=⎝radicalbigg π 2λ⎝bracketleftbig f/prime/prime xx(x)+λ2f(x)⎝bracketrightbig . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 471). 20.⎝integraldisplay ⎝integraldisplayx a(x–t)3/2J1/2(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.24 with n=1 .L e t f(a)=f/prime x(a)=f/prime/prime xx(a)=0 .T h e nt h e solution is y(x)=√ π 2√ 2λ⎝parenleftbiggd2 dx2+λ2⎝parenrightbigg2⎝integraldisplayx af(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472). 21.⎝integraldisplay ⎝integraldisplayx a(x–t)3/2J3/2(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.23 with n=2 .I f f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 then the solution is y(x)=√ π (2λ)3/2⎝parenleftbiggd2 dx2+λ2⎝parenrightbigg2 f(x). Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 471). 22.⎝integraldisplay ⎝integraldisplayx a(x–t)5/2J3/2(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.24 with n=2 . L e t f(a)=f/prime x(a)=···=f/prime/prime/prime/prime xxxx(a)=0 . Then the solution is y(x)=√ π 4(2λ)3/2⎝parenleftbiggd2 dx2+λ2⎝parenrightbigg3⎝integraldisplayx af(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472). 23.⎝integraldisplayx a(x–t)2n–1 2J2n–1 2(λ(x–t))y(t)dt=f(x), n=1 ,2 ,3 , ... Letf(a)=f/prime x(a)=···=f(2n–1)(a) = 0. Then the solution is y(x)=√ π (2λ)2n–1 2(n–1 ) !⎝parenleftbiggd2 dx2+λ2⎝parenrightbiggn f(x). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 471). 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 91 24.⎝integraldisplayx a(x–t)2n+1 2J2n–1 2(λ(x–t))y(t)dt=f(x), n=1 ,2 ,3 , ... Letf(a)=f/prime x(a)=···=f(2n)(a) = 0. Then the solution is y(x)=√ π 2(2λ)n–1/2n!⎝parenleftbiggd2 dx2+λ2⎝parenrightbiggn+1⎝integraldisplayx af(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472). 25.⎝integraldisplay ⎝integraldisplayx a[Jν(λx)–Jν(λt)]y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=Jν(λx), where Jν(z) is the Bessel function (see Supplement 11.6-1). Solution: y(x)=d dx⎝bracketleftbiggxf/prime x(x) νJν(λx)–λxJν+1(λx)⎝bracketrightbigg . 26.⎝integraldisplay ⎝integraldisplayx a[AJν(λx)+BJν(λt)]y(t)dt=f(x). ForB=–A, see equation 1.8.25. We consider the interval [ a,x]i nw h i c h Jν(λx) does not change its sign. Solution with B≠–A: y(x)=±1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingleJν(λx)⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingleJν(λt)⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . Here the sign of Jν(λx) should be taken. 27.⎝integraldisplay ⎝integraldisplayx a[AJν(λx)+BJµ(βt)]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=AJν(λx)a n d h(t)=BJµ(βt). 28.⎝integraldisplay ⎝integraldisplayx a(x–t)νJν(λ(x–t))y(t)dt=f(x). 1◦.L e t R e ν>– 1/2a n df(a)=f/prime x(a)=...=f(2n–1) x (a)=0 ,w h e r e n=[ R eν+1/2] + 1 and [A] stands for the integer part of the number A. Then the solution is y(x)=π(2λ)1–n Γ(ν+1/2)Γ(n–ν–1/2)⎝integraldisplayx a(x–t)n–ν–1Jn–ν–1(λ(x–t))⎝parenleftbiggd2 dt2+λ2⎝parenrightbiggn f(t)dt. 2◦.F o rν=nandν=n–1/2(nis an integer) see equations 1.8.17 and 1.8.23. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 471), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 29.⎝integraldisplay ⎝integraldisplayx a(x–t)ν+1Jν(λ(x–t))y(t)dt=f(x). 1◦.L e t R e ν>– 1a n d f(a)=f/prime x(a)=···=f(2n–2)(a)=0 ,w h e r e n=[ R eν+3/2] + 1 and [A] stands for the integer part of the number A. Then the solution is y(x)=21–nλ2–nπ Γ(ν+3/2)Γ(n–ν–3/2)⎝integraldisplayx a(x–t)n–ν–2Jn–ν–2(λ(x–t))⎝parenleftbiggd2 dt2+λ2⎝parenrightbiggn F(t)dt, where F(t)=⎝integraldisplayt af(s)ds. 2◦.F o rν=nandν=n–1/2(nis an integer) see equations 1.8.18 and 1.8.24. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 92 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 30.⎝integraldisplay ⎝integraldisplayx 0Jν(λ(x–t)) x–ty(t)dt=f(x), Re ν>0 . 1◦.I fν=nis a positive integer number and f(0) =f/prime x(0) =···=f(n) x(0) = 0 then y(x)=n λn[n/2]⎝summationdisplay k=0C2k n⎝parenleftbiggd dx⎝parenrightbiggn–2k⎝parenleftbiggd2 dx2+λ2⎝parenrightbiggk f(x) +n λn⎝integraldisplayx 0J0(λ(x–t))[(n–1)/2]⎝summationdisplay k=0C2k+1 n⎝parenleftbiggd dt⎝parenrightbiggn–2k–1⎝parenleftbiggd2 dt2+λ2⎝parenrightbiggk+1 f(t)dt, where [ A] stands for the integer part of the number AandCk n=n! k!(n–k)!are binomial coefﬁcients (0! = 1). 2◦.I fνis not an integer, [Re ν]+1= m>1 ,a n d f(0) =f/prime x(0) =···=f(m) x(0) = 0 then y(x)=ν λm⎝integraldisplayx 0Jm–ν(λ(x–t))[(m–1)/2]⎝summationdisplay k=0C2k+1 m⎝parenleftbiggd dt⎝parenrightbiggm–2k–1⎝parenleftbiggd2 dt2+λ2⎝parenrightbiggk+1 f(t)dt +ν(m–ν) λm⎝integraldisplayx 0Jm–ν(λ(x–t)) x–t[m/2]⎝summationdisplay k=0C2k m⎝parenleftbiggd dt⎝parenrightbiggm–2k⎝parenleftbiggd2 dt2+λ2⎝parenrightbiggk f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 470–471). 31.⎝integraldisplay ⎝integraldisplayx aJ0⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.8.38 with n=0 .I f f(a)=f/prime x(a)=0t h e nt h es o l u t i o ni s y(x)=d2 dx2⎝integraldisplayx aI0⎝parenleftbig λ√ x–t⎝parenrightbig f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472). 32.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AJν⎝parenleftbig⎝parenleftbig λ√ x⎝parenrightbig⎝parenrightbig +BJν⎝parenleftbig⎝parenleftbig λ√ t⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). We consider the interval [ a,x]i nw h i c h Jν⎝parenleftbig λ√ x⎝parenrightbig does not change its sign. Solution with B≠–A: y(x)=±1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingleJν⎝parenleftbig λ√ x⎝parenrightbig⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingleJν⎝parenleftbig λ√ t⎝parenrightbig⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . Here the sign Jν⎝parenleftbig λ√ x⎝parenrightbig should be taken. 33.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AJν⎝parenleftbig⎝parenleftbig λ√ x⎝parenrightbig⎝parenrightbig +BJµ⎝parenleftbig⎝parenleftbig β√ t⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=AJν⎝parenleftbig λ√ x⎝parenrightbig andh(t)=BJµ⎝parenleftbig β√ t⎝parenrightbig . 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 93 34.⎝integraldisplay ⎝integraldisplayx a√ x–tJ 1⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.8.38 with n= 1. If the conditions f(a)=f/prime x(a)=f/prime/prime xx(a)=0 are satisﬁed, then the solution is y(x)=2 λd3 dx3⎝integraldisplayx aI0⎝parenleftbig λ√ x–t⎝parenrightbig f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472). 35.⎝integraldisplay ⎝integraldisplayx a(x–t)1/4J1/2⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.8.39 with n= 1. If the conditions f(a)=f/prime x(a)=0a r e satisﬁed, then the solution is y(x)=⎝radicalbigg 2 πλd2 dx2⎝integraldisplayx acosh⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. 36.⎝integraldisplay ⎝integraldisplayx a(x–t)3/4J3/2⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.8.39 with n= 2. If the conditions f(a)=f/prime x(a)=f/prime/prime xx(a)=0 are satisﬁed, then the solution is y(x)=23/2 √ πλ3/2d3 dx3⎝integraldisplayx acosh⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. 37.⎝integraldisplay ⎝integraldisplayx a(x–t)–1/ 4J–1/ 2⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.8.39 with n= 0. If the condition f(a) = 0 is satisﬁed, then the solution is y(x)=⎝radicalbigg λ 2πd dx⎝integraldisplayx acosh⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. 38.⎝integraldisplay ⎝integraldisplayx a(x–t)n/2Jn⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x), n=0 ,1 ,2 , ... This is a special case of equation 1.8.40 with ν=nandm=n+ 2. If the conditions f(a)=f/prime x(a)=···=f(n+1) x(a) = 0 are satisﬁed, then the solution is y(x)=⎝parenleftBig2 λ⎝parenrightBigndn+2 dxn+2⎝integraldisplayx aI0⎝parenleftbig λ√ x–t⎝parenrightbig f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472). 39.⎝integraldisplayx a(x–t)2n–1 4J2n–1 2⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x), n=0 ,1 ,2 , ... This is a special case of equation 1.8.40 with ν=n–1/2a n d m=n+ 1. If the conditions f(a)=f/prime x(a)=···=f(n) x(a) = 0 are satisﬁed, then the solution is y(x)=1 √ π⎝parenleftbigg2 λ⎝parenrightbigg2n–1 2dn+1 dxn+1⎝integraldisplayx acosh⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. 94 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 40.⎝integraldisplay ⎝integraldisplayx a(x–t)ν/2Jν⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x), Re ν> –1. 1◦.L e tf(a)=f/prime x(a)=···=f(m–1) x (a)=0 ,w h e r e m=[ R eν+1 ]+1a n d [ A] stands for the integer part of the number A. Then the solution is y(x)=⎝parenleftBig2 λ⎝parenrightBigm–2dm dxm⎝integraldisplayx a⎝parenleftbig x–t⎝parenrightbigm–ν–2 2Im–ν–2⎝parenleftbig λ√ x–t⎝parenrightbig f(t)dt. 2◦.F o rν=nandν=n–1/2(nis an integer) see equations 1.8.38 and 1.8.39. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472). 41.⎝integraldisplay ⎝integraldisplay∞ x(t–x)ν/2Jν⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig y(t)dt=f(x), Re ν> –1. Solution: y(x)=⎝parenleftbiggλ 2⎝parenrightbigg2–m⎝parenleftbigg –d dx⎝parenrightbiggm⎝integraldisplay∞ x(t–x)(m–ν)/2–1Im–ν–2⎝parenleftbig λ√ t–x⎝parenrightbig f(t)dt, where m=[ R eν+1 ]+1a n d [ A] stands for the integer part of the number A. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 474), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 42.⎝integraldisplay ⎝integraldisplayx 0(x–t)ν/2Jν⎝parenleftbig⎝parenleftbig λ⎝radicalbig t(x–t)⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=λ 2x–1/2⎝integraldisplayx 0(x–t)–(ν+1)/2J–ν–1⎝parenleftbig λ⎝radicalbig x(x–t)⎝parenrightbig tν+1d(t–νf(t)), where –1 < Re ν<0 . References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 473), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 43.⎝integraldisplay ⎝integraldisplayx 0(x–t)ν/2Jν⎝parenleftbig⎝parenleftbig λ⎝radicalbig x(x–t)⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=λ 2x–νd dx⎝parenleftbigg xν+1⎝integraldisplayx 0tν/2(x–t)–(ν+1)/2I–ν–1⎝parenleftbig λ⎝radicalbig t(x–t)⎝parenrightbig f(t)dt⎝parenrightbigg , where –1 < Re ν<0 . References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 473), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 44.⎝integraldisplay ⎝integraldisplayx 0J0⎝parenleftbig⎝parenleftbig λ√ x2–t2⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=f/prime x(x)+λd dx⎝integraldisplayx 0t √ x2–t2I1⎝parenleftbig λ√ x2–t2⎝parenrightbig f(t)dt. Reference: S. Feny ¨o and H. W. Stolle (1984, p. 328). 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 95 45.⎝integraldisplay ⎝integraldisplayx 0⎝parenleftbig⎝parenleftbig x2–t2⎝parenrightbig⎝parenrightbig–1/ 4J–1/ 2⎝parenleftbig⎝parenleftbig λ√ x2–t2⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=⎝radicalbigg 2λ πd dx⎝integraldisplayx 0tcosh⎝parenleftbig λ√ x2–t2⎝parenrightbig √ x2–t2f(t)dt. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 46.⎝integraldisplay ⎝integraldisplay∞ x⎝parenleftbig⎝parenleftbig t2–x2⎝parenrightbig⎝parenrightbig–1/ 4J–1/ 2⎝parenleftbig⎝parenleftbig λ√ t2–x2⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=–⎝radicalbigg 2λ πd dx⎝integraldisplay∞ xtcosh⎝parenleftbig λ√ t2–x2⎝parenrightbig √ t2–x2f(t)dt. 47.⎝integraldisplay ⎝integraldisplayx 0⎝parenleftbig⎝parenleftbig x2–t2⎝parenrightbig⎝parenrightbigν/2Jν⎝parenleftbig⎝parenleftbig λ√ x2–t2⎝parenrightbig⎝parenrightbig y(t)dt=f(x), –1 < ν<0 . Solution: y(x)=λd dx⎝integraldisplayx 0t⎝parenleftbig x2–t2⎝parenrightbig–(ν+1)/2I–ν–1⎝parenleftbig λ√ x2–t2⎝parenrightbig f(t)dt. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 48.⎝integraldisplay ⎝integraldisplay∞ x⎝parenleftbig⎝parenleftbig t2–x2⎝parenrightbig⎝parenrightbigν/2Jν⎝parenleftbig⎝parenleftbig λ√ t2–x2⎝parenrightbig⎝parenrightbig y(t)dt=f(x), –1 < ν<0 . Solution: y(x)=–λd dx⎝integraldisplay∞ xt⎝parenleftbig t2–x2⎝parenrightbig–(ν+1)/2I–ν–1⎝parenleftbig λ√ t2–x2⎝parenrightbig f(t)dt. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 49.⎝integraldisplay ⎝integraldisplayx a[AtkJν(λx)+BxmJµ(λt)]y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=AJν(λx),h1(t)=tk,g2(x)=Bxm,a n d h2(t)=Jµ(λt). 50.⎝integraldisplay ⎝integraldisplayx a[AJ2 ν(λx)+BJ2 ν(λt)]y(t)dt=f(x). Solution with B≠–A: y(x)=1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingleJν(λx)⎝vextendsingle⎝vextendsingle–2A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingleJν(λt)⎝vextendsingle⎝vextendsingle–2B A+Bf/prime t(t)dt⎝bracerightbigg . 51.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AJk ν(λx)+BJm µ(βt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=AJk ν(λx)a n d h(t)=BJm µ(βt). 96 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 52.⎝integraldisplay ⎝integraldisplayx 0⎝parenleftbigg ⎝parenleftbiggx–t x–t+γ⎝parenrightbigg ⎝parenrightbiggν/2 Jν(λ⎝radicalbig (x–t)(x –t+γ))y(t)dt=f(x). Let –1 < Re ν<m+1<2 n+1(nandmare the minimal integer numbers), and f(0) =f/prime x(0) = ···=f(2n+m+1) x (0) = 0. Then y(x)=√ πλ–m⎝integraldisplayx 0⎝parenleftbiggx–t x–t–γ⎝parenrightbigg(m–ν)/2 Jm–ν⎝parenleftbig λ⎝radicalbig (x–t)(x–t–γ)⎝parenrightbig ×⎝integraldisplayt 0m⎝summationdisplay j=0Cj m Γ(n–j/2)⎝parenleftbiggt–s 2λ⎝parenrightbiggn–(j+1)/2 Jn–(j+1)/2(λ(t–s))⎝parenleftbiggd2 ds2+λ2⎝parenrightbiggn+1⎝parenleftbiggd ds⎝parenrightbiggm–j f(s)dsdt, where Ck nare binomial coefﬁcients. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 473), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 53.⎝integraldisplay ⎝integraldisplayx a[Y0(λx)–Y0(λt)]y(t)dt=f(x). Solution: y(x)=–d dx⎝bracketleftbiggf/prime x(x) λY1(λx)⎝bracketrightbigg . 54.⎝integraldisplay ⎝integraldisplayx a[Yν(λx)–Yν(λt)]y(t)dt=f(x). Solution: y(x)=d dx⎝bracketleftbiggxf/prime x(x) νYν(λx)–λxYν+1(λx)⎝bracketrightbigg . 55.⎝integraldisplay ⎝integraldisplayx a[AYν(λx)+BYν(λt)]y(t)dt=f(x). ForB=–A, see equation 1.8.54. We consider the interval [ a,x]i nw h i c h Yν(λx) does not change its sign. Solution with B≠–A: y(x)=±1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingleYν(λx)⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingleYν(λt)⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . Here the sign of Yν(λx) should be taken. 56.⎝integraldisplay ⎝integraldisplayx a[AtkYν(λx)+BxmYµ(λt)]y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=AYν(λx),h1(t)=tk,g2(x)=Bxm,a n d h2(t)=Yµ(λt). 57.⎝integraldisplay ⎝integraldisplayx a[AJν(λx)Yµ(βt)+BJν(λt)Yµ(βx)]y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=AJν(λx),h1(t)=Yµ(βt),g2(x)= BYµ(βx), and h2(t)=Jν(λt). 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 97 1.8-6. Kernels Containing Modiﬁed Bessel Functions. 58.⎝integraldisplay ⎝integraldisplayx aI0(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.64 with n=0a n d I0(z) is the modiﬁed Bessel function (see Supplement 11.7-1). If f(a)=f/prime x(a) = 0 then the solution is y(x)=⎝integraldisplayx aI0(λ(x–t))⎝parenleftbiggd2 dt2–λ2⎝parenrightbigg f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481). 59.⎝integraldisplay ⎝integraldisplayx a[I0(λx)–I0(λt)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . Solution: y(x)=d dx⎝bracketleftbiggf/prime x(x) λI1(λx)⎝bracketrightbigg . 60.⎝integraldisplay ⎝integraldisplayx a[AI 0(λx)+BI 0(λt)]y(t)dt=f(x). ForB=–A, see equation 1.8.59. Solution with B≠–A: y(x)=±1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingleI0(λx)⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingleI0(λt)⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . Here the sign of Iν(λx) should be taken. 61.⎝integraldisplay ⎝integraldisplayx a(x–t)I0(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.65 with n=0 . I f f(a)=f/prime x(a)=f/prime/prime xx(a)=0t h e nt h e solution is y(x)=⎝integraldisplayx aI0(λ(x–t))⎝parenleftbiggd2 dt2–λ2⎝parenrightbigg2 F(t)dt,F(t)=⎝integraldisplayt af(s)ds. 62.⎝integraldisplay ⎝integraldisplayx a(x–t)I1(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.64 with n=1 .I f f(a)=f/prime x(a)=0t h e nt h es o l u t i o ni s y(x)=λ–1f/prime x(x)+λ–1⎝integraldisplayx aI0(λ(x–t))⎝parenleftbiggd2 dt2–λ2⎝parenrightbigg f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481). 63.⎝integraldisplay ⎝integraldisplayx a(x–t)2I1(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.65 with n=1 .I f f(a)=f/prime x(a)=···=f/prime/prime/prime/prime xxxx(a)=0t h e n the solution is y(x)=1 3λ⎝integraldisplayx aI0(λ(x–t))⎝parenleftbiggd2 dt2–λ2⎝parenrightbigg3 F(t)dt,F(t)=⎝integraldisplayt af(s)ds. 98 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 64.⎝integraldisplay ⎝integraldisplayx a(x–t)nIn(λ(x–t))y(t)dt=f(x), n=0 ,1 ,2 , ... Iff(a)=f/prime x(a)=···=f(2n+1) x (a)=0t h e nt h es o l u t i o ni s y(x)=2nn! (2n)!λn⎝integraldisplayx aI0(λ(x–t))⎝parenleftbiggd2 dt2–λ2⎝parenrightbiggn+1 f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 65.⎝integraldisplay ⎝integraldisplayx a(x–t)n+1In(λ(x–t))y(t)dt=f(x), n=0 ,1 ,2 , ... This is a special case of equation 1.8.78 with ν=nandm=n+2 . I f f(a)=f/prime x(a)=···= f(2n+2) x (a) = 0 then the solution is y(x)=2n+1(n+1 ) ! (2n+2 ) !λn⎝integraldisplayx aI0(λ(x–t))⎝parenleftbiggd2 dt2–λ2⎝parenrightbiggn+2 F(t)dt,F(t)=⎝integraldisplayt af(s)ds. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 482), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 66.⎝integraldisplay ⎝integraldisplayx a(x–t)1/2I1/2(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.70 with n=1 .I f f(a)=f/prime x(a)=0t h e nt h es o l u t i o ni s y(x)=⎝radicalbigg π 2λ⎝bracketleftbig f/prime/prime xx(x)–λ2f(x)⎝bracketrightbig . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481). 67.⎝integraldisplay ⎝integraldisplayx a(x–t)3/2I1/2(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.71 with n=1 . I f f(a)=f/prime x(a)=f/prime/prime xx(a)=0t h e nt h e solution is y(x)=√ π 2(2λ)1/2⎝parenleftbiggd2 dx2–λ2⎝parenrightbigg2⎝integraldisplayx af(t)dt. 68.⎝integraldisplay ⎝integraldisplayx a(x–t)3/2I3/2(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.70 with n=2 .I f f(a)=f/prime x(a)=f/prime/prime xx(a)=f/prime/prime/prime xxx(a)=0 then the solution is y(x)=√ π (2λ)3/2⎝parenleftbiggd2 dx2–λ2⎝parenrightbigg2 f(x). Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481). 69.⎝integraldisplay ⎝integraldisplayx a(x–t)5/2I3/2(λ(x–t))y(t)dt=f(x). This is a special case of equation 1.8.71 with n=2 .I f f(a)=f/prime x(a)=···=f/prime/prime/prime/prime xxxx(a)=0t h e n the solution is y(x)=√ π 4(2λ)3/2⎝parenleftbiggd2 dx2–λ2⎝parenrightbigg3⎝integraldisplayx af(t)dt. 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 99 70.⎝integraldisplay ⎝integraldisplayx a(x–t)2n–1 2I2n–1 2(λ(x–t))y(t)dt=f(x), n=1 ,2 ,3 , ... Iff(a)=f/prime x(a)=···=f(2n–1) x (a) = 0 then the solution is y(x)=√ π (2λ)n–1/2(n–1 ) !⎝parenleftbiggd2 dx2–λ2⎝parenrightbiggn f(x). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 71.⎝integraldisplayx a(x–t)2n+1 2I2n–1 2(λ(x–t))y(t)dt=f(x), n=0 ,1 ,2 , ... Iff(a)=f/prime x(a)=···=f(2n)(a)=0t h e nt h es o l u t i o ni s y(x)=√ π 2(2λ)n–1/2n!⎝parenleftbiggd2 dx2–λ2⎝parenrightbiggn+1⎝integraldisplayx af(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 482). 72.⎝integraldisplay ⎝integraldisplayx a[Iν(λx)–Iν(λt)]y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=Iν(λx), where Iν(z) is the modiﬁed Bessel function (see Supplement 11.7-1). 73.⎝integraldisplay ⎝integraldisplayx a[AIν(λx)+BIν(λt)]y(t)dt=f(x). Solution with B≠–A: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig Iν(λx)⎝bracketrightbig–A A+B⎝integraldisplayx a⎝bracketleftbig Iν(λt)⎝bracketrightbig–B A+Bf/prime t(t)dt⎝bracerightbigg . 74.⎝integraldisplay ⎝integraldisplayx a[AIν(λx)+BIµ(βt)]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=AIν(λx)a n d h(t)=BIµ(βt). 75.⎝integraldisplay ⎝integraldisplayx 0Iν(λ(x–t))y(t)dt=f(x). 1◦.L e t – 1 < R e ν<1a n d f(0) =f/prime x(0) = 0. Then the solution is y(x)=⎝integraldisplayx 0I–ν(λ(x–t))⎝parenleftbiggd2 dt2–λ2⎝parenrightbigg f(t)dt. 2◦.L e tν=n≥0(nis an integer number) and f(0) =f/prime x(0) =···=f(n+1) x(0) = 0. Then the solution is y(x)=λ–n[(n–1)/2]⎝summationdisplay k=0C2k+1 n⎝parenleftbiggd dx⎝parenrightbiggn–2k–1⎝parenleftbiggd2 dx2–λ2⎝parenrightbiggk+1 f(x) +λ–n⎝integraldisplayx 0I0(λ(x–t))[n/2]⎝summationdisplay k=0C2k n⎝parenleftbiggd dt⎝parenrightbiggn–2k⎝parenleftbiggd2 dt2–λ2⎝parenrightbiggk+1 f(t)dt, where [ A] stands for the integer part of the number AandCk nare binomial coefﬁcients. 100 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 3◦.L e t R e ν>– 1a n d f(0) =f/prime x(0) =···=f(m+1) x (0) = 0, where m=[ R eν+1 ] . T h e nt h e solution is y(x)=m–ν λm⎝integraldisplayx 0Im–ν(λ(x–t)) x–t[(m–1)/2]⎝summationdisplay k=0C2k+1 m⎝parenleftbiggd dt⎝parenrightbiggm–2k–1⎝parenleftbiggd2 dt2–λ2⎝parenrightbiggk+1 f(t)dt +λ–m⎝integraldisplayx 0Im–ν(λ(x–t))[m/2]⎝summationdisplay k=0C2k m⎝parenleftbiggd dt⎝parenrightbiggm–2k⎝parenleftbiggd2 dt2–λ2⎝parenrightbiggk+1 f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 479–480), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 76.⎝integraldisplay ⎝integraldisplayx 0(x–t)–1Iν(λ(x–t))y(t)dt=f(x). 1◦.L e t R e ν>0a n d f(0) =f/prime x(0) =···=f(m) x(0) = 0, where m=[ R eν]+1a n d [ A] stands for the integer part of the number A. Then the solution is y(x)=νλ–m⎝integraldisplayx 0Im–ν(λ(x–t))[(m–1)/2]⎝summationdisplay k=0C2k+1 m⎝parenleftbiggd dt⎝parenrightbiggm–2k–1⎝parenleftbiggd2 dt2–λ2⎝parenrightbiggk+1 f(t)dt +ν(m–ν)λ–m⎝integraldisplayx 0(x–t)–1Im–ν(λ(x–t))[m/2]⎝summationdisplay k=0C2k m⎝parenleftbiggd dt⎝parenrightbiggm–2k⎝parenleftbiggd2 dt2–λ2⎝parenrightbiggk f(t)dt, where Ck nare binomial coefﬁcients. 2◦.I fν=n>0(nis an integer number) and f(0) =f/prime x(0) =···=f(n) x(0) = 0 then y(x)=nλ–n[n/2]⎝summationdisplay k=0C2k n⎝parenleftbiggd dx⎝parenrightbiggn–2k⎝parenleftbiggd2 dx2–λ2⎝parenrightbiggk f(x) +nλ–n⎝integraldisplayx 0I0(λ(x–t))[(n–1)/2]⎝summationdisplay k=0C2k+1 n⎝parenleftbiggd dt⎝parenrightbiggn–2k–1⎝parenleftbiggd2 dt2–λ2⎝parenrightbiggk+1 f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 480–481). 77.⎝integraldisplay ⎝integraldisplayx a(x–t)νIν(λ(x–t))y(t)dt=f(x), Re ν>– 1/2. 1◦.L e tf(a)=f/prime x(a)=···=f(2m–1) x (a)=0 ,w h e r e m=[ R eν+1/2] + 1 and [ A] stands for the integer part of the number A. Then the solution is y(x)=(2λ)1–mπ Γ(ν+1/2)Γ(m–ν–1/2)⎝integraldisplayx a(x–t)m–ν–1Im–ν–1(λ(x–t))⎝parenleftbiggd2 dt2–λ2⎝parenrightbiggm f(t)dt. 2◦.L e tf(a)=f/prime x(a)=···=f(m–1) x (a)=0 ,w h e r e m=[ 2R e ν+1 ]+1 . T h e nt h es o l u t i o ni s y(x)=√ πλΓ(–ν–1 ) 22ν+1Γ(ν+1/2)e–λxdm dxm⎝bracketleftbigg eλx⎝integraldisplayx a(x–t)m–ν–1 ×m⎝summationdisplay k=0(–m)k(–2ν–2 )k Γ(m+k–2ν–1 )k!(k–ν–1 )Ik–ν–1(λ(x–t))f(t)dt⎝bracketrightbigg , where ( a)k=a(a+1 )...(a+k– 1) is the Pochhammer symbol. 3◦.F o rν=nandν=n–1/2(nis an integer) see equations 1.8.64 and 1.8.70. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 101 78.⎝integraldisplay ⎝integraldisplayx a(x–t)ν+1Iν(λ(x–t))y(t)dt=f(x). 1◦.L e t R e ν>– 1a n d f(a)=f/prime x(a)=···=f(2m–2)(a)=0 ,w h e r e m=[ R eν+3/2] + 1 and [A] stands for the integer part of the number A. Then the solution is y(x)=21–mλ2–mπ Γ(ν+3/2)Γ(m–ν–3/2)⎝integraldisplayx a(x–t)m–ν–2Im–ν–2(λ(x–t))⎝parenleftbiggd2 dt2–λ2⎝parenrightbiggm F(t)dt, where F(t)=⎝integraldisplayt af(s)ds. 2◦.F o rν=nandν=n–1/2(nis an integer) see equations 1.8.65 and 1.8.71. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 482), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 79.⎝integraldisplay ⎝integraldisplayx aI0⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.8.86 with n= 0. If the conditions f(a)=f/prime x(a)=0a r e satisﬁed, then the solution is y(x)=d2 dx2⎝integraldisplayx aJ0⎝parenleftbig λ√ x–t⎝parenrightbig f(t)dt. 80.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AIν⎝parenleftbig⎝parenleftbig λ√ x⎝parenrightbig⎝parenrightbig +BIν⎝parenleftbig⎝parenleftbig λ√ t⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution with B≠–A: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig Iν⎝parenleftbig λ√ x⎝parenrightbig⎝bracketrightbig–A A+B⎝integraldisplayx a⎝bracketleftbig Iν⎝parenleftbig λ√ t⎝parenrightbig⎝bracketrightbig–B A+Bf/prime t(t)dt⎝bracerightbigg . 81.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AIν⎝parenleftbig⎝parenleftbig λ√ x⎝parenrightbig⎝parenrightbig +BIµ⎝parenleftbig⎝parenleftbig β√ t⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=AIν⎝parenleftbig λ√ x⎝parenrightbig andh(t)=BIµ⎝parenleftbig β√ t⎝parenrightbig . 82.⎝integraldisplay ⎝integraldisplayx a√ x–tI 1⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.8.86 with n= 1. If the conditions f(a)=f/prime x(a)=f/prime/prime xx(a)=0 are satisﬁed, then the solution is y(x)=2 λd3 dx3⎝integraldisplayx aJ0⎝parenleftbig λ√ x–t⎝parenrightbig f(t)dt. 83.⎝integraldisplay ⎝integraldisplayx a(x–t)1/4I1/2⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.8.87 with n= 1. If the conditions f(a)=f/prime x(a)=0a r e satisﬁed, then the solution is y(x)=⎝radicalbigg 2 πλd2 dx2⎝integraldisplayx acos⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. 102 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 84.⎝integraldisplay ⎝integraldisplayx a(x–t)3/4I3/2⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.8.87 with n= 2. If the conditions f(a)=f/prime x(a)=f/prime/prime xx(a)=0 are satisﬁed, then the solution is y(x)=23/2 √ πλ3/2d3 dx3⎝integraldisplayx acos⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. 85.⎝integraldisplay ⎝integraldisplayx a(x–t)–1/ 4I–1/ 2⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 1.8.87 with n= 0. If the condition f(a) = 0 is satisﬁed, then the solution is y(x)=⎝radicalbigg λ 2πd dx⎝integraldisplayx acos⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. 86.⎝integraldisplay ⎝integraldisplayx a(x–t)n/2In⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x), n=0 ,1 ,2 , ... This is a special case of equation 1.8.88 with ν=nandm=n+ 2. If the conditions f(a)=f/prime x(a)=···=f(n+1) x(a) = 0 are satisﬁed, then the solution is y(x)=⎝parenleftBig2 λ⎝parenrightBigndn+2 dxn+2⎝integraldisplayx aJ0⎝parenleftbig λ√ x–t⎝parenrightbig f(t)dt. 87.⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig x–t⎝parenrightbig⎝parenrightbig2n–1 4I2n–1 2⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x), n=0 ,1 ,2 , ... This is a special case of equation 1.8.88 with ν=n–1/2a n d m=n+ 1. If the conditions f(a)=f/prime x(a)=···=f(n) x(a) = 0 are satisﬁed, then the solution is y(x)=1 √ π⎝parenleftbigg2 λ⎝parenrightbigg2n–1 2dn+1 dxn+1⎝integraldisplayx acos⎝parenleftbig λ√ x–t⎝parenrightbig √ x–tf(t)dt. 88.⎝integraldisplay ⎝integraldisplayx a(x–t)ν/2Iν⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x), Re ν> –1. 1◦.L e tf(a)=f/prime x(a)=···=f(m–1) x (a)=0 ,w h e r e m=[ R eν+1 ]+1a n d [ A] stands for the integer part of the number A. Then the solution is y(x)=⎝parenleftBig2 λ⎝parenrightBigm–2dm dxm⎝integraldisplayx a⎝parenleftbig x–t⎝parenrightbigm–ν–2 2Jm–ν–2⎝parenleftbig λ√ x–t⎝parenrightbig f(t)dt. 2◦.F o rν=nandν=n–1/2(nis an integer) see equations 1.8.86 and 1.8.87. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 482), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 89.⎝integraldisplay ⎝integraldisplay∞ x(t–x)ν/2Iν(λ√ t–x)y(t)dt=f(x), Re ν> –1. Solution: y(x)=⎝parenleftbiggλ 2⎝parenrightbigg2–m⎝parenleftbigg –d dx⎝parenrightbiggm⎝integraldisplay∞ x(t–x)(m–ν)/2–1Jm–ν–2(λ√ t–x)f(t)dt, where m=[ R eν+1 ]+1a n d [ A] stands for the integer part of the number A. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 484), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 103 90.⎝integraldisplay ⎝integraldisplayx 0(x–t)ν/2Iν⎝parenleftbig⎝parenleftbig λ⎝radicalbig t(x–t)⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=λ 2x–1/2⎝integraldisplayx 0(x–t)–(ν+1)/2J–ν–1⎝parenleftbig λ⎝radicalbig x(x–t)⎝parenrightbig tν+1d⎝parenleftbig t–νf(t)⎝parenrightbig , where –1 < Re ν<0 . References: K. Soni (1968), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 483), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 91.⎝integraldisplay ⎝integraldisplayx 0(x–t)ν/2Iν⎝parenleftbig⎝parenleftbig λ⎝radicalbig x(x–t)⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=λ 2x–νd dx⎝bracketleftbigg xν+1⎝integraldisplayx 0tν/2(x–t)–(ν+1)/2J–ν–1⎝parenleftbig λ⎝radicalbig t(x–t)⎝parenrightbig f(t)dt⎝bracketrightbigg , where –1 < Re ν<0 . References: K. Soni (1968), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 483), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 92.⎝integraldisplay ⎝integraldisplayx 0⎝parenleftbig⎝parenleftbig x2–t2⎝parenrightbig⎝parenrightbig–1/ 4I–1/ 2⎝parenleftbig⎝parenleftbig λ√ x2–t2⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=⎝radicalbigg 2λ πd dx⎝integraldisplayx 0tcos⎝parenleftbig λ√ x2–t2⎝parenrightbig √ x2–t2f(t)dt. 93.⎝integraldisplay ⎝integraldisplay∞ x⎝parenleftbig⎝parenleftbig t2–x2⎝parenrightbig⎝parenrightbig–1/ 4I–1/ 2⎝parenleftbig⎝parenleftbig λ√ t2–x2⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=–⎝radicalbigg 2λ πd dx⎝integraldisplay∞ xtcos⎝parenleftbig λ√ t2–x2⎝parenrightbig √ t2–x2f(t)dt. 94.⎝integraldisplay ⎝integraldisplayx 0⎝parenleftbig⎝parenleftbig x2–t2⎝parenrightbig⎝parenrightbigν/2Iν⎝parenleftbig⎝parenleftbig λ√ x2–t2⎝parenrightbig⎝parenrightbig y(t)dt=f(x), –1 < ν<0 . Solution: y(x)=λd dx⎝integraldisplayx 0t⎝parenleftbig x2–t2⎝parenrightbig–(ν+1)/2J–ν–1⎝parenleftbig λ√ x2–t2⎝parenrightbig f(t)dt. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 95.⎝integraldisplay ⎝integraldisplay∞ x(t2–x2)ν/2Iν⎝parenleftbig⎝parenleftbig λ√ t2–x2⎝parenrightbig⎝parenrightbig y(t)dt=f(x), –1 < ν<0 . Solution: y(x)=–λd dx⎝integraldisplay∞ xt(t2–x2)–(ν+1)/2J–ν–1⎝parenleftbig λ√ t2–x2⎝parenrightbig f(t)dt. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 104 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 96.⎝integraldisplay ⎝integraldisplayx 0⎝parenleftbigg ⎝parenleftbiggx–t x–t+γ⎝parenrightbigg ⎝parenrightbiggν/2 Iν⎝parenleftbig⎝parenleftbig λ⎝radicalbig (x–t)(x –t+γ)⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Let –1 < Re ν<m+1<2 n+1(nandmare the minimal integer numbers), and f(0) =f/prime x(0) = ···=f(2n+m+1) x (0) = 0. Then y(x)=√ πλ–m⎝integraldisplayx 0⎝parenleftbiggx–t x–t+γ⎝parenrightbigg(m–ν)/2 Im–ν⎝parenleftbig λ⎝radicalbig (x–t)(x–t+γ)⎝parenrightbig ×⎝integraldisplayt 0m⎝summationdisplay j=0Cj m Γ(n–j/2)⎝parenleftbiggt–s 2λ⎝parenrightbiggn–(j+1)/2 In–(j+1)/2(λ(t–s))⎝parenleftbiggd2 ds2–λ2⎝parenrightbiggn+1⎝parenleftbiggd ds⎝parenrightbiggm–j f(s)dsdt, where Ck nare binomial coefﬁcients. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 483–484), S. G. Samko, A. A. Kil- bas, and O. I. Marichev (1993). 97.⎝integraldisplay ⎝integraldisplayx a[AtkIν(λx)+BxsIµ(λt)]y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=AIν(λx),h1(t)=tk,g2(x)=Bxs,a n d h2(t)=Iµ(λt). 98.⎝integraldisplay ⎝integraldisplayx a[AI2 ν(λx)+BI2 ν(λt)]y(t)dt=f(x). Solution with B≠–A: y(x)=1 A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingleIν(λx)⎝vextendsingle⎝vextendsingle–2A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingleIν(λt)⎝vextendsingle⎝vextendsingle–2B A+Bf/prime t(t)dt⎝bracerightbigg . 99.⎝integraldisplay ⎝integraldisplayx a[AIk ν(λx)+BIs µ(βt)]y(t)dt=f(x). This is a special case of equation 1.9.6 with g(x)=AIk ν(λx)a n d h(t)=BIs µ(βt). 100.⎝integraldisplay ⎝integraldisplayx a[K0(λx)–K0(λt)]y(t)dt=f(x). Solution: y(x)=–d dx⎝bracketleftbiggf/prime x(x) λK 1(λx)⎝bracketrightbigg . 101.⎝integraldisplay ⎝integraldisplayx a[Kν(λx)–Kν(λt)]y(t)dt=f(x). This is a special case of equation 1.9.2 with g(x)=Kν(λx). 102.⎝integraldisplay ⎝integraldisplayx a[AK ν(λx)+BK ν(λt)]y(t)dt=f(x). Solution with B≠–A: y(x)=1 A+Bd dx⎝braceleftbigg⎝bracketleftbig Kν(λx)⎝bracketrightbig–A A+B⎝integraldisplayx a⎝bracketleftbig Kν(λt)⎝bracketrightbig–B A+Bf/prime t(t)dt⎝bracerightbigg . 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 105 103.⎝integraldisplay ⎝integraldisplayx a[AtkKν(λx)+BxsKµ(λt)]y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=AKν(λx),h1(t)=tk,g2(x)=Bxs,a n d h2(t)=Kµ(λt). 104.⎝integraldisplay ⎝integraldisplayx a[AIν(λx)Kµ(βt)+BIν(λt)Kµ(βx)]y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=AIν(λx),h1(t)=Kµ(βt),g2(x)= BKµ(βx), and h2(t)=Iν(λt). 1.8-7. Kernels Containing Legendre Polynomials. 105.⎝integraldisplayx 1Pn⎝parenleftbigg ⎝parenleftbiggx t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) = 0, x≥1. HerePn(x) is the Legendre polynomial (see Supplement 11.11-1). Solution: y(x)=xn+1 (n–1 ) !⎝parenleftbigg1 xd dx⎝parenrightbiggn+1⎝integraldisplayx 1(x–t)n–1f(t)dt, where n=1 ,2 ,3 , ... Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 495–496). 106.⎝integraldisplayx 1Pn⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) =f/prime x(1) = 0, x≥1. Solution: y(x)=⎝integraldisplayx 1t2–nPn–2⎝parenleftbiggx t⎝parenrightbigg⎝parenleftbigg1 td dt⎝parenrightbigg2⎝bracketleftbig tnf(t)⎝bracketrightbig dt, where n=2 ,3 ,4 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 496). 107.⎝integraldisplay1 xPn⎝parenleftbigg ⎝parenleftbiggx t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) =f/prime x(1) = 0, 0 < x≤1. Solution: y(x)=x–2⎝integraldisplay1 xtn+2Pn–2⎝parenleftbiggt x⎝parenrightbigg⎝parenleftbigg1 td dt⎝parenrightbigg2⎝bracketleftbig t2–nf(t)⎝bracketrightbig dt, where n=2 ,3 ,4 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 496). 108.⎝integraldisplay1 xPn⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) =f/prime x(1) = 0, 0 < x≤1. Solution: y(x)=⎝integraldisplay1 xt2–nPn–2⎝parenleftbiggx t⎝parenrightbigg⎝parenleftbigg1 td dt⎝parenrightbigg2⎝bracketleftbig tnf(t)⎝bracketrightbig dt, where n=2 ,3 ,4 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 496). 106 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 109.⎝integraldisplayx 0Pn⎝parenleftbigg ⎝parenleftbigg 2x t–1⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(0) = 0, x>0 . Solution: y(x)=xn (n–1 ) !dn+1 dxn+1⎝bracketleftbigg x–n⎝integraldisplayx 0(x–t)n–1f(t)dt⎝bracketrightbigg , where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 497). 110.⎝integraldisplay1 xPn⎝parenleftbigg ⎝parenleftbigg 2x t–1⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) = 0, x≤1. Solution: y(x)=xn⎝parenleftbigg –d dx⎝parenrightbiggn+1⎝bracketleftbigg x–n⎝integraldisplay1 x(t–x)n–1 (n–1 ) !f(t)dt⎝bracketrightbigg , where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 497). 111.⎝integraldisplay1 xPn⎝parenleftbigg ⎝parenleftbigg 2t x–1⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) = 0, x≤1. Solution: y(x)=⎝parenleftbigg –d dx⎝parenrightbiggn+1⎝bracketleftbigg xn+1⎝integraldisplay1 x(t–x)n–1 (n–1 ) !t–n–1f(t)dt⎝bracketrightbigg , where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 498). 112.⎝integraldisplayx 0Pn(cosh( x–t))y(t)dt=f(x), f(0) =f/prime x(0) = 0. Solution: y(x)=⎝parenleftbiggd2 dx2–(n+1 )2⎝parenrightbigg⎝integraldisplayx 0Pn+1(cosh(x–t))f(t)dt, where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 498). 113.⎝integraldisplayx 0Pn(cos(x–t))y(t)dt=f(x), f(0) =f/prime x(0) = 0. Solution: y(x)=⎝parenleftbiggd2 dx2+(n+1 )2⎝parenrightbigg⎝integraldisplayx 0Pn+1(cos(x–t))f(t)dt, where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 498). 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 107 1.8-8. Kernels Containing Associated Legendre Functions. 114.⎝integraldisplay ⎝integraldisplayx a(x2–t2)–µ/2Pµ ν⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig y(t)dt=f(x), 0 ≤a<x. Here Pµ ν(x) is the modiﬁed associated Legendre function (see Supplement 11.11-3). Let 1 – n<R eµ<1(n=1 ,2 ,... )a n df(a)=f/prime x(a)=···=f(n–1) x(a) = 0. Then the solution is y(x)=xn+µ–1dn dxn⎝bracketleftbigg x1–µ⎝integraldisplayx a(x2–t2)n+µ–2 2t–nP2–n–µ ν⎝parenleftbiggt x⎝parenrightbigg f(t)dt⎝bracketrightbigg . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 515), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 115.⎝integraldisplay ⎝integraldisplayx a(x2–t2)–µ/2Pµ ν⎝parenleftBig ⎝parenleftBigt x⎝parenrightBig ⎝parenrightBig y(t)dt=f(x), 0 ≤a<x. Let 1 – n<R eµ<1(n=1 ,2 , ...)a n df(a)=f/prime x(a)=···=f(n–1) x(a)=0 .T h e nt h es o l u t i o n is y(x)=dn dxn⎝integraldisplayx a(x2–t2)n+µ–2 2 P2–n–µ ν⎝parenleftBigx t⎝parenrightBig f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 515), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 116.⎝integraldisplay ⎝integraldisplay∞ x(t2–x2)–µ/2Pµ ν⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig y(t)dt=f(x). Let 1 – n<R eµ<1(n=1 ,2 , ...). Then the solution is y(x) = (–1)nxn+µ–1dn dxn⎝bracketleftbigg x1–µ⎝integraldisplayb x(t2–x2)n+µ–2 2t–nP2–n–µ ν⎝parenleftbiggt x⎝parenrightbigg f(t)dt⎝bracketrightbigg . References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 516), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 117.⎝integraldisplay ⎝integraldisplay∞ x(t2–x2)–µ/2Pµ ν⎝parenleftBig ⎝parenleftBigt x⎝parenrightBig ⎝parenrightBig y(t)dt=f(x). Let 1 – n<R eµ<1(n=1 ,2 , ...). Then the solution is y(x) = (–1)ndn dxn⎝integraldisplayb x(t2–x2)n+µ–2 2 P2–n–µ ν⎝parenleftBigx t⎝parenrightBig f(t)dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 516), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 1.8-9. Kernels Containing Conﬂuent Hypergeometric Functions. 118.⎝integraldisplay ⎝integraldisplayx s(x–t)b–1Φ⎝parenleftbig⎝parenleftbig a,b;λ(x–t)⎝parenrightbig⎝parenrightbig y(t)dt=f(x). HereΦ(a,b;z) is the Kummer conﬂuent hypergeometric function (see Supplement 11.9-1). Let 0 < Re b<n(n=1 ,2 , ...)a n df(s)=f/prime x(s)=···=f(n–1) x(s) = 0. Then the solution is y(x)=dn dxn⎝integraldisplayx s(x–t)n–b–1 Γ(b)Γ(n–b)Φ⎝parenleftbig –a,n–b;λ(x–t)⎝parenrightbig f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 530), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 108 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 119.⎝integraldisplay ⎝integraldisplay∞ x(t–x)b–1Φ⎝parenleftbig⎝parenleftbig a,b;λ(x–t)⎝parenrightbig⎝parenrightbig y(t)dt=f(x). HereΦ(a,b;z) is the Kummer conﬂuent hypergeometric function (see Supplement 11.9-1). If 0 < Re b<n(n=1 ,2 , ...) then the solution is y(x)=⎝integraldisplay∞ x(t–x)n–b–1 Γ(b)Γ(n–b)Φ⎝parenleftbig –a,n–b;λ(x–t)⎝parenrightbig f(n) t(t)dt. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 530), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 120.⎝integraldisplay ⎝integraldisplayx 0(x–t)ν–1/ 2Mµ,ν⎝parenleftbig⎝parenleftbig λ(x–t)⎝parenrightbig⎝parenrightbig y(t)dt=f(x). HereMµ,ν(z) is the Whittaker conﬂuent hypergeometric function (see Supplement 11.9-3). Let –1 /2<R e ν<(n–1 )/2a n df (0) =f/prime x(0) =···=f(n–1) x(0) = 0. Then solution is y(x)=λ–n/2 Γ(2ν+1 )e–λx/ 2dn dxn⎝bracketleftbigg eλx/ 2⎝integraldisplayx 0(x–t)(ν–3)/2–ν Γ((ν–1 )/2–ν)Mn/2–µ,n/2–ν–1⎝parenleftbig λ(x–t)⎝parenrightbig f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 522). 121.⎝integraldisplay ⎝integraldisplay∞ x(t–x)ν–1/ 2Mµ,ν⎝parenleftbig⎝parenleftbig λ(t–x)⎝parenrightbig⎝parenrightbig y(t)dt=f(x). HereMµ,ν(z) is the Whittaker conﬂuent hypergeometric function (see Supplement 11.9-3). Let –1 /2<R e ν<(n–1 )/2. Then solution is y(x)=λ–n/2 Γ(2ν+1 )⎝integraldisplay∞ x(t–x)(ν–3)/2–ν Γ((ν–1 )/2–ν)eλt/2Mn/2–µ,n/2–ν–1⎝parenleftbig λ(t–x)⎝parenrightbigdn dtn⎝bracketleftbig e–λt/2f(t)⎝bracketrightbig dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 522). 1.8-10. Kernels Containing Hermite Polynomials. 122.⎝integraldisplay ⎝integraldisplayx 0(x–t)–1/ 2H2n⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x), f(0) = 0. HereHm(x) is the Hermite polynomial (see Supplement 11.17-3). Solution: y(x)=(–1)nn! √ π(2n)!⎝parenleftbiggd dx⎝parenrightbiggm⎝integraldisplayx 0(x–t)m–3/2 Γ(m–1/2)F⎝parenleftbig n,m–1 2;λ2(x–t)⎝parenrightbig f(t)dt, where m≥1a n dF (a,b;x) is the Kummer conﬂuent hypergeometric function (see Supple- ment 11.9-1). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 556). 123.⎝integraldisplay ⎝integraldisplayx 0H2n+1⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x), f(0) =f/prime x(0) = 0. Solution: y(x)=(–1)nn! λ√ π(2n+1 ) !⎝parenleftbiggd dx⎝parenrightbiggm⎝integraldisplayx 0(x–t)m–5/2 Γ(m–3/2)F⎝parenleftbig n,m–3 2;λ2(x–t)⎝parenrightbig f(t)dt, where m≥2a n dF (a,b;x) is the Kummer conﬂuent hypergeometric function (see Supple- ment 11.9-1). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 556). 1.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 109 1.8-11. Kernels Containing Chebyshev Polynomials. 124.⎝integraldisplayx 1(x2–t2)–1/ 2Tn⎝parenleftbigg ⎝parenleftbiggx t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) = 0, x≥1. HereTn(x) is the Chebyshev polynomials of the ﬁrst kind (see Supplement 11.17-2). Solution: y(x)=2 π⎝integraldisplayx 1tn(x2–t2)–1/2Tn–1⎝parenleftbiggt x⎝parenrightbiggd dt⎝bracketleftbig t1–nf(t)⎝bracketrightbig dt, where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 499). 125.⎝integraldisplayx 1(x2–t2)–1/ 2Tn⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) = 0, x≥1. Solution: y(x)=2 πxn+1d dx⎝bracketleftbigg x–n⎝integraldisplayx 1(x2–t2)–1/2Tn+1⎝parenleftbiggx t⎝parenrightbigg f(t)dt⎝bracketrightbigg , where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 499). 126.⎝integraldisplay1 x(t2–x2)–1/ 2Tn⎝parenleftbigg ⎝parenleftbiggx t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) = 0, 0 < x≤1. Solution: y(x)=–2 πx–nd dx⎝bracketleftbigg xn+1⎝integraldisplay1 x(t2–x2)–1/2Tn+1⎝parenleftbiggt x⎝parenrightbigg f(t)dt⎝bracketrightbigg , where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 499). 127.⎝integraldisplay1 x(t2–x2)–1/ 2Tn⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) = 0, 0 < x≤1. Solution: y(x)=–2 π⎝integraldisplay1 xt1–n(t2–x2)–1/2Tn–1⎝parenleftbiggx t⎝parenrightbiggd dt[tnf(t)]dt, where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 500). 128.⎝integraldisplayx 0(x–t)–1/ 2Tn⎝parenleftbigg ⎝parenleftbigg 2x t–1⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(0) = 0, x>0 . Solution: y(x)=xn √ πΓ(n–1/2)dn dxn⎝bracketleftbigg x–n⎝integraldisplayx 0(x–t)n–3/2f(t)dt⎝bracketrightbigg , where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 500). 110 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 129.⎝integraldisplay1 x(t–x)–1/ 2Tn⎝parenleftbigg ⎝parenleftbigg 2x t–1⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) = 0, x≤1. Solution: y(x)=xn √ π⎝parenleftbigg –d dx⎝parenrightbiggn⎝bracketleftbigg x–n⎝integraldisplay1 x(t–x)n–3/2 Γ(n–1/2)f(t)dt⎝bracketrightbigg , where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 501). 130.⎝integraldisplay1 x(t–x)–1/ 2Tn⎝parenleftbigg ⎝parenleftbigg 2t x–1⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) = 0, x≤1. Solution: y(x)=1 √ πx⎝parenleftbigg –d dx⎝parenrightbiggn⎝bracketleftbigg xn+1/2⎝integraldisplay1 x(t–x)n–3/2 Γ(n–1/2)t–nf(t)dt⎝bracketrightbigg , where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 501). 1.8-12. Kernels Containing Laguerre Polynomials. 131.⎝integraldisplay ⎝integraldisplayx 0Ln(λ(x–t))y(t)dt=f(x), f(0) =f/prime x(0) = 0, x>0 . HereLn(x) is the Laguerre polynomial (see Supplement 11.17-1). Solution: y(x)=eλx⎝integraldisplayx 0Ln–1(λ(t–x))e–λtf/prime/prime tt(t)dt, where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 504). 132.⎝integraldisplay ⎝integraldisplay∞ xLn(λ(t–x))y(t)dt=f(x), f(0) =f/prime x(0) = 0, x>0 . HereLn(x) is the Laguerre polynomial (see Supplement 11.17-1). Solution: y(x)=e–λx⎝integraldisplay∞ xLn+1(λ(x–t))eλtf/prime/prime tt(t)dt, where n=1 ,2 ,3 , ... References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 505). 1.8-13. Kernels Containing Jacobi Theta Functions. 133.⎝integraldisplay ⎝integraldisplayx 0ϑ2(0,x–t)y(t)dt=f(x), f(0) = 0. Hereϑ2(v,q) is the Jacobi theta function (see Supplement 11.15-1). Solution: y(x)=1 π⎝integraldisplayx 0ϑ3(0,x–t)f/prime t(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 551). 1.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 111 134.⎝integraldisplay ⎝integraldisplayx 0ϑ3(0,x–t)y(t)dt=f(x), f(0) = 0. Hereϑ3(v,q) is the Jacobi theta function (see Supplement 11.15-1). Solution: y(x)=1 π⎝integraldisplayx 0ϑ2(0,x–t)f/prime t(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 551). 1.8-14. Kernels Containing Other Special Functions. 135.⎝integraldisplay ⎝integraldisplayx s(x–t)c–1F⎝parenleftbigg ⎝parenleftbigg a,b,c;1 –x t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). HereΦ(a,b,c;z) is the Gaussian hypergeometric function (see Supplement 11.10-1). Solution: y(x)=x–adn dxn⎝braceleftBig xa⎝integraldisplayx s(x–t)n–c–1 Γ(c)Γ(n–c)F⎝parenleftBig –a,n–b,n–c;1–t x⎝parenrightBig f(t)dt⎝bracerightBig , where 0 < c<nandn=1 ,2 , ... If the right-hand side of the equation is differentiable sufﬁciently many times and the conditions f(s)=f/prime x(s)=···=f(n–1) x(s) = 0 are satisﬁed, then the solution of the integral equation can be written in the form y(x)=⎝integraldisplayx s(x–t)n–c–1 Γ(c)Γ(n–c)F⎝parenleftBig –a,–b,n–c;1–t x⎝parenrightBig f(n) t(t)dt. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 136.⎝integraldisplay ⎝integraldisplayx 0(x–t)–(ν+1)/2Dν⎝parenleftbig⎝parenleftbig λ√ x–t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). HereDν(z) is the parabolic cylinder function (see Supplement 11.12-1) and –1 < Re ν<1 . Solution: y(x)=1 π⎝integraldisplayx 0(x–t)(ν–1)/2eλ2t/4Dν⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenleftbiggd dt+λ2 2⎝parenrightbigg⎝parenleftBig e–λ2t/4f(t)⎝parenrightBig dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 464). 1.9. Equations Whose Kernels Contain Arbitrary Functions 1.9-1. Equations with Degenerate Kernel: K(x,t)=g1(x)h1(t)+g2(x)h2(t). 1.⎝integraldisplay ⎝integraldisplayx ag(x)h(t)y(t)dt=f(x). Solution: y=1 h(x)d dx⎝bracketleftbiggf(x) g(x)⎝bracketrightbigg =1 g(x)h(x)f/prime x(x)–g/prime x(x) g2(x)h(x)f(x). 112 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 2.⎝integraldisplay ⎝integraldisplayx a[g(x)–g(t)]y(t)dt=f(x). It is assumed that f(a)=f/prime x(a)=0a n d f/prime x/g/prime x≠const. Solution: y(x)=d dx⎝bracketleftbiggf/prime x(x) g/primex(x)⎝bracketrightbigg . 3.⎝integraldisplay ⎝integraldisplayx a[g(x)–g(t)+b]y(t)dt=f(x). Forb= 0, see equation 1.9.2. Assume that b≠0. Differentiation with respect to xyields an equation of the form 2.9.2: y(x)+1 bg/prime x(x)⎝integraldisplayx ay(t)dt=1 bf/prime x(x). Solution: y(x)=1 bf/prime x(x)–1 b2g/prime x(x)⎝integraldisplayx aexp⎝bracketleftBigg(t)–g(x) b⎝bracketrightBig f/prime t(t)dt. 4.⎝integraldisplay ⎝integraldisplayx a[Ag(x)+Bg(t)]y(t)dt=f(x). ForB=–A, see equation 1.9.2. Assume that B≠–A. Solution with B≠–A: y(x)=signg(x) A+Bd dx⎝braceleftbigg⎝vextendsingle⎝vextendsingleg(x)⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingleg(t)⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . 5.⎝integraldisplay ⎝integraldisplayx a[Ag(x)+Bg(t)+C]y(t)dt=f(x). ForB=–A, see equation 1.9.3. Assume that B≠–Aand (A +B)g(x)+C>0 . Solution: y(x)=d dx⎝braceleftbigg⎝vextendsingle⎝vextendsingle(A+B)g(x)+C⎝vextendsingle⎝vextendsingle–A A+B⎝integraldisplayx a⎝vextendsingle⎝vextendsingle(A+B)g(t)+C⎝vextendsingle⎝vextendsingle–B A+Bf/prime t(t)dt⎝bracerightbigg . 6.⎝integraldisplay ⎝integraldisplayx a[g(x)+h(t)]y(t)dt=f(x). Solution: y(x)=d dx⎝bracketleftbiggΦ(x) g(x)+h(x)⎝integraldisplayx af/prime t(t)dt Φ(t)⎝bracketrightbigg ,Φ(x)=e x p⎝bracketleftbigg⎝integraldisplayx ah/prime t(t)dt g(t)+h(t)⎝bracketrightbigg . 7.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(x)+(x–t)h(x)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=g(x)+xh(x),h1(t)=1 , g2(x)=h(x), andh2(t)=–t. Solution: y(x)=d dx⎝braceleftbigg Φ(x)h(x) g(x)⎝integraldisplayx a⎝bracketleftbiggf(t) h(t)⎝bracketrightbigg/prime tdt Φ(t)⎝bracerightbigg ,Φ(t)=e x p⎝bracketleftbigg –⎝integraldisplayx ah(t) g(t)dt⎝bracketrightbigg . 1.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 113 8.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(t)+(x–t)h(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=x,h1(t)=h(t),g2(x)=1 ,a n d h2(t)= g(t)–th(t). 9.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(x)+(Axλ+Btµ)h(x)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=g(x)+Axλh(x),h1(t)=1 ,g 2(x)=h(x), andh2(t)=Btµ. 10.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(t)+(Axλ+Btµ)h(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=Axλ,h1(t)=h(t),g2(x)=1 ,a n d h2(t)=g(t)+Btµh(t). 11.⎝integraldisplay ⎝integraldisplayx a[g(x)h(t)–h(x)g(t)]y(t)dt=f(x), f(a)=f/prime x(a)=0 . Forg= const or h= const, see equation 1.9.2. Solution: y(x)=1 hd dx⎝bracketleftbigg(f/h)/prime x (g/h)/primex⎝bracketrightbigg ,w h e r e f=f(x),g=g(x),h=h(x). HereAf+Bg+Ch/ ≡0, with A,B,a n dCbeing some constants. 12.⎝integraldisplay ⎝integraldisplayx a[Ag(x)h(t)+Bg(t)h(x)]y(t)dt=f(x). ForB=–A, see equation 1.9.11. Solution with B≠–A: y(x)=1 (A+B)h(x)d dx⎝braceleftBigg⎝bracketleftbiggh(x) g(x)⎝bracketrightbiggA A+B⎝integraldisplayx a⎝bracketleftbiggh(t) g(t)⎝bracketrightbiggB A+Bd dt⎝bracketleftbiggf(t) h(t)⎝bracketrightbigg dt⎝bracerightBigg . 13.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig 1+[g(t)–g(x)]h(x)⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=1– g(x)h(x),h1(t)=1 , g2(x)=h(x), andh2(t)=g(t). Solution: y(x)=d dx⎝braceleftbigg h(x)Φ(x)⎝integraldisplayx a⎝bracketleftbiggf(t) h(t)⎝bracketrightbigg/prime tdt Φ(t)⎝bracerightbigg ,Φ(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag/prime t(t)h(t)dt⎝bracketrightbigg . 14.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig e–λ(x–t)+⎝bracketleftbig⎝bracketleftbig eλxg(t)–eλtg(x)⎝bracketrightbig⎝bracketrightbig h(x)⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 1.9.15 with g1(x)=eλxh(x),h1(t)=g(t),g2(x)=e–λx– g(x)h(x), and h2(t)=eλt. 114 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 15.⎝integraldisplay ⎝integraldisplayx a[g1(x)h1(t)+g2(x)h2(t)]y(t)dt=f(x). Forg2/g1= const or h2/h1= const, see equation 1.9.1. 1◦. Solution with g1(x)h1(x)+g2(x)h2(x)/ ≡0a n df (x)/ ≡constg2(x): y(x)=1 h1(x)d dx⎝braceleftbiggg2(x)h1(x)Φ(x) g1(x)h1(x)+g2(x)h2(x)⎝integraldisplayx a⎝bracketleftbiggf(t) g2(t)⎝bracketrightbigg/prime tdt Φ(t)⎝bracerightbigg ,( 1 ) where Φ(x)=e x p⎝braceleftbigg⎝integraldisplayx a⎝bracketleftbiggh2(t) h1(t)⎝bracketrightbigg/prime tg2(t)h1(t)dt g1(t)h1(t)+g2(t)h2(t)⎝bracerightbigg .( 2) Iff(x)≡constg2(x), the solution is given by formulas (1) and (2) in which the subscript 1 must be changed by 2 and vice versa. 2◦. Solution with g1(x)h1(x)+g2(x)h2(x)≡0: y(x)=1 h1d dx⎝bracketleftbigg(f/g 2)/prime x (g1/g2)/primex⎝bracketrightbigg =–1 h1d dx⎝bracketleftbigg(f/g 2)/prime x (h2/h1)/primex⎝bracketrightbigg , where f=f(x),g2=g2(x),h1=h1(x), and h2=h2(x). 1.9-2. Equations with Difference Kernel: K(x,t)=K(x–t). 16.⎝integraldisplay ⎝integraldisplayx aK(x–t)y(t)dt=f(x). 1◦.L e tK(0) = 1 and f(a) = 0. Differentiating the equation with respect to xyields a V olterra equation of the second kind: y(x)+⎝integraldisplayx aK/prime x(x–t)y(t)dt=f/prime x(x). The solution of this equation can be represented in the form y(x)=f/prime x(x)+⎝integraldisplayx aR(x–t)f/prime t(t)dt. Here the resolvent R(x) is related to the kernel K(x) of the original equation by R(x)=L–1⎝bracketleftbigg1 p˜K(p)–1⎝bracketrightbigg , ˜K(p)=L⎝bracketleftbig K(x)⎝bracketrightbig , where LandL–1are the operators of the direct and inverse Laplace transforms, respectively. ˜K(p)=L⎝bracketleftbig K(x)⎝bracketrightbig =⎝integraldisplay∞ 0e–pxK(x)dx,R(x)=L–1⎝bracketleftbig˜R(p)⎝bracketrightbig =1 2πi⎝integraldisplayc+i∞ c–i∞epx˜R(p)dp. 2◦.L e tK(x) have an integrable power-law singularity at x= 0. Denote by w=w(x)t h e solution of the simpler auxiliary equation (compared with the original equation) with a=0 and constant right-hand side f≡1,⎝integraldisplayx 0K(x–t)w(t)dt=1 . ( 1 ) Then the solution of the original integral equation with arbitrary right-hand side is expressed in terms of was follows: y(x)=d dx⎝integraldisplayx aw(x–t)f(t)dt=f(a)w(x–a)+⎝integraldisplayx aw(x–t)f/prime t(t)dt.( 2 ) Remark. The integral equation and its solution (2) form the Sonine transform pair . References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 426), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 1.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 115 17.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=f(x). Solution: y(x)=–d dx⎝integraldisplay∞ xH(t–x)f(t)dt, where ⎝integraldisplayx 0K(t)H(x–t)dt=1 . 18.⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Axn,n= 0 ,1 ,2 , ... This is a special case of equation 1.9.20 with λ=0 . 1◦. Solution with n=0 : y(x)=A B,B=⎝integraldisplay∞ 0K(z)dz. 2◦. Solution with n=1 : y(x)=A Bx+AC B2,B=⎝integraldisplay∞ 0K(z)dz,C=⎝integraldisplay∞ 0zK(z)dz. 3◦. Solution with n=2 : y2(x)=A Bx2+2AC B2x+2AC2 B3–AD B2, B=⎝integraldisplay∞ 0K(z)dz,C=⎝integraldisplay∞ 0zK(z)dz,D=⎝integraldisplay∞ 0z2K(z)dz. 4◦. Solution with n=3 ,4 , ...is given by: yn(x)=A⎝braceleftbigg∂n ∂λn⎝bracketleftBigeλx B(λ)⎝bracketrightBig⎝bracerightbigg λ=0,B(λ)=⎝integraldisplay∞ 0K(z)e–λzdz. 19.⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Aeλx. Solution: y(x)=A Beλx,B=⎝integraldisplay∞ 0K(z)e–λzdz=L{K(z),λ}. 20.⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Axneλx,n=1 , 2 , ... 1◦. Solution with n=1 : y1(x)=A Bxeλx+AC B2eλx, B=⎝integraldisplay∞ 0K(z)e–λzdz,C=⎝integraldisplay∞ 0zK(z)e–λzdz. It is convenient to calculate the coefﬁcients BandCusing tables of Laplace transforms according to the formulas B=L{K(z),λ}andC=L{zK(z),λ}. 116 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 2◦. Solution with n=2 : y2(x)=A Bx2eλx+2AC B2xeλx+⎝parenleftbigg 2AC2 B3–AD B2⎝parenrightbigg eλx, B=⎝integraldisplay∞ 0K(z)e–λzdz,C=⎝integraldisplay∞ 0zK(z)e–λzdz,D=⎝integraldisplay∞ 0z2K(z)e–λzdz. 3◦. Solution with n=3 ,4 , ...is given by: yn(x)=∂ ∂λyn–1(x)=A∂n ∂λn⎝bracketleftbiggeλx B(λ)⎝bracketrightbigg ,B(λ)=⎝integraldisplay∞ 0K(z)e–λzdz. 21.⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Acosh(λx ). Solution: y(x)=A 2B–eλx+A 2B+e–λx=1 2⎝parenleftBigA B–+A B+⎝parenrightBig cosh(λx)+1 2⎝parenleftBigA B––A B+⎝parenrightBig sinh(λx), B–=⎝integraldisplay∞ 0K(z)e–λzdz,B+=⎝integraldisplay∞ 0K(z)eλzdz. 22.⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Asinh(λx). Solution: y(x)=A 2B–eλx–A 2B+e–λx=1 2⎝parenleftBigA B––A B+⎝parenrightBig cosh(λx)+1 2⎝parenleftBigA B–+A B+⎝parenrightBig sinh(λx), B–=⎝integraldisplay∞ 0K(z)e–λzdz,B+=⎝integraldisplay∞ 0K(z)eλzdz. 23.⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Acos(λx ). Solution: y(x)=A B2c+B2s⎝bracketleftbig Bccos(λx)– Bssin(λx)⎝bracketrightbig , Bc=⎝integraldisplay∞ 0K(z)c o s (λz)dz,Bs=⎝integraldisplay∞ 0K(z)s i n (λz)dz. 24.⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Asin(λx). Solution: y(x)=A B2c+B2s⎝bracketleftbig Bcsin(λx)+Bscos(λx)⎝bracketrightbig , Bc=⎝integraldisplay∞ 0K(z)c o s (λz)dz,Bs=⎝integraldisplay∞ 0K(z)s i n (λz)dz. 1.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 117 25.⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Aeµxcos(λx ). Solution: y(x)=A B2c+B2seµx⎝bracketleftbig Bccos(λx)– Bssin(λx)⎝bracketrightbig , Bc=⎝integraldisplay∞ 0K(z)e–µzcos(λz )dz,Bs=⎝integraldisplay∞ 0K(z)e–µzsin(λz)dz. 26.⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Aeµxsin(λx). Solution: y(x)=A B2c+B2seµx⎝bracketleftbig Bcsin(λx)+Bscos(λx)⎝bracketrightbig , Bc=⎝integraldisplay∞ 0K(z)e–µzcos(λz )dz,Bs=⎝integraldisplay∞ 0K(z)e–µzsin(λz)dz. 27.⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=f(x). 1◦. For a polynomial right-hand side of the equation, f(x)=n⎝summationtext k=0Akxk, the solution has the form y(x)=n⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermined coefﬁcients. The solution can also be obtained by the formula given in 1.9.18 (item 4◦). 2◦.F o rf(x)=eλxn⎝summationtext k=0Akxk, the solution has the form y(x)=eλxn⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermined coefﬁcients. The solution can also be obtained by the formula given in 1.9.20 (item 3◦). 3◦.F o rf(x)=n⎝summationtext k=0Akexp(λkx), the solution has the form y(x)=n⎝summationdisplay k=0Ak Bkexp(λkx), Bk=⎝integraldisplay∞ 0K(z)e x p ( – λkz)dz. 4◦.F o rf(x)=c o s ( λx)n⎝summationtext k=0Akxk, the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 118 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 5◦.F o rf(x)=s i n ( λx)n⎝summationtext k=0Akxk, the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 6◦.F o rf(x)=n⎝summationtext k=0Akcos(λ kx), the solution has the form y(x)=n⎝summationdisplay k=0Ak B2 ck+B2 sk⎝bracketleftbig Bckcos(λ kx)–Bsksin(λkx)⎝bracketrightbig , Bck=⎝integraldisplay∞ 0K(z)c o s (λkz)dz,Bsk=⎝integraldisplay∞ 0K(z)s i n (λkz)dz. 7◦.F o rf(x)=n⎝summationtext k=0Aksin(λkx), the solution has the form y(x)=n⎝summationdisplay k=0Ak B2 ck+B2 sk⎝bracketleftbig Bcksin(λkx)+Bskcos(λ kx)⎝bracketrightbig , Bck=⎝integraldisplay∞ 0K(z)c o s (λkz)dz,Bsk=⎝integraldisplay∞ 0K(z)s i n (λkz)dz. 28.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=f(x). Solution: y(x)=–d dx⎝integraldisplay∞ xH(t–x)f(t)dt, where ⎝integraldisplayx 0K(t)H(x–t)dt=1 . References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 426), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 29.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Axn,n= 0 ,1 ,2 , ... This is a special case of equation 1.9.31 with λ=0 . 1◦. Solution with n=0 : y(x)=A B,B=⎝integraldisplay∞ 0K(–z)dz. 2◦. Solution with n=1 : y(x)=A Bx–AC B2,B=⎝integraldisplay∞ 0K(–z)dz,C=⎝integraldisplay∞ 0zK(–z)dz. 3◦. Solution with n=2 : y2(x)=A Bx2–2AC B2x+2AC2 B3–AD B2, B=⎝integraldisplay∞ 0K(–z)dz,C=⎝integraldisplay∞ 0zK(–z)dz,D=⎝integraldisplay∞ 0z2K(–z)dz. 4◦. Solution with n=3 ,4 , ...is given by yn(x)=A⎝braceleftbigg∂n ∂λn⎝bracketleftbiggeλx B(λ)⎝bracketrightbigg⎝bracerightbigg λ=0,B(λ)=⎝integraldisplay∞ 0K(–z)eλzdz. 1.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 119 30.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Aeλx. Solution: y(x)=A Beλx,B=⎝integraldisplay∞ 0K(–z)eλzdz. The expression for Bis the Laplace transform of the function K(–z) with parameter p=–λand can be calculated with the aid of tables of Laplace transforms given (e.g., see Supplement 5). 31.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Axneλx,n=1 , 2 , ... 1◦. Solution with n=1 : y1(x)=A Bxeλx–AC B2eλx, B=⎝integraldisplay∞ 0K(–z)eλzdz,C=⎝integraldisplay∞ 0zK(–z)eλzdz. It is convenient to calculate the coefﬁcients BandCusing tables of Laplace transforms with parameter p=–λ. 2◦. Solution with n=2 : y2(x)=A Bx2eλx–2AC B2xeλx+⎝parenleftbigg 2AC2 B3–AD B2⎝parenrightbigg eλx, B=⎝integraldisplay∞ 0K(–z)eλzdz,C=⎝integraldisplay∞ 0zK(–z)eλzdz,D=⎝integraldisplay∞ 0z2K(–z)eλzdz. 3◦. Solution with n=3 ,4 , ...is given by: yn(x)=∂ ∂λyn–1(x)=A∂n ∂λn⎝bracketleftbiggeλx B(λ)⎝bracketrightbigg ,B(λ)=⎝integraldisplay∞ 0K(–z)eλzdz. 32.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Acosh(λx ). Solution: y(x)=A 2B+eλx+A 2B–e–λx=1 2⎝parenleftBigA B++A B–⎝parenrightBig cosh(λx)+1 2⎝parenleftBigA B+–A B–⎝parenrightBig sinh(λx), B+=⎝integraldisplay∞ 0K(–z)eλzdz,B–=⎝integraldisplay∞ 0K(–z)e–λzdz. 33.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Asinh(λx). Solution: y(x)=A 2B+eλx–A 2B–e–λx=1 2⎝parenleftBigA B+–A B–⎝parenrightBig cosh(λx)+1 2⎝parenleftBigA B++A B–⎝parenrightBig sinh(λx), B+=⎝integraldisplay∞ 0K(–z)eλzdz,B–=⎝integraldisplay∞ 0K(–z)e–λzdz. 120 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 34.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Acos(λx ). Solution: y(x)=A B2c+B2s⎝bracketleftbig Bccos(λx)+ Bssin(λx)⎝bracketrightbig , Bc=⎝integraldisplay∞ 0K(–z)c o s (λz)dz,Bs=⎝integraldisplay∞ 0K(–z)s i n (λz)dz. 35.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Asin(λx). Solution: y(x)=A B2c+B2s⎝bracketleftbig Bcsin(λx)–Bscos(λx)⎝bracketrightbig , Bc=⎝integraldisplay∞ 0K(–z)c o s (λz)dz,Bs=⎝integraldisplay∞ 0K(–z)s i n (λz)dz. 36.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Aeµxcos(λx ). Solution: y(x)=A B2c+B2seµx⎝bracketleftbig Bccos(λx)+ Bssin(λx)⎝bracketrightbig , Bc=⎝integraldisplay∞ 0K(–z)eµzcos(λz )dz,Bs=⎝integraldisplay∞ 0K(–z)eµzsin(λz)dz. 37.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Aeµxsin(λx). Solution: y(x)=A B2c+B2seµx⎝bracketleftbig Bcsin(λx)–Bscos(λx)⎝bracketrightbig , Bc=⎝integraldisplay∞ 0K(–z)eµzcos(λz )dz,Bs=⎝integraldisplay∞ 0K(–z)eµzsin(λz)dz. 38.⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=f(x). 1◦. For a polynomial right-hand side of the equation, f(x)=n⎝summationtext k=0Akxk, the solution has the form y(x)=n⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermined coefﬁcients. The solution can also be obtained by the formula given in 1.9.29 (item 4◦). 1.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 121 2◦.F o rf(x)=eλxn⎝summationtext k=0Akxk, the solution has the form y(x)=eλxn⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermined coefﬁcients. The solution can also be obtained by the formula given in 1.9.31 (item 3◦). 3◦.F o rf(x)=n⎝summationtext k=0Akexp(λkx), the solution has the form y(x)=n⎝summationdisplay k=0Ak Bkexp(λkx), Bk=⎝integraldisplay∞ 0K(–z)e x p (λkz)dz. 4◦.F o rf(x)=c o s ( λx)n⎝summationtext k=0Akxk, the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 5◦.F o rf(x)=s i n ( λx)n⎝summationtext k=0Akxk, the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 6◦.F o rf(x)=n⎝summationtext k=0Akcos(λ kx), the solution has the form y(x)=n⎝summationdisplay k=0Ak B2 ck+B2 sk⎝bracketleftbig Bckcos(λ kx)+Bsksin(λkx)⎝bracketrightbig , Bck=⎝integraldisplay∞ 0K(–z)c o s (λkz)dz,Bsk=⎝integraldisplay∞ 0K(–z)s i n (λkz)dz. 7◦.F o rf(x)=n⎝summationtext k=0Aksin(λkx), the solution has the form y(x)=n⎝summationdisplay k=0Ak B2 ck+B2 sk⎝bracketleftbig Bcksin(λkx)–Bskcos(λ kx)⎝bracketrightbig , Bck=⎝integraldisplay∞ 0K(–z)c o s (λkz)dz,Bsk=⎝integraldisplay∞ 0K(–z)s i n (λkz)dz. 8◦. For arbitrary right-hand side f=f(x), the solution of the integral equation can be calculated by the formula y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞˜f(p) ˜k(–p)epxdp, ˜f(p)=⎝integraldisplay∞ 0f(x)e–pxdx, ˜k(–p)=⎝integraldisplay∞ 0K(–z)epzdz. To calculate ˜f(p)a n d ˜k(–p), it is convenient to use tables of Laplace transforms, and to determine y(x), tables of inverse Laplace transforms. 122 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 1.9-3. Other Equations. 39.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(x)–g(t)⎝bracketrightbig⎝bracketrightbigny(t)dt=f(x), n=1 , 2 , ... The right-hand side of the equation is assumed to satisfy the conditions f(a)=f/prime x(a)=···= f(n) x(a)=0 . Solution: y(x)=1 n!g/prime x(x)⎝parenleftbigg1 g/primex(x)d dx⎝parenrightbiggn+1 f(x). 40.⎝integraldisplay ⎝integraldisplayx a⎝radicalbig g(x)–g(t)y(t)dt=f(x), f(a)=0 . Solution: y(x)=2 πg/prime x(x)⎝parenleftbigg1 g/primex(x)d dx⎝parenrightbigg2⎝integraldisplayx af(t)g/prime t(t)dt √ g(x)–g(t). 41.⎝integraldisplay ⎝integraldisplayx ay(t)dt √ g(x)–g(t)=f(x), g/prime x>0 . Solution: y(x)=1 πd dx⎝integraldisplayx af(t)g/prime t(t)dt √ g(x)–g(t). 42.⎝integraldisplay ⎝integraldisplayx aeλ(x–t)y(t)dt √ g(x)–g(t)=f(x), g/prime x>0 . Solution: y(x)=1 πeλxd dx⎝integraldisplayx ae–λtf(t)g/prime t(t) √ g(x)–g(t)dt. 43.⎝integraldisplay ⎝integraldisplayx a[g(x)–g(t)]λy(t)dt=f(x), f(a)=0 , 0< λ<1 . Solution: y(x)=kg/prime x(x)⎝parenleftbigg1 g/primex(x)d dx⎝parenrightbigg2⎝integraldisplayx ag/prime t(t)f(t)dt [g(x)–g(t)]λ,k=sin(πλ) πλ. 44.⎝integraldisplay ⎝integraldisplayx ah(t)y(t)dt [g(x)–g(t)]λ=f(x), g/prime x>0 , 0< λ<1 . Solution: y(x)=sin(πλ) πh(x)d dx⎝integraldisplayx af(t)g/prime t(t)dt [g(x)–g(t)]1–λ. 45.⎝integraldisplay ⎝integraldisplayx 0K⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=Axλ+Bxµ. Solution: y(x)=A Iλxλ–1+B Iµxµ–1,Iλ=⎝integraldisplay1 0K(z)zλ–1dz,Iµ=⎝integraldisplay1 0K(z)zµ–1dz. 1.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 123 46.⎝integraldisplay ⎝integraldisplayx 0K⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=Pn(x), Pn(x)=xλn⎝summationdisplay m=0Amxm. Solution: y(x)=xλn⎝summationdisplay m=0Am Imxm–1,Im=⎝integraldisplay1 0K(z)zλ+m–1dz. The integral I0is supposed to converge. 47.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig g1(x)⎝bracketleftbig⎝bracketleftbig h1(t)–h1(x)⎝bracketrightbig⎝bracketrightbig +g2(x)⎝bracketleftbig⎝bracketleftbig h2(t)–h2(x)⎝bracketrightbig⎝bracketrightbig⎝bracerightbig⎝bracerightbig y(t)dt=f(x). This is a special case of equation 1.9.52 with g3(x)=–g1(x)h1(x)–g2(x)h2(x)a n dh3(t)=1 . The substitution Y(x)=⎝integraldisplayx ay(t)dtfollowed by integration by parts leads to an integral equation of the form 1.9.15: ⎝integraldisplayx a⎝braceleftBig g1(x)⎝bracketleftbig h1(t)⎝bracketrightbig/prime t+g2(x)⎝bracketleftbig h2(t)⎝bracketrightbig/prime t⎝bracerightBig Y(t)dt=–f(x). 48.⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig g1(x)⎝bracketleftbig⎝bracketleftbig h1(t)–eλ(x–t)h1(x)⎝bracketrightbig⎝bracketrightbig +g2(x)⎝bracketleftbig⎝bracketleftbig h2(t)–eλ(x–t)h2(x)⎝bracketrightbig⎝bracketrightbig⎝bracerightbig⎝bracerightbig y(t)dt=f(x). This is a special case of equation 1.9.52 with g3(x)=–eλx⎝bracketleftbig g1(x)h1(x)+g2(x)h2(x)⎝bracketrightbig ,a n d h3(t)=e–λt. The substitution Y(x)=⎝integraldisplayx ae–λty(t)dtfollowed by integration by parts leads to an integral equation of the form 1.9.15: ⎝integraldisplayx a⎝braceleftBig g1(x)⎝bracketleftbig eλth1(t)⎝bracketrightbig/prime t+g2(x)⎝bracketleftbig eλth2(t)⎝bracketrightbig/prime t⎝bracerightBig Y(t)dt=–f(x). 49.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Agλ(x)gµ(t)+Bgλ+β(x)gµ–β(t)–(A+B)gλ+γ(x)gµ–γ(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.52 with g1(x)=Agλ(x),h1(t)=gµ(t),g2(x)=Bgλ+β(x), h2(t)=gµ–β(t),g3(x)=– (A+B)gλ+γ(x), and h3(t)=gµ–γ(t). 50.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Agλ(x)h(x)gµ(t)+Bgλ+β(x)h(x)gµ–β(t) –(A+B)gλ+γ(x)gµ–γ(t)h(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.52 with g1(x)=Agλ(x)h(x),h1(t)=gµ(t),g2(x)= Bgλ+β(x)h(x),h2(t)=gµ–β(t),g3(x)=– (A+B)gλ+γ(x), and h3(t)=gµ–γ(t)h(t). 51.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Agλ(x)h(x)gµ(t)+Bgλ+β(x)h(t)gµ–β(t) –(A+B)gλ+γ(x)gµ–γ(t)h(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 1.9.52 with g1(x)=Agλ(x)h(x),h1(t)=gµ(t),g2(x)= Bgλ+β(x),h2(t)=gµ–β(t)h(t),g3(x)=– (A+B)gλ+γ(x), and h3(t)=gµ–γ(t)h(t). 124 LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 52.⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g1(x)h1(t)+g2(x)h2(t)+g3(x)h3(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), where g1(x)h1(x)+g2(x)h2(x)+g3(x)h3(x)≡0. The substitution Y(x)=⎝integraldisplayx ah3(t)y(t)dtfollowed by integration by parts leads to an integral equation of the form 1.9.15: ⎝integraldisplayx a⎝braceleftbigg g1(x)⎝bracketleftbiggh1(t) h3(t)⎝bracketrightbigg/prime t+g2(x)⎝bracketleftbiggh2(t) h3(t)⎝bracketrightbigg/prime t⎝bracerightbigg Y(t)dt=–f(x). 53.⎝integraldisplay ⎝integraldisplayx –∞Q(x–t)eαty(ξ)dt=Aepx,ξ=eβtg(x–t). Solution: y(ξ)=A qξp–α β,q=⎝integraldisplay∞ 0Q(z)[g(z)]p–α βe–pzdz. 1.10. Some Formulas and Transformations 1.Let the solution of the integral equation ⎝integraldisplayx aK(x,t)y(t)dt=f(x)( 1) have the form y(x)=F⎝bracketleftbig f(x)⎝bracketrightbig ,( 2) whereFis some linear integro-differential operator. Then the solution of the more complicated integral equation⎝integraldisplayx aK(x,t)g(x)h(t)y(t)dt=f(x)( 3 ) has the form y(x)=1 h(x)F⎝bracketleftBigf(x) g(x)⎝bracketrightBig .( 4) Below are formulas for the solutions of integral equations of the form (3) for some speciﬁc functions g(x)a n dh(t). In all cases, it is assumed that the solution of equation (1) is known and is determined by formula (2). (a) The solution of the equation ⎝integraldisplayx aK(x,t)(x/t )λy(t)dt=f(x) has the form y(x)=xλF⎝bracketleftbig x–λf(x)⎝bracketrightbig . (b) The solution of the equation ⎝integraldisplayx aK(x,t)eλ(x–t)y(t)dt=f(x) has the form y(x)=eλxF⎝bracketleftbig e–λxf(x)⎝bracketrightbig . 1.10. S OME FORMULAS AND TRANSFORMATIONS 125 2.Let the solution of the integral equation (1) have the form y(x)=L1⎝parenleftBig x,d dx⎝parenrightBig f(x)+L2⎝parenleftBig x,d dx⎝parenrightBig⎝integraldisplayx aR(x,t)f(t)dt,( 5) where L1andL2are some linear differential operators. The solution of the more complicated integral equation ⎝integraldisplayx aK⎝parenleftbig ϕ(x),ϕ(t)⎝parenrightbig y(t)dt=f(x), (6) where ϕ(x) is an arbitrary monotone function (differentiable sufﬁciently many times, ϕ/prime x>0 ) ,i s determined by the formula y(x)=ϕ/prime x(x)L1⎝parenleftbigg ϕ(x),1 ϕ/primex(x)d dx⎝parenrightbigg f(x) +ϕ/prime x(x)L2⎝parenleftbigg ϕ(x),1 ϕ/primex(x)d dx⎝parenrightbigg⎝integraldisplayx aR⎝parenleftbig ϕ(x),ϕ(t)⎝parenrightbig ϕ/prime t(t)f(t)dt.(7) Below are formulas for the solutions of integral equations of the form (6) for some speciﬁc functions ϕ(x). In all cases, it is assumed that the solution of equation (1) is known and is determined by formula (5). (a) For ϕ(x)=xλ, y(x)=λxλ–1L1⎝parenleftbigg xλ,1 λxλ–1d dx⎝parenrightbigg f(x)+λ2xλ–1L2⎝parenleftbigg xλ,1 λxλ–1d dx⎝parenrightbigg⎝integraldisplayx aR⎝parenleftbig xλ,tλ⎝parenrightbig tλ–1f(t)dt. (b) For ϕ(x)=eλx, y(x)=λeλxL1⎝parenleftbigg eλx,1 λeλxd dx⎝parenrightbigg f(x)+λ2eλxL2⎝parenleftbigg eλx,1 λeλxd dx⎝parenrightbigg⎝integraldisplayx aR⎝parenleftbig eλx,eλt⎝parenrightbig eλtf(t)dt. (c) For ϕ(x)=l n ( λx), y(x)=1 xL1⎝parenleftbigg ln(λx),xd dx⎝parenrightbigg f(x)+1 xL2⎝parenleftbigg ln(λx),xd dx⎝parenrightbigg⎝integraldisplayx a1 tR⎝parenleftbig ln(λx), ln( λt)⎝parenrightbig f(t)dt. (d) For ϕ(x)=c o s ( λx), y(x)=–λsin(λx)L1⎝parenleftbigg cos(λx),–1 λsin(λx)d dx⎝parenrightbigg f(x) +λ2sin(λx)L2⎝parenleftbigg cos(λx),–1 λsin(λx)d dx⎝parenrightbigg⎝integraldisplayx aR⎝parenleftbig cos(λx), cos(λt )⎝parenrightbig sin(λt)f(t)dt. (e) For ϕ(x)=s i n ( λx), y(x)=λcos(λx) L1⎝parenleftbigg sin(λx),1 λcos(λx)d dx⎝parenrightbigg f(x) +λ2cos(λx) L2⎝parenleftbigg sin(λx),1 λcos(λx)d dx⎝parenrightbigg⎝integraldisplayx aR⎝parenleftbig sin(λx), sin( λt)⎝parenrightbig cos(λt )f(t)dt. Chapter 2 Linear Equations of the Second Kind with Variable Limit of Integration /trianglerightsld Notation: f=f(x),g=g(x),h=h(x),K=K(x), andM=M(x)are arbitrary functions (these may be composite functions of the argument depending on two variables xandt);A,B,C,D,a, b,c,α,β,γ,λ, andµare free parameters; and mandnare nonnegative integers. 2.1. Equations Whose Kernels Contain Power-Law Functions 2.1-1. Kernels Linear in the Arguments xandt. 1. y(x)–λ⎝integraldisplay ⎝integraldisplayx ay(t)dt=f(x). Solution: y(x)=f(x)+λ⎝integraldisplayx aeλ(x–t)f(t)dt. 2. y(x)+λx⎝integraldisplay ⎝integraldisplayx ay(t)dt=f(x). Solution: y(x)=f(x)–λ⎝integraldisplayx axexp⎝bracketleftbig1 2λ(t2–x2)⎝bracketrightbig f(t)dt. 3. y(x)+λ⎝integraldisplay ⎝integraldisplayx aty(t)dt=f(x). Solution: y(x)=f(x)–λ⎝integraldisplayx atexp⎝bracketleftbig1 2λ(t2–x2)⎝bracketrightbig f(t)dt. 4. y(x)+λ⎝integraldisplay ⎝integraldisplayx a(x–t)y(t)dt=f(x). This is a special case of equation 2.1.34 with n=1 . 1◦. Solution with λ>0 : y(x)=f(x)–k⎝integraldisplayx asin[k(x–t)]f(t)dt,k=√ λ. 127 128 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2◦. Solution with λ<0 : y(x)=f(x)+k⎝integraldisplayx asinh[k(x–t)]f(t)dt,k=√ –λ. 5. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig A+B(x–t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). 1◦. Solution with A2>4B: y(x)=f(x)–⎝integraldisplayx aR(x–t)f(t)dt, R(x)=e x p⎝parenleftbig –1 2Ax⎝parenrightbig⎝bracketleftbigg Acosh(βx )+2B–A2 2βsinh(βx)⎝bracketrightbigg ,β=⎝radicalBig 1 4A2–B. 2◦. Solution with A2<4B: y(x)=f(x)–⎝integraldisplayx aR(x–t)f(t)dt, R(x)=e x p⎝parenleftbig –1 2Ax⎝parenrightbig⎝bracketleftbigg Acos(βx )+2B–A2 2βsin(βx)⎝bracketrightbigg ,β=⎝radicalBig B–1 4A2. 3◦. Solution with A2=4B: y(x)=f(x)–⎝integraldisplayx aR(x–t)f(t)dt,R(x)=e x p⎝parenleftbig –1 2Ax⎝parenrightbig⎝parenleftbig A–1 4A2x⎝parenrightbig . 6. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ax +Bt +C⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.6 with g(x)=–Axandh(t)=–Bt–C.F o rB=–Asee equation 2.1.5. By differentiation followed by the substitution Y(x)=⎝integraldisplayx ay(t)dt, the original equation can be reduced to the second-order linear ordinary differential equation Y/prime/prime xx–⎝bracketleftbig (A+B)x+C⎝bracketrightbig Y/prime x–AY=f/prime x(x)( 1) under the initial conditions Y(a)=0 , Y/prime x(a)=f(a). (2) A fundamental system of solutions of the homogeneous equation (1) with f≡0h a st h e form Y1(x)=Φ⎝parenleftbig α,1 2;kz2⎝parenrightbig ,Y2(x)=Ψ⎝parenleftbig α,1 2;kz2⎝parenrightbig , α=A 2(A+B),k=A+B 2,z=x+C A+B, whereΦ⎝parenleftbig α,β;x⎝parenrightbig andΨ⎝parenleftbig α,β;x⎝parenrightbig are degenerate hypergeometric functions. Solving the homogeneous equation (1) under conditions (2) for an arbitrary function f=f(x) and taking into account the relation y(x)=Y/prime x(x), we thus obtain the solution of the integral equation in the form y(x)=f(x)–⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=∂2 ∂x∂t⎝bracketleftbiggY1(x)Y2(t)–Y2(x)Y1(t) W(t)⎝bracketrightbigg ,W(t)=2√ πk Γ(α)exp⎝bracketleftbigg k⎝parenleftBig t+C A+B⎝parenrightBig2⎝bracketrightbigg . 2.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 129 2.1-2. Kernels Quadratic in the Arguments xandt. 7. y(x)+A⎝integraldisplay ⎝integraldisplayx ax2y(t)dt=f(x). This is a special case of equation 2.1.50 with λ=2a n d µ=0 . Solution: y(x)=f(x)–A⎝integraldisplayx ax2exp⎝bracketleftbig1 3A(t3–x3)⎝bracketrightbig f(t)dt. 8. y(x)+A⎝integraldisplay ⎝integraldisplayx axty (t)dt=f(x). This is a special case of equation 2.1.50 with λ=1a n d µ=1 . Solution: y(x)=f(x)–A⎝integraldisplayx axtexp⎝bracketleftbig1 3A(t3–x3)⎝bracketrightbig f(t)dt. 9. y(x)+A⎝integraldisplay ⎝integraldisplayx at2y(t)dt=f(x). This is a special case of equation 2.1.50 with λ=0a n d µ=2 . Solution: y(x)=f(x)–A⎝integraldisplayx at2exp⎝bracketleftbig1 3A(t3–x3)⎝bracketrightbig f(t)dt. 10. y(x)+λ⎝integraldisplay ⎝integraldisplayx a(x–t)2y(t)dt=f(x). This is a special case of equation 2.1.34 with n=2 . Solution: y(x)=f(x)–⎝integraldisplayx aR(x–t)f(t)dt, R(x)=2 3ke–2kx–2 3kekx⎝bracketleftBig cos⎝parenleftbig√ 3kx⎝parenrightbig –√ 3s i n⎝parenleftbig√ 3kx⎝parenrightbig⎝bracketrightBig ,k=⎝parenleftbig1 4λ⎝parenrightbig1/3. 11. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x2–t2)y(t)dt=f(x). This is a special case of equation 2.9.5 with g(x)=Ax2. Solution: y(x)=f(x)+1 W⎝integraldisplayx a⎝bracketleftbig u/prime 1(x)u/prime 2(t)–u/prime 2(x)u/prime 1(t)⎝bracketrightbig f(t)dt, where the primes denote differentiation with respect to the argument speciﬁed in the parenthe- ses;u1(x),u2(x) is a fundamental system of solutions of the second-order linear homogeneous ordinary differential equation u/prime/prime xx+2Axu = 0; and the functions u1(x)a n du2(x)a r ee x - pressed in terms of Bessel functions or modiﬁed Bessel functions, depending on the sign of the parameter A: ForA>0 , W=3/π,u1(x)=√ xJ1/3⎝parenleftBig⎝radicalBig 8 9Ax3/2⎝parenrightBig ,u2(x)=√ xY 1/3⎝parenleftBig⎝radicalBig 8 9Ax3/2⎝parenrightBig . ForA<0 , W=–3 2,u1(x)=√ xI1/3⎝parenleftBig⎝radicalBig 8 9|A|x3/2⎝parenrightBig ,u2(x)=√ xK 1/3⎝parenleftBig⎝radicalBig 8 9|A|x3/2⎝parenrightBig . 130 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 12. y(x)+A⎝integraldisplay ⎝integraldisplayx a(xt–t2)y(t)dt=f(x). This is a special case of equation 2.9.4 with g(t)=At. Solution: y(x)=f(x)+A W⎝integraldisplayx at⎝bracketleftbig y1(x)y2(t)–y2(x)y1(t)⎝bracketrightbig f(t)dt, where y1(x),y2(x) is a fundamental system of solutions of the second-order linear homo- geneous ordinary differential equation y/prime/prime xx+Axy = 0; the functions y1(x)a n d y2(x)a r e expressed in terms of Bessel functions or modiﬁed Bessel functions, depending on the sign of the parameter A: ForA>0 , W=3/π,y1(x)=√ xJ1/3⎝parenleftbig2 3√ Ax3/2⎝parenrightbig ,y2(x)=√ xY 1/3⎝parenleftbig2 3√ Ax3/2⎝parenrightbig . ForA<0 , W=–3 2,y1(x)=√ xI1/3⎝parenleftbig2 3⎝radicalbig |A|x3/2⎝parenrightbig ,y2(x)=√ xK 1/3⎝parenleftbig2 3⎝radicalbig |A|x3/2⎝parenrightbig . 13. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x2–xt)y(t)dt=f(x). This is a special case of equation 2.9.3 with g(x)=Ax. Solution: y(x)=f(x)+A W⎝integraldisplayx ax⎝bracketleftbig y1(x)y2(t)–y2(x)y1(t)⎝bracketrightbig f(t)dt, where y1(x),y2(x) is a fundamental system of solutions of the second-order linear homo- geneous ordinary differential equation y/prime/prime xx+Axy = 0; the functions y1(x)a n d y2(x)a r e expressed in terms of Bessel functions or modiﬁed Bessel functions, depending on the sign of the parameter A: ForA>0 , W=3/π,y1(x)=√ xJ1/3⎝parenleftbig2 3√ Ax3/2⎝parenrightbig ,y2(x)=√ xY 1/3⎝parenleftbig2 3√ Ax3/2⎝parenrightbig . ForA<0 , W=–3 2,y1(x)=√ xI1/3⎝parenleftbig2 3⎝radicalbig |A|x3/2⎝parenrightbig ,y2(x)=√ xK 1/3⎝parenleftbig2 3⎝radicalbig |A|x3/2⎝parenrightbig . 14. y(x)+A⎝integraldisplay ⎝integraldisplayx a(t2–3x2)y(t)dt=f(x). This is a special case of equation 2.1.55 with λ=1a n d µ=2 . 15. y(x)+A⎝integraldisplay ⎝integraldisplayx a(2xt–3x2)y(t)dt=f(x). This is a special case of equation 2.1.55 with λ=2a n d µ=1 . 16. y(x)–⎝integraldisplay ⎝integraldisplayx a(ABxt –ABx2+Ax +B)y(t)dt=f(x). This is a special case of equation 2.9.16 with g(x)=Axandh(x)=B. Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=(Ax+B)e x p⎝bracketleftbig1 2A(x2–t2)⎝bracketrightbig +B2⎝integraldisplayx texp⎝bracketleftbig1 2A(s2–t2)+B(x–s)⎝bracketrightbig ds. 2.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 131 17. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ax2–At2+Bx –Ct +D⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.6 with g(x)=Ax2+Bx+Dandh(t)=–At2–Ct. Solution: y(x)=f(x)+⎝integraldisplayx a∂2 ∂x∂t⎝bracketleftbiggY1(x)Y2(t)–Y2(x)Y1(t) W(t)⎝bracketrightbigg f(t)dt. HereY1(x),Y2(x) is a fundamental system of solutions of the second-order homogeneous ordinary differential equation Y/prime/prime xx+⎝bracketleftbig (B–C)x+D⎝bracketrightbig Y/prime x+( 2Ax+B)Y= 0 (see A. D. Polyanin and V . F. Zaitsev (2003) for details about this equation): Y1(x)=e x p ( – kx)Φ⎝parenleftbig α,1 2;1 2(C–B)z2⎝parenrightbig ,Y2(x)=e x p ( – kx)Ψ⎝parenleftbig α,1 2;1 2(C–B)z2⎝parenrightbig , W(x)=–√ 2π(C–B) Γ(α)exp⎝bracketleftbig1 2(C–B)z2–2kx⎝bracketrightbig ,k=2A B–C, α=–4A2+2AD(C–B)+B(C–B)2 2(C–B)3,z=x–4A+(C–B)D (C–B)2, where Φ⎝parenleftbig α,β;x⎝parenrightbig andΨ⎝parenleftbig α,β;x⎝parenrightbig are degenerate hypergeometric functions and Γ(α)i st h e gamma function. 18. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Ax +B+(Cx +D)(x –t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.11 with g(x)=Ax+Bandh(x)=Cx+D. Solution with A≠0: y(x)=f(x)+⎝integraldisplayx a⎝bracketleftbig Y/prime/prime 2(x)Y1(t)–Y/prime/prime 1(x)Y2(t)⎝bracketrightbigf(t) W(t)dt. HereY1(x),Y2(x) is a fundamental system of solutions of the second-order homogeneous ordinary differential equation Y/prime/prime xx–(Ax+B)Y/prime x–(Cx+D)Y= 0 (see A. D. Polyanin and V . F. Zaitsev (2003) for details about this equation): Y1(x)=e x p ( – kx)Φ⎝parenleftbig α,1 2;1 2Az2⎝parenrightbig ,Y2(x)=e x p ( – kx)Ψ⎝parenleftbig α,1 2;1 2Az2⎝parenrightbig , W(x)=–√ 2πA⎝bracketleftbig Γ(α)⎝bracketrightbig–1exp⎝parenleftbig1 2Az2–2kx⎝parenrightbig ,k=C/A , α=1 2(A2D–ABC –C2)A–3,z=x+(AB+2C)A–2, whereΦ⎝parenleftbig α,β;x⎝parenrightbig andΨ⎝parenleftbig α,β;x⎝parenrightbig are degenerate hypergeometric functions, Γ(α) is the gamma function. 19. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig At+B+(Ct +D)(t–x)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.12 with g(t)=–At–Bandh(t)=–Ct–D. Solution with A≠0: y(x)=f(x)–⎝integraldisplayx a⎝bracketleftbig Y1(x)Y/prime/prime 2(t)–Y/prime/prime 1(t)Y2(x)⎝bracketrightbigf(t) W(x)dt. HereY1(x),Y2(x) is a fundamental system of solutions of the second-order homogeneous ordinary differential equation Y/prime/prime xx–(Ax+B)Y/prime x–(Cx+D)Y= 0 (see A. D. Polyanin and V . F. Zaitsev (2003) for details about this equation): Y1(x)=e x p ( – kx)Φ⎝parenleftbig α,1 2;1 2Az2⎝parenrightbig ,Y2(x)=e x p ( – kx)Ψ⎝parenleftbig α,1 2;1 2Az2⎝parenrightbig , W(x)=–√ 2πA⎝bracketleftbig Γ(α)⎝bracketrightbig–1exp⎝parenleftbig1 2Az2–2kx⎝parenrightbig ,k=C/A , α=1 2(A2D–ABC –C2)A–3,z=x+(AB+2C)A–2, where Φ⎝parenleftbig α,β;x⎝parenrightbig andΨ⎝parenleftbig α,β;x⎝parenrightbig are degenerate hypergeometric functions and Γ(α)i st h e gamma function. 132 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2.1-3. Kernels Cubic in the Arguments xandt. 20. y(x)+A⎝integraldisplay ⎝integraldisplayx ax3y(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx ax3exp⎝bracketleftbig1 4A(t4–x4)⎝bracketrightbig f(t)dt. 21. y(x)+A⎝integraldisplay ⎝integraldisplayx ax2ty(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx ax2texp⎝bracketleftbig1 4A(t4–x4)⎝bracketrightbig f(t)dt. 22. y(x)+A⎝integraldisplay ⎝integraldisplayx axt2y(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx axt2exp⎝bracketleftbig1 4A(t4–x4)⎝bracketrightbig f(t)dt. 23. y(x)+A⎝integraldisplay ⎝integraldisplayx at3y(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx at3exp⎝bracketleftbig1 4A(t4–x4)⎝bracketrightbig f(t)dt. 24. y(x)+λ⎝integraldisplay ⎝integraldisplayx a(x–t)3y(t)dt=f(x). This is a special case of equation 2.1.34 with n=3 . Solution: y(x)=f(x)–⎝integraldisplayx aR(x–t)f(t)dt, where R(x)=⎝braceleftBigg k⎝bracketleftbig cosh(kx )s i n (kx)–s i n h ( kx)c o s (kx)⎝bracketrightbig ,k=⎝parenleftbig3 2λ⎝parenrightbig1/4forλ>0 , 1 2s⎝bracketleftbig sin(sx)–s i n h ( sx)⎝bracketrightbig ,s=( – 6λ)1/4forλ<0 . 25. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x3–t3)y(t)dt=f(x). This is a special case of equation 2.1.52 with λ=3 . 26. y(x)–A⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig 4x3–t3⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.1.55 with λ=1a n d µ=3 . 27. y(x)+A⎝integraldisplay ⎝integraldisplayx a(xt2–t3)y(t)dt=f(x). This is a special case of equation 2.1.49 with λ=2 . 2.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 133 28. y(x)+A⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig x2t–t3⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The transformation z=x2,τ=t2,y(x)=w(z) leads to an equation of the form 2.1.4: w(z)+1 2A⎝integraldisplayz a2(z–τ)w(τ)dτ=F(z), F(z)=f(x). 29. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Ax2t+Bt3⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The transformation z=x2,τ=t2,y(x)=w(z) leads to an equation of the form 2.1.6: w(z)+⎝integraldisplayz a2⎝parenleftbig1 2Az+1 2Bτ⎝parenrightbig w(τ)dτ=F(z), F(z)=f(x). 30. y(x)+B⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig 2x3–xt2⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.1.55 with λ=2 ,µ=2 ,a n d B=– 2A. 31. y(x)–A⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig 4x3–3x2t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.1.55 with λ=3a n d µ=1 . 32. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig ABx3–ABx2t–Ax2–B⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with g(x)=Ax2andλ=B. Solution: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x,t)=(Ax2+B)e x p⎝bracketleftbig1 3A(x3–t3)⎝bracketrightbig +B2⎝integraldisplayx texp⎝bracketleftbig1 3A(s3–t3)+B(x–s)⎝bracketrightbig ds. 33. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig ABxt2–ABt3+At2+B⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with g(t)=At2andλ=B. Solution: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x,t)=– ( At2+B)e x p⎝bracketleftbig1 3A(t3–x3)⎝bracketrightbig +B2⎝integraldisplayx texp⎝bracketleftbig1 3A(s3–x3)+B(t–s)⎝bracketrightbig ds. 2.1-4. Kernels Containing Higher-Order Polynomials in xandt. 34. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)ny(t)dt=f(x), n=1 , 2 , ... 1◦. Differentiating the equation n+ 1 times with respect to xyields an (n + 1)st-order linear ordinary differential equation with constant coefﬁcients for y=y(x): y(n+1) x +An!y=f(n+1) x(x). This equation under the initial conditions y(a)=f(a),y/prime x(a)=f/prime x(a),...,y(n) x(a)=f(n) x(a) determines the solution of the original integral equation. 134 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2◦. Solution: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x)=1 n+1n⎝summationdisplay k=0exp(σkx)⎝bracketleftbig σkcos(β kx)–βksin(βkx)⎝bracketrightbig , where the coefﬁcients σkandβkare given by σk=|An!|1 n+1cos⎝parenleftBig2πk n+1⎝parenrightBig ,βk=|An!|1 n+1sin⎝parenleftBig2πk n+1⎝parenrightBig forA<0 , σk=|An!|1 n+1cos⎝parenleftBig2πk+π n+1⎝parenrightBig ,βk=|An!|1 n+1sin⎝parenleftBig2πk+π n+1⎝parenrightBig forA>0 . 35. y(x)+A⎝integraldisplay ⎝integraldisplay∞ x(t–x)ny(t)dt=f(x), n=1 , 2 , ... The Picard–Goursat equation. This is a special case of equation 2.9.62 with K(z)=A(–z)n. 1◦. A solution of the homogeneous equation (f ≡0) is y(x)=Ce–λx,λ=⎝parenleftbig –An!⎝parenrightbig1 n+1, where Cis an arbitrary constant and A< 0. This is a unique solution for n=0 ,1 ,2 ,3 . The general solution of the homogeneous equation for any sign of Ahas the form y(x)=s⎝summationdisplay k=1Ckexp(–λkx). (1) HereCkare arbitrary constants and λkare the roots of the algebraic equation λn+1+An!=0 that satisfy the condition Re λk> 0. The number of terms in (1) is determined by the inequality s≤2⎝bracketleftbign 4⎝bracketrightbig +1 ,w h e r e[ a] stands for the integral part of a number a. For more details about the solution of the homogeneous Picard–Goursat equation, see Subsection 11.11-1 (Example 1). 2◦.F o rf(x)=m⎝summationtext k=1akexp(–βkx), where βk> 0, a solution of the equation has the form y(x)=m⎝summationdisplay k=1akβn+1 k βn+1 k+An!exp(–βkx), (2) where βn+1 k+An!≠0. For A> 0, this formula can also be used for arbitrary f(x) expandable into a convergent exponential series (which corresponds to m=∞). 3◦.F o rf(x)=e–βxm⎝summationtext k=1akxk,w h e r e β> 0, a solution of the equation has the form y(x)=e–βxm⎝summationdisplay k=0Bkxk,( 3) where the constants Bkare found by the method of undetermined coefﬁcients. The solution can also be constructed using the formulas given in item 3◦, equation 2.9.55. 2.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 135 4◦.F o rf(x)=c o s ( βx)m⎝summationtext k=1akexp(–µkx), a solution of the equation has the form y(x)=c o s ( βx)m⎝summationdisplay k=1Bkexp(–µkx)+s i n ( βx)m⎝summationdisplay k=1Ckexp(–µkx), (4) where the constants BkandCkare found by the method of undetermined coefﬁcients. The solution can also be constructed using the formulas given in 2.9.60. 5◦.F o rf(x)=s i n ( βx)m⎝summationtext k=1akexp(–µkx), a solution of the equation has the form y(x)=c o s ( βx)m⎝summationdisplay k=1Bkexp(–µkx)+s i n ( βx)m⎝summationdisplay k=1Ckexp(–µkx), (5) where the constants BkandCkare found by the method of undetermined coefﬁcients. The solution can also be constructed using the formulas given in 2.9.61. 6◦. To obtain the general solution in item 2◦–5◦, the solution (1) of the homogeneousequation must be added to each right-hand side of (2)–(5). 36. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)tny(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 2.1.49 with λ=n. 37. y(x)+A⎝integraldisplay ⎝integraldisplayx a(xn–tn)y(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 2.1.52 with λ=n. 38. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig ABxn+1–ABxnt–Axn–B⎝parenrightbig⎝parenrightbig y(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 2.9.7 with g(x)=Axnandλ=B. Solution: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x,t)=(Axn+B)e x p⎝bracketleftbiggA n+1⎝parenleftbig xn+1–tn+1⎝parenrightbig⎝bracketrightbigg +B2⎝integraldisplayx texp⎝bracketleftbiggA n+1⎝parenleftbig sn+1–tn+1⎝parenrightbig +B(x–s)⎝bracketrightbigg ds. 39. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig ABxtn–ABtn+1+Atn+B⎝parenrightbig⎝parenrightbig y(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 2.9.8 with g(t)=Atnandλ=B. Solution: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x,t)=– (Atn+B)e x p⎝bracketleftbiggA n+1⎝parenleftbig tn+1–xn+1⎝parenrightbig⎝bracketrightbigg +B2⎝integraldisplayx texp⎝bracketleftbiggA n+1⎝parenleftbig sn+1–xn+1⎝parenrightbig +B(t–s)⎝bracketrightbigg ds. 136 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2.1-5. Kernels Containing Rational Functions. 40. y(x)+x–3⎝integraldisplay ⎝integraldisplayx at⎝bracketleftbig⎝bracketleftbig 2Ax +( 1– A)t⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This equation can be obtained by differentiating the equation ⎝integraldisplayx a⎝bracketleftbig Ax2t+( 1– A)xt2⎝bracketrightbig y(t)dt=F(x), F(x)=⎝integraldisplayx at3f(t)dt, which has the form 1.1.17: Solution: y(x)=1 xd dx⎝bracketleftbigg x–A⎝integraldisplayx atA–1ϕ/prime t(t)dt⎝bracketrightbigg ,ϕ(x)=1 x⎝integraldisplayx at3f(t)dt. 41. y(x)–λ⎝integraldisplay ⎝integraldisplayx 0y(t)dt x+t=f(x). Dixon’s equation. This is a special case of equation 2.1.62 with a=b=1a n d µ=0 . 1◦. The solution of the homogeneous equation ( f≡0) is y(x)=Cxβ(β> –1, λ> 0). (1) HereCis an arbitrary constant, and β=β(λ) is determined by the transcendental equation λI(β)=1 , w h e r e I(β)=⎝integraldisplay1 0zβdz 1+z.( 2) 2◦. For a polynomial right-hand side, f(x)=N⎝summationdisplay n=0Anxn the solution bounded at zero is given by y(x)=⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩N⎝summationdisplay n=0An 1–(λ/λn)xnforλ<λ0, N⎝summationdisplay n=0An 1–(λ/λn)xn+Cxβforλ>λ0andλ≠λn, λn=1 I(n),I(n) = (–1)n⎝bracketleftbigg ln 2 +n⎝summationdisplay m=1(–1)m m⎝bracketrightbigg , where Cis an arbitrary constant,and β=β(λ) is determined by the transcendental equation (2). For special λ=λn(n=1 ,2 , ...), the solution differs in one term and has the form y(x)=n–1⎝summationdisplay m=0Am 1–(λn/λm)xm+N⎝summationdisplay m=n+1Am 1–(λn/λm)xm–An¯λn λnxnlnx+Cxn, where ¯λn= (–1)n+1⎝bracketleftbiggπ2 12+n⎝summationtext k=1(–1)k k2⎝bracketrightbigg–1 . 2.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 137 Remark. For arbitrary f(x), expandable into power series, the formulas of item 2◦can be used, in which one should set N=∞. In this case, the radius of convergence of the solution y(x) is equal to the radius of convergence of f(x). 3◦. For logarithmic-polynomial right-hand side, f(x)=l nx⎝parenleftbiggN⎝summationdisplay n=0Anxn⎝parenrightbigg , the solution with logarithmic singularity at zero is given by y(x)=⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩lnx N⎝summationdisplay n=0An 1–(λ/λn)xn+N⎝summationdisplay n=0AnDnλ [ 1–(λ/λn)]2xnforλ<λ0, lnxN⎝summationdisplay n=0An 1–(λ/λn)xn+N⎝summationdisplay n=0AnDnλ [ 1–(λ/λn)]2xn+Cxβforλ>λ0andλ≠λn, λn=1 I(n),I(n) = (–1)n⎝bracketleftbigg ln 2 +n⎝summationdisplay k=1(–1)k k⎝bracketrightbigg ,Dn= (–1)n+1⎝bracketleftbiggπ2 12+n⎝summationdisplay k=1(–1)k k2⎝bracketrightbigg . 4◦. For arbitrary f(x), the transformation x=1 2e2z,t=1 2e2τ,y(x)=e–zw(z),f(x)=e–zg(z) leads to an integral equation with difference kernel of the form 2.9.51: w(z)–λ⎝integraldisplayz –∞w(τ)dτ cosh(z –τ)=g(z). 42. y(x)–λ⎝integraldisplay ⎝integraldisplayx ax+b t+by(t)dt=f(x). This is a special case of equation 2.9.1 with g(x)=x+b. Solution: y(x)=f(x)+λ⎝integraldisplayx ax+b t+beλ(x–t)f(t)dt. 43. y(x)=2 (1 –λ2)x2⎝integraldisplay ⎝integraldisplayx λxt 1+ty(t)dt. This equation is encountered in nuclear physics and describes deceleration of neutrons in matter. 1◦. Solution with λ=0 : y(x)=C (1 +x)2, where Cis an arbitrary constant. 2◦.F o rλ≠0, the solution can be found in the series form y(x)=∞⎝summationdisplay n=0Anxn. Reference: I. Sneddon (1995). 138 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2.1-6. Kernels Containing Square Roots and Fractional Powers. 44. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)√ ty(t)dt=f(x). This is a special case of equation 2.1.49 with λ=1 2. 45. y(x)+A⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig√ x–√ t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.1.52 with λ=1 2. 46. y(x)+λ⎝integraldisplay ⎝integraldisplayx ay(t)dt √ x–t=f(x). Abel’s equation of the second kind. This equation is encountered in problems of heat and mass transfer. Solution: y(x)=F(x)+πλ2⎝integraldisplayx aexp[πλ2(x–t)]F(t)dt, where F(x)=f(x)–λ⎝integraldisplayx af(t)dt √ x–t. References: H. Brakhage, K. Nickel, and P. Rieder (1965), Yu. I. Babenko (1986). 47. y(x)–λ⎝integraldisplay ⎝integraldisplayx 0y(t)dt √ ax2+bt2=f(x), a>0 , b>0 . 1◦. The solution of the homogeneous equation ( f≡0) is y(x)=Cxβ(β> –1, λ> 0). (1) HereCis an arbitrary constant, and β=β(λ) is determined by the transcendental equation λI(β)=1 , w h e r e I(β)=⎝integraldisplay1 0zβdz √ a+bz2.( 2) 2◦. For a polynomial right-hand side, f(x)=N⎝summationdisplay n=0Anxn the solution bounded at zero is given by y(x)=⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩N⎝summationdisplay n=0An 1–(λ/λn)xnforλ<λ0, N⎝summationdisplay n=0An 1–(λ/λn)xn+Cxβforλ>λ0andλ≠λn, λ0=√ b Arsinh⎝parenleftbig⎝radicalbig b/a⎝parenrightbig,λn=1 I(n),I(n)=⎝integraldisplay1 0zndz √ a+bz2. HereCis an arbitrary constant, and β=β(λ) is determined by the transcendental equation (2). 2.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 139 3◦. For special λ=λn(n=1 ,2 , ...), the solution differs in one term and has the form y(x)=n–1⎝summationdisplay m=0Am 1–(λn/λm)xm+N⎝summationdisplay m=n+1Am 1–(λn/λm)xm–An¯λn λnxnlnx+Cxn, where ¯λn=⎝bracketleftbigg⎝integraldisplay1 0znlnzd z √ a+bz2⎝bracketrightbigg–1 . 4◦. For arbitrary f(x), expandable into power series, the formulas of item 2◦can be used, in which one should set N=∞. In this case, the radius of convergence of the solution y(x)i s equal to the radius of convergence of f(x). 48. y(x)+λ⎝integraldisplay ⎝integraldisplayx ay(t)dt (x–t)3/4=f(x). This equation admits solution by quadratures (see equation 2.1.60 and Example 2 in Subsection 11.4-2). 2.1-7. Kernels Containing Arbitrary Powers. 49. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)tλy(t)dt=f(x). This is a special case of equation 2.9.4 with g(t)=Atλ. Solution: y(x)=f(x)+A W⎝integraldisplayx a⎝bracketleftbig y1(x)y2(t)–y2(x)y1(t)⎝bracketrightbig tλf(t)dt, where y1(x),y2(x) is a fundamental system of solutions of the second-order linear homo- geneous ordinary differential equation y/prime/prime xx+Axλy= 0; the functions y1(x)a n dy2(x)a r e expressed in terms of Bessel functions or modiﬁed Bessel functions, depending on the signofA: ForA>0 , W=2q π,y1(x)=√ xJ 1 2q⎝parenleftbigg√ A qxq⎝parenrightbigg ,y2(x)=√ xY 1 2q⎝parenleftbigg√ A qxq⎝parenrightbigg ,q=λ+2 2, ForA<0 , W=–q,y1(x)=√ xI 1 2q⎝parenleftbigg√ |A| qxq⎝parenrightbigg ,y2(x)=√ xK 1 2q⎝parenleftbigg√ |A| qxq⎝parenrightbigg ,q=λ+2 2. 50. y(x)+A⎝integraldisplay ⎝integraldisplayx axλtµy(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Axλandh(t)=tµ(λandµare arbitrary numbers). Solution: y(x)=f(x)–⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=⎧ ⎨ ⎩Axλtµexp⎝bracketleftBigA λ+µ+1⎝parenleftbig tλ+µ+1–xλ+µ+1⎝parenrightbig⎝bracketrightBig forλ+µ+1≠0, Axλ–Atµ+Aforλ+µ+1=0 . 140 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 51. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)xλtµy(t)dt=f(x). The substitution u(x)=x–λy(x) leads to an equation of the form 2.1.49: u(x)+A⎝integraldisplayx a(x–t)tλ+µu(t)dt=f(x)x–λ. 52. y(x)+A⎝integraldisplay ⎝integraldisplayx a(xλ–tλ)y(t)dt=f(x). This is a special case of equation 2.9.5 with g(x)=Axλ. Solution: y(x)=f(x)+1 W⎝integraldisplayx a⎝bracketleftbig u/prime 1(x)u/prime 2(t)–u/prime 2(x)u/prime 1(t)⎝bracketrightbig f(t)dt, where the primes denote differentiation w ith respect to the argument speciﬁed in the paren- theses, and u1(x),u2(x) is a fundamental system of solutions of the second-order linear ho- mogeneous ordinary differential equation u/prime/prime xx+Aλxλ–1u= 0; the functions u1(x)a n du2(x) are expressed in terms of Bessel functions or modiﬁed Bessel functions, depending on the sign of A: ForAλ>0 , W=2q π,u1(x)=√ xJ 1 2q⎝parenleftbigg√ Aλ qxq⎝parenrightbigg ,u2(x)=√ xY 1 2q⎝parenleftbigg√ Aλ qxq⎝parenrightbigg ,q=λ+1 2, ForAλ<0 , W=–q,u1(x)=√ xI 1 2q⎝parenleftbigg√ |Aλ| qxq⎝parenrightbigg ,u2(x)=√ xλ K 1 2q⎝parenleftbigg√ |Aλ| qxq⎝parenrightbigg ,q=λ+1 2. 53. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλtλ–1+Bt2λ–1⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The transformation z=xλ,τ=tλ,y(x)=Y(z) leads to an equation of the form 2.1.6: Y(z)–⎝integraldisplayz b⎝parenleftbiggA λz+B λτ⎝parenrightbigg Y(τ)dτ=F(z), F(z)=f(x),b=aλ. 54. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Axλ+µtλ–µ–1+Bxµt2λ–µ–1⎝parenrightbig⎝parenrightbig y(t)dt=f(x). The substitution y(x)=xµw(x) leads to an equation of the form 2.1.53: w(x)–⎝integraldisplayx a⎝parenleftbig Axλtλ–1+Bt2λ–1⎝parenrightbig w(t)dt=x–µf(x). 55. y(x)+A⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig λxλ–1tµ–(λ+µ)xλ+µ–1⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This equation can be obtained by differentiating equation 1.1.52: ⎝integraldisplayx a⎝bracketleftbig 1+A(xλtµ–xλ+µ)⎝bracketrightbig y(t)dt=F(x), F(x)=⎝integraldisplayx af(x)dx. Solution: y(x)=d dx⎝braceleftbiggxλ Φ(x)⎝integraldisplayx a⎝bracketleftbig t–λF(t)⎝bracketrightbig/prime tΦ(t)dt⎝bracerightbigg ,Φ(x)=e x p⎝parenleftbigg –Aµ µ+λxµ+λ⎝parenrightbigg . 2.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 141 56. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig ABxλ+1–ABxλt–Axλ–B⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.7. Solution: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x,t)=(Axλ+B)e x p⎝bracketleftbiggA λ+1⎝parenleftbig xλ+1–tλ+1⎝parenrightbig⎝bracketrightbigg +B2⎝integraldisplayx texp⎝bracketleftbiggA λ+1⎝parenleftbig sλ+1–tλ+1⎝parenrightbig +B(x–s)⎝bracketrightbigg ds. 57. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig ABxtλ–ABtλ+1+Atλ+B⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.8. Solution: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x,t)=– (Atλ+B)e x p⎝bracketleftbiggA λ+1⎝parenleftbig tλ+1–xλ+1⎝parenrightbig⎝bracketrightbigg +B2⎝integraldisplayx texp⎝bracketleftbiggA λ+1⎝parenleftbig sλ+1–xλ+1⎝parenrightbig +B(t–s)⎝bracketrightbigg ds. 58. y(x)–λ⎝integraldisplay ⎝integraldisplayx a⎝parenleftBig ⎝parenleftBigx+b t+b⎝parenrightBig ⎝parenrightBigµ y(t)dt=f(x). This is a special case of equation 2.9.1 with g(x)=(x+b)µ. Solution: y(x)=f(x)+λ⎝integraldisplayx a⎝parenleftBigx+b t+b⎝parenrightBigµ eλ(x–t)f(t)dt. 59. y(x)–λ⎝integraldisplay ⎝integraldisplayx axµ+b tµ+by(t)dt=f(x). This is a special case of equation 2.9.1 with g(x)=xµ+b. Solution: y(x)=f(x)+λ⎝integraldisplayx axµ+b tµ+beλ(x–t)f(t)dt. 60. y(x)–λ⎝integraldisplay ⎝integraldisplayx 0y(t)dt (x–t)α=f(x), 0 < α<1 . Generalized Abel equation of the second kind. 1◦. Assume that the number αcan be represented in the form α=1–m n,w h e r e m=1 ,2 , ...,n=2 ,3 , ... (m<n). In this case, the solution of the generalized Abel equation of the second kind can be written in closed form (in quadratures): y(x)=f(x)+⎝integraldisplayx 0R(x–t)f(t)dt, 142 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION where R(x)=n–1⎝summationdisplay ν=1λνΓν(m/n ) Γ(νm/n )x(νm/n )–1+b mm–1⎝summationdisplay µ=0εµexp⎝parenleftbig εµbx⎝parenrightbig +b mn–1⎝summationdisplay ν=1λνΓν(m/n ) Γ(νm/n )⎝bracketleftbiggm–1⎝summationdisplay µ=0εµexp⎝parenleftbig εµbx⎝parenrightbig⎝integraldisplayx 0t(νm/n )–1exp⎝parenleftbig –εµbt⎝parenrightbig dt⎝bracketrightbigg , b=λn/mΓn/m(m/n ),εµ=e x p⎝parenleftBig2πµi m⎝parenrightBig ,i2= –1, µ=0 ,1 , ...,m–1 . 2◦. Solution with any αfrom 0 < α<1 : y(x)=f(x)+⎝integraldisplayx 0R(x–t)f(t)dt,w h e r e R(x)=∞⎝summationdisplay n=1⎝bracketleftbig λΓ(1 –α)x1–α⎝bracketrightbign xΓ⎝bracketleftbig n(1 –α)⎝bracketrightbig. References: H. Brakhage, K. Nickel, and P. Rieder (1965), V . I. Smirnov (1974). 61. y(x)–λ xα⎝integraldisplay ⎝integraldisplayx 0y(t)dt (x–t)1–α=f(x), 0 < α≤1. 1◦. The solution of the homogeneous equation ( f≡0) is y(x)=Cxβ(β> –1, λ> 0). (1) HereCis an arbitrary constant, and β=β(λ) is determined by the transcendental equation λB(α,β+1 )=1 , ( 2 ) where B(p,q)=⎝integraltext1 0zp–1(1 –z)q–1dzis the beta function. 2◦. For a polynomial right-hand side, f(x)=N⎝summationdisplay n=0Anxn the solution bounded at zero is given by y(x)=⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩N⎝summationdisplay n=0An 1–(λ/λn)xnforλ<α, N⎝summationdisplay n=0An 1–(λ/λn)xn+Cxβforλ>αandλ≠λn, λn=(α)n+1 n!,( α)n+1=α(α+1 )...(α+n). HereCis an arbitrary constant, and β=β(λ) is determined by the transcendental equation (2). For special λ=λn(n=1 ,2 , ...), the solution differs in one term and has the form y(x)=n–1⎝summationdisplay m=0Am 1–(λn/λm)xm+N⎝summationdisplay m=n+1Am 1–(λn/λm)xm–An¯λn λnxnlnx+Cxn, where ¯λn=⎝bracketleftbigg⎝integraldisplay1 0(1 –z)α–1znlnzd z⎝bracketrightbigg–1 . 2.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 143 3◦. For arbitrary f(x), expandable into power series, the formulas of item 2◦can be used, in which one should set N=∞. In this case, the radius of convergence of the solution y(x)i s equal to the radius of convergence of f(x). 4◦.F o r f(x)=l n ( kx)N⎝summationdisplay n=0Anxn, a solution has the form y(x)=l n ( kx)N⎝summationdisplay n=0Bnxn+N⎝summationdisplay n=0Dnxn, where the constants BnandDnare found by the method of undetermined coefﬁcients. To obtain the general solution we must add the solution (1) of the homogeneous equation. In Mikhailov (1966), solvability conditions for the integral equation in question were investigated for various classes of f(x). 62. y(x)–λ xµ⎝integraldisplay ⎝integraldisplayx 0y(t)dt (ax+bt)1–µ=f(x). Herea>0 ,b>0 ,a n d µis an arbitrary number. 1◦. The solution of the homogeneous equation ( f≡0) is y(x)=Cxβ(β> –1, λ> 0). (1) HereCis an arbitrary constant, and β=β(λ) is determined by the transcendental equation λI(β)=1 , w h e r e I(β)=⎝integraldisplay1 0zβ(a+bz)µ–1dz.( 2) 2◦. For a polynomial right-hand side, f(x)=N⎝summationdisplay n=0Anxn the solution bounded at zero is given by y(x)=⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩N⎝summationdisplay n=0An 1–(λ/λn)xnforλ<λ0, N⎝summationdisplay n=0An 1–(λ/λn)xn+Cxβforλ>λ0andλ≠λn, λn=1 I(n),I(n)=⎝integraldisplay1 0zn(a+bz)µ–1dz. HereCis an arbitrary constant, and β=β(λ) is determined by the transcendental equation (2). 3◦. For special λ=λn(n=1 ,2 , ...), the solution differs in one term and has the form y(x)=n–1⎝summationdisplay m=0Am 1–(λn/λm)xm+N⎝summationdisplay m=n+1Am 1–(λn/λm)xm–An¯λn λnxnlnx+Cxn, where ¯λn=⎝bracketleftbigg⎝integraldisplay1 0zn(a+bz)µ–1lnzd z⎝bracketrightbigg–1 . 4◦. For arbitrary f(x) expandable into power series, the formulas of item 2◦can be used, in which one should set N=∞. In this case, the radius of convergence of the solution y(x)i s equal to the radius of convergence of f(x). 144 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2.2. Equations Whose Kernels Contain Exponential Functions 2.2-1. Kernels Containing Exponential Functions. 1. y(x)+A⎝integraldisplay ⎝integraldisplayx aeλ(x–t)y(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx ae(λ–A)(x–t)f(t)dt. 2. y(x)+A⎝integraldisplay ⎝integraldisplayx aeλx+βty(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Aeλxandh(t)=eβt.F o rβ=–λ,s e e equation 2.2.1. Solution: y(x)=f(x)–⎝integraldisplayx aR(x,t)f(t)dt,R(x,t)=Aeλx+βtexp⎝braceleftbiggA λ+β⎝bracketleftbig e(λ+β)t–e(λ+β)x⎝bracketrightbig⎝bracerightbigg . 3. y(x)+A⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig eλ(x–t)–1⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). 1◦. Solution with D≡λ(λ–4A)>0 : y(x)=f(x)–2Aλ √ D⎝integraldisplayx aR(x–t)f(t)dt,R(x)=e x p⎝parenleftbig1 2λx⎝parenrightbig sinh⎝parenleftbig1 2√ Dx⎝parenrightbig . 2◦. Solution with D≡λ(λ–4A)<0 : y(x)=f(x)–2Aλ √ |D|⎝integraldisplayx aR(x–t)f(t)dt,R(x)=e x p⎝parenleftbig1 2λx⎝parenrightbig sin⎝parenleftbig1 2⎝radicalbig |D|x⎝parenrightbig . 3◦. Solution with λ=4A: y(x)=f(x)–4A2⎝integraldisplayx a(x–t)e x p⎝bracketleftbig 2A(x–t)⎝bracketrightbig f(t)dt. 4. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeλ(x–t)+B⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.2.10 with A1=A,A2=B,λ1=λ,a n dλ2=0 . 1◦. The structure of the solution depends on the sign of the discriminant D≡(A–B–λ)2+4AB (1) of the square equation µ2+(A+B–λ)µ–Bλ=0 . ( 2 ) 2◦.I fD> 0, then equation (2) has the real different roots µ1=1 2(λ–A–B)+1 2√ D,µ2=1 2(λ–A–B)–1 2√ D. 2.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 145 In this case, the original integ ral equation has the solution y(x)=f(x)+⎝integraldisplayx a⎝bracketleftbig E1eµ1(x–t)+E2eµ2(x–t)⎝bracketrightbig f(t)dt, where E1=Aµ1 µ2–µ1+Bµ1–λ µ2–µ1,E2=Aµ2 µ1–µ2+Bµ2–λ µ1–µ2. 3◦.I fD< 0, then equation (2) has the complex conjugate roots µ1=σ+iβ,µ2=σ–iβ,σ=1 2(λ–A–B),β=1 2√ –D. In this case, the original integ ral equation has the solution y(x)=f(x)+⎝integraldisplayx a⎝braceleftBig E1eσ(x–t)cos[β (x–t)] +E2eσ(x–t)sin[β(x–t)]⎝bracerightBig f(t)dt, where E1=–A–B,E2=1 β(–Aσ–Bσ+Bλ). 5. y(x)+A⎝integraldisplay ⎝integraldisplayx a(eλx–eλt)y(t)dt=f(x). This is a special case of equation 2.9.5 with g(x)=Aeλx. Solution: y(x)=f(x)+1 W⎝integraldisplayx a⎝bracketleftbig u/prime 1(x)u/prime 2(t)–u/prime 2(x)u/prime 1(t)⎝bracketrightbig f(t)dt, where the primes denote differentiation w ith respect to the argument speciﬁed in the paren- theses, and u1(x),u2(x) is a fundamental system of solutions of the second-order linear homogeneous ordinary differential equation u/prime/prime xx+Aλeλxu= 0; the functions u1(x)a n du2(x) are expressed in terms of Bessel functions or modiﬁed Bessel functions, depending on the sign of A: ForAλ>0 , W=λ π,u1(x)=J0⎝parenleftbigg2√ Aλ λeλx/ 2⎝parenrightbigg ,u2(x)=Y0⎝parenleftbigg2√ Aλ λeλx/ 2⎝parenrightbigg , ForAλ<0 , W=–λ 2,u1(x)=I0⎝parenleftbigg2√ |Aλ| λeλx/ 2⎝parenrightbigg ,u2(x)=K0⎝parenleftbigg2√ |Aλ| λeλx/ 2⎝parenrightbigg . 6. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig Aeλx+Beλt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.6 with g(x)=Aeλxandh(t)=Beλt.F o rB=–A,s e e equation 2.2.5. Differentiating the original integral equation followed by substituting Y(x)=⎝integraldisplayx ay(t)dt yields the second-order linear ordinary differential equation Y/prime/prime xx+(A+B)eλxY/prime x+AλeλxY=f/prime x(x)( 1) 146 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION under the initial conditions Y(a)=0 , Y/prime x(a)=f(a). (2) A fundamental system of solutions of the homogeneous equation (1) with f≡0h a st h e form Y1(x)=Φ⎝parenleftBigA m,1 ;–m λeλx⎝parenrightBig ,Y2(x)=Ψ⎝parenleftBigA m,1 ;–m λeλx⎝parenrightBig ,m=A+B, whereΦ⎝parenleftbig α,β;x⎝parenrightbig andΨ⎝parenleftbig α,β;x⎝parenrightbig are degenerate hypergeometric functions. Solving the homogeneous equation (1) under conditions (2) for an arbitrary function f=f(x) and taking into account the relation y(x)=Y/prime x(x), we thus obtain the solution of the integral equation in the form y(x)=f(x)–⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=Γ(A/m ) λ∂2 ∂x∂t⎝braceleftbigg exp⎝parenleftBigm λeλt⎝parenrightBig⎝bracketleftbig Y1(x)Y2(t)–Y2(x)Y1(t)⎝bracketrightbig⎝bracerightbigg . 7. y(x)+A⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig eλ(x+t)–e2λt⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). The transformation z=eλx,τ=eλtleads to an equation of the form 2.1.4. 1◦. Solution with Aλ>0 : y(x)=f(x)–λk⎝integraldisplayx aeλtsin⎝bracketleftbig k(eλx–eλt)⎝bracketrightbig f(t)dt,k=⎝radicalbig A/λ. 2◦. Solution with Aλ<0 : y(x)=f(x)+λk⎝integraldisplayx aeλtsinh⎝bracketleftbig k(eλx–eλt)⎝bracketrightbig f(t)dt,k=⎝radicalbig |A/λ|. 8. y(x)+A⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig eλx+µt–e(λ+µ)t⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). The transformation z=eµx,τ=eµt,Y(z)=y(x) leads to an equation of the form 2.1.52: Y(z)+A µ⎝integraldisplayz b(zk–τk)Y(τ)dτ=F(z), F(z)=f(x), where k=λ/µ,b=eµa. 9. y(x)+A⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig λeλx+µt–(λ+µ)e(λ+µ)x⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This equation can be obtained by differentiating an equation of the form 1.2.22: ⎝integraldisplayx a⎝bracketleftbig 1+Aeλx(eµt–eµx)⎝bracketrightbig y(t)dt=F(x), F(x)=⎝integraldisplayx af(t)dt. Solution: y(x)=d dx⎝braceleftbigg eλxΦ(x)⎝integraldisplayx a⎝bracketleftbiggF(t) eλt⎝bracketrightbigg/prime tdt Φ(t)⎝bracerightbigg ,Φ(x)=e x p⎝bracketleftbiggAµ λ+µe(λ+µ)x⎝bracketrightbigg . 2.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 147 10. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig A1eλ1(x–t)+A2eλ2(x–t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). 1◦. Introduce the notation I1=⎝integraldisplayx aeλ1(x–t)y(t)dt,I2=⎝integraldisplayx aeλ2(x–t)y(t)dt. Differentiating the integral equation twice yi elds (the ﬁrst line is the original equation) y+A1I1+A2I2=f,f=f(x), (1) y/prime x+(A1+A2)y+A1λ1I1+A2λ2I2=f/prime x,( 2) y/prime/prime xx+(A1+A2)y/prime x+(A1λ1+A2λ2)y+A1λ2 1I1+A2λ2 2I2=f/prime/prime xx.( 3 ) Eliminating I1andI2, we arrive at the second-order linear ordinary differential equation with constant coefﬁcients y/prime/prime xx+(A1+A2–λ1–λ2)y/prime x+(λ1λ2–A1λ2–A2λ1)y=f/prime/prime xx–(λ1+λ2)f/prime x+λ1λ2f.( 4 ) Substituting x=ainto (1) and (2) yields the initial conditions y(a)=f(a), y/prime x(a)=f/prime x(a)–(A1+A2)f(a). (5) Solving the differential equation (4) under conditions (5), we can ﬁnd the solution of the integral equation. 2◦. Consider the characteristic equation µ2+(A1+A2–λ1–λ2)µ+λ1λ2–A1λ2–A2λ1=0 ( 6 ) which corresponds to the homogeneousdifferential equation (4) (with f(x)≡0). The structure of the solution of the integral equation depends on the sign of the discriminant D≡(A1–A2–λ1+λ2)2+4A1A2 of the quadratic equation (6). IfD> 0, the quadratic equation (6) has the real different roots µ1=1 2(λ1+λ2–A1–A2)+1 2√ D,µ2=1 2(λ1+λ2–A1–A2)–1 2√ D. In this case, the solution of the origin al integral equation has the form y(x)=f(x)+⎝integraldisplayx a⎝bracketleftbig B1eµ1(x–t)+B2eµ2(x–t)⎝bracketrightbig f(t)dt, where B1=A1µ1–λ2 µ2–µ1+A2µ1–λ1 µ2–µ1,B2=A1µ2–λ2 µ1–µ2+A2µ2–λ1 µ1–µ2. IfD< 0, the quadratic equation (6) has the complex conjugate roots µ1=σ+iβ,µ2=σ–iβ,σ=1 2(λ1+λ2–A1–A2),β=1 2√ –D. In this case, the solution of the origin al integral equation has the form y(x)=f(x)+⎝integraldisplayx a⎝braceleftbig B1eσ(x–t)cos[β (x–t)] +B2eσ(x–t)sin[β(x–t)]⎝bracerightbig f(t)dt, where B1=–A1–A2,B2=1 β⎝bracketleftbig A1(λ2–σ)+A2(λ1–σ)⎝bracketrightbig . 148 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 11. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeλ(x+t)–Ae2λt+Beλt⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). The transformation z=eλx,τ=eλt,Y(z)=y(x) leads to an equation of the form 2.1.5: Y(z)+⎝integraldisplayz b⎝bracketleftbig B1(z–τ)+A1⎝bracketrightbig Y(τ)dτ=F(z), F(z)=f(x), where A1=B/λ ,B1=A/λ,b=eλa. 12. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aeλ(x+t)+Be2λt+Ceλt⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). The transformation z=eλx,τ=eλt,Y(z)=y(x) leads to an equation of the form 2.1.6: Y(z)–⎝integraldisplayz b(A1z+B1τ+C1)Y(τ)dτ=F(z), F(z)=f(x), where A1=–A/λ,B1=–B/λ ,C1=–C/λ,b=eλa. 13. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig λeλ(x–t)+A⎝parenleftbig⎝parenleftbig µeµx+λt–λeλx+µt⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.23 with h(t)=A. Solution: y(x)=1 eλxd dx⎝braceleftbigg Φ(x)⎝integraldisplayx a⎝bracketleftbiggF(t) eλt⎝bracketrightbigg/prime te2λt Φ(t)dt⎝bracerightbigg , Φ(x)=e x p⎝bracketleftbigg Aλ–µ λ+µe(λ+µ)x⎝bracketrightbigg ,F(x)=⎝integraldisplayx af(t)dt. 14. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig λe–λ(x–t)+A⎝parenleftbig⎝parenleftbig µeλx+µt–λeµx+λt⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.24 with h(x)=A. Assume that f(a) = 0. Solution: y(x)=⎝integraldisplayx aw(t)dt,w(x)=e–λxd dx⎝braceleftbigge2λx Φ(x)⎝integraldisplayx a⎝bracketleftbiggf(t) eλt⎝bracketrightbigg/prime tΦ(t)dt⎝bracerightbigg , Φ(x)=e x p⎝bracketleftbigg Aλ–µ λ+µe(λ+µ)x⎝bracketrightbigg . 15. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig λeλ(x–t)+Aeβt⎝parenleftbig⎝parenleftbig µeµx+λt–λeλx+µt⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.23 with h(t)=Aeβt. Solution: y(x)=e–(λ+β)xd dx⎝braceleftbigg Φ(x)⎝integraldisplayx a⎝bracketleftbiggF(t) eλt⎝bracketrightbigg/prime te(2λ+β)t Φ(t)dt⎝bracerightbigg , Φ(x)=e x p⎝bracketleftbigg Aλ–µ λ+µ+βe(λ+µ+β)x⎝bracketrightbigg ,F(x)=⎝integraldisplayx af(t)dt. 2.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 149 16. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig λe–λ(x–t)+Aeβx⎝parenleftbig⎝parenleftbig µeλx+µt–λeµx+λt⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.24 with h(x)=Aeβx. Assume that f(a) = 0. Solution: y(x)=⎝integraldisplayx aw(t)dt,w(x)=e–λxd dx⎝braceleftbigge(2λ+β)x Φ(x)⎝integraldisplayx a⎝bracketleftbiggf(t) e(λ+β)t⎝bracketrightbigg/prime tΦ(t)dt⎝bracerightbigg , Φ(x)=e x p⎝bracketleftbigg Aλ–µ λ+µ+βe(λ+µ+β)x⎝bracketrightbigg . 17. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig ABe(λ+1)x+t–ABeλx+2t–Aeλx+t–Bet⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). The transformation z=ex,τ=et,Y(z)=y(x) leads to an equation of the form 2.1.56: Y(z)+⎝integraldisplayz b⎝parenleftbig ABzλ+1–ABzλτ–Azλ–B⎝parenrightbig Y(τ)dτ=F(z), where F(z)=f(x)a n db=ea. 18. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig ABex+λt–ABe(λ+1)t+Aeλt+Bet⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). The transformation z=ex,τ=et,Y(z)=y(x) leads to an equation of the form 2.1.57 (in which λis substituted by λ–1 ) : Y(z)+⎝integraldisplayz b⎝parenleftbig ABzτλ–1–ABτλ+Aτλ–1+B⎝parenrightbig Y(τ)dτ=F(z), where F(z)=f(x)a n db=ea. 19. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1Akeλk(x–t)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). 1◦. This integral equation can be reduced to an nth-order linear nonhomogeneous ordinary differential equation with constant coefﬁcients. Set Ik(x)=⎝integraldisplayx aeλk(x–t)y(t)dt.( 1) Differentiating (1) with respect to xyields I/prime k=y(x)+λk⎝integraldisplayx aeλk(x–t)y(t)dt,( 2) where the prime stands for differentiation with respect to x. From the comparison of (1) with (2) we see that I/prime k=y(x)+λkIk,Ik=Ik(x). (3) The integral equation can be written in terms of Ik(x) as follows: y(x)+n⎝summationdisplay k=1AkIk=f(x). (4) 150 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION Differentiating (4) with respect to xand taking account of (3), we obtain y/prime x(x)+σny(x)+n⎝summationdisplay k=1AkλkIk=f/prime x(x), σn=n⎝summationdisplay k=1Ak.( 5) Eliminating the integral Infrom (4) and (5), we ﬁnd that y/prime x(x)+⎝parenleftbig σn–λn)y(x)+n–1⎝summationdisplay k=1Ak(λk–λn)Ik=f/prime x(x)–λnf(x). (6) Differentiating (6) with respect to xand eliminating In–1from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear differential operator (acting on y) with constant coefﬁcients plus the sumn–2⎝summationtext k=1A1 kIk.I f w e proceed with successively eliminating In–2,In–3,...,I1with the aid of differentiation and formula (3), then we will ﬁnally arrive at an nth-order linear nonhomogeneous ordinary differential equation with constant coefﬁcients. The initial conditions for y(x) can be obtained by setting x=ain the integral equation and all its derivative equations. 2◦. The solution of the equation can be represented in the form y(x)=f(x)+⎝integraldisplayx a⎝bracketleftbiggn⎝summationdisplay k=1Bkeµk(x–t)⎝bracketrightbigg f(t)dt.( 7) The unknown constants µkare the roots of the algebraic equation n⎝summationdisplay k=1Ak z–λk+1=0 , ( 8 ) which is reduced (by separating the numerator) to the problem of ﬁnding the roots of an nth-order characteristic polynomial. After the µkhave been calculated, the coefﬁcients Bkcan be found from the following linear system of algebraic equations: n⎝summationdisplay k=1Bk λm–µk+1=0 , m=1 ,...,n.( 9 ) Another way of determining the Bkis presented in item 3◦below. If all the roots µkof equation (8) are real and different, then the solution of the original integral equation can be calculated by formula (7). To a pair of complex conjugate roots µk,k+1=α±iβof the characteristic polynomial (8) there corresponds a pair of complex conjugate coefﬁcients Bk,k+1in equation (9). In this case, the corresponding terms Bkeµk(x–t)+Bk+1eµk+1(x–t)in solution (7) can be written in the form Bkeα(x–t)⎝bracketleftbig cosβ(x–t)⎝bracketrightbig + Bk+1eα(x–t)⎝bracketleftbig sinβ(x–t)⎝bracketrightbig ,w h e r e Bkand Bk+1are real coefﬁcients. 3◦.F o ra= 0, the solution of the original integral equation is given by y(x)=f(x)–⎝integraldisplayx 0R(x–t)f(t)dt,R(x)=L–1⎝bracketleftbig R(p)⎝bracketrightbig , (10) 2.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 151 where L–1⎝bracketleftbig R(p)⎝bracketrightbig is the inverse Laplace transform of the function R(p)= K(p) 1+ K(p), K(p)=n⎝summationdisplay k=1Ak p–λk. (11) The transform R(p)o ft h er e s o l v e n t R(x) can be represented as a regular fractional function: R(p)=Q(p) P(p),P(p)=(p–µ1)(p–µ2)...(p–µn), where Q(p) is a polynomial in pof degree < n. The roots µkof the polynomial P(p) coincide with the roots of equation (8). If all µkare real and different, then the resolvent can be determined by the formula R(x)=n⎝summationdisplay k=1Bkeµkx,Bk=Q(µk) P/prime(µk), where the prime stands for differentiation. 2.2-2. Kernels Containing Power-Law and Exponential Functions. 20. y(x)+A⎝integraldisplay ⎝integraldisplayx axeλ(x–t)y(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx axexp⎝bracketleftbig1 2A(t2–x2)+λ(x–t)⎝bracketrightbig f(t)dt. 21. y(x)+A⎝integraldisplay ⎝integraldisplayx ateλ(x–t)y(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx atexp⎝bracketleftbig1 2A(t2–x2)+λ(x–t)⎝bracketrightbig f(t)dt. 22. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)eλty(t)dt=f(x). This is a special case of equation 2.9.4 with g(t)=Aeλt. Solution: y(x)=f(x)+A W⎝integraldisplayx a⎝bracketleftbig u1(x)u2(t)–u2(x)u1(t)⎝bracketrightbig eλtf(t)dt, where u1(x),u2(x) is a fundamental system of solutions of the second-order linear homo- geneous ordinary differential equation u/prime/prime xx+Aeλxu= 0; the functions u1(x)a n du2(x)a r e expressed in terms of Bessel functions or modiﬁed Bessel functions, depending on sign A: W=λ π,u1(x)=J0⎝parenleftbigg2√ A λeλx/ 2⎝parenrightbigg ,u2(x)=Y0⎝parenleftbigg2√ A λeλx/ 2⎝parenrightbigg forA>0 , W=–λ 2,u1(x)=I0⎝parenleftbigg2√ |A| λeλx/ 2⎝parenrightbigg ,u2(x)=K0⎝parenleftbigg2√ |A| λeλx/ 2⎝parenrightbigg forA<0 . 152 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 23. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)eλ(x–t)y(t)dt=f(x). 1◦. Solution with A>0 : y(x)=f(x)–k⎝integraldisplayx aeλ(x–t)sin[k(x–t)]f(t)dt,k=√ A. 2◦. Solution with A<0 : y(x)=f(x)+k⎝integraldisplayx aeλ(x–t)sinh[k(x–t)]f(t)dt,k=√ –A. 24. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)eλx+µty(t)dt=f(x). The substitution u(x)=e–λxy(x) leads to an equation of the form 2.2.22: u(x)+A⎝integraldisplayx a(x–t)e(λ+µ)tu(t)dt=f(x)e–λx. 25. y(x)–⎝integraldisplay ⎝integraldisplayx a(Ax +Bt +C)eλ(x–t)y(t)dt=f(x). The substitution u(x)=e–λxy(x) leads to an equation of the form 2.1.6: u(x)–⎝integraldisplayx a(Ax+Bt+C)u(t)dt=f(x)e–λx. 26. y(x)+A⎝integraldisplay ⎝integraldisplayx ax2eλ(x–t)y(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx ax2exp⎝bracketleftbig1 3A(t3–x3)+λ(x–t)⎝bracketrightbig f(t)dt. 27. y(x)+A⎝integraldisplay ⎝integraldisplayx axteλ(x–t)y(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx axtexp⎝bracketleftbig1 3A(t3–x3)+λ(x–t)⎝bracketrightbig f(t)dt. 28. y(x)+A⎝integraldisplay ⎝integraldisplayx at2eλ(x–t)y(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx at2exp⎝bracketleftbig1 3A(t3–x3)+λ(x–t)⎝bracketrightbig f(t)dt. 29. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)2eλ(x–t)y(t)dt=f(x). Solution: y(x)=f(x)–⎝integraldisplayx aR(x–t)f(t)dt, R(x)=2 3ke(λ–2k)x–2 3ke(λ+k)x⎝bracketleftbig cos⎝parenleftbig√ 3kx⎝parenrightbig –√ 3s i n⎝parenleftbig√ 3kx⎝parenrightbig⎝bracketrightbig ,k=⎝parenleftbig1 4A⎝parenrightbig1/3. 2.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 153 30. y(x)+A⎝integraldisplay ⎝integraldisplayx 0(x2–t2)eλ(x–t)y(t)dt=f(x). The substitution u(x)=e–λxy(x) leads to an equation of the form 2.1.11: u(x)+A⎝integraldisplayx 0(x2–t2)u(t)dt=f(x)e–λx. 31. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)neλ(x–t)y(t)dt=f(x), n=1 , 2 , ... Solution: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x)=1 n+1eλxn⎝summationdisplay k=0exp(σkx)⎝bracketleftbig σkcos(β kx)–βksin(βkx)⎝bracketrightbig , where σk=|An!|1 n+1cos⎝parenleftBig2πk n+1⎝parenrightBig ,βk=|An!|1 n+1sin⎝parenleftBig2πk n+1⎝parenrightBig forA<0 , σk=|An!|1 n+1cos⎝parenleftBig2πk+π n+1⎝parenrightBig ,βk=|An!|1 n+1sin⎝parenleftBig2πk+π n+1⎝parenrightBig forA>0 . 32. y(x)+b⎝integraldisplay ⎝integraldisplayx aexp[λ(x–t)] √ x–ty(t)dt=f(x). Solution: y(x)=eλx⎝braceleftbigg F(x)+πb2⎝integraldisplayx aexp[πb2(x–t)]F(t)dt⎝bracerightbigg , where F(x)=e–λxf(x)–b⎝integraldisplayx ae–λtf(t) √ x–tdt. 33. y(x)+A⎝integraldisplay ⎝integraldisplayx a(x–t)tkeλ(x–t)y(t)dt=f(x). The substitution u(x)=e–λxy(x) leads to an equation of the form 2.1.49: u(x)+A⎝integraldisplayx a(x–t)tku(t)dt=f(x)e–λx. 34. y(x)+A⎝integraldisplay ⎝integraldisplayx a(xk–tk)eλ(x–t)y(t)dt=f(x). The substitution u(x)=e–λxy(x) leads to an equation of the form 2.1.52: u(x)+A⎝integraldisplayx a(xk–tk)u(t)dt=f(x)e–λx. 35. y(x)–λ⎝integraldisplay ⎝integraldisplayx 0eµ(x–t) (x–t)αy(t)dt=f(x), 0 < α<1 . Solution: y(x)=f(x)+⎝integraldisplayx 0R(x–t)f(t)dt,w h e r e R(x)=eµx∞⎝summationdisplay n=1⎝bracketleftbig λΓ(1 –α)x1–α⎝bracketrightbign xΓ⎝bracketleftbig n(1 –α)⎝bracketrightbig. 154 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 36. y(x)+A⎝integraldisplay ⎝integraldisplayx aexp⎝bracketleftbig⎝bracketleftbig λ(x2–t2)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=f(x)–A⎝integraldisplayx aexp⎝bracketleftbig λ(x2–t2)–A(x–t)⎝bracketrightbig f(t)dt. 37. y(x)+A⎝integraldisplay ⎝integraldisplayx aexp⎝parenleftbig⎝parenleftbig λx2+βt2⎝parenrightbig⎝parenrightbig y(t)dt=f(x). In the case β=–λ, see equation 2.2.36. This is a special case of equation 2.9.2 with g(x)=–Aexp⎝parenleftbig λx2)a n dh(t)=e x p⎝parenleftbig βt2⎝parenrightbig . 38. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xexp⎝parenleftbig⎝parenleftbig –λ√ t–x⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=Aexp⎝parenleftbig –λ√ –x⎝parenrightbig . 39. y(x)+A⎝integraldisplay ⎝integraldisplayx aexp⎝bracketleftbig⎝bracketleftbig λ(xµ–tµ)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), µ>0 . This is a special case of equation 2.9.2 with g(x)=–Aexp⎝parenleftbig λxµ⎝parenrightbig andh(t)=e x p⎝parenleftbig –λtµ⎝parenrightbig . Solution: y(x)=f(x)–A⎝integraldisplayx aexp⎝bracketleftbig λ(xµ–tµ)–A(x–t)⎝bracketrightbig f(t)dt. 40. y(x)+k⎝integraldisplay ⎝integraldisplayx 01 xexp⎝parenleftBig ⎝parenleftBig –λt x⎝parenrightBig ⎝parenrightBig y(t)dt=g(x). This is a special case of equation 2.9.71 with f(z)=ke–λz. For a polynomial right-hand side, g(x)=N⎝summationtext n=0Anxn, a solution is given by y(x)=N⎝summationdisplay n=0An 1+kBnxn,Bn=n! λn+1–e–λn⎝summationdisplay k=0n! k!1 λn–k+1. 2.3. Equations Whose Kernels Contain Hyperbolic Functions 2.3-1. Kernels Containing Hyperbolic Cosine. 1. y(x)–A⎝integraldisplay ⎝integraldisplayx acosh(λx )y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acosh(λx)a n d h(t)=1 . Solution: y(x)=f(x)+A⎝integraldisplayx acosh(λx)e x p⎝braceleftBigA λ⎝bracketleftbig sinh(λx)–s i n h ( λt)⎝bracketrightbig⎝bracerightBig f(t)dt. 2.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 155 2. y(x)–A⎝integraldisplay ⎝integraldisplayx acosh(λt )y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=c o s h ( λt). Solution: y(x)=f(x)+A⎝integraldisplayx acosh(λt)e x p⎝braceleftBigA λ⎝bracketleftbig sinh(λx)–s i n h ( λt)⎝bracketrightbig⎝bracerightBig f(t)dt. 3. y(x)+A⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)]y(t)dt=f(x). This is a special case of equation 2.9.28 with g(t)=A. Therefore, solving the original integral equation is reduced to solving the second-order linear nonhomogeneous ordinary differential equation with constant coefﬁcients y/prime/prime xx+Ay/prime x–λ2y=f/prime/prime xx–λ2f,f=f(x), under the initial conditions y(a)=f(a),y/prime x(a)=f/prime x(a)–Af(a). Solution: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x)=e x p⎝parenleftbig –1 2Ax⎝parenrightbig⎝bracketleftbiggA2 2ksinh(kx)–Acosh(kx )⎝bracketrightbigg ,k=⎝radicalBig λ2+1 4A2. 4. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝braceleftbigg ⎝braceleftbiggn⎝summationdisplay k=1Akcosh[λ k(x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x). This equation can be reduced to an equation of the form 2.2.19 by using the identity coshz≡1 2⎝parenleftbig ez+e–z⎝parenrightbig . Therefore, the integral equation in question can be reduced to a linear nonhomogeneous ordinary differential equation of order 2 nwith constant coefﬁcients. 5. y(x)–A⎝integraldisplay ⎝integraldisplayx acosh(λx ) cosh(λt )y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)cosh(λx) cosh(λt )f(t)dt. 6. y(x)–A⎝integraldisplay ⎝integraldisplayx acosh(λt ) cosh(λx )y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)cosh(λt ) cosh(λx)f(t)dt. 7. y(x)–A⎝integraldisplay ⎝integraldisplayx acoshk(λx)c o s hm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acoshk(λx)a n d h(t)=c o s hm(µt). 156 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 8. y(x)+A⎝integraldisplay ⎝integraldisplayx atcosh[λ (x–t)]y(t)dt=f(x). This is a special case of equation 2.9.28 with g(t)=At. 9. y(x)+A⎝integraldisplay ⎝integraldisplayx atkcoshm(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Acoshm(λx)a n d h(t)=tk. 10. y(x)+A⎝integraldisplay ⎝integraldisplayx axkcoshm(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Axkandh(t)=c o s hm(λt). 11. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosh(kx )+B–AB (x–t)c o s h ( kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Acosh(kx). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[Acosh(kx )+B]G(x) G(t)+B2 G(t)⎝integraldisplayx teB(x–s)G(s)ds,G(x)=e x p⎝bracketleftbiggA ksinh(kx)⎝bracketrightbigg . 12. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acosh(kt )+B+AB (x–t)c o s h ( kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Acosh(kt ). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=– [ Acosh(kt )+B]G(t) G(x)+B2 G(x)⎝integraldisplayx teB(t–s)G(s)ds,G(x)=e x p⎝bracketleftbiggA ksinh(kx)⎝bracketrightbigg . 13. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xcosh⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=Acosh⎝parenleftbig λ√ –x⎝parenrightbig . 2.3-2. Kernels Containing Hyperbolic Sine. 14. y(x)–A⎝integraldisplay ⎝integraldisplayx asinh(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asinh(λx)a n d h(t)=1 . Solution: y(x)=f(x)+A⎝integraldisplayx asinh(λx)e x p⎝braceleftbiggA λ⎝bracketleftbig cosh(λx)–c o s h ( λt)⎝bracketrightbig⎝bracerightbigg f(t)dt. 2.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 157 15. y(x)–A⎝integraldisplay ⎝integraldisplayx asinh(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t) = sinh(λt ). Solution: y(x)=f(x)+A⎝integraldisplayx asinh(λt)e x p⎝braceleftbiggA λ⎝bracketleftbig cosh(λx)–c o s h ( λt)⎝bracketrightbig⎝bracerightbigg f(t)dt. 16. y(x)+A⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]y(t)dt=f(x). This is a special case of equation 2.9.30 with g(x)=A. 1◦. Solution with λ(A–λ)>0 : y(x)=f(x)–Aλ k⎝integraldisplayx asin[k(x–t)]f(t)dt,w h e r e k=⎝radicalbig λ(A–λ). 2◦. Solution with λ(A–λ)<0 : y(x)=f(x)–Aλ k⎝integraldisplayx asinh[k(x–t)]f(t)dt,w h e r e k=⎝radicalbig λ(λ–A). 3◦. Solution with A=λ: y(x)=f(x)–λ2⎝integraldisplayx a(x–t)f(t)dt. 17. y(x)+A⎝integraldisplay ⎝integraldisplayx asinh3[λ(x–t)]y(t)dt=f(x). Using the formula sinh3β=1 4sinh 3β–3 4sinhβ, we arrive at an equation of the form 2.3.18: y(x)+⎝integraldisplayx a⎝braceleftbig1 4Asinh⎝bracketleftbig 3λ(x–t)⎝bracketrightbig –3 4Asinh[λ(x–t)]⎝bracerightbig y(t)dt=f(x). 18. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A1sinh[λ1(x–t)] +A2sinh[λ2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). 1◦. Introduce the notation I1=⎝integraldisplayx asinh[λ1(x–t)]y(t)dt,I2=⎝integraldisplayx asinh[λ2(x–t)]y(t)dt, J1=⎝integraldisplayx acosh[λ 1(x–t)]y(t)dt,J2=⎝integraldisplayx acosh[λ 2(x–t)]y(t)dt. Successively differentiating the integral equation four times yields (the ﬁrst line is the original equation) y+A1I1+A2I2=f,f=f(x), (1) y/prime x+A1λ1J1+A2λ2J2=f/prime x,( 2) y/prime/prime xx+(A1λ1+A2λ2)y+A1λ2 1I1+A2λ2 2I2=f/prime/prime xx,( 3) y/prime/prime/prime xxx+(A1λ1+A2λ2)y/prime x+A1λ3 1J1+A2λ3 2J2=f/prime/prime/prime xxx,( 4) y/prime/prime/prime/prime xxxx +(A1λ1+A2λ2)y/prime/prime xx+(A1λ3 1+A2λ3 2)y+A1λ4 1I1+A2λ4 2I2=f/prime/prime/prime/prime xxxx.( 5 ) 158 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION Eliminating I1andI2from (1), (3), and (5), we arrive at a fourth-order linear ordinary differential equation with constant coefﬁcients: y/prime/prime/prime/prime xxxx –(λ2 1+λ2 2–A1λ1–A2λ2)y/prime/prime xx+(λ2 1λ22–A1λ1λ2 2–A2λ2 1λ2)y= f/prime/prime/prime/prime xxxx –(λ2 1+λ2 2)f/prime/prime xx+λ2 1λ22f.(6) The initial conditions can be obtained by setting x=ain (1)–(4): y(a)=f(a),y/prime x(a)=f/prime x(a), y/prime/prime xx(a)=f/prime/prime xx(a)–(A1λ1+A2λ2)f(a), y/prime/prime/prime xxx(a)=f/prime/prime/prime xxx(a)–(A1λ1+A2λ2)f/prime x(a).(7) On solving the differential equation (6) under conditions (7), we thus ﬁnd the solution of the integral equation. 2◦. Consider the characteristic equation z2–(λ2 1+λ2 2–A1λ1–A2λ2)z+λ2 1λ22–A1λ1λ2 2–A2λ2 1λ2=0 , ( 8 ) whose roots, z1andz2, determine the solution structure of the integral equation. Assume that the discriminant of equation (8) is positive: D≡(A1λ1–A2λ2–λ2 1+λ2 2)2+4A1A2λ1λ2>0 . In this case, the quadratic equation (8) has the real (different) roots z1=1 2(λ2 1+λ2 2–A1λ1–A2λ2)+1 2√ D,z2=1 2(λ2 1+λ2 2–A1λ1–A2λ2)–1 2√ D. Depending on the signs of z1andz2the following three cases are possible. Case 1 .I fz1>0a n d z2> 0, then the solution of the integral equation has the form (i=1 ,2 ) : y(x)=f(x)+⎝integraldisplayx a{B1sinh[µ1(x–t)] +B2sinh⎝bracketleftbig µ2(x–t)⎝bracketrightbig⎝bracerightbig f(t)dt,µi=√ zi, where B1=A1λ1(µ2 1–λ2 2) µ1(µ2 2–µ2 1)+A2λ2(µ2 1–λ2 1) µ1(µ2 2–µ2 1),B2=A1λ1(µ2 2–λ2 2) µ2(µ2 1–µ2 2)+A2λ2(µ2 2–λ2 1) µ2(µ2 1–µ2 2). Case 2 .I fz1<0a n d z2< 0, then the solution of the integral equation has the form y(x)=f(x)+⎝integraldisplayx a{B1sin[µ1(x–t)] +B2sin⎝bracketleftbig µ2(x–t)⎝bracketrightbig⎝bracerightbig f(t)dt,µi=⎝radicalbig |zi|, where the coefﬁcients B1andB2are found by solving the following system of linear algebraic equations: B1µ1 λ2 1+µ2 1+B2µ2 λ2 1+µ2 2+1=0 ,B1µ1 λ2 2+µ2 1+B2µ2 λ2 2+µ2 2+1=0 . Case 3 .I fz1>0a n d z2< 0, then the solution of the integral equation has the form y(x)=f(x)+⎝integraldisplayx a{B1sinh[µ1(x–t)] +B2sin⎝bracketleftbig µ2(x–t)⎝bracketrightbig⎝bracerightbig f(t)dt,µi=⎝radicalbig |zi|, where B1andB2are determined from the following sy stem of linear algebraic equations: B1µ1 λ2 1–µ2 1+B2µ2 λ2 1+µ2 2+1=0 ,B1µ1 λ2 2–µ2 1+B2µ2 λ2 2+µ2 2+1=0 . 2.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 159 19. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝braceleftbigg ⎝braceleftbiggn⎝summationdisplay k=1Aksinh[λk(x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x). 1◦. This equation can be reduced to an equation of the form 2.2.19 with the aid of the formula sinhz=1 2⎝parenleftbig ez–e–z⎝parenrightbig . Therefore, the original integral equation can be reduced to a linear nonhomogeneous ordinary differential equation of order 2 nwith constant coefﬁcients. 2◦. Let us ﬁnd the roots zkof the algebraic equation n⎝summationdisplay k=1λkAk z–λ2 k+1=0 . ( 1 ) By reducing it to a common denominator, we arrive at the problem of determining the roots of annth-degree characteristic polynomial. Assume that all zkare real, different, and nonzero. Let us divide the roots into two groups z1>0 , z2>0 , ...,zs> 0 (positive roots); zs+1<0 , zs+2<0 , ...,zn< 0 (negative roots). Then the solution of the integral equation can be written in the form y(x)=f(x)+⎝integraldisplayx a⎝braceleftbiggs⎝summationdisplay k=1Bksinh⎝bracketleftbig µk(x–t)⎝bracketrightbig +n⎝summationdisplay k=s+1Cksin⎝bracketleftbig µk(x–t)⎝bracketrightbig⎝bracerightbigg f(t)dt,µk=⎝radicalbig |zk|.( 2 ) The coefﬁcients BkandCkare determined from the following system of linear algebraic equations: s⎝summationdisplay k=0Bkµk λ2m–µ2 k+n⎝summationdisplay k=s+1Ckµk λ2m+µ2 k+1=0 , µk=⎝radicalbig |zk|,m=1 ,...,n.( 3 ) In the case of a nonzero root zs= 0, we can introduce the new constant D=Bsµsand proceed to the limit µs→0. As a result, the term D(x–t) appears in solution (2) instead of Bssinh⎝bracketleftbig µs(x–t)⎝bracketrightbig and the corresponding terms Dλ–2 mappear in system (3). 20. y(x)–A⎝integraldisplay ⎝integraldisplayx asinh(λx) sinh(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)sinh(λx) sinh(λt)f(t)dt. 21. y(x)–A⎝integraldisplay ⎝integraldisplayx asinh(λt) sinh(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)sinh(λt) sinh(λx)f(t)dt. 22. y(x)–A⎝integraldisplay ⎝integraldisplayx asinhk(λx)s i n hm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asinhk(λx)a n d h(t)=s i n hm(µt). 160 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 23. y(x)+A⎝integraldisplay ⎝integraldisplayx atsinh[λ(x–t)]y(t)dt=f(x). This is a special case of equation 2.9.30 with g(t)=At. Solution: y(x)=f(x)+Aλ W⎝integraldisplayx at⎝bracketleftbig u1(x)u2(t)–u2(x)u1(t)⎝bracketrightbig f(t)dt, where u1(x),u2(x) is a fundamental system of solutions of the second-order linear ordinary differential equation u/prime/prime xx+λ(Ax–λ)u=0 ,a n d Wis the Wronskian. The functions u1(x)a n du2(x) are expressed in terms of Bessel functions or modiﬁed Bessel functions, depending on the sign of Aλ, as follows: ifAλ>0 ,t h e n u1(x)=ξ1/2J1/3⎝parenleftbig2 3√ Aλ ξ3/2⎝parenrightbig ,u2(x)=ξ1/2Y1/3⎝parenleftbig2 3√ Aλξ3/2⎝parenrightbig , W=3/π,ξ=x–(λ/A); ifAλ<0 ,t h e n u1(x)=ξ1/2I1/3⎝parenleftbig2 3√ –Aλξ3/2⎝parenrightbig ,u2(x)=ξ1/2K1/3⎝parenleftbig2 3√ –Aλξ3/2⎝parenrightbig , W=–3 2,ξ=x–(λ/A). 24. y(x)+A⎝integraldisplay ⎝integraldisplayx axsinh[λ(x–t)]y(t)dt=f(x). This is a special case of equation 2.9.31 with g(x)=Axandh(t)=1 . Solution: y(x)=f(x)+Aλ W⎝integraldisplayx ax⎝bracketleftbig u1(x)u2(t)–u2(x)u1(t)⎝bracketrightbig f(t)dt, where u1(x),u2(x) is a fundamental system of solutions of the second-order linear ordinary differential equation u/prime/prime xx+λ(Ax–λ)u=0 ,a n d Wis the Wronskian. The functions u1(x),u2(x), and Ware speciﬁed in 2.3.23. 25. y(x)+A⎝integraldisplay ⎝integraldisplayx atksinhm(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Asinhm(λx)a n d h(t)=tk. 26. y(x)+A⎝integraldisplay ⎝integraldisplayx axksinhm(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Axkandh(t)=s i n hm(λt). 27. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinh(kx)+B–AB (x–t) sinh( kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Asinh(kx). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[Asinh(kx)+B]G(x) G(t)+B2 G(t)⎝integraldisplayx teB(x–s)G(s)ds,G(x)=e x p⎝bracketleftbiggA kcosh(kx )⎝bracketrightbigg . 2.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 161 28. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asinh(kt)+B+AB (x–t)s i n h ( kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Asinh(kt). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t) = –[sinh( kt)+B]G(t) G(x)+B2 G(x)⎝integraldisplayx teB(t–s)G(s)ds,G(x)=e x p⎝bracketleftbiggA kcosh(kx )⎝bracketrightbigg . 29. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xsinh⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=Asinh⎝parenleftbig λ√ –x⎝parenrightbig . 2.3-3. Kernels Containing Hyperbolic Tangent. 30. y(x)–A⎝integraldisplay ⎝integraldisplayx atanh(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atanh(λx)a n d h(t)=1 . Solution: y(x)=f(x)+A⎝integraldisplayx atanh(λx)⎝bracketleftbiggcosh(λx) cosh(λt)⎝bracketrightbiggA/λ f(t)dt. 31. y(x)–A⎝integraldisplay ⎝integraldisplayx atanh(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=t a n h ( λt). Solution: y(x)=f(x)+A⎝integraldisplayx atanh(λt)⎝bracketleftbiggcosh(λx) cosh(λt )⎝bracketrightbiggA/λ f(t)dt. 32. y(x)+A⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tanh(λx) – tanh( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.5 with g(x)=Atanh(λx). Solution: y(x)=f(x)+1 W⎝integraldisplayx a⎝bracketleftbig Y/prime 1(x)Y/prime 2(t)–Y/prime 2(x)Y/prime 1(t)⎝bracketrightbig f(t)dt, where Y1(x),Y2(x) is a fundamental system of solutions of the second-order linear ordinary differential equation cosh2(λx)Y/prime/prime xx+AλY =0 ,Wis the Wronskian, and the primes stand for the differentiation with respect to the argument speciﬁed in the parentheses. As shown in A. D. Polyanin and V . F. Zaitsev (2003), the functions Y1(x)a n dY2(x) can be represented in the form Y1(x)=F⎝parenleftBig α,β,1 ;eλx 1+eλx⎝parenrightBig ,Y2(x)=Y1(x)⎝integraldisplayx adξ Y2 1(ξ),W=1 , where F(α,β,γ;z) is the hypergeometric function, in which αandβare determined from the algebraic system α+β=1 ,αβ=–A/λ. 162 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 33. y(x)–A⎝integraldisplay ⎝integraldisplayx atanh(λx) tanh(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)tanh(λx) tanh(λt)f(t)dt. 34. y(x)–A⎝integraldisplay ⎝integraldisplayx atanh(λt) tanh(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)tanh(λt) tanh(λx)f(t)dt. 35. y(x)–A⎝integraldisplay ⎝integraldisplayx atanhk(λx)t a n hm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atanhk(λx)a n d h(t)=t a n hm(µt). 36. y(x)+A⎝integraldisplay ⎝integraldisplayx atktanhm(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Atanhm(λx)a n d h(t)=tk. 37. y(x)+A⎝integraldisplay ⎝integraldisplayx axktanhm(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Axkandh(t)=t a n hm(λt). 38. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xtanh[λ(t–x)]y(t)dt=f(x). This is a special case of equation 2.9.62 with K(z)=Atanh(– λz). 39. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xtanh⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.62 with K(z)=Atanh⎝parenleftbig λ√ –z⎝parenrightbig . 40. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanh(kx)+B–AB (x–t) tanh( kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Atanh(kx). 41. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atanh(kt)+B+AB (x–t)t a n h ( kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Atanh(kt). 2.3-4. Kernels Containing Hyperbolic Cotangent. 42. y(x)–A⎝integraldisplay ⎝integraldisplayx acoth(λx )y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acoth(λx)a n d h(t)=1 . Solution: y(x)=f(x)+A⎝integraldisplayx acoth(λx)⎝bracketleftBigsinh(λx) sinh(λt)⎝bracketrightBigA/λ f(t)dt. 2.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 163 43. y(x)–A⎝integraldisplay ⎝integraldisplayx acoth(λt )y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=c o t h ( λt). Solution: y(x)=f(x)+A⎝integraldisplayx acoth(λt)⎝bracketleftBigsinh(λx) sinh(λt)⎝bracketrightBigA/λ f(t)dt. 44. y(x)–A⎝integraldisplay ⎝integraldisplayx acoth(λt ) coth(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)coth(λt) coth(λx)f(t)dt. 45. y(x)–A⎝integraldisplay ⎝integraldisplayx acoth(λx) coth(λt )y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)coth(λx) coth(λt)f(t)dt. 46. y(x)–A⎝integraldisplay ⎝integraldisplayx acothk(λx)c o t hm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acothk(λx)a n d h(t)=c o t hm(µt). 47. y(x)+A⎝integraldisplay ⎝integraldisplayx atkcothm(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Acothm(λx)a n d h(t)=tk. 48. y(x)+A⎝integraldisplay ⎝integraldisplayx axkcothm(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Axkandh(t)=c o t hm(λt). 49. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xcoth[λ(t–x)]y(t)dt=f(x). This is a special case of equation 2.9.62 with K(z)=Acoth(– λz). 50. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xcoth⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.62 with K(z)=Acoth⎝parenleftbig λ√ –z⎝parenrightbig . 51. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoth(kx )+B–AB (x–t)c o t h ( kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Acoth(kx). 52. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acoth(kt )+B+AB (x–t)c o t h ( kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Acoth(kt). 164 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2.3-5. Kernels Containing Combinations of Hyperbolic Functions. 53. y(x)–A⎝integraldisplay ⎝integraldisplayx acoshk(λx)s i n hm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acoshk(λx)a n d h(t)=s i n hm(µt). 54. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A+Bcosh(λx )+B(x–t)[λsinh(λx)–Acosh(λx )]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.32 with b=Bandg(x)=A. 55. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A+Bsinh(λx)+B(x–t)[λcosh(λx )–Asinh(λx)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.33 with b=Bandg(x)=A. 56. y(x)–A⎝integraldisplay ⎝integraldisplayx atanhk(λx)c o t hm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atanhk(λx)a n d h(t)=c o t hm(µt). 2.4. Equations Whose Kernels Contain Logarithmic Functions 2.4-1. Kernels Containing Logarithmic Functions. 1. y(x)–A⎝integraldisplay ⎝integraldisplayx aln(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aln(λx)a n d h(t)=1 . Solution: y(x)=f(x)+A⎝integraldisplayx aln(λx)e–A(x–t)(λx)Ax (λt)Atf(t)dt. 2. y(x)–A⎝integraldisplay ⎝integraldisplayx aln(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=l n ( λt). Solution: y(x)=f(x)+A⎝integraldisplayx aln(λt)e–A(x–t)(λx)Ax (λt)Atf(t)dt. 3. y(x)+A⎝integraldisplay ⎝integraldisplayx a(lnx–l nt)y(t)dt=f(x). This is a special case of equation 2.9.5 with g(x)=Alnx. Solution: y(x)=f(x)+1 W⎝integraldisplayx a⎝bracketleftbig u/prime 1(x)u/prime 2(t)–u/prime 2(x)u/prime 1(t)⎝bracketrightbig f(t)dt, where the primes denote differentiation w ith respect to the argument speciﬁed in the paren- theses; and u1(x),u2(x) is a fundamental system of solutions of the second-order linear homogeneous ordinary differential equation u/prime/prime xx+Ax–1u=0 ,w i t h u1(x)a n du2(x) expressed in terms of Bessel functions or modiﬁed Bessel functions, depending on the sign of A: W=1 π,u1(x)=√ xJ1⎝parenleftbig 2√ Ax⎝parenrightbig ,u2(x)=√ xY 1⎝parenleftbig 2√ Ax⎝parenrightbig forA>0 , W=–1 2,u1(x)=√ xI1⎝parenleftbig 2√ –Ax⎝parenrightbig ,u2(x)=√ xK 1⎝parenleftbig 2√ –Ax⎝parenrightbig forA<0 . 2.4. E QUATIONS WHOSE KERNELS CONTAIN LOGARITHMIC FUNCTIONS 165 4. y(x)–A⎝integraldisplay ⎝integraldisplayx aln(λx) ln(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)ln(λx) ln(λt)f(t)dt. 5. y(x)–A⎝integraldisplay ⎝integraldisplayx aln(λt) ln(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)ln(λt) ln(λx)f(t)dt. 6. y(x)–A⎝integraldisplay ⎝integraldisplayx alnk(λx)l nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Alnk(λx)a n d h(t)=l nm(µt). 7. y(x)+a⎝integraldisplay ⎝integraldisplay∞ xln(t–x)y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=aln(–x). Forf(x)=m⎝summationtext k=1Akexp(–λkx), where λk> 0, a solution of the equation has the form y(x)=m⎝summationdisplay k=1Ak Bkexp(–λkx), Bk=1–a λk(lnλk+C), whereC= 0.5772 ...is the Euler constant. 8. y(x)+a⎝integraldisplay ⎝integraldisplay∞ xln2(t–x)y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=aln2(–x). Forf(x)=m⎝summationtext k=1Akexp(–λkx), where λk> 0, a solution of the equation has the form y(x)=m⎝summationdisplay k=1Ak Bkexp(–λkx), Bk=1+a λk⎝bracketleftbig1 6π2+( l nλk+C)2⎝bracketrightbig , whereC= 0.5772 ...is the Euler constant. 2.4-2. Kernels Containing Power-Law and Logarithmic Functions. 9. y(x)–A⎝integraldisplay ⎝integraldisplayx axklnm(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Axkandh(t)=l nm(λt). 10. y(x)–A⎝integraldisplay ⎝integraldisplayx atklnm(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Alnm(λx)a n d h(t)=tk. 166 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 11. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aln(kx)+B–AB (x–t)l n (kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Aln(kx). 12. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aln(kt)+B+AB (x–t)l n (kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Aln(kt). 13. y(x)+a⎝integraldisplay ⎝integraldisplay∞ x(t–x)nln(t–x)y(t)dt=f(x), n=1 , 2 , ... Forf(x)=m⎝summationtext k=1Akexp(–λkx), where λk> 0, a solution of the equation has the form y(x)=m⎝summationdisplay k=1Ak Bkexp(–λkx), Bk=1+an! λn+1 k⎝parenleftbig 1+1 2+1 3+···+1 n–l nλk–C⎝parenrightbig , whereC= 0.5772 ...is the Euler constant. 14. y(x)+a⎝integraldisplay ⎝integraldisplay∞ xln(t–x) √ t–xy(t)dt=f(x). This is a special case of equation 2.9.62 with K(–x)=ax–1/2lnx. Forf(x)=m⎝summationtext k=1Akexp(–λkx), where λk> 0, a solution of the equation has the form y(x)=m⎝summationdisplay k=1Ak Bkexp(–λkx), Bk=1–a⎝radicalbigg π λk⎝bracketleftbig ln(4λk)+C⎝bracketrightbig , whereC= 0.5772 ...is the Euler constant. 2.5. Equations Whose Kernels Contain Trigonometric Functions 2.5-1. Kernels Containing Cosine. 1. y(x)–A⎝integraldisplay ⎝integraldisplayx acos(λx )y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acos(λx)a n d h(t)=1 . Solution: y(x)=f(x)+A⎝integraldisplayx acos(λx)e x p⎝braceleftBigA λ⎝bracketleftbig sin(λx)–s i n ( λt)⎝bracketrightbig⎝bracerightBig f(t)dt. 2. y(x)–A⎝integraldisplay ⎝integraldisplayx acos(λt )y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=c o s ( λt). Solution: y(x)=f(x)+A⎝integraldisplayx acos(λt )e x p⎝braceleftBigA λ⎝bracketleftbig sin(λx)–s i n ( λt)⎝bracketrightbig⎝bracerightBig f(t)dt. 2.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 167 3. y(x)+A⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)]y(t)dt=f(x). This is a special case of equation 2.9.34 with g(t)=A. Therefore, solving this integral equation is reduced to solving the following second-order linear nonhomogeneous ordinary differential equation with constant coefﬁcients: y/prime/prime xx+Ay/prime x+λ2y=f/prime/prime xx+λ2f,f=f(x), with the initial conditions y(a)=f(a),y/prime x(a)=f/prime x(a)–Af(a). 1◦. Solution with |A|>2|λ|: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x)=e x p⎝parenleftbig –1 2Ax⎝parenrightbig⎝bracketleftbiggA2 2ksinh(kx)–Acosh(kx )⎝bracketrightbigg ,k=⎝radicalBig 1 4A2–λ2. 2◦. Solution with |A|<2|λ|: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x)=e x p⎝parenleftbig –1 2Ax⎝parenrightbig⎝bracketleftBigA2 2ksin(kx)–Acos(kx )⎝bracketrightBig ,k=⎝radicalBig λ2–1 4A2. 3◦. Solution with λ=±1 2A: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt,R(x)=e x p⎝parenleftbig –1 2Ax⎝parenrightbig⎝parenleftbig1 2A2x–A⎝parenrightbig . 4. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝braceleftbigg ⎝braceleftbiggn⎝summationdisplay k=1Akcos[λ k(x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x). This integral equation is reduced to a linear nonhomogeneous ordinary differential equation of order 2 nwith constant coefﬁcients. Set Ik(x)=⎝integraldisplayx acos[λ k(x–t)]y(t)dt.( 1 ) Differentiating (1) with respect to xtwice yields I/prime k=y(x)–λk⎝integraldisplayx asin[λk(x–t)]y(t)dt, I/prime/prime k=y/prime x(x)–λ2 k⎝integraldisplayx acos[λ k(x–t)]y(t)dt,(2) where the primes stand for differentiation with respect to x. Comparing (1) and (2), we see that I/prime/prime k=y/prime x(x)–λ2 kIk,Ik=Ik(x). (3) 168 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION With the aid of (1), the integral equation can be rewritten in the form y(x)+n⎝summationdisplay k=1AkIk=f(x). (4) Differentiating (4) with respect to xtwice taking into account (3) yields y/prime/prime xx(x)+σny/prime x(x)–n⎝summationdisplay k=1Akλ2 kIk=f/prime/prime xx(x), σn=n⎝summationdisplay k=1Ak.( 5) Eliminating the integral Infrom (4) and (5), we obtain y/prime/prime xx(x)+σny/prime x(x)+λ2 ny(x)+n–1⎝summationdisplay k=1Ak(λ2 n–λ2 k)Ik=f/prime/prime xx(x)+λ2 nf(x). (6) Differentiating (6) with respect to xtwice followed by eliminating In–1from the resulting expression with the aid of (6) yields a similar e quation whose left-hand side is a fourth- order differential operator (acting on y) with constant coefﬁcients plus the sumn–2⎝summationtext k=1BkIk. Successively eliminating the terms In–2,In–3,...using double differentiation and formula (3), we ﬁnally arrive at a linear nonhomogeneous ordinary differential equation of order 2 nwith constant coefﬁcients. The initial conditions for y(x) can be obtained by setting x=ain the integral equation and all its derivative equations. 5. y(x)–A⎝integraldisplay ⎝integraldisplayx acos(λx ) cos(λt )y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)cos(λx) cos(λt )f(t)dt. 6. y(x)–A⎝integraldisplay ⎝integraldisplayx acos(λt ) cos(λx )y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)cos(λt ) cos(λx)f(t)dt. 7. y(x)–A⎝integraldisplay ⎝integraldisplayx acosk(λx)c o sm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acosk(λx)a n d h(t)=c o sm(µt). 8. y(x)+A⎝integraldisplay ⎝integraldisplayx atcos[λ (x–t)]y(t)dt=f(x). This is a special case of equation 2.9.34 with g(t)=At. 9. y(x)+A⎝integraldisplay ⎝integraldisplayx atkcosm(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Acosm(λx)a n d h(t)=tk. 2.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 169 10. y(x)+A⎝integraldisplay ⎝integraldisplayx axkcosm(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Axkandh(t)=c o sm(λt). 11. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos(kx )+B–AB (x–t)c o s (kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Acos(kx ). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[Acos(kx )+B]G(x) G(t)+B2 G(t)⎝integraldisplayx teB(x–s)G(s)ds,G(x)=e x p⎝bracketleftbiggA ksin(kx)⎝bracketrightbigg . 12. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acos(kt )+B+AB (x–t)c o s (kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Acos(kt ). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=– [ Acos(kt )+B]G(t) G(x)+B2 G(x)⎝integraldisplayx teB(t–s)G(s)ds,G(x)=e x p⎝bracketleftbiggA ksin(kx)⎝bracketrightbigg . 13. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xcos⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=Acos⎝parenleftbig λ√ –x⎝parenrightbig . 2.5-2. Kernels Containing Sine. 14. y(x)–A⎝integraldisplay ⎝integraldisplayx asin(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asin(λx)a n d h(t)=1 . Solution: y(x)=f(x)+A⎝integraldisplayx asin(λx)e x p⎝braceleftBigA λ⎝bracketleftbig cos(λt )–c o s ( λx)⎝bracketrightbig⎝bracerightBig f(t)dt. 15. y(x)–A⎝integraldisplay ⎝integraldisplayx asin(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=s i n ( λt). Solution: y(x)=f(x)+A⎝integraldisplayx asin(λt)e x p⎝braceleftBigA λ⎝bracketleftbig cos(λt )–c o s ( λx)⎝bracketrightbig⎝bracerightBig f(t)dt. 170 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 16. y(x)+A⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)]y(t)dt=f(x). This is a special case of equation 2.9.36 with g(t)=A. 1◦. Solution with λ(A+λ)>0 : y(x)=f(x)–Aλ k⎝integraldisplayx asin[k(x–t)]f(t)dt,w h e r e k=⎝radicalbig λ(A+λ). 2◦. Solution with λ(A+λ)<0 : y(x)=f(x)–Aλ k⎝integraldisplayx asinh[k(x–t)]f(t)dt,w h e r e k=⎝radicalbig –λ(λ+A). 3◦. Solution with A=–λ: y(x)=f(x)+λ2⎝integraldisplayx a(x–t)f(t)dt. 17. y(x)+A⎝integraldisplay ⎝integraldisplayx asin3[λ(x–t)]y(t)dt=f(x). Using the formula sin3β=–1 4sin 3β+3 4sinβ, we arrive at an equation of the form 2.5.18: y(x)+⎝integraldisplayx a⎝braceleftbig –1 4Asin[3λ(x–t)] +3 4Asin[λ(x–t)]⎝bracerightbig y(t)dt=f(x). 18. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A1sin[λ1(x–t)] +A2sin[λ2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This equation can be solved by the same method as equation 2.3.18, by reducing it to a fourth-order linear ordinary differential equation with constant coefﬁcients. Consider the characteristic equation z2+(λ2 1+λ2 2+A1λ1+A2λ2)z+λ2 1λ22+A1λ1λ2 2+A2λ2 1λ2=0 , ( 1 ) whose roots, z1andz2, determine the solution structure of the integral equation. Assume that the discriminant of equation (1) is positive: D≡(A1λ1–A2λ2+λ2 1–λ2 2)2+4A1A2λ1λ2>0 . In this case, the quadratic equation (1) has the real (different) roots z1=–1 2(λ2 1+λ2 2+A1λ1+A2λ2)+1 2√ D,z2=–1 2(λ2 1+λ2 2+A1λ1+A2λ2)–1 2√ D. Depending on the signs of z1andz2the following three cases are possible. Case 1 .I fz1>0a n d z2> 0, then the solution of the integral equation has the form (i=1 ,2 ) : y(x)=f(x)+⎝integraldisplayx a{B1sinh[µ1(x–t)] +B2sinh⎝bracketleftbig µ2(x–t)⎝bracketrightbig⎝bracerightbig f(t)dt,µi=√ zi, where the coefﬁcients B1andB2are determined from the following system of linear algebraic equations: B1µ1 λ2 1+µ2 1+B2µ2 λ2 1+µ2 2–1=0 ,B1µ1 λ2 2+µ2 1+B2µ2 λ2 2+µ2 2–1=0 . 2.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 171 Case 2 .I fz1<0a n d z2< 0, then the solution of the integral equation has the form y(x)=f(x)+⎝integraldisplayx a{B1sin[µ1(x–t)] +B2sin⎝bracketleftbig µ2(x–t)⎝bracketrightbig⎝bracerightbig f(t)dt,µi=⎝radicalbig |zi|, where B1andB2are determined from the system B1µ1 λ2 1–µ2 1+B2µ2 λ2 1–µ2 2–1=0 ,B1µ1 λ2 2–µ2 1+B2µ2 λ2 2–µ2 2–1=0 . Case 3 .I fz1>0a n d z2< 0, then the solution of the integral equation has the form y(x)=f(x)+⎝integraldisplayx a{B1sinh[µ1(x–t)] +B2sin⎝bracketleftbig µ2(x–t)⎝bracketrightbig⎝bracerightbig f(t)dt,µi=⎝radicalbig |zi|, where B1andB2are determined from the system B1µ1 λ2 1+µ2 1+B2µ2 λ2 1–µ2 2–1=0 ,B1µ1 λ2 2+µ2 1+B2µ2 λ2 2–µ2 2–1=0 . Remark. The solution of the original integral equation can be obtained from the solution of equation 2.3.18 by performing the following change of parameters: λk→iλk,µk→iµk,Ak→–iAk,Bk→–iBk,i2=– 1 ( k=1 ,2 ) . 19. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝braceleftbigg ⎝braceleftbiggn⎝summationdisplay k=1Aksin[λk(x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x). 1◦. This integral equation can be reduced to a linear nonhomogeneous ordinary differential equation of order 2 nwith constant coefﬁcients. Set Ik(x)=⎝integraldisplayx asin[λk(x–t)]y(t)dt.( 1) Differentiating (1) with respect to xtwice yields I/prime k=λk⎝integraldisplayx acos[λ k(x–t)]y(t)dt,I/prime/prime k=λky(x)–λ2 k⎝integraldisplayx asin[λk(x–t)]y(t)dt,( 2 ) where the primes stand for differentiation with respect to x. Comparing (1) and (2), we see that I/prime/prime k=λky(x)–λ2 kIk,Ik=Ik(x). (3) With aid of (1), the integral equation can be rewritten in the form y(x)+n⎝summationdisplay k=1AkIk=f(x). (4) Differentiating (4) with respect to xtwice taking into account (3) yields y/prime/prime xx(x)+σny(x)–n⎝summationdisplay k=1Akλ2 kIk=f/prime/prime xx(x), σn=n⎝summationdisplay k=1Akλk.( 5 ) 172 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION Eliminating the integral Infrom (4) and (5), we obtain y/prime/prime xx(x)+(σn+λ2 n)y(x)+n–1⎝summationdisplay k=1Ak(λ2 n–λ2 k)Ik=f/prime/prime xx(x)+λ2 nf(x). (6) Differentiating (6) with respect to xtwice followed by eliminating In–1from the resulting expression with the aid of (6) yields a similar e quation whose left-hand side is a fourth- order differential operator (acting on y) with constant coefﬁcients plus the sumn–2⎝summationtext k=1BkIk. Successively eliminating the terms In–2,In–3,...using double differentiation and formula (3), we ﬁnally arrive at a linear nonhomogeneous ordinary differential equation of order 2 nwith constant coefﬁcients. The initial conditions for y(x) can be obtained by setting x=ain the integral equation and all its derivative equations. 2◦. Let us ﬁnd the roots zkof the algebraic equation n⎝summationdisplay k=1λkAk z+λ2 k+1=0 . ( 7 ) By reducing it to a common denominator, we arrive at the problem of determining the roots of annth-degree characteristic polynomial. Assume that all zkare real, different, and nonzero. Let us divide the roots into two groups z1>0 , z2>0 , ...,zs> 0 (positive roots); zs+1<0 , zs+2<0 , ...,zn< 0 (negative roots). Then the solution of the integral equation can be written in the form y(x)=f(x)+⎝integraldisplayx a⎝braceleftbiggs⎝summationdisplay k=1Bksinh⎝bracketleftbig µk(x–t)⎝bracketrightbig +n⎝summationdisplay k=s+1Cksin⎝bracketleftbig µk(x–t)⎝bracketrightbig⎝bracerightbigg f(t)dt,µk=⎝radicalbig |zk|.( 8 ) The coefﬁcients BkandCkare determined from the following system of linear algebraic equations: s⎝summationdisplay k=0Bkµk λ2m+µ2 k+n⎝summationdisplay k=s+1Ckµk λ2m–µ2 k–1=0 , µk=⎝radicalbig |zk|m=1 ,2 , ...,n.( 9 ) In the case of a nonzero root zs= 0, we can introduce the new constant D=Bsµsand proceed to the limit µs→0. As a result, the term D(x–t) appears in solution (8) instead of Bssinh⎝bracketleftbig µs(x–t)⎝bracketrightbig and the corresponding terms Dλ–2 mappear in system (9). 20. y(x)–A⎝integraldisplay ⎝integraldisplayx asin(λx) sin(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)sin(λx) sin(λt)f(t)dt. 21. y(x)–A⎝integraldisplay ⎝integraldisplayx asin(λt) sin(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)sin(λt) sin(λx)f(t)dt. 2.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 173 22. y(x)–A⎝integraldisplay ⎝integraldisplayx asink(λx)s i nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asink(λx)a n d h(t)=s i nm(µt). 23. y(x)+A⎝integraldisplay ⎝integraldisplayx atsin[λ(x–t)]y(t)dt=f(x). This is a special case of equation 2.9.36 with g(t)=At. Solution: y(x)=f(x)+Aλ W⎝integraldisplayx at⎝bracketleftbig u1(x)u2(t)–u2(x)u1(t)⎝bracketrightbig f(t)dt, where u1(x),u2(x) is a fundamental system of solutions of the second-order linear ordinary differential equation u/prime/prime xx+λ(Ax+λ)u=0 ,a n d Wis the Wronskian. Depending on the sign of Aλ, the functions u1(x)a n du2(x) are expressed in terms of Bessel functions or modiﬁed Bessel functions as follows: ifAλ>0 ,t h e n u1(x)=ξ1/2J1/3⎝parenleftbig2 3√ Aλ ξ3/2⎝parenrightbig ,u2(x)=ξ1/2Y1/3⎝parenleftbig2 3√ Aλξ3/2⎝parenrightbig , W=3/π,ξ=x+(λ/A); ifAλ<0 ,t h e n u1(x)=ξ1/2I1/3⎝parenleftbig2 3√ –Aλξ3/2⎝parenrightbig ,u2(x)=ξ1/2K1/3⎝parenleftbig2 3√ –Aλξ3/2⎝parenrightbig , W=–3 2,ξ=x+(λ/A). 24. y(x)+A⎝integraldisplay ⎝integraldisplayx axsin[λ(x–t)]y(t)dt=f(x). This is a special case of equation 2.9.37 with g(x)=Axandh(t)=1 . Solution: y(x)=f(x)+Aλ W⎝integraldisplayx ax⎝bracketleftbig u1(x)u2(t)–u2(x)u1(t)⎝bracketrightbig f(t)dt, where u1(x),u2(x) is a fundamental system of solutions of the second-order linear ordinary differential equation u/prime/prime xx+λ(Ax+λ)u=0 ,a n d Wis the Wronskian. The functions u1(x),u2(x), and Ware speciﬁed in 2.5.23. 25. y(x)+A⎝integraldisplay ⎝integraldisplayx atksinm(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Asinm(λx)a n d h(t)=tk. 26. y(x)+A⎝integraldisplay ⎝integraldisplayx axksinm(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Axkandh(t)=s i nm(λt). 27. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asin(kx)+B–AB (x–t)s i n (kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Asin(kx). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[Asin(kx)+B]G(x) G(t)+B2 G(t)⎝integraldisplayx teB(x–s)G(s)ds,G(x)=e x p⎝bracketleftbigg –A kcos(kx )⎝bracketrightbigg . 174 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 28. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Asin(kt)+B+AB (x–t)s i n (kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Asin(kt). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=– [ Asin(kt)+B]G(t) G(x)+B2 G(x)⎝integraldisplayx teB(t–s)G(s)ds,G(x)=e x p⎝bracketleftbigg –A kcos(kx )⎝bracketrightbigg . 29. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xsin⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=Asin⎝parenleftbig λ√ –x⎝parenrightbig . 2.5-3. Kernels Containing Tangent. 30. y(x)–A⎝integraldisplay ⎝integraldisplayx atan(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atan(λx)a n d h(t)=1 . Solution: y(x)=f(x)+A⎝integraldisplayx atan(λx)⎝vextendsingle⎝vextendsingle⎝vextendsinglecos(λt ) cos(λx)⎝vextendsingle⎝vextendsingle⎝vextendsingleA/λ f(t)dt. 31. y(x)–A⎝integraldisplay ⎝integraldisplayx atan(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=t a n ( λt). Solution: y(x)=f(x)+A⎝integraldisplayx atanh(λt)⎝vextendsingle⎝vextendsingle⎝vextendsinglecos(λt ) cos(λx)⎝vextendsingle⎝vextendsingle⎝vextendsingleA/λ f(t)dt. 32. y(x)+A⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig tan(λx)–t a n ( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.5 with g(x)=Atan(λx). Solution: y(x)=f(x)+1 W⎝integraldisplayx a⎝bracketleftbig Y/prime 1(x)Y/prime 2(t)–Y/prime 2(x)Y/prime 1(t)⎝bracketrightbig f(t)dt, where Y1(x),Y2(x) is a fundamental system of solutions of the second-order linear ordinary differential equation cos2(λx)Y/prime/prime xx+AλY =0 ,Wis the Wronskian, and the primes stand for the differentiation with respect to the argument speciﬁed in the parentheses. As shown in A. D. Polyanin and V . F. Zaitsev (2003), the functions Y1(x)a n dY2(x) can be expressed via the hypergeometric function. 33. y(x)–A⎝integraldisplay ⎝integraldisplayx atan(λx) tan(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)tan(λx) tan(λt)f(t)dt. 2.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 175 34. y(x)–A⎝integraldisplay ⎝integraldisplayx atan(λt) tan(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)tan(λt) tan(λx)f(t)dt. 35. y(x)–A⎝integraldisplay ⎝integraldisplayx atank(λx)t a nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atank(λx)a n d h(t)=t a nm(µt). 36. y(x)+A⎝integraldisplay ⎝integraldisplayx atktanm(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Atanm(λx)a n d h(t)=tk. 37. y(x)+A⎝integraldisplay ⎝integraldisplayx axktanm(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Axkandh(t)=t a nm(λt). 38. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atan(kx)+B–AB (x–t)t a n (kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Atan(kx). 39. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Atan(kt)+B+AB (x–t)t a n (kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Atan(kt). 2.5-4. Kernels Containing Cotangent. 40. y(x)–A⎝integraldisplay ⎝integraldisplayx acot(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acot(λx)a n d h(t)=1 . Solution: y(x)=f(x)+A⎝integraldisplayx acot(λx)⎝vextendsingle⎝vextendsingle⎝vextendsinglesin(λx) sin(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingleA/λ f(t)dt. 41. y(x)–A⎝integraldisplay ⎝integraldisplayx acot(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=c o t ( λt). Solution: y(x)=f(x)+A⎝integraldisplayx acoth(λt)⎝vextendsingle⎝vextendsingle⎝vextendsinglesin(λx) sin(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingleA/λ f(t)dt. 42. y(x)–A⎝integraldisplay ⎝integraldisplayx acot(λx) cot(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)cot(λx) cot(λt)f(t)dt. 176 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 43. y(x)–A⎝integraldisplay ⎝integraldisplayx acot(λt) cot(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)cot(λt) cot(λx)f(t)dt. 44. y(x)+A⎝integraldisplay ⎝integraldisplayx atkcotm(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Acotm(λx)a n d h(t)=tk. 45. y(x)+A⎝integraldisplay ⎝integraldisplayx axkcotm(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=–Axkandh(t)=c o tm(λt). 46. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acot(kx)+B–AB (x–t)c o t (kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Acot(kx). 47. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Acot(kt)+B+AB (x–t)c o t (kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Acot(kt). 2.5-5. Kernels Containing Combinations of Trigonometric Functions. 48. y(x)–A⎝integraldisplay ⎝integraldisplayx acosk(λx)s i nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acosk(λx)a n d h(t)=s i nm(µt). 49. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A+Bcos(λx )–B(x–t)[λsin(λx)+Acos(λx )]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.38 with b=Bandg(x)=A. 50. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig A+Bsin(λx)+B(x–t)[λcos(λx )–Asin(λx)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.39 with b=Bandg(x)=A. 51. y(x)–A⎝integraldisplay ⎝integraldisplayx atank(λx)c o tm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atank(λx)a n d h(t)=c o tm(µt). 2.6. Equations Whose Kernels Contain Inverse Trigonometric Functions 2.6-1. Kernels Containing Arccosine. 1. y(x)–A⎝integraldisplay ⎝integraldisplayx aarccos( λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aarccos( λx)a n d h(t)=1 . 2.6. E QUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS 177 2. y(x)–A⎝integraldisplay ⎝integraldisplayx aarccos( λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t) = arccos( λt). 3. y(x)–A⎝integraldisplay ⎝integraldisplayx aarccos( λx) arccos( λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)arccos( λx) arccos( λt)f(t)dt. 4. y(x)–A⎝integraldisplay ⎝integraldisplayx aarccos( λt) arccos( λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)arccos( λt) arccos( λx)f(t)dt. 5. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarccos( kx)+B–AB (x–t) arccos( kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Aarccos( kx). 6. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarccos( kt)+B+AB (x–t) arccos( kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Aarccos( kt). 2.6-2. Kernels Containing Arcsine. 7. y(x)–A⎝integraldisplay ⎝integraldisplayx aarcsin( λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aarcsin( λx)a n d h(t)=1 . 8. y(x)–A⎝integraldisplay ⎝integraldisplayx aarcsin( λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t) = arcsin( λt). 9. y(x)–A⎝integraldisplay ⎝integraldisplayx aarcsin( λx) arcsin( λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)arcsin( λx) arcsin( λt)f(t)dt. 10. y(x)–A⎝integraldisplay ⎝integraldisplayx aarcsin( λt) arcsin( λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)arcsin( λt) arcsin( λx)f(t)dt. 178 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 11. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarcsin( kx)+B–AB (x–t)a r c s i n ( kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Aarcsin( kx). 12. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarcsin( kt)+B+AB (x–t)a r c s i n ( kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Aarcsin( kt). 2.6-3. Kernels Containing Arctangent. 13. y(x)–A⎝integraldisplay ⎝integraldisplayx aarctan( λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aarctan( λx)a n d h(t)=1 . 14. y(x)–A⎝integraldisplay ⎝integraldisplayx aarctan( λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t) = arctan( λt). 15. y(x)–A⎝integraldisplay ⎝integraldisplayx aarctan( λx) arctan( λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)arctan( λx) arctan( λt)f(t)dt. 16. y(x)–A⎝integraldisplay ⎝integraldisplayx aarctan( λt) arctan( λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)arctan( λt) arctan( λx)f(t)dt. 17. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xarctan[ λ(t–x)]y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=Aarctan(– λx). 18. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarctan( kx)+B–AB (x–t)a r c t a n ( kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Aarctan( kx). 19. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarctan( kt)+B+AB (x–t)a r c t a n ( kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Aarctan( kt). 2.6-4. Kernels Containing Arccotangent. 20. y(x)–A⎝integraldisplay ⎝integraldisplayx aarccot( λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aarccot(λx)a n d h(t)=1 . 2.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 179 21. y(x)–A⎝integraldisplay ⎝integraldisplayx aarccot( λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t) = arccot( λt). 22. y(x)–A⎝integraldisplay ⎝integraldisplayx aarccot( λx) arccot( λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)arccot(λx) arccot(λt )f(t)dt. 23. y(x)–A⎝integraldisplay ⎝integraldisplayx aarccot( λt) arccot( λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)arccot(λt ) arccot(λx)f(t)dt. 24. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xarccot[ λ(t–x)]y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=Aarccot(– λx). 25. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarccot( kx)+B–AB (x–t) arccot( kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=Aarccot(kx ). 26. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig Aarccot( kt)+B+AB (x–t) arccot( kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=Aarccot(kt ). 2.7. Equations Whose Kernels Contain Combinations of Elementary Functions 2.7-1. Kernels Containing Exponential and Hyperbolic Functions. 1. y(x)+A⎝integraldisplay ⎝integraldisplayx aeµ(x–t)cosh[λ (x–t)]y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x)=e x p⎝bracketleftbig (µ–1 2A)x⎝bracketrightbig⎝bracketleftbiggA2 2ksinh(kx)–Acosh(kx )⎝bracketrightbigg ,k=⎝radicalBig λ2+1 4A2. 180 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2. y(x)+A⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sinh[λ(x–t)]y(t)dt=f(x). 1◦. Solution with λ(A–λ)>0 : y(x)=f(x)–Aλ k⎝integraldisplayx aeµ(x–t)sin[k(x–t)]f(t)dt,w h e r e k=⎝radicalbig λ(A–λ). 2◦. Solution with λ(A–λ)<0 : y(x)=f(x)–Aλ k⎝integraldisplayx aeµ(x–t)sinh[k(x–t)]f(t)dt,w h e r e k=⎝radicalbig λ(λ–A). 3◦. Solution with A=λ: y(x)=f(x)–λ2⎝integraldisplayx a(x–t)eµ(x–t)f(t)dt. 3. y(x)+⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝braceleftbig ⎝braceleftbig A1sinh[λ1(x–t)] +A2sinh[λ2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 2.3.18: w(x)+⎝integraldisplayx a⎝braceleftbig A1sinh[λ1(x–t)] +A2sinh[λ2(x–t)]⎝bracerightbig w(t)dt=e–µxf(x). 4. y(x)+A⎝integraldisplay ⎝integraldisplayx ateµ(x–t)sinh[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 2.3.23: w(x)+A⎝integraldisplayx atsinh[λ(x–t)]w(t)dt=e–µxf(x). 2.7-2. Kernels Containing Exponential and Logarithmic Functions. 5. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµtln(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aln(λx)a n d h(t)=eµt. 6. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµxln(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aeµxandh(t)=l n ( λt). 7. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµ(x–t)ln(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx ae(µ–A)(x–t)ln(λx)(λx)Ax (λt)Atf(t)dt. 2.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 181 8. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµ(x–t)ln(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx ae(µ–A)(x–t)ln(λt)(λx)Ax (λt)Atf(t)dt. 9. y(x)+A⎝integraldisplay ⎝integraldisplayx aeµ(x–t)(lnx–l nt)y(t)dt=f(x). Solution: y(x)=f(x)+1 W⎝integraldisplayx aeµ(x–t)⎝bracketleftbig u/prime 1(x)u/prime 2(t)–u/prime 2(x)u/prime 1(t)⎝bracketrightbig f(t)dt, where the primes stand for the differentiation with respect to the argument speciﬁed in the parentheses, and u1(x),u2(x) is a fundamental system of solutions of the second-order linear homogeneous ordinary differential equation u/prime/prime xx+Ax–1u=0 ,w i t h u1(x)a n du2(x) expressed in terms of Bessel functions or modiﬁed Bessel functions, depending on the sign of A: W=1 π,u1(x)=√ xJ1⎝parenleftbig 2√ Ax⎝parenrightbig ,u2(x)=√ xY 1⎝parenleftbig 2√ Ax⎝parenrightbig forA>0 , W=–1 2,u1(x)=√ xI1⎝parenleftbig 2√ –Ax⎝parenrightbig ,u2(x)=√ xK 1⎝parenleftbig 2√ –Ax⎝parenrightbig forA<0 . 10. y(x)+a⎝integraldisplay ⎝integraldisplay∞ xeλ(x–t)ln(t–x)y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=aeλxln(–x). 2.7-3. Kernels Containing Exponential and Trigonometric Functions. 11. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµtcos(λx )y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acos(λx)a n d h(t)=eµt. 12. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµxcos(λt )y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aeµxandh(t)=c o s ( λt). 13. y(x)+A⎝integraldisplay ⎝integraldisplayx aeµ(x–t)cos[λ (x–t)]y(t)dt=f(x). 1◦. Solution with |A|>2|λ|: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x)=e x p⎝bracketleftbig (µ–1 2A)x⎝bracketrightbig⎝bracketleftbiggA2 2ksinh(kx)–Acosh(kx)⎝bracketrightbigg ,k=⎝radicalBig 1 4A2–λ2. 2◦. Solution with |A|<2|λ|: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt, R(x)=e x p⎝bracketleftbig (µ–1 2A)x⎝bracketrightbig⎝bracketleftbiggA2 2ksin(kx)–Acos(kx )⎝bracketrightbigg ,k=⎝radicalBig λ2–1 4A2. 3◦. Solution with λ=±1 2A: y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt,R(x)=⎝parenleftbig1 2A2x–A⎝parenrightbig exp⎝bracketleftbig⎝parenleftbig µ–1 2A⎝parenrightbig x⎝bracketrightbig . 182 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 14. y(x)–⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝bracketleftbig⎝bracketleftbig Acos(kx )+B–AB (x–t)c o s (kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx aeµ(x–t)M(x,t)f(t)dt, M(x,t)=[Acos(kx )+B]G(x) G(t)+B2 G(t)⎝integraldisplayx teB(x–s)G(s)ds,G(x)=e x p⎝bracketleftbiggA ksin(kx)⎝bracketrightbigg . 15. y(x)+⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝bracketleftbig⎝bracketleftbig Acos(kt )+B+AB (x–t)c o s (kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx aeµ(x–t)M(x,t)f(t)dt, M(x,t)=– [ Acos(kt )+B]G(t) G(x)+B2 G(x)⎝integraldisplayx teB(t–s)G(s)ds,G(x)=e x p⎝bracketleftbiggA ksin(kx)⎝bracketrightbigg . 16. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµtsin(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asin(λx)a n d h(t)=eµt. 17. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµxsin(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aeµxandh(t)=s i n ( λt). 18. y(x)+A⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sin[λ(x–t)]y(t)dt=f(x). 1◦. Solution with λ(A+λ)>0 : y(x)=f(x)–Aλ k⎝integraldisplayx aeµ(x–t)sin[k(x–t)]f(t)dt,w h e r e k=⎝radicalbig λ(A+λ). 2◦. Solution with λ(A+λ)<0 : y(x)=f(x)–Aλ k⎝integraldisplayx aeµ(x–t)sinh[k(x–t)]f(t)dt,w h e r e k=⎝radicalbig –λ(λ+A). 3◦. Solution with A=–λ: y(x)=f(x)+λ2⎝integraldisplayx a(x–t)eµ(x–t)f(t)dt. 19. y(x)+A⎝integraldisplay ⎝integraldisplayx aeµ(x–t)sin3[λ(x–t)]y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 2.5.17: w(x)+A⎝integraldisplayx asin3[λ(x–t)]w(t)dt=e–µxf(x). 2.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 183 20. y(x)+⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝braceleftbig ⎝braceleftbig A1sin[λ1(x–t)] +A2sin[λ2(x–t)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 2.5.18: w(x)+⎝integraldisplayx a⎝braceleftbig A1sin[λ1(x–t)] +A2sin[λ2(x–t)]⎝bracerightbig w(t)dt=e–µxf(x). 21. y(x)+⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝braceleftbigg ⎝braceleftbiggn⎝summationdisplay k=1Aksin[λk(x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 2.5.19: w(x)+⎝integraldisplayx a⎝braceleftbiggn⎝summationdisplay k=1Aksin[λk(x–t)]⎝bracerightbigg w(t)dt=e–µxf(x). 22. y(x)+A⎝integraldisplay ⎝integraldisplayx ateµ(x–t)sin[λ(x–t)]y(t)dt=f(x). Solution: y(x)=f(x)+Aλ W⎝integraldisplayx ateµ(x–t)⎝bracketleftbig u1(x)u2(t)–u2(x)u1(t)⎝bracketrightbig f(t)dt, where u1(x),u2(x) is a fundamental system of solutions of the second-order linear ordinary differential equation u/prime/prime xx+λ(Ax+λ)u=0 ,a n d Wis the Wronskian. Depending on the sign of Aλ, the functions u1(x)a n du2(x) are expressed in terms of Bessel functions or modiﬁed Bessel functions as follows: ifAλ>0 ,t h e n u1(x)=ξ1/2J1/3⎝parenleftbig2 3√ Aλ ξ3/2⎝parenrightbig ,u2(x)=ξ1/2Y1/3⎝parenleftbig2 3√ Aλξ3/2⎝parenrightbig , W=3/π,ξ=x+(λ/A); ifAλ<0 ,t h e n u1(x)=ξ1/2I1/3⎝parenleftbig2 3√ –Aλξ3/2⎝parenrightbig ,u2(x)=ξ1/2K1/3⎝parenleftbig2 3√ –Aλξ3/2⎝parenrightbig , W=–3 2,ξ=x+(λ/A). 23. y(x)+A⎝integraldisplay ⎝integraldisplayx axeµ(x–t)sin[λ(x–t)]y(t)dt=f(x). Solution: y(x)=f(x)+Aλ W⎝integraldisplayx axeµ(x–t)⎝bracketleftbig u1(x)u2(t)–u2(x)u1(t)⎝bracketrightbig f(t)dt, where u1(x),u2(x) is a fundamental system of solutions of the second-order linear ordinary differential equation u/prime/prime xx+λ(Ax+λ)u=0 ,a n d Wis the Wronskian. The functions u1(x),u2(x), and Ware speciﬁed in 2.7.22. 24. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xeµ(t–x)sin⎝parenleftbig⎝parenleftbig λ√ t–x⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=Ae–µxsin⎝parenleftbig λ√ –x⎝parenrightbig . 184 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 25. y(x)–⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝bracketleftbig⎝bracketleftbig Asin(kx)+B–AB (x–t)s i n (kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx aeµ(x–t)M(x,t)f(t)dt, M(x,t)=[Asin(kx)+B]G(x) G(t)+B2 G(t)⎝integraldisplayx teB(x–s)G(s)ds,G(x)=e x p⎝bracketleftbigg –A kcos(kx )⎝bracketrightbigg . 26. y(x)+⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝bracketleftbig⎝bracketleftbig Asin(kt)+B+AB (x–t)s i n (kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx aeµ(x–t)M(x,t)f(t)dt, M(x,t)=– [ Asin(kt)+B]G(t) G(x)+B2 G(x)⎝integraldisplayx teB(t–s)G(s)ds,G(x)=e x p⎝bracketleftbigg –A kcos(kx )⎝bracketrightbigg . 27. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµttan(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atan(λx)a n d h(t)=eµt. 28. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµxtan(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aeµxandh(t)=t a n ( λt). 29. y(x)+A⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝bracketleftbig⎝bracketleftbig tan(λx)–t a n ( λt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 2.5.32: w(x)+A⎝integraldisplayx a⎝bracketleftbig tan(λx)–t a n ( λt)⎝bracketrightbig w(t)dt=e–µxf(x). 30. y(x)–⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝bracketleftbig⎝bracketleftbig Atan(kx)+B–AB (x–t)t a n (kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 2.9.7 with λ=Band g(x)=Atan(kx): w(x)–⎝integraldisplayx a⎝bracketleftbig Atan(kx)+B–AB(x–t)t a n (kx)⎝bracketrightbig w(t)dt=e–µxf(x). 31. y(x)+⎝integraldisplay ⎝integraldisplayx aeµ(x–t)⎝bracketleftbig⎝bracketleftbig Atan(kt)+B+AB (x–t)t a n (kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). The substitution w(x)=e–µxy(x) leads to an equation of the form 2.9.8 with λ=Band g(t)=Atan(kt): w(x)+⎝integraldisplayx a⎝bracketleftbig Atan(kt)+B+AB(x–t)t a n (kt)⎝bracketrightbig w(t)dt=e–µxf(x). 2.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 185 32. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµtcot(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acot(λx)a n d h(t)=eµt. 33. y(x)–A⎝integraldisplay ⎝integraldisplayx aeµxcot(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aeµxandh(t)=c o t ( λt). 2.7-4. Kernels Containing Hyperbolic and Logarithmic Functions. 34. y(x)–A⎝integraldisplay ⎝integraldisplayx acoshk(λx)l nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acoshk(λx)a n d h(t)=l nm(µt). 35. y(x)–A⎝integraldisplay ⎝integraldisplayx acoshk(λt)l nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Alnm(µx)a n dh(t)=c o s hk(λt). 36. y(x)–A⎝integraldisplay ⎝integraldisplayx asinhk(λx)l nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asinhk(λx)a n d h(t)=l nm(µt). 37. y(x)–A⎝integraldisplay ⎝integraldisplayx asinhk(λt)l nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Alnm(µx)a n dh(t)=s i n hk(λt). 38. y(x)–A⎝integraldisplay ⎝integraldisplayx atanhk(λx)l nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atanhk(λx)a n d h(t)=l nm(µt). 39. y(x)–A⎝integraldisplay ⎝integraldisplayx atanhk(λt)l nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Alnm(µx)a n dh(t)=t a n hk(λt). 40. y(x)–A⎝integraldisplay ⎝integraldisplayx acothk(λx)l nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acothk(λx)a n d h(t)=l nm(µt). 41. y(x)–A⎝integraldisplay ⎝integraldisplayx acothk(λt)l nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Alnm(µx)a n dh(t)=c o t hk(λt). 186 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2.7-5. Kernels Containing Hyperbolic and Trigonometric Functions. 42. y(x)–A⎝integraldisplay ⎝integraldisplayx acoshk(λx)c o sm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acoshk(λx)a n d h(t)=c o sm(µt). 43. y(x)–A⎝integraldisplay ⎝integraldisplayx acoshk(λt)c o sm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acosm(µx)a n dh(t)=c o s hk(λt). 44. y(x)–A⎝integraldisplay ⎝integraldisplayx acoshk(λx)s i nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acoshk(λx)a n d h(t)=s i nm(µt). 45. y(x)–A⎝integraldisplay ⎝integraldisplayx acoshk(λt)s i nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asinm(µx)a n dh(t)=c o s hk(λt). 46. y(x)–A⎝integraldisplay ⎝integraldisplayx asinhk(λx)c o sm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asinhk(λx)a n d h(t)=c o sm(µt). 47. y(x)–A⎝integraldisplay ⎝integraldisplayx asinhk(λt)c o sm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acosm(µx)a n dh(t)=s i n hk(λt). 48. y(x)–A⎝integraldisplay ⎝integraldisplayx asinhk(λx)s i nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asinhk(λx)a n d h(t)=s i nm(µt). 49. y(x)–A⎝integraldisplay ⎝integraldisplayx asinhk(λt)s i nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asinm(µx)a n dh(t)=s i n hk(λt). 50. y(x)–A⎝integraldisplay ⎝integraldisplayx atanhk(λx)c o sm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atanhk(λx)a n d h(t)=c o sm(µt). 51. y(x)–A⎝integraldisplay ⎝integraldisplayx atanhk(λt)c o sm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acosm(µx)a n dh(t)=t a n hk(λt). 52. y(x)–A⎝integraldisplay ⎝integraldisplayx atanhk(λx)s i nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atanhk(λx)a n d h(t)=s i nm(µt). 53. y(x)–A⎝integraldisplay ⎝integraldisplayx atanhk(λt)s i nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asinm(µx)a n dh(t)=t a n hk(λt). 2.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 187 2.7-6. Kernels Containing Logarithmic and Trigonometric Functions. 54. y(x)–A⎝integraldisplay ⎝integraldisplayx acosk(λx)l nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acosk(λx)a n d h(t)=l nm(µt). 55. y(x)–A⎝integraldisplay ⎝integraldisplayx acosk(λt)l nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Alnm(µx)a n dh(t)=c o sk(λt). 56. y(x)–A⎝integraldisplay ⎝integraldisplayx asink(λx)l nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Asink(λx)a n d h(t)=l nm(µt). 57. y(x)–A⎝integraldisplay ⎝integraldisplayx asink(λt)l nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Alnm(µx)a n dh(t)=s i nk(λt). 58. y(x)–A⎝integraldisplay ⎝integraldisplayx atank(λx)l nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Atank(λx)a n d h(t)=l nm(µt). 59. y(x)–A⎝integraldisplay ⎝integraldisplayx atank(λt)l nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Alnm(µx)a n dh(t)=t a nk(λt). 60. y(x)–A⎝integraldisplay ⎝integraldisplayx acotk(λx)l nm(µt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Acotk(λx)a n d h(t)=l nm(µt). 61. y(x)–A⎝integraldisplay ⎝integraldisplayx acotk(λt)l nm(µx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Alnm(µx)a n dh(t)=c o tk(λt). 2.8. Equations Whose Kernels Contain Special Functions 2.8-1. Kernels Containing Bessel Functions. 1. y(x)–λ⎝integraldisplay ⎝integraldisplayx 0J0(x–t)y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx 0R(x–t)f(t)dt, where R(x)=λcos⎝parenleftbig√ 1–λ2x⎝parenrightbig +λ2 √ 1–λ2sin⎝parenleftbig√ 1–λ2x⎝parenrightbig +λ √ 1–λ2⎝integraldisplayx 0sin⎝bracketleftbig√ 1–λ2(x–t)⎝bracketrightbigJ1(t) tdt. Reference: V . I. Smirnov (1974). 188 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2. y(x)–A⎝integraldisplay ⎝integraldisplayx aJν(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=AJν(λx)a n d h(t)=1 . 3. y(x)–A⎝integraldisplay ⎝integraldisplayx aJν(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=Jν(λt). 4. y(x)–A⎝integraldisplay ⎝integraldisplayx aJν(λx) Jν(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)Jν(λx) Jν(λt)f(t)dt. 5. y(x)–A⎝integraldisplay ⎝integraldisplayx aJν(λt) Jν(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)Jν(λt) Jν(λx)f(t)dt. 6. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xJν(λ(t–x))y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=AJν(–λx). 7. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AJν(kx)+B–AB (x–t)Jν(kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=AJν(kx). 8. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AJν(kt)+B+AB (x–t)Jν(kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=AJν(kt). 9. y(x)–λ⎝integraldisplay ⎝integraldisplayx 0eµ(x–t)J0(x–t)y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx 0R(x–t)f(t)dt, where R(x)=eµx⎝braceleftbigg λcos⎝parenleftbig√ 1–λ2x⎝parenrightbig +λ2 √ 1–λ2sin⎝parenleftbig√ 1–λ2x⎝parenrightbig + λ √ 1–λ2⎝integraldisplayx 0sin⎝bracketleftbig√ 1–λ2(x–t)⎝bracketrightbigJ1(t) tdt⎝bracerightbigg . 10. y(x)–A⎝integraldisplay ⎝integraldisplayx aYν(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=AYν(λx)a n d h(t)=1 . 2.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 189 11. y(x)–A⎝integraldisplay ⎝integraldisplayx aYν(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=Yν(λt). 12. y(x)–A⎝integraldisplay ⎝integraldisplayx aYν(λx) Yν(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)Yν(λx) Yν(λt)f(t)dt. 13. y(x)–A⎝integraldisplay ⎝integraldisplayx aYν(λt) Yν(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)Yν(λt) Yν(λx)f(t)dt. 14. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xYν(λ(t–x))y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=AYν(–λx). 15. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AYν(kx)+B–AB (x–t)Yν(kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=AYν(kx). 16. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AYν(kt)+B+AB (x–t)Yν(kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=AYν(kt). 2.8-2. Kernels Containing Modiﬁed Bessel Functions. 17. y(x)–A⎝integraldisplay ⎝integraldisplayx aIν(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=AIν(λx)a n d h(t)=1 . 18. y(x)–A⎝integraldisplay ⎝integraldisplayx aIν(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=Iν(λt). 19. y(x)–A⎝integraldisplay ⎝integraldisplayx aIν(λx) Iν(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)Iν(λx) Iν(λt)f(t)dt. 20. y(x)–A⎝integraldisplay ⎝integraldisplayx aIν(λt) Iν(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)Iν(λt) Iν(λx)f(t)dt. 190 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 21. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xIν(λ(t–x))y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=AIν(–λx). 22. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AIν(kx)+B–AB (x–t)Iν(kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=AIν(kx). 23. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AIν(kt)+B+AB (x–t)Iν(kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=AIν(kt). 24. y(x)–A⎝integraldisplay ⎝integraldisplayx aKν(λx)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=AKν(λx)a n d h(t)=1 . 25. y(x)–A⎝integraldisplay ⎝integraldisplayx aKν(λt)y(t)dt=f(x). This is a special case of equation 2.9.2 with g(x)=Aandh(t)=Kν(λt). 26. y(x)–A⎝integraldisplay ⎝integraldisplayx aKν(λx) Kν(λt)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)Kν(λx) Kν(λt)f(t)dt. 27. y(x)–A⎝integraldisplay ⎝integraldisplayx aKν(λt) Kν(λx)y(t)dt=f(x). Solution: y(x)=f(x)+A⎝integraldisplayx aeA(x–t)Kν(λt) Kν(λx)f(t)dt. 28. y(x)+A⎝integraldisplay ⎝integraldisplay∞ xKν(λ(t–x))y(t)dt=f(x). This is a special case of equation 2.9.62 with K(x)=AKν(–λx). 29. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AK ν(kx)+B–AB (x–t)Kν(kx)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.7 with λ=Bandg(x)=AKν(kx). 30. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig AK ν(kt)+B+AB (x–t)Kν(kt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.8 with λ=Bandg(t)=AKν(kt). 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 191 2.9. Equations Whose Kernels Contain Arbitrary Functions 2.9-1. Equations with Degenerate Kernel: K(x,t)=g1(x)h1(t)+···+gn(x)hn(t). 1. y(x)–λ⎝integraldisplay ⎝integraldisplayx ag(x) g(t)y(t)dt=f(x). Solution: y(x)=f(x)+λ⎝integraldisplayx aeλ(x–t)g(x) g(t)f(t)dt. 2. y(x)–⎝integraldisplay ⎝integraldisplayx ag(x)h(t)y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt,w h e r e R(x,t)=g(x)h(t)e x p⎝bracketleftbigg⎝integraldisplayx tg(s)h(s)ds⎝bracketrightbigg . 3. y(x)+⎝integraldisplay ⎝integraldisplayx a(x–t)g(x)y(t)dt=f(x). This is a special case of equation 2.9.11. 1◦. Solution: y(x)=f(x)+1 W⎝integraldisplayx a⎝bracketleftbig Y1(x)Y2(t)–Y2(x)Y1(t)⎝bracketrightbig g(x)f(t)dt,( 1) where Y1=Y1(x)a n dY2=Y2(x) are two linearly independent solutions ( Y1/Y2/ ≡const) of the second-order linear homogeneous differential equation Y/prime/prime xx+g(x)Y= 0. In this case, the Wronskian is a constant: W=Y1(Y2)/prime x–Y2(Y1)/prime x≡const. 2◦. Given only one nontrivial solution Y1=Y1(x) of the linear homogeneous differential equation Y/prime/prime xx+g(x)Y= 0, one can obtain the solution of the integral equation by formula (1) with W=1 , Y2(x)=Y1(x)⎝integraldisplayx bdξ Y2 1(ξ), where bis an arbitrary number. 4. y(x)+⎝integraldisplay ⎝integraldisplayx a(x–t)g(t)y(t)dt=f(x). This is a special case of equation 2.9.12. 1◦. Solution: y(x)=f(x)+1 W⎝integraldisplayx a⎝bracketleftbig Y1(x)Y2(t)–Y2(x)Y1(t)⎝bracketrightbig g(t)f(t)dt,( 1 ) where Y1=Y1(x)a n dY2=Y2(x) are two linearly independent solutions ( Y1/Y2/ ≡const) of the second-order linear homogeneous differential equation Y/prime/prime xx+g(x)Y= 0. In this case, the Wronskian is a constant: W=Y1(Y2)/prime x–Y2(Y1)/prime x≡const. 2◦. Given only one nontrivial solution Y1=Y1(x) of the linear homogeneous differential equation Y/prime/prime xx+g(x)Y= 0, one can obtain the solution of the integral equation by formula (1) with W=1 , Y2(x)=Y1(x)⎝integraldisplayx bdξ Y2 1(ξ), where bis an arbitrary number. 192 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 5. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(x)–g(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). 1◦. Differentiating the equation with respect to xyields y/prime x(x)+g/prime x(x)⎝integraldisplayx ay(t)dt=f/prime x(x). (1) Introducing the new variable Y(x)=⎝integraldisplayx ay(t)dt, we obtain the second-order linear ordinary differential equation Y/prime/prime xx+g/prime x(x)Y=f/prime x(x), (2) which must be supplemented by the initial conditions Y(a)=0 , Y/prime x(a)=f(a). (3) Conditions (3) follow from the original equation and the deﬁnition of Y(x). For exact solutions of second-order linear ordinary differential equations (2) with vari- ousf(x), see E. Kamke (1977), G. M. Murphy (1960), and A. D. Polyanin and V . F. Zaitsev (2003). 2◦.L e tY1=Y1(x)a n dY2=Y2(x) be two linearly independent solutions (Y 1/Y2/ ≡const) of the second-order linear homogeneous differential equation Y/prime/prime xx+g/prime x(x)Y= 0, which follows from (2) for f(x)≡0. In this case, the Wronskian is a constant: W=Y1(Y2)/prime x–Y2(Y1)/prime x≡const . Solving the nonhomogeneous equation (2) under the initial conditions (3) with arbitrary f=f(x) and taking into account y(x)=Y/prime x(x), we obtain the solution of the original integral equation in the form y(x)=f(x)+1 W⎝integraldisplayx a⎝bracketleftbig Y/prime 1(x)Y/prime 2(t)–Y/prime 2(x)Y/prime 1(t)⎝bracketrightbig f(t)dt,( 4 ) where the primes stand for the differentiation with respect to the argument speciﬁed in the parentheses. 3◦. Given only one nontrivial solution Y1=Y1(x) of the linear homogeneous differential equation Y/prime/prime xx+g/prime x(x)Y= 0, one can obtain the solution of the nonhomogeneous equation (2) under the initial conditions (3) by formula (4) with W=1 , Y2(x)=Y1(x)⎝integraldisplayx bdξ Y2 1(ξ), where bis an arbitrary number. 6. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(x)+h(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). 1◦. Differentiating the equation with respect to xyields y/prime x(x)+⎝bracketleftbig g(x)+h(x)⎝bracketrightbig y(x)+g/prime x(x)⎝integraldisplayx ay(t)dt=f/prime x(x). 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 193 Introducing the new variable Y(x)=⎝integraldisplayx ay(t)dt, we obtain the second-order linear ordinary differential equation Y/prime/prime xx+⎝bracketleftbig g(x)+h(x)⎝bracketrightbig Y/prime x+g/prime x(x)Y=f/prime x(x), (1) which must be supplemented by the initial conditions Y(a)=0 , Y/prime x(a)=f(a). (2) Conditions (3) follow from the original equation and the deﬁnition of Y(x). For exact solutions of second-order linear ordinary differential equations (1) with vari- ousf(x), see E. Kamke (1977), G. M. Murphy (1960), and A. D. Polyanin and V . F. Zaitsev (2003). 2◦.L e tY1=Y1(x)a n dY2=Y2(x) be two linearly independent solutions ( Y1/Y2/ ≡const) of the second-order linear homogeneous differential equation Y/prime/prime xx+⎝bracketleftbig g(x)+h(x)⎝bracketrightbig Y/prime x+g/prime x(x)Y=0 , which follows from (1) for f(x)≡0. Solving the nonhomogeneous equation (1) under the initial conditions (2) with arbitrary f=f(x) and taking into account y(x)=Y/prime x(x), we obtain the solution of the original integral equation in the form y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=∂2 ∂x∂t⎝bracketleftbiggY1(x)Y2(t)–Y2(x)Y1(t) W(t)⎝bracketrightbigg ,W(x)=Y1(x)Y/prime 2(x)–Y2(x)Y/prime 1(x), where W(x) is the Wronskian and the primes stand for the differentiation with respect to the argument speciﬁed in the parentheses. 7. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(x)+λ–λ(x–t)g(x)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.16 with h(x)=λ. Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[g(x)+λ]G(x) G(t)+λ2 G(t)⎝integraldisplayx teλ(x–s)G(s)ds,G(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg . 8. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(t)+λ+λ(x–t)g(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=– [ g(t)+λ]G(t) G(x)+λ2 G(x)⎝integraldisplayx teλ(t–s)G(s)ds,G(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg . 9. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g1(x)+g2(x)t⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This equation can be rewritten in the form of equation 2.9.11 with g1(x)=g(x)+xh(x)a n d g2(x)=–h(x). 194 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 10. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g1(t)+g2(t)x⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This equation can be rewritten in the form of equation 2.9.12 with g1(t)=g(t)+th(t)a n d g2(t)=–h(t). 11. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(x)+h(x)(x –t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). 1◦. The solution of the integral equation can be represented in the form y(x)=Y/prime/prime xx,w h e r e Y=Y(x) is the solution of the second-order linear nonhomogeneous ordinary differential equation Y/prime/prime xx–g(x)Y/prime x–h(x)Y=f(x), (1) under the initial conditions Y(a)=Y/prime x(a)=0 . ( 2 ) 2◦.L e tY1=Y1(x)a n dY2=Y2(x) be two nontrivial linearly independent solutions of the second-order linear homogeneous differential equation Y/prime/prime xx–g(x)Y/prime x–h(x)Y=0, which follows from (1) for f(x)≡0. Then the solution of the nonhomogeneous differential equation (1) under conditions (2) is given by Y(x)=⎝integraldisplayx a⎝bracketleftbig Y2(x)Y1(t)–Y1(x)Y2(t)⎝bracketrightbigf(t) W(t)dt,W(t)=Y1(t)Y/prime 2(t)–Y2(t)Y/prime 1(t), (3) where W(t) is the Wronskian and the primes denote the derivatives. Substituting (3) into (1), we obtain the solution of the original integral equation in the form y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt,R(x,t)=1 W(t)[Y/prime/prime 2(x)Y1(t)–Y/prime/prime 1(x)Y2(t)]. (4) 3◦.L e tY1=Y1(x) be a nontrivial particular solution of the homogeneous differential equa- tion (1) (with f≡0) satisfying the initial condition Y1(a)≠0. Then the function Y2(x)=Y1(x)⎝integraldisplayx aW(t) [Y1(t)]2dt,W(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg (5) is another nontrivial solution of the homogeneous equation. Substituting (5) into (4) yields the solution of the original integral equation in the form y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=g(x)W(x) Y1(x)Y1(t) W(t)+[g(x)Y/prime 1(x)+h(x)Y1(x)]Y1(t) W(t)⎝integraldisplayx tW(s) [Y1(s)]2ds, where W(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg . 12. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig g(t)+h(t)(t–x)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=g(t)Y(x)W(x) Y(t)W(t)+Y(x)W(x)[g(t)Y/prime t(t)+h(t)Y(t)]⎝integraldisplayt xds W(s)[Y(s)]2, W(t)=e x p⎝bracketleftbigg⎝integraldisplayt bg(t)dt⎝bracketrightbigg , 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 195 where Y=Y(x) is an arbitrary nontrivial solution of the second-order homogeneous differ- ential equation Y/prime/prime xx+g(x)Y/prime x+h(x)Y=0 satisfying the condition Y(a)≠0. 13. y(x)+⎝integraldisplay ⎝integraldisplayx a(x–t)g(x)h(t)y(t)dt=f(x). The substitution y(x)=g(x)u(x) leads to an equation of the form 2.9.4: u(x)+⎝integraldisplayx a(x–t)g(t)h(t)u(t)dt=f(x)/g(x). 14. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig g(x)+λxn+λ(x–t)xn–1[n–xg(x)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.16 with h(x)=λxn. Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[g(x)+λxn]G(x) G(t)+λ(λx2n+nxn–1)H(x) G(t)⎝integraldisplayx tG(s) H(s)ds, where G(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg andH(x)=e x p⎝parenleftBigλ n+1xn+1⎝parenrightBig . 15. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig g(x)+λ+(x–t)[g/prime x(x)–λg(x)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.16. Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[g(x)+λ]eλ(x–t)+⎝braceleftbig [g(x)]2+g/prime x(x)⎝bracerightbig G(x)⎝integraldisplayx teλ(s–t) G(s)ds, where G(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg . 16. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig g(x)+h(x)+(x–t)[h/prime x(x)–g(x)h(x)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[g(x)+h(x)]G(x) G(t)+{[h(x)]2+h/prime x(x)}H(x) G(t)⎝integraldisplayx tG(s) H(s)ds, where G(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg andH(x)=e x p⎝bracketleftbigg⎝integraldisplayx ah(s)ds⎝bracketrightbigg . 196 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 17. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝braceleftbigg ⎝braceleftbiggϕ/prime x(x) ϕ(t)+⎝bracketleftbig⎝bracketleftbig ϕ(t)g/prime x(x)–ϕ/prime x(x)g(t)⎝bracketrightbig⎝bracketrightbig h(t)⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x). 1◦. This equation is equivalent to the equation ⎝integraldisplayx a⎝braceleftbiggϕ(x) ϕ(t)+⎝bracketleftbig ϕ(t)g(x)–ϕ(x)g(t)⎝bracketrightbig h(t)⎝bracerightbigg y(t)dt=F(x),F(x)=⎝integraldisplayx af(x)dx,( 1 ) obtained by differentiating the original equation with respect to x. Equation (1) is a special case of equation 1.9.15 with g1(x)=g(x),h1(t)=ϕ(t)h(t),g2(x)=ϕ(x),h2(t)=1 ϕ(t)–g(t)h(t). 2◦. Solution: y(x)=1 ϕ(x)h(x)d dx⎝braceleftbigg Ξ(x)⎝integraldisplayx a⎝bracketleftbiggF(t) ϕ(t)⎝bracketrightbigg/prime tϕ2(t)h(t) Ξ(t)dt⎝bracerightbigg , F(x)=⎝integraldisplayx af(x)dx,Ξ(x)=e x p⎝braceleftbigg –⎝integraldisplayx a⎝bracketleftbiggg(t) ϕ(t)⎝bracketrightbigg/prime tϕ2(t)h(t)dt⎝bracerightbigg . 18. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbigg ⎝braceleftbiggϕ/prime t(t) ϕ(x)+⎝bracketleftbig⎝bracketleftbig ϕ(x)g/prime t(t)–ϕ/prime t(t)g(x)⎝bracketrightbig⎝bracketrightbig h(x)⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x). 1◦.L e tf(a) = 0. The change y(x)=⎝integraldisplayx aw(t)dt (1) followed by the integration by parts leads to the equation ⎝integraldisplayx a⎝braceleftbiggϕ(t) ϕ(x)+⎝bracketleftbig ϕ(x)g(t)–ϕ(t)g(x)⎝bracketrightbig h(x)⎝bracerightbigg w(t)dt=f(x), (2) which is a special case of equation 1.9.15 with g1(x)=1 ϕ(x)–g(x)h(x),h1(t)=ϕ(t),g2(x)=ϕ(x)h(x),h2(t)=g(t). The solution of equation (2) is given by y(x)=1 ϕ(x)d dx⎝braceleftbigg ϕ2(x)h(x)Φ(x)⎝integraldisplayx a⎝bracketleftbiggf(t) ϕ(t)h(t)⎝bracketrightbigg/prime tdt Φ(t)⎝bracerightbigg , Φ(x)=e x p⎝braceleftbigg⎝integraldisplayx a⎝bracketleftbiggg(t) ϕ(t)⎝bracketrightbigg/prime tϕ2(t)h(t)dt⎝bracerightbigg . 2◦.L e tf(a)≠0. The substitution y(x)= ¯y(x)+f(a) leads to the integral equation ¯ y(x) with the right-hand side ¯f(x) satisfying the condition ¯f(a) = 0. Thus we obtain case 1◦. 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 197 19. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1gk(x)(x –t)k–1⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). The solution can be represented in the form y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt.( 1) Here the resolvent R(x,t)i sg i v e nb y R(x,t)=w(n) x,w(n) x=dnw dxn,( 2) where wis the solution of the nth-order linear homogeneous ordinary differential equation w(n) x–g1(x)w(n–1) x –g2(x)w(n–2) x –2g3(x)w(n–3) x –···–(n–1 ) !gn(x)w=0 ( 3 ) satisfying the following initial conditions at x=t: w⎝vextendsingle⎝vextendsingle x=t=w/prime x⎝vextendsingle⎝vextendsingle x=t=···=w(n–2) x⎝vextendsingle⎝vextendsingle x=t=0 , w(n–1) x⎝vextendsingle⎝vextendsingle x=t=1 . ( 4 ) Note that the differential equation (3) implicitly depends on tvia the initial conditions (4). References: E. Goursat (1923), A. F. Verlan’ and V . S. Sizikov (1987). 20. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1gk(t)(t–x)k–1⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). The solution can be represented in the form y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt.( 1) Here the resolvent R(x,t)i sg i v e nb y R(x,t)=–u(n) t,u(n) t=dnu dtn,( 2) where uis the solution of the nth-order linear homogeneous ordinary differential equation u(n) t+g1(t)u(n–1) t +g2(t)u(n–2) t +2g3(t)u(n–3) t +···+(n–1 ) !gn(t)u=0 , ( 3 ) satisfying the following initial conditions at t=x: u⎝vextendsingle⎝vextendsingle t=x=u/prime t⎝vextendsingle⎝vextendsingle t=x=···=u(n–2) t⎝vextendsingle⎝vextendsingle t=x=0 , u(n–1) t⎝vextendsingle⎝vextendsingle t=x=1 . ( 4 ) Note that the differential equation (3) implicitly depends on xvia the initial conditions (4). References: E. Goursat (1923), A. F. Verlan’ and V . S. Sizikov (1987). 21. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig eλx+µt–eµx+λt⎝parenrightbig⎝parenrightbig g(t)y(t)dt=f(x). Let us differentiate the equation twice and then eliminate the integral terms from the resulting relations and the original equation. As a result, we arrive at the second-order linear ordinary differential equation y/prime/prime xx–(λ+µ)y/prime x+⎝bracketleftbig (λ–µ)e(λ+µ)xg(x)+λµ⎝bracketrightbig y=f/prime/prime xx(x)–(λ+µ)f/prime x(x)+λµf(x), which must be supplemented by the initial conditions y(a)=f(a),y/prime x(a)=f/prime x(a). 198 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 22. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig eλxg(t)+eµxh(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Let us differentiate the equation twice and then eliminate the integral terms from the resulting relations and the original equation. As a result, we arrive at the second-order linear ordinarydifferential equation y /prime/prime xx+⎝bracketleftbig eλxg(x)+eµxh(x)–λ–µ⎝bracketrightbig y/prime x+⎝bracketleftbig eλxg/prime x(x)+eµxh/prime x(x) +(λ–µ)eλxg(x)+(µ–λ)eµxh(x)+λµ⎝bracketrightbig y=f/prime/prime xx(x)–(λ+µ)f/prime x(x)+λµf(x), which must be supplemented by the initial conditions y(a)=f(a), y/prime x(a)=f/prime x(a)–⎝bracketleftbig eλag(a)+eµah(a)⎝bracketrightbig f(a). Example. The Arutyunyan equation, y(x)–⎝integraldisplayx aϕ(t)∂ ∂t⎝braceleftbigg1 ϕ(t)+ψ(t)⎝bracketleftBig 1–e–λ(x–t)⎝bracketrightBig⎝bracerightbigg y(t)dt=f(x), can be reduced to the above equation. The former is encountered in the theory of viscoelasticity for aging solids. The solution of the Arutyunyan equation is given by y(x)=f(x)–⎝integraldisplayx a1 ϕ(t)∂ ∂t⎝bracketleftbigg ϕ(t)–λψ(t)ϕ2(t)eη(t)⎝integraldisplayx te–η(s)ds⎝bracketrightbigg f(t)dt, where η(x)=x⎝integraldisplay a⎝braceleftbigg λ⎝bracketleftbig 1+ψ(t)ϕ(t)⎝bracketrightbig –ϕ/prime(t) ϕ(t)⎝bracerightbigg dt. Reference: N. Kh. Arutyunyan (1966). 23. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig λeλ(x–t)+⎝parenleftbig⎝parenleftbig µeµx+λt–λeλx+µt⎝parenrightbig⎝parenrightbig h(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.17 with ϕ(x)=eλxandg(x)=eµx. Solution: y(x)=1 eλxh(x)d dx⎝braceleftbigg Φ(x)⎝integraldisplayx a⎝bracketleftbiggF(t) eλt⎝bracketrightbigg/prime te2λth(t) Φ(t)dt⎝bracerightbigg , F(x)=⎝integraldisplayx af(t)dt,Φ(x)=e x p⎝bracketleftbigg (λ–µ)⎝integraldisplayx ae(λ+µ)th(t)dt⎝bracketrightbigg . 24. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝bracketleftbig⎝bracketleftbig λe–λ(x–t)+⎝parenleftbig⎝parenleftbig µeλx+µt–λeµx+λt⎝parenrightbig⎝parenrightbig h(x)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 2.9.18 with ϕ(x)=eλxandg(x)=eµx. Assume that f(a) = 0. Solution: y(x)=⎝integraldisplayx aw(t)dt,w(x)=e–λxd dx⎝braceleftbigge2λxh(x) Φ(x)⎝integraldisplayx a⎝bracketleftbiggf(t) eλth(t)⎝bracketrightbigg/prime tΦ(t)dt⎝bracerightbigg , Φ(x)=e x p⎝bracketleftbigg (λ–µ)⎝integraldisplayx ae(λ+µ)th(t)dt⎝bracketrightbigg . 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 199 25. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig g(x)+beλx+b(x–t)eλx[λ–g(x)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.16 with h(x)=beλx. Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[g(x)+beλx]G(x) G(t)+(b2e2λx+bλeλx)H(x) G(t)⎝integraldisplayx tG(s) H(s)ds, where G(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg andH(x)=e x p⎝parenleftbiggb λeλx⎝parenrightbigg . 26. y(x)+⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig λeλ(x–t)+⎝bracketleftbig⎝bracketleftbig eλtg/prime x(x)–λeλxg(t)⎝bracketrightbig⎝bracketrightbig h(t)⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.17 with ϕ(x)=eλx. 27. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig λe–λ(x–t)+⎝bracketleftbig⎝bracketleftbig eλxg/prime t(t)–λeλtg(x)⎝bracketrightbig⎝bracketrightbig h(x)⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.18 with ϕ(x)=eλx. 28. y(x)+⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)]g(t)y(t)dt=f(x). Differentiating the equation with respect to xtwice yields y/prime x(x)+g(x)y(x)+λ⎝integraldisplayx asinh[λ(x–t)]g(t)y(t)dt=f/prime x(x), (1) y/prime/prime xx(x)+⎝bracketleftbig g(x)y(x)⎝bracketrightbig/prime x+λ2⎝integraldisplayx acosh[λ(x–t)]g(t)y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order linear ordinary differential equation y/prime/prime xx+⎝bracketleftbig g(x)y⎝bracketrightbig/prime x–λ2y=f/prime/prime xx(x)–λ2f(x). (3) By setting x=ain the original equation and (1), we obtain the initial conditions for y=y(x): y(a)=f(a), y/prime x(a)=f/prime x(a)–f(a)g(a). (4) Equation (3) under conditions (4) determines the solution of the original integral equation. 29. y(x)+⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)]g(x)h(t)y(t)dt=f(x). The substitution y(x)=g(x)u(x) leads to an equation of the form 2.9.28: u(x)+⎝integraldisplayx acosh[λ(x–t)]g(t)h(t)u(t)dt=f(x)/g(x). 200 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 30. y(x)+⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]g(t)y(t)dt=f(x). 1◦. Differentiating the equation with respect to xtwice yields y/prime x(x)+λ⎝integraldisplayx acosh[λ (x–t)]g(t)y(t)dt=f/prime x(x), (1) y/prime/prime xx(x)+λg(x)y(x)+λ2⎝integraldisplayx asinh[λ(x–t)]g(t)y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order linear ordinary differential equation y/prime/prime xx+λ⎝bracketleftbig g(x)–λ⎝bracketrightbig y=f/prime/prime xx(x)–λ2f(x). (3) By setting x=ain the original equation and (1), we obtain the initial conditions for y=y(x): y(a)=f(a), y/prime x(a)=f/prime x(a). (4) For exact solutions of second-order linear ordinary differential equations (3) with vari- ousg(x), see E. Kamke (1977), G. M. Murphy (1960), and A. D. Polyanin and V . F. Zaitsev (2003). 2◦.L e ty1=y1(x)a n dy2=y2(x) be two linearly independent solutions (y 1/y2/ ≡const) of the homogeneous differential equation y/prime/prime xx+λ⎝bracketleftbig g(x)–λ⎝bracketrightbig y= 0, which follows from (3) for f(x)≡0. In this case, the Wronskian is a constant: W=y1(y2)/prime x–y2(y1)/prime x≡const . The solution of the nonhomogeneous equation (3) under conditions (4) with arbitrary f=f(x) has the form y(x)=f(x)+λ W⎝integraldisplayx a⎝bracketleftbig y1(x)y2(t)–y2(x)y1(t)⎝bracketrightbig g(t)f(t)dt (5) and determines the solution of the original integral equation. 3◦. Given only one nontrivial solution y1=y1(x) of the linear homogeneous differential equation y/prime/prime xx+λ⎝bracketleftbig g(x)–λ⎝bracketrightbig y=0, one can obtain the solution of the nonhomogeneous equation (3) under the initial conditions (4) by formula (5) with W=1 , y2(x)=y1(x)⎝integraldisplayx bdξ y2 1(ξ), where bis an arbitrary number. 31. y(x)+⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]g(x)h(t)y(t)dt=f(x). The substitution y(x)=g(x)u(x) leads to an equation of the form 2.9.30: u(x)+⎝integraldisplayx asinh[λ(x–t)]g(t)h(t)u(t)dt=f(x)/g(x). 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 201 32. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig g(x)+bcosh(λx )+b(x–t)[λsinh(λx)–c o s h ( λx)g(x)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.16 with h(x)=bcosh(λx). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[g(x)+bcosh(λx)]G(x) G(t)+⎝bracketleftbig b2cosh2(λx)+bλsinh(λx)⎝bracketrightbigH(x) G(t)⎝integraldisplayx tG(s) H(s)ds, where G(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg andH(x)=e x p⎝bracketleftbiggb λsinh(λx)⎝bracketrightbigg . 33. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig g(x)+bsinh(λx)+b(x–t)[λcosh(λx ) – sinh( λx)g(x)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.16 with h(x)=bsinh(λx). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[g(x)+bsinh(λx)]G(x) G(t)+⎝bracketleftbig b2sinh2(λx)+bλcosh(λx)⎝bracketrightbigH(x) G(t)⎝integraldisplayx tG(s) H(s)ds, where G(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg andH(x)=e x p⎝bracketleftbiggb λcosh(λx)⎝bracketrightbigg . 34. y(x)+⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)]g(t)y(t)dt=f(x). Differentiating the equation with respect to xtwice yields y/prime x(x)+g(x)y(x)–λ⎝integraldisplayx asin[λ(x–t)]g(t)y(t)dt=f/prime x(x), (1) y/prime/prime xx(x)+⎝bracketleftbig g(x)y(x)⎝bracketrightbig/prime x–λ2⎝integraldisplayx acos[λ (x–t)]g(t)y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order linear ordinary differential equation y/prime/prime xx+⎝bracketleftbig g(x)y⎝bracketrightbig/prime x+λ2y=f/prime/prime xx(x)+λ2f(x). (3) By setting x=ain the original equation and (1), we obtain the initial conditions for y=y(x): y(a)=f(a), y/prime x(a)=f/prime x(a)–f(a)g(a). (4) 35. y(x)+⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)]g(x)h(t)y(t)dt=f(x). The substitution y(x)=g(x)u(x) leads to an equation of the form 2.9.34: u(x)+⎝integraldisplayx acos[λ (x–t)]g(t)h(t)u(t)dt=f(x)/g(x). 202 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 36. y(x)+⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)]g(t)y(t)dt=f(x). 1◦. Differentiating the equation with respect to xtwice yields y/prime x(x)+λ⎝integraldisplayx acos[λ (x–t)]g(t)y(t)dt=f/prime x(x), (1) y/prime/prime xx(x)+λg(x)y(x)–λ2⎝integraldisplayx asin[λ(x–t)]g(t)y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order linear ordinary differential equation y/prime/prime xx+λ⎝bracketleftbig g(x)+λ⎝bracketrightbig y=f/prime/prime xx(x)+λ2f(x). (3) By setting x=ain the original equation and (1), we obtain the initial conditions for y=y(x): y(a)=f(a), y/prime x(a)=f/prime x(a). (4) For exact solutions of second-order linear ordinary differential equations (3) with vari- ousf(x), see E. Kamke (1977) and A. D. Polyanin and V . F. Zaitsev (2003). 2◦.L e ty1=y1(x)a n dy2=y2(x) be two linearly independent solutions (y 1/y2/ ≡const) of the homogeneous differential equation y/prime/prime xx+λ⎝bracketleftbig g(x)–λ⎝bracketrightbig y= 0, which follows from (3) for f(x)≡0. In this case, the Wronskian is a constant: W=y1(y2)/prime x–y2(y1)/prime x≡const . The solution of the nonhomogeneous equation (3) under conditions (4) with arbitrary f=f(x) has the form y(x)=f(x)+λ W⎝integraldisplayx a⎝bracketleftbig y1(x)y2(t)–y2(x)y1(t)⎝bracketrightbig g(t)f(t)dt (5) and determines the solution of the original integral equation. 3◦. Given only one nontrivial solution y1=y1(x) of the linear homogeneous differential equa- tiony/prime/prime xx+λ⎝bracketleftbig g(x)+λ⎝bracketrightbig y= 0, one can obtain the solution of the nonhomogeneous equation (3) under the initial conditions (4) by formula (5) with W=1 , y2(x)=y1(x)⎝integraldisplayx bdξ y2 1(ξ), where bis an arbitrary number. 37. y(x)+⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)]g(x)h(t)y(t)dt=f(x). The substitution y(x)=g(x)u(x) leads to an equation of the form 2.9.36: u(x)+⎝integraldisplayx asin[λ(x–t)]g(t)h(t)u(t)dt=f(x)/g(x). 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 203 38. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig g(x)+bcos(λx )–b(x–t)[λsin(λx)+c o s ( λx)g(x)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.16 with h(x)=bcos(λx). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[g(x)+bcos(λx)]G(x) G(t)+⎝bracketleftbig b2cos2(λx)–bλsin(λx)⎝bracketrightbigH(x) G(t)⎝integraldisplayx tG(s) H(s)ds, where G(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg andH(x)=e x p⎝bracketleftbiggb λsin(λx)⎝bracketrightbigg . 39. y(x)–⎝integraldisplay ⎝integraldisplayx a⎝braceleftbig ⎝braceleftbig g(x)+bsin(λx)+b(x–t)[λcos(λx )–s i n ( λx)g(x)]⎝bracerightbig ⎝bracerightbig y(t)dt=f(x). This is a special case of equation 2.9.16 with h(x)=bsin(λx). Solution: y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, R(x,t)=[g(x)+bsin(λx)]G(x) G(t)+⎝bracketleftbig b2sin2(λx)+bλcos(λx)⎝bracketrightbigH(x) G(t)⎝integraldisplayx tG(s) H(s)ds, where G(x)=e x p⎝bracketleftbigg⎝integraldisplayx ag(s)ds⎝bracketrightbigg andH(x)=e x p⎝bracketleftbigg –b λcos(λx)⎝bracketrightbigg . 2.9-2. Equations with Difference Kernel: K(x,t)=K(x–t). 40. y(x)+⎝integraldisplay ⎝integraldisplayx aK(x–t)y(t)dt=f(x). Renewal equation. 1◦. To solve this integral equation, direct and inverse Laplace transforms are used. The solution can be represented in the form y(x)=f(x)–⎝integraldisplayx aR(x–t)f(t)dt.( 1 ) Here the resolvent R(x) is expressed via the kernel K(x) of the original equation as follows: R(x)=1 2πi⎝integraldisplayc+i∞ c–i∞˜R(p)epxdp, ˜R(p)=˜K(p) 1+˜K(p), ˜K(p)=⎝integraldisplay∞ 0K(x)e–pxdx. References: R. Bellman and K. L. Cooke (1963), M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971), V . I. Smirnov (1974). 2◦.L e tw=w(x) be the solution of the simpler auxiliary equation with a=0a n d f≡1: w(x)+⎝integraldisplayx 0K(x–t)w(t)dt=1 . ( 2 ) Then the solution of the original integral equation with arbitrary f=f(x) is expressed via the solution of the auxiliary equation (2) as y(x)=d dx⎝integraldisplayx aw(x–t)f(t)dt=f(a)w(x–a)+⎝integraldisplayx aw(x–t)f/prime t(t)dt. Reference: R. Bellman and K. L. Cooke (1963). 204 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 41. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=0 . Eigenfunctions of this integral equation are determined by the roots of the following tran- scendental (algebraic) equation for the parameter λ:⎝integraldisplay∞ 0K(z)e–λzdz= –1. (1) The left-hand side of this equation is the Laplace transform of the kernel of the integral equation. 1◦. For a real simple root λkof equation (1) there is a corresponding eigenfunction yk(x)=e x p ( λkx). 2◦. For a real root λkof multiplicity rthere are corresponding reigenfunctions yk1(x)=e x p ( λkx),yk2(x)=xexp(λkx),...,ykr(x)=xr–1exp(λkx). 3◦. For a complex simple root λk=αk+iβkof equation (1) there is a corresponding eigenfunction pair y(1) k(x)=e x p ( αkx)c o s (βkx),y(2) k(x)=e x p ( αkx)s i n (βkx). 4◦. For a complex root λk=αk+iβkof multiplicity rthere are corresponding reigenfunction pairs y(1) k1(x)=e x p ( αkx)c o s (βkx), y(1) k2(x)=xexp(αkx)c o s (βkx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(1) kr(x)=xr–1exp(αkx)c o s (βkx),y(2) k1(x)=e x p ( αkx)s i n (βkx), y(2) k2(x)=xexp(αkx)s i n (βkx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(2) kr(x)=xr–1exp(αkx)s i n (βkx). The general solution is the combination (with arbitrary constants) of the eigenfunctions of the homogeneous integral equation. /trianglerightsldF or equations 2.9.42–2.9.51, only particular solutions are given. T o obtain the general solu- tion, one must add the general solution of the corresponding homogeneous equation 2.9.41 to the particular solution. 42. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Axn,n= 0 ,1 ,2 , ... This is a special case of equation 2.9.44 with λ=0 . 1◦. A solution with n=0 : y(x)=A B,B=1+⎝integraldisplay∞ 0K(z)dz. 2◦. A solution with n=1 : y(x)=A Bx+AC B2,B=1+⎝integraldisplay∞ 0K(z)dz,C=⎝integraldisplay∞ 0zK(z)dz. 3◦. A solution with n=2 : y2(x)=A Bx2+2AC B2x+2AC2 B3–AD B2, B=1+⎝integraldisplay∞ 0K(z)dz,C=⎝integraldisplay∞ 0zK(z)dz,D=⎝integraldisplay∞ 0z2K(z)dz. 4◦. A solution with n=3 ,4 , ...is given by: yn(x)=A⎝braceleftbigg∂n ∂λn⎝bracketleftBigeλx B(λ)⎝bracketrightBig⎝bracerightbigg λ=0,B(λ)=1+⎝integraldisplay∞ 0K(z)e–λzdz. 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 205 43. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Aeλx. A solution: y(x)=A Beλx,B=1+⎝integraldisplay∞ 0K(z)e–λzdz. The integral term in the expression for Bis the Laplace transform of K(z), which may be calculated using tables of Lapl ace transforms (e.g., see Supplement 5). 44. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Axneλx,n=1 , 2 , ... 1◦. A solution with n=1 : y1(x)=A Bxeλx+AC B2eλx, B=1+⎝integraldisplay∞ 0K(z)e–λzdz,C=⎝integraldisplay∞ 0zK(z)e–λzdz. It is convenient to calculate BandCusing tables of Laplace transforms. 2◦. A solution with n=2 : y2(x)=A Bx2eλx+2AC B2xeλx+⎝parenleftbigg 2AC2 B3–AD B2⎝parenrightbigg eλx, B=1+⎝integraldisplay∞ 0K(z)e–λzdz,C=⎝integraldisplay∞ 0zK(z)e–λzdz,D=⎝integraldisplay∞ 0z2K(z)e–λzdz. 3◦. A solution with n=3 ,4 , ...is given by: yn(x)=∂ ∂λyn–1(x)=A∂n ∂λn⎝bracketleftbiggeλx B(λ)⎝bracketrightbigg ,B(λ)=1+⎝integraldisplay∞ 0K(z)e–λzdz. 45. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Acosh(λx ). A solution: y(x)=A 2B–eλx+A 2B+e–λx=1 2⎝parenleftBigA B–+A B+⎝parenrightBig cosh(λx)+1 2⎝parenleftBigA B––A B+⎝parenrightBig sinh(λx), B–=1+⎝integraldisplay∞ 0K(z)e–λzdz,B+=1+⎝integraldisplay∞ 0K(z)eλzdz. 46. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Asinh(λx). A solution: y(x)=A 2B–eλx–A 2B+e–λx=1 2⎝parenleftBigA B––A B+⎝parenrightBig cosh(λx)+1 2⎝parenleftBigA B–+A B+⎝parenrightBig sinh(λx), B–=1+⎝integraldisplay∞ 0K(z)e–λzdz,B+=1+⎝integraldisplay∞ 0K(z)eλzdz. 206 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 47. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Acos(λx ). A solution: y(x)=A B2c+B2s⎝bracketleftbig Bccos(λx)– Bssin(λx)⎝bracketrightbig , Bc=1+⎝integraldisplay∞ 0K(z)c o s (λz)dz,Bs=⎝integraldisplay∞ 0K(z)s i n (λz)dz. 48. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Asin(λx). A solution: y(x)=A B2c+B2s⎝bracketleftbig Bcsin(λx)+Bscos(λx)⎝bracketrightbig , Bc=1+⎝integraldisplay∞ 0K(z)c o s (λz)dz,Bs=⎝integraldisplay∞ 0K(z)s i n (λz)dz. 49. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Aeµxcos(λx ). A solution: y(x)=A B2c+B2seµx⎝bracketleftbig Bccos(λx)– Bssin(λx)⎝bracketrightbig , Bc=1+⎝integraldisplay∞ 0K(z)e–µzcos(λz )dz,Bs=⎝integraldisplay∞ 0K(z)e–µzsin(λz)dz. 50. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=Aeµxsin(λx). A solution: y(x)=A B2c+B2seµx⎝bracketleftbig Bcsin(λx)+Bscos(λx)⎝bracketrightbig , Bc=1+⎝integraldisplay∞ 0K(z)e–µzcos(λz )dz,Bs=⎝integraldisplay∞ 0K(z)e–µzsin(λz)dz. 51. y(x)+⎝integraldisplay ⎝integraldisplayx –∞K(x–t)y(t)dt=f(x). 1◦. For a polynomial right-hand side, f(x)=n⎝summationtext k=0Akxk, a solution has the form y(x)=n⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermi ned coefﬁcients. One can also make use of the formula given in item 4◦of equation 2.9.42 to construct the solution. 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 207 2◦.F o rf(x)=eλxn⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=eλxn⎝summationdisplay k=0Bkxk, where the Bkare found by the method of undetermined coefﬁcients. One can also make use of the formula given in item 3◦of equation 2.9.44 to construct the solution. 3◦.F o rf(x)=n⎝summationtext k=0Akexp(λkx), a solution of the equation has the form y(x)=n⎝summationdisplay k=0Ak Bkexp(λkx), Bk=1+⎝integraldisplay∞ 0K(z)e x p ( – λkz)dz. 4◦.F o rf(x)=c o s ( λx)n⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 5◦.F o rf(x)=s i n ( λx)n⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 6◦.F o rf(x)=n⎝summationtext k=0Akcos(λ kx), the solution of a equation has the form y(x)=n⎝summationdisplay k=0Ak B2 ck+B2 sk⎝bracketleftbig Bckcos(λ kx)–Bsksin(λkx)⎝bracketrightbig , Bck=1+⎝integraldisplay∞ 0K(z)c o s (λkz)dz,Bsk=⎝integraldisplay∞ 0K(z)s i n (λkz)dz. 7◦.F o rf(x)=n⎝summationtext k=0Aksin(λkx), a solution of the equation has the form y(x)=n⎝summationdisplay k=0Ak B2 ck+B2 sk⎝bracketleftbig Bcksin(λkx)+Bskcos(λ kx)⎝bracketrightbig , Bck=1+⎝integraldisplay∞ 0K(z)c o s (λkz)dz,Bsk=⎝integraldisplay∞ 0K(z)s i n (λkz)dz. 8◦.F o rf(x)=c o s ( λx)n⎝summationtext k=0Akexp(µkx), a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=0AkBck B2 ck+B2 skexp(µkx)–s i n ( λx)n⎝summationdisplay k=0AkBsk B2 ck+B2 skexp(µkx), Bck=1+⎝integraldisplay∞ 0K(z)e x p ( – µkz)c o s (λz)dz,Bsk=⎝integraldisplay∞ 0K(z)e x p ( – µkz)s i n (λz)dz. 208 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 9◦.F o rf(x)=s i n ( λx)n⎝summationtext k=0Akexp(µkx), a solution of the equation has the form y(x)=s i n ( λx)n⎝summationdisplay k=0AkBck B2 ck+B2 skexp(µkx)+c o s ( λx)n⎝summationdisplay k=0AkBsk B2 ck+B2 skexp(µkx), Bck=1+⎝integraldisplay∞ 0K(z)e x p ( – µkz)c o s (λz)dz,Bsk=⎝integraldisplay∞ 0K(z)e x p ( – µkz)s i n (λz)dz. 52. y(x)+⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=0 . Eigenfunctions of this integral equation are determined by the roots of the following tran- scendental (algebraic) equation for the parameter λ: ⎝integraldisplay∞ 0K(–z)eλzdz= –1. (1) The left-hand side of this equation is the Laplace transform of the function K(–z) with parameter – λ. 1◦. For a real simple root λkof equation (1) there is a corresponding eigenfunction yk(x)=e x p ( λkx). 2◦. For a real root λkof multiplicity rthere are corresponding reigenfunctions yk1(x)=e x p ( λkx),yk2(x)=xexp(λkx),...,ykr(x)=xr–1exp(λkx). 3◦. For a complex simple root λk=αk+iβkof equation (1) there is a corresponding eigenfunction pair y(1) k(x)=e x p ( αkx)c o s (βkx),y(2) k(x)=e x p ( αkx)s i n (βkx). 4◦. For a complex root λk=αk+iβkof multiplicity rthere are corresponding reigenfunction pairs y(1) k1(x)=e x p ( αkx)c o s (βkx), y(1) k2(x)=xexp(αkx)c o s (βkx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(1) kr(x)=xr–1exp(αkx)c o s (βkx),y(2) k1(x)=e x p ( αkx)s i n (βkx), y(2) k2(x)=xexp(αkx)s i n (βkx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(2) kr(x)=xr–1exp(αkx)s i n (βkx). The general solution is the combination (with arbitrary constants) of the eigenfunctions of the homogeneous integral equation. /trianglerightsldF or equations 2.9.53–2.9.62, only particular solutions are given. T o obtain the general solu- tion, one must add the general solution of the corresponding homogeneous equation 2.9.52 to theparticular solution. 53. y(x)+⎝integraldisplay ⎝integraldisplay ∞ xK(x–t)y(t)dt=Axn,n=0 , 1 , 2 , ... This is a special case of equation 2.9.55 with λ=0 . 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 209 1◦. A solution with n=0 : y(x)=A B,B=1+⎝integraldisplay∞ 0K(–z)dz. 2◦. A solution with n=1 : y(x)=A Bx–AC B2,B=1+⎝integraldisplay∞ 0K(–z)dz,C=⎝integraldisplay∞ 0zK(–z)dz. 3◦. A solution with n=2 : y2(x)=A Bx2–2AC B2x+2AC2 B3–AD B2, B=1+⎝integraldisplay∞ 0K(–z)dz,C=⎝integraldisplay∞ 0zK(–z)dz,D=⎝integraldisplay∞ 0z2K(–z)dz. 4◦. A solution with n=3 ,4 , ...is given by: yn(x)=A⎝braceleftbigg∂n ∂λn⎝bracketleftBigeλx B(λ)⎝bracketrightBig⎝bracerightbigg λ=0,B(λ)=1+⎝integraldisplay∞ 0K(–z)eλzdz. 54. y(x)+⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Aeλx. A solution: y(x)=A Beλx,B=1+⎝integraldisplay∞ 0K(–z)eλzdz=1+ L{K(–z), –λ}. The integral term in the expression for Bis the Laplace transform of K(–z) with parameter – λ, which may be calculated using tables of Laplace transforms (e.g., see H. Bateman andA. Erd ´elyi (vol. 1, 1954), V . A. Ditkin and A. P. Prudnikov (1965), and Supplement 5). 55. y(x)+⎝integraldisplay ⎝integraldisplay ∞ xK(x–t)y(t)dt=Axneλx,n=1 , 2 , ... 1◦. A solution with n=1 : y1(x)=A Bxeλx–AC B2eλx, B=1+⎝integraldisplay∞ 0K(–z)eλzdz,C=⎝integraldisplay∞ 0zK(–z)eλzdz. It is convenient to calculate BandCusing tables of Laplace transforms (with parameter – λ). 2◦. A solution with n=2 : y2(x)=A Bx2eλx–2AC B2xeλx+⎝parenleftbigg 2AC2 B3–AD B2⎝parenrightbigg eλx, B=1+⎝integraldisplay∞ 0K(–z)eλzdz,C=⎝integraldisplay∞ 0zK(–z)eλzdz,D=⎝integraldisplay∞ 0z2K(–z)eλzdz. 3◦. A solution with n=3 ,4 , ...is given by yn(x)=∂ ∂λyn–1(x)=A∂n ∂λn⎝bracketleftbiggeλx B(λ)⎝bracketrightbigg ,B(λ)=1+⎝integraldisplay∞ 0K(–z)eλzdz. 210 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 56. y(x)+⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Acosh(λx ). A solution: y(x)=A 2B+eλx+A 2B–e–λx=1 2⎝parenleftBigA B++A B–⎝parenrightBig cosh(λx)+1 2⎝parenleftBigA B+–A B–⎝parenrightBig sinh(λx), B+=1+⎝integraldisplay∞ 0K(–z)eλzdz,B–=1+⎝integraldisplay∞ 0K(–z)e–λzdz. 57. y(x)+⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Asinh(λx). A solution: y(x)=A 2B+eλx–A 2B–e–λx=1 2⎝parenleftBigA B+–A B–⎝parenrightBig cosh(λx)+1 2⎝parenleftBigA B++A B–⎝parenrightBig sinh(λx), B+=1+⎝integraldisplay∞ 0K(–z)eλzdz,B–=1+⎝integraldisplay∞ 0K(–z)e–λzdz. 58. y(x)+⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Acos(λx ). A solution: y(x)=A B2c+B2s⎝bracketleftbig Bccos(λx)+ Bssin(λx)⎝bracketrightbig , Bc=1+⎝integraldisplay∞ 0K(–z)c o s (λz)dz,Bs=⎝integraldisplay∞ 0K(–z)s i n (λz)dz. 59. y(x)+⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Asin(λx). A solution: y(x)=A B2c+B2s⎝bracketleftbig Bcsin(λx)–Bscos(λx)⎝bracketrightbig , Bc=1+⎝integraldisplay∞ 0K(–z)c o s (λz)dz,Bs=⎝integraldisplay∞ 0K(–z)s i n (λz)dz. 60. y(x)+⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Aeµxcos(λx ). A solution: y(x)=A B2c+B2seµx⎝bracketleftbig Bccos(λx)+ Bssin(λx)⎝bracketrightbig , Bc=1+⎝integraldisplay∞ 0K(–z)eµzcos(λz )dz,Bs=⎝integraldisplay∞ 0K(–z)eµzsin(λz)dz. 61. y(x)+⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=Aeµxsin(λx). A solution: y(x)=A B2c+B2seµx⎝bracketleftbig Bcsin(λx)–Bscos(λx)⎝bracketrightbig , Bc=1+⎝integraldisplay∞ 0K(–z)eµzcos(λz )dz,Bs=⎝integraldisplay∞ 0K(–z)eµzsin(λz)dz. 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 211 62. y(x)+⎝integraldisplay ⎝integraldisplay∞ xK(x–t)y(t)dt=f(x). 1◦. For a polynomial right-hand side, f(x)=n⎝summationtext k=0Akxk, a solution has the form y(x)=n⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermi ned coefﬁcients. One can also make use of the formula given in item 4◦of equation 2.9.53 to construct the solution. 2◦.F o rf(x)=eλxn⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=eλxn⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermi ned coefﬁcients. One can also make use of the formula given in item 3◦of equation 2.9.55 to construct the solution. 3◦.F o rf(x)=n⎝summationtext k=0Akexp(λkx), a solution of the equation has the form y(x)=n⎝summationdisplay k=0Ak Bkexp(λkx), Bk=1+⎝integraldisplay∞ 0K(–z)e x p (λkz)dz. 4◦.F o rf(x)=c o s ( λx)n⎝summationtext k=0Akxka solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 5◦.F o rf(x)=s i n ( λx)n⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the BkandCkare found by the method of undetermined coefﬁcients. 6◦.F o rf(x)=n⎝summationtext k=0Akcos(λ kx), a solution of the equation has the form y(x)=n⎝summationdisplay k=0Ak B2 ck+B2 sk⎝bracketleftbig Bckcos(λ kx)+Bsksin(λkx)⎝bracketrightbig , Bck=1+⎝integraldisplay∞ 0K(–z)c o s (λkz)dz,Bsk=⎝integraldisplay∞ 0K(–z)s i n (λkz)dz. 212 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 7◦.F o rf(x)=n⎝summationtext k=0Aksin(λkx), a solution of the equation has the form y(x)=n⎝summationdisplay k=0Ak B2 ck+B2 sk⎝bracketleftbig Bcksin(λkx)–Bskcos(λ kx)⎝bracketrightbig , Bck=1+⎝integraldisplay∞ 0K(–z)c o s (λkz)dz,Bsk=⎝integraldisplay∞ 0K(–z)s i n (λkz)dz. 8◦.F o rf(x)=c o s ( λx)n⎝summationtext k=0Akexp(µkx), a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=0AkBck B2 ck+B2 skexp(µkx)+s i n ( λx)n⎝summationdisplay k=0AkBsk B2 ck+B2 skexp(µkx), Bck=1+⎝integraldisplay∞ 0K(–z)e x p (µkz)c o s (λz)dz,Bsk=⎝integraldisplay∞ 0K(–z)e x p (µkz)s i n (λz)dz. 9◦.F o rf(x)=s i n ( λx)n⎝summationtext k=0Akexp(µkx), a solution of the equation has the form y(x)=s i n ( λx)n⎝summationdisplay k=0AkBck B2 ck+B2 skexp(µkx)–c o s ( λx)n⎝summationdisplay k=0AkBsk B2 ck+B2 skexp(µkx), Bck=1+⎝integraldisplay∞ 0K(–z)e x p (µkz)c o s (λz)dz,Bsk=⎝integraldisplay∞ 0K(–z)e x p (µkz)s i n (λz)dz. 10◦. In the general case of arbitrary right-hand side f=f(x), the solution of the integral equation can be represented in the form y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞˜f(p) 1+˜k(–p)epxdp, ˜f(p)=⎝integraldisplay∞ 0f(x)e–pxdx, ˜k(–p)=⎝integraldisplay∞ 0K(–z)epzdz. To calculate ˜f(p)a n d ˜k(–p), it is convenient to use tables of Laplace transforms, and to determine y(x), tables of inverse Laplace transforms. 2.9-3. Other Equations. 63. y(x)+⎝integraldisplay ⎝integraldisplayx 01 xf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=0 . Eigenfunctions of this integral equation are determined by the roots of the following tran- scendental (algebraic) equation for the parameter λ: ⎝integraldisplay1 0f(z)zλdz= –1. (1) 1◦. For a real simple root λkof equation (1) there is a corresponding eigenfunction yk(x)=xλk. 2.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 213 2◦. For a real root λkof multiplicity rthere are corresponding reigenfunctions yk1(x)=xλk,yk2(x)=xλklnx,...,ykr(x)=xλklnr–1x. 3◦. For a complex simple root λk=αk+iβkof equation (1) there is a corresponding eigenfunction pair y(1) k(x)=xαkcos(β klnx),y(2) k(x)=xαksin(βklnx). 4◦. For a complex root λk=αk+iβkof multiplicity rthere are corresponding reigenfunction pairs y(1) k1(x)=xαkcos(β klnx), y(1) k2(x)=xαklnxcos(β klnx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(1) kr(x)=xαklnr–1xcos(β klnx),y(2) k1(x)=xαksin(βklnx), y(2) k2(x)=xαklnxsin(βklnx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(2) kr(x)=xαklnr–1xsin(βklnx). The general solution is the combination (with arbitrary constants) of the eigenfunctions of the homogeneous integral equation. /trianglerightsldF or equations 2.9.64–2.9.71, only particular solutions are given. T o obtain the general solu- tion, one must add the general solution of the corresponding homogeneous equation 2.9.63 to the particular solution. 64. y(x)+⎝integraldisplay ⎝integraldisplayx 01 xf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=Ax +B. A solution: y(x)=A 1+I1x+B 1+I0,I0=⎝integraldisplay1 0f(t)dt,I1=⎝integraldisplay1 0tf(t)dt. 65. y(x)+⎝integraldisplay ⎝integraldisplayx 01 xf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=Axβ. A solution: y(x)=A Bxβ,B=1+⎝integraldisplay1 0f(t)tβdt. 66. y(x)+⎝integraldisplay ⎝integraldisplayx 01 xf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=Alnx+B. A solution: y(x)=plnx+q, where p=A 1+I0,q=B 1+I0–AIl (1 +I0)2,I0=⎝integraldisplay1 0f(t)dt,Il=⎝integraldisplay1 0f(t)l ntd t. 67. y(x)+⎝integraldisplay ⎝integraldisplayx 01 xf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=Axβlnx. A solution: y(x)=pxβlnx+qxβ, where p=A 1+I1,q=–AI2 (1 +I1)2,I1=⎝integraldisplay1 0f(t)tβdt,I2=⎝integraldisplay1 0f(t)tβlntd t. 214 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 68. y(x)+⎝integraldisplay ⎝integraldisplayx 01 xf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=Acos(ln x). A solution: y(x)=AIc I2c+I2scos(ln x)+AIs I2c+I2ssin(lnx), Ic=1+⎝integraldisplay1 0f(t)c o s ( l n t)dt,Is=⎝integraldisplay1 0f(t)s i n ( l n t)dt. 69. y(x)+⎝integraldisplay ⎝integraldisplayx 01 xf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=Asin(ln x). A solution: y(x)=–AIs I2c+I2scos(ln x)+AIc I2c+I2ssin(lnx), Ic=1+⎝integraldisplay1 0f(t)c o s ( l n t)dt,Is=⎝integraldisplay1 0f(t)s i n ( l n t)dt. 70. y(x)+⎝integraldisplay ⎝integraldisplayx 01 xf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=Axβcos(ln x)+Bxβsin(ln x). A solution: y(x)=pxβcos(ln x)+qxβsin(lnx), where p=AIc–BIs I2c+I2s,q=AIs+BIc I2c+I2s, Ic=1+⎝integraldisplay1 0f(t)tβcos(ln t)dt,Is=⎝integraldisplay1 0f(t)tβsin(lnt)dt. 71. y(x)+⎝integraldisplay ⎝integraldisplayx 01 xf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=g(x). 1◦. For a polynomial right-hand side, g(x)=N⎝summationdisplay n=0Anxn a solution bounded at zero is given by y(x)=N⎝summationdisplay n=0An 1+fnxn,fn=⎝integraldisplay1 0f(z)zndz. Here it is assumed that f0<∞andfn≠–1 (n=0 ,1 ,2 , ...). If for some nthe relation fn= –1 holds, then a solution differs from the above case in one term and has the form y(x)=n–1⎝summationdisplay m=0Am 1+fmxm+N⎝summationdisplay m=n+1Am 1+fmxm+An ¯fnxnlnx, ¯fn=⎝integraldisplay1 0f(z)znlnzd z. For arbitrary g(x) expandable into power series, the formulas of item 1◦can be used, in which one should set N=∞. In this case, the convergenceradius of the obtained solution y(x) is equal to that of the function g(x). 2.10. S OME FORMULAS AND TRANSFORMATIONS 215 2◦.F o rg(x)=l nxn⎝summationtext k=0Akxk, a solution has the form y(x)=l nxn⎝summationdisplay k=0Bkxk+n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 3◦.F o rg(x)=n⎝summationtext k=0Ak⎝parenleftbig lnx)k, a solution of the equation has the form y(x)=n⎝summationdisplay k=0Bk⎝parenleftbig lnx)k, where the Bkare found by the method of undetermined coefﬁcients. 4◦.F o rg(x)=n⎝summationtext k=1Akcos(λ klnx), a solution of the equation has the form y(x)=n⎝summationdisplay k=1Bkcos(λ klnx)+n⎝summationdisplay k=1Cksin(λklnx), where the BkandCkare found by the method of undetermined coefﬁcients. 5◦.F o rg(x)=n⎝summationtext k=1Aksin(λklnx) a solution of the equation has the form y(x)=n⎝summationdisplay k=1Bkcos(λ klnx)+n⎝summationdisplay k=1Cksin(λklnx), where the BkandCkare found by the method of undetermined coefﬁcients. 6◦. For arbitrary right-hand side g(x), the transformation x=e–z,t=e–τ,y(x)=ezw(z),f(ξ)=F(lnξ),g(x)=ezG(z) leads to an equation with difference kernel of the form 2.9.62: w(z)+⎝integraldisplay∞ zF(z–τ)w(τ)dτ=G(z). 7◦. For arbitrary right-hand side g(x), the solution of the integral equation can be expressed via the inverse Mellin transform (see Example 2 in Subsection 11.6-4). 2.10. Some Formulas and Transformations Let the solution of the integral equation y(x)+⎝integraldisplayx aK(x,t)y(t)dt=f(x)( 1) have the form y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt.( 2) 216 LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION Then the solution of the more complicated integral equation y(x)+⎝integraldisplayx aK(x,t)g(x) g(t)y(t)dt=f(x)( 3) has the form y(x)=f(x)+⎝integraldisplayx aR(x,t)g(x) g(t)f(t)dt.( 4) Below are formulas for the solutions of integral equations of the form (3) for some speciﬁc func- tionsg(x). In all cases, it is assumed that the solution of equation (1) is known and is given by (2). 1◦. The solution of the equation y(x)+⎝integraldisplayx aK(x,t)(x/t )λy(t)dt=f(x) has the form y(x)=f(x)+⎝integraldisplayx aR(x,t)(x/t )λf(t)dt. 2◦. The solution of the equation y(x)+⎝integraldisplayx aK(x,t)eλ(x–t)y(t)dt=f(x) has the form y(x)=f(x)+⎝integraldisplayx aR(x,t)eλ(x–t)f(t)dt. Chapter 3 Linear Equations of the First Kind with Constant Limits of Integration /trianglerightsld Notation: f=f(x),g=g(x),h=h(x),K=K(x), andM=M(x)are arbitrary functions (these may be composite functions of the argument depending on two variables xandt);A,B,C,a,b,c, k,α,β,γ,λ, andµare free parameters; and nis a nonnegative integer . 3.1. Equations Whose Kernels Contain Power-Law Functions 3.1-1. Kernels Linear in the Arguments xandt. 1.⎝integraldisplay ⎝integraldisplay1 0|x–t|y(t)dt=f(x). 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayx 0(x–t)y(t)dt+⎝integraldisplay1 x(t–x)y(t)dt=f(x). (1) Differentiating (1) with respect to xyields ⎝integraldisplayx 0y(t)dt–⎝integraldisplay1 xy(t)dt=f/prime x(x). (2) Differentiating (2) yields the solution y(x)=1 2f/prime/prime xx(x). (3) 2◦. Let us demonstrate that the right-hand side f(x) of the integral equation must satisfy certain relations. By setting x=0a n d x= 1 in (1), we obtain two corollaries⎝integraldisplay1 0ty(t)dt=f(0) and⎝integraldisplay1 0(1 –t)y(t)dt=f(1), which can be rewritten in the form ⎝integraldisplay1 0ty(t)dt=f(0),⎝integraldisplay1 0y(t)dt=f(0) +f(1). (4) In Section 3.1, we mean that kernels of the integral equations discussed may contain power-law functions or modulus of power-law functions. 217 218 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION Substitute y(x) of (3) into (4). Integration by parts yields f/prime x(1) =f(1)+f(0) and f/prime x(1)–f/prime x(0) = 2f(1) + 2f (0). Hence, we obtain the desired constraints for f(x): f/prime x(1) =f(0) +f(1), f/prime x(0) +f/prime x(1) = 0. (5) Conditions (5) make it possible to ﬁnd the admissible general form of the right-hand side of the integral equation: f(x)=F(x)+Ax+B, A=–1 2⎝bracketleftbig F/prime x(1) +F/prime x(0)⎝bracketrightbig ,B=1 2⎝bracketleftbig F/prime x(1) –F(1) –F(0)⎝bracketrightbig , where F(x) is an arbitrary bounded twice differentiable function with bounded ﬁrst derivative. 2.⎝integraldisplay ⎝integraldisplayb a|x–t|y(t)dt=f(x), 0 ≤a<b<∞. This is a special case of equation 3.8.3 with g(x)=x. Solution: y(x)=1 2f/prime/prime xx(x). The right-hand side f(x) of the integral equation must satisfy certain relations. The general form of f(x)i sa sf o l l o w s : f(x)=F(x)+Ax+B, A=–1 2⎝bracketleftbig F/prime x(a)+F/prime x(b)⎝bracketrightbig ,B=1 2⎝bracketleftbig aF/prime x(a)+bF/prime x(b)–F(a)–F(b)⎝bracketrightbig , where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive). 3.⎝integraldisplay ⎝integraldisplaya 0|λx –t|y(t)dt=f(x), λ>0 . Here 0 ≤x≤aand 0 ≤t≤a. 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayλx 0(λx–t)y(t)dt+⎝integraldisplaya λx(t–λx)y(t)dt=f(x). (1) Differentiating (1) with respect to x,w eﬁ n dt h a t λ⎝integraldisplayλx 0y(t)dt–λ⎝integraldisplaya λxy(t)dt=f/prime x(x). (2) Differentiating (2) yields 2 λ2y(λx)= f/prime/prime xx(x). Hence, we obtain the solution y(x)=1 2λ2f/prime/prime xx⎝parenleftBigx λ⎝parenrightBig .( 3) 2◦. Let us demonstrate that the right-hand side f(x) of the integral equation must satisfy certain relations. By setting x= 0 in (1) and (2), we obtain two corollaries⎝integraldisplaya 0ty(t)dt=f(0), λ⎝integraldisplaya 0y(t)dt=–f/prime x(0). (4) Substitute y(x) from (3) into (4). Integrating by parts yields the desired constraints for f(x): (a/λ)f/prime x(a/λ)=f(0) +f(a/λ), f/prime x(0) +f/prime x(a/λ)=0 . ( 5 ) Conditions (5) make it possible to establish the admissible general form of the right-hand side of the integral equation: f(x)=F(z)+Az+B,z=λx; A=–1 2⎝bracketleftbig F/prime z(a)+F/prime z(0)⎝bracketrightbig ,B=1 2⎝bracketleftbig aF/prime z(a)–F(a)–F(0)⎝bracketrightbig , where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive). 3.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 219 4.⎝integraldisplay ⎝integraldisplaya 0|x–λt|y(t)dt=f(x), λ>0 . Here 0 ≤x≤aand 0 ≤t≤a. Solution: y(x)=1 2λf/prime/prime xx(λx). The right-hand side f(x) of the integral equation must satisfy the relations aλf/prime x(aλ)=f(0) +f(aλ), f/prime x(0) +f/prime x(aλ)=0 . Hence, it follows the general form of the right-hand side:f(x)=F(x)+Ax+B,A=– 1 2⎝bracketleftbig F/prime x(λa)+F/prime x(0)⎝bracketrightbig ,B=1 2⎝bracketleftbig aλF/prime x(aλ)–F(λa)–F(0)⎝bracketrightbig , where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive). 3.1-2. Kernels Quadratic in the Arguments xandt. 5.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleAx +Bx2–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), A>0 , B>0 . This is a special case of equation 3.8.5 with g(x)=Ax+Bx2. 6.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–At–Bt2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), A>0 , B>0 . This is a special case of equation 3.8.6 with g(x)=At+Bt2. 7.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglext–t2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), 0 ≤a<b<∞. The substitution w(t)=ty(t) leads to an equation of the form 3.1.2: ⎝integraldisplayb a|x–t|w(t)dt=f(x). 8.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex2–t2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=x2. Solution: y(x)=d dx⎝bracketleftbiggf/prime x(x) 4x⎝bracketrightbigg . The right-hand side f(x) of the equation must satisfy certain constraints, given in 3.8.3. 9.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex2–βt2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), β>0 . This is a special case of equation 3.8.4 with g(x)=x2andβ=λ2. 10.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleAx +Bx2–Aλt –Bλ2t2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), λ>0 . This is a special case of equation 3.8.4 with g(x)=Ax+Bx2. 220 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.1-3. Kernels Containing Integer Powers of xandtor Rational Functions. 11.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle3y(t)dt=f(x). Let us remove the modulus in the integrand: ⎝integraldisplayx a(x–t)3y(t)dt+⎝integraldisplayb x(t–x)3y(t)dt=f(x). (1) Differentiating (1) twice yields 6⎝integraldisplayx a(x–t)y(t)dt+6⎝integraldisplayb x(t–x)y(t)dt=f/prime/prime xx(x). This equation can be rewritten in the form 3.1.2: ⎝integraldisplayb a|x–t|y(t)dt=1 6f/prime/prime xx(x). (2) Therefore the solution of the integral equation is given by y(x)=1 12y/prime/prime/prime/prime xxxx(x). (3) The right-hand side f(x) of the equation must satisfy cert ain conditions. To obtain these conditions, one must substitute solution (3) into (1) with x=aandx=band into (2) with x=aandx=b, and then integrate the four resulting relations by parts. 12.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex3–t3⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=x3. 13.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglext2–t3⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x)0 ≤a<b<∞. The substitution w(t)=t2y(t) leads to an equation of the form 3.1.2: ⎝integraldisplayb a|x–t|w(t)dt=f(x). 14.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex2t–t3⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). The substitution w(t)=|t|y(t) leads to an equation of the form 3.1.8: ⎝integraldisplayb a⎝vextendsingle⎝vextendsinglex2–t2⎝vextendsingle⎝vextendsinglew(t)dt=f(x). 15.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex3–βt3⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), β>0 . This is a special case of equation 3.8.4 with g(x)=x3andβ=λ3. 3.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 221 16.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle2n+1y(t)dt=f(x), n= 0 ,1 ,2 , ... Solution: y(x)=1 2(2n+1 ) !f(2n+2) x (x). (1) The right-hand side f(x) of the equation must satisfy cert ain conditions. To obtain these conditions, one must substitute solution (1) into the relations ⎝integraldisplayb a(t–a)2n+1y(t)dt=f(a),⎝integraldisplayb a(t–a)2n–ky(t)dt=(–1)k+1 Akf(k+1) x(a), Ak=( 2n+ 1)(2n )...(2n+1–k); k=0 ,1 , ...,2n, and then integrate the resulting equations by parts. 17.⎝integraldisplay ⎝integraldisplay∞ 0y(t)dt x+t=f(x). The left-hand side of this equation is the Stieltjes transform . 1◦. By setting x=ez,t=eτ,y(t)=e–τ/2w(τ),f(x)=e–z/2g(z), we obtain an integral equation with difference kernel of the form 3.8.15: ⎝integraldisplay∞ –∞w(τ)dτ 2c o s h⎝bracketleftbig1 2(z–τ)⎝bracketrightbig=g(z), whose solution is given by w(z)=1 √ 2π3⎝integraldisplay∞ –∞cosh(πu )˜g(u)eiuxdu,˜g(u)=1 √ 2π⎝integraldisplay∞ –∞g(z)e–iuzdz,i2= –1. 2◦. Solution: y(x)=1 2πilim ε→+0⎝bracketleftbig f(–x–iε)–f(–x+iε)⎝bracketrightbig =1 π√ x∞⎝summationdisplay k=0(–1)k (2k)!⎝parenleftbiggπ xd dx⎝parenrightbigg2k⎝bracketleftbig√ xf(x)⎝bracketrightbig . 3◦. Under some assumptions, the solution of the original equation can be represented in the form y(x) = lim n→∞(–1)n (n+ 1)!(n–1 )⎝bracketleftbig x2n+1f(n) x(x)⎝bracketrightbig(n+1) x,( 1) which is the real inversion of the Stieltjes transform. An alternative form of the solution is y(x) = lim n→∞(–1)n 2π⎝parenleftBige n⎝parenrightBig2n⎝bracketleftbig x2nf(n) x(x)⎝bracketrightbig(n) x.( 2) To obtain an approximate solution of the integral equation, one restricts oneself to a speciﬁc value of nin (1) or (2) instead of taking the limit. References: E. A. C. Paley and N. Wiener (1934), D. V . Widder (1939, 1971), I. I. Hirschman and D. V . Widder (1955), P. P. Zabreyko, A. I. Koshelev, et al. (1975), E. C. Titchmarsh (1986), Yu. A. Brychkov and A. P. Prudnikov(1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 428). 222 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.1-4. Kernels Containing Square Roots. 18.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle√ x–√ t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), 0 < a<∞. This is a special case of equation 3.8.3 with g(x)=√ x. Solution: y(x)=d dx⎝bracketleftbig√ xf/prime x(x)⎝bracketrightbig . The right-hand side f(x) of the equation must satisfy certain conditions. The general form of the right-hand side is f(x)=F(x)+Ax+B,A=–F/prime x(a),B=1 2⎝bracketleftbig aF/prime x(a)–F(a)–F(0)⎝bracketrightbig , where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive). 19.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle√ x–β√ t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), β>0 . This is a special case of equation 3.8.4 with g(x)=√ xandβ=√ λ. 20.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle√ x–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.5 with g(x)=√ x(see item 3◦of 3.8.5). 21.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–√ t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.6 with g(t)=√ t(see item 3◦of 3.8.6). 22.⎝integraldisplay ⎝integraldisplaya 0y(t) √ |x–t|dt=f(x), 0 < a≤∞. This is a special case of equation 3.1.30 with k=1 2. Solution: y(x)=–A x1/4d dx⎝bracketleftbigg⎝integraldisplaya xdt (t–x)1/4⎝integraldisplayt 0f(s)ds s1/4(t–s)1/4⎝bracketrightbigg ,A=1 √ 8πΓ2(3/4). 23.⎝integraldisplay ⎝integraldisplay∞ –∞y(t) √ |x–t|dt=f(x). This is a special case of equation 3.1.35 with λ=1 2. Solution: y(x)=1 4π⎝integraldisplay∞ –∞f(x)–f(t) |x–t|3/2dt. 24.⎝integraldisplay ⎝integraldisplay1 –1y(t)dt √ 1+x2–2xt=f(x). Solution: y(x)=1 2∞⎝summationdisplay n=02n+1 n!f(n) x(0)Pn(x), where Pn(x) are the Legendre polynomials (see Supplement 11.11-1) Pn(x)=1 n!2ndn dxn(x2–1 )n. P. M. Morse and H. Feshbach (1953). 3.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 223 3.1-5. Kernels Containing Arbitrary Powers. 25.⎝integraldisplay ⎝integraldisplaya 0|xk–tk|y(t)dt=f(x), 0 < k<1 , 0< a<∞. 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayx 0(xk–tk)y(t)dt+⎝integraldisplaya x(tk–xk)y(t)dt=f(x). (1) Differentiating (1) with respect to xyields kxk–1⎝integraldisplayx 0y(t)dt–kxk–1⎝integraldisplaya xy(t)dt=f/prime x(x). (2) Let us divide both sides of (2) by kxk–1and differentiate the resulting equation. As a result, we obtain the solution y(x)=1 2kd dx⎝bracketleftbig x1–kf/prime x(x)⎝bracketrightbig .( 3) 2◦. Let us demonstrate that the right-hand side f(x) of the integral equation must satisfy certain relations. By setting x=0and x=a,in (1), we obtain two corollaries⎝integraldisplaya 0tky(t)dt=f(0) and⎝integraldisplaya 0(ak–tk)y(t)dt=f(a), which can be rewritten in the form ⎝integraldisplaya 0tky(t)dt=f(0), ak⎝integraldisplaya 0y(t)dt=f(0) +f(a). (4) Substitute y(x) of (3) into (4). Integrating by parts yields the relations af/prime x(a)=kf(a)+kf(0) andaf/prime x(a)=2kf(a)+2kf(0). Hence, the desired constraints for f(x) have the form f(0) +f(a)=0 , f/prime x(a)=0 . ( 5 ) Conditions (5) make it possible to ﬁnd the admissible general form of the right-hand side of the integral equation: f(x)=F(x)+Ax+B,A=–F/prime x(a),B=1 2⎝bracketleftbig aF/prime x(a)–F(a)–F(0)⎝bracketrightbig , where F(x) is an arbitrary bounded twice differentiable function with bounded ﬁrst derivative. The ﬁrst derivative may be unbounded at x= 0, in which case the conditions⎝bracketleftbig x1–kF/prime x⎝bracketrightbig x=0=0 must hold. 26.⎝integraldisplay ⎝integraldisplaya 0|xk–βtk|y(t)dt=f(x), 0 < k<1 , β>0 . This is a special case of equation 3.8.4 with g(x)=xkandβ=λk. 27.⎝integraldisplay ⎝integraldisplaya 0|xktm–tk+m|y(t)dt=f(x), 0 < k<1 , 0< a<∞. The substitution w(t)=tmy(t) leads to an equation of the form 3.1.25: ⎝integraldisplaya 0|xk–tk|w(t)dt=f(x). 224 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 28.⎝integraldisplay ⎝integraldisplay1 0|xk–tm|y(t)dt=f(x), k>0 , m>0 . The transformation z=xk,τ=tm,w(τ)=τ1–m my(t) leads to an equation of the form 3.1.1: ⎝integraldisplay1 0|z–τ|w(τ)dτ=F(z), F(z)=mf(z1/k). 29.⎝integraldisplay ⎝integraldisplayb a|x–t|1+λy(t)dt=f(x), 0 ≤λ<1 . Forλ= 0, see equation 3.1.2. Assume that 0 < λ<1 . 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayx a(x–t)1+λy(t)dt+⎝integraldisplayb x(t–x)1+λy(t)dt=f(x). (1) Let us differentiate (1) with respect to xtwice and then divide both the sides by λ(λ+1 ) .A s a result, we obtain ⎝integraldisplayx a(x–t)λ–1y(t)dt+⎝integraldisplayb x(t–x)λ–1y(t)dt=1 λ(λ+1 )f/prime/prime xx(x). (2) Rewrite equation (2) in the form ⎝integraldisplayb ay(t)dt |x–t|k=1 λ(λ+1 )f/prime/prime xx(x), k=1–λ.( 3) See 3.1.30 and 3.1.31 for the solutions of equation (3) for various aandb. 2◦. The right-hand side f(x) of the integral equation must satisfy certain relations. By setting x=aandx=bin (1), we obtain two corollaries ⎝integraldisplayb a(t–a)1+λy(t)dt=f(a),⎝integraldisplayb a(b–t)1+λy(t)dt=f(b). (4) On substituting the solution y(x) of (3) into (4) and then integrating by parts, we obtain the desired constraints for f(x). 30.⎝integraldisplay ⎝integraldisplaya 0y(t) |x–t|kdt=f(x), 0 < k<1 , 0< a≤∞. 1◦. Solution: y(x)=–Axk–1 2d dx⎝bracketleftBigg⎝integraldisplaya xt1–2k 2dt (t–x)1–k 2⎝integraldisplayt 0f(s)ds s1–k 2(t–s)1–k 2⎝bracketrightBigg , A=1 2πcos⎝parenleftBigπk 2⎝parenrightBig Γ(k)⎝bracketleftbigg Γ⎝parenleftbigg1+k 2⎝parenrightbigg⎝bracketrightbigg–2 , whereΓ(k) is the gamma function. 2◦. The transformation x=z2,t=ξ2,w(ξ)=2ξy(t) leads to an equation of the form 3.1.32: ⎝integraldisplay√ a 0w(ξ) |z2–ξ2|kdξ=f⎝parenleftbig z2⎝parenrightbig . 3.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 225 31.⎝integraldisplay ⎝integraldisplayb ay(t) |x–t|kdt=f(x), 0 < k<1 . It is assumed that |a|+|b|<∞. Solution: y(x)=1 2πcot(1 2πk)d dx⎝integraldisplayx af(t)dt (x–t)1–k–1 π2cos2(1 2πk)⎝integraldisplayx aZ(t)F(t) (x–t)1–kdt, where Z(t)=(t–a)1+k 2(b–t)1–k 2,F(t)=d dt⎝bracketleftbigg⎝integraldisplayt adτ (t–τ)k⎝integraldisplayb τf(s)ds Z(s)(s–τ)1–k⎝bracketrightbigg . Reference: F. D. Gakhov (1977). 32.⎝integraldisplay ⎝integraldisplaya 0y(t) |x2–t2|kdt=f(x), 0 < k<1 , 0< a≤∞. Solution: y(x)=–2Γ(k)c o s⎝parenleftbig1 2πk⎝parenrightbig π⎝bracketleftbig Γ⎝parenleftbig1+k 2⎝parenrightbig⎝bracketrightbig2xk–1d dx⎝integraldisplaya xt2–2kF(t)dt (t2–x2)1–k 2,F(t)=⎝integraldisplayt 0skf(s)ds (t2–s2)1–k 2. Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975). 33.⎝integraldisplay ⎝integraldisplayb ay(t) |xλ–tλ|kdt=f(x), 0 < k<1 , λ>0 . 1◦. The transformation z=xλ,τ=tλ,w(τ)=τ1–λ λy(t) leads to an equation of the form 3.1.31: ⎝integraldisplayB Aw(τ) |z–τ|kdτ=F(z), where A=aλ,B=bλ,F(z)=λf(z1/λ). 2◦. Solution with a=0 : y(x)=–Axλ(k–1) 2d dx⎝bracketleftBigg⎝integraldisplayb xtλ(3–2k )–2 2dt (tλ–xλ)1–k 2⎝integraldisplayt 0sλ(k+1)–2 2f(s)ds (tλ–sλ)1–k 2⎝bracketrightBigg , A=λ2 2πcos⎝parenleftbiggπk 2⎝parenrightbigg Γ(k)⎝bracketleftbigg Γ⎝parenleftbigg1+k 2⎝parenrightbigg⎝bracketrightbigg–2 , whereΓ(k) is the gamma function. 34.⎝integraldisplay ⎝integraldisplay1 0y(t) |xλ–tm|kdt=f(x), 0 < k<1 , λ>0 , m>0 . The transformation z=xλ,τ=tm,w(τ)=τ1–m my(t) leads to an equation of the form 3.1.31: ⎝integraldisplay1 0w(τ) |z–τ|kdτ=F(z), F(z)=mf(z1/λ). 226 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 35.⎝integraldisplay ⎝integraldisplay∞ –∞y(t) |x–t|1–λdt=f(x), 0 < Re λ<1 . Solution: y(x)=λ 2πtan⎝parenleftBigπλ 2⎝parenrightBig⎝integraldisplay∞ –∞f(x)–f(t) |x–t|1+λdt =λ 2πtan⎝parenleftBigπλ 2⎝parenrightBig⎝integraldisplay∞ 02f(x)–f(x+t)–f(x–t) t1+λdt. It is assumed that the condition⎝integraldisplay∞ –∞|f(x)|pdx<∞is satisﬁed for some p,1<p<1/λ. The integral equation and its solution form the Riesz transform pair (the Riesz potential ). References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 428), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 36.⎝integraldisplay ⎝integraldisplay∞ –∞y(t) |x3–t|1–λdt=f(x), 0 < λ<1 . The substitution z=x3leads to an equation of the form 3.1.35: ⎝integraldisplay∞ –∞y(t) |z–t|1–λdt=f⎝parenleftbig z1/3⎝parenrightbig . 37.⎝integraldisplay ⎝integraldisplay∞ –∞y(t) |x3–t3|1–λdt=f(x), 0 < λ<1 . The transformation z=x3,τ=t3,w(τ)=τ–2/3y(t) leads to an equation of the form 3.1.35: ⎝integraldisplay∞ –∞w(τ) |z–τ|1–λdτ=F(z), F(z)=3f⎝parenleftbig z1/3⎝parenrightbig . 38.⎝integraldisplay ⎝integraldisplay∞ –∞sign(x–t) |x–t|1–λy(t)dt=f(x), 0 < Re λ<1 . Solution: y(x)=λ 2πcot⎝parenleftBigπλ 2⎝parenrightBig⎝integraldisplay∞ –∞f(x)–f(t) |x–t|1+λsign(x–t)dt =λ 2πcot⎝parenleftBigπλ 2⎝parenrightBig⎝integraldisplay∞ 0f(x+t)–f(x–t) t1+λdt =λ 2πcot⎝parenleftBigπλ 2⎝parenrightBigd dx⎝integraldisplay∞ –∞f(t) |x–t|λdt. The integral equation and its solution form the Feller transform pair (the Feller potential ). References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 428), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 39.⎝integraldisplay ⎝integraldisplay∞ –∞a+bsign(x–t) |x–t|1–λy(t)dt=f(x), 0 < Re λ<1 . Solution: y(x)=Cλ⎝integraldisplay∞ –∞a+bsign(x–t) |x–t|1+λ⎝bracketleftbig f(x)–f(t)⎝bracketrightbig dt =Cλ⎝integraldisplay∞ 0t–1–λ⎝bracketleftbig 2af(x)–(a+b)f(x–t)–(a–b)f(x+t)⎝bracketrightbig dt =Cd dx⎝integraldisplay∞ –∞b+asign(x–t) |x–t|λf(t)dt, 3.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 227 where C=sin(πλ) 4π⎝bracketleftbig a2cos2⎝parenleftbig1 2πλ⎝parenrightbig +b2sin2⎝parenleftbig1 2πλ⎝parenrightbig⎝bracketrightbig. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 431), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 40.⎝integraldisplay ⎝integraldisplay∞ 0y(t)dt (ax +bt)k=f(x), a>0 , b>0 , k>0 . By setting x=1 2ae2z,t=1 2be2τ,y(t)=be(k–2)τw(τ),f(x)=e–kzg(z), we obtain an integral equation with the difference kernel of the form 3.8.15: ⎝integraldisplay∞ –∞w(τ)dτ coshk(z–τ)=g(z). 41.⎝integraldisplay ⎝integraldisplay∞ 0tz–1y(t)dt=f(z). The left-hand side of this equation is the Mellin transform of y(t)(zis treated as a complex variable). Solution: y(t)=1 2πi⎝integraldisplayc+i∞ c–i∞t–zf(z)dz,i2= –1. For speciﬁc f(z), one can use tables of Mellin and Laplace integral transforms to calculate the integral. References: H. Bateman and A. Erd ´elyi (vol. 2, 1954), V . A. Ditkin and A. P. Prudnikov (1965). 3.1-6. Equations Containing the Unknown Function of a Complicated Argument. 42.⎝integraldisplay ⎝integraldisplay1 0y(xt)dt=f(x). Solution: y(x)=xf/prime x(x)+f(x). The function f(x) is assumed to satisfy the condition⎝bracketleftbig xf(x)⎝bracketrightbig x=0=0 . 43.⎝integraldisplay ⎝integraldisplay1 0tλy(xt)dt=f(x). The substitution ξ=xtleads to equation⎝integraldisplayx 0ξλy(ξ)dξ=xλ+1f(x). Differentiating with respect to xyields the solution y(x)=xf/prime x(x)+(λ+1 )f(x). The function f(x) is assumed to satisfy the condition⎝bracketleftbig xλ+1f(x)⎝bracketrightbig x=0=0 . 228 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 44.⎝integraldisplay ⎝integraldisplay1 0⎝parenleftbig⎝parenleftbig Axk+Btm)y(xt)dt=f(x). The substitution ξ=xtleads to an equation of the form 1.1.51: ⎝integraldisplayx 0⎝parenleftbig Axk+m+Bξm⎝parenrightbig y(ξ)dξ=xm+1f(x). 45.⎝integraldisplay ⎝integraldisplay1 0y(xt)dt √ 1–t=f(x). The substitution ξ=xtleads to Abel’s equation 1.1.36: ⎝integraldisplayx 0y(ξ)dξ √ x–ξ=√ xf(x). 46.⎝integraldisplay ⎝integraldisplay1 0y(xt)dt (1 –t)λ=f(x), 0 < λ<1 . The substitution ξ=xtleads to the generalized Abel equation 1.1.47: ⎝integraldisplayx 0y(ξ)dξ (x–ξ)λ=x1–λf(x). 47.⎝integraldisplay ⎝integraldisplay1 0tµy(xt) (1 –t)λdt=f(x), 0 < λ<1 . The transformation ξ=xt,w(ξ)=ξµy(ξ) leads to the generalized Abel equation 1.1.47: ⎝integraldisplayx 0w(ξ)dξ (x–ξ)λ=x1+µ–λf(x). 48.⎝integraldisplay ⎝integraldisplay∞ 0y(x+t)–y(x–t) tdt=f(x). Solution: y(x)=–1 π2⎝integraldisplay∞ 0f(x+t)–f(x–t) tdt. References: V . A. Ditkin and A. P. Prudnikov (1965), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 427). 3.1-7. Singular Equations. In this subsection, all singular integrals are understood in the sense of the Cauchy principal value. 49.⎝integraldisplay ⎝integraldisplay∞ –∞y(t)dt t–x=f(x). Solution: y(x)=–1 π2⎝integraldisplay∞ –∞f(t)dt t–x. The integral equation and its solution form a Hilbert transform pair (in the asymmetric form). References: V . A. Ditkin and A. P. Prudnikov (1965), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 427). 3.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 229 50.⎝integraldisplay ⎝integraldisplay∞ 0y(t)dt t–x=f(x). Solution: y(x)=–√ x π2⎝integraldisplay∞ 0f(t) √ t(t–x)dt. The integral equation and its solution form a Hilbert transform pair on the semiaxis (in the asymmetric form). References: D. Hilbert (1953), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 427), I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004, p. 8). 51.⎝integraldisplay ⎝integraldisplayb ay(t)dt t–x=f(x). This equation is encountered in hydrodynamics in solving the problem on the ﬂow of an ideal inviscid ﬂuid around a thin proﬁle ( a≤x≤b). It is assumed that |a|+|b|<∞. 1◦. The solution bounded at the endpoints is y(x)=–1 π2⎝radicalbig (x–a)(b–x)⎝integraldisplayb af(t) √ (t–a)(b–t)dt t–x, provided that⎝integraldisplayb af(t)dt √ (t–a)(b–t)=0 . 2◦. The solution bounded at the endpoint x=aand unbounded at the endpoint x=bis y(x)=–1 π2⎝radicalbigg x–a b–x⎝integraldisplayb a⎝radicalbigg b–t t–af(t) t–xdt. 3◦. The solution unbounded at the endpoints is y(x)=–1 π2√ (x–a)(b–x)⎝bracketleftbigg⎝integraldisplayb a√ (t–a)(b–t) t–xf(t)dt+C⎝bracketrightbigg , where Cis an arbitrary constant. The formula⎝integraldisplayb ay(t)dt=C/π holds. Solutions that have a singularity point x=sinside the interval [ a,b] can be found in Subsection 14.4-3. Reference: F. D. Gakhov (1977). 52.⎝integraldisplay1 –1⎝parenleftbigg ⎝parenleftbigg1 t–x+1 x+t+2⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), –1 < x<1 . Solution for f(x)=πq= const: y(t)=q1+t √ (1 –t)(3 + t). Reference: H. F. Bueckner (1966). 230 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 53.⎝integraldisplay ⎝integraldisplay1 0⎝parenleftbigg ⎝parenleftbigg1 t–x+λ t+x⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), 0 < x<1 . Solution for f(x)=πq= const: y(x)=q 2s i n (1 2πβ)⎝bracketleftbigg⎝parenleftbiggx 1+√ 1–x2⎝parenrightbiggβ⎝parenleftbiggβ √ 1–x2+1⎝parenrightbigg +⎝parenleftbiggx 1+√ 1–x2⎝parenrightbigg–β⎝parenleftbiggβ √ 1–x2–1⎝parenrightbigg⎝bracketrightbigg , where βis given by cos(πβ )=–λ,0 < β<1 . We assume that the following necessary cond ition holds ⎝integraldisplay1 0y(t)dt=0 . References: H. F. Bueckner (1966), P. S. Theocaric and N. I. Ioakimidis (1977). 54.1 πi⎝integraldisplay ⎝integraldisplaya –a⎝parenleftbigg ⎝parenleftbigg1 t–x–λx xt–a2⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), – a<x<a (i2= –1). 1◦. Solution: y(x)=⎝parenleftbigga–x –a–x⎝parenrightbiggβ1 2πi⎝integraldisplaya –a⎝parenleftbigga–t –a–t⎝parenrightbigg–β⎝parenleftbigg1 t–x–x xt–a2⎝parenrightbigg f(t)dt +⎝parenleftbigga–x –a–x⎝parenrightbigg–β1 2πi⎝integraldisplaya –a⎝parenleftbigga–t –a–t⎝parenrightbiggβ⎝parenleftbigg1 t–x–x xt–a2⎝parenrightbigg f(t)dt, where λ=c o sθandβ=1–θ π. We assume that the following necessary cond ition holds 1 2πi⎝integraldisplaya –a⎝bracketleftbigg e–πiβ⎝parenleftbigga–t –a–t⎝parenrightbiggβ –eπiβ⎝parenleftbigga–t –a–t⎝parenrightbigg–β⎝bracketrightbiggf(t) tdt=0 . 2◦. Solution for f(x)≡0: y(x)=C1Λ1(x)+C2Λ2(x)+C3Λ3(x), where C1,C2,a n dC3are arbitrary constants, and Λ1(x)=( 1+ λ)eiπβ⎝parenleftbigga–t –a–t⎝parenrightbigg1–β +( 1– λ)e–iπβ⎝parenleftbigga–t –a–t⎝parenrightbiggβ , Λ2(x)=( 1+ λ)e–iπβ⎝parenleftbigga–t –a–t⎝parenrightbigg–1+β +( 1– λ)eiπβ⎝parenleftbigga–t –a–t⎝parenrightbigg–β , Λ3(x)=eiπβ⎝parenleftbigga–t –a–t⎝parenrightbigg1–β +e–iπβ⎝parenleftbigga–t –a–t⎝parenrightbigg–1+β . Reference: D. I. Sherman (1969). 3.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 231 55.⎝integraldisplayb ay(t) (x–t)2dt=f(x), a≤x≤b. The simple hypersingular equation of the ﬁrst kind with Cauchy-type kernel. This equation governs circulation-free ﬂow of an ideal incompressible ﬂuid past the segment [ a,b]. Let the conditions y(a)=y(b) = 0 be satisﬁed. Then the solution is y(x)=1 π2⎝integraldisplayb aln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle√ (b–t)(x–a)–√ (b–x)(t–a) √ (b–t)(x–a)+√ (b–x)(t–a)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglef /prime t(t)dt. This equation is discussed in Subsection 14.6-3 in detail. Reference: I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004, p. 7). 56.1 π2⎝integraldisplay ⎝integraldisplay1 –1⎝integraldisplay ⎝integraldisplay1 –1u(x,y)dx dy (x0–x)(y 0–y)=f(x0,y0). A two-dimensional singular equation . A solution, which is bounded on the lines x=±1a n dy=±1 but which is unbounded on the line x=q(–1 <q< 1), is given by the formula u(x0,y0)=⎝radicalBig (1 –x2 0)(1 –y2 0) π2⎝integraldisplay1 –1⎝integraldisplay1 –1f(x,y)dx dy ⎝radicalbig (1 –x2)(1 –y2)(x–x0)(y–y0) –⎝radicalBig (1 –x2 0)(1 –y2 0) π2(q–x0)⎝integraldisplay1 –1dx √ 1–x2⎝parenleftbigg1 π2⎝integraldisplay1 –1f(x,y)dy ⎝radicalbig 1–y2(y–y0)⎝parenrightbigg , provided that⎝integraldisplay1 –1f(x0,y)dy ⎝radicalbig 1–y2=0 , – 1 ≤x0≤1. Reference: I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004, pp. 16–20). 3.2. Equations Whose Kernels Contain Exponential Functions 3.2-1. Kernels Containing Exponential Functions of the Form eλ|x–t|. 1.⎝integraldisplay ⎝integraldisplay∞ –∞e–λ|x–t|y(t)dt=f(x), f(±∞ )=0 . Solution: y(x)=1 2λ⎝bracketleftbig λ2f(x)–f/prime/prime xx(x)⎝bracketrightbig . References: I. I. Hirschman and D. V . Widder (1955), F. D. Gakhov and Yu. I. Cherskii (1978), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 433). 2.⎝integraldisplay ⎝integraldisplay∞ 0e–λ|x–t|y(t)dt=f(x), f(∞)=0 . 1◦. Solution: y(x)=1 2λe–λxd dxe2λxd dxe–λxf(x). 2◦.I ff/prime x(0) –λf(0) = 0 then y(x)=1 2λ⎝bracketleftbig λ2f(x)–f/prime/prime xx(x)⎝bracketrightbig . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 433). 232 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.⎝integraldisplay ⎝integraldisplayb aeλ|x–t|y(t)dt=f(x), – ∞<a<b<∞. 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayx aeλ(x–t)y(t)dt+⎝integraldisplayb xeλ(t–x)y(t)dt=f(x). (1) Differentiating (1) with respect to xtwice yields 2λy(x)+λ2⎝integraldisplayx aeλ(x–t)y(t)dt+λ2⎝integraldisplayb xeλ(t–x)y(t)dt=f/prime/prime xx(x). (2) By eliminating the integral terms from (1) and (2), we obtain the solution y(x)=1 2λ⎝bracketleftbig f/prime/prime xx(x)–λ2f(x)⎝bracketrightbig .( 3 ) 2◦. The right-hand side f(x) of the integral equation must satisfy certain relations. By setting x=aandx=bin (1), we obtain two corollaries ⎝integraldisplayb aeλty(t)dt=eλaf(a),⎝integraldisplayb ae–λty(t)dt=e–λbf(b). (4) On substituting the solution y(x) of (3) into (4) and then integrating by parts, we see that eλbf/prime x(b)–eλaf/prime x(a)=λeλaf(a)+λeλbf(b), e–λbf/prime x(b)–e–λaf/prime x(a)=λe–λaf(a)+λe–λbf(b). Hence, we obtain the desired constraints for f(x): f/prime x(a)+λf(a)=0 , f/prime x(b)–λf(b)=0 . ( 5 ) The general form of the right-hand side satisfying conditions (5) is given by f(x)=F(x)+Ax+B, A=1 bλ–aλ–2⎝bracketleftbig F/prime x(a)+F/prime x(b)+λF(a)–λF(b)⎝bracketrightbig ,B=–1 λ⎝bracketleftbig F/prime x(a)+λF(a)+Aaλ +A⎝bracketrightbig , where F(x) is an arbitrary bounded, twice differentiable function. 4.⎝integraldisplay ⎝integraldisplayb a⎝parenleftbig⎝parenleftbig Aeλ|x–t|+Beµ|x–t|⎝parenrightbig⎝parenrightbig y(t)dt=f(x), – ∞<a<b<∞. Let us remove the modulus in the integrand and differentiate the resulting equation with respect to xtwice to obtain 2(Aλ+Bµ)y(x)+⎝integraldisplayb a⎝parenleftbig Aλ2eλ|x–t|+Bµ2eµ|x–t|⎝parenrightbig y(t)dt=f/prime/prime xx(x). (1) Eliminating the integral term with eµ|x–t|from (1) with the aid of the original integral equation, we ﬁnd that 2(Aλ+Bµ)y(x)+A(λ2–µ2)⎝integraldisplayb aeλ|x–t|y(t)dt=f/prime/prime xx(x)–µ2f(x). (2) ForAλ+Bµ= 0, this is an equation of the form 3.2.3, and for Aλ+Bµ≠0, this is an equation of the form 4.2.15. The right-hand side f(x) must satisfy certain relations, which can be obtained by setting x=aandx=bin the original equation (a similar procedure is used in 3.2.3). 3.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 233 5.⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1Akexp⎝parenleftbig⎝parenleftbig λk|x–t|⎝parenrightbig⎝parenrightbig⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), – ∞<a<b<∞. 1◦. Let us remove the modulus in the kth summand of the integrand: Ik(x)=⎝integraldisplayb aexp⎝parenleftbig λk|x–t|⎝parenrightbig y(t)dt=⎝integraldisplayx aexp[λk(x–t)]y(t)dt+⎝integraldisplayb xexp[λk(t–x)]y(t)dt.( 1 ) Differentiating (1) with respect to xtwice yields I/prime k=λk⎝integraldisplayx aexp[λk(x–t)]y(t)dt–λk⎝integraldisplayb xexp[λk(t–x)]y(t)dt, I/prime/prime k=2λky(x)+λ2 k⎝integraldisplayx aexp[λk(x–t)]y(t)dt+λ2 k⎝integraldisplayb xexp[λk(t–x)]y(t)dt,(2) where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we ﬁnd the relation between I/prime/prime kandIk: I/prime/prime k=2λky(x)+λ2 kIk,Ik=Ik(x). (3) 2◦. With the aid of (1), the integral equation can be rewritten in the form n⎝summationdisplay k=1AkIk=f(x). (4) Differentiating (4) with respect to xtwice and taking into account (3), we obtain σ1y(x)+n⎝summationdisplay k=1Akλ2 kIk=f/prime/prime xx(x), σ1=2n⎝summationdisplay k=1Akλk.( 5) Eliminating the integral Infrom (4) and (5) yields σ1y(x)+n–1⎝summationdisplay k=1Ak(λ2 k–λ2 n)Ik=f/prime/prime xx(x)–λ2 nf(x). (6) Differentiating (6) with respect to xtwice and eliminating In–1from the resulting equation with the aid of (6), we obtain a similar equation whose right-hand side is a second-order linear differential operator (acting on y) with constant coefﬁcients plus the sumn–2⎝summationtext k=1BkIk.I f we successively eliminate In–2,In–3,...,I1with the aid of double differentiation, then we ﬁnally arrive at a linear nonhomogeneous ordinary differential equation of order 2( n– 1) with constant coefﬁcients. 3◦. The right-hand side f(x) must satisfy certain conditions. To ﬁnd these conditions, one must set x=ain the integral equation and its derivati ves. (Alternatively, these conditions can be found by setting x=aandx=bin the integral equation and all its derivatives obtained by means of double differentiation.) 234 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.2-2. Kernels Containing Exponential Functions of the Forms eλxandeµt. 6.⎝integraldisplay ⎝integraldisplayb a|eλx–eλt|y(t)dt=f(x), λ>0 . This is a special case of equation 3.8.3 with g(x)=eλx. Solution: y(x)=1 2λd dx⎝bracketleftbig e–λxf/prime x(x)⎝bracketrightbig . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3). 7.⎝integraldisplay ⎝integraldisplaya 0|eβx–eµt|y(t)dt=f(x), β>0 , µ>0 . This is a special case of equation 3.8.4 with g(x)=eβxandλ=µ/β. 8.⎝integraldisplay ⎝integraldisplayb ay(t)dt |eλx–eλt|k=f(x), 0 < k<1 . The transformation z=eλx,τ=eλt,w(τ)=e–λty(t) leads to an equation of the form 3.1.31: ⎝integraldisplayB Aw(τ) |z–τ|kdτ=F(z), where A=eλa,B=eλb,F(z)=λf⎝parenleftbig1 λlnz⎝parenrightbig . 9.⎝integraldisplay ⎝integraldisplay∞ 0y(t)dt (eλx+eλt)k=f(x), λ>0 , k>0 . This equation can be rewritten as an equation with difference kernel in the form 3.8.16: ⎝integraldisplay∞ 0w(t)dt coshk⎝bracketleftbig1 2λ(x–t)⎝bracketrightbig=g(x), where w(t)=2–kexp⎝parenleftbig –1 2λkt⎝parenrightbig y(t)a n d g(x)=e x p⎝parenleftbig1 2λkx⎝parenrightbig f(x). 3.2-3. Kernels Containing Exponential Functions of the Form eλxt. 10.⎝integraldisplay ⎝integraldisplay∞ –∞e–xty(t)dt=f(x). Solution: y(t)=1 2πi⎝integraldisplayc+i∞ c–i∞estf(s)ds =1 √ 2π3⎝integraldisplay∞ 0e–ξ2/2dξ⎝integraldisplay∞ –∞e–x2/2cos⎝parenleftbig ξ(x+t)⎝parenrightbig f(x)dx. The integral equation and its solution form a two-side Laplace transform pair . References: B. Van der Pol and H. Bremmer (1955), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 433). 3.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 235 11.⎝integraldisplay ⎝integraldisplay∞ –∞eλxty(t)dt=f(x), λ≠0. 1◦. The transformation x=–1 λz,f(x)=F(z) leads to an equation of the form 3.2.10: ⎝integraldisplay∞ –∞e–zty(t)dt=F(z). 2◦. The transformation y(t)=e x p ( – t2)Y(t),x=2 λζ,f(x)=e x p ( ζ2)Φ(ζ) leads to an equation of the form 3.2.17: ⎝integraldisplay∞ –∞e–(ζ–t)2Y(t)dt=Φ(ζ). 12.⎝integraldisplay∞ –∞e–ixty(t)dt=f(x), i2= –1. Solution: y(t)=1 2π⎝integraldisplay∞ –∞eixtf(x)dx. Up to constant factors, the function f(x) and the solution y(t)a r et h e F ourier transform pair. References: V . A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), B. Davis (1978), F. Oberhettinger (1980), Yu. A. Brychkov and A. P. Prudnikov (1989), W. H. Beyer (1991), I. Sneddon (1995), A. Pinkus and S. Zafrany (1997), R. Bracewell (1999), A. D. Poularikas (2000), R. J. Beerends, H. G. ter Morschem, and J. C. van den Berg (2003), L. Debnath and D. Bhatta (2007). 13.⎝integraldisplay ⎝integraldisplay∞ 0e–zty(t)dt=f(z). The left-hand side of the equation is the Laplace transform ofy(t)(zis treated as a complex variable). 1◦. Solution: y(t)=1 2πi⎝integraldisplayc+i∞ c–i∞eztf(z)dz,i2= –1. For speciﬁc functions f(z), one may use tables of inverse Laplace transforms to calculate the integral (e.g., see Supplement 6). 2◦. For real z=x, under some assumptions the solution of the original equation can be represented in the form y(x) = lim n→∞(–1)n n!⎝parenleftBign x⎝parenrightBign+1 f(n) x⎝parenleftBign x⎝parenrightBig , which is the real inversion of the Laplace transform. To calculate the solution approximately, one should restrict oneself to a speciﬁc value of nin this formula instead of taking the limit. References: G. Doetsch (1950, 1956, 1958, 1974), H. Bateman and A. Erd ´elyi (vol. 1, 1954), I. I. Hirschman and D. V . Widder (1955), V . A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), F. Oberhettinger (1973), B. Davis (1978), W. R. LePage (1980), R. Bellman and R. Roth (1984), Yu. A. Brychkov and A. P. Prudnikov (1989), W. H. Beyer (1991), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, V ols 4 and 5), R. J. Beerends,H. G. ter Morschem, and J. C. van den Berg (2003). 236 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.2-4. Kernels Containing Power-Law and Exponential Functions. 14.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglekeλx–k–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.5 with g(x)=keλx–k. 15.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–keλt–k⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.6 with g(t)=keλt+k. 16.⎝integraldisplay ⎝integraldisplay∞ –∞t–ix–1/ 2exp⎝parenleftbigg ⎝parenleftbigg2x–i 4π⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), i2= –1. Solution: y(x)=1 4π⎝integraldisplay∞ –∞xit–1/2exp⎝parenleftbigg2t+i 4π⎝parenrightbiggf(t) cosh(πt )dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463). 3.2-5. Kernels Containing Exponential Functions of the Form eλ(x±t)2. 17.⎝integraldisplay ⎝integraldisplay∞ –∞e–(x–t)2y(t)dt=f(x). 1◦. The transformation Y(t)=e x p ( – t2)y(t),z=– 2x,F(z)=e x p ( x2)f(x) leads to an equation of the form 3.2.10: ⎝integraldisplay∞ –∞e–ztY(t)dt=F(z). 2◦. Solution: y(t)=1 π3/2⎝integraldisplay∞ 0es2/4ds⎝integraldisplay∞ –∞cos⎝parenleftbig s(t–x)⎝parenrightbig f(x)dx =e x p⎝bracketleftbigg –1 4√ πd2 dt2f(t)⎝bracketrightbigg ≡∞⎝summationdisplay k=01 k!⎝parenleftbigg –1 4√ π⎝parenrightbiggkd2kf(t) dt2k. (See equation 3.2.18 for λ=1 . ) 3◦. Solution: y(x)=1 √ π∞⎝summationdisplay n=0f(n) x(0) 2nn!Hn(x), where Hn(x) are the Hermite polynomials (see Supplement 11.17-3) Hm(x)=( – 1 )mexp⎝parenleftbig x2⎝parenrightbigdm dxmexp⎝parenleftbig –x2⎝parenrightbig . References: P. M. Morse and H. Feshbach (1953), I. I. Hirschman and D. V . Widder (1955), P. G. Rooney (1963), M. L. Krasnov (1975). 3.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 237 18.1 √ πλ⎝integraldisplay ⎝integraldisplay∞ –∞exp⎝bracketleftbigg ⎝bracketleftbigg –(x–t)2 λ⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). It is the Gauss transform (the Weierstrass transform forλ=4 ) . Solution: y(t)=1 π⎝integraldisplay∞ 0eλs2/4ds⎝integraldisplay∞ –∞cos⎝parenleftbig s(t–x)⎝parenrightbig f(x)dx =e x p⎝bracketleftbigg –λ 4d2 dt2f(t)⎝bracketrightbigg ≡∞⎝summationdisplay k=01 k!⎝parenleftbigg –λ 4⎝parenrightbiggkd2kf(t) dt2k. References: I. I. Hirschman and D. V . Widder (1955), P. G. Rooney (1963), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 435). 19.⎝integraldisplay ⎝integraldisplay∞ –∞ei(x+t)2y(t)dt=f(x), i2= –1. Solution: y(x)=1 π⎝integraldisplay∞ –∞e–i(x+t)2f(t)dt. References: E. A. C. Paley and N. Wiener (1934), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 435). 3.2-6. Other Kernels. 20.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleexp(λx2)–e x p ( λt2)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), λ>0 . This is a special case of equation 3.8.3 with g(x)=e x p ( λx2). Solution: y(x)=1 4λd dx⎝bracketleftbigg1 xexp(–λx2)f/prime x(x)⎝bracketrightbigg . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3). 21.1 √ πx⎝integraldisplay ⎝integraldisplay∞ 0exp⎝parenleftbigg ⎝parenleftbigg –t2 4x⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). Applying the Laplace transformation to the equation, we obtain ˜y(√ p) √ p=˜f(p), ˜f(p)=⎝integraldisplay∞ 0e–ptf(t)dt. Substituting pbyp2and solving for the transform ˜ y,w eﬁ n dt h a t ˜ y(p)=p˜f(p2). The inverse Laplace transform provides the solution of the original integral equation: y(t)=L–1{p˜f(p2)}, L–1{g(p)}≡1 2πi⎝integraldisplayc+i∞ c–i∞eptg(p)dp. 238 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.3. Equations Whose Kernels Contain Hyperbolic Functions 3.3-1. Kernels Containing Hyperbolic Cosine. 1.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecosh(λx )–c o s h ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=c o s h ( λx). Solution: y(x)=1 2λd dx⎝bracketleftbiggf/prime x(x) sinh(λx)⎝bracketrightbigg . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3). 2.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecosh(βx )–c o s h ( µt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), β>0 , µ>0 . This is a special case of equation 3.8.4 with g(x)=c o s h ( βx)a n dλ=µ/β. 3.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecoshkx–c o s hkt|y(t)dt=f(x), 0 < k<1 . This is a special case of equation 3.8.3 with g(x)=c o s hkx. Solution: y(x)=1 2kd dx⎝bracketleftbiggf/prime x(x) sinhxcoshk–1x⎝bracketrightbigg . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3). 4.⎝integraldisplay ⎝integraldisplayb ay(t) |cosh(λx )–c o s h ( λt)|kdt=f(x), 0 < k<1 . This is a special case of equation 3.8.7 with g(x)=c o s h ( λx)+ β,w h e r e βis an arbitrary number. 3.3-2. Kernels Containing Hyperbolic Sine. 5.⎝integraldisplay ⎝integraldisplayb asinh⎝parenleftbig⎝parenleftbig λ|x–t|⎝parenrightbig⎝parenrightbig y(t)dt=f(x), – ∞<a<b<∞. 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayx asinh[λ(x–t)]y(t)dt+⎝integraldisplayb xsinh[λ(t–x)]y(t)dt=f(x). (1) Differentiating (1) with respect to xtwice yields 2λy(x)+λ2⎝integraldisplayx asinh[λ(x–t)]y(t)dt+λ2⎝integraldisplayb xsinh[λ(t–x)]y(t)dt=f/prime/prime xx(x). (2) 3.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 239 Eliminating the integral terms from (1) and (2), we obtain the solution y(x)=1 2λ⎝bracketleftbig f/prime/prime xx(x)–λ2f(x)⎝bracketrightbig .( 3 ) 2◦. The right-hand side f(x) of the integral equation must satisfy certain relations. By setting x=aandx=bin (1), we obtain two corollaries ⎝integraldisplayb asinh[λ(t–a)]y(t)dt=f(a),⎝integraldisplayb asinh[λ(b–t)]y(t)dt=f(b). (4) Substituting solution (3) into (4) and integrating by parts yields the desired conditions for f(x): sinh[λ(b–a)]f/prime x(b)–λcosh[λ (b–a)]f(b)=λf(a), sinh[λ(b–a)]f/prime x(a)+λcosh[λ (b–a)]f(a)=–λf(b).(5) The general form of the right-hand side is given by f(x)=F(x)+Ax+B,( 6) where F(x) is an arbitrary bounded twice differentiable function, and the coefﬁcients AandB are expressed in terms of F(a),F(b),F/prime x(a), andF/prime x(b) and can be determined by substituting formula (6) into conditions (5). 6.⎝integraldisplay ⎝integraldisplayb a⎝braceleftBig ⎝braceleftBig Asinh⎝parenleftbig⎝parenleftbig λ|x–t|⎝parenrightbig⎝parenrightbig +Bsinh⎝parenleftbig⎝parenleftbig µ|x–t|⎝parenrightbig⎝parenrightbig⎝bracerightBig ⎝bracerightBig y(t)dt=f(x), – ∞<a<b<∞. Let us remove the modulus in the integrand and differentiate the equation with respect to x twice to obtain 2(Aλ+Bµ)y(x)+⎝integraldisplayb a⎝braceleftbig Aλ2sinh⎝parenleftbig λ|x–t|⎝parenrightbig +Bµ2sinh⎝parenleftbig µ|x–t|⎝parenrightbig⎝bracerightbig y(t)dt=f/prime/prime xx(x). (1) Eliminating the integral term with sinh⎝parenleftbig µ|x–t|⎝parenrightbig from (1) yields 2(Aλ+Bµ)y(x)+A(λ2–µ2)⎝integraldisplayb asinh⎝parenleftbig λ|x–t|⎝parenrightbig y(t)dt=f/prime/prime xx(x)–µ2f(x). (2) ForAλ+Bµ= 0, this is an equation of the form 3.3.5, and for Aλ+Bµ≠0, this is an equation of the form 4.3.26. The right-hand side f(x) must satisfy certain relations, which can be obtained by setting x=aandx=bin the original equation (a similar procedure is used in 3.3.5). 7.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesinh(λx) – sinh( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x) = sinh(λx). Solution: y(x)=1 2λd dx⎝bracketleftbiggf/prime x(x) cosh(λx)⎝bracketrightbigg . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3). 240 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 8.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesinh(βx)–s i n h ( µt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), β>0 , µ>0 . This is a special case of equation 3.8.4 with g(x) = sinh(βx )a n dλ=µ/β. 9.⎝integraldisplay ⎝integraldisplayb asinh3⎝parenleftbig⎝parenleftbig λ|x–t|⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Using the formula sinh3β=1 4sinh 3β–3 4sinhβ, we arrive at an equation of the form 3.3.6: ⎝integraldisplayb a⎝bracketleftbig1 4Asinh⎝parenleftbig 3λ|x–t|⎝parenrightbig –3 4Asinh⎝parenleftbig λ|x–t|⎝parenrightbig⎝bracketrightbig y(t)dt=f(x). 10.⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1Aksinh⎝parenleftbig⎝parenleftbig λk|x–t|⎝parenrightbig⎝parenrightbig⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), – ∞<a<b<∞. 1◦. Let us remove the modulus in the kth summand of the integrand: Ik(x)=⎝integraldisplayb asinh⎝parenleftbig λk|x–t|⎝parenrightbig y(t)dt=⎝integraldisplayx asinh[λk(x–t)]y(t)dt+⎝integraldisplayb xsinh[λk(t–x)]y(t)dt.( 1 ) Differentiating (1) with respect to xtwice yields I/prime k=λk⎝integraldisplayx acosh[λ k(x–t)]y(t)dt–λk⎝integraldisplayb xcosh[λ k(t–x)]y(t)dt, I/prime/prime k=2λky(x)+λ2 k⎝integraldisplayx asinh[λk(x–t)]y(t)dt+λ2 k⎝integraldisplayb xsinh[λk(t–x)]y(t)dt,(2) where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we ﬁnd the relation between I/prime/prime kandIk: I/prime/prime k=2λky(x)+λ2 kIk,Ik=Ik(x). (3) 2◦. With the aid of (1), the integral equation can be rewritten in the form n⎝summationdisplay k=1AkIk=f(x). (4) Differentiating (4) with respect to xtwice and taking into account (3), we ﬁnd that σ1y(x)+n⎝summationdisplay k=1Akλ2 kIk=f/prime/prime xx(x), σ1=2n⎝summationdisplay k=1Akλk.( 5) Eliminating the integral Infrom (4) and (5) yields σ1y(x)+n–1⎝summationdisplay k=1Ak(λ2 k–λ2 n)Ik=f/prime/prime xx(x)–λ2 nf(x). (6) Differentiating (6) with respect to xtwice and eliminating In–1from the resulting equation with the aid of (6), we obtain a similar equation whose right-hand side is a second-order linear differential operator (acting on y) with constant coefﬁcients plus the sumn–2⎝summationtext k=1BkIk. If we successively eliminate In–2,In–3,..., with the aid of double differentiation, then we ﬁnally arrive at a linear nonhomogeneous ordinary differential equation of order 2( n– 1) with constant coefﬁcients. 3◦. The right-hand side f(x) must satisfy certain conditions. To ﬁnd these conditions, one should set x=ain the integral equation and its derivati ves. (Alternatively, these conditions can be found by setting x=aandx=bin the integral equation and all its derivatives obtained by means of double differentiation.) 3.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 241 11.⎝integraldisplay ⎝integraldisplayb 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesinhkx–s i n hkt⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), 0 < k<1 . This is a special case of equation 3.8.3 with g(x)=s i n hkx. Solution: y(x)=1 2kd dx⎝bracketleftbiggf/prime x(x) coshxsinhk–1x⎝bracketrightbigg . The right-hand side f(x) must satisfy certain conditions. As follows from item 3◦of equation 3.8.3, the admissible general form of the right-hand side is given by f(x)=F(x)+Ax+B,A=–F/prime x(b), B=1 2⎝bracketleftbig bF/prime x(b)–F(0) –F(b)⎝bracketrightbig , where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive). 12.⎝integraldisplay ⎝integraldisplayb ay(t) |sinh(λx)–s i n h ( λt)|kdt=f(x), 0 < k<1 . This is a special case of equation 3.8.7 with g(x) = sinh(λx)+ β,w h e r e βis an arbitrary number. 13.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleksinh(λx)–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.5 with g(x)=ksinh(λx). 14.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–ksinh(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.6 with g(x)=ksinh(λt). 3.3-3. Kernels Containing Hyperbolic Tangent. 15.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingletanh(λx)–t a n h ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=t a n h ( λx). Solution: y(x)=1 2λd dx⎝bracketleftbig cosh2(λx)f/prime x(x)⎝bracketrightbig . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦of equation 3.8.3). 16.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingletanh(βx)–t a n h ( µt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), β>0 , µ>0 . This is a special case of equation 3.8.4 with g(x)=t a n h ( βx)a n dλ=µ/β. 242 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 17.⎝integraldisplay ⎝integraldisplayb 0|tanhkx–t a n hkt|y(t)dt=f(x), 0 < k<1 . This is a special case of equation 3.8.3 with g(x)=t a n hkx. Solution: y(x)=1 2kd dx⎝bracketleftbig cosh2xcothk–1xf/prime x(x)⎝bracketrightbig . The right-hand side f(x) must satisfy certain conditions. As follows from item 3◦of equation 3.8.3, the admissible general form of the right-hand side is given by f(x)=F(x)+Ax+B,A=–F/prime x(b),B=1 2⎝bracketleftbig bF/prime x(b)–F(0) –F(b)⎝bracketrightbig , where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive). 18.⎝integraldisplay ⎝integraldisplayb ay(t) |tanh(λx) – tanh( λt)|kdt=f(x), 0 < k<1 . This is a special case of equation 3.8.7 with g(x)=t a n h ( λx)+ β,w h e r e βis an arbitrary number. 19.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglektanh(λx)–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.5 with g(x)=ktanh(λx). 20.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–ktanh(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.6 with g(x)=ktanh(λt). 3.3-4. Kernels Containing Hyperbolic Cotangent. 21.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecoth(λx )–c o t h ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=c o t h ( λx). 22.⎝integraldisplay ⎝integraldisplayb 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecothkx–c o t hkt⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), 0 < k<1 . This is a special case of equation 3.8.3 with g(x)=c o t hkx. 3.4. Equations Whose Kernels Contain Logarithmic Functions 3.4-1. Kernels Containing Logarithmic Functions. 1.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleln(x/t )⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=l nx. Solution: y(x)=1 2d dx⎝bracketleftbig xf/prime x(x)⎝bracketrightbig . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦of equation 3.8.3). 3.4. E QUATIONS WHOSE KERNELS CONTAIN LOGARITHMIC FUNCTIONS 243 2.⎝integraldisplay ⎝integraldisplayb aln|x–t|y(t)dt=f(x). Carleman’s equation. 1◦. Solution with b–a≠4: y(x)=1 π2√ (x–a)(b–x)⎝bracketleftbigg⎝integraldisplayb a√ (t–a)(b–t)f/prime t(t)dt t–x+1 ln⎝bracketleftbig1 4(b–a)⎝bracketrightbig⎝integraldisplayb af(t)dt √ (t–a)(b–t)⎝bracketrightbigg . 2◦.I fb–a= 4, then for the equation to be solvable, the condition ⎝integraldisplayb af(t)(t–a)–1/2(b–t)–1/2dt=0 must be satisﬁed. In this case, the solution has the form y(x)=1 π2√ (x–a)(b–x)⎝bracketleftbigg⎝integraldisplayb a√ (t–a)(b–t)f/prime t(t)dt t–x+C⎝bracketrightbigg , where Cis an arbitrary constant. Reference: F. D. Gakhov (1977). 3.⎝integraldisplay ⎝integraldisplayb a⎝parenleftbig⎝parenleftbig ln|x–t|+β⎝parenrightbig⎝parenrightbig y(t)dt=f(x). By setting x=e–βz,t=e–βτ,y(t)=Y(τ),f(x)=e–βg(z), we arrive at an equation of the form 3.4.2: ⎝integraldisplayB Aln|z–τ|Y(τ)dτ=g(z), A=aeβ,B=beβ. 4.⎝integraldisplay ⎝integraldisplaya –a⎝parenleftBig ⎝parenleftBig lnA |x–t|⎝parenrightBig ⎝parenrightBig y(t)dt=f(x), – a≤x≤a. This is a special case of equation 3.4.3 with b=–a. Solution with 0 < a<2A: y(x)=1 2M/prime(a)⎝bracketleftbiggd da⎝integraldisplaya –aw(t,a)f(t)dt⎝bracketrightbigg w(x,a) –1 2⎝integraldisplaya |x|w(x,ξ)d dξ⎝bracketleftbigg1 M/prime(ξ)d dξ⎝integraldisplayξ –ξw(t,ξ)f(t)dt⎝bracketrightbigg dξ –1 2d dx⎝integraldisplaya |x|w(x,ξ) M/prime(ξ)⎝bracketleftbigg⎝integraldisplayξ –ξw(t,ξ)df(t)⎝bracketrightbigg dξ, where M(ξ)=⎝parenleftbigg ln2A ξ⎝parenrightbigg–1 ,w(x,ξ)=M(ξ) π⎝radicalbig ξ2–x2, and the prime stands for the derivative. Reference: I. C. Gohberg and M. G. Krein (1967). 244 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 5.⎝integraldisplay ⎝integraldisplaya 0ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex+t x–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). Solution: y(x)=–2 π2d dx⎝integraldisplaya xF(t)dt √ t2–x2,F(t)=d dt⎝integraldisplayt 0sf(s)ds √ t2–s2. Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975). 6.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleln1+λx 1+λt⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=l n ( 1+ λx). Solution: y(x)=1 2λd dx⎝bracketleftbig (1 +λx)f/prime x(x)⎝bracketrightbig . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3). 7.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglelnβx–l nβt⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), 0 < β<1 . This is a special case of equation 3.8.3 with g(x)=l nβx. 8.⎝integraldisplay ⎝integraldisplayb ay(t) |ln(x/t )|βdt=f(x), 0 < β<1 . This is a special case of equation 3.8.7 with g(x)=l nx+A,w h e r e Ais an arbitrary number. 3.4-2. Kernels Containing Power-Law and Logarithmic Functions. 9.⎝integraldisplay ⎝integraldisplay1 0⎝parenleftbig⎝parenleftbig ln|x–t|+βtk⎝parenrightbig⎝parenrightbig y(t)dt=f(x). See Example 3 in Subsection 12.6-2 with ψ(t)=βtk. 10.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglekln(1 + λx)–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.5 with g(x)=kln(1 + λx). 11.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–kln(1 + λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.6 with g(x)=kln(1 + λt). 12.⎝integraldisplay ⎝integraldisplay∞ 01 tln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex+t x–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). Solution: y(x)=x π2d dx⎝integraldisplay∞ 0df(t) dtln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle1–x2 t2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingledt. Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975). 3.4. E QUATIONS WHOSE KERNELS CONTAIN LOGARITHMIC FUNCTIONS 245 13.⎝integraldisplay ⎝integraldisplay∞ 0lnx–l nt x–ty(t)dt=f(x). The left-hand side of this equation is the iterated Stieltjes transform. Under some assumptions, the solution of the integral equation can be represented in the form y(x)=1 4π2lim n→∞⎝parenleftBige n⎝parenrightBig4n Dnx2nD2nx2nDnf(x),D=d dx. To calculate the solution approximately, one should restrict oneself to a speciﬁc value of nin this formula instead of taking the limit. Reference: I. I. Hirschman and D. V . Widder (1955). 14.⎝integraldisplay ⎝integraldisplayb aln|xβ–tβ|y(t)dt=f(x), β>0 . The transformation z=xβ,τ=tβ,w(τ)=t1–βy(t) leads to Carleman’s equation 3.4.2: ⎝integraldisplayB Aln|z–τ|w(τ)dτ=F(z), A=aβ,B=bβ, where F(z)=βf⎝parenleftbig z1/β⎝parenrightbig . 15.⎝integraldisplay ⎝integraldisplay1 0ln|xβ–tµ|y(t)dt=f(x), β>0 ,µ>0 . The transformation z=xβ,τ=tµ,w(τ)=t1–µy(t) leads to an equation of the form 3.4.2: ⎝integraldisplay1 0ln|z–τ|w(τ)dτ=F(z), F(z)=µf⎝parenleftbig z1/β⎝parenrightbig . 16.⎝integraldisplay ⎝integraldisplay∞ 01 √ xtln(xt)y(t)dt=f(x). Solution: y(x)=–1 π2⎝integraldisplay∞ 01 √ xtln(xt)f(t)dt. References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 450). 17.d dx⎝integraldisplay ⎝integraldisplay∞ –∞ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle1–x t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). Solution: y(x)=–1 π2d dx⎝integraldisplay∞ –∞ln⎝vextendsingle⎝vextendsingle⎝vextendsingle1–x t⎝vextendsingle⎝vextendsingle⎝vextendsinglef(t)dt. References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 450). 18.⎝integraldisplay ⎝integraldisplay∞ 0(xt)–[1+iln(xt)]/2y(t)dt=f(x), i2= –1. Solution: y(x)=1 2π⎝integraldisplay∞ 0(xt)–[1–i ln(xt)]/2f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 452). 246 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.4-3. Equation Containing the Unknown Function of a Complicated Argument. 19.⎝integraldisplay ⎝integraldisplay1 0⎝parenleftbig⎝parenleftbig Alnt+B)y(xt)dt=f(x). The substitution ξ=xtleads to an equation of the form 1.9.3 with g(x)=–Alnx: ⎝integraldisplayx 0⎝parenleftbig Alnξ–Alnx+B⎝parenrightbig y(ξ)dξ=xf(x). 3.5. Equations Whose Kernels Contain Trigonometric Functions 3.5-1. Kernels Containing Cosine. 1.⎝integraldisplay ⎝integraldisplay∞ 0cos(xt )y(t)dt=f(x). Solution: y(x)=2 π⎝integraldisplay∞ 0cos(xt )f(t)dt. Up to constant factors, the function f(x) and the solution y(t)a r et h e F ourier cosine transform pair . References: E. A. C. Paley and N. Wiener (1934), S. Bochner and K. C. Chandrasekharan (1949), G. N. Watson (1952), H. Bateman and A. Erd ´elyi (V ol. 1, 1954), S. Bochner (1959), V . A. Ditkin and A. P. Prudnikov (1965), B. Davis (1978), F. Oberhettinger (1980), E. C. Titchmarsh (1986), Ya. A. Brychkov and A. P. Prudnikov (1989),A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 440), I. Sneddon (1995), A. D. Poularikas (2000). 2.⎝integraldisplay ⎝integraldisplayb acos(xt )y(t)dt=f(x), 0 ≤x<∞. Solution: y(t)=⎝braceleftBigg2 π⎝integraldisplay∞ 0cos(xt )f(x)dx ifa<t<b, 0i f0<t<aort>b, where 0 ≤a≤b≤∞. 3.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecos(λx )–c o s ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=c o s ( λx). Solution: y(x)=–1 2λd dx⎝bracketleftbiggf/prime x(x) sin(λx)⎝bracketrightbigg . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦of equation 3.8.3). 4.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecos(βx )–c o s ( µt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), β>0 , µ>0 . This is a special case of equation 3.8.4 with g(x)=c o s ( βx)a n dλ=µ/β. 3.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 247 5.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecoskx–c o skt⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), 0 < k<1 . This is a special case of equation 3.8.3 with g(x)=c o skx. Solution: y(x)=–1 2kd dx⎝bracketleftbiggf/prime x(x) sinxcosk–1x⎝bracketrightbigg . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦of equation 3.8.3). 6.⎝integraldisplay ⎝integraldisplayb ay(t) |cos(λx )–c o s ( λt)|kdt=f(x), 0 < k<1 . This is a special case of equation 3.8.7 with g(x)=c o s ( λx)+ β,w h e r e βis an arbitrary number. 7.⎝integraldisplay ⎝integraldisplay∞ 0t–ix–1/ 2cos⎝parenleftbigg ⎝parenleftbigg1+2ix 4π⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), i2= –1. Solution: y(t)=1 π⎝integraldisplay∞ –∞tix–1/2cos⎝parenleftbigg1–2ix 4π⎝parenrightbiggf(x) cosh(πx )dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463). 3.5-2. Kernels Containing Sine. 8.⎝integraldisplay ⎝integraldisplay∞ 0sin(xt)y(t)dt=f(x). Solution: y(x)=2 π⎝integraldisplay∞ 0sin(xt)f(t)dt. Up to constant factors, the function f(x) and the solution y(t)a r et h e F ourier sine transform pair . References: E. A. C. Paley and N. Wiener (1934), S. Bochner and K. C. Chandrasekharan (1949), G. N. Watson (1952), H. Bateman and A. Erd ´elyi (V ol. 1, 1954), S. Bochner (1959), V . A. Ditkin and A. P. Prudnikov (1965), B. Davis (1978), F. Oberhettinger (1980), E. C. Titchmarsh (1986), Ya. A. Brychkov and A. P. Prudnikov (1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 440), I. Sneddon (1995), A. D. Poularikas (2000). 9.⎝integraldisplay ⎝integraldisplayb asin(xt)y(t)dt=f(x), 0 ≤x<∞. Solution: y(t)=⎝braceleftBigg2 π⎝integraldisplay∞ 0sin(xt)f(x)dx ifa<t<b, 0i f0<t<aort>b, where 0 ≤a≤b≤∞. 10.⎝integraldisplay ⎝integraldisplay∞ –∞sin⎝parenleftbig⎝parenleftbig λ|x–t|⎝parenrightbig⎝parenrightbig y(t)dt=f(x), f(±∞ )=0 . Solution: y(x)=1 2λ⎝bracketleftbig f/prime/prime xx(x)+λ2f(x)⎝bracketrightbig . 248 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 11.⎝integraldisplay ⎝integraldisplayb asin⎝parenleftbig⎝parenleftbig λ|x–t|⎝parenrightbig⎝parenrightbig y(t)dt=f(x), – ∞<a<b<∞. 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayx asin[λ(x–t)]y(t)dt+⎝integraldisplayb xsin[λ(t–x)]y(t)dt=f(x). (1) Differentiating (1) with respect to xtwice yields 2λy(x)–λ2⎝integraldisplayx asin[λ(x–t)]y(t)dt–λ2⎝integraldisplayb xsin[λ(t–x)]y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral terms from (1) and (2), we obtain the solution y(x)=1 2λ⎝bracketleftbig f/prime/prime xx(x)+λ2f(x)⎝bracketrightbig .( 3) 2◦. The right-hand side f(x) of the integral equation must satisfy certain relations. By setting x=aandx=bin (1), we obtain two corollaries ⎝integraldisplayb asin[λ(t–a)]y(t)dt=f(a),⎝integraldisplayb asin[λ(b–t)]y(t)dt=f(b). (4) Substituting solution (3) into (4) followed by integrating by parts yields the desired conditions forf(x): sin[λ(b–a)]f/prime x(b)–λcos[λ (b–a)]f(b)=λf(a), sin[λ(b–a)]f/prime x(a)+λcos[λ (b–a)]f(a)=–λf(b).(5) The general form of the right-hand side of the integral equation is given by f(x)=F(x)+Ax+B,( 6 ) where F(x) is an arbitrary bounded twice differentiable function, and the coefﬁcients AandB are expressed in terms of F(a),F(b),F/prime x(a), andF/prime x(b) and can be determined by substituting formula (6) into conditions (5). 12.⎝integraldisplay ⎝integraldisplayb a⎝braceleftbig ⎝braceleftbig Asin⎝parenleftbig⎝parenleftbig λ|x–t|⎝parenrightbig⎝parenrightbig +Bsin⎝parenleftbig⎝parenleftbig µ|x–t|⎝parenrightbig⎝parenrightbig⎝bracerightbig⎝bracerightbig y(t)dt=f(x), – ∞<a<b<∞. Let us remove the modulus in the integrand and differentiate the equation with respect to x twice to obtain 2(Aλ+Bµ)y(x)–⎝integraldisplayb a⎝braceleftbig Aλ2sin⎝parenleftbig λ|x–t|⎝parenrightbig +Bµ2sin⎝parenleftbig µ|x–t|⎝parenrightbig⎝bracerightbig y(t)dt=f/prime/prime xx(x). (1) Eliminating the integral term with sin⎝parenleftbig µ|x–t|⎝parenrightbig from (1) with the aid of the original equation, we ﬁnd that 2(Aλ+Bµ)y(x)+A(µ2–λ2)⎝integraldisplayb asin⎝parenleftbig λ|x–t|⎝parenrightbig y(t)dt=f/prime/prime xx(x)+µ2f(x). (2) ForAλ+Bµ= 0, this is an equation of the form 3.5.11 and for Aλ+Bµ≠0, this is an equation of the form 4.5.29. The right-hand side f(x) must satisfy certain relations, which can be obtained by setting x=aandx=bin the original equation (a similar procedure is used in 3.5.11). 3.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 249 13.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesin(λx)–s i n ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=s i n ( λx). Solution: y(x)=1 2λd dx⎝bracketleftbiggf/prime x(x) cos(λx)⎝bracketrightbigg . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦of equation 3.8.3). 14.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesin(βx)–s i n ( µt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), β>0 , µ>0 . This is a special case of equation 3.8.4 with g(x)=s i n ( βx)a n dλ=µ/β. 15.⎝integraldisplay ⎝integraldisplayb asin3⎝parenleftbig⎝parenleftbig λ|x–t|⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Using the formula sin3β=–1 4sin 3β+3 4sinβ, we arrive at an equation of the form 3.5.12: ⎝integraldisplayb a⎝bracketleftbig –1 4Asin⎝parenleftbig 3λ|x–t|⎝parenrightbig +3 4Asin⎝parenleftbig λ|x–t|⎝parenrightbig⎝bracketrightbig y(t)dt=f(x). 16.⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1Aksin⎝parenleftbig⎝parenleftbig λk|x–t|⎝parenrightbig⎝parenrightbig⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), – ∞<a<b<∞. 1◦. Let us remove the modulus in the kth summand of the integrand: Ik(x)=⎝integraldisplayb asin⎝parenleftbig λk|x–t|⎝parenrightbig y(t)dt=⎝integraldisplayx asin[λk(x–t)]y(t)dt+⎝integraldisplayb xsin[λk(t–x)]y(t)dt.( 1 ) Differentiating (1) with respect to xyields I/prime k=λk⎝integraldisplayx acos[λ k(x–t)]y(t)dt–λk⎝integraldisplayb xcos[λ k(t–x)]y(t)dt, I/prime/prime k=2λky(x)–λ2 k⎝integraldisplayx asin[λk(x–t)]y(t)dt–λ2 k⎝integraldisplayb xsin[λk(t–x)]y(t)dt,(2) where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we ﬁnd the relation between I/prime/prime kandIk: I/prime/prime k=2λky(x)–λ2 kIk,Ik=Ik(x). (3) 2◦. With the aid of (1), the integral equation can be rewritten in the form n⎝summationdisplay k=1AkIk=f(x). (4) Differentiating (4) with respect to xtwice and taking into account (3), we ﬁnd that σ1y(x)–n⎝summationdisplay k=1Akλ2 kIk=f/prime/prime xx(x), σ1=2n⎝summationdisplay k=1Akλk.( 5) 250 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION Eliminating the integral Infrom (4) and (5) yields σ1y(x)+n–1⎝summationdisplay k=1Ak(λ2 n–λ2 k)Ik=f/prime/prime xx(x)+λ2 nf(x). (6) Differentiating (6) with respect to xtwice and eliminating In–1from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear differential operator (acting on y) with constant coefﬁcients plus the sumn–2⎝summationtext k=1BkIk. If we successively eliminate In–2,In–3,..., with the aid of double differentiation, then we ﬁnally arrive at a linear nonhomogeneous ordinary differential equation of order 2( n– 1) with constant coefﬁcients. 3◦. The right-hand side f(x) must satisfy certain conditions. To ﬁnd these conditions, one should set x=ain the integral equation and its derivati ves. (Alternatively, these conditions can be found by setting x=aandx=bin the integral equation and all its derivatives obtained by means of double differentiation.) 17.⎝integraldisplay ⎝integraldisplayb 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesinkx–s i nkt⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), 0 < k<1 . This is a special case of equation 3.8.3 with g(x)=s i nkx. Solution: y(x)=1 2kd dx⎝bracketleftbiggf/prime x(x) cosxsink–1x⎝bracketrightbigg . The right-hand side f(x) must satisfy certain conditions. As follows from item 3◦of equation 3.8.3, the admissible general form of the right-hand side is given by f(x)=F(x)+Ax+B,A=–F/prime x(b),B=1 2⎝bracketleftbig bF/prime x(b)–F(0) –F(b)⎝bracketrightbig , where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive). 18.⎝integraldisplay ⎝integraldisplayb ay(t) |sin(λx)–s i n ( λt)|kdt=f(x), 0 < k<1 . This is a special case of equation 3.8.7 with g(x)=s i n ( λx)+β,w h e r e βis an arbitrary number. 19.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleksin(λx)–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.5 with g(x)=ksin(λx). 20.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–ksin(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.6 with g(t)=ksin(λt). 21.⎝integraldisplay ⎝integraldisplay∞ 0sint t2[y(x+t)–y(x–t)]dt=f(x). Solution: y(x)=1 π⎝integraldisplay∞ 0⎝bracketleftbiggcost t+S i (t)⎝bracketrightbigg [f(x–t)–f(x+t)]dt, where Si( t) is sine integral (see Supplement 11.3-1). The integral equation and its solution form the Boas transform pair . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 442). 3.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 251 22.⎝integraldisplay ⎝integraldisplay∞ 0t–ix–1/ 2sin⎝parenleftbigg ⎝parenleftbigg1+2ix 4π⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), i2= –1. Solution: y(t)=1 π⎝integraldisplay∞ –∞tix–1/2sin⎝parenleftbigg1–2ix 4π⎝parenrightbiggf(x) cosh(πx )dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463). 3.5-3. Kernels Containing Tangent. 23.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingletan(λx)–t a n ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=t a n ( λx). Solution: y(x)=1 2λd dx⎝bracketleftbigg cos2(λx)f/prime x(x)⎝bracketrightbigg . The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2◦of equation 3.8.3). 24.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingletan(βx)–t a n ( µt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), β>0 , µ>0 . This is a special case of equation 3.8.4 with g(x)=t a n ( βx)a n dλ=µ/β. 25.⎝integraldisplay ⎝integraldisplayb 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingletankx–t a nkt⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), 0 < k<1 . This is a special case of equation 3.8.3 with g(x)=t a nkx. Solution: y(x)=1 2kd dx⎝bracketleftbigg cos2xcotk–1xf/prime x(x)⎝bracketrightbigg . The right-hand side f(x) must satisfy certain conditions. As follows from item 3◦of equation 3.8.3, the admissible general form of the right-hand side is given by f(x)=F(x)+Ax+B,A=–F/prime x(b),B=1 2⎝bracketleftbig bF/prime x(b)–F(0) –F(b)⎝bracketrightbig , where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive). 26.⎝integraldisplay ⎝integraldisplayb ay(t) |tan(λx)–t a n ( λt)|kdt=f(x), 0 < k<1 . This is a special case of equation 3.8.7 with g(x)=t a n ( λx)+β,w h e r e βis an arbitrary number. 27.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglektan(λx)–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.5 with g(x)=ktan(λx). 28.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–ktan(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.6 with g(t)=ktan(λt). 252 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.5-4. Kernels Containing Cotangent. 29.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecot(λx)–c o t ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=c o t ( λx). 30.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecotkx–c o tkt⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), 0 < k<1 . This is a special case of equation 3.8.3 with g(x)=c o tkx. 3.5-5. Kernels Containing a Combination of Trigonometric Functions. 31.⎝integraldisplay ⎝integraldisplay∞ –∞⎝bracketleftbig⎝bracketleftbig cos(xt )+s i n ( xt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=1 2π⎝integraldisplay∞ –∞⎝bracketleftbig cos(xt )+s i n ( xt)⎝bracketrightbig f(t)dt. Up to constant factors, the function f(x) and the solution y(t)a r et h e Hartley transform pair . Reference: D. Zwillinger (1989). 32.⎝integraldisplay ⎝integraldisplay∞ 0⎝bracketleftbig⎝bracketleftbig sin(xt)–xtcos(xt )⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This equation can be reduced to a special case of equation 3.7.17 with ν=3 2. Solution: y(x)=2 π⎝integraldisplay∞ 0sin(xt)–xtcos(xt ) x2t2f(t)dt. 33.⎝integraldisplay ⎝integraldisplay∞ 0[sin(xt )+xtcos(xt )]y(t)dt=f(x). Solution: y(x)=–2 π⎝integraldisplay∞ 0si(xt)y(t)dt, where si( z) is the sine integral (see Supplement 11.3-1). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 457). 34.⎝integraldisplay∞ 0[1 – cos( xt)+xtsin(xt)]y(t)dt=f(x). Solution: y(x)=2 π⎝integraldisplay∞ 0ci(xt)f(t)dt, where ci( z) is the cosine integral (see Supplement 11.3-2). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 457). 3.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 253 35.⎝integraldisplay ⎝integraldisplay∞ 0(xt)1/2⎝bracketleftbigg ⎝bracketleftbiggsin(xt) xt+2c o s ( xt)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). Solution: y(x)=⎝radicalbigg 2 π⎝integraldisplay∞ 0⎝bracketleftbigg1 2–S(xt)⎝bracketrightbigg f(t)dt. where S(z) is the Fresnel sine integral (see Supplement 11.3-3). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 459). 36.⎝integraldisplay ⎝integraldisplay∞ 0(xt)1/2⎝bracketleftbigg ⎝bracketleftbiggcos(xt )–1 xt–2s i n ( xt)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). Solution: y(x)=⎝radicalbigg 2 π⎝integraldisplay∞ 0⎝bracketleftbigg1 2–C(xt)⎝bracketrightbigg f(t)dt, where C(z) is the Fresnel cosine integral (see Supplement 11.3-3). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 460). 37.⎝integraldisplay ⎝integraldisplay∞ 0(1 –ν)s i n (xt)+xtcos(xt ) (xt)νy(t)dt=f(x). Solution: y(x)=2 π⎝integraldisplay∞ 0S(xt,ν)f(t)dt, where S(z,ν) is the generalized Fresnel sine integral (see Supplement 11.3-3). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 461). 38.⎝integraldisplay ⎝integraldisplay∞ 0(1 –ν)c o s (xt)–xtsin(xt) (xt)νy(t)dt=f(x). Solution: y(x)=2 π⎝integraldisplay∞ 0C(xt,ν)y(t)dt, where C(z,ν) is the generalized Fresnel cosine integral (see Supplement 11.3-3). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 461). 39.⎝integraldisplay ⎝integraldisplayπ 0⎝bracketleftbigg ⎝bracketleftbiggasin(x+t) 1–2acos(x +t)+a2+asin(x–t) 1–2acos(x –t)+a2⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), 0 < a<1 . Solution: y(x)=C+2 π2∞⎝summationdisplay n=1fn ancos(nx ), fn=⎝integraldisplayπ 0f(x)s i n (nx)dx, where Cis an arbitrary constant. Remark. The kernel of the integral equation can be represented as a series in powers of a: K(x,t)=asin(x+t) 1–2acos(x +t)+a2+asin(x–t) 1–2acos(x –t)+a2=2∞⎝summationdisplay n=1ansin(nx)c o s (nt). References: W. Schmeidler (1950, p. 169), S. Feny ¨o and H. W. Stolle (1984, pp. 18–19). 254 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.5-6. Equations Containing the Unknown Function of a Complicated Argument. 40.⎝integraldisplay ⎝integraldisplayπ/2 0y(ξ)dt=f(x), ξ=xsint. Schl ¨omilch equation. Solution: y(x)=2 π⎝bracketleftbigg f(0) +x⎝integraldisplayπ/2 0f/prime ξ(ξ)dt⎝bracketrightbigg ,ξ=xsint. References: E. T. Whittaker and G. N. Watson (1958), F. D. Gakhov (1977). 41.⎝integraldisplay ⎝integraldisplayπ/2 0y(ξ)dt=f(x), ξ=xsinkt. Generalized Schl ¨omilch equation. This is a special case of equation 3.5.43 for λ=0a n d m=0 . Solution: y(x)=2k πxk–1 kd dx⎝bracketleftbigg x1 k⎝integraldisplayx 0sintf(ξ)dt⎝bracketrightbigg ,ξ=xsinkt. 42.⎝integraldisplay ⎝integraldisplayπ/2 0sinλty(ξ)dt=f(x), ξ=xsinkt. This is a special case of equation 3.5.43 for m=0 . Solution: y(x)=2k πxk–λ–1 kd dx⎝bracketleftbigg xλ+1 k⎝integraldisplayx 0sinλ+1tf(ξ)dt⎝bracketrightbigg ,ξ=xsinkt. 43.⎝integraldisplay ⎝integraldisplayπ/2 0sinλtcosmty(ξ)dt=f(x), ξ=xsinkt. 1◦.L e tλ> –1,m> –1, and k> 0. The transformation z=x2 k,ζ=zsin2t,w(ζ)=ζλ–1 2y⎝parenleftbig ζk 2⎝parenrightbig leads to an equation of the form 1.1.44: ⎝integraldisplayz 0(z–ζ)m–1 2w(ζ)dζ=F(z), F(z)=2zλ+m 2f⎝parenleftbig zk 2⎝parenrightbig . 2◦. Solution with –1 < m<1 : y(x)=2k πsin⎝bracketleftBigπ(1 –m) 2⎝bracketrightBig xk–λ–1 kd dx⎝bracketleftbigg xλ+1 k⎝integraldisplayπ/2 0sinλ+1ttanmtf(ξ)dt⎝bracketrightbigg , where ξ=xsinkt. 3.6. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 255 3.5-7. Singular Equations. 44.⎝integraldisplay ⎝integraldisplay2π 0cot⎝parenleftBig ⎝parenleftBigt–x 2⎝parenrightBig ⎝parenrightBig y(t)dt=f(x), 0 ≤x≤2π. Here the integral is understood in the sense of the Cauchy principal value and the right-hand side is assumed to satisfy the condition⎝integraldisplay2π 0f(t)dt=0 . Solution: y(x)=–1 4π2⎝integraldisplay2π 0cot⎝parenleftBigt–x 2⎝parenrightBig f(t)dt+C, where Cis an arbitrary constant. It follows from the solution that⎝integraldisplay2π 0y(t)dt=2πC. The equation and its solution form a Hilbert transform pair (in the asymmetric form). Reference: F. D. Gakhov (1977). 45.⎝integraldisplay ⎝integraldisplayπ –π⎝bracketleftbigg ⎝bracketleftbigg 1+c o t⎝parenleftBig ⎝parenleftBigx–t 2⎝parenrightBig ⎝parenrightBig⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), – π≤x≤π. Hilbert–Plessner equation. Solution: y(x)=1 4π2⎝integraldisplayπ –π⎝bracketleftbigg 1+c o t⎝parenleftBigx–t 2⎝parenrightBig⎝bracketrightbigg f(t)dt. Reference: S. Feny ¨o and H. W. Stolle (1984, pp. 36–38). 46.⎝integraldisplay2π 0⎝bracketleftBig⎝bracketleftBig sin⎝parenleftBig ⎝parenleftBigξ–x 2⎝parenrightBig⎝parenrightBig⎝bracketrightBig⎝bracketrightBig–2 y(ξ)dξ=f(x), 0 ≤x≤2π. The simple hypersingular equation of the ﬁrst kind with Hilbert-type kernel . Let the periodic conditions y(0) =y(2π) be satisﬁed. Then the solution is y(x)=–1 4π2⎝integraldisplay2π 0f(ξ)l n⎝vextendsingle⎝vextendsingle⎝vextendsinglesin⎝parenleftBigξ–x 2⎝parenrightBig⎝vextendsingle⎝vextendsingle⎝vextendsingledξ+C, where Cis an arbitrary constant. This equation is discussed in Subsection 14.6-4 in detail. Reference: I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004, p. 8). 3.6. Equations Whose Kernels Contain Combinations of Elementary Functions 3.6-1. Kernels Containing Hyperbolic and Logarithmic Functions. 1.⎝integraldisplay ⎝integraldisplayb aln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecosh(λx )–c o s h ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.9 with g(x)=c o s h ( λx). 2.⎝integraldisplay ⎝integraldisplayb aln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesinh(λx) – sinh( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.9 with g(x) = sinh(λx). 256 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.⎝integraldisplay ⎝integraldisplaya –aln⎝bracketleftbigg ⎝bracketleftbiggsinh⎝parenleftbig⎝parenleftbig1 2A⎝parenrightbig⎝parenrightbig 2s i n h⎝parenleftbig⎝parenleftbig1 2|x–t|⎝parenrightbig⎝parenrightbig⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), – a≤x≤a. Solution with 0 < a<A: y(x)=1 2M/prime(a)⎝bracketleftbiggd da⎝integraldisplaya –aw(t,a)f(t)dt⎝bracketrightbigg w(x,a) –1 2⎝integraldisplaya |x|w(x,ξ)d dξ⎝bracketleftbigg1 M/prime(ξ)d dξ⎝integraldisplayξ –ξw(t,ξ)f(t)dt⎝bracketrightbigg dξ –1 2d dx⎝integraldisplaya |x|w(x,ξ) M/prime(ξ)⎝bracketleftbigg⎝integraldisplayξ –ξw(t,ξ)df(t)⎝bracketrightbigg dξ, where the prime stands for the derivative with respect to the argument and M(ξ)=⎝bracketleftbigg ln⎝parenleftbiggsinh⎝parenleftbig1 2A⎝parenrightbig sinh⎝parenleftbig1 2ξ⎝parenrightbig⎝parenrightbigg⎝bracketrightbigg–1 ,w(x,ξ)=cosh⎝parenleftbig1 2x⎝parenrightbig M(ξ) π√ 2c o s h ξ–2c o s h x. Reference: I. C. Gohberg and M. G. Krein (1967). 4.⎝integraldisplay ⎝integraldisplayb aln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingletanh(λx)–t a n h ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.9 with g(x)=t a n h ( λx). 5.⎝integraldisplay ⎝integraldisplaya –aln⎝bracketleftbig⎝bracketleftbig coth⎝parenleftbig⎝parenleftbig1 4|x–t|⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), – a≤x≤a. Solution: y(x)=1 2M/prime(a)⎝bracketleftbiggd da⎝integraldisplaya –aw(t,a)f(t)dt⎝bracketrightbigg w(x,a) –1 2⎝integraldisplaya |x|w(x,ξ)d dξ⎝bracketleftbigg1 M/prime(ξ)d dξ⎝integraldisplayξ –ξw(t,ξ)f(t)dt⎝bracketrightbigg dξ –1 2d dx⎝integraldisplaya |x|w(x,ξ) M/prime(ξ)⎝bracketleftbigg⎝integraldisplayξ –ξw(t,ξ)df(t)⎝bracketrightbigg dξ, where the prime stands for the derivative with respect to the argument and M(ξ)=P–1/2(coshξ) Q–1/2(coshξ),w(x,ξ)=1 πQ –1/2(coshξ)√ 2c o s h ξ–2c o s h x, andP–1/2(coshξ)a n dQ–1/2(coshξ) are the Legendre functions of the ﬁrst and second kind, respectively. Reference: I. C. Gohberg and M. G. Krein (1967). 3.6-2. Kernels Containing Logarithmic and Trigonometric Functions. 6.⎝integraldisplay ⎝integraldisplayb aln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecos(λx )–c o s ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.9 with g(x)=c o s ( λx). 3.6. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 257 7.⎝integraldisplay ⎝integraldisplayb aln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesin(λx)–s i n ( λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.9 with g(x)=s i n ( λx). 8.⎝integraldisplay ⎝integraldisplayπ 0ln1–c o s ( x+t) 1–c o s ( x–t)y(t)dt=f(x), 0 ≤x≤π. Solution: y(x)=2 π2∞⎝summationdisplay n=1nfnsin(nx), fn=⎝integraldisplayπ 0f(x)s i n (nx)dx. Reference: S. Feny ¨o and H. W. Stolle (1984, p. 44). 9.⎝integraldisplay ⎝integraldisplaya –aln⎝bracketleftbigg ⎝bracketleftbiggsin⎝parenleftbig⎝parenleftbig1 2A⎝parenrightbig⎝parenrightbig 2s i n⎝parenleftbig⎝parenleftbig1 2|x–t|⎝parenrightbig⎝parenrightbig⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), – a≤x≤a. Solution with 0 < a<A: y(x)=1 2M/prime(a)⎝bracketleftbiggd da⎝integraldisplaya –aw(t,a)f(t)dt⎝bracketrightbigg w(x,a) –1 2⎝integraldisplaya |x|w(x,ξ)d dξ⎝bracketleftbigg1 M/prime(ξ)d dξ⎝integraldisplayξ –ξw(t,ξ)f(t)dt⎝bracketrightbigg dξ –1 2d dx⎝integraldisplaya |x|w(x,ξ) M/prime(ξ)⎝bracketleftbigg⎝integraldisplayξ –ξw(t,ξ)df(t)⎝bracketrightbigg dξ, where the prime stands for the derivative with respect to the argument and M(ξ)=⎝bracketleftbigg ln⎝parenleftbiggsin⎝parenleftbig1 2A⎝parenrightbig sin⎝parenleftbig1 2ξ⎝parenrightbig⎝parenrightbigg⎝bracketrightbigg–1 ,w(x,ξ)=cos⎝parenleftbig1 2ξ⎝parenrightbig M(ξ) π√ 2c o sx–2c o s ξ. Reference: I. C. Gohberg and M. G. Krein (1967). 10.d dx⎝integraldisplay ⎝integraldisplayπ –πln⎝parenleftbigg ⎝parenleftbigg 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesinx–t 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). Solution: y(x)=–1 π2d dx⎝integraldisplayπ –πln⎝parenleftbigg 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesinx–t 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝parenrightbigg f(t)dt,⎝integraldisplay π –πy(t)dt=0 . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 452). 3.6-3. Kernels Containing Combinations of Exponential and Other Elementary Functions. 11.⎝integraldisplay ⎝integraldisplayb a⎝parenleftbig⎝parenleftbig ln|x–t|+Ae–αx –βt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 3.8.28 with ϕ(x)=Ae–αxandψ(t)=e–βt. 12.⎝integraldisplay ⎝integraldisplay∞ 0[sin(xt )+Ae–αx –βt]y(t)dt=f(x). This is a special case of equation 3.8.29 with ϕ(x)=Ae–αxandψ(t)=e–βt. 13.⎝integraldisplay ⎝integraldisplay∞ 0[cos(xt)+Ae–αx –βt]y(t)dt=f(x). This is a special case of equation 3.8.30 with ϕ(x)=Ae–αxandψ(t)=e–βt. 258 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.7. Equations Whose Kernels Contain Special Functions∗ 3.7-1. Kernels Containing Error Function, Exponential Integral or Logarithmic Integral. 1.⎝integraldisplay ⎝integraldisplay∞ 0⎝bracketleftbig⎝bracketleftbig exp(i(x+t)2) erf(eπi/ 4(x+t)) + exp( i(x–t)2) erf(eπi/ 4(x–t))⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Here erf zis the error function (see Supplement 11.2-1) and i2= –1. Solution: y(x)=–1 π⎝integraldisplay∞ 0⎝bracketleftbig exp(–i(t+x)2)e r f (e3πi/4(t+x)) + exp(– i(t–x)2)e r f (e3πi/4(t–x))⎝bracketrightbig f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 459). 2.⎝integraldisplay ⎝integraldisplay∞ 0e–ixtEi(ixt)y(t)dt=f(x), i2= –1. Here Ei(z ) is the exponential integral (see Supplement 11.2-2). Solution: y(t)=1 2π2⎝integraldisplay∞ –∞⎝bracketleftbigg eixterf(eπi/4√ xt)–1+i √ 2πxt⎝bracketrightbigg f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 456). 3.⎝integraldisplay ⎝integraldisplay∞ 1li⎝parenleftbigg ⎝parenleftbiggx t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), f(1) =f/prime(1) = 0. Here li( z) is the logarithmic integral (see Supplement 11.2-3). Solution: y(t)=–⎝integraldisplayx 1t–2ν⎝parenleftBig lnt x⎝parenrightBig⎝bracketleftbigg⎝parenleftbigg td dt⎝parenrightbigg2 –td dt⎝bracketrightbigg f(t)dt, where ν(z)=⎝integraldisplay∞ 0zξdξ Γ(ξ+1 ). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 457). 3.7-2. Kernels Containing Sine Integrals, Cosine Integrals, or Fresnel Integrals. 4.⎝integraldisplay ⎝integraldisplay∞ 0si(xt)y(t)dt=f(x). Here si( z) is the sine integral (see Supplement 11.3-1). Solution: y(x)=–2 π⎝integraldisplay∞ 0[sin(xt)+xtcos(xt )]f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 457). * For notation and properties of special functions, see Supplement 11. 3.7. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 259 5.⎝integraldisplay ⎝integraldisplay∞ 0ci(xt)y(t)dt=f(x). Here ci( z) is the cosine integral (see Supplement 11.3-2). Solution: y(x)=2 π⎝integraldisplay∞ 0[1 – cos( xt)+xtsin(xt)]f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 457). 6.⎝integraldisplay ⎝integraldisplay∞ 0⎝bracketleftbigg ⎝bracketleftbigg1 2–S(xt)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). HereS(z) is the Fresnel sine integral (see Supplement 11.3-3). Solution: y(x)=⎝radicalbigg 2 π⎝integraldisplay∞ 0(xt)1/2⎝bracketleftbiggsin(xt) xt+2c o s ( xt)⎝bracketrightbigg f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 459). 7.⎝integraldisplay ⎝integraldisplay∞ 0⎝bracketleftbigg ⎝bracketleftbigg1 2–C(xt)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). HereC(z) is the Fresnel cosine integral (see Supplement 11.3-3). Solution: y(x)=⎝radicalbigg 2 π⎝integraldisplay∞ 0(xt)1/2⎝bracketleftbiggcos(xt )–1 xt–2s i n ( xt)⎝bracketrightbigg f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 460). 8.⎝integraldisplay ⎝integraldisplay∞ 0S(xt,ν)y(t)dt=f(x). HereS(z,ν) is the generalized Fresnel sine integral (see Supplement 11.3-3). Solution: y(x)=2 π⎝integraldisplay∞ 0(1 –ν)s i n (xt)+xtcos(xt ) (xt)νf(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 461). 9.⎝integraldisplay ⎝integraldisplay∞ 0C(xt,ν)y(t)dt=f(x). HereC(z,ν) is the generalized Fresnel cosine integral (see Supplement 11.3-3). Solution: y(x)=2 π⎝integraldisplay∞ 0(1 –ν)c o s (xt)–xtsin(xt) (xt)νf(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 461). 260 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.7-3. Kernels Containing Gamma Functions. 10.⎝integraldisplay ⎝integraldisplay∞ 0(xt)–(π+1)/2Γ(±iln(xt))y(t)dt=f(x), i2= –1. HereΓ(z) is the incomplete gamma function (see Supplement 11.4-1). Solution: y(x)=1 4π2⎝integraldisplay∞ 0(xt)–(π+1)/2Γ(∓iln(xt))f(t)dt. The integral equation and its solution form a Paley–Wiener transform pair (in the asym- metric form). References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 453). 11.⎝integraldisplay ⎝integraldisplay∞ –∞e–π(x+t)/2Γ(±i(x+t))y(t)dt=f(x). Solution: y(x)=1 4π2⎝integraldisplay∞ –∞e–π(x+t)/2Γ(∓i(x+t))f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 453). 12.⎝integraldisplay ⎝integraldisplay∞ –∞Γ(α+i(x+t))Γ(α–i(x+t))y(t)dt=f(x). Solution: y(x)=–αsin(2πα) 2π3⎝integraldisplay∞ –∞Γ(–α+i(x+t))Γ(–α–i(x+t))f(t)dt, where Re α<0( 2α≠–1, –2, ...). References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 453). 3.7-4. Kernels Containing Incomplete Gamma Functions. 13.⎝integraldisplay ⎝integraldisplay∞ –∞(t–x)α–1γ(1 –α,2i(t–x))y(t)dt=f(x), i2= –1. Hereγ(ν,z) is the incomplete gamma function (see Supplement 11.5-1). Solution: y(x)=–1 4π2⎝integraldisplay∞ –∞(t–x)–α–1γ(1 +α,2i(t–x))f(t)dt, where –1 /2<R e α≤0. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 462). 14.⎝integraldisplay ⎝integraldisplay∞ –∞⎝bracketleftbigg ⎝bracketleftbigg exp⎝parenleftbigg ⎝parenleftbigg2x–i 4π⎝parenrightbigg ⎝parenrightbigg t–ix–1/ 2+(b–a)aix–1/ 2eiatΓ⎝parenleftbigg ⎝parenleftbigg1 2–ix,iat⎝parenrightbigg⎝parenrightbigg⎝bracketrightbigg⎝bracketrightbigg y(t)dt=f(x). Solution: y(x)=1 4π⎝integraldisplay∞ –∞⎝bracketleftbigg exp⎝parenleftbigg2t+i 4π⎝parenrightbigg xit–1/2+(a–b)b–it–1/2e–ibxΓ⎝parenleftbigg1 2+it,–ibx⎝parenrightbigg⎝bracketrightbiggf(t) cosh(πt )dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463). 3.7. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 261 15.⎝integraldisplay ⎝integraldisplay∞ 0⎝braceleftbigg ⎝braceleftbigg t–ix–1/ 2sin⎝parenleftbigg ⎝parenleftbigg1+2ix 4π⎝parenrightbigg ⎝parenrightbigg +i 2(b–a)aix–1/ 2⎝bracketleftbigg ⎝bracketleftbigg e–iatΓ⎝parenleftbigg ⎝parenleftbigg1 2–ix,–iat⎝parenrightbigg ⎝parenrightbigg –eiatΓ⎝parenleftbigg ⎝parenleftbigg1 2–ix,iat⎝parenrightbigg⎝parenrightbigg⎝bracketrightbigg⎝bracketrightbigg⎝bracerightbigg⎝bracerightbigg y(t)dt=f(x). Solution: y(t)=1 π⎝integraldisplay∞ –∞⎝braceleftbigg tix–1/2sin⎝parenleftbigg1–2ix 4π⎝parenrightbigg +i 2(a–b)b–ix–1/2⎝bracketleftbigg e–ibtΓ⎝parenleftbigg1 2+ix,–ibt⎝parenrightbigg –eibtΓ⎝parenleftbigg1 2+ix,ibt⎝parenrightbigg⎝bracketrightbigg⎝bracerightbiggf(x) cosh(πx )dx, where a,b∉(–∞, 0) are complex numbers. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463). 16.⎝integraldisplay ⎝integraldisplay∞ 0⎝braceleftbigg ⎝braceleftbigg t–ix–1/ 2cos⎝parenleftbigg ⎝parenleftbigg1+2ix 4π⎝parenrightbigg ⎝parenrightbigg +1 2(b–a)aix–1/ 2⎝bracketleftbigg ⎝bracketleftbigg e–iatΓ⎝parenleftbigg ⎝parenleftbigg1 2–ix,–iat⎝parenrightbigg ⎝parenrightbigg +eiatΓ⎝parenleftbigg ⎝parenleftbigg1 2–ix,iat⎝parenrightbigg⎝parenrightbigg⎝bracketrightbigg⎝bracketrightbigg⎝bracerightbigg⎝bracerightbigg y(t)dt=f(x). Solution: y(t)=1 π⎝integraldisplay∞ –∞⎝braceleftbigg tix–1/2cos⎝parenleftbigg1–2ix 4π⎝parenrightbigg +1 2(a–b)b–ix–1/2⎝bracketleftbigg e–ibtΓ⎝parenleftbigg1 2+ix,–ibt⎝parenrightbigg +eibtΓ⎝parenleftbigg1 2+ix,ibt⎝parenrightbigg⎝bracketrightbigg⎝bracerightbiggf(x) cosh(πx )dx, where a,b∉(–∞, 0) are complex numbers. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463). 3.7-5. Kernels Containing Bessel Functions of the First Kind. 17.⎝integraldisplay ⎝integraldisplay∞ 0tJν(xt)y(t)dt=f(x). HereJν(z) is the Bessel function of the ﬁrst kind (see Supplement 11.6-1). Solution: y(x)=⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩⎝integraldisplay ∞ 0tJν(xt)f(t)dt if Reν≥–1 or ν= –2, –3, ..., ⎝integraldisplay∞ 0t⎝bracketleftbigg Jν(xt)–n–1⎝summationdisplay k=0(–1)k(xt/2)2k+ν k!Γ(ν+k+1 )⎝bracketrightbigg f(t)dtif Reν<– 1a n d ν≠–2, –3, ..., where – n–1<R e ν<–n,n=1 ,2 , ... The functions f(x)a n dy(x)a r et h e Hankel transform pair . References: E. C. Titchmarsh (1923), J. L. Grifﬁth (1958), V . A. Ditkin and A. P. Prudnikov (1965), F. Oberhet- tinger (1972), I. Sneddon (1972), H. M. Srivastava and R. G. Buschman (1977), B. Davis (1978), A. P. Prudnikov,Yu. A. Brychkov, and O. I. Marichev (1992, p. 468), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993), I. Sneddon (1995). 262 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 18.⎝integraldisplay ⎝integraldisplayb atJν(xt)y(t)dt=f(x), 0 ≤x<∞. Solution: y(t)=⎝braceleftBigg⎝integraldisplay∞ 0xJν(xt)f(x)dx ifa<t<b, 0i f0<t<aort>b, where 0 ≤a≤b≤∞and Re ν> –1. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 468), I. N. Sneddon (1995). 19.⎝integraldisplay ⎝integraldisplay∞ 0tJ0(xt)y(t)dt=0 , a≤x<∞. Homogeneous integral equation of the ﬁrst kind. Solution: y(t)=⎝integraldisplaya 0cos(xt )ϕ(x)dx, where ϕ(x) is an arbitrary continuously differentiable function. Reference: Ya. S. Uﬂyand (1977). 20.⎝integraldisplay ⎝integraldisplay∞ 0tJν(xt)y(t)dt=0 , a≤x<∞. Homogeneous integral equation of the ﬁrst kind, Re ν>– 1/2. Solution: y(t)=⎝radicalbigg πt 2⎝integraldisplaya 0√ xJν–1/2(xt)ϕ(x)dx, where ϕ(x) is an arbitrary continuously differentiable function. Reference: Ya. S. Uﬂyand (1977). 21.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleJν(λx)–Jν(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=Jν(λx), where Jν(z) is the Bessel function of the ﬁrst kind. 22.⎝integraldisplay ⎝integraldisplay∞ 0Jν(λ(x–t))y(t)dt=f(x). 1◦.I f|Reν|<1a n d f(0) =f/prime(0) = 0 then y(x)=⎝integraldisplayx 0J–ν(λ(x–t))⎝parenleftbiggd2 dt2+λ2⎝parenrightbigg f(t)dt. 2◦.I fν=nis a positive integer number and f(0) =f/prime(0) =···=f(n+1)(0) = 0 then y(x)=1 λn[(n–1)/2]⎝summationdisplay k=0C2k+1 n⎝parenleftbiggd dx⎝parenrightbiggn–2k–1⎝parenleftbiggd2 dx2+λ2⎝parenrightbiggk+1 f(x) +1 λn⎝integraldisplayx 0J0(λ(x–t))[n/2]⎝summationdisplay k=0C2k n⎝parenleftbiggd dt⎝parenrightbiggn–2k⎝parenleftbiggd2 dt2+λ2⎝parenrightbiggk+1 f(t)dt, where [ A] stands for the integer part of the number AandCk n=n! k!(n–k)!are binomial coefﬁcients (0! = 1). 3.7. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 263 3◦.I fνis not an integer, m–1<R e ν<m(m=0 ,1 ,2 , ...), and f(0) =f/prime(0) =···= f(m+1)(0) = 0 then y(x)=m–ν λm⎝integraldisplayx 0Jm–ν(λ(x–t)) x–t[(m–1)/2]⎝summationdisplay k=0C2k+1 m⎝parenleftbiggd dt⎝parenrightbiggm–2k–1⎝parenleftbiggd2 dt2+λ2⎝parenrightbiggk+1 f(t)dt +1 λm⎝integraldisplayx 0Jm–ν(λ(x–t))[m/2]⎝summationdisplay k=0C2k m⎝parenleftbiggd dt⎝parenrightbiggm–2k⎝parenleftbiggd2 dt2+λ2⎝parenrightbiggk+1 f(t)dt. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 470), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 23.⎝integraldisplay ⎝integraldisplay∞ –∞|x–t|νJν(λ|x–t|)y(t)dt=f(x). Solution: y(x)=–λcos(νπ ) 4s i n2(νπ)⎝integraldisplay∞ –∞sign(t–x) |t–x|2ν+1d dt⎝bracketleftBig |t–x|ν+1J–ν–1(λ|t–x|)f(t)⎝bracketrightBig dt, where 0 < Re ν<1/2. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 469), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 24.⎝integraldisplay ⎝integraldisplay∞ 0Jn/2–1(2πxt )G(x,t)y(t)dt=f(x), G(x,t)=2πx(t/x )n/2,n=1 , 2 , ... Solution: y(x)=⎝integraldisplay∞ 0Jn/2–1(2πxt)G(x,t)f(t)dt. The functions f(x)a n dy(t)a r et h e Bochner transform pair . Reference: Yu. A. Brychkov and A. P. Prudnikov (1979). 25.⎝integraldisplay ⎝integraldisplay∞ 0d dx⎝bracketleftBig⎝bracketleftBig xJ2 ν(xt)⎝bracketrightBig⎝bracketrightBig ty(t)dt=f(x). Solution: y(x)=– 2π⎝integraldisplay∞ 0tJν(x,t)Yν(xt)f(t)dt =π⎝integraldisplay∞ 0t⎝braceleftBig sin(2νπ)[J2 –ν(xt)–Y2 ν(xt)] – 2 cos(2 νπ)J–ν(xt)Y–ν(xt)⎝bracerightBig f(t)dt. References: I. I. Hirschman and D. V . Widder (1955), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 474). 26.⎝integraldisplay ⎝integraldisplay∞ 0t⎝bracketleftbig⎝bracketleftbig J–µ(xt)J–ν(xt)±Jµ(xt)Jν(xt)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=π 2c o s⎝bracketleftbigπ 2(ν±µ)⎝bracketrightbig sin⎝bracketleftbigπ 2(ν∓µ)⎝bracketrightbig⎝integraldisplay∞ 0td dt⎝bracketleftBig t⎝parenleftbig Jµ(xt)J–ν(xt)∓J–µ(xt)Jν(xt)⎝parenrightbig⎝bracketrightBig f(t)dt, where Re( µ+ν)<3/2. References: I. I. Hirschman and D. V . Widder (1955), E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 475). 264 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 27.⎝integraldisplay ⎝integraldisplay∞ 0[Jix(t)+J–ix(t)]y(t)dt=f(x), i2= –1. Solution: y(x)=1 2x⎝integraldisplay∞ 0t[Jit(x)+J–it(x)] sinh(πt)f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 469). 28.⎝integraldisplay ⎝integraldisplay∞ 0[Jit(x)+J–it(x)]y(t)dt=f(x), i2= –1. Solution: y(x)=x 2s i n h ( πx)⎝integraldisplay∞ 0Jix(t)+J–ix(t) tf(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 469). 3.7-6. Kernels Containing Bessel Functions of the Second Kind. 29.⎝integraldisplay ⎝integraldisplay∞ 0tYν(xt)y(t)dt=f(x). HereYν(z) is the Bessel function of the second kind (see Supplement 11.6-1). 1◦.I f|Reν|<1t h e n y(x)=⎝integraldisplay∞ 0tHν(xt)f(t)dt, where Hν(x) is the Struve function, which is deﬁned as Hν(x)=∞⎝summationdisplay j=0(–1)j(x/2)ν+2j+1 Γ⎝parenleftbig j+3 2⎝parenrightbig Γ⎝parenleftbig ν+j+3 2⎝parenrightbig. The function f(x) and the solution y(x)a r et h e Yν-transform pair . 2◦.I f 1 < |Reν|<3t h e n y(x)=⎝integraldisplay∞ 0t⎝bracketleftbigg Hν(xt)–(xt)ν–1 2ν–1√ πΓ(ν+1/2)⎝bracketrightbigg f(t)dt. References: E. C. Titchmarsh (1948), G. N. Watson (1952), J. L. Grifﬁth (1958), F. Oberhettinger (1972), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 475). 30.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleYν(λx)–Yν(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=Yν(λx), where Yν(z) is the Bessel function of the second kind. 3.7. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 265 3.7-7. Kernels Containing Combinations of the Bessel Functions. 31.⎝integraldisplay∞ 0[cos(pπ)Jν(xt)+s i n ( pπ)Yν(xt)]ty (t)dt=f(x). Solution: y(x)=⎝integraldisplay∞ 0Φ(xt)tf(t)dt,Φ(z)=∞⎝summationdisplay n=0(–1)n(z/2)ν+2p+2n Γ(p+n+1 )Γ(ν+p+n+1 ). The functions f(x)a n dy(x)a r et h e Hardy transform pair . Reference: Yu. A. Brychkov and A. P. Prudnikov (1989). 32.⎝integraldisplay ⎝integraldisplay∞ 0tJν(xt)Yν(xt)y(t)dt=f(x). Solution: y(x)=2π⎝integraldisplay∞ 0td dt⎝bracketleftBig tJ2 ν(xt)⎝bracketrightBig f(t)dt, where Re ν>– 1/4. References: E. C. Titchmarsh (1948), I. I. Hirschman and D. V . Widder (1955), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 476). 33.⎝integraldisplay ⎝integraldisplay∞ at[Jν(ax)Yν(xt)–Yν(ax)Jν(xt)]y(t)dt=f(x). Solution: y(x)=⎝integraldisplay∞ 0t[Jν(at)Yν(xt)–Yν(at)Jν(xt)] J2ν(at)+Y2ν(at)f(t)dt. The function f(x) and the solution y(x)a r et h e W eber transform pair . References: G. N. Watson (1952), Yu. A. Brychkov and A. P. Prudnikov (1979, 1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 477). 34.⎝integraldisplay ⎝integraldisplay∞ 0t[Jν(at)Yν(xt)–Yν(at)Jν(xt)]y(t)dt=f(x). Solution: y(x)=x J2ν(ax)+Y2ν(ax)⎝integraldisplay∞ 0t[Jν(ax)Yν(xt)–Yν(ax)Jν(xt)]f(t)dt. References: G. N. Watson (1952), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 477). 35.⎝integraldisplay ⎝integraldisplay∞ –∞e±π(x–t)/2H(1) i(t–x)(a)y(t)dt=f(x), i2= –1. HereH(1) ν(z)=Jν(z)+iYν(z) is the Hankel function of the ﬁrst kind (see Supplement 11.6-5). Solution: y(x)=1 4⎝integraldisplay∞ –∞e±π(t–x)/2H(1) i(t–x)(a)f(t)dt, where a>0 . References: Vu Kim Tuan (1988), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 479). 266 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 36.⎝integraldisplay ⎝integraldisplay∞ –∞e±π(x–t)/2H(2) i(t–x)(a)y(t)dt=f(x). HereH(2) ν(z)=Jν(z)–iYν(z) is the Hankel function of the second kind (see Supplement 11.6-5). Solution: y(x)=1 4⎝integraldisplay∞ –∞e±π(t–x)/2H(2) i(t–x)(a)f(t)dt, where a>0 . References: Vu Kim Tuan (1988), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 479). 3.7-8. Kernels Containing Modiﬁed Bessel Functions of the First Kind. 37.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleIν(λx)–Iν(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=Iν(λx), where Iν(z) is the modiﬁed Bessel function of the ﬁrst kind (see Supplement 11.7-1). 38.⎝integraldisplay ⎝integraldisplay∞ 0d dxI2 it(x)y(t)dt=f(x), i2= –1. Solution: y(x)=2i πx⎝integraldisplay∞ 0K2 ix(t)f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 485). 39.⎝integraldisplay ⎝integraldisplay∞ –∞Ai(x+t)y(t)dt=f(x). Here Ai(x )=1 3√ x⎝bracketleftbig I–1/3(z)–I1/3(z)⎝bracketrightbig is the Airy function (see Supplement 11.8-1). Solution: y(x)=⎝integraldisplay∞ –∞Ai(x+t)f(t)dt. References: Vu Kim Tuan (1988), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 485). 3.7-9. Kernels Containing Modiﬁed Bessel Functions of the Second Kind. 40.⎝integraldisplay ⎝integraldisplay∞ –∞K0⎝parenleftbig⎝parenleftbig |x–t|⎝parenrightbig⎝parenrightbig y(t)dt=f(x). HereK0(z) is the modiﬁed Bessel function of the second kind (the MacDonald function), see Supplement 11.7-1. Solution: y(x)=–1 π2⎝parenleftbiggd2 dx2–1⎝parenrightbigg⎝integraldisplay∞ –∞K0⎝parenleftbig |x–t|⎝parenrightbig f(t)dt. Reference: D. Naylor (1986). 41.⎝integraldisplay ⎝integraldisplayb a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleKν(λx)–Kν(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). This is a special case of equation 3.8.3 with g(x)=Kν(λx). 3.7. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 267 42.⎝integraldisplay ⎝integraldisplay∞ 0√ ztK ν(zt)y(t)dt=f(z). HereKν(z) is the modiﬁed Bessel function of the second kind. Up to a constant factor, the left-hand side of this equation is the Meijer transform of y(t) (zis treated as a complex variable). Solution: y(t)=1 πi⎝integraldisplayc+i∞ c–i∞√ ztIν(zt)f(z)dz. For speciﬁc f(z), one may use tables of Meijer integral transforms to calculate the integral. Reference: V . A. Ditkin and A. P. Prudnikov (1965). 43.⎝integraldisplay ⎝integraldisplay∞ 0Kix(t)y(t)dt=f(x), i2= –1. Solution: y(x)=2 π2x⎝integraldisplay∞ 0tsinh(πt)Kit(x)f(t)dt. The function f(x) and the solution y(x)a r et h e Kontorovich-Lebedev transform pair . References: V . A. Ditkin and A. P. Prudnikov (1965), F. Oberhettinger (1972), Yu. A. Brychkov and A. P. Prud- nikov (1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 487). 44.⎝integraldisplay ⎝integraldisplay∞ 0Kit(x)y(t)dt=f(x). Solution: y(x)=2xsinh(πx) π2⎝integraldisplay∞ 0Kix(t) tf(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 487). 45.⎝integraldisplay ⎝integraldisplay∞ 0K2 it(x)y(t)dt=f(x). Solution: y(x)=4xsinh(πx) π2⎝integraldisplay∞ 0d dt⎝braceleftBig [Iix(t)+I–ix(t)]K ix(t)⎝bracerightBig f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 492). 46.⎝integraldisplay ⎝integraldisplay∞ 0ReKix+1/2(t)y(t)dt=f(x). Solution: y(x)=4 π2⎝integraldisplay∞ 0cosh(πt)R eKit+1/2(x)f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 488). 47.⎝integraldisplay ⎝integraldisplay∞ 0ImKix+1/2(t)y(t)dt=f(x). Solution: y(x)=4 π2⎝integraldisplay∞ 0cosh(πt)I mKit+1/2(x)f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 488). 268 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 48.⎝integraldisplay ⎝integraldisplay∞ 0ReKit+1/2(x)y(t)dt=f(x). Solution: y(x)=4 π2cosh(πx )⎝integraldisplay∞ 0ReKix+1/2(t)f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 488). 49.⎝integraldisplay ⎝integraldisplay∞ 0ImKit+1/2(x)y(t)dt=f(x). Solution: y(x)=4 π2cosh(πx )⎝integraldisplay∞ 0ImKix+1/2(t)f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 488). 50.⎝integraldisplay ⎝integraldisplay∞ –∞eπ(x+t)/2Ki(x+t)(a)y(t)dt=f(x). Solution: y(x)=1 π2⎝integraldisplay∞ –∞eπ(x+t)/2Ki(x+t)(a)f(t)dt, where a> 0. The function f(x) and the solution y(x)a r ea Crum transform pair (in the asymmetric form). Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 488). 51.⎝integraldisplay ⎝integraldisplay∞ –∞Ki(x+t)(±ia)y(t)dt=f(x). Solution: y(x)=1 π2⎝integraldisplay∞ –∞Ki(x+t)(∓ia)f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 488–489). 52.⎝integraldisplay ⎝integraldisplay∞ –∞t–1 4(2ix+1)K 1 2+ix⎝parenleftbig⎝parenleftbig 2iλ√ t⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(x)=λ π2⎝integraldisplay∞ –∞x1 4(2it–1)K 1 2–it⎝parenleftbig 2iλ√ x⎝parenrightbig f(t)dt, where λ>0a n d√ x=–i√ |x|forx<0 . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 489). 53.⎝integraldisplay ⎝integraldisplay∞ 0⎝bracketleftBig⎝bracketleftBig (a+t)–1 4(2ix+1)K 1 2+ix(2iλ√ a+t) +(a–t)–1 4(2ix+1)K 1 2+ix(2iλ√ a–t)⎝bracketrightBig⎝bracketrightBig y(t)dt=f(x). Solution: y(t)=λ π2⎝integraldisplay∞ –∞⎝bracketleftBig (a+t)1 4(2ix–1)K 1 2–ix(–2iλ√ a+t)+(a–t)1 4(2ix–1)K 1 2–ix(–2iλ√ a–t)⎝bracketrightBig f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 489). 3.7. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 269 54.⎝integraldisplay ⎝integraldisplay∞ –∞x1 4(2it–1)K 1 2–it⎝parenleftbig⎝parenleftbig 2iλ√ x⎝parenrightbig⎝parenrightbig y(t)dt=f(x),λ>0 . Solution: y(x)=λ π2⎝integraldisplay∞ –∞t–1 4(2ix+1)K 1 2+ix⎝parenleftbig 2iλ√ t⎝parenrightbig f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 489). 55.⎝integraldisplay ⎝integraldisplay∞ –∞⎝bracketleftBig⎝bracketleftBig (a+t)1 4(2it–1)K 1 2–it(–2iλ√ a+x) +(a–t)1 4(2it–1)K 1 2–it(–2iλ√ a–x)⎝bracketrightBig⎝bracketrightBig y(t)dt=f(x). Solution: y(t)=λ π2⎝integraldisplay∞ 0⎝bracketleftBig (a+x)–1 4(2it+1)K 1 2+it(2iλ√ a+x)+(a–x)–1 4(2it+1)K 1 2+it(2iλ√ a–x)⎝bracketrightBig f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 490). 56.⎝integraldisplay ⎝integraldisplay∞ –∞exp⎝parenleftBig ⎝parenleftBigπx 2signt⎝parenrightBig ⎝parenrightBig Kix(|t|)y(t)dt=f(x). Solution: y(x)=1 π2x⎝integraldisplay∞ –∞texp⎝parenleftBigπt 2signx⎝parenrightBig Kit(|x|)f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 490). 3.7-10. Kernels Containing a Combination of Bessel and Modiﬁed Bessel Functions. 57.⎝integraldisplay ⎝integraldisplay∞ 0[Iix(t)+I–ix(t)]Kix(t)y(t)dt=f(x). Solution: y(x)=–4 π2d dx⎝integraldisplay ⎝integraldisplay∞ 0tsinh(πt)K2 it(x)f(t)dt. The integral equation and its solution form the Lebedev transform pair . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 493). 58.⎝integraldisplay ⎝integraldisplay∞ 0[Kit(a)Iit(x)–Iit(a)Kit(x)]y(t)dt=f(x), 0 < x<a. Solution: y(t)=2tsinh(πt) π2|Iia(a)|2⎝integraldisplay ⎝integraldisplaya 0x–1[Kit(a)Iit(x)–Iit(a)Kit(x)]f(x)dx,t>0 . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 494). 59.⎝integraldisplay ⎝integraldisplay∞ 0t⎝bracketleftbigg ⎝bracketleftbigg Y0(xt)–2 πK0(xt)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). Solution: y(x)=⎝integraldisplay ⎝integraldisplay∞ 0t⎝bracketleftbigg Y0(xt)–2 πK0(xt)⎝bracketrightbigg f(t)dt. The integral equation and its solution form the divisor transform pair . References: F. Oberhettinger (1973), E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 492). 270 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 60.⎝integraldisplay ⎝integraldisplay∞ 0t⎝bracketleftbigg ⎝bracketleftbigg Y2n+1(xt)±2 πK2n+1(xt)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), n=1 ,2 , ... Solution: y(x)=⎝integraldisplay ⎝integraldisplay∞ 0t⎝bracketleftbigg Y2n+1(xt)∓2 πK2n+1(xt)⎝bracketrightbigg f(t)dt. References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 493). 61.⎝integraldisplay ⎝integraldisplay∞ 0t⎝bracketleftbigg ⎝bracketleftbigg Y2n(xt)+2 πK2n(xt)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), n=1 ,2 , ... Solution: y(x)=⎝integraldisplay ⎝integraldisplay∞ 0t⎝bracketleftbigg Y2n(xt)+2 πK2n(xt)⎝bracketrightbigg f(t)dt. References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 493). 3.7-11. Kernels Containing Legendre Functions. 62.⎝integraldisplay∞ 1P–1 2+ix(t)y(t)dt=f(x), 0 ≤x<∞. HerePν(x) is the Legendre function of the ﬁrst kind (see Supplement 11.11-3) and i2= –1. Solution: y(t)=⎝integraldisplay∞ 0xtanh(πx)Pix–1/2(t)f(x)dx. The functions f(x)a n dy(t)a r et h e Mehler–F ock transform pair . Remark. The Legendre function of the ﬁrst kind can be represented in the form P–1 2+ix(t)=2 πcosh(πx )⎝integraldisplay∞ 0cos(xs)ds √ 2(t+c o s h s),1 ≤t<∞. References: N. N. Lebedev (1965), V . A. Ditkin and A. P. Prudnikov (1965), Yu. A. Brychkov and A. P. Prudnikov (1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 512). 63.⎝integraldisplay∞ 0P–1 2+it(x)y(t)dt=f(x), 1 ≤x<∞. Solution: y(t)=ttanh(πt)⎝integraldisplay∞ 1P–1 2+it(x)f(x)dx. References: N. N. Lebedev (1965), V . A. Ditkin and A. P. Prudnikov (1965), Yu. A. Brychkov and A. P. Prudnikov (1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 513). 64.⎝integraldisplay∞ 0[P–1 2+ix(it)±P–1 2+ix(–it)]y(t)dt=f(x). Solution: y(t)=1 2⎝integraldisplay∞ 0sinh(πx) cosh2(πx)[P–1 2+ix(–it)±P–1 2+ix(it)]f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 513). 3.7. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 271 65.⎝integraldisplay∞ 0[P–1 2+it(ix)±P–1 2+it(–ix)]y(t)dt=f(x). Solution: y(t)=tsinh(πt) 2c o s h2(πt)⎝integraldisplay∞ 0[P–1 2+it(–ix)±P–1 2+it(ix)]f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 514). 66.⎝integraldisplay∞ –∞[ie–iπxP–1 2+x(cost)+P–1 2+x(– cos t)]y(t)dt=f(x). Solution: y(t)=1 2sint⎝integraldisplay∞ –∞x sinh(2 πx)[ieiπxP–1 2+x(cost)+P–1 2+x(– cos t)]f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 513). 67.⎝integraldisplay∞ 0[P–1 2+it(x)]2y(t)dt=f(x), 1 ≤x<∞. Solution: y(t)=ttanh(πt)⎝integraldisplay∞ 1P–1 2+it(x)⎝bracketleftbig Q–1 2+it(x)+Q–1 2–it(x)⎝bracketrightbig (x2–1 )1/2d dx⎝bracketleftbig (x2–1 )1/2f(x)⎝bracketrightbig dx, where Qν(x) is the Legendre function of the second kind. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 514). 68.⎝integraldisplay∞ 1P–1 2+ix(t)⎝bracketleftbig⎝bracketleftbig Q–1 2+ix(t)+Q–1 2–ix(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), 0 ≤x<∞. HereQν(x) is the Legendre function of the second kind. Solution: y(t)=(t2–1 )1/2d dt⎝bracketleftbigg (t2–1 )1/2⎝integraldisplay∞ 0xtanh(πx)[P–1 2+ix(t)]2f(x)dx⎝bracketrightbigg . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 519). 3.7-12. Kernels Containing Associated Legendre Functions. 69.⎝integraldisplay∞ 1Pµ –1 2+ix(t)y(t)dt=f(x), 0 ≤x<∞. HerePµ ν(x) is the associated Legendre function of the ﬁrst kind (see Supplement 11.11-3) andi2= –1. Solution: y(t)=1 π⎝integraldisplay∞ 0xsinh(πx)Γ⎝parenleftbig1 2–µ+ix⎝parenrightbig Γ⎝parenleftbig1 2–µ–ix⎝parenrightbig Pµ ix–1/2(t)f(x)dx. The functions f(x)a n dy(t)a r et h e generalized Mehler–F ock transform pair . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 518). 272 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 70.⎝integraldisplay∞ 0Pµ –1 2+it(x)y(t)dt=f(x), 1 ≤x<∞. Solution: y(t)=1 πtsinh(πt)Γ⎝parenleftbig1 2–µ+it⎝parenrightbig Γ⎝parenleftbig1 2–µ–it⎝parenrightbig⎝integraldisplay∞ 1Pµ it–1/2(x)f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 519). 71.⎝integraldisplay1 –1Pix –1 2+ia(±t)y(t)dt=f(x), – ∞<x<∞. Solution: y(t)=1 2πi(1 –t)⎝integraldisplay∞ –∞xΓ⎝parenleftbig1 2+ia–ix⎝parenrightbig Γ⎝parenleftbig1 2–ia–ix⎝parenrightbig Pix –1 2+ia(∓t)f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 518). 72.⎝integraldisplay∞ 0⎝bracketleftbig⎝bracketleftbig (x+t–1 )2–4xt⎝bracketrightbig⎝bracketrightbig–1/ 2Q1 ν–1 2⎝parenleftbigg ⎝parenleftbiggx+t–1 2√ xt⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), Re ν> –1. HereQµ ν(x) is the associated Legendre function of the second kind (see Supplement 11.11-3). Solution: y(t)=1 4π2⎝integraldisplay∞ 0(xt)–1/2⎝bracketleftbig (x+t–1 )2–4xt⎝bracketrightbig–1/2Q1 ν–1 2⎝parenleftbiggx+t–1 2√ xt⎝parenrightbigg f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 520). 3.7-13. Kernels Containing Kummer Conﬂuent Hypergeometric Functions. 73.⎝integraldisplay ⎝integraldisplay∞ 0F(a,b;ixt)y(t)dt=f(x). HereF(a,b;x) is the Kummer conﬂuent hypergeometric function (see Supplement 11.9-1) andi2= –1. Let Re( b–a)<n<R eb–1/2. Then the solution is y(t)=Γ(a) 2πΓ(b)tb–1⎝parenleftbiggd dt⎝parenrightbiggn⎝bracketleftbigg tn–b+1⎝integraldisplay∞ –∞e–ixtΨ(n+a–b,n–b+2 ;ixt)f(x)dx⎝bracketrightbigg . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 530). 74.⎝integraldisplay ⎝integraldisplay∞ 0F⎝parenleftbig⎝parenleftbig1 2b±ix,b;–it⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(t)=tb–1 2πΓ2(b)⎝integraldisplay∞ –∞e∓πxΓ⎝parenleftbig1 2b+ix⎝parenrightbig Γ⎝parenleftbig1 2b–ix⎝parenrightbig F⎝parenleftbig1 2b∓ix,b;it⎝parenrightbig f(x)dx, where Re b>0 . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 531). 3.7. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 273 75.⎝integraldisplay ⎝integraldisplay∞ 0tixF⎝parenleftbig⎝parenleftbig1 2+ix,b+ix;iαt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(t)=tb–1 2π⎝parenleftbigg –d dt⎝parenrightbiggn⎝integraldisplay∞ –∞tn–b–ixe–iαtΓ⎝parenleftbig1 2+ix⎝parenrightbig Γ⎝parenleftbig b+ix⎝parenrightbigΨ⎝parenleftbig n–b+1 2,n–b+1–ix;iαt⎝parenrightbig f(x)dx, where Im α=0a n d0<R e b–1/2<n<R eb. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 531). 76.⎝integraldisplay ⎝integraldisplay∞ –∞F⎝parenleftbig⎝parenleftbig a,b;iβ(x–t)⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(t)=β2(a–1 ) (a–b+1 )s i n ( πb) 4π(b–1 ) (b–2 ) (b–3 )s i n ( πa)s i n [π(b–a)]⎝integraldisplay∞ –∞F⎝parenleftbig 2–a ,b–a–1 ;iβ(x–t)⎝parenrightbig f(x)dx, where 1 < Re a<3/2 and –1 < Re(b –a)<– 1/2. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 531). 77.⎝integraldisplay ⎝integraldisplay∞ –∞F⎝parenleftbig⎝parenleftbig1 2±ia,1 2;±i(x–t)2⎝parenrightbig⎝parenrightbig y(t)dt=f(x), a>0 . Solution: y(t)=eπa πcosh(πa )⎝integraldisplay∞ –∞F⎝parenleftbig1 2∓ia,1 2;∓i(x–t)2⎝parenrightbig f(x)dx. References: Vu Kim Tuan (1988), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 532). 78.⎝integraldisplay ⎝integraldisplay∞ –∞F⎝parenleftbig⎝parenleftbig1 2b±it,b;ix⎝parenrightbig⎝parenrightbig y(t)dt=f(x), Re b>0 . Solution: y(t)=e±πt 2πΓ2(b)Γ⎝parenleftbig1 2b+it⎝parenrightbig Γ⎝parenleftbig1 2b–it⎝parenrightbig⎝integraldisplay∞ 0xb–1F⎝parenleftbig1 2b∓it,b;–ix⎝parenrightbig f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 532). 79.⎝integraldisplay ⎝integraldisplay∞ –∞F⎝parenleftbig⎝parenleftbig1 2b±it,b;–ix⎝parenrightbig⎝parenrightbig y(t)dt=f(x), Re b>0 . Solution: y(t)=e∓πt 2πΓ2(b)Γ⎝parenleftbig1 2b+it⎝parenrightbig Γ⎝parenleftbig1 2b–it⎝parenrightbig⎝integraldisplay∞ 0xb–1F⎝parenleftbig1 2b∓it,b;ix⎝parenrightbig f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 532). 80.⎝integraldisplay ⎝integraldisplay∞ –∞x–itF⎝parenleftbig⎝parenleftbig1 2–it,b–it;iβx⎝parenrightbig⎝parenrightbig y(t)dt=f(x). Solution: y(t)=Γ⎝parenleftbig1 2(1 –it)⎝parenrightbig 2πΓ⎝parenleftbig b–1 2it⎝parenrightbig⎝integraldisplay∞ 0xn–b+ite–iβxΨ⎝parenleftbig n+1 2–b,n+1–b+it;iβx⎝parenrightbig⎝parenleftbiggd dx⎝parenrightbiggn⎝bracketleftbig xb–1f(x)⎝bracketrightbig dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 533). 274 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.7-14. Kernels Containing Tricomi Conﬂuent Hypergeometric Functions. 81.⎝integraldisplay ⎝integraldisplay∞ 0tixΨ(a+ix,2ix+1 ;t)y(t)dt=f(x). HereΨ(a,b;x) is the Tricomi conﬂuent hypergeometric function (see Supplement 11.9-1) andi2= –1. Solution: y(t)=e–t π2t⎝integraldisplay∞ 0xsinh(2 πx)Γ(a–ix)Γ(a+ix)tixΨ(a+ix,2ix+1 ;t)f(x)dx. References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 534). 82.⎝integraldisplay ⎝integraldisplay∞ 0xitΨ(a+it,2ix+1 ;t)y(t)dt=f(x). Solution: y(t)=t π2sinh(2πt )Γ(a–it)Γ(a+it)⎝integraldisplay∞ 0x–1+ite–xΨ(a+it,2it+1 ;x)f(x)dx. References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 535). 83.⎝integraldisplay ⎝integraldisplay∞ –∞Ψ⎝parenleftbig⎝parenleftbig1 2+ix,3 2–iβ+ix;±it⎝parenrightbig⎝parenrightbig y(t)dt=f(x), Im β=0 . Solution: y(t)=1 4π⎝integraldisplay∞ –∞1 cosh(πx )Ψ⎝parenleftbig1 2–ix,3 2+iβ–ix;∓it⎝parenrightbig f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 536). 3.7-15. Kernels Containing Whittaker Conﬂuent Hypergeometric Functions. 84.⎝integraldisplay ⎝integraldisplay∞ 0M±ix,ν(it)y(t)dt=f(x), Re ν>–1 2. HereMµ,ν(z) is the Whittaker conﬂuent hypergeometric function (see Supplement 11.9-3) andi2= –1. Solution: y(t)=1 2πΓ2(2ν+1 )t⎝integraldisplay∞ –∞e∓πxΓ⎝parenleftbig1 2+ν+ix⎝parenrightbig Γ⎝parenleftbig1 2+ν–ix⎝parenrightbig M±ix,ν(–it)f(x)dx. The integral equation and its solution form the Buchholz transform pair . References: H. Buchholz (1969), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 523). 85.⎝integraldisplay ⎝integraldisplay∞ 0M±ix,ν(–it)y(t)dt=f(x), Re ν>–1 2. Solution: y(t)=1 2πΓ2(2ν+1 )t⎝integraldisplay∞ –∞e±πxΓ⎝parenleftbig1 2+ν+ix⎝parenrightbig Γ⎝parenleftbig1 2+ν–ix⎝parenrightbig M∓ix,ν(it)f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 523–524). 3.7. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 275 86.⎝integraldisplay ⎝integraldisplay∞ –∞M±it,ν(ix)y(t)dt=f(x), Re ν>–1 2. Solution: y(t)=e∓πt 2πΓ2(2ν+1 )Γ⎝parenleftbig1 2+ν+it⎝parenrightbig Γ⎝parenleftbig1 2+ν–it⎝parenrightbig⎝integraldisplay∞ 0x–1M∓it,ν(–ix)f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 524). 87.⎝integraldisplay ⎝integraldisplay∞ –∞M±it,ν(–ix)y(t)dt=f(x), Re ν>–1 2. Solution: y(t)=e±πt 2πΓ2(2ν+1 )Γ⎝parenleftbig1 2+ν+it⎝parenrightbig Γ⎝parenleftbig1 2+ν–it⎝parenrightbig⎝integraldisplay∞ 0x–1M∓it,ν(ix)f(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 524–525). 88.⎝integraldisplay ⎝integraldisplay∞ –∞Γ⎝parenleftbig⎝parenleftbig1 2+ν+ix–it⎝parenrightbig⎝parenrightbig Γ⎝parenleftbig⎝parenleftbig1 2+ν–ix+it⎝parenrightbig⎝parenrightbig Mit–ix,ν(a)y(t)dt=f(x). Solution: y(t)=(2ν+1 )s i n ( 2 πν) 4π3⎝integraldisplay∞ –∞Γ⎝parenleftbig –1 2–ν+ix–it⎝parenrightbig Γ⎝parenleftbig –1 2–ν–ix+it⎝parenrightbig Mit–ix,–ν–1(a)f(x)dx. References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 526). 89.⎝integraldisplay ⎝integraldisplay∞ 0Wµ,ix(t)y(t)dt=f(x). HereWµ,ν(z) is the Whittaker conﬂuent hypergeometric function (see Supplement 11.9-3). Solution: y(t)=1 π2t2⎝integraldisplay∞ 0xsinh(2πx )Γ⎝parenleftbig1 2–µ–ix⎝parenrightbig Γ⎝parenleftbig1 2–µ+ix⎝parenrightbig Wµ,ix(t)f(x)dx. References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 527). 90.⎝integraldisplay ⎝integraldisplay∞ 0Wµ,it(x)y(t)dt=f(x). Solution: y(t)=t π2sinh(2 πt)Γ⎝parenleftbig1 2–µ–it⎝parenrightbig Γ⎝parenleftbig1 2–µ+it⎝parenrightbig⎝integraldisplay∞ 0x–2Wµ,it(x)f(x)dx. References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 527). 91.⎝integraldisplay ⎝integraldisplay∞ –∞e–ixt/ 2Wµ,ν(ixt)y(t)dt=f(x). Solution: y(t)=Γ⎝parenleftbig3 2–µ–ν⎝parenrightbig 2πΓ(1 +n–2ν)(it)–n/2–1 ×⎝integraldisplay∞ 0x(n–1)/2–νeixt/ 2Wµ+n/2–1,n/2–ν(ixt)⎝parenleftbiggd dx⎝parenrightbiggn⎝bracketleftbig xν–1/2f(x)⎝bracketrightbig dx, where Re µ<R eν+1/2<3/4. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 528). 276 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.7-16. Kernels Containing Gauss Hypergeometric Functions. 92.⎝integraldisplay ⎝integraldisplaya 0F⎝parenleftbigg ⎝parenleftbiggβ 2,β+1 2,µ;4x2t2 (x2+t2)2⎝parenrightbigg ⎝parenrightbiggy(t)dt (x2+t2)β=f(x). Here 0 < a≤∞,0< β<µ<β+1 ,a n d F(a,b,c;z) is the Gauss hypergeometric function (see Supplement 11.10-1). 1◦. Solution: y(x)=x2µ–2 Γ(1 +β–µ)d dx⎝integraldisplaya xtg(t)dt (t2–x2)µ–β, g(t)=2Γ(β)s i n [ (β–µ)π] πΓ(µ)t1–2βd dt⎝integraldisplayt 0s2µ–1f(s)ds (t2–s2)µ–β. 2◦.I fa=∞andf(x) is a differentiable function, then the solution can be represented in the form y(x)=Ad dt⎝integraldisplay∞ 0(xt)2µf/prime t(t) (x2+t2)2µ–βF⎝parenleftbigg µ–β 2,µ+1–β 2,µ+1 ;4x2t2 (x2+t2)2⎝parenrightbigg dt, where A=Γ(β)Γ(2µ–β)s i n [ (β–µ)π] πΓ(µ)Γ(1 +µ). Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975). 93.⎝integraldisplay ⎝integraldisplay∞ 0F(a+ix,a–ix,c;–t)y(t)dt=f(x), a,c>0 . Solution: y(t)=tc–1(1 +t)2a–c π2Γ2(c)⎝integraldisplay∞ 0xsinh(2 πx)|Γ(a+ix)Γ(c–a+ix)|2F(a+ix,a–ix,c;–t)f(x)dx. The integral equation and its solution form the Olevskii transform pair . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 538). 3.7-17. Kernels Containing Parabolic Cylinder Functions. 94.⎝integraldisplay ⎝integraldisplay∞ –∞D–ix–1/ 2(±e–πi/ 4t)y(t)dt=f(x), i2= –1. HereDν(z) is the parabolic cylinder function (see Supplement 11.12-1). Solution: y(x)=1 4π⎝integraldisplay∞ –∞e–πt/2 cosh(πt )Dit–1/2(±eπi/4x)f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 467). 95.⎝integraldisplay ⎝integraldisplay∞ –∞exp⎝bracketleftbigg ⎝bracketleftbigg ±i(x–t)2 4⎝bracketrightbigg ⎝bracketrightbigg⎝bracketleftbig⎝bracketleftbig D±iα(e∓πi/ 4(t–x)) –D±iα(e∓πi/ 4(x–t))⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). Solution: y(x)=eπα/ 2 8πcosh2(πα/2)⎝integraldisplay∞ –∞exp⎝bracketleftbigg ∓i(x–t)2 4⎝bracketrightbigg⎝bracketleftbig D∓iα(e±πi/4(t–x)) +D∓iα(e±πi/4(x–t))⎝bracketrightbig f(t)dt, where α>0 . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 466). 3.7. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 277 96.⎝integraldisplay ⎝integraldisplay∞ 0⎝braceleftbigg ⎝braceleftbigg exp⎝bracketleftbigg ⎝bracketleftbiggi(x+t)2 4⎝bracketrightbigg ⎝bracketrightbigg⎝bracketleftbig⎝bracketleftbig D2iα(e3πi/ 4(x+t)) –D2iα(e–πi/ 4(x+t))⎝bracketrightbig⎝bracketrightbig +e x p⎝bracketleftbigg ⎝bracketleftbiggi(x–t)2 4⎝bracketrightbigg ⎝bracketrightbigg⎝bracketleftbig⎝bracketleftbig D2iα(e3πi/ 4(x–t)) –D2iα(e–πi/ 4(x–t))⎝bracketrightbig⎝bracketrightbig⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x). Solution: y(x)=eπα 8πsinh2(πα)⎝integraldisplay∞ 0⎝braceleftbigg exp⎝bracketleftbigg –i(x+t)2 4⎝bracketrightbigg⎝bracketleftbig D–2iα(–eπi/4(x+t)) –D–2iα(eπi/4(x+t))⎝bracketrightbig +e x p⎝bracketleftbigg –i(t–x)2 4⎝bracketrightbigg⎝bracketleftbig D–2iα(–eπi/4(t–x)) –D–2iα(eπi/4(t–x))⎝bracketrightbig⎝bracerightbigg f(t)dt. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 465–466). 3.7-18. Kernels Containing Other Special Functions. 97.⎝integraldisplay ⎝integraldisplaya 0K⎝parenleftbigg ⎝parenleftbigg2√ xt x+t⎝parenrightbigg ⎝parenrightbiggy(t)dt x+t=f(x). Here K(z)=⎝integraldisplay1 0dt ⎝radicalbig (1 –t2)(1 –z2t2)is the complete elliptic integral of the ﬁrst kind (see Supplement 11.13-1). Solution: y(x)=–4 π2d dx⎝integraldisplaya xtF(t)dt √ t2–x2,F(t)=d dt⎝integraldisplayt 0sf(s)ds √ t2–s2. Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975). 98.⎝integraldisplay ⎝integraldisplay∞ 0⎝bracketleftbigg ⎝bracketleftbigg ζ⎝parenleftbigg ⎝parenleftbigg1 2+ix,it⎝parenrightbigg ⎝parenrightbigg –ζ⎝parenleftbigg ⎝parenleftbigg1 2+ix,1 2+it⎝parenrightbigg⎝parenrightbigg⎝bracketrightbigg⎝bracketrightbigg y(t)dt=f(x). Here ζ(z,v)=∞⎝summationdisplay k=01 (v+k)zis the generalized Riemann zeta function (Re z>1 ;v≠ 0, –1, –2, ...). Solution: y(t)=eπi/4 4π√ t⎝integraldisplay∞ –∞eπx/ 2 cosh(πx )⎝bracketleftbigg 1+⎝parenleftbigg 1+i 2t⎝parenrightbiggix–1/2⎝bracketrightbigg tixf(x)dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 454). 99.⎝integraldisplay ⎝integraldisplay∞ 0⎝braceleftbigg ⎝braceleftbigg t–ix–1/ 2sin(1 + 2ix)π 4+2–ix–3/ 2eπx⎝bracketleftbigg ⎝bracketleftbigg ζ⎝parenleftbigg ⎝parenleftbigg1 2+ix,1–it 2⎝parenrightbigg ⎝parenrightbigg –ζ⎝parenleftbigg ⎝parenleftbigg1 2+ix,1+it 2⎝parenrightbigg ⎝parenrightbigg –ζ⎝parenleftbigg ⎝parenleftbigg1 2+ix,–it 2⎝parenrightbigg ⎝parenrightbigg +ζ⎝parenleftbigg ⎝parenleftbigg1 2+ix,it 2⎝parenrightbigg⎝parenrightbigg⎝bracketrightbigg⎝bracketrightbigg⎝bracerightbigg⎝bracerightbigg y(t)dt=f(x). Hereζ(z,v) is the generalized Riemann zeta function (see Eq. 3.7.98). Solution: y(t)=1 π⎝integraldisplay∞ –∞⎝braceleftbigg tix–1/2sin(1 – 2ix)π 4+s in⎝bracketleftbigg⎝parenleftbigg1 2–ix⎝parenrightbigg arctan t⎝bracketrightbigg (t2+1)ix/2–1/4⎝bracerightbiggf(x) cosh(πx )dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 454). 278 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 100.⎝integraldisplay ⎝integraldisplay∞ 0⎝braceleftbigg ⎝braceleftbigg t–ix–1/ 2cos(1 + 2ix)π 4–i2–ix–3/ 2eπx⎝bracketleftbigg ⎝bracketleftbigg ζ⎝parenleftbigg ⎝parenleftbigg1 2+ix,1–it 2⎝parenrightbigg ⎝parenrightbigg +ζ⎝parenleftbigg ⎝parenleftbigg1 2+ix,1+it 2⎝parenrightbigg ⎝parenrightbigg –ζ⎝parenleftbigg ⎝parenleftbigg1 2+ix,–it 2⎝parenrightbigg ⎝parenrightbigg –ζ⎝parenleftbigg ⎝parenleftbigg1 2+ix,it 2⎝parenrightbigg⎝parenrightbigg⎝bracketrightbigg⎝bracketrightbigg⎝bracerightbigg⎝bracerightbigg y(t)dt=f(x). Hereζ(z,v) is the generalized Riemann zeta function (see Eq. 3.7.98). Solution: y(t)=1 π⎝integraldisplay∞ –∞⎝braceleftbigg tix–1/2cos( 1–2ix)π 4+cos⎝bracketleftbigg⎝parenleftbigg1 2–ix⎝parenrightbigg arctan t⎝bracketrightbigg (t2+1)ix/2–1/4⎝bracerightbiggf(x) cosh(πx )dx. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 455). 3.8. Equations Whose Kernels Contain Arbitrary Functions 3.8-1. Equations with Degenerate Kernel. 1.⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig g1(x)h1(t)+g2(x)h2(t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This integral equation has solutions only if its right-hand side is representable in the form f(x)=A1g1(x)+A2g2(x), A1= const, A2= const . (1) In this case, any function y=y(x) satisfying the normalization type conditions ⎝integraldisplayb ah1(t)y(t)dt=A1,⎝integraldisplayb ah2(t)y(t)dt=A2 (2) is a solution of the integral equation. Otherwise, the equation has no solutions. 2.⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=0gk(x)hk(t)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). This integral equation has solutions only if its right-hand side is representable in the form f(x)=n⎝summationdisplay k=0Akgk(x), (1) where the Akare some constants. In this case, any function y=y(x) satisfying the normal- ization type conditions ⎝integraldisplayb ahk(t)y(t)dt=Ak (k=1 ,...,n)( 2) is a solution of the integral equation. Otherwise, the equation has no solutions. 3.8. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 279 3.8-2. Equations Containing Modulus. 3.⎝integraldisplay ⎝integraldisplayb a|g(x)–g(t)|y(t)dt=f(x). Leta≤x≤banda≤t≤b; it is assumed in items 1◦and 2◦that 0 < g/prime x(x)<∞. 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayx a⎝bracketleftbig g(x)–g(t)⎝bracketrightbig y(t)dt+⎝integraldisplayb x⎝bracketleftbig g(t)–g(x)⎝bracketrightbig y(t)dt=f(x). (1) Differentiating (1) with respect to xyields g/prime x(x)⎝integraldisplayx ay(t)dt–g/prime x(x)⎝integraldisplayb xy(t)dt=f/prime x(x). (2) Divide both sides of (2) by g/prime x(x) and differentiate the resulting equation to obtain the solution y(x)=1 2d dx⎝bracketleftbiggf/prime x(x) g/primex(x)⎝bracketrightbigg .( 3) 2◦. Let us demonstrate that the right-hand side f(x) of the integral equation must satisfy certain relations. By setting x=aandx=b, in (1), we obtain two corollaries ⎝integraldisplayb a⎝bracketleftbig g(t)–g(a)⎝bracketrightbig y(t)dt=f(a),⎝integraldisplayb a⎝bracketleftbig g(b)–g(t)⎝bracketrightbig y(t)dt=f(b). (4) Substitute y(x) of (3) into (4). Integrating by parts yields the desired constraints for f(x): ⎝bracketleftbig g(b)–g(a)⎝bracketrightbigf/prime x(b) g/primex(b)=f(a)+f(b), ⎝bracketleftbig g(a)–g(b)⎝bracketrightbigf/prime x(a) g/primex(a)=f(a)+f(b).(5) Let us point out a useful property of these constraints: f/prime x(b)g/prime x(a)+f/prime x(a)g/prime x(b)=0 . Conditions (5) make it possible to ﬁnd the admissible general form of the right-hand side of the integral equation: f(x)=F(x)+Ax+B,( 6) where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive), and the coefﬁcients AandBare given by A=–g/prime x(a)F/prime x(b)+g/prime x(b)F/prime x(a) g/primex(a)+g/primex(b), B=–1 2A(a+b)–1 2⎝bracketleftbig F(a)+F(b)⎝bracketrightbig –g(b)–g(a) 2g/primex(a)⎝bracketleftbig A+F/prime x(a)⎝bracketrightbig . 3◦.I fg(x) is representable in the form g(x)=O(x–a)kwith 0 < k< 1 in the vicinity of the point x=a(in particular, the derivative g/prime xis unbounded as x→a), then the solution of the integral equation is given by formula (3) as well. In this case, the right-hand side of the integral equation must satisfy the conditions f(a)+f(b)=0 , f/prime x(b)=0 . ( 7 ) 280 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION As before, the right-hand side of the integral equation is given by (6), with A=–F/prime x(b), B=1 2⎝bracketleftbig (a+b)F/prime x(b)–F(a)–F(b)⎝bracketrightbig . 4◦.F o rg/prime x(a) = 0, the right-hand side of the inte gral equation must satisfy the conditions f/prime x(a)=0 ,⎝bracketleftbig g(b)–g(a)⎝bracketrightbig f/prime x(b)=⎝bracketleftbig f(a)+f(b)⎝bracketrightbig g/prime x(b). As before, the right-hand side of the integral equation is given by (6), with A=–F/prime x(a), B=1 2⎝bracketleftbig (a+b)F/prime x(a)–F(a)–F(b)⎝bracketrightbig +g(b)–g(a) 2g/primex(b)⎝bracketleftbig F/prime x(b)–F/prime x(a)⎝bracketrightbig . 4.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleg(x)–g(λt)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x), λ>0 . Assume that 0 ≤x≤a,0≤t≤a,a n d0< g/prime x(x)<∞. 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayx/λ 0⎝bracketleftbig g(x)–g(λt)⎝bracketrightbig y(t)dt+⎝integraldisplaya x/λ⎝bracketleftbig g(λt)–g(x)⎝bracketrightbig y(t)dt=f(x). (1) Differentiating (1) with respect to xyields g/prime x(x)⎝integraldisplayx/λ 0y(t)dt–g/prime x(x)⎝integraldisplaya x/λy(t)dt=f/prime x(x). (2) Let us divide both sides of (2) by g/prime x(x) and differentiate the resulting equation to obtain y(x/λ)=1 2λ⎝bracketleftbig f/prime x(x)/g/prime x(x)⎝bracketrightbig/prime x. Substituting xbyλxyields the solution y(x)=λ 2d dz⎝bracketleftbiggf/prime z(z) g/primez(z)⎝bracketrightbigg ,z=λx.( 3) 2◦. Let us demonstrate that the right-hand side f(x) of the integral equation must satisfy certain relations. By setting x= 0 in (1) and (2), we obtain two corollaries ⎝integraldisplaya 0⎝bracketleftbig g(λt)–g(0)⎝bracketrightbig y(t)dt=f(0), g/prime x(0)⎝integraldisplaya 0y(t)dt=–f/prime x(0). (4) Substitute y(x) of (3) into (4). Integrating by parts yields the desired constraints for f(x): f/prime x(0)g/prime x(λa)+f/prime x(λa)g/prime x(0) = 0, ⎝bracketleftbig g(λa)–g(0)⎝bracketrightbigf/prime x(λa) g/primex(λa)=f(0) +f(λa).(5) Conditions (5) make it possible to ﬁnd the admissible general form of the right-hand side of the integral equation: f(x)=F(x)+Ax+B,( 6) where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive), and the coefﬁcients AandBare given by A=–g/prime x(0)F/prime x(λa)+g/prime x(λa)F/prime x(0) g/primex(0) +g/primex(λa), B=–1 2Aaλ –1 2⎝bracketleftbig F(0) +F(λa)⎝bracketrightbig –g(λa)–g(0) 2g/primex(0)⎝bracketleftbig A+F/prime x(0)⎝bracketrightbig . 3.8. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 281 3◦.I fg(x) is representable in the form g(x)=O(x)kwith 0 < k< 1 in the vicinity of the pointx= 0 (in particular, the derivative g/prime xis unbounded as x→0), then the solution of the integral equation is given by formula (3) as well. In this case, the right-hand side of the integral equation must satisfy the conditions f(0) +f(λa)=0 , f/prime x(λa)=0 . ( 7 ) As before, the right-hand side of the integral equation is given by (6), with A=–F/prime x(λa), B=1 2⎝bracketleftbig aλF/prime x(λa)–F(0) –F(λa)⎝bracketrightbig . 5.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleg(x)–t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). Assume that 0 ≤x≤a,0≤t≤a;g(0) = 0, and 0 < g/prime x(x)<∞. 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayg(x) 0⎝bracketleftbig g(x)–t⎝bracketrightbig y(t)dt+⎝integraldisplaya g(x)⎝bracketleftbig t–g(x)⎝bracketrightbig y(t)dt=f(x). (1) Differentiating (1) with respect to xyields g/prime x(x)⎝integraldisplayg(x) 0y(t)dt–g/prime x(x)⎝integraldisplaya g(x)y(t)dt=f/prime x(x). (2) Let us divide both sides of (2) by g/prime x(x) and differentiate the resulting equation to obtain 2g/prime x(x)y⎝parenleftbig g(x)⎝parenrightbig =⎝bracketleftbig f/prime x(x)/g/prime x(x)⎝bracketrightbig/prime x. Hence, we ﬁnd the solution: y(x)=1 2g/primez(z)d dz⎝bracketleftbiggf/prime z(z) g/primez(z)⎝bracketrightbigg ,z=g–1(x), (3) where g–1is the inverse of g. 2◦. Let us demonstrate that the right-hand side f(x) of the integral equation must satisfy certain relations. By setting x= 0 in (1) and (2), we obtain two corollaries ⎝integraldisplaya 0ty(t)dt=f(0), g/prime x(0)⎝integraldisplaya 0y(t)dt=–f/prime x(0). (4) Substitute y(x) of (3) into (4). Integrating by parts yields the desired constraints for f(x): f/prime x(0)g/prime x(xa)+f/prime x(xa)g/prime x(0) = 0, xa=g–1(a); g(xa)f/prime x(xa) g/primex(xa)=f(0) +f(xa).(5) Conditions (5) make it possible to ﬁnd the admissible general form of the right-hand side of the integral equation in question: f(x)=F(x)+Ax+B,( 6) 282 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive), and the coefﬁcients AandBare given by A=–g/prime x(0)F/prime x(xa)+g/prime x(xa)F/prime x(0) g/primex(0) +g/primex(xa),xa=g–1(a), B=–1 2Axa–1 2⎝bracketleftbig F(0) +F(xa)⎝bracketrightbig –g(xa) 2g/primex(0)⎝bracketleftbig A+F/prime x(0)⎝bracketrightbig . 3◦.I fg(x) is representable in the vicinity of the point x= 0 in the form g(x)=O(x)kwith 0<k< 1 (i.e., the derivative g/prime xis unbounded as x→0), then the solution of the integral equation is given by formula (3) as well. In this case, the right-hand side of the integral equation must satisfy the conditions f(0) +f(xa)=0 , f/prime x(xa)=0 . ( 7 ) As before, the right-hand side of the integral equation is given by (6), with A=–F/prime x(xa), B=1 2⎝bracketleftbig xaF/prime x(xa)–F(0) –F(xa)⎝bracketrightbig . 6.⎝integraldisplay ⎝integraldisplaya 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglex–g(t)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=f(x). Assume that 0 ≤x≤a,0≤t≤a;g(0) = 0, and 0 < g/prime x(x)<∞. 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayg–1(x) 0⎝bracketleftbig x–g(t)⎝bracketrightbig y(t)dt+⎝integraldisplaya g–1(x)⎝bracketleftbig g(t)–x⎝bracketrightbig y(t)dt=f(x), (1) where g–1is the inverse of g. Differentiating (1) with respect to xyields ⎝integraldisplayg–1(x) 0y(t)dt–⎝integraldisplaya g–1(x)y(t)dt=f/prime x(x). (2) Differentiating the resulting equation yields 2 y⎝parenleftbig g–1(x)⎝parenrightbig =g/prime x(x)f/prime/prime xx(x). Hence, we obtain the solution y(x)=1 2g/prime z(z)f/prime/prime zz(z), z=g(x). (3) 2◦. Let us demonstrate that the right-hand side f(x) of the integral equation must satisfy certain relations. By setting x= 0 in (1) and (2), we obtain two corollaries ⎝integraldisplaya 0g(t)y(t)dt=f(0),⎝integraldisplaya 0y(t)dt=–f/prime x(0). (4) Substitute y(x) of (3) into (4). Integrating by parts yields the desired constraints for f(x): xaf/prime x(xa)=f(0) +f(xa), f/prime x(0) +f/prime x(xa)=0 , xa=g(a). (5) Conditions (5) make it possible to ﬁnd the admissible general form of the right-hand side of the integral equation: f(x)=F(x)+Ax+B, A=–1 2⎝bracketleftbig F/prime x(0) +F/prime x(xa)⎝bracketrightbig ,B=1 2⎝bracketleftbig xaF/prime x(0) –F(xa)–F(0)⎝bracketrightbig ,xa=g(a), where F(x) is an arbitrary bounded twice differentiable function (with bounded ﬁrst deriva- tive). 3.8. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 283 7.⎝integraldisplay ⎝integraldisplayb ay(t) |g(x)–g(t)|kdt=f(x), 0 < k<1 . Letg/prime x≠0. The transformation z=g(x),τ=g(t),w(τ)=1 g/prime t(t)y(t) leads to an equation of the form 3.1.31: ⎝integraldisplayB Aw(τ) |z–τ|kdτ=F(z), A=g(a),B=g(b), where F=F(z) is the function which is obtained from z=g(x)a n dF=f(x) by eliminating x. 8.⎝integraldisplay ⎝integraldisplay1 0y(t) |g(x)–h(t)|kdt=f(x), 0 < k<1 . Letg(0) = 0, g(1) = 1, g/prime x>0 ;h(0) = 0, h(1) = 1, and h/prime t>0 . The transformation z=g(x),τ=h(t),w(τ)=1 h/prime t(t)y(t) leads to an equation of the form 3.1.30: ⎝integraldisplay1 0w(τ) |z–τ|kdτ=F(z), where F=F(z) is the function which is obtained from z=g(x)a n dF=f(x) by eliminating x. 9.⎝integraldisplay ⎝integraldisplayb ay(t)l n|g(x)–g(t)|dt=f(x). Letg/prime x≠0. The transformation z=g(x),τ=g(t),w(τ)=1 g/prime t(t)y(t) leads to Carleman’s equation 3.4.2: ⎝integraldisplayB Aln|z–τ|w(τ)dτ=F(z), A=g(a),B=g(b), where F=F(z) is the function which is obtained from z=g(x)a n dF=f(x) by eliminating x. 10.⎝integraldisplay ⎝integraldisplay1 0y(t)l n|g(x)–h(t)|dt=f(x). Letg(0) = 0, g(1) = 1, g/prime x>0 ;h(0) = 0, h(1) = 1, and h/prime t>0 . The transformation z=g(x),τ=h(t),w(τ)=1 h/prime t(t)y(t) leads to an equation of the form 3.4.2: ⎝integraldisplay1 0ln|z–τ|w(τ)dτ=F(z), where F=F(z) is the function which is obtained from z=g(x)a n dF=f(x) by eliminating x. 284 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.8-3. Equations with Difference Kernel: K(x,t)=K(x–t). 11.⎝integraldisplay ⎝integraldisplay∞ –∞K(x–t)y(t)dt=Axn,n= 0 ,1 ,2 , ... 1◦. Solution with n=0 : y(x)=A B,B=⎝integraldisplay∞ –∞K(x)dx. 2◦. Solution with n=1 : y(x)=A Bx+AC B2,B=⎝integraldisplay∞ –∞K(x)dx,C=⎝integraldisplay∞ –∞xK(x). 3◦. Solution with n≥2: y(x)=⎝braceleftbiggdn dλn⎝bracketleftbiggAeλx B(λ)⎝bracketrightbigg⎝bracerightbigg λ=0,B(λ)=⎝integraldisplay∞ –∞K(x)e–λxdx. 12.⎝integraldisplay ⎝integraldisplay∞ –∞K(x–t)y(t)dt=Aeλx. Solution: y(x)=A Beλx,B=⎝integraldisplay∞ –∞K(x)e–λxdx. 13.⎝integraldisplay ⎝integraldisplay∞ –∞K(x–t)y(t)dt=Axneλx,n=1 , 2 , ... 1◦. Solution with n=1 : y(x)=A Bxeλx+AC B2eλx, B=⎝integraldisplay∞ –∞K(x)e–λxdx,C=⎝integraldisplay∞ –∞xK(x)e–λxdx. 2◦. Solution with n≥2: y(x)=dn dλn⎝bracketleftbiggAeλx B(λ)⎝bracketrightbigg ,B(λ)=⎝integraldisplay∞ –∞K(x)e–λxdx. 14.⎝integraldisplay ⎝integraldisplay∞ –∞K(x–t)y(t)dt=Acos(λx )+Bsin(λx). Solution: y(x)=AIc+BIs I2c+I2scos(λx)+BIc–AIs I2c+I2ssin(λx), Ic=⎝integraldisplay∞ –∞K(z)c o s (λz)dz,Is=⎝integraldisplay∞ –∞K(z)s i n (λz)dz. 3.8. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 285 15.⎝integraldisplay ⎝integraldisplay∞ –∞K(x–t)y(t)dt=f(x). The Fourier transform is used to solve this equation. 1◦. Solution: y(x)=1 2π⎝integraldisplay∞ –∞˜f(u) ˜K(u)eiuxdu, ˜f(u)=1 √ 2π⎝integraldisplay∞ –∞f(x)e–iuxdx,˜K(u)=1 √ 2π⎝integraldisplay∞ –∞K(x)e–iuxdx. The following statement is valid. Let f(x)∈L2(–∞,∞)a n d K(x)∈L1(–∞,∞). Then for a solution y(x)∈L2(–∞,∞) of the integral equation to exist, it is necessary and sufﬁcient that ˜f(u)/˜K(u)∈L2(–∞,∞). 2◦. Let the function P(s) deﬁned by the formula 1 P(s)=⎝integraldisplay∞ –∞e–stK(t)dt be a polynomial of degree nwith real roots of the form P(s)=⎝parenleftBig 1–s a1⎝parenrightBig⎝parenleftBig 1–s a2⎝parenrightBig ...⎝parenleftBig 1–s an⎝parenrightBig . Then the solution of the integral equation is given by y(x)=P(D)f(x),D=d dx. References: I. I. Hirschman and D. V . Widder (1955), V . A. Ditkin and A. P. Prudnikov (1965). 16.⎝integraldisplay ⎝integraldisplay∞ 0K(x–t)y(t)dt=f(x). The Wiener–Hopf equation of the ﬁrst kind. This equation is discussed in Subsection 12.8-1 in detail. 3.8-4. Other Equations of the Form⎝integraltextb aK(x,t)y(t)dt=F(x). 17.⎝integraldisplay ⎝integraldisplay∞ –∞K(ax–t)y(t)dt=Aeλx. Solution: y(x)=A Bexp⎝parenleftBigλ ax⎝parenrightBig ,B=⎝integraldisplay∞ –∞K(z)e x p⎝parenleftBig –λ az⎝parenrightBig dz. 18.⎝integraldisplay ⎝integraldisplay∞ –∞K(ax–t)y(t)dt=f(x). The substitution z=axleads to an equation of the form 3.8.15: ⎝integraldisplay∞ –∞K(z–t)y(t)dt=f(z/a). 286 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 19.⎝integraldisplay ⎝integraldisplay∞ –∞K(ax+t)y(t)dt=Aeλx. Solution: y(x)=A Bexp⎝parenleftBig –λ ax⎝parenrightBig ,B=⎝integraldisplay∞ –∞K(z)e x p⎝parenleftBig –λ az⎝parenrightBig dz. 20.⎝integraldisplay ⎝integraldisplay∞ –∞K(ax+t)y(t)dt=f(x). The transformation τ=–t,z=ax,y(t)=Y(τ) leads to an equation of the form 3.8.15: ⎝integraldisplay∞ –∞K(z–τ)Y(τ)dt=f(z/a). 21.⎝integraldisplay ⎝integraldisplay∞ –∞[eβtK(ax+t)+eµtM(ax –t)]y(t)dt=Aeλx. Solution: y(x)=AIk(q)epx–Im(p)eqx Ik(p)Ik(q)–Im(p)Im(q),p=–λ a–β,q=λ a–µ, where Ik(q)=⎝integraldisplay∞ –∞K(z)e(β+q)zdz,Im(q)=⎝integraldisplay∞ –∞M(z)e–(µ+q)zdz. 22.⎝integraldisplay ⎝integraldisplay∞ 0g(xt)y(t)dt=f(x). By setting x=ez,t=e–τ,y(t)=eτw(τ),g(ξ)=G(lnξ),f(ξ)=F(lnξ), we arrive at an integral equation with difference kernel of the form 3.8.15: ⎝integraldisplay∞ –∞G(z–τ)w(τ)dτ=F(z). 23.⎝integraldisplay ⎝integraldisplay∞ 0g⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig y(t)dt=f(x). By setting x=ez,t=eτ,y(t)=e–τw(τ),g(ξ)=G(lnξ),f(ξ)=F(lnξ), we arrive at an integral equation with difference kernel of the form 3.8.15: ⎝integraldisplay∞ –∞G(z–τ)w(τ)dτ=F(z). 24.⎝integraldisplay ⎝integraldisplay∞ 0g⎝parenleftbig⎝parenleftbig xβtλ⎝parenrightbig⎝parenrightbig y(t)dt=f(x), β>0 , λ>0 . By setting x=ez/β,t=e–τ/λ,y(t)=eτ/λw(τ),g(ξ)=G(lnξ),f(ξ)=1 λF(βlnξ), we arrive at an integral equation with difference kernel of the form 3.8.15: ⎝integraldisplay∞ –∞G(z–τ)w(τ)dτ=F(z). 3.8. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 287 25.⎝integraldisplay ⎝integraldisplay∞ 0g⎝parenleftbigg ⎝parenleftbiggxβ tλ⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), β>0 , λ>0 . By setting x=ez/β,t=eτ/λ,y(t)=e–τ/λw(τ),g(ξ)=G(lnξ),f(ξ)=1 λF(βlnξ), we arrive at an integral equation with difference kernel of the form 3.8.15: ⎝integraldisplay∞ –∞G(z–τ)w(τ)dτ=F(z). 26.⎝integraldisplay ⎝integraldisplaya 0⎝bracketleftbigg ⎝bracketleftbigg1 |x–t|k+ϕ(x)ψ(t)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), 0 < k<1 . The solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.1.30. 27.⎝integraldisplay ⎝integraldisplay∞ 0exp[–g(x)t2]y(t)dt=f(x). Assume that g(0) =∞,g(∞)=0 ,a n d g/prime x<0 . The substitution z=1 4g(x)leads to equation 3.2.21: 1 √ πz⎝integraldisplay∞ 0exp⎝parenleftbigg –t2 4z⎝parenrightbigg y(t)dt=F(z), where the function F(z) is determined by the relations F=2 √ πf(x)⎝radicalbig g(x)a n dz=1 4g(x) by means of eliminating x. 28.⎝integraldisplay ⎝integraldisplayb a⎝bracketleftBig⎝bracketleftBig ln|x–t|+ϕ(x)ψ(t)⎝bracketrightBig⎝bracketrightBig y(t)dt=f(x). The solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.4.2. See also Example 3 in Subsection 12.6-2. 29.⎝integraldisplay ⎝integraldisplay∞ 0[sin(xt )+ϕ(x)ψ(t)]y(t)dt=f(x). The solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.5.8. Solution: y(t)=yf(t)+Ayϕ(t), where yf(t)=2 π⎝integraldisplay∞ 0sin(xt)f(x)dx,yϕ(t)=2 π⎝integraldisplay∞ 0sin(xt)ϕ(x)dx,A=–⎝integraldisplay∞ 0ψ(t)yf(t)dt 1+⎝integraldisplay∞ 0ψ(t)yϕ(t)dt. 288 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 30.⎝integraldisplay ⎝integraldisplay∞ 0[cos(xt)+ϕ(x)ψ(t)]y(t)dt=f(x). The solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.5.1. Solution: y(t)=yf(t)+Ayϕ(t), where yf(t)=2 π⎝integraldisplay∞ 0cos(xt )f(x)dx,yϕ(t)=2 π⎝integraldisplay∞ 0cos(xt )ϕ(x)dx,A=–⎝integraldisplay∞ 0ψ(t)yf(t)dt 1+⎝integraldisplay∞ 0ψ(t)yϕ(t)dt. 31.⎝integraldisplay ⎝integraldisplay∞ 0ta–1cos⎝bracketleftbig⎝bracketleftbig ϕ(x)ta⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), a>0 . Transformation z=ϕ(x),τ=ta,Y(τ)=y(t),F(z)=af(x) leads to an equation of the form 3.5.1: ⎝integraldisplay ⎝integraldisplay∞ 0cos(zτ )Y(τ)dτ=F(z). 32.⎝integraldisplay ⎝integraldisplay∞ 0ta–1sin⎝bracketleftbig⎝bracketleftbig ϕ(x)ta⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x), a>0 . Transformation z=ϕ(x),τ=ta,Y(τ)=y(t),F(z)=af(x) leads to an equation of the form 3.5.8: ⎝integraldisplay ⎝integraldisplay∞ 0sin(zτ)Y(τ)dτ=F(z). 33.⎝integraldisplay ⎝integraldisplay∞ 0[tJν(xt)+ϕ(x)ψ(t)]y(t)dt=f(x), ν> –1. HereJν(z) is the Bessel function of the ﬁrst kind. The solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, withϕ(x) = 0, coincides with equation 3.7.17. Solution: y(t)=y f(t)+Ayϕ(t), where yf(t)=⎝integraldisplay∞ 0xJν(xt)f(x)dx,yϕ(t)=⎝integraldisplay∞ 0xJν(xt)ϕ(x)dx,A=–⎝integraldisplay∞ 0ψ(t)yf(t)dt 1+⎝integraldisplay∞ 0ψ(t)yϕ(t)dt. 3.8. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 289 3.8-5. Equations of the Form⎝integraltextb aK(x,t)y(···)dt=F(x). 34.⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Ax +B. Solution: y(x)=A I1x+B I0,I0=⎝integraldisplayb af(t)dt,I1=⎝integraldisplayb atf(t)dt. 35.⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Axβ. Solution: y(x)=A Bxβ,B=⎝integraldisplayb af(t)tβdt. 36.⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Alnx+B. Solution: y(x)=plnx+q, where p=A I0,q=B I0–AIl I2 0,I0=⎝integraldisplayb af(t)dt,Il=⎝integraldisplayb af(t)l ntd t. 37.⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Axβlnx. Solution: y(x)=pxβlnx+qxβ, where p=A I1,q=–AI2 I2 1,I1=⎝integraldisplayb af(t)tβdt,I2=⎝integraldisplayb af(t)tβlntd t. 38.⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Acos(ln x). Solution: y(x)=AIc I2c+I2scos(ln x)+AIs I2c+I2ssin(lnx), Ic=⎝integraldisplayb af(t)c o s ( l n t)dt,Is=⎝integraldisplayb af(t)s i n ( l n t)dt. 39.⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Asin(ln x). Solution: y(x)=–AIs I2c+I2scos(ln x)+AIc I2c+I2ssin(lnx), Ic=⎝integraldisplayb af(t)c o s ( l n t)dt,Is=⎝integraldisplayb af(t)s i n ( l n t)dt. 290 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 40.⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Axβcos(ln x)+Bxβsin(ln x). Solution: y(x)=pxβcos(ln x)+qxβsin(lnx), where p=AIc–BIs I2c+I2s,q=AIs+BIc I2c+I2s, Ic=⎝integraldisplayb af(t)tβcos(ln t)dt,Is=⎝integraldisplayb af(t)tβsin(lnt)dt. 41.⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=Ax +B. Solution: y(x)=px+q, where p=A I0,q=AI1 I2 0+B I0,I0=⎝integraldisplayb af(t)dt,I1=⎝integraldisplayb atf(t)dt. 42.⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=Aeλx. Solution: y(x)=A Beλx,B=⎝integraldisplayb af(t)e x p ( – λt)dt. 43.⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=Acos(λx ). Solution: y(x)=–AIs I2c+I2ssin(λx)+AIc I2c+I2scos(λx), Ic=⎝integraldisplayb af(t)c o s ( λt)dt,Is=⎝integraldisplayb af(t)s i n (λt)dt. 44.⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=Asin(λx). Solution: y(x)=AIc I2c+I2ssin(λx)+AIs I2c+I2scos(λx), Ic=⎝integraldisplayb af(t)c o s ( λt)dt,Is=⎝integraldisplayb af(t)s i n (λt)dt. 3.8. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 291 45.⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=eµx(Asinλx +Bcosλx). Solution: y(x)=eµx(psinλx+qcosλx), where p=AIc–BIs I2c+I2s,q=AIs+BIc I2c+I2s, Ic=⎝integraldisplayb af(t)e–µtcos(λt )dt,Is=⎝integraldisplayb af(t)e–µtsin(λt)dt. 46.⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=g(x). 1◦.F o rg(x)=n⎝summationtext k=1Akexp(λkx), the solution of the equation has the form y(x)=n⎝summationdisplay k=1Ak Bkexp(λkx), Bk=⎝integraldisplayb af(t)e x p ( – λkt)dt. 2◦. For a polynomial right-hand side, g(x)=n⎝summationtext k=0Akxk, the solution has the form y(x)=n⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermined coefﬁcients. 3◦.F o rg(x)=eλxn⎝summationtext k=0Akxk, the solution has the form y(x)=eλxn⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermined coefﬁcients. 4◦.F o rg(x)=n⎝summationtext k=1Akcos(λ kx), the solution has the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 5◦.F o rg(x)=n⎝summationtext k=1Aksin(λkx), the solution has the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 292 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 6◦.F o rg(x)=c o s ( λx)n⎝summationtext k=0Akxk, the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 7◦.F o rg(x)=s i n ( λx)n⎝summationtext k=0Akxk, the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 8◦.F o rg(x)=eµxn⎝summationtext k=1Akcos(λ kx), the solution has the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 9◦.F o rg(x)=eµxn⎝summationtext k=1Aksin(λkx), the solution has the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 10◦.F o rg(x)=c o s ( λx)n⎝summationtext k=1Akexp(µkx), the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Bkexp(µkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 11◦.F o rg(x)=s i n ( λx)n⎝summationtext k=1Akexp(µkx), the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Bkexp(µkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 47.⎝integraldisplay ⎝integraldisplayb af(t)y(x+βt)dt=Ax +B. Solution: y(x)=px+q, where p=A I0,q=B I0–AI1β I2 0,I0=⎝integraldisplayb af(t)dt,I1=⎝integraldisplayb atf(t)dt. 3.8. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 293 48.⎝integraldisplay ⎝integraldisplayb af(t)y(x+βt)dt=Aeλx. Solution: y(x)=A Beλx,B=⎝integraldisplayb af(t)e x p ( λβt)dt. 49.⎝integraldisplay ⎝integraldisplayb af(t)y(x+βt)dt=Asinλx +Bcosλx. Solution: y(x)=psinλx+qcosλx, where p=AIc+BIs I2c+I2s,q=BIc–AIs I2c+I2s, Ic=⎝integraldisplayb af(t)c o s ( λβt)dt,Is=⎝integraldisplayb af(t)s i n (λβt)dt. 50.⎝integraldisplay ⎝integraldisplay1 0y(ξ)dt=f(x), ξ=g(x)t. Assume that g(0) = 0, g(1) = 1, and g/prime x≥0. 1◦. The substitution z=g(x) leads to an equation of the form 3.1.42:⎝integraldisplay1 0y(zt)dt=F(z), where the function F(z) is obtained from z=g(x)a n dF=f(x) by eliminating x. 2◦. Solution y=y(z) in the parametric form: y(z)=g(x) g/primex(x)f/prime x(x)+f(x), z=g(x). 51.⎝integraldisplay ⎝integraldisplay1 0tλy(ξ)dt=f(x), ξ=g(x)t. Assume that g(0) = 0, g(1) = 1, and g/prime x≥0. 1◦. The substitution z=g(x) leads to an equation of the form 3.1.43:⎝integraldisplay1 0tλy(zt)dt=F(z), where the function F(z) is obtained from z=g(x)a n dF=f(x) by eliminating x. 2◦. Solution y=y(z) in the parametric form: y(z)=g(x) g/primex(x)f/prime x(x)+(λ+1 )f(x), z=g(x). 52.⎝integraldisplay ⎝integraldisplayb af(t)y(ξ)dt=Axβ,ξ=xϕ(t). Solution: y(x)=A Bxβ,B=⎝integraldisplayb af(t)⎝bracketleftbig ϕ(t)⎝bracketrightbigβdt.( 1) 294 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 53.⎝integraldisplay ⎝integraldisplayb af(t)y(ξ)dt=g(x), ξ=xϕ(t). 1◦.F o rg(x)=n⎝summationtext k=0Akxk, the solution of the equation has the form y(x)=n⎝summationdisplay k=0Ak Bkxk,Bk=⎝integraldisplayb af(t)⎝bracketleftbig ϕ(t)⎝bracketrightbigkdt. 2◦.F o rg(x)=n⎝summationtext k=0Akxλk, the solution has the form y(x)=n⎝summationdisplay k=0Ak Bkxλk,Bk=⎝integraldisplayb af(t)⎝bracketleftbig ϕ(t)⎝bracketrightbigλkdt. 3◦.F o rg(x)=l nxn⎝summationtext k=0Akxk, the solution has the form y(x)=l nxn⎝summationdisplay k=0Bkxk+n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 4◦.F o rg(x)=n⎝summationtext k=0Ak⎝parenleftbig lnx)k, the solution has the form y(x)=n⎝summationdisplay k=0Bk⎝parenleftbig lnx)k, where the constants Bkare found by the method of undetermined coefﬁcients. 5◦.F o rg(x)=n⎝summationtext k=1Akcos(λ klnx), the solution has the form y(x)=n⎝summationdisplay k=1Bkcos(λ klnx)+n⎝summationdisplay k=1Cksin(λklnx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 6◦.F o rg(x)=n⎝summationtext k=1Aksin(λklnx), the solution has the form y(x)=n⎝summationdisplay k=1Bkcos(λ klnx)+n⎝summationdisplay k=1Cksin(λklnx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 3.8. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 295 54.⎝integraldisplay ⎝integraldisplayb af(t)y(ξ)dt=g(x), ξ=x+ϕ(t). 1◦.F o rg(x)=n⎝summationtext k=1Akexp(λkx), the solution of the equation has the form y(x)=n⎝summationdisplay k=1Ak Bkexp(λkx), Bk=⎝integraldisplayb af(t)e x p⎝bracketleftbig λkϕ(t)⎝bracketrightbig dt. 2◦. For a polynomial right-hand side, g(x)=n⎝summationtext k=0Akxk, the solution has the form y(x)=n⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermined coefﬁcients. 3◦.F o rg(x)=eλxn⎝summationtext k=0Akxk, the solution has the form y(x)=eλxn⎝summationdisplay k=0Bkxk, where the constants Bkare found by the method of undetermined coefﬁcients. 4◦.F o rg(x)=n⎝summationtext k=1Akcos(λ kx) the solution has the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 5◦.F o rg(x)=n⎝summationtext k=1Aksin(λkx), the solution has the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 6◦.F o rg(x)=c o s ( λx)n⎝summationtext k=0Akxk, the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 7◦.F o rg(x)=s i n ( λx)n⎝summationtext k=0Akxk, the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkare found by the method of undetermined coefﬁcients. 296 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 8◦.F o rg(x)=eµxn⎝summationtext k=1Akcos(λ kx), the solution has the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 9◦.F o rg(x)=eµxn⎝summationtext k=1Aksin(λkx), the solution has the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 10◦.F o rg(x)=c o s ( λx)n⎝summationtext k=1Akexp(µkx), the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Bkexp(µkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 11◦.F o rg(x)=s i n ( λx)n⎝summationtext k=1Akexp(µkx), the solution has the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Bkexp(µkx), where the constants BkandCkare found by the method of undetermined coefﬁcients. 3.9. Dual Integral Equations of the First Kind 3.9-1. Kernels Containing Trigonometric Functions. 1.⎝integraldisplay ⎝integraldisplay∞ 0cos(xt )y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0sin(xt)y(t)dt=0 f o r 1<x <∞. Solution: y(x)=2x π⎝integraldisplay1 0tJ0(xt)⎝bracketleftbigg⎝integraldisplayt 0f(s)ds √ t2–s2⎝bracketrightbigg dt. References: C. Nasim and B. D. Aggarwala (1984), B. N. Mandal and N. Mandal (1999, pp. 134–136). 2.⎝integraldisplay ⎝integraldisplay∞ 0cos(xt )y(t)dt=0 f o r 0<x <1 , ⎝integraldisplay ⎝integraldisplay∞ 0sin(xt)y(t)dt=f(x)f o r1 < x<∞. Solution: y(x)=2x π⎝integraldisplay∞ 1tJ0(xt)⎝bracketleftbigg⎝integraldisplay∞ tf(s)ds √ s2–t2⎝bracketrightbigg dt. References: C. Nasim and B. D. Aggarwala (1984), B. N. Mandal and N. Mandal (1999, pp. 136–137). 3.9. D UALINTEGRAL EQUATIONS OF THE FIRST KIND 297 3.⎝integraldisplay ⎝integraldisplay∞ 0cos(xt )y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0tcos(xt )y(t)dt=0 f o r 1< x<∞. Solution: y(x)=2x π⎝integraldisplay1 0tJ0(xt)⎝bracketleftbigg⎝integraldisplayt 0f(s)ds √ t2–s2⎝bracketrightbigg dt–2 πJ1(x)⎝integraldisplay1 0f(s)ds √ 1–s2. References: I. W. Busbridge (1938), B. N. Mandal and N. Mandal (1999, pp. 138–139). 4.⎝integraldisplay ⎝integraldisplay∞ 0tcos(xt )y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0cos(xt )y(t)dt=0 f o r 1<x <∞. Solution: y(x)=2 π⎝integraldisplay1 0tJ0(xt)⎝bracketleftbigg⎝integraldisplayt 0f(s)ds √ t2–s2⎝bracketrightbigg dt. References: I. W. (1937, p. 339), B. N. Mandal and N. Mandal (1999, pp. 139–140). 5.⎝integraldisplay ⎝integraldisplay∞ 0sin(xt)y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0tsin(xt)y(t)dt=0 f o r 1< x<∞. It is assumed that f(0) = 0. Solution: y(x)=2 π⎝integraldisplay1 0tJ0(xt)⎝bracketleftbigg⎝integraldisplayt 0f/prime s(s)ds √ t2–s2⎝bracketrightbigg dt. References: I. W. Busbridge (1938), B. N. Mandal and N. Mandal (1999, pp. 140–141). 6.⎝integraldisplay ⎝integraldisplay∞ 0tsin(xt)y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0sin(xt)y(t)dt=0 f o r 1<x <∞. Solution: y(x)=2 π⎝integraldisplay1 0J1(xt)⎝bracketleftbigg⎝integraldisplayt 0sf(s)ds √ t2–s2⎝bracketrightbigg dt. References: B. Noble (1963), B. N. Mandal and N. Mandal (1999, pp. 141–142). 7.⎝integraldisplay ⎝integraldisplay∞ 0[asin(xt)+tcos(xt )]y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0t[asin(xt)+tcos(xt )]y(t)dt=0 f o r 1< x<∞. Solution: y(x)=⎝integraldisplay1 0tJ0(xt)F(t)dt+F(1) K0(a)⎝integraldisplay∞ 1tJ0(xt)⎝bracketleftbigg⎝integraldisplay∞ te–asds √ s2–t2⎝bracketrightbigg dt, 298 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION where F(t)=2 πd dt⎝integraldisplayt 0zϕ(z)dz √ t2–z2=2 π⎝integraldisplayt 0ϕ/prime z(z)dz √ t2–z2,ϕ(z)=e–az⎝integraldisplayz 0easf(s)ds(0 <z<1 ) , andK0(x) is the modiﬁed Bessel functions of the second kind. References: B. D. Aggarwala and C. Nasim (1996), B. N. Mandal and N. Mandal (1999, pp. 143–145). 8.⎝integraldisplay ⎝integraldisplay∞ 0t[asin(xt)+tcos(xt )]y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0[asin(xt)+tcos(xt )]y(t)dt=g(x)f o r 1 < x<∞. Solution for a≠0: y(x)=⎝integraldisplay1 0J1(xt)F(t)dt–⎝integraldisplay∞ 1tJ1(xt)G(t)dt+2Da π⎝integraldisplay∞ tJ1(xt)K1(at)dt,D=F(1) +G(1) aK 1(a), where F(t)=2 π⎝integraldisplayt 0zϕ(z)dz √ t2–z2, ϕ(z)=e–az⎝integraldisplayz 0easf(s)ds(0 <z<1 ) , G(t)=2 πd dt⎝integraldisplay∞ tψ(z)dz √ z2–t2,ψ(z)=e–az⎝integraldisplayz 1easg(s)ds(1 <z<∞), andK0(x) is the modiﬁed Bessel functions of the second kind. References: B. D. Aggarwala and C. Nasim (1996), B. N. Mandal and N. Mandal (1999, pp. 143, 147–148). 3.9-2. Kernels Containing Bessel Functions of the First Kind. 9.⎝integraldisplay ⎝integraldisplay∞ 0J0(xt)y(t)dt=f(x)f o r0 < x<a, ⎝integraldisplay ⎝integraldisplay∞ 0tJ0(xt)y(t)dt=0 f o r a<x<∞. Solution: y(x)=2 π⎝integraldisplaya 0cos(xt )⎝bracketleftbiggd dt⎝integraldisplayt 0sf(s)ds √ t2–s2⎝bracketrightbigg dt. 10.⎝integraldisplay ⎝integraldisplay∞ 0tJ0(xt)y(t)dt=f(x)f o r0 < x<a, ⎝integraldisplay ⎝integraldisplay∞ 0J0(xt)y(t)dt=0 f o r a<x<∞. Solution: y(x)=2 π⎝integraldisplaya 0sin(xt)⎝bracketleftbiggd dt⎝integraldisplayt 0sf(s)ds √ t2–s2⎝bracketrightbigg dt. 11.⎝integraldisplay ⎝integraldisplay∞ 0tJµ(xt)y(t)dt=f(x)f o r0 < x<a, ⎝integraldisplay ⎝integraldisplay∞ 0Jµ(xt)y(t)dt=0 f o r a<x<∞. Solution: y(x)=⎝radicalbigg 2x π⎝integraldisplaya 0t3/2Jµ+1 2(xt)⎝bracketleftbigg⎝integraldisplayπ/2 0sinµ+1θf(tsinθ)dθ⎝bracketrightbigg dt. 3.9. D UALINTEGRAL EQUATIONS OF THE FIRST KIND 299 12.⎝integraldisplay ⎝integraldisplay∞ 0Jµ(xt)y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0t2Jµ(xt)y(t)dt=0 f o r 1< x<∞. Solution: y(x)=f(1)Jµ–1(x)+x⎝integraldisplay1 0tJµ(xt)f(t)dt. Reference: B. N. Mandal and N. Mandal (1999, p. 31). 13.⎝integraldisplay ⎝integraldisplay∞ 0t2βJµ(xt)y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0Jµ(xt)y(t)dt=0 f o r 1< x<∞. 1◦. Solution for β>0 : y(x)=(2x)1–β Γ(β)⎝integraldisplay1 0t1+βJµ+β(xt)F(t)dt,F(t)=⎝integraldisplay1 0f(tζ)ζµ+1(1 –ζ2)β–1dζ.( 1 ) 2◦. Solution for β> –1: y(x)=(2x)–β Γ(1 +β)⎝bracketleftbigg x1+βJµ+β(x)⎝integraldisplay1 0tµ+1(1 –t2)βf(t)dt+⎝integraldisplay1 0tµ+1(1 –t2)βΦ(x,t)dt⎝bracketrightbigg ,( 2 ) Φ(x,t)=⎝integraldisplay1 0(xξ)2+βJµ+β+1(xξ)f(ξt)dξ. Formula (2) holds for β>– 1a n df o r– µ–1 2<2β<µ+3 2. It can be shown that for β>0t h e solution of Eq. (2) can be reduced to the form (1). 14.⎝integraldisplay ⎝integraldisplay∞ 0t–2αJµ(xt)y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0t–2βJµ(xt)y(t)dt=g(x)f o r 1 < x<∞. Solution for 0 < β–α<1 : y(x)=21+α–βx1+α+β Γ(β–α)⎝integraldisplay1 0t1+α–β–µJµ+β–α(xt)⎝bracketleftbigg⎝integraldisplayt 0s1+µ(t2–s2)β–α–1f(s)ds⎝bracketrightbigg dt –2β–α Γ(1 +α–β)x1+α+β⎝integraldisplay∞ 1tµ+β–αJµ+β–α(xt)⎝bracketleftbiggd dt⎝integraldisplay∞ ts1–µ(s2–t2)α–βg(s)ds⎝bracketrightbigg dt. References: C. Nasim and B. D. Aggarwala (1984), B. N. Mandal and N. Mandal (1999, pp. 40–44). 15.⎝integraldisplay ⎝integraldisplay∞ 0J0(xt)y(t)dt=f(x)f o r 0 < x<a, ⎝integraldisplay ⎝integraldisplay∞ 0cos(xt )y(t)dt=g(x)f o r a<x<∞. Solution: y(x)=2 π⎝integraldisplaya 0cos(xt )⎝bracketleftbiggd dt⎝integraldisplayt 0sf(s)ds √ t2–s2⎝bracketrightbigg dt+2 π⎝integraldisplay∞ acos(xt )g(t)dt. Reference: B. N. Mandal and N. Mandal (1999, pp. 194–195). 300 LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 16.⎝integraldisplay ⎝integraldisplay∞ 0tJ0(xt)y(t)dt=f(x)f o r0 < x<a, ⎝integraldisplay ⎝integraldisplay∞ 0sin(xt)y(t)dt=g(x)f o r a<x<∞. Solution: y(x)=2 π⎝integraldisplaya 0sin(xt)⎝bracketleftbigg⎝integraldisplayt 0sf(s)ds √ t2–s2⎝bracketrightbigg dt+2 π⎝integraldisplay∞ asin(xt)g(t)dt. Reference: B. N. Mandal and N. Mandal (1999, pp. 195–196). 3.9-3. Kernels Containing Bessel Functions of the Second Kind. 17.⎝integraldisplay ⎝integraldisplay∞ 0t–2αYµ(xt)y(t)dt=f(x)f o r0 < x<1 , ⎝integraldisplay ⎝integraldisplay∞ 0t–2βYν(xt)y(t)dt=g(x)f o r 1 < x<∞. Let 2( α–β)=ν–µ>0 , |µ|<1 2,|ν|<1 2. 1◦. Solution for 0 < ν–µ<1 : y(x)=–2ν–µ Γ(1 +µ–ν)x2β+1⎝integraldisplay1 0tνHν(xt)⎝bracketleftbiggd dt⎝integraldisplay1 ts1–µ(s2–t2)µ–νf(s)ds⎝bracketrightbigg dt +2µ–ν Γ(µ–ν)⎝integraldisplay∞ 1t1+µHµ(xt)⎝bracketleftbigg⎝integraldisplay∞ ts1–ν(s2–t2)ν–µ–1g(s)ds⎝bracketrightbigg dt, where Hµ(x) is the Struve function, which is deﬁned as Hµ(x)=∞⎝summationdisplay j=0(–1)j(x/2)µ+2j+1 Γ⎝parenleftbig j+3 2⎝parenrightbig Γ⎝parenleftbig µ+j+3 2⎝parenrightbig. 2◦. Solution for –1 < ν–µ<0 : y(x)=21–ν–µ Γ(µ–ν)x2β+1⎝integraldisplay1 0tν+1Hν(xt)⎝bracketleftbigg⎝integraldisplay1 ts1–µ(s2–t2)µ–ν–1f(s)ds⎝bracketrightbigg dt +2µ–ν Γ(1 –µ+ν)x2α+1⎝integraldisplay∞ 1tµHµ(xt)⎝bracketleftbigg⎝integraldisplay∞ ts1–ν(s2–t2)ν–µg(s)ds⎝bracketrightbigg dt. References: C. Nasim and B. D. Aggarwala (1984), B. N. Mandal and N. Mandal (1999, pp. 58–59). 3.9-4. Kernels Containing Legendre Spherical Functions of the First Kind, i2= –1. 18.⎝integraldisplay ⎝integraldisplay∞ 0tP–1 2+it(coshx)y(t)dt=f(x)f o r 0 < x<a, ⎝integraldisplay ⎝integraldisplay∞ 0tanh(πt)P–1 2+it(coshx)y(t)dt=0 f o r a<x<∞. Solution: y(x)=√ 2 π⎝integraldisplaya 0sin(xt)⎝bracketleftbigg⎝integraldisplayt 0f(s)s i n hs √ cosht–c o s h sds⎝bracketrightbigg dt. Note that P–1 2+it(coshx)=√ 2 π⎝integraldisplayx 0cos(ts ) √ coshx–c o s h sds,x>0 , where the integral on the right-hand side is called the Meler integral . Chapter 4 Linear Equations of the Second Kind with Constant Limits of Integration /trianglerightsld Notation: f=f(x),g=g(x),h=h(x),v=v(x),w=w(x),K=K(x)are arbitrary functions; A,B,C,D,E,a,b,c,l,α,β,γ,δ,µ, andνare arbitrary parameters; nis a nonnegative integer; andiis the imaginary unit. /trianglerightsldPreliminary remarks. A number λis called a characteristic value of the integral equation y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt=f(x) if there exist nontrivial solutions of the corresponding homogeneous equation (with f(x)≡0). The nontrivial solutions themselves are called the eigenfunctions of the integral equation correspondingto the characteristic value λ.I fλis a characteristic value, the number 1 /λis called an eigenvalue of the integral equation. A value of the parameter λis said to be regular if for this value the homogeneous equation has only the trivial solution. Sometimes the characteristic values and the eigenfunctions of a Fredholm integral equation are called the characteristic values and the eigenfunctions of the kernel K(x,t). In the above equation, it is usually assumed that a≤x≤b. 4.1. Equations Whose Kernels Contain Power-Law Functions 4.1-1. Kernels Linear in the Arguments xandt. 1. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)y(t)dt=f(x). Solution: y(x)=f(x)+λ(A1x+A2), where A1=12f 1+6λ(f1∆2–2f2∆1) λ2∆4 1+1 2,A2=–12f 2+2λ(3f2∆2–2f1∆3) λ2∆4 1+1 2, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb axf(x)dx,∆n=bn–an. 301 302 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 2. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x+t)y(t)dt=f(x). The characteristic values of the equation: λ1=6(b+a)+4⎝radicalbig 3(a2+ab+b2) (a–b)3,λ2=6(b+a)–4⎝radicalbig 3(a2+ab+b2) (a–b)3. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1x+A2), where A1=12f 1–6λ(f1∆2–2f2∆1) 12 – 12λ ∆2–λ2∆4 1,A2=12f 2–2λ(3f2∆2–2f1∆3) 1 2–1 2 λ∆2–λ2∆4 1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb axf(x)dx,∆n=bn–an. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1: y1(x)=x+1 λ1(b–a)–b+a 2. 3◦. Solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. The equation has no multiple characteristic values. 3. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax +Bt)y(t)dt=f(x). The characteristic values of the equation: λ1,2=3(A+B)(b+a)±⎝radicalbig 9(A–B)2(b+a)2+4 8AB(a2+ab+b2) AB(a–b)3. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1x+A2), where the constants A1andA2are given by A1=12Af 1–6ABλ (f1∆2–2f2∆1) 1 2–6 ( A+B)λ∆2–ABλ2∆4 1,A2=12Bf 2–2ABλ (3f2∆2–2f1∆3) 1 2–6 ( A+B)λ∆2–ABλ2∆4 1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb axf(x)dx,∆n=bn–an. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1: y1(x)=x+1 λ1A(b–a)–b+a 2. 3◦. Solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 303 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A≠±B, where the characteristic valueλ∗=4 (A+B)(b2–a2)is double: y(x)=f(x)+Cy∗(x), where Cis an arbitrary constant and y∗(x) is an eigenfunction of the equation corresponding toλ∗: y∗(x)=x–(A–B)(b+a) 4A. The equation has no multiple characteristic values if A=±B. 4. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=1 . Solution: y(x)=f(x)+λ(A1+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.8. 5. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax +Bt +C)y(t)dt=f(x). This is a special case of equation 4.9.7 with g(x)=xandh(t)=1 . Solution: y(x)=f(x)+λ(A1x+A2), where A1andA2are the constants determined by the formulas presented in 4.9.7. 6. y(x)+A⎝integraldisplay ⎝integraldisplayb a|x–t|y(t)dt=f(x). This is a special case of equation 4.9.36 with g(t)=A. 1◦. The function y=y(x) obeys the following second-order linear nonhomogeneousordinary differential equation with constant coefﬁcients: y/prime/prime xx+2Ay=f/prime/prime xx(x). (1) The boundary conditions for (1) have the form (see 4.9.36) y/prime x(a)+y/prime x(b)=f/prime x(a)+f/prime x(b), y(a)+y(b)+(b–a)y/prime x(a)=f(a)+f(b)+(b–a)f/prime x(a).(2) Equation (1) under the boundary conditions (2) determines the solution of the original integral equation. 2◦.F o rA< 0, the general solution of equation (1) is given by y(x)=C1cosh(kx )+C2sinh(kx)+f(x)+k⎝integraldisplayx asinh[k(x–t)]f(t)dt,k=√ –2A,( 3 ) where C1andC2are arbitrary constants. ForA> 0, the general solution of equation (1) is given by y(x)=C1cos(kx )+C2sin(kx)+f(x)–k⎝integraldisplayx asin[k(x–t)]f(t)dt,k=√ 2A.( 4 ) The constants C1andC2in solutions (3) and (4) are determined by conditions (2). 304 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 3◦. In the special case a=0a n d A> 0, the solution of the integral equation is given by formula (4) with C1=kIs(1 + cos λ)–Ic(λ+s i nλ) 2+2c o s λ+λsinλ,C2=kIssinλ+Ic(1 + cos λ) 2+2c o s λ+λsinλ, k=√ 2A,λ=bk,Is=⎝integraldisplayb 0sin[k(b–t)]f(t)dt,Ic=⎝integraldisplayb 0cos[k (b–t)]f(t)dt. 4.1-2. Kernels Quadratic in the Arguments xandt. 7. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x2+t2)y(t)dt=f(x). The characteristic values of the equation: λ1=1 1 3(b3–a3)+⎝radicalBig 1 5(b5–a5)(b–a),λ2=1 1 3(b3–a3)–⎝radicalBig 1 5(b5–a5)(b–a). 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1x2+A2), where the constants A1andA2are given by A1=f1–λ⎝parenleftbig1 3f1∆3–f2∆1⎝parenrightbig λ2⎝parenleftbig1 9∆2 3–1 5∆1∆5⎝parenrightbig –2 3λ∆3+1,A2=f2–λ⎝parenleftbig1 3f2∆3–1 5f1∆5⎝parenrightbig λ2⎝parenleftbig1 9∆2 3–1 5∆1∆5⎝parenrightbig –2 3λ∆3+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb ax2f(x)dx,∆n=bn–an. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=x2+⎝radicalBigg b5–a5 5(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. Solution with λ=λ2≠λ1andf1=f2=0 : y(x)=f(x)+Cy2(x), y2(x)=x2–⎝radicalBigg b5–a5 5(b–a), where Cis an arbitrary constant and y2(x) is an eigenfunction of the equation corresponding to the characteristic value λ2. 4◦. The equation has no multiple characteristic values. 4.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 305 8. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x2–t2)y(t)dt=f(x). The characteristic values of the equation: λ1,2=±1 ⎝radicalBig 1 9(b3–a3)2–1 5(b5–a5)(b–a). 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1x2+A2), where the constants A1andA2are given by A1=f1+λ⎝parenleftbig1 3f1∆3–f2∆1⎝parenrightbig λ2⎝parenleftbig1 5∆1∆5–1 9∆2 2⎝parenrightbig +1,A2=–f2+λ⎝parenleftbig1 3f2∆3–1 5f1∆5⎝parenrightbig λ2⎝parenleftbig1 5∆1∆5–1 9∆2 2⎝parenrightbig +1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb ax2f(x)dx,∆n=bn–an. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=x2+3–λ 1(b3–a3) 3λ1(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. The equation has no multiple characteristic values. 9. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax2+Bt2)y(t)dt=f(x). The characteristic values of the equation: λ1,2=1 3(A+B)∆3±⎝radicalBig 1 9(A–B)2∆2 3+4 5AB∆1∆5 2AB⎝parenleftbig1 9∆2 3–1 5∆1∆5⎝parenrightbig ,∆n=bn–an. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1x2+A2), where the constants A1andA2are given by A1=Af1–ABλ⎝parenleftbig1 3f1∆3–f2∆1⎝parenrightbig ABλ2⎝parenleftbig1 9∆2 3–1 5∆1∆5⎝parenrightbig –1 3(A+B)λ∆3+1, A2=Bf2–ABλ⎝parenleftbig1 3f2∆3–1 5f1∆5⎝parenrightbig ABλ2⎝parenleftbig1 9∆2 3–1 5∆1∆5⎝parenrightbig –1 3(A+B)λ∆3+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb ax2f(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=x2+3–λ 1A(b3–a3) 3λ1A(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 306 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A≠±B,w h e r e λ∗= 6 (A+B)(b3–a3)is the double characteristic value: y(x)=f(x)+C1y∗(x), where C1is an arbitrary constant and y∗(x) is an eigenfunction of the equation corresponding toλ∗: y∗(x)=x2–(A–B)(b3–a3) 6A(b–a). The equation has no multiple characteristic values if A=±B. 10. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(xt–t2)y(t)dt=f(x). This is a special case of equation 4.9.8 with A=0 ,B=1 ,a n d h(t)=t. Solution: y(x)=f(x)+λ(A1+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.8. 11. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x2–xt)y(t)dt=f(x). This is a special case of equation 4.9.10 with A=0 ,B=1 ,a n d h(x)=x. Solution: y(x)=f(x)+λ(E1x2+E2x), where E1andE2are the constants determined by the formulas presented in 4.9.10. 12. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Bxt +Ct2)y(t)dt=f(x). This is a special case of equation 4.9.9 with A=0a n d h(t)=t. Solution: y(x)=f(x)+λ(A1+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.9. 13. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Bx2+Cxt )y(t)dt=f(x). This is a special case of equation 4.9.11 with A=0a n d h(x)=x. Solution: y(x)=f(x)+λ(A1x2+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.11. 14. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Axt +Bx2+Cx +D)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=Bx2+Cx+D,h1(t)=1 , g2(x)=x, andh2(t)=At. Solution: y(x)=f(x)+λ[A1(Bx2+Cx+D)+A2x], where A1andA2are the constants determined by the formulas presented in 4.9.18. 4.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 307 15. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax2+Bt2+Cx +Dt +E)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=Ax2+Cx,h1(t)=1 , g2(x)=1 ,a n d h2(t)=Bt2+Dt+E. Solution: y(x)=f(x)+λ[A1(Ax2+Cx)+A2], where A1andA2are the constants determined by the formulas presented in 4.9.18. 16. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Ax +B+(Cx +D)(x –t)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=Cx2+(A+D)x+B,h1(t)=1 , g2(x)=Cx+D,a n dh2(t)=–t. Solution: y(x)=f(x)+λ[A1(Cx2+Ax+Dx+B)+A2(Cx+D)], where A1andA2are the constants determined by the formulas presented in 4.9.18. 17. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[At+B+(Ct +D)(t–x)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=1 ,h1(t)=Ct2+(A+D)t+B,g2(x)=x, andh2(t)=– ( Ct+D). Solution: y(x)=f(x)+λ(A1+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.18. 18. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)2y(t)dt=f(x). This is a special case of equation 4.9.19 with g(x)=x,h(t)=–t,a n d m=2 . 19. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax +Bt)2y(t)dt=f(x). This is a special case of equation 4.9.19 with g(x)=Ax,h(t)=Bt,a n dm=2 . 4.1-3. Kernels Cubic in the Arguments xandt. 20. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x3+t3)y(t)dt=f(x). The characteristic values of the equation: λ1=1 1 4(b4–a4)+⎝radicalBig 1 7(b7–a7)(b–a),λ2=1 1 4(b4–a4)–⎝radicalBig 1 7(b7–a7)(b–a). 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1x3+A2), 308 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION where the constants A1andA2are given by A1=f1–λ⎝parenleftbig1 4f1∆4–f2∆1⎝parenrightbig λ2⎝parenleftbig1 16∆2 4–1 7∆1∆7⎝parenrightbig –1 2λ∆4+1,A2=f2–λ⎝parenleftbig1 4f2∆4–1 7f1∆7⎝parenrightbig λ2⎝parenleftbig1 16∆2 4–1 7∆1∆7⎝parenrightbig –1 2λ∆4+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb ax3f(x)dx,∆n=bn–an. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=x3+⎝radicalBigg b7–a7 7(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. Solution with λ=λ2≠λ1andf1=f2=0 : y(x)=f(x)+Cy2(x), y2(x)=x3–⎝radicalBigg b7–a7 7(b–a), where Cis an arbitrary constant and y2(x) is an eigenfunction of the equation corresponding to the characteristic value λ2. 4◦. The equation has no multiple characteristic values. 21. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x3–t3)y(t)dt=f(x). The characteristic values of the equation: λ1,2=±1 ⎝radicalBig 1 4(a4–b4)2–1 7(a7–b7)(b–a). 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1x3+A2), where the constants A1andA2are given by A1=f1+λ⎝parenleftbig1 4f1∆4–f2∆1⎝parenrightbig λ2⎝parenleftbig1 7∆1∆7–1 16∆2 4⎝parenrightbig +1,A2=–f2+λ⎝parenleftbig1 4f2∆4–1 7f1∆7⎝parenrightbig λ2⎝parenleftbig1 7∆1∆7–1 16∆2 4⎝parenrightbig +1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb ax3f(x)dx,∆n=bn–an. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=x3+4–λ 1(b4–a4) 4λ1(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. The equation has no multiple characteristic values. 4.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 309 22. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax3+Bt3)y(t)dt=f(x). The characteristic values of the equation: λ1,2=1 4(A+B)∆4±⎝radicalBig 1 16(A–B)2∆2 4+4 7AB∆1∆7 2AB⎝parenleftbig1 16∆2 4–1 7∆1∆7⎝parenrightbig ,∆n=bn–an. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1x3+A2), where the constants A1andA2are given by A1=Af1–ABλ⎝parenleftbig1 4f1∆4–f2∆1⎝parenrightbig ABλ2⎝parenleftbig1 16∆2 4–1 7∆1∆7⎝parenrightbig –1 4λ(A+B)∆4+1, A2=Bf2–ABλ⎝parenleftbig1 4f2∆4–1 7f1∆7⎝parenrightbig ABλ2⎝parenleftbig1 16∆2 4–1 7∆1∆7⎝parenrightbig –1 4λ(A+B)∆4+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb ax3f(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=x3+4–λ 1A(b4–a4) 4λ1A(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A≠±B,w h e r e λ∗= 8 (A+B)(b4–a4)is the double characteristic value: y(x)=f(x)+Cy∗(x), y∗(x)=x3–(A–B)(b4–a4) 8A(b–a), where Cis an arbitrary constant and y∗(x) is an eigenfunction of the equation corresponding toλ∗. The equation has no multiple characteristic values if A=±B. 23. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(xt2–t3)y(t)dt=f(x). This is a special case of equation 4.9.8 with A=0 ,B=1 ,a n d h(t)=t2. Solution: y(x)=f(x)+λ(A1+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.8. 310 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 24. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Bxt2+Ct3)y(t)dt=f(x). This is a special case of equation 4.9.9 with A=0a n d h(t)=t2. Solution: y(x)=f(x)+λ(A1+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.9. 25. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax2t+Bxt2)y(t)dt=f(x). This is a special case of equation 4.9.17 with g(x)=x2andh(x)=x. Solution: y(x)=f(x)+λ(A1x2+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.17. 26. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax3+Bxt2)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=x3,h1(t)=A,g2(x)=x,a n dh2(t)=Bt2. Solution: y(x)=f(x)+λ(A1x3+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.18. 27. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax3+Bx2t+Cx2+D)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=Ax3+Cx2+D,h1(t)=1 , g2(x)=x2, andh2(t)=Bt. Solution: y(x)=f(x)+λ[A1(Ax3+Cx2+D)+A2x2], where A1andA2are the constants determined by the formulas presented in 4.9.18. 28. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Axt2+Bt3+Ct2+D)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=x,h1(t)=At2,g2(x)=1 ,a n d h2(t)= Bt3+Ct2+D. Solution: y(x)=f(x)+λ(A1x+A2), where A1andA2are the constants determined by the formulas presented in 4.9.18. 29. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)3y(t)dt=f(x). This is a special case of equation 4.9.19 with g(x)=x,h(t)=–t,a n d m=3 . 30. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax +Bt)3y(t)dt=f(x). This is a special case of equation 4.9.19 with g(x)=Ax,h(t)=Bt,a n dm=3 . 4.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 311 4.1-4. Kernels Containing Higher-Order Polynomials in xandt. 31. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(xn+tn)y(t)dt=f(x), n=1 , 2 , ... The characteristic values of the equation: λ1,2=1 ∆n±√ ∆0∆2n,w h e r e ∆n=1 n+1(bn+1–an+1). 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1xn+A2), where the constants A1andA2are given by A1=f1–λ(f1∆n–f2∆0) λ2(∆2n–∆0∆2n)–2λ∆n+1,A2=f2–λ(f2∆n–f1∆2n) λ2(∆2n–∆0∆2n)–2λ∆n+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb axnf(x)dx,∆n=1 n+1(bn+1–an+1). 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=xn+⎝radicalbig ∆2n/∆0, where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. Solution with λ=λ2≠λ1andf1=f2=0 : y(x)=f(x)+Cy2(x), y2(x)=xn–⎝radicalbig ∆2n/∆0, where Cis an arbitrary constant and y2(x) is an eigenfunction of the equation corresponding to the characteristic value λ2. 4◦. The equation has no multiple characteristic values. 32. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(xn–tn)y(t)dt=f(x), n=1 , 2 , ... The characteristic values of the equation: λ1,2=±⎝bracketleftbigg1 (n+1 )2(bn+1–an+1)2–1 2n+1(b2n+1–a2n+1)(b–a)⎝bracketrightbigg–1/2 . 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1xn+A2), where the constants A1andA2are given by A1=f1+λ(f1∆n–f2∆0) λ2(∆0∆2n–∆2n)+1,A2=–f2+λ(f2∆n–f1∆2n) λ2(∆0∆2n–∆2n)+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb axnf(x)dx,∆n=1 n+1(bn+1–an+1). 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=xn+1–λ 1∆n λ1∆0, where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. The equation has no multiple characteristic values. 312 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 33. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Axn+Btn)y(t)dt=f(x), n=1 , 2 , ... The characteristic values of the equation: λ1,2=(A+B)∆n±⎝radicalbig (A–B)2∆2n+4AB∆0∆2n 2AB(∆2n–∆0∆2n),∆n=1 n+1(bn+1–an+1). 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1xn+A2), where the constants A1andA2are given by A1=Af1–ABλ (f1∆n–f2∆0) ABλ2(∆2n–∆0∆2n)–(A+B)λ∆n+1, A2=Bf2–ABλ (f2∆n–f1∆2n) ABλ2(∆2n–∆0∆2n)–(A+B)λ∆n+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb axnf(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=xn+1–Aλ 1∆n Aλ 1∆0, where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A≠±B, where the characteristic valueλ∗=2/[(A+B)∆n] is double: y(x)=f(x)+Cy∗(x), y∗(x)=xn–(A–B)∆n 2A∆0. HereCis an arbitrary constant and y∗(x) is an eigenfunction of the equation corresponding toλ∗. The equation has no multiple characteristic values if A=±B. 34. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)tmy(t)dt=f(x), m=1 , 2 , ... This is a special case of equation 4.9.8 with A=0 ,B=1 ,a n d h(t)=tm. Solution: y(x)=f(x)+λ(A1+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.8. 35. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)xmy(t)dt=f(x), m=1 , 2 , ... This is a special case of equation 4.9.10 with A=0 ,B=1 ,a n d h(x)=xm. Solution: y(x)=f(x)+λ(A1xm+1+A2xm), where A1andA2are the constants determined by the formulas presented in 4.9.10. 4.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 313 36. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Axm+1+Bxmt+Cxm+D)y(t)dt=f(x), m=1 , 2 , ... This is a special case of equation 4.9.18 with g1(x)=Axm+1+Cxm+D,h1(t)=1 , g2(x)=xm, andh2(t)=Bt. Solution: y(x)=f(x)+λ[A1(Axm+1+Cxm+D)+A2xm], where A1andA2are the constants determined by the formulas presented in 4.9.18. 37. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Axtm+Btm+1+Ctm+D)y(t)dt=f(x), m=1 , 2 , ... This is a special case of equation 4.9.18 with g1(x)=x,h1(t)=Atm,g2(x)=1 ,a n d h2(t)=Btm+1+Ctm+D. Solution: y(x)=f(x)+λ(A1x+A2), where A1andA2are the constants determined by the formulas presented in 4.9.18. 38. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Axntn+Bxmtm)y(t)dt=f(x), n,m=1 , 2 , ...,n≠m. This is a special case of equation 4.9.14 with g(x)=xnandh(t)=tm. Solution: y(x)=f(x)+λ(A1xn+A2xm), where A1andA2are the constants determined by the formulas presented in 4.9.14. 39. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Axntm+Bxmtn)y(t)dt=f(x), n,m=1 , 2 , ...,n≠m. This is a special case of equation 4.9.17 with g(x)=xnandh(t)=tm. Solution: y(x)=f(x)+λ(A1xn+A2xm), where A1andA2are the constants determined by the formulas presented in 4.9.17. 40. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)my(t)dt=f(x), m=1 , 2 , ... This is a special case of equation 4.9.19 with g(x)=xandh(t)=–t. 41. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax +Bt)my(t)dt=f(x), m=1 , 2 , ... This is a special case of equation 4.9.19 with g(x)=Axandh(t)=Bt. 42. y(x)+A⎝integraldisplay ⎝integraldisplayb a|x–t|tky(t)dt=f(x). This is a special case of equation 4.9.36 with g(t)=Atk. Solving the integral equation is reduced to solving the ordinary differential equation y/prime/prime xx+2Axky=f/prime/prime xx(x), the general solution of which can be expressed via Bessel functions or modiﬁed Bessel functions (the boundary conditions are given in 4.9.36). 314 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 43. y(x)+A⎝integraldisplay ⎝integraldisplayb a|x–t|2n+1y(t)dt=f(x), n= 0 ,1 ,2 , ... Let us remove the modulus in the integrand: y(x)+A⎝integraldisplayx a(x–t)2n+1y(t)dt+A⎝integraldisplayb x(t–x)2n+1y(t)dt=f(x). (1) Thek-fold differentiation of (1) with respect to xyields y(k) x(x)+ABk⎝integraldisplayx a(x–t)2n+1–ky(t)dt+ (–1)kABk⎝integraldisplayb x(t–x)2n+1–ky(t)dt=f(k) x(x), Bk=( 2n+ 1)(2n )...(2n+2–k), k=1 ,2 , ...,2n+1 .(2) Differentiating (2) with k=2n+ 1, we arrive at the following linear nonhomogeneous differential equation with constant coefﬁcients for y=y(x): y(2n+2) x +2 ( 2n+1 ) !Ay=f(2n+2) x (x). (3) Equation (3) must satisfy the initial conditions which can be obtained by setting x=ain (1) and (2): y(a)+A⎝integraldisplayb a(t–a)2n+1y(t)dt=f(a), y(k) x(a)+( – 1 )kABk⎝integraldisplayb a(t–a)2n+1–ky(t)dt=f(k) x(a),k=1 ,2 , ...,2n+1 .(4) These conditions can be reduced to a more habitual form containing no integrals. To this end, ymust be expressed from equation (3) in terms of y(2n+2) x andf(2n+2) x and substituted into (4), and then one must integrate the resulting expressions by parts (sufﬁciently many times). 4.1-5. Kernels Containing Rational Functions. 44. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝parenleftbigg ⎝parenleftbigg1 x+1 t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 4.9.2 with g(x)=1/x. Solution: y(x)=f(x)+λ⎝parenleftbiggA1 x+A2⎝parenrightbigg , where A1andA2are the constants determined by the formulas presented in 4.9.2. 45. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝parenleftbigg ⎝parenleftbigg1 x–1 t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 4.9.3 with g(x)=1/x. Solution: y(x)=f(x)+λ⎝parenleftbiggA1 x+A2⎝parenrightbigg , where A1andA2are the constants determined by the formulas presented in 4.9.3. 4.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 315 46. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝parenleftbigg ⎝parenleftbiggA x+B t⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 4.9.4 with g(x)=1/x. Solution: y(x)=f(x)+λ⎝parenleftbiggA1 x+A2⎝parenrightbigg , where A1andA2are the constants determined by the formulas presented in 4.9.4. 47. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝parenleftbigg ⎝parenleftbiggA x+α+B t+β⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 4.9.5 with g(x)=A x+αandh(t)=B t+β. Solution: y(x)=f(x)+λ⎝parenleftbigg A1A x+α+A2⎝parenrightbigg , where A1andA2are the constants determined by the formulas presented in 4.9.5. 48. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝parenleftbigg ⎝parenleftbiggx t–t x⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 4.9.16 with g(x)=xandh(t)=1/t. Solution: y(x)=f(x)+λ⎝parenleftbigg A1x+A2 x⎝parenrightbigg , where A1andA2are the constants determined by the formulas presented in 4.9.16. 49. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝parenleftbigg ⎝parenleftbiggAx t+Bt x⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 4.9.17 with g(x)=xandh(t)=1/t. Solution: y(x)=f(x)+λ⎝parenleftbigg A1x+A2 x⎝parenrightbigg , where A1andA2are the constants determined by the formulas presented in 4.9.17. 50. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝parenleftbigg ⎝parenleftbigg Ax+α t+β+Bt+α x+β⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 4.9.17 with g(x)=x+αandh(t)=1 t+β. Solution: y(x)=f(x)+λ⎝bracketleftbigg A1(x+α)+A2 x+β⎝bracketrightbigg , where A1andA2are the constants determined by the formulas presented in 4.9.17. 51. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbigg A(x+α)n (t+β)m+B(t+α)n (x+β)m⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), n,m=0 , 1 , 2 , ... This is a special case of equation 4.9.17 with g(x)=(x+α)nandh(t)=(t+β)–m. Solution: y(x)=f(x)+λ⎝bracketleftbigg A1(x+α)n+A2 (x+β)m⎝bracketrightbigg , where A1andA2are the constants determined by the formulas presented in 4.9.17. 316 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 52. y(x)–λ⎝integraldisplay ⎝integraldisplay∞ 1y(t) x+tdt=f(x), 1 ≤x<∞,–∞<πλ <1 . Solution: y(x)=⎝integraldisplay∞ 0τsinh(πτ)F(τ) cosh(πτ )–πλP–1 2+iτ(x)dτ, F(τ)=⎝integraldisplay∞ 1f(x)P–1 2+iτ(x)dx, where Pν(x)=F⎝parenleftbig –ν,ν+1 ,1 ;1 2(1 –x)⎝parenrightbig is the Legendre spherical function of the ﬁrst kind, for which the integral representation P–1 2+iτ(coshα)=2 π⎝integraldisplayα 0cos(τs)ds √ 2(cosh α–c o s h s)(α≥0) can be used. Reference: V . A. Ditkin and A. P. Prudnikov (1965). 53. ( x2+b2)y(x)=λ π⎝integraldisplay ⎝integraldisplay∞ –∞a3y(t) a2+(x–t)2dt. This equation is encountered in atomic and nuclear physics. We seek the solution in the form y(x)=∞⎝summationdisplay m=0Amx x2+(am+b)2.( 1) The coefﬁcients Amobey the equations mAm⎝parenleftbiggm+2b a⎝parenrightbigg +λAm–1=0 ,∞⎝summationdisplay m=0Am=0 . ( 2 ) Using the ﬁrst equation of (2) to express all AmviaA0(A0can be chosen arbitrarily), substituting the result into the second equation of (2), and dividing by A0, we obtain 1+∞⎝summationdisplay m=1(–λ)m m!1 (1 + 2b/a)(2 + 2 b/a)...(m+2b/a)=0 . ( 3 ) It follows from the deﬁnitions of the Bessel f unctions of the ﬁrst kind that equation (3) can be rewritten in the form λ–b/aJ2b/a⎝parenleftbig 2√ λ⎝parenrightbig =0 . ( 4 ) In this sort of problem, aandλare usually assumed to be given and b,which is proportional to the system energy, to be unknown. The quantity bcan be determined by tables of zeros of Bessel functions. In some cases, bandaare given and λis unknown. Reference: I. Sneddon (1995). 54.⎝integraldisplay ⎝integraldisplay1 –1y(x)–y(t) |x–t|dt=λy(x). The characteristic values of the equation: λn=2⎝parenleftBig 1+1 2+···+1 n⎝parenrightBig ,w h e r e n=1 ,2 , ... The eigenfunctions of the equation: yn(x)=Pn(x), where n=1 ,2 , ... HerePn(x)=1 n!2ndn dxn(x2–1 )nare the Legendre polynomials. Reference: A. G. Petrov (1986). 4.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 317 4.1-6. Kernels Containing Arbitrary Powers. 55. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)tµy(t)dt=f(x). This is a special case of equation 4.9.8 with A=0 ,B=1 ,a n d h(t)=tµ. Solution: y(x)=f(x)+λ(A1+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.8. 56. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)xνy(t)dt=f(x). This is a special case of equation 4.9.10 with A=0 ,B=1 ,a n d h(x)=xν. Solution: y(x)=f(x)+λ(E1xν+1+E2xν), where E1andE2are the constants determined by the formulas presented in 4.9.10. 57. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(xµ–tµ)y(t)dt=f(x). This is a special case of equation 4.9.3 with g(x)=xµ. Solution: y(x)=f(x)+λ(A1xµ+A2), where A1andA2are the constants determined by the formulas presented in 4.9.3. 58. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Axν+Btν)tµy(t)dt=f(x). This is a special case of equation 4.9.6 with g(x)=xνandh(t)=tµ. Solution: y(x)=f(x)+λ(A1xν+A2), where A1andA2are the constants determined by the formulas presented in 4.9.6. 59. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Dxν+Etµ)xγy(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=xν+γ,h1(t)=D,g2(x)=xγ,a n d h2(t)=Etµ. Solution: y(x)=f(x)+λ(A1xν+γ+A2xγ), where A1andA2are the constants determined by the formulas presented in 4.9.18. 60. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Axνtµ+Bxγtδ)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=xν,h1(t)=Atµ,g2(x)=xγ,a n d h2(t)=Btδ. Solution: y(x)=f(x)+λ(A1xν+A2xγ), where A1andA2are the constants determined by the formulas presented in 4.9.18. 318 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 61. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(A+Bxtµ+Ctµ+1)y(t)dt=f(x). This is a special case of equation 4.9.9 with h(t)=tµ. Solution: y(x)=f(x)+λ(A1+A2x), where A1andA2are the constants determined by the formulas presented in 4.9.9. 62. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Atα+Bxβtµ+Ctµ+γ)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=1 ,h1(t)=Atα+Ctµ+γ,g2(x)=xβ,a n d h2(t)=Btµ. Solution: y(x)=f(x)+λ(A1+A2xβ), where A1andA2are the constants determined by the formulas presented in 4.9.18. 63. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Axαtγ+Bxβtγ+Cxµtν)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=Axα+Bxβ,h1(t)=tγ,g2(x)=xµ,a n d h2(t)=Ctν. Solution: y(x)=f(x)+λ[A1(Axα+Bxβ)+A2xµ], where A1andA2are the constants determined by the formulas presented in 4.9.18. 64. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbigg A(x+p1)β (t+q1)γ+B(x+p2)µ (t+q2)δ⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=(x+p1)β,h1(t)=A(t+q1)–γ,g2(x)= (x+p2)µ,a n dh2(t)=B(t+q2)–δ. Solution: y(x)=f(x)+λ⎝bracketleftbig A1(x+p1)β+A2(x+p2)µ⎝bracketrightbig , where A1andA2are the constants determined by the formulas presented in 4.9.18. 65. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝parenleftbigg ⎝parenleftbigg Axµ+a tν+b+Bxγ+c tδ+d⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=xµ+a,h1(t)=A tν+b,g2(x)=xγ+c, andh2(t)=B tδ+d. Solution: y(x)=f(x)+λ[A1(xµ+a)+A2(xγ+c)], where A1andA2are the constants determined by the formulas presented in 4.9.18. 4.1. E QUATIONS WHOSE KERNELS CONTAIN POWER -LAWFUNCTIONS 319 4.1-7. Singular Equations. In this subsection, all singular integrals are understood in the sense of the Cauchy principal value. 66. Ay(x)+B π⎝integraldisplay ⎝integraldisplay1 –1y(t)dt t–x=f(x), –1 < x<1 . HereAandBare real numbers such that B≠0,A±B≠0, and A2+B2=1 . 1◦. The solution bounded at the endpoints: y(x)=Af(x)–B π⎝integraldisplay1 –1g(x) g(t)f(t)dt t–x,g(x)=( 1+ x)α(1 –x)1–α,( 1 ) where αis the solution of the trigonometric equation A+Bcot(πα)=0 ( 2 ) on the interval 0 < α< 1. This solution y(x) exists if and only if⎝integraldisplay1 –1f(t) g(t)dt=0 . 2◦. The solution bounded at the endpoint x= 1 and unbounded at the endpoint x= –1: y(x)=Af(x)–B π⎝integraldisplay1 –1g(x) g(t)f(t)dt t–x,g(x)=( 1+ x)α(1 –x)–α,( 3 ) where αis the solution of the trigonometric equation (2) on the interval –1 < α<0 . 3◦. The solution unbounded at the endpoints: y(x)=Af(x)–B π⎝integraldisplay1 –1g(x) g(t)f(t)dt t–x+Cg(x), g(x)=( 1+ x)α(1 –x)–1–α,( 4 ) where Cis an arbitrary constant and αis the solution of the trigonometric equation (2) on the interval –1 < α<0 . References: N. I. Muskhelishvili (1992), I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004, pp. 6–7). 67. y(x)–λ⎝integraldisplay ⎝integraldisplay∞ –∞y(t)dt t–x=f(x). Solution: y(x)=1 1+π2λ2⎝bracketleftbigg f(x)+λ⎝integraldisplay∞ –∞f(t)dt t–x⎝bracketrightbigg . Reference: M. L. Krasnov (1975). 68. y(x)–λ⎝integraldisplay ⎝integraldisplay1 0⎝parenleftbigg ⎝parenleftbigg1 t–x–1 x+t–2xt⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), 0 < x<1 . Tricomi’s equation. Solution: y(x)=1 1+λ2π2⎝bracketleftbigg f(x)+⎝integraldisplay1 0tα(1 –x)α xα(1 –t)α⎝parenleftbigg1 t–x–1 x+t–2xt⎝parenrightbigg f(t)dt⎝bracketrightbigg +C(1 –x)β x1+β, α=2 πarctan( λπ)( – 1 < α<1 ) , t a nβπ 2=λπ(–2 <β<0 ) , where Cis an arbitrary constant. References: P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. G. Tricomi (1985). 320 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 69. y(x)+λ⎝integraldisplay ⎝integraldisplay1 –1⎝parenleftbigg ⎝parenleftbigg1+t 1+x⎝parenrightbigg ⎝parenrightbigg2 n+2⎝parenleftbigg ⎝parenleftbigg1 t–x–1 1–xt⎝parenrightbigg ⎝parenrightbigg y(t)dt=f(x), 0 < x<1 . Tricomi–Gellerstedt equation. In the class of functions y(x) for which integrals⎝integraltext1 –1|y(x)|ln|x±1|dxare ﬁnite the unique solution of the equation has the form y(x)=1 1+λ2π2⎝bracketleftbigg f(x)–λ⎝integraldisplay1 –1⎝parenleftbigg1–t2 1–x2⎝parenrightbigg1 n+2⎝parenleftbigg1 t–x–1 1–xt⎝parenrightbigg f(t)dt⎝bracketrightbigg . Reference: S. G. Mikhlin (1967). 4.2. Equations Whose Kernels Contain Exponential Functions 4.2-1. Kernels Containing Exponential Functions. 1. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(eβx+eβt)y(t)dt=f(x). The characteristic values of the equation: λ1,2=β eβb–eβa±⎝radicalBig 1 2β(b–a)(e2βb–e2βa). 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1eβx+A2), where the constants A1andA2are given by A1=f1–λ⎝bracketleftbig f1∆β–(b–a)f2⎝bracketrightbig λ2⎝bracketleftbig ∆2 β–(b–a)∆2β⎝bracketrightbig –2λ∆β+1,A2=f2–λ(f2∆β–f1∆2β) λ2⎝bracketleftbig ∆2 β–(b–a)∆2β⎝bracketrightbig –2λ∆β+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb af(x)eβxdx,∆β=1 β(eβb–eβa). 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=eβx+⎝radicalBigg e2βb–e2βa 2β(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. Solution with λ=λ2≠λ1andf1=f2=0 : y(x)=f(x)+Cy2(x), y2(x)=eβx–⎝radicalBigg e2βb–e2βa 2β(b–a), where Cis an arbitrary constant and y2(x) is an eigenfunction of the equation corresponding to the characteristic value λ2. 4◦. The equation has no multiple characteristic values. 4.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 321 2. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(eβx–eβt)y(t)dt=f(x). The characteristic values of the equation: λ1,2=±β ⎝radicalBig (eβb–eβa)2–1 2β(b–a)(e2βb–e2βa). 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1eβx+A2), where the constants A1andA2are given by A1=f1+λ⎝bracketleftbig f1∆β–(b–a)f2⎝bracketrightbig λ2⎝bracketleftbig (b–a)∆2β–∆2 β⎝bracketrightbig +1,A2=–f2+λ(f2∆β–f1∆2β) λ2⎝bracketleftbig (b–a)∆2β–∆2 β⎝bracketrightbig +1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb af(x)eβxdx,∆β=1 β(eβb–eβa). 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=eβx+1–λ1∆β λ1(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. The equation has no multiple characteristic values. 3. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Aeβx+Beβt)y(t)dt=f(x). The characteristic values of the equation: λ1,2=(A+B)∆β±⎝radicalBig (A–B)2∆2 β+4AB(b–a)∆2β 2AB⎝bracketleftbig ∆2 β–(b–a)∆2β⎝bracketrightbig ,∆β=1 β(eβb–eβa). 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1eβx+A2), where the constants A1andA2are given by A1=Af1–ABλ⎝bracketleftbig f1∆β–(b–a)f2⎝bracketrightbig ABλ2⎝bracketleftbig ∆2 β–(b–a)∆2β⎝bracketrightbig –(A+B)λ∆β+1, A2=Bf2–ABλ (f2∆β–f1∆2β) ABλ2⎝bracketleftbig ∆2 β–(b–a)∆2β⎝bracketrightbig –(A+B)λ∆β+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb af(x)eβxdx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=eβx+1–Aλ 1∆β A(b–a)λ1, where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 322 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A≠±B, where the characteristic valueλ∗=2 (A+B)∆βis double: y(x)=f(x)+Cy∗(x), y∗(x)=eβx–(A–B)∆β 2A(b–a), where Cis an arbitrary constant and y∗(x) is an eigenfunction of the equation corresponding toλ∗. The equation has no multiple characteristic values if A=±B. 4. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig Aeβ(x–t)+B⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=eβx,h1(t)=Ae–βt,g2(x)=1 ,a n d h2(t)=B. Solution: y(x)=f(x)+λ(A1eβx+A2), where A1andA2are the constants determined by the formulas presented in 4.9.18. 5. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig Aeβx+µt+Be(β+µ)t⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 4.9.6 with g(x)=eβxandh(t)=eµt. Solution: y(x)=f(x)+λ(A1eβx+A2), where A1andA2are the constants determined by the formulas presented in 4.9.6. 6. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig Aeα(x+t)+Beβ(x+t)⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 4.9.14 with g(x)=eαxandh(t)=eβt. Solution: y(x)=f(x)+λ(A1eαx+A2eβx), where A1andA2are the constants determined by the formulas presented in 4.9.14. 7. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝parenleftbig⎝parenleftbig Aeαx +βt+Beβx+αt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). This is a special case of equation 4.9.17 with g(x)=eαxandh(t)=eβt. Solution: y(x)=f(x)+λ(A1eαx+A2eβx), where A1andA2are the constants determined by the formulas presented in 4.9.17. 8. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig De(γ+µ)x+Eeνt+µx⎝bracketrightbig⎝bracketrightbig y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=e(γ+µ)x,h1(t)=D,g2(x)=eµx,a n d h2(t)=Eeνt. Solution: y(x)=f(x)+λ[A1e(γ+µ)x+A2eµx], where A1andA2are the constants determined by the formulas presented in 4.9.18. 4.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 323 9. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Aeαx +βt+Beγx+δt)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=eαx,h1(t)=Aeβt,g2(x)=eγx,a n d h2(t)=Beδt. Solution: y(x)=f(x)+λ(A1eαx+A2eγx), where A1andA2are the constants determined by the formulas presented in 4.9.18. 10. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1Akeγk(x–t)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x). This is a special case of equation 4.9.20 with gk(x)=eγkxandhk(t)=Ake–γkt. 11. y(x)–1 2⎝integraldisplay ⎝integraldisplay∞ 0e–|x–t|y(t)dt=Aeµx,0 < µ<1 . Solution: y(x)=C(1 +x)+Aµ–2⎝bracketleftbig (µ2–1 )eµx–µ+1⎝bracketrightbig , where Cis an arbitrary constant. Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975). 12. y(x)+λ⎝integraldisplay ⎝integraldisplay∞ 0e–|x–t|y(t)dt=f(x). Solution: y(x)=f(x)–λ √ 1+2λ⎝integraldisplay∞ 0exp⎝parenleftbig –√ 1+2λ|x–t|⎝parenrightbig f(t)dt +⎝parenleftbigg 1–λ+1 √ 1+2λ⎝parenrightbigg⎝integraldisplay∞ 0exp⎝bracketleftbig –√ 1+2λ(x+t)⎝bracketrightbig f(t)dt, where λ>–1 2. Reference: F. D. Gakhov and Yu. I. Cherskii (1978). 13. y(x)–λ⎝integraldisplay ⎝integraldisplay∞ –∞e–|x–t|y(t)dt=0 , λ>0 . The Lalesco–Picard equation. Solution: y(x)=⎧ ⎪⎨ ⎪⎩C1exp⎝parenleftbig x√ 1–2λ⎝parenrightbig +C2exp⎝parenleftbig –x√ 1–2λ⎝parenrightbig for 0 < λ<1 2, C1+C2x forλ=1 2, C1cos⎝parenleftbig x√ 2λ–1⎝parenrightbig +C2sin⎝parenleftbig x√ 2λ–1⎝parenrightbig forλ>1 2, where C1andC2are arbitrary constants. Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971). 324 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 14. y(x)+λ⎝integraldisplay ⎝integraldisplay∞ –∞e–|x–t|y(t)dt=f(x). 1◦. Solution with λ>–1 2: y(x)=f(x)–λ √ 1+2λ⎝integraldisplay∞ –∞exp⎝parenleftbig –√ 1+2λ|x–t|⎝parenrightbig f(t)dt. 2◦.I fλ≤–1 2, for the equation to be solvable the conditions ⎝integraldisplay∞ –∞f(x)c o s (ax)dx=0 ,⎝integraldisplay∞ –∞f(x)s i n (ax)dx=0 , where a=√ –1 – 2λ , must be satisﬁed. In this case, the solution has the form y(x)=f(x)–a2+1 2a⎝integraldisplay∞ 0sin(at)f(x+t)dt,( – ∞<x<∞). In the class of solutions not belonging to L2(–∞,∞), the homogeneous equation (with f(x)≡0) has a nontrivial solution. In this case, the general solution of the corresponding nonhomogeneous equation with λ≤–1 2has the form y(x)=C1sin(ax)+C2cos(ax )+f(x)–a2+1 4a⎝integraldisplay∞ –∞sin(a|x–t|)f(t)dt. Reference: F. D. Gakhov and Yu. I. Cherskii (1978). 15. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ|x–t|y(t)dt=f(x). This is a special case of equation 4.9.37 with g(t)=A. 1◦. The function y=y(x) obeys the following second-order linear nonhomogeneousordinary differential equation with constant coefﬁcients: y/prime/prime xx+λ(2A–λ)y=f/prime/prime xx(x)–λ2f(x). (1) The boundary conditions for (1) have the form (see 4.9.37) y/prime x(a)+λy(a)=f/prime x(a)+λf(a), y/prime x(b)–λy(b)=f/prime x(b)–λf(b).(2) Equation (1) under the boundary conditions (2) determines the solution of the original integral equation. 2◦.F o rλ(2A–λ) < 0, the general solution of equation (1) is given by y(x)=C1cosh(kx )+C2sinh(kx)+f(x)–2Aλ k⎝integraldisplayx asinh[k(x–t)]f(t)dt, k=⎝radicalbig λ(λ–2A),(3) where C1andC2are arbitrary constants. Forλ(2A–λ) > 0, the general solution of equation (1) is given by y(x)=C1cos(kx )+C2sin(kx)+f(x)–2Aλ k⎝integraldisplayx asin[k(x–t)]f(t)dt, k=⎝radicalbig λ(2A–λ).(4) Forλ=2A, the general solution of equation (1) is given by y(x)=C1+C2x+f(x)–4A2⎝integraldisplayx a(x–t)f(t)dt.( 5 ) The constants C1andC2in solutions (3)–(5) are determined by conditions (2). 4.2. E QUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS 325 3◦. In the special case a=0a n d λ(2A–λ) > 0, the solution of the integral equation is given by formula (4) with C1=A(kIc–λIs) (λ–A)s i nµ–kcosµ,C2=–λ kA(kIc–λIs) (λ–A)s i nµ–kcosµ, k=⎝radicalbig λ(2A–λ),µ=bk,Is=⎝integraldisplayb 0sin[k(b–t)]f(t)dt,Ic=⎝integraldisplayb 0cos[k (b–t)]f(t)dt. 16. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1Akexp(λk|x–t|)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), – ∞<a<b<∞. 1◦. Let us remove the modulus in the kth summand of the integrand: Ik(x)=⎝integraldisplayb aexp(λk|x–t|)y(t)dt=⎝integraldisplayx aexp[λk(x–t)]y(t)dt+⎝integraldisplayb xexp[λk(t–x)]y(t)dt.( 1 ) Differentiating (1) with respect to xtwice yields I/prime k=λk⎝integraldisplayx aexp[λk(x–t)]y(t)dt–λk⎝integraldisplayb xexp[λk(t–x)]y(t)dt, I/prime/prime k=2λky(x)+λ2 k⎝integraldisplayx aexp[λk(x–t)]y(t)dt+λ2 k⎝integraldisplayb xexp[λk(t–x)]y(t)dt,(2) where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we ﬁnd the relation between I/prime/prime kandIk: I/prime/prime k=2λky(x)+λ2 kIk,Ik=Ik(x). (3) 2◦. With the aid of (1), the integral equation can be rewritten in the form y(x)+n⎝summationdisplay k=1AkIk=f(x). (4) Differentiating (4) with respect to xtwice and taking into account (3), we ﬁnd that y/prime/prime xx(x)+σny(x)+n⎝summationdisplay k=1Akλ2 kIk=f/prime/prime xx(x), σn=2n⎝summationdisplay k=1Akλk.( 5 ) Eliminating the integral Infrom (4) and (5) yields y/prime/prime xx(x)+(σn–λ2 n)y(x)+n–1⎝summationdisplay k=1Ak(λ2 k–λ2 n)Ik=f/prime/prime xx(x)–λ2 nf(x). (6) Differentiating (6) with respect to xtwice and eliminating In–1from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear differential operator (acting on y) with constant coefﬁcients plus the sumn–2⎝summationtext k=1BkIk.I f w e successively eliminate In–2,In–3,..., with the aid of double differentiation, then we ﬁnally arrive at a linear nonhomogeneous ordinary differential equation of order 2 nwith constant coefﬁcients. 3◦. The boundary conditions for y(x) can be found by setting x=ain the integral equation and all its derivatives. (Alternatively, these conditions can be found by setting x=aandx=b in the integral equation and all its derivatives obtained by means of double differentiation.) 326 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 4.2-2. Kernels Containing Power-Law and Exponential Functions. 17. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)eγty(t)dt=f(x). This is a special case of equation 4.9.8 with A=0 ,B=1 ,a n d h(t)=eγt. 18. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)eγxy(t)dt=f(x). This is a special case of equation 4.9.10 with A=0 ,B=1 ,a n d h(x)=eγx. 19. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)eγx+µty(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=xeγx,h1(t)=eµt,g2(x)=eγx,a n d h2(t)=–teµt. 20. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+(Bx +Ct)eγx]y(t)dt=f(x). This is a special case of equation 4.9.11 with h(x)=eγx. 21. y(x)–λ⎝integraldisplay ⎝integraldisplayb 0(x2+t2)eγ(x+t)y(t)dt=f(x). This is a special case of equation 4.9.15 with g(x)=x2eγxandh(t)=eγt. 22. y(x)–λ⎝integraldisplay ⎝integraldisplayb 0(x2–t2)eγ(x–t)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=x2eγx,h1(t)=e–γt,g2(x)=eγx,a n d h2(t)=–t2e–γt. 23. y(x)–λ⎝integraldisplay ⎝integraldisplayb 0(Axn+Btn)eαx +βty(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 4.9.18 with g1(x)=xneαx,h1(t)=Aeβt,g2(x)=eαx,a n d h2(t)=Btneβt. 24. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1Aktνkeαkx+βkt⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 4.9.20 with gk(x)=eαkxandhk(t)=Aktνkeβkt. 25. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1Akxνkeαkx+βkt⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 4.9.20 with gk(x)=Akxνkeαkxandhk(t)=eβkt. 26. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)neγ(x–t)y(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 4.9.20. 4.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 327 27. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(x–t)neαx +βty(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 4.9.20. 28. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(Ax +Bt)neαx +βty(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 4.9.20. 29. y(x)+A⎝integraldisplay ⎝integraldisplayb ateλ|x–t|y(t)dt=f(x). This is a special case of equation 4.9.37 with g(t)=At. The solution of the integral equation can be written via the Bessel functions (or modiﬁed Bessel functions) of order 1/3. 30. y(x)+⎝integraldisplay ⎝integraldisplay∞ 0(a+b|x–t|)e x p ( – |x–t|)y(t)dt=f(x). Let the biquadratic polynomial P(k)=k4+2 (a–b+1 )k2+2a+2b+ 1 have no real roots and letk=α+iβbe a root of the equation P(k)=0s u c ht h a t α>0a n d β> 0. In this case, the solution has the form y(x)=f(x)+ρ⎝integraldisplay∞ 0exp(–β|x–t|)c o s (θ+α|x–t|)f(t)dt +[α+(β–1 )2]2 4α2β⎝integraldisplay∞ 0exp[–β(x+t)] cos[α (x–t)]f(t)dt +R 4α2⎝integraldisplay∞ 0exp[–β(x+t)] cos[ψ +α(x+t)]f(t)dt, where the parameters ρ,θ,R,a n dψare determined from the system of algebraic equations obtained by separating real and i maginary parts in the relations ρeiθ=µ β–iα,Reiψ=(β–1–iα )4 8α2(β–iα). Reference: F. D. Gakhov and Yu. I. Cherskii (1978). 4.3. Equations Whose Kernels Contain Hyperbolic Functions 4.3-1. Kernels Containing Hyperbolic Cosine. 1. y(x)–λ⎝integraldisplay ⎝integraldisplayb acosh(βx )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s h ( βx)a n dh(t)=1 . 2. y(x)–λ⎝integraldisplay ⎝integraldisplayb acosh(βt )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=c o s h ( βt). 328 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 3. y(x)–λ⎝integraldisplay ⎝integraldisplayb acosh[β (x–t)]y(t)dt=f(x). This is a special case of equation 4.9.13 with g(x)=c o s h ( βx)a n dh(t)=s i n h ( βt). Solution: y(x)=f(x)+λ⎝bracketleftbig A1cosh(βx )+A2sinh(βx)⎝bracketrightbig , where A1andA2are the constants determined by the formulas presented in 4.9.13. 4. y(x)–λ⎝integraldisplay ⎝integraldisplayb acosh[β (x+t)]y(t)dt=f(x). This is a special case of equation 4.9.12 with g(x)=c o s h ( βx)a n dh(t)=s i n h ( βt). Solution: y(x)=f(x)+λ⎝bracketleftbig A1cosh(βx )+A2sinh(βx)⎝bracketrightbig , where A1andA2are the constants determined by the formulas presented in 4.9.12. 5. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝braceleftbigg ⎝braceleftbiggn⎝summationdisplay k=1Akcosh[β k(x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 4.9.20. 6. y(x)–λ⎝integraldisplay ⎝integraldisplayb acosh(βx ) cosh(βt )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s h ( βx)a n dh(t)=1 cosh(βt ). 7. y(x)–λ⎝integraldisplay ⎝integraldisplayb acosh(βt ) cosh(βx )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 cosh(βx )andh(t)=c o s h ( βt). 8. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoshk(βx)c o s hm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s hk(βx)a n dh(t)=c o s hm(µt). 9. y(x)–λ⎝integraldisplay ⎝integraldisplayb atkcoshm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s hm(βx)a n dh(t)=tk. 10. y(x)–λ⎝integraldisplay ⎝integraldisplayb axkcoshm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t)=c o s hm(βt). 11. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)c o s h ( βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=c o s h ( βx). 4.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 329 12. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)c o s h ( βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=c o s h ( βt). 13. y(x)+λ⎝integraldisplay ⎝integraldisplay∞ –∞y(t)dt cosh[b (x–t)]=f(x). Solution with b>π|λ|: y(x)=f(x)–2λb √ b2–π2λ2⎝integraldisplay∞ –∞sinh[2 k(x–t)] sinh[2b (x–t)]f(t)dt,k=b πarccos⎝parenleftBigπλ b⎝parenrightBig . Reference: F. D. Gakhov and Yu. I. Cherskii (1978). 4.3-2. Kernels Containing Hyperbolic Sine. 14. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinh(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = sinh(βx )a n dh(t)=1 . 15. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinh(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t) = sinh(βt ). 16. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinh[β(x–t)]y(t)dt=f(x). This is a special case of equation 4.9.16 with g(x)=s i n h ( βx)a n dh(t)=c o s h ( βt). Solution: y(x)=f(x)+λ⎝bracketleftbig A1sinh(βx)+A2cosh(βx )⎝bracketrightbig , where A1andA2are the constants determined by the formulas presented in 4.9.16. 17. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinh[β(x+t)]y(t)dt=f(x). This is a special case of equation 4.9.15 with g(x)=s i n h ( βx)a n dh(t)=c o s h ( βt). Solution: y(x)=f(x)+λ⎝bracketleftbig A1sinh(βx)+A2cosh(βx )⎝bracketrightbig , where A1andA2are the constants determined by the formulas presented in 4.9.15. 18. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝braceleftbigg ⎝braceleftbiggn⎝summationdisplay k=1Aksinh[βk(x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 4.9.20. 19. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinh(βx) sinh(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = sinh(βx )a n dh(t)=1 sinh(βt). 330 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 20. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinh(βt) sinh(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 sinh(βx)andh(t)=s i n h( βt). 21. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinhk(βx)s i n hm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i n hk(βx)a n dh(t)=s i n hm(µt). 22. y(x)–λ⎝integraldisplay ⎝integraldisplayb atksinhm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i n hm(βx)a n dh(t)=tk. 23. y(x)–λ⎝integraldisplay ⎝integraldisplayb axksinhm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t)=s i n hm(βt). 24. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)s i n h ( βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t) = sinh(βt ). 25. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)s i n h ( βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x) = sinh(βx ). 26. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λ|x–t|)y(t)dt=f(x). This is a special case of equation 4.9.38 with g(t)=A. 1◦. The function y=y(x) obeys the following second-order linear nonhomogeneousordinary differential equation with constant coefﬁcients: y/prime/prime xx+λ(2A–λ)y=f/prime/prime xx(x)–λ2f(x). (1) The boundary conditions for (1) have the form (see 4.9.38) sinh[λ(b–a)]ϕ/prime x(b)–λcosh[λ(b–a)]ϕ(b)=λϕ(a), sinh[λ(b–a)]ϕ/prime x(a)+λcosh[λ (b–a)]ϕ(a)=–λϕ(b),ϕ(x)=y(x)–f(x). (2) Equation (1) under the boundary conditions (2) determines the solution of the original integral equation. 2◦.F o rλ(2A–λ)=–k2< 0, the general solution of equation (1) is given by y(x)=C1cosh(kx )+C2sinh(kx)+f(x)–2Aλ k⎝integraldisplayx asinh[k(x–t)]f(t)dt,( 3 ) where C1andC2are arbitrary constants. Forλ(2A–λ)=k2> 0, the general solution of equation (1) is given by y(x)=C1cos(kx )+C2sin(kx)+f(x)–2Aλ k⎝integraldisplayx asin[k(x–t)]f(t)dt.( 4 ) Forλ=2A, the general solution of equation (1) is given by y(x)=C1+C2x+f(x)–4A2⎝integraldisplayx a(x–t)f(t)dt.( 5 ) The constants C1andC2in solutions (3)–(5) are determined by conditions (2). 4.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 331 27. y(x)+A⎝integraldisplay ⎝integraldisplayb atsinh(λ|x–t|)y(t)dt=f(x). This is a special case of equation 4.9.38 with g(t)=At. The solution of the integral equation can be written via the Bessel functions (or modiﬁed Bessel functions) of order 1/3. 28. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh3(λ|x–t|)y(t)dt=f(x). Using the formula sinh3β=1 4sinh 3β–3 4sinhβ, we arrive at an equation of the form 4.3.29 withn=2 : y(x)+⎝integraldisplayb a⎝bracketleftbig1 4Asinh(3λ |x–t|)–3 4Asinh(λ|x–t|)⎝bracketrightbig y(t)dt=f(x). 29. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1Aksinh(λk|x–t|)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), – ∞<a<b<∞. 1◦. Let us remove the modulus in the kth summand of the integrand: Ik(x)=⎝integraldisplayb asinh(λk|x–t|)y(t)dt=⎝integraldisplayx asinh[λk(x–t)]y(t)dt+⎝integraldisplayb xsinh[λk(t–x)]y(t)dt.( 1 ) Differentiating (1) with respect to xtwice yields I/prime k=λk⎝integraldisplayx acosh[λ k(x–t)]y(t)dt–λk⎝integraldisplayb xcosh[λ k(t–x)]y(t)dt, I/prime/prime k=2λky(x)+λ2 k⎝integraldisplayx asinh[λk(x–t)]y(t)dt+λ2 k⎝integraldisplayb xsinh[λk(t–x)]y(t)dt,(2) where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we ﬁnd the relation between I/prime/prime kandIk: I/prime/prime k=2λky(x)+λ2 kIk,Ik=Ik(x). (3) 2◦. With the aid of (1), the integral equation can be rewritten in the form y(x)+n⎝summationdisplay k=1AkIk=f(x). (4) Differentiating (4) with respect to xtwice and taking into account (3), we ﬁnd that y/prime/prime xx(x)+σny(x)+n⎝summationdisplay k=1Akλ2 kIk=f/prime/prime xx(x), σn=2n⎝summationdisplay k=1Akλk.( 5 ) Eliminating the integral Infrom (4) and (5) yields y/prime/prime xx(x)+(σn–λ2 n)y(x)+n–1⎝summationdisplay k=1Ak(λ2 k–λ2 n)Ik=f/prime/prime xx(x)–λ2 nf(x). (6) Differentiating (6) with respect to xtwice and eliminating In–1from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear differential operator (acting on y) with constant coefﬁcients plus the sumn–2⎝summationtext k=1BkIk.I f w e successively eliminate In–2,In–3,..., with the aid of double differentiation, then we ﬁnally arrive at a linear nonhomogeneous ordinary differential equation of order 2 nwith constant coefﬁcients. 3◦. The boundary conditions for y(x) can be found by setting x=ain the integral equation and its derivatives. (Alternatively, these conditions can be found by setting x=aandx=b in the integral equation and all its derivatives obtained by means of double differentiation.) 332 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 4.3-3. Kernels Containing Hyperbolic Tangent. 30. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanh(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n h ( βx)a n dh(t)=1 . 31. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanh(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=t a n h ( βt). 32. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Atanh(βx)+Btanh(βt)]y(t)dt=f(x). This is a special case of equation 4.9.4 with g(x)=t a n h ( βx). 33. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanh(βx) tanh(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n h ( βx)a n dh(t)=1 tanh(βt). 34. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanh(βt) tanh(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 tanh(βx)andh(t)=t a n h ( βt). 35. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanhk(βx)t a n hm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n hk(βx)a n dh(t)=t a n hm(µt). 36. y(x)–λ⎝integraldisplay ⎝integraldisplayb atktanhm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n hm(βx)a n dh(t)=tk. 37. y(x)–λ⎝integraldisplay ⎝integraldisplayb axktanhm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t)=t a n hm(βt). 38. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)t a n h ( βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=t a n h ( βt). 39. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)t a n h ( βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=t a n h ( βx). 4.3. E QUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS 333 4.3-4. Kernels Containing Hyperbolic Cotangent. 40. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoth(βx )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o t h ( βx)a n dh(t)=1 . 41. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoth(βt )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=c o t h ( βt). 42. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Acoth(βx )+Bcoth(βt )]y(t)dt=f(x). This is a special case of equation 4.9.4 with g(x)=c o t h ( βx). 43. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoth(βx ) coth(βt )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o t h ( βx)a n dh(t)=1 coth(βt). 44. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoth(βt ) coth(βx )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 coth(βx)andh(t)=c o t h ( βt). 45. y(x)–λ⎝integraldisplay ⎝integraldisplayb acothk(βx)c o t hm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o t hk(βx)a n dh(t)=c o t hm(µt). 46. y(x)–λ⎝integraldisplay ⎝integraldisplayb atkcothm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o t hm(βx)a n dh(t)=tk. 47. y(x)–λ⎝integraldisplay ⎝integraldisplayb axkcothm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t)=c o t hm(βt). 48. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)c o t h ( βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=c o t h ( βt). 49. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)c o t h ( βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=c o t h ( βx). 334 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 4.3-5. Kernels Containing Combination of Hyperbolic Functions. 50. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoshk(βx)s i n hm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s hk(βx)a n dh(t)=s i n hm(µt). 51. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Asinh(αx)c o s h ( βt)+Bsinh(γx)c o s h ( δt)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=s i n h ( αx),h1(t)=Acosh(βt ),g2(x)= sinh(γx), and h2(t)=Bcosh(δt). 52. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanhk(γx)c o t hm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n hk(γx)a n dh(t)=c o t hm(µt). 53. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Atanh(αx)c o t h ( βt)+Btanh(γx)c o t h ( δt)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=t a n h ( αx),h1(t)=Acoth(βt),g2(x)= tanh(γx), and h2(t)=Bcoth(δt). 4.4. Equations Whose Kernels Contain Logarithmic Functions 4.4-1. Kernels Containing Logarithmic Functions. 1. y(x)–λ⎝integraldisplay ⎝integraldisplayb aln(γx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l n ( γx)a n dh(t)=1 . 2. y(x)–λ⎝integraldisplay ⎝integraldisplayb aln(γt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=l n ( γt). 3. y(x)–λ⎝integraldisplay ⎝integraldisplayb a(lnx–l nt)y(t)dt=f(x). This is a special case of equation 4.9.3 with g(x)=l nx. 4. y(x)–λ⎝integraldisplay ⎝integraldisplayb aln(γx) ln(γt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l n ( γx)a n dh(t)=1 ln(γt). 5. y(x)–λ⎝integraldisplay ⎝integraldisplayb aln(γt) ln(γx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 ln(γx)andh(t)=l n ( γt). 4.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 335 6. y(x)–λ⎝integraldisplay ⎝integraldisplayb alnk(γx)l nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l nk(γx)a n dh(t)=l nm(µt). 4.4-2. Kernels Containing Power-Law and Logarithmic Functions. 7. y(x)–λ⎝integraldisplay ⎝integraldisplayb atklnm(γx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l nm(γx)a n dh(t)=tk. 8. y(x)–λ⎝integraldisplay ⎝integraldisplayb axklnm(γt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t)=l nm(γt). 9. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)l n (γt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=l n ( γt). 10. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)l n (γx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=l n ( γx). 11. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+(Bx +Ct)l n (γt)]y(t)dt=f(x). This is a special case of equation 4.9.9 with h(t)=l n ( γt). 12. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+(Bx +Ct)l n (γx)]y(t)dt=f(x). This is a special case of equation 4.9.11 with h(x)=l n ( γx). 13. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Atnlnm(βx)+Bxklnl(γt)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=l nm(βx),h1(t)=Atn,g2(x)=xk,a n d h2(t)=Blnl(γt). 4.5. Equations Whose Kernels Contain Trigonometric Functions 4.5-1. Kernels Containing Cosine. 1. y(x)–λ⎝integraldisplay ⎝integraldisplayb acos(βx )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s ( βx)a n dh(t)=1 . 336 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 2. y(x)–λ⎝integraldisplay ⎝integraldisplayb acos(βt )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=c o s ( βt). 3. y(x)–λ⎝integraldisplay ⎝integraldisplayb acos[β (x–t)]y(t)dt=f(x). This is a special case of equation 4.9.12 with g(x)=c o s ( βx)a n dh(t)=s i n ( βt). Solution: y(x)=f(x)+λ⎝bracketleftbig A1cos(βx )+A2sin(βx)⎝bracketrightbig , where A1andA2are the constants determined by the formulas presented in 4.9.12. 4. y(x)–λ⎝integraldisplay ⎝integraldisplayb acos[β (x+t)]y(t)dt=f(x). This is a special case of equation 4.9.13 with g(x)=c o s ( βx)a n dh(t)=s i n ( βt). Solution: y(x)=f(x)+λ⎝bracketleftbig A1cos(βx )+A2sin(βx)⎝bracketrightbig , where A1andA2are the constants determined by the formulas presented in 4.9.13. 5. y(x)–λ⎝integraldisplay ⎝integraldisplay∞ 0cos(xt )y(t)dt=0 . Characteristic values: λ=±⎝radicalbig 2/π. For the characteristic values, the integral equation has inﬁnitely many linearly independent eigenfunctions. Eigenfunctions for λ=+⎝radicalbig 2/πhave the form y+(x)=f(x)+⎝radicalbigg 2 π⎝integraldisplay∞ 0f(t)c o s ( xt)dt,( 1 ) where f=f(x) is any continuous function absolutely integrable on the interval [0, ∞). Eigenfunctions for λ=–⎝radicalbig 2/πhave the form y–(x)=f(x)–⎝radicalbigg 2 π⎝integraldisplay∞ 0f(t)c o s ( xt)dt,( 2 ) where f=f(x) is any continuous function absolutely integrable on the interval [0, ∞). In particular, from (1) and (2) with f(x)=e–axwe obtain y+(x)=e–ax+⎝radicalbigg 2 πa a2+x2forλ=+⎝radicalbigg 2 π, y–(x)=e–ax–⎝radicalbigg 2 πa a2+x2forλ=–⎝radicalbigg 2 π, where ais any positive number. Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971). 6. y(x)–λ⎝integraldisplay ⎝integraldisplay∞ 0cos(xt )y(t)dt=f(x). Solution: y(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0cos(xt )f(t)dt, where λ≠±⎝radicalbig 2/π. Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971). 4.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 337 7. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝braceleftbigg ⎝braceleftbiggn⎝summationdisplay k=1Akcos[β k(x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x), n=1 , 2 , ... This equation can be reduced to a special case of equation 4.9.20; the formula cos[ β(x–t)] = cos(βx )c o s (βt)+s i n ( βx)s i n (βt) must be used. 8. y(x)–λ⎝integraldisplay ⎝integraldisplayb acos(βx ) cos(βt )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s ( βx)a n dh(t)=1 cos(βt ). 9. y(x)–λ⎝integraldisplay ⎝integraldisplayb acos(βt ) cos(βx )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 cos(βx )andh(t)=c o s ( βt). 10. y(x)–λ⎝integraldisplay ⎝integraldisplayb acosk(βx)c o sm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o sk(βx)a n dh(t)=c o sm(µt). 11. y(x)–λ⎝integraldisplay ⎝integraldisplayb atkcosm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o sm(βx)a n dh(t)=tk. 12. y(x)–λ⎝integraldisplay ⎝integraldisplayb axkcosm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t)=c o sm(βt). 13. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)c o s (βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=c o s ( βx). 14. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)c o s (βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=c o s ( βt). 4.5-2. Kernels Containing Sine. 15. y(x)–λ⎝integraldisplay ⎝integraldisplayb asin(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i n ( βx)a n dh(t)=1 . 16. y(x)–λ⎝integraldisplay ⎝integraldisplayb asin(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=s i n ( βt). 338 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 17. y(x)–λ⎝integraldisplay ⎝integraldisplayb asin[β(x–t)]y(t)dt=f(x). This is a special case of equation 4.9.16 with g(x)=s i n ( βx)a n dh(t)=c o s ( βt). Solution: y(x)=f(x)+λ⎝bracketleftbig A1sin(βx)+A2cos(βx )⎝bracketrightbig , where A1andA2are the constants determined by the formulas presented in 4.9.16. 18. y(x)–λ⎝integraldisplay ⎝integraldisplayb asin[β(x+t)]y(t)dt=f(x). This is a special case of equation 4.9.15 with g(x)=s i n ( βx)a n dh(t)=c o s ( βt). Solution: y(x)=f(x)+λ⎝bracketleftbig A1sin(βx)+A2cos(βx )⎝bracketrightbig , where A1andA2are the constants determined by the formulas presented in 4.9.15. Example. Let us consider the case of a=0 ,b=π,β= 1 in detail. 1◦. Solution for λ≠±2/π: y(x)=f(x)+λAsinx+λBcosx, (1) where A=f1+1 2πλf 2 1–1 4π2λ2,B=1 2πλf 1+f2 1–1 4π2λ2,f1=⎝integraldisplayπ 0f(t)c o std t,f2=⎝integraldisplayπ 0f(t)s i ntd t. (2) 2◦. Characteristic values and normed eigenfunctions of the homogeneous equation for f(x)≡0 are given by the formulas λ1=–2 π,y1(x)=1 √ π(sinx–c o sx); λ2=2 π,y2(x)=1 √ π(sinx+c o sx). 3◦.I fλ=– 2/πandf1=f2(values of f1andf2can be found using formulas of Item 1◦). In this case the solution can be obtained with the help of formula (1) in which B=f1–Awhere Ais an arbitrary constant. Ifλ=2/πandf1=–f2then the solution can be found using formula (1) in which B=A–f1where Ais an arbitrary constant. 4◦.I fλ=– 2/πandf1≠f2orλ=2/πandf1≠–f2, then the equation under consideration has no solutions. 19. y(x)–λ⎝integraldisplay ⎝integraldisplay∞ 0sin(xt)y(t)dt=0 . Characteristic values: λ=±⎝radicalbig 2/π. For the characteristic values, the integral equation has inﬁnitely many linearly independent eigenfunctions. Eigenfunctions for λ=+⎝radicalbig 2/πhave the form y+(x)=f(x)+⎝radicalbigg 2 π⎝integraldisplay∞ 0f(t)s i n (xt)dt, where f=f(x) is any continuous function absolutely integrable on the interval [0, ∞). Eigenfunctions for λ=–⎝radicalbig 2/πhave the form y–(x)=f(x)–⎝radicalbigg 2 π⎝integraldisplay∞ 0f(t)s i n (xt)dt, where f=f(x) is any continuous function absolutely integrable on the interval [0, ∞). Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971). 4.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 339 20. y(x)–λ⎝integraldisplay ⎝integraldisplay∞ 0sin(xt)y(t)dt=f(x). Solution: y(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0sin(xt)f(t)dt, where λ≠±⎝radicalbig 2/π. References: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971), F. D. Gakhov and Yu. I. Cherskii (1978). 21. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝braceleftbigg ⎝braceleftbiggn⎝summationdisplay k=1Aksin[βk(x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x), n=1 , 2 , ... This equation can be reduced to a special case of equation 4.9.20; the formula sin[ β(x–t)] = sin(βx)c o s (βt)–s i n ( βt)c o s (βx) must be used. 22. y(x)–λ⎝integraldisplay ⎝integraldisplayb asin(βx) sin(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i n ( βx)a n dh(t)=1 sin(βt). 23. y(x)–λ⎝integraldisplay ⎝integraldisplayb asin(βt) sin(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 sin(βx)andh(t)=s i n( βt). 24. y(x)–λ⎝integraldisplay ⎝integraldisplayb asink(βx)s i nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i nk(βx)a n dh(t)=s i nm(µt). 25. y(x)–λ⎝integraldisplay ⎝integraldisplayb atksinm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i nm(βx)a n dh(t)=tk. 26. y(x)–λ⎝integraldisplay ⎝integraldisplayb axksinm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t)=s i nm(βt). 27. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)s i n (βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=s i n ( βt). 28. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)s i n (βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=s i n ( βx). 340 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 29. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λ|x–t|)y(t)dt=f(x). This is a special case of equation 4.9.39 with g(t)=A. 1◦. The function y=y(x) obeys the following second-order linear nonhomogeneousordinary differential equation with constant coefﬁcients: y/prime/prime xx+λ(2A+λ)y=f/prime/prime xx(x)+λ2f(x). (1) The boundary conditions for (1) have the form (see 4.9.39) sin[λ(b–a)]ϕ/prime x(b)–λcos[λ (b–a)]ϕ(b)=λϕ(a), sin[λ(b–a)]ϕ/prime x(a)+λcos[λ (b–a)]ϕ(a)=–λϕ(b),ϕ(x)=y(x)–f(x). (2) Equation (1) under the boundary conditions (2) determines the solution of the original integral equation. 2◦.F o rλ(2A+λ)=–k2< 0, the general solution of equation (1) is given by y(x)=C1cosh(kx )+C2sinh(kx)+f(x)–2Aλ k⎝integraldisplayx asinh[k(x–t)]f(t)dt,( 3 ) where C1andC2are arbitrary constants. Forλ(2A+λ)=k2> 0, the general solution of equation (1) is given by y(x)=C1cos(kx )+C2sin(kx)+f(x)–2Aλ k⎝integraldisplayx asin[k(x–t)]f(t)dt.( 4 ) Forλ=2A, the general solution of equation (1) is given by y(x)=C1+C2x+f(x)+4A2⎝integraldisplayx a(x–t)f(t)dt.( 5 ) The constants C1andC2in solutions (3)–(5) are determined by conditions (2). 30. y(x)+A⎝integraldisplay ⎝integraldisplayb atsin(λ|x–t|)y(t)dt=f(x). This is a special case of equation 4.9.39 with g(t)=At. The solution of the integral equation can be written via the Bessel functions (or modiﬁed Bessel functions) of order 1/3. 31. y(x)+A⎝integraldisplay ⎝integraldisplayb asin3(λ|x–t|)y(t)dt=f(x). Using the formula sin3β=–1 4sin 3β+3 4sinβ, we arrive at an equation of the form 4.5.32 withn=2 : y(x)+⎝integraldisplayb a⎝bracketleftbig –1 4Asin(3λ|x–t|)+3 4Asin(λ|x–t|)⎝bracketrightbig y(t)dt=f(x). 4.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 341 32. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1Aksin(λk|x–t|)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x), – ∞<a<b<∞. 1◦. Let us remove the modulus in the kth summand of the integrand: Ik(x)=⎝integraldisplayb asin(λk|x–t|)y(t)dt=⎝integraldisplayx asin[λk(x–t)]y(t)dt+⎝integraldisplayb xsin[λk(t–x)]y(t)dt.( 1 ) Differentiating (1) with respect to xtwice yields I/prime k=λk⎝integraldisplayx acos[λ k(x–t)]y(t)dt–λk⎝integraldisplayb xcos[λ k(t–x)]y(t)dt, I/prime/prime k=2λky(x)–λ2 k⎝integraldisplayx asin[λk(x–t)]y(t)dt–λ2 k⎝integraldisplayb xsin[λk(t–x)]y(t)dt,(2) where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we ﬁnd the relation between I/prime/prime kandIk: I/prime/prime k=2λky(x)–λ2 kIk,Ik=Ik(x). (3) 2◦. With the aid of (1), the integral equation can be rewritten in the form y(x)+n⎝summationdisplay k=1AkIk=f(x). (4) Differentiating (4) with respect to xtwice and taking into account (3), we ﬁnd that y/prime/prime xx(x)+σny(x)–n⎝summationdisplay k=1Akλ2 kIk=f/prime/prime xx(x), σn=2n⎝summationdisplay k=1Akλk.( 5 ) Eliminating the integral Infrom (4) and (5) yields y/prime/prime xx(x)+(σn+λ2 n)y(x)+n–1⎝summationdisplay k=1Ak(λ2 n–λ2 k)Ik=f/prime/prime xx(x)+λ2 nf(x). (6) Differentiating (6) with respect to xtwice and eliminating In–1from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear differential operator (acting on y) with constant coefﬁcients plus the sumn–2⎝summationtext k=1BkIk.I f w e successively eliminate In–2,In–3,..., with the aid of double differentiation, then we ﬁnally arrive at a linear nonhomogeneous ordinary differential equation of order 2 nwith constant coefﬁcients. 3◦. The boundary conditions for y(x) can be found by setting x=ain the integral equation and all its derivatives. (Alternatively, these conditions can be found by setting x=aandx=b in the integral equation and all its derivatives obtained by means of double differentiation.) 33. y(x)–λ⎝integraldisplay ⎝integraldisplay∞ –∞sin(x–t) x–ty(t)dt=f(x). Solution: y(x)=f(x)+λ √ 2π–πλ⎝integraldisplay∞ –∞sin(x–t) x–tf(t)dt,λ≠⎝radicalbigg 2 π. Reference: F. D. Gakhov and Yu. I. Cherskii (1978). 342 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 4.5-3. Kernels Containing Tangent. 34. y(x)–λ⎝integraldisplay ⎝integraldisplayb atan(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n ( βx)a n dh(t)=1 . 35. y(x)–λ⎝integraldisplay ⎝integraldisplayb atan(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=t a n ( βt). 36. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Atan(βx)+Btan(βt)]y(t)dt=f(x). This is a special case of equation 4.9.4 with g(x)=t a n ( βx). 37. y(x)–λ⎝integraldisplay ⎝integraldisplayb atan(βx) tan(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n ( βx)a n dh(t)=1 tan(βt). 38. y(x)–λ⎝integraldisplay ⎝integraldisplayb atan(βt) tan(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 tan(βx)andh(t)=t a n ( βt). 39. y(x)–λ⎝integraldisplay ⎝integraldisplayb atank(βx)t a nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a nk(βx)a n dh(t)=t a nm(µt). 40. y(x)–λ⎝integraldisplay ⎝integraldisplayb atktanm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a nm(βx)a n dh(t)=tk. 41. y(x)–λ⎝integraldisplay ⎝integraldisplayb axktanm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t)=t a nm(βt). 42. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)t a n (βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=t a n ( βt). 43. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)t a n (βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=t a n ( βx). 4.5. E QUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS 343 4.5-4. Kernels Containing Cotangent. 44. y(x)–λ⎝integraldisplay ⎝integraldisplayb acot(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o t ( βx)a n dh(t)=1 . 45. y(x)–λ⎝integraldisplay ⎝integraldisplayb acot(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=c o t ( βt). 46. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Acot(βx)+Bcot(βt)]y(t)dt=f(x). This is a special case of equation 4.9.4 with g(x)=c o t ( βx). 47. y(x)–λ⎝integraldisplay ⎝integraldisplayb acot(βx) cot(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o t ( βx)a n dh(t)=1 cot(βt). 48. y(x)–λ⎝integraldisplay ⎝integraldisplayb acot(βt) cot(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 cot(βx)andh(t)=c o t ( βt). 49. y(x)–λ⎝integraldisplay ⎝integraldisplayb acotk(βx)c o tm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o tk(βx)a n dh(t)=c o tm(µt). 50. y(x)–λ⎝integraldisplay ⎝integraldisplayb atkcotm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o tm(βx)a n dh(t)=tk. 51. y(x)–λ⎝integraldisplay ⎝integraldisplayb axkcotm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t)=c o tm(βt). 52. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)c o t (βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=c o t ( βt). 53. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)c o t (βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=c o t ( βx). 344 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 4.5-5. Kernels Containing Combinations of Trigonometric Functions. 54. y(x)–λ⎝integraldisplay ⎝integraldisplayb acosk(βx)s i nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o sk(βx)a n dh(t)=s i nm(µt). 55. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Asin(αx)c o s (βt)+Bsin(γx)c o s (δt)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=s i n ( αx),h1(t)=Acos(βt ),g2(x)=s i n ( γx), andh2(t)=Bcos(δt ). 56. y(x)–λ⎝integraldisplay ⎝integraldisplayb atank(γx)c o tm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a nk(γx)a n dh(t)=c o tm(µt). 57. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Atan(αx)c o t (βt)+Btan(γx)c o t (δt)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=t an ( αx),h1(t)=Acot(βt),g2(x)=t an ( γx), andh2(t)=Bcot(δt). 4.5-6. Singular Equation. 58. Ay(x)–B 2π⎝integraldisplay ⎝integraldisplay2π 0cot⎝parenleftBig ⎝parenleftBigt–x 2⎝parenrightBig ⎝parenrightBig y(t)dt=f(x), 0 ≤x≤2π. Here the integral is understood in the sense of the Cauchy principal value. Without loss of generality we may assume that A2+B2=1 . Solution: y(x)=Af(x)+B 2π⎝integraldisplay2π 0cot⎝parenleftBigt–x 2⎝parenrightBig f(t)dt+B2 2πA⎝integraldisplay2π 0f(t)dt. Reference: I. K. Lifanov (1996). 4.6. Equations Whose Kernels Contain Inverse Trigonometric Functions 4.6-1. Kernels Containing Arccosine. 1. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarccos( βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arccos( βx)a n dh(t)=1 . 2. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarccos( βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t) = arccos( βt). 4.6. E QUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS 345 3. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarccos( βx) arccos( βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arccos( βx)a n dh(t)=1 arccos( βt). 4. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarccos( βt) arccos( βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 arccos( βx)andh(t) = arccos( βt). 5. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarccosk(βx) arccosm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arccosk(βx)a n dh(t) = arccosm(µt). 6. y(x)–λ⎝integraldisplay ⎝integraldisplayb atkarccosm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arccosm(βx)a n dh(t)=tk. 7. y(x)–λ⎝integraldisplay ⎝integraldisplayb axkarccosm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t) = arccosm(βt). 8. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t) arccos( βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x) = arccos( βx). 9. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t) arccos( βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t) = arccos( βt). 4.6-2. Kernels Containing Arcsine. 10. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarcsin( βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arcsin( βx)a n dh(t)=1 . 11. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarcsin( βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t) = arcsin( βt). 12. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarcsin( βx) arcsin( βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arcsin( βx)a n dh(t)=1 arcsin( βt). 346 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 13. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarcsin( βt) arcsin( βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 arcsin( βx)andh(t) = arcsin ( βt). 14. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarcsink(βx)a r c s i nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arcsink(βx)a n dh(t) = arcsinm(µt). 15. y(x)–λ⎝integraldisplay ⎝integraldisplayb atkarcsinm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arcsinm(βx)a n dh(t)=tk. 16. y(x)–λ⎝integraldisplay ⎝integraldisplayb axkarcsinm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t) = arcsinm(βt). 17. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)a r c s i n ( βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t) = arcsin( βt). 18. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)a r c s i n ( βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x) = arcsin( βx). 4.6-3. Kernels Containing Arctangent. 19. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarctan( βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arctan( βx)a n dh(t)=1 . 20. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarctan( βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t) = arctan( βt). 21. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Aarctan( βx)+Barctan( βt)]y(t)dt=f(x). This is a special case of equation 4.9.4 with g(x) = arctan( βx). 22. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarctan( βx) arctan( βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arctan( βx)a n dh(t)=1 arctan( βt). 4.6. E QUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS 347 23. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarctan( βt) arctan( βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 arctan( βx)andh(t) = arctan( βt). 24. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarctank(βx)a r c t a nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arctank(βx)a n dh(t) = arctanm(µt). 25. y(x)–λ⎝integraldisplay ⎝integraldisplayb atkarctanm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arctanm(βx)a n dh(t)=tk. 26. y(x)–λ⎝integraldisplay ⎝integraldisplayb axkarctanm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t) = arctanm(βt). 27. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)a r c t a n ( βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t) = arctan( βt). 28. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)a r c t a n ( βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x) = arctan( βx). 4.6-4. Kernels Containing Arccotangent. 29. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarccot( βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arccot( βx)a n dh(t)=1 . 30. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarccot( βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t) = arccot( βt). 31. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Aarccot( βx)+Barccot( βt)]y(t)dt=f(x). This is a special case of equation 4.9.4 with g(x) = arccot( βx). 32. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarccot( βx) arccot( βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arccot( βx)a n dh(t)=1 arccot(βt ). 348 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 33. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarccot( βt) arccot( βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1 arccot(βx )andh(t) = arccot( βt). 34. y(x)–λ⎝integraldisplay ⎝integraldisplayb aarccotk(βx) arccotm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arccotk(βx)a n dh(t) = arccotm(µt). 35. y(x)–λ⎝integraldisplay ⎝integraldisplayb atkarccotm(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x) = arccotm(βx)a n dh(t)=tk. 36. y(x)–λ⎝integraldisplay ⎝integraldisplayb axkarccotm(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=xkandh(t) = arccotm(βt). 37. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t) arccot( βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t) = arccot( βt). 38. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t) arccot( βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x) = arccot( βx). 4.7. Equations Whose Kernels Contain Combinations of Elementary Functions 4.7-1. Kernels Containing Exponential and Hyperbolic Functions. 1. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµ(x–t)cosh[β (x–t)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=eµxcosh(βx ),h1(t)=e–µtcosh(βt ), g2(x)=eµxsinh(βx), and h2(t)=–e–µtsinh(βt). 2. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµ(x–t)sinh[β(x–t)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=eµxsinh(βx),h1(t)=e–µtcosh(βt ), g2(x)=eµxcosh(βx ), and h2(t)=–e–µtsinh(βt). 3. y(x)–λ⎝integraldisplay ⎝integraldisplayb ateµ(x–t)sinh[β(x–t)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=eµxsinh(βx),h1(t)=te–µtcosh(βt ), g2(x)=eµxcosh(βx ), and h2(t)=–te–µtsinh(βt). 4.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 349 4.7-2. Kernels Containing Exponential and Logarithmic Functions. 4. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµtln(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l n ( βx)a n dh(t)=eµt. 5. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµxln(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=eµxandh(t)=l n ( βt). 6. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµ(x–t)ln(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=eµxln(βx)a n dh(t)=e–µt. 7. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµ(x–t)ln(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=eµxandh(t)=e–µtln(βt). 8. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµ(x–t)(lnx–l nt)y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=eµxlnx,h1(t)=e–µt,g2(x)=eµx,a n d h2(t)=–e–µtlnt. 9. y(x)+b2–a2 2a⎝integraldisplay ⎝integraldisplay∞ 01 texp⎝parenleftBig ⎝parenleftBig –a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglelnx t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝parenrightBig ⎝parenrightBig y(t)dt=f(x). Solution with a>0 ,b>0 ,a n d x>0 : y(x)=f(x)+a2–b2 2b⎝integraldisplay∞ 01 texp⎝parenleftBig –b⎝vextendsingle⎝vextendsingle⎝vextendsinglelnx t⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝parenrightBig f(t)dt. Reference: F. D. Gakhov and Yu. I. Cherskii (1978). 4.7-3. Kernels Containing Exponential and Trigonometric Functions. 10. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµtcos(βx )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s ( βx)a n dh(t)=eµt. 11. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµxcos(βt )y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=eµxandh(t)=c o s ( βt). 12. y(x)–λ⎝integraldisplay ⎝integraldisplay∞ 0eµ(x–t)cos(xt )y(t)dt=f(x). Solution: y(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0eµ(x–t)cos(xt )f(t)dt,λ≠±⎝radicalbig 2/π. 350 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 13. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµ(x–t)cos[β (x–t)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=eµxcos(βx ),h1(t)=e–µtcos(βt ), g2(x)=eµxsin(βx), and h2(t)=e–µtsin(βt). 14. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµtsin(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i n ( βx)a n dh(t)=eµt. 15. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµxsin(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=eµxandh(t)=s i n ( βt). 16. y(x)–λ⎝integraldisplay ⎝integraldisplay∞ 0eµ(x–t)sin(xt)y(t)dt=f(x). Solution: y(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0eµ(x–t)sin(xt)f(t)dt,λ≠±⎝radicalbig 2/π. 17. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµ(x–t)sin[β(x–t)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=eµxsin(βx),h1(t)=e–µtcos(βt ), g2(x)=eµxcos(βx ), and h2(t)=–e–µtsin(βt). 18. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµ(x–t)⎝braceleftbigg ⎝braceleftbiggn⎝summationdisplay k=1Aksin[βk(x–t)]⎝bracerightbigg ⎝bracerightbigg y(t)dt=f(x), n=1 , 2 , ... This is a special case of equation 4.9.20. 19. y(x)–λ⎝integraldisplay ⎝integraldisplayb ateµ(x–t)sin[β(x–t)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=eµxsin(βx),h1(t)=te–µtcos(βt ), g2(x)=eµxcos(βx ), and h2(t)=–te–µtsin(βt). 20. y(x)–λ⎝integraldisplay ⎝integraldisplayb axeµ(x–t)sin[β(x–t)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=xeµxsin(βx),h1(t)=e–µtcos(βt ), g2(x)=xeµxcos(βx ), and h2(t)=–e–µtsin(βt). 21. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµttan(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n ( βx)a n dh(t)=eµt. 22. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµxtan(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=eµxandh(t)=t a n ( βt). 4.7. E QUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS 351 23. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµ(x–t)[tan(βx )–t a n ( βt)]y(t)dt=f(x). This is a special case of equation 4.9.18 with g1(x)=eµxtan(βx),h1(t)=e–µt,g2(x)=eµx, andh2(t)=–e–µttan(βt). 24. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµtcot(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o t ( βx)a n dh(t)=eµt. 25. y(x)–λ⎝integraldisplay ⎝integraldisplayb aeµxcot(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=eµxandh(t)=c o t ( βt). 4.7-4. Kernels Containing Hyperbolic and Logarithmic Functions. 26. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoshk(βx)l nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s hk(βx)a n dh(t)=l nm(µt). 27. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoshk(βt)l nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l nm(µx)a n dh(t)=c o s hk(βt). 28. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinhk(βx)l nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i n hk(βx)a n dh(t)=l nm(µt). 29. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinhk(βt)l nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l nm(µx)a n dh(t)=s i n hk(βt). 30. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanhk(βx)l nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n hk(βx)a n dh(t)=l nm(µt). 31. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanhk(βt)l nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l nm(µx)a n dh(t)=t a n hk(βt). 32. y(x)–λ⎝integraldisplay ⎝integraldisplayb acothk(βx)l nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o t hk(βx)a n dh(t)=l nm(µt). 33. y(x)–λ⎝integraldisplay ⎝integraldisplayb acothk(βt)l nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l nm(µx)a n dh(t)=c o t hk(βt). 352 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 4.7-5. Kernels Containing Hyperbolic and Trigonometric Functions. 34. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoshk(βx)c o sm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s hk(βx)a n dh(t)=c o sm(µt). 35. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoshk(βt)c o sm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o sm(µx)a n dh(t)=c o s hk(βt). 36. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoshk(βx)s i nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o s hk(βx)a n dh(t)=s i nm(µt). 37. y(x)–λ⎝integraldisplay ⎝integraldisplayb acoshk(βt)s i nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i nm(µx)a n dh(t)=c o s hk(βt). 38. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinhk(βx)c o sm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i n hk(βx)a n dh(t)=c o sm(µt). 39. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinhk(βt)c o sm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o sm(µx)a n dh(t)=s i n hk(βt). 40. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinhk(βx)s i nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i n hk(βx)a n dh(t)=s i nm(µt). 41. y(x)–λ⎝integraldisplay ⎝integraldisplayb asinhk(βt)s i nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i nm(µx)a n dh(t)=s i n hk(βt). 42. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanhk(βx)c o sm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n hk(βx)a n dh(t)=c o sm(µt). 43. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanhk(βt)c o sm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o sm(µx)a n dh(t)=t a n hk(βt). 44. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanhk(βx)s i nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a n hk(βx)a n dh(t)=s i nm(µt). 4.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 353 45. y(x)–λ⎝integraldisplay ⎝integraldisplayb atanhk(βt)s i nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i nm(µx)a n dh(t)=t a n hk(βt). 4.7-6. Kernels Containing Logarithmic and Trigonometric Functions. 46. y(x)–λ⎝integraldisplay ⎝integraldisplayb acosk(βx)l nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o sk(βx)a n dh(t)=l nm(µt). 47. y(x)–λ⎝integraldisplay ⎝integraldisplayb acosk(βt)l nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l nm(µx)a n dh(t)=c o sk(βt). 48. y(x)–λ⎝integraldisplay ⎝integraldisplayb asink(βx)l nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=s i nk(βx)a n dh(t)=l nm(µt). 49. y(x)–λ⎝integraldisplay ⎝integraldisplayb asink(βt)l nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l nm(µx)a n dh(t)=s i nk(βt). 50. y(x)–λ⎝integraldisplay ⎝integraldisplayb atank(βx)l nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=t a nk(βx)a n dh(t)=l nm(µt). 51. y(x)–λ⎝integraldisplay ⎝integraldisplayb atank(βt)l nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l nm(µx)a n dh(t)=t a nk(βt). 52. y(x)–λ⎝integraldisplay ⎝integraldisplayb acotk(βx)l nm(µt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=c o tk(βx)a n dh(t)=l nm(µt). 53. y(x)–λ⎝integraldisplay ⎝integraldisplayb acotk(βt)l nm(µx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=l nm(µx)a n dh(t)=c o tk(βt). 4.8. Equations Whose Kernels Contain Special Functions 4.8-1. Kernels Containing Bessel Functions. 1. y(x)–λ⎝integraldisplay ⎝integraldisplayb aJν(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=Jν(βx)a n dh(t)=1 . 354 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 2. y(x)–λ⎝integraldisplay ⎝integraldisplayb aJν(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=Jν(βt). 3. y(x)+λ⎝integraldisplay ⎝integraldisplay∞ 0tJν(xt)y(t)dt=0 , ν> –1. Characteristic values: λ=±1. For the characteristic values, the integral equation has inﬁnitely many linearly independent eigenfunctions. Eigenfunctions for λ= +1 have the form y+(x)=f(x)–⎝integraldisplay∞ 0tJν(xt)f(t)dt, where f=f(x) is an arbitrary function. Eigenfunctions for λ= –1 have the form y–(x)=f(x)+⎝integraldisplay∞ 0tJν(xt)f(t)dt, where f=f(x) is an arbitrary function. 4. y(x)+λ⎝integraldisplay ⎝integraldisplay∞ 0tJν(xt)y(t)dt=f(x), ν> –1. Solution: y(x)=f(x) 1–λ2–λ 1–λ2⎝integraldisplay∞ 0tJν(xt)f(t)dt,λ≠±1. 5. y(x)+λ⎝integraldisplay ⎝integraldisplay∞ 0Jν⎝parenleftbig⎝parenleftbig 2√ xt⎝parenrightbig⎝parenrightbig y(t)dt=f(x). By setting x=1 2z2,t=1 2τ2,y(x)=Y(z), and f(x)=F(z), we arrive at an equation of the form 4.8.4: Y(z)+λ⎝integraldisplay∞ 0τJν(zτ)Y(τ)dτ=F(z). 6. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)Jν(βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=Jν(βt). 7. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)Jν(βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=Jν(βx). 8. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AJµ(αx)+BJν(βt)]y(t)dt=f(x). This is a special case of equation 4.9.5 with g(x)=AJµ(αx)a n dh(t)=BJν(βt). 9. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AJµ(x)Jν(t)+BJν(x)Jµ(t)]y(t)dt=f(x). This is a special case of equation 4.9.17 with g(x)=Jµ(x)a n dh(t)=Jν(t). 4.8. E QUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 355 10. y(x)–λ⎝integraldisplay ⎝integraldisplayb aYν(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=Yν(βx)a n dh(t)=1 . 11. y(x)–λ⎝integraldisplay ⎝integraldisplayb aYν(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=Yν(βt). 12. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)Yν(βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=Yν(βt). 13. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)Yν(βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=Yν(βx). 14. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AYµ(αx)+BYν(βt)]y(t)dt=f(x). This is a special case of equation 4.9.5 with g(x)=AYµ(αx)a n dh(t)=BYν(βt). 15. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AYµ(x)Yµ(t)+BYν(x)Yν(t)]y(t)dt=f(x). This is a special case of equation 4.9.14 with g(x)=Yµ(x)a n dh(t)=Yν(t). 16. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AYµ(x)Yν(t)+BYν(x)Yµ(t)]y(t)dt=f(x). This is a special case of equation 4.9.17 with g(x)=Yµ(x)a n dh(t)=Yν(t). 4.8-2. Kernels Containing Modiﬁed Bessel Functions. 17. y(x)–λ⎝integraldisplay ⎝integraldisplayb aIν(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=Iν(βx)a n dh(t)=1 . 18. y(x)–λ⎝integraldisplay ⎝integraldisplayb aIν(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=Iν(βt). 19. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)Iν(βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=Iν(βt). 20. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)Iν(βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=Iν(βx). 356 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 21. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AIµ(αx)+BIν(βt)]y(t)dt=f(x). This is a special case of equation 4.9.5 with g(x)=AIµ(αx)a n dh(t)=BIν(βt). 22. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AIµ(x)Iµ(t)+BIν(x)Iν(t)]y(t)dt=f(x). This is a special case of equation 4.9.14 with g(x)=Iµ(x)a n dh(t)=Iν(t). 23. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AIµ(x)Iν(t)+BIν(x)Iµ(t)]y(t)dt=f(x). This is a special case of equation 4.9.17 with g(x)=Iµ(x)a n dh(t)=Iν(t). 24. y(x)–λ⎝integraldisplay ⎝integraldisplayb aKν(βx)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=Kν(βx)a n dh(t)=1 . 25. y(x)–λ⎝integraldisplay ⎝integraldisplayb aKν(βt)y(t)dt=f(x). This is a special case of equation 4.9.1 with g(x)=1a n d h(t)=Kν(βt). 26. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)Kν(βt)]y(t)dt=f(x). This is a special case of equation 4.9.8 with h(t)=Kν(βt). 27. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)Kν(βx)]y(t)dt=f(x). This is a special case of equation 4.9.10 with h(x)=Kν(βx). 28. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AK µ(αx)+BK ν(βt)]y(t)dt=f(x). This is a special case of equation 4.9.5 with g(x)=AKµ(αx)a n dh(t)=BKν(βt). 29. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AK µ(x)Kµ(t)+BK ν(x)Kν(t)]y(t)dt=f(x). This is a special case of equation 4.9.14 with g(x)=Kµ(x)a n dh(t)=Kν(t). 30. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[AK µ(x)Kν(t)+BK ν(x)Kµ(t)]y(t)dt=f(x). This is a special case of equation 4.9.17 with g(x)=Kµ(x)a n dh(t)=Kν(t). 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 357 4.9. Equations Whose Kernels Contain Arbitrary Functions 4.9-1. Equations with Degenerate Kernel: K(x,t)=g1(x)h1(t)+···+gn(x)hn(t). 1. y(x)–λ⎝integraldisplay ⎝integraldisplayb ag(x)h(t)y(t)dt=f(x). 1◦. Assume that λ≠⎝parenleftBig⎝integraldisplayb ag(t)h(t)dt⎝parenrightBig–1 . Solution: y(x)=f(x)+λkg(x), where k=⎝parenleftbigg 1–λ⎝integraldisplayb ag(t)h(t)dt⎝parenrightbigg–1⎝integraldisplayb ah(t)f(t)dt. 2◦. Assume that λ=⎝parenleftBig⎝integraldisplayb ag(t)h(t)dt⎝parenrightBig–1 . For⎝integraldisplayb ah(t)f(t)dt= 0, the solution has the form y=f(x)+Cg(x), where Cis an arbitrary constant. For⎝integraldisplayb ah(t)f(t)dt≠0, there is no solution. The limits of integration may take the values a=–∞and/or b=∞, provided that the corresponding improper integral converges. 2. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[g(x)+g(t)]y(t)dt=f(x). The characteristic values of the equation: λ1=1 g1+√ (b–a)g2,λ2=1 g1–√ (b–a)g2, where g1=⎝integraldisplayb ag(x)dx,g2=⎝integraldisplayb ag2(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2], where the constants A1andA2are given by A1=f1–λ[f1g1–(b–a)f2] [g2 1–(b–a)g2]λ2–2g1λ+1,A2=f2–λ(f2g1–f1g2) [g2 1–(b–a)g2]λ2–2g1λ+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb af(x)g(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+⎝radicalbigg g2 b–a, where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 358 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 3◦. Solution with λ=λ2≠λ1andf1=f2=0 : y(x)=f(x)+Cy2(x),y2(x)=g(x)–⎝radicalbigg g2 b–a, where Cis an arbitrary constant and y2(x) is an eigenfunction of the equation corresponding to the characteristic value λ2. 4◦. The equation has no multiple characteristic values. 3. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[g(x)–g(t)]y(t)dt=f(x). The characteristic values of the equation: λ1=1 ⎝radicalbig g2 1–(b–a)g2,λ2=–1 ⎝radicalbig g2 1–(b–a)g2, where g1=⎝integraldisplayb ag(x)dx,g2=⎝integraldisplayb ag2(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2], where the constants A1andA2are given by A1=f1+λ[f1g1–(b–a)f2] [(b–a)g2–g2 1]λ2+1,A2=–f2+λ(f2g1–f1g2) [(b–a)g2–g2 1]λ2+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb af(x)g(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+1–λ 1g1 λ1(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. The equation has no multiple characteristic values. 4. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Ag(x)+Bg(t)]y(t)dt=f(x). The characteristic values of the equation: λ1,2=(A+B)g1±⎝radicalbig (A–B)2g2 1+4AB(b–a)g2 2AB[g2 1–(b–a)g2], 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 359 where g1=⎝integraldisplayb ag(x)dx,g2=⎝integraldisplayb ag2(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2], where the constants A1andA2are given by A1=Af1–λAB [f1g1–(b–a)f2] AB[g2 1–(b–a)g2]λ2–(A+B)g1λ+1,A2=Bf2–λAB (f2g1–f1g2) AB[g2 1–(b–a)g2]λ2–(A+B)g1λ+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb af(x)g(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+1–λ 1Ag1 λ1A(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A≠±B, where the characteristic valueλ∗=2 (A+B)g1is double: y(x)=f(x)+Cy∗(x), y∗(x)=g(x)–(A–B)g1 2A(b–a). HereCis an arbitrary constant and y∗(x) is an eigenfunction of the equation corresponding toλ∗. The equation has no multiple characteristic values if A=±B. 5. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[g(x)+h(t)]y(t)dt=f(x). The characteristic values of the equation: λ1,2=s1+s3±⎝radicalbig (s1–s3)2+4 (b–a)s2 2[s1s3–(b–a)s2], where s1=⎝integraldisplayb ag(x)dx,s2=⎝integraldisplayb ag(x)h(x)dx,s3=⎝integraldisplayb ah(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2], where the constants A1andA2are given by A1=f1–λ[f1s3–(b–a)f2] [s1s3–(b–a)s2]λ2–(s1+s3)λ+1,A2=f2–λ(f2s1–f1s2) [s1s3–(b–a)s2]λ2–(s1+s3)λ+1, f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb af(x)h(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+1–λ 1s1 λ1(b–a), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 360 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that s1≠±s3, where the characteristic valueλ∗=2 s1+s3is double: y(x)=f(x)+Cy∗(x), y∗(x)=g(x)–s1–s3 2(b–a). HereCis an arbitrary constant and y∗(x) is an eigenfunction of the equation corresponding toλ∗. The equation has no multiple characteristic values if s1=±s3. 6. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Ag(x)+Bg(t)]h(t)y(t)dt=f(x). The characteristic values of the equation: λ1,2=(A+B)s1±⎝radicalbig (A–B)2s2 1+4ABs0s2 2AB(s2 1–s0s2), where s0=⎝integraldisplayb ah(x)dx,s1=⎝integraldisplayb ag(x)h(x)dx,s2=⎝integraldisplayb ag2(x)h(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2], where the constants A1andA2are given by A1=Af1–ABλ (f1s1–f2s0) AB(s2 1–s0s2)λ2–(A+B)s1λ+1,A2=Bf2–ABλ (f2s1–f1s2) AB(s2 1–s0s2)λ2–(A+B)s1λ+1, f1=⎝integraldisplayb af(x)h(x)dx,f2=⎝integraldisplayb af(x)g(x)h(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+1–λ 1As1 λ1As0, where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A≠±B, where the characteristic valueλ∗=2 (A+B)s1is double: y(x)=f(x)+Cy∗(x), where Cis an arbitrary constant and y∗(x)=g(x)–(A–B)s1 2As0 is an eigenfunction of the equation corresponding to λ∗. The equation has no multiple characteristic values if A=±B. 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 361 7. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Ag(x)+Bg(t)+C]h(t)y(t)dt=f(x). The characteristic values of the equation: λ1,2=(A+B)s1+Cs0±⎝radicalBig (A–B)2s2 1+2 (A+B)Cs1s0+C2s2 0+4ABs0s2 2AB(s2 1–s0s2), where s0=⎝integraldisplayb ah(x)dx,s1=⎝integraldisplayb ag(x)h(x)dx,s2=⎝integraldisplayb ag2(x)h(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2], where the constants A1andA2are given by A1=Af1–ABλ (f1s1–f2s0) AB(s2 1–s0s2)λ2–[ (A+B)s1+Cs0]λ+1, A2=C1f1+Bf2–ABλ (f2s1–f1s2) AB(s2 1–s0s2)λ2–[ (A+B)s1+Cs0]λ+1, f1=⎝integraldisplayb af(x)h(x)dx,f2=⎝integraldisplayb af(x)g(x)h(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+⎝tildewideCy1(x), y1(x)=g(x)+1–λ 1As1 λ1As0, where⎝tildewideCis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that ( A±B)s1±Cs0=≠0, where the characteristic value λ∗=2 (A+B)s1+Cs0is double: y(x)=f(x)+⎝tildewideCy∗(x), where⎝tildewideCis an arbitrary constant and y∗(x)=g(x)–(A–B)s1–Cs0 2As0 is an eigenfunction of the equation corresponding to λ∗. The equation has no multiple characteristic values if ( A±B)s1±Cs0=0 . 362 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 8. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)h(t)]y(t)dt=f(x). The characteristic values of the equation: λ1,2=A(b–a)±⎝radicalbig [A(b–a)–2Bh 1]2+2Bh 0[A(b2–a2)–2Bh 2] B⎝braceleftbig A(b–a)[2h1–(b+a)h0]–2B(h2 1–h0h2)⎝bracerightbig , where h0=⎝integraldisplayb ah(x)dx,h1=⎝integraldisplayb axh(x)dx,h2=⎝integraldisplayb ax2h(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1+A2x), where the constants A1andA2are given by A1=f1–λ⎝bracketleftbig B(f1h1+f2h2)–1 2Af2(b2–a2)⎝bracketrightbig B⎝braceleftbig A(b–a)⎝bracketleftbig h1–1 2(b+a)h0⎝bracketrightbig –B(h2 1–h0h2)⎝bracerightbig λ2+A(b–a)λ+1, A2=f2–λ[A(b–a)f2–B(f1h0+f2h1)] B⎝braceleftbig A(b–a)⎝bracketleftbig h1–1 2(b+a)h0⎝bracketrightbig –B(h2 1–h0h2)⎝bracerightbig λ2+A(b–a)λ+1, f1=A⎝integraldisplayb af(x)dx–B⎝integraldisplayb axf(x)h(x)dx,f2=B⎝integraldisplayb af(x)h(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=1+2–2λ1[A(b–a)–Bh 1] λ1[A(b2–a2)–2Bh 2]x, where Cis an arbitrary constant, and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A≠0o r2Bh 1–A(b–a)≠0, where the characteristic value λ∗=2 A(b–a)is double: y(x)=f(x)+Cy∗(x), where Cis an arbitrary constant, and y∗(x)=1–A(b–a)–2Bh 1 A(b2–a2)–2Bh 2x is an eigenfunction of the equation corresponding to λ∗. The equation has no multiple characteristic values if A=0o r2 Bh 1–A(b–a)=0 . 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 363 9. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+(Bx +Ct)h(t)]y(t)dt=f(x). The characteristic values of the equation: λ1,2=A(b–a)+(C+B)h1±√ D B⎝braceleftbig A(b–a)[2h1–(b+a)h0]+2C(h2 1–h0h2)⎝bracerightbig, D=[A(b–a)+(C–B)h1]2+2Bh 0[A(b2–a2)+2Ch 2], where h0=⎝integraldisplayb ah(x)dx,h1=⎝integraldisplayb axh(x)dx,h2=⎝integraldisplayb ax2h(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ(A1+A2x), where the constants A1andA2are given by A1=∆–1⎝braceleftbig f1–λ⎝bracketleftbig Bf1h1–Cf2h2–1 2A(b2–a2)f2⎝bracketrightbig⎝bracerightbig , A2=∆–1⎝braceleftbig f2–λ⎝bracketleftbig A(b–a)f2–Bf1h0+Cf2h1⎝bracketrightbig⎝bracerightbig , ∆=B⎝braceleftbig A(b–a)⎝bracketleftbig h1–1 2(b+a)h0⎝bracketrightbig +C(h2 1–h0h2)⎝bracerightbig λ2+[A(b–a)+(B+C)h1]λ+1 , f1=A⎝integraldisplayb af(x)dx+C⎝integraldisplayb axf(x)h(x)dx,f2=B⎝integraldisplayb af(x)h(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+⎝tildewideCy1(x), y1(x)=1+2–2λ1[A(b–a)+Ch 1] λ1[A(b2–a2)+2Ch 2]x, where⎝tildewideCis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that ±A(b–a)+(B±C)h1≠0, where the characteristic value λ∗=2 A(b–a)+(B+C)h1is double: y(x)=f(x)+⎝tildewideCy∗(x), where⎝tildewideCis an arbitrary constant and y∗(x)=1–A(b–a)+(C–B)h1 A(b2–a2)+2Ch 2x is an eigenfunction of the equation corresponding to λ∗. The equation has no multiple characteristic values if ±A(b–a)+(B±C)h1=0 . 364 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 10. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+B(x–t)h(x)]y(t)dt=f(x). The characteristic values of the equation: λ1,2=A(b–a)±⎝radicalbig [A(b–a)+2Bh 1]2–4Bh 0[A(b–a)+Bh 2] 2B{h0[A(b–a)+Bh 2]–h1[A(b–a)+Bh 1]}, where h0=⎝integraldisplayb ah(x)dx,h1=⎝integraldisplayb axh(x)dx,h2=⎝integraldisplayb ax2h(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ⎝bracketleftbig AE 1+(BE 1x+E2)h(x)⎝bracketrightbig , where the constants E1andE2are given by E1=∆–1⎝bracketleftbig f1+λB(f1h1–f2h0)⎝bracketrightbig , E2=∆–1⎝braceleftbig f2–λf2⎝bracketleftbig A(b–a)+Bh 1⎝bracketrightbig –λf1⎝bracketleftbig A(b–a)+Bh 2⎝bracketrightbig⎝bracerightbig , ∆=B{h0[A(b–a)+Bh 2]–h1[A(b–a)+Bh 1]}λ2–A(b–a)λ+1 , f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb axf(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=A+Bxh (x)+1–λ 1[A(b–a)+Bh 1] λ1h0h(x), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A≠0o rA(b–a)+4Bh 1≠0, where the characteristic value λ∗=2 A(b–a)is double: y(x)=f(x)+Cy∗(x), where Cis an arbitrary constant and y∗(x)=A+Bxh (x)–A(b–a)+2Bh 1 2h0h(x) is an eigenfunction of the equation corresponding to λ∗. The equation has no multiple characteristic values if A=0o r A(b–a)+4Bh 1=0 . 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 365 11. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[A+(Bx +Ct)h(x)]y(t)dt=f(x). The characteristic values of the equation: λ1,2=A(b–a)+(B+C)h1±√ D 2C{h1[A(b–a)+Bh 1]–h0[A(b–a)+Bh 2]}, D=[A(b–a)+(B–C)h1]2+4Ch 0[A(b–a)+Bh 2], where h0=⎝integraldisplayb ah(x)dx,h1=⎝integraldisplayb axh(x)dx,h2=⎝integraldisplayb ax2h(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ⎝bracketleftbig AE 1+(BE 1x+E2)h(x)⎝bracketrightbig , where the constants E1andE2are given by E1=∆–1[f1–λC(f1h1–f2h0)], E2=C∆–1⎝braceleftbig f2–λf2⎝bracketleftbig A(b–a)+Bh 1⎝bracketrightbig –λf1⎝bracketleftbig A(b–a)+Bh 2⎝bracketrightbig⎝bracerightbig , ∆=C⎝braceleftbig h1[A(b–a)+Bh 1]–h0⎝bracketleftbig A(b–a)+Bh 2⎝bracketrightbig⎝bracerightbig λ2–[A(b–a)+(B+C)h1]λ+1 , f1=⎝integraldisplayb af(x)dx,f2=⎝integraldisplayb axf(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+⎝tildewideCy1(x), y1(x)=A+Bxh (x)+1–λ1[A(b–a)+Bh 1] λ1h0h(x), where⎝tildewideCis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A(b–a)+(B±C)h1≠0, where the characteristic value λ∗=2 A(b–a)+(B+C)h1is double: y(x)=f(x)+⎝tildewideCy∗(x), where⎝tildewideCis an arbitrary constant and y∗(x)=A+Bxh (x)–A(b–a)+(B–C)h1 2h0h(x) is an eigenfunction of the equation corresponding to λ∗. The equation has no multiple characteristic values if A(b–a)+(B±C)h1=0 . 366 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 12. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[g(x)g(t)+h(x)h(t)]y(t)dt=f(x). The characteristic values of the equation: λ1=s1+s3+⎝radicalbig (s1–s3)2+4s2 2 2(s1s3–s2 2),λ2=s1+s3–⎝radicalbig (s1–s3)2+4s2 2 2(s1s3–s2 2), where s1=⎝integraldisplayb ag2(x)dx,s2=⎝integraldisplayb ag(x)h(x)dx,s3=⎝integraldisplayb ah2(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2h(x)], where the constants A1andA2are given by A1=f1–λ(f1s3–f2s2) (s1s3–s2 2)λ2–(s1+s3)λ+1,A2=f2–λ(f2s1–f1s2) (s1s3–s2 2)λ2–(s1+s3)λ+1, f1=⎝integraldisplayb af(x)g(x)dx,f2=⎝integraldisplayb af(x)h(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+( 1– λ1s1)h(x)/(λ1s2), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that s1≠±s3, where the characteristic valueλ∗=1/s1is double: y(x)=f(x)+⎝tildewideC1g(x)+⎝tildewideC2h(x), where⎝tildewideC1and⎝tildewideC2are arbitrary constants. The equation has no multiple characteristic values if s1=±s3. 13. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[g(x)g(t)–h(x)h(t)]y(t)dt=f(x). The characteristic values of the equation: λ1=s1–s3+⎝radicalbig (s1+s3)2–4s2 2 2(s2 2–s1s3),λ2=s1–s3–⎝radicalbig (s1+s3)2–4s2 2 2(s2 2–s1s3), where s1=⎝integraldisplayb ag2(x)dx,s2=⎝integraldisplayb ag(x)h(x)dx,s3=⎝integraldisplayb ah2(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2h(x)], 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 367 where the constants A1andA2are given by A1=f1+λ(f1s3–f2s2) (s2 2–s1s3)λ2–(s1–s3)λ+1,A2=–f2+λ(f2s1–f1s2) (s2 2–s1s3)λ2–(s1–s3)λ+1, f1=⎝integraldisplayb af(x)g(x)dx,f2=⎝integraldisplayb af(x)h(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+1–λ 1s1 λ1s2h(x), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that s1≠±s3, where the characteristic valueλ∗=2 s1–s3is double: y(x)=f(x)+Cy∗(x), y∗(x)=g(x)–s1+s3 2s2h(x), where Cis an arbitrary constant and y∗(x) is an eigenfunction of the equation corresponding toλ∗. The equation has no multiple characteristic values if s1=±s3. 14. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Ag(x)g(t)+Bh (x)h(t)]y(t)dt=f(x). The characteristic values of the equation: λ1,2=As1+Bs3±⎝radicalbig (As1–Bs3)2+4ABs2 2 2AB(s1s3–s2 2), where s1=⎝integraldisplayb ag2(x)dx,s2=⎝integraldisplayb ag(x)h(x)dx,s3=⎝integraldisplayb ah2(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2h(x)], where the constants A1andA2are given by A1=Af1–λAB (f1s3–f2s2) AB(s1s3–s2 2)λ2–(As1+Bs3)λ+1,A2=Bf2–λAB (f2s1–f1s2) AB(s1s3–s2 2)λ2–(As1+Bs3)λ+1, f1=⎝integraldisplayb af(x)g(x)dx,f2=⎝integraldisplayb af(x)h(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+1–λ 1As1 λ1As2h(x), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 368 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that As1≠±Bs3,w h e r et h e characteristic value λ∗=2 As1+Bs3is double: y(x)=f(x)+Cy∗(x), where Cis an arbitrary constant and y∗(x)=g(x)–As1–Bs3 2As2h(x) is an eigenfunction of the equation corresponding to λ∗. The equation has no multiple characteristic values if As1=±Bs3. 15. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[g(x)h(t)+h(x)g(t)]y(t)dt=f(x). The characteristic values of the equation: λ1=1 s1+√ s2s3,λ2=1 s1–√ s2s3, where s1=⎝integraldisplayb ah(x)g(x)dx,s2=⎝integraldisplayb ah2(x)dx,s3=⎝integraldisplayb ag2(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2h(x)], where the constants A1andA2are given by A1=f1–λ(f1s1–f2s2) (s2 1–s2s3)λ2–2s1λ+1,A2=f2–λ(f2s1–f1s3) (s2 1–s2s3)λ2–2s1λ+1, f1=⎝integraldisplayb af(x)h(x)dx,f2=⎝integraldisplayb af(x)g(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+⎝radicalbigg s3 s2h(x), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. Solution with λ=λ2≠λ1andf1=f2=0 : y(x)=f(x)+Cy2(x), y2(x)=g(x)–⎝radicalbigg s3 s2h(x), where Cis an arbitrary constant and y2(x) is an eigenfunction of the equation corresponding to the characteristic value λ2. 4◦. The equation has no multiple characteristic values. 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 369 16. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[g(x)h(t)–h(x)g(t)]y(t)dt=f(x). The characteristic values of the equation: λ1=1 ⎝radicalbig s2 1–s2s3,λ2=–1 ⎝radicalbig s2 1–s2s3, where s1=⎝integraldisplayb ah(x)g(x)dx,s2=⎝integraldisplayb ah2(x)dx,s3=⎝integraldisplayb ag2(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2h(x)], where the constants A1andA2are given by A1=f1+λ(f1s1–f2s2) (s2s3–s2 1)λ2+1,A2=–f2+λ(f2s1–f1s3) (s2s3–s2 1)λ2+1, f1=⎝integraldisplayb af(x)h(x)dx,f2=⎝integraldisplayb af(x)g(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+⎝radicalbig s2 1–s2s3–s1 s2h(x), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. Solution with λ=λ2≠λ1andf1=f2=0 : y(x)=f(x)+Cy2(x), y2(x)=g(x)–⎝radicalbig s2 1–s2s3+s1 s2h(x), where Cis an arbitrary constant and y2(x) is an eigenfunction of the equation corresponding to the characteristic value λ2. 4◦. The equation has no multiple characteristic values. 17. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[Ag(x)h(t)+Bh (x)g(t)]y(t)dt=f(x). The characteristic values of the equation: λ1,2=(A+B)s1±⎝radicalbig (A–B)2s2 1+4ABs2s3 2AB(s2 1–s2s3), where s1=⎝integraldisplayb ah(x)g(x)dx,s2=⎝integraldisplayb ah2(x)dx,s3=⎝integraldisplayb ag2(x)dx. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g(x)+A2h(x)], 370 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION where the constants A1andA2are given by A1=Af1–λAB (f1s1–f2s2) AB(s2 1–s2s3)λ2–(A+B)s1λ+1,A2=Bf2–λAB (f2s1–f1s3) AB(s2 1–s2s3)λ2–(A+B)s1λ+1, f1=⎝integraldisplayb af(x)h(x)dx,f2=⎝integraldisplayb af(x)g(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), y1(x)=g(x)+1–λ 1As1 λ1As2h(x), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1. 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that A≠±B, where the characteristic valueλ∗=2 (A+B)s1is double: y(x)=f(x)+Cy∗(x), y∗(x)=g(x)–(A–B)s1 2As2h(x). HereCis an arbitrary constant and y∗(x) is an eigenfunction of the equation corresponding toλ∗. The equation has no multiple characteristic values if A=±B. 18. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[g1(x)h1(t)+g2(x)h2(t)]y(t)dt=f(x). The characteristic values of the equation λ1andλ2are given by λ1,2=s11+s22±⎝radicalbig (s11–s22)2+4s12s21 2(s11s22–s12s21), provided that the integrals s11=⎝integraldisplayb ah1(x)g1(x)dx,s12=⎝integraldisplayb ah1(x)g2(x)dx,s21=⎝integraldisplayb ah2(x)g1(x)dx,s22=⎝integraldisplayb ah2(x)g2(x)dx are convergent. 1◦. Solution with λ≠λ1,2: y(x)=f(x)+λ[A1g1(x)+A2g2(x)], where the constants A1andA2are given by A1=f1–λ(f1s22–f2s12) (s11s22–s12s21)λ2–(s11+s22)λ+1,A2=f2–λ(f2s11–f1s21) (s11s22–s12s21)λ2–(s11+s22)λ+1, f1=⎝integraldisplayb af(x)h1(x)dx,f2=⎝integraldisplayb af(x)h2(x)dx. 2◦. Solution with λ=λ1≠λ2andf1=f2=0 : y(x)=f(x)+Cy1(x), where Cis an arbitrary constant and y1(x) is an eigenfunction of the equation corresponding to the characteristic value λ1: y1(x)=g1(x)+1–λ 1s11 λ1s12g2(x)=g1(x)+λ1s21 1–λ1s22g2(x). 3◦. The solution with λ=λ2≠λ1andf1=f2= 0 is given by the formulas of item 2◦in which one must replace λ1andy1(x)b yλ2andy2(x), respectively. 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 371 4◦. Solution with λ=λ1,2=λ∗andf1=f2= 0 provided that s11≠±s22, where the characteristic valueλ∗=2 s11+s22is double: y(x)=f(x)+Cy∗(x), where Cis an arbitrary constant and y∗(x)=g1(x)–s11–s22 2s12g2(x) is an eigenfunction of the equation corresponding to λ∗. The equation has no multiple characteristic values if s11=±s22. 19. y(x)–λ⎝integraldisplay ⎝integraldisplayb a[g(x)+h(t)]my(t)dt=f(x), m=1 , 2 , ... This is a special case of equation 4.9.20 with gk(x)=gk(x),hk(t)=Ck mhm–k(t), and k=1 ,...,m. Solution: y(x)=f(x)+λm⎝summationdisplay k=0Akgk(x), where the Akare constants that can be determined from 4.9.20. 20. y(x)–λ⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1gk(x)hk(t)⎝bracketrightbigg ⎝bracketrightbigg y(t)dt=f(x),n=2 , 3 , ... The characteristic values of the integral equation (counting the multiplicity, we have exactly nof them) are the roots of the algebraic equation ∆(λ)=0 , where ∆(λ)=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle1–λs 11 –λs12 –λs13··· –λs1n –λs21 1–λs22 –λs23··· –λs2n –λs31 –λs32 1–λs33··· –λs3n ............... –λs n1 –λsn2 –λsn3··· 1–λsnn⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle =( –λ) n⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingles 11–λ–1s12 s13 ··· s1n s21 s22–λ–1s23 ··· s2n s31 s32 s33–λ–1··· s3n ............... sn1 sn2 sn3···snn–λ–1⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle, and the integrals s mk=⎝integraldisplayb ahm(x)gk(x)dx;m,k=1 ,...,n, are assumed to be convergent. 372 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION Solution with regular λ: y(x)=f(x)+λn⎝summationdisplay k=1Akgk(x), where the constants Akform the solution of the following system of algebraic equations: Am–λn⎝summationdisplay k=1smkAk=fm,fm=⎝integraldisplayb af(x)hm(x)dx,m=1 ,...,n. TheAkcan be calculated by Cramer’s rule: Ak=∆k(λ)/∆(λ), where ∆k(λ)=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle1–λs 11··· –λs1k–1f1–λs1k+1··· –λs1n –λs21··· –λs2k–1f2–λs2k+1··· –λs2n ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ –λsn1··· –λsnk–1fn–λsnk+1··· 1–λsnn⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle. For solutions of the equation in the case in which λis a characteristic value, see Subsec- tion 13.2-2. Reference: S. G. Mikhlin (1960). 4.9-2. Equations with Difference Kernel: K(x,t)=K(x–t). 21. y(x)=λ⎝integraldisplay ⎝integraldisplayπ –πK(x–t)y(t)dt,K(x)=K(–x). Characteristic values: λn=1 πan,an=1 π⎝integraldisplayπ –πK(x)c o s (nx)dx (n=0 ,1 ,2 ,... ). The corresponding eigenfunctions are y0(x)=1 , y(1) n(x)=c o s ( nx),y(2) n(x)=s i n ( nx)(n=1 ,2 , ...). For each value λnwithn≠0, there are two corresponding linearly independent eigenfunctions y(1) n(x)a n dy(2) n(x). Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971). 22. y(x)+⎝integraldisplay ⎝integraldisplay∞ –∞K(x–t)y(t)dt=Aeλx. Solution: y(x)=A 1+qeλx,q=⎝integraldisplay∞ –∞K(x)e–λxdx. 23. y(x)+⎝integraldisplay ⎝integraldisplay∞ –∞K(x–t)y(t)dt=Acos(λx )+Bsin(λx). Solution: y(x)=AIc+BIs I2c+I2scos(λx)+BIc–AIs I2c+I2ssin(λx), Ic=1+⎝integraldisplay∞ –∞K(z)c o s (λz)dz,Is=⎝integraldisplay∞ –∞K(z)s i n (λz)dz. 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 373 24. y(x)–⎝integraldisplay ⎝integraldisplay∞ –∞K(x–t)y(t)dt=f(x). Here –∞<x<∞,f(x)∈L1(–∞,∞), and K(x)∈L1(–∞,∞). For the integral equation to be solvable (in L1), it is necessary and sufﬁcient that 1–√ 2π⎝tildewideK(u)≠0, –∞ <u<∞,( 1) where⎝tildewideK(u)=1 √ 2π⎝integraldisplay∞ –∞K(x)e–iuxdxis the Fourier transform of K(x). In this case, the equation has a unique solution, which is given by y(x)=f(x)+⎝integraldisplay∞ –∞R(x–t)f(t)dt, R(x)=1 √ 2π⎝integraldisplay∞ –∞⎝tildewideR(u)eiuxdu,⎝tildewideR(u)=⎝tildewideK(u) 1–√ 2π⎝tildewideK(u). Reference: V . A. Ditkin and A. P. Prudnikov (1965). 25. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0K(x–t)y(t)dt=f(x). The Wiener–Hopf equation of the second kind. * Here 0 ≤x<∞,K(x)∈L1(–∞,∞),f(x)∈L1(0,∞), and y(x)∈L1(0,∞). For the integral equation to be solvable, it is necessary and sufﬁcient that Ω(u)=1– ˇK(u)≠0, –∞ <u<∞,( 1) where ˇK(u)=⎝integraldisplay∞ –∞K(x)eiuxdxis the Fourier transform (in the asymmetric form) of K(x). In this case, the index of the equation can be introduced, ν=– i n d Ω(u)=–1 2π⎝bracketleftbig argΩ(u)⎝bracketrightbig∞ –∞. 1◦. Solution with ν=0 : y(x)=f(x)+⎝integraldisplay∞ 0R(x,t)f(t)dt, where R(x,t)=R+(x–t)+R–(t–x)+⎝integraldisplay∞ 0R+(x–s)R–(t–s)ds, and the functions R+(x)a n dR–(x) satisfy the conditions R+(x)=0a n d R–(x)=0f o r x<0 and are uniquely deﬁned by their Fourier transforms as follows: 1+⎝integraldisplay∞ 0R±(t)e±iutdt=e x p⎝bracketleftbigg –1 2lnΩ(u)∓1 2πi⎝integraldisplay∞ –∞lnΩ(t) t–udt⎝bracketrightbigg . Alternatively, R+(x)a n d R–(x) can be obtained by constructing the solutions of the equations R+(x)+⎝integraldisplay∞ 0K(x–t)R+(t)dt=K(x), 0 ≤x≤∞, R–(x)+⎝integraldisplay∞ 0K(t–x)R–(t)dt=K(–x), 0 ≤x≤∞. * A comprehensive discussion of this equation is given in Subsection 13.10-1, Section 13.11, and Section 13.12. 374 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 2◦. Solution with ν>0 : y(x)=f(x)+ν⎝summationdisplay m=1Cmxm–1e–x+⎝integraldisplay∞ 0R◦(x,t)⎝bracketleftbigg f(t)+ν⎝summationdisplay m=1Cmtm–1e–t⎝bracketrightbigg dt, where the Cmare arbitrary constants, R◦(x,t)=R(0) +(x–t)+R(1) –(t–x)+⎝integraldisplay∞ 0R(0) +(x–s)R(1) –(t–s)ds, and the functions R(0) +(x)a n dR(1) –(x) are uniquely deﬁned by their Fourier transforms: 1+⎝integraldisplay∞ 0R(1) ±(t)e±iutdt=⎝parenleftbiggu–i u+i⎝parenrightbiggν⎝bracketleftbigg 1+⎝integraldisplay∞ 0R(0) ±(t)e±iutdt⎝bracketrightbigg , 1+⎝integraldisplay∞ 0R(0) ±(t)e±iutdt=e x p⎝bracketleftbigg –1 2lnΩ◦(u)∓1 2πi⎝integraldisplay∞ –∞lnΩ◦(t) t–udt⎝bracketrightbigg , Ω◦(u)(u+i)ν=Ω(u)(u–i)ν. 3◦.F o rν< 0, the solution exists only if the conditions ⎝integraldisplay∞ 0f(x)ψm(x)dx=0 , m=1 ,2 , ...,–ν, are satisﬁed. Here ψ1(x),...,ψν(x) is the system of linearly independent solutions of the transposed homogeneous equation ψ(x)–⎝integraldisplay∞ 0K(t–x)ψ(t)dt=0 . Then y(x)=f(x)+⎝integraldisplay∞ 0R∗(x,t)f(t)dt, where R∗(x,t)=R(1) +(x–t)+R(0) –(t–x)+⎝integraldisplay∞ 0R(1) +(x–s)R(0) –(t–s)ds, and the functions R(1) +(x)a n dR(0) –(x) are uniquely deﬁned in item 2◦by their Fourier trans- forms. References: V . I. Smirnov (1974), F. D. Gakhov and Yu. I. Cherskii (1978), I. M. Vinogradov (1979). 4.9-3. Other Equations of the Form y(x)+⎝integraltextb aK(x,t)y(t)dt=F(x). 26. y(x)–⎝integraldisplay ⎝integraldisplay∞ –∞K(x+t)y(t)dt=f(x). The Fourier transform is used to solve this equation. Solution: y(x)=1 √ 2π⎝integraldisplay∞ –∞⎝tildewidef(u)+√ 2π⎝tildewidef(–u)⎝tildewideK(u) 1–√ 2π⎝tildewideK(u)⎝tildewideK(–u)eiuxdu, where ⎝tildewidef(u)=1 √ 2π⎝integraldisplay∞ –∞f(x)e–iuxdx,⎝tildewideK(u)=1 √ 2π⎝integraldisplay∞ –∞K(x)e–iuxdx. Reference: V . A. Ditkin and A. P. Prudnikov (1965). 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 375 27. y(x)+⎝integraldisplay ⎝integraldisplay∞ –∞eβtK(x+t)y(t)dt=Aeλx. Solution: y(x)=eλx–k(λ)e–(β+λ)x 1–k (λ)k(–β–λ),k(λ)=⎝integraldisplay∞ –∞K(x)e(λ+β)xdx. 28. y(x)+⎝integraldisplay ⎝integraldisplay∞ –∞[eβtK(x+t)+M(x–t)]y(t)dt=Aeλx. Solution: y(x)=AIk(λ)epx–[ 1+ Im(p)]eλx Ik(λ)Ik(p)–[ 1+ Im(λ)][1 + Im(p)],p=–λ–β, where Ik(λ)=⎝integraldisplay∞ –∞K(z)e(β+λ)zdz,Im(λ)=⎝integraldisplay∞ –∞M(z)e–λzdz. 29. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0K(xt)y(t)dt=f(x). The solution can be obtained with the aid of the inverse Mellin transform: y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞⎝tildewidef(s)+⎝tildewideK(s)⎝tildewidef(1 –s) 1–⎝tildewideK(s)⎝tildewideK(1 –s)x–sds, where⎝tildewidefand⎝tildewideKstand for the Mellin transforms of the right-hand side and of the kernel of the integral equation, ⎝tildewidef(s)=⎝integraldisplay∞ 0f(x)xs–1dx,⎝tildewideK(s)=⎝integraldisplay∞ 0K(x)xs–1dx. Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971). 30. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0K(xt)tβy(t)dt=Axλ. Solution: y(x)=Axλ+Iβ+λx–β–λ–1 1–I β+λI–λ–1,Iµ=⎝integraldisplay∞ 0K(ξ)ξµdξ. It is assumed that all improper integrals are convergent. 31. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0K(xt)tβy(t)dt=f(x). The solution can be obtained with the aid of the inverse Mellin transform as follows: y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞⎝tildewidef(s)+⎝tildewideK(s)⎝tildewidef(1 +β–s) 1–⎝tildewideK(s)⎝tildewideK(1 +β–s)x–sds, where⎝tildewidefand⎝tildewideKstand for the Mellin transforms of the right-hand side and of the kernel of the integral equation, ⎝tildewidef(s)=⎝integraldisplay∞ 0f(x)xs–1dx,⎝tildewideK(s)=⎝integraldisplay∞ 0K(x)xs–1dx. 376 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 32. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0g(xt)xλtµy(t)dt=f(x). This equation can be rewritten in the form of equation 4.9.31 by setting K(z)=zλg(z)a n d β=µ–λ. 33. y(x)–⎝integraldisplay ⎝integraldisplay∞ 01 tK⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig y(t)dt=0 . Eigenfunctions of this integral equation are determined by the roots of the following tran- scendental (algebraic) equation for the parameter λ: ⎝integraldisplay∞ 0K⎝parenleftBig1 z⎝parenrightBig zλ–1dz=1 . ( 1 ) 1◦. For a real simple root λnof equation (1), there is a corresponding eigenfunction yn(x)=xλn. 2◦. For a real root λnof multiplicity r, there are corresponding reigenfunctions yn1(x)=xλn,yn2(x)=xλnlnx,...,ynr(x)=xλnlnr–1x. 3◦. For a complex simple root λn=αn+iβnof equation (1), there is a corresponding pair of eigenfunctions y(1) n(x)=xαncos(β nlnx),y(2) n(x)=xαnsin(βnlnx). 4◦. For a complex root λn=αn+iβnof multiplicity r, there are corresponding reigenfunction pairs y(1) n1(x)=xαncos(β nlnx), y(1) n2(x)=xαnlnxcos(β nlnx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(1) nr(x)=xαnlnr–1xcos(β nlnx),y(2) n1(x)=xαnsin(βnlnx), y(2) n2(x)=xαnlnxsin(βnlnx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(2) nr(x)=xαnlnr–1xsin(βnlnx). The general solution is the linear combination (with arbitrary constants) of the eigenfunc- tions of the homogeneous integral equation. 34. y(x)–⎝integraldisplay ⎝integraldisplay∞ 01 tK⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig y(t)dt=Axb. A solution: y(x)=A Bxb,B=1–⎝integraldisplay∞ 0K⎝parenleftBig1 ξ⎝parenrightBig ξb–1dξ. It is assumed that the improper integral is convergent and B≠0. The general solution of the integral equations is the sum of the above solution and the solution of the homogeneous equation 4.9.33. 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 377 35. y(x)–⎝integraldisplay ⎝integraldisplay∞ 01 tK⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig y(t)dt=f(x). The solution can be obtained with the aid of the inverse Mellin transform: y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞⎝tildewidef(s) 1–⎝tildewideK(s)x–sds, where⎝tildewidefand⎝tildewideKstand for the Mellin transforms of the right-hand side and the kernel of the integral equation, ⎝tildewidef(s)=⎝integraldisplay∞ 0f(x)xs–1dx,⎝tildewideK(s)=⎝integraldisplay∞ 0K(x)xs–1dx. Example. Forf(x)=Ae–λxandK(x)=1 2e–x, the solution of the integral equation has the form y(x)=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩4A ( 3–2C)(λx )3forλx>1 , –2A∞⎝summationdisplay k=11 (λx)skψ(sk)forλx<1 . HereC= 0.5772 ...is the Euler constant, ψ(z)=[ l nΓ(z)]/prime zis the logarithmic derivative of the gamma function, and the skare the negative roots of the transcendental equation Γ(sk)=2 ,w h e r e Γ(z) is the gamma function. Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971). 36. y(x)+⎝integraldisplay ⎝integraldisplayb a|x–t|g(t)y(t)dt=f(x), a≤x≤b. 1◦. Let us remove the modulus in the integrand, y(x)+⎝integraldisplayx a(x–t)g(t)y(t)dt+⎝integraldisplayb x(t–x)g(t)y(t)dt=f(x). (1) Differentiating (1) with respect to xyields y/prime x(x)+⎝integraldisplayx ag(t)y(t)dt–⎝integraldisplayb xg(t)y(t)dt=f/prime x(x). (2) Differentiating (2), we arrive at a second-order ordinary differential equation for y=y(x), y/prime/prime xx+2g(x)y=f/prime/prime xx(x). (3) 2◦. Let us derive the boundary conditions for equation (3). We assume that the limits of integration satisfy the conditions –∞ <a<b<∞. By setting x=aandx=bin (1), we obtain two consequences y(a)+⎝integraldisplayb a(t–a)g(t)y(t)dt=f(a), y(b)+⎝integraldisplayb a(b–t)g(t)y(t)dt=f(b).(4) Let us express g(x)yfrom (3) via y/prime/prime xxandf/prime/prime xxand substitute the result into (4). Integrating by parts yields the desired boundary conditions for y(x), y(a)+y(b)+(b–a)[f/prime x(b)–y/prime x(b)] =f(a)+f(b), y(a)+y(b)+(a–b)[f/prime x(a)–y/prime x(a)] =f(a)+f(b).(5) Note a useful consequence of (5), y/prime x(a)+y/prime x(b)=f/prime x(a)+f/prime x(b), (6) which can be used together with one of conditions (5). Equation (3) under the boundary conditions (5) determines the solution of the original integral equation. Conditions (5) make it possible to calculate the constants of integration that occur in the solution of the differential equation (3). 378 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 37. y(x)+⎝integraldisplay ⎝integraldisplayb aeλ|x–t|g(t)y(t)dt=f(x), a≤x≤b. 1◦. Let us remove the modulus in the integrand: y(x)+⎝integraldisplayx aeλ(x–t)g(t)y(t)dt+⎝integraldisplayb xeλ(t–x)g(t)y(t)dt=f(x). (1) Differentiating (1) with respect to xtwice yields y/prime/prime xx(x)+2λg(x)y(x)+λ2⎝integraldisplayx aeλ(x–t)g(t)y(t)dt+λ2⎝integraldisplayb xeλ(t–x)g(t)y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary differential equation for y=y(x), y/prime/prime xx+2λg(x)y–λ2y=f/prime/prime xx(x)–λ2f(x). (3) 2◦. Let us derive the boundary conditions for equation (3). We assume that the limits of integration satisfy the conditions –∞ <a<b<∞. By setting x=aandx=bin (1), we obtain two consequences y(a)+e–λa⎝integraldisplayb aeλtg(t)y(t)dt=f(a), y(b)+eλb⎝integraldisplayb ae–λtg(t)y(t)dt=f(b).(4) Let us express g(x)yfrom (3) via y/prime/prime xxandf/prime/prime xxand substitute the result into (4). Integrating by parts yields the conditions eλbϕ/prime x(b)–eλaϕ/prime x(a)=λeλaϕ(a)+λeλbϕ(b), e–λbϕ/prime x(b)–e–λaϕ/prime x(a)=λe–λaϕ(a)+λe–λbϕ(b),ϕ(x)=y(x)–f(x). Finally, after some manipulations, we arrive at the desired boundary conditions for y(x): ϕ/prime x(a)+λϕ(a)=0 , ϕ/prime x(b)–λϕ(b)=0 ; ϕ(x)=y(x)–f(x). (5) Equation (3) under the boundary conditions (5) determines the solution of the original integral equation. Conditions (5) make it possible to calculate the constants of integrationthat occur in solving the differential equation (3). 38. y(x)+⎝integraldisplay ⎝integraldisplay b asinh(λ|x–t|)g(t)y(t)dt=f(x), a≤x≤b. 1◦. Let us remove the modulus in the integrand: y(x)+⎝integraldisplayx asinh[λ(x–t)]g(t)y(t)dt+⎝integraldisplayb xsinh[λ(t–x)]g(t)y(t)dt=f(x). (1) Differentiating (1) with respect to xtwice yields y/prime/prime xx(x)+2λg(x)y(x)+λ2⎝integraldisplayx asinh[λ(x–t)]g(t)y(t)dt +λ2⎝integraldisplayb xsinh[λ(t–x)]g(t)y(t)dt=f/prime/prime xx(x). (2) 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 379 Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary differential equation for y=y(x), y/prime/prime xx+2λg(x)y–λ2y=f/prime/prime xx(x)–λ2f(x). (3) 2◦. Let us derive the boundary conditions for equation (3). We assume that the limits of integration satisfy the conditions –∞ <a<b<∞. By setting x=aandx=bin (1), we obtain two corollaries y(a)+⎝integraldisplayb asinh[λ(t–a)]g(t)y(t)dt=f(a), y(b)+⎝integraldisplayb asinh[λ(b–t)]g(t)y(t)dt=f(b).(4) Let us express g(x)yfrom (3) via y/prime/prime xxandf/prime/prime xxand substitute the result into (4). Integrating by parts yields the desired boundary conditions for y(x), sinh[λ(b–a)]ϕ/prime x(b)–λcosh[λ (b–a)]ϕ(b)=λϕ(a), sinh[λ(b–a)]ϕ/prime x(a)+λcosh[λ(b–a)]ϕ(a)=–λϕ(b);ϕ(x)=y(x)–f(x).(5) Equation (3) under the boundary conditions (5) determines the solution of the original integral equation. Conditions (5) make it possible to calculate the constants of integration that occur in solving the differential equation (3). 39. y(x)+⎝integraldisplay ⎝integraldisplayb asin(λ|x–t|)g(t)y(t)dt=f(x), a≤x≤b. 1◦. Let us remove the modulus in the integrand: y(x)+⎝integraldisplayx asin[λ(x–t)]g(t)y(t)dt+⎝integraldisplayb xsin[λ(t–x)]g(t)y(t)dt=f(x). (1) Differentiating (1) with respect to xtwice yields y/prime/prime xx(x)+2λg(x)y(x)–λ2⎝integraldisplayx asin[λ(x–t)]g(t)y(t)dt –λ2⎝integraldisplayb xsin[λ(t–x)]g(t)y(t)dt=f/prime/prime xx(x). (2) Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary differential equation for y=y(x), y/prime/prime xx+2λg(x)y+λ2y=f/prime/prime xx(x)+λ2f(x). (3) 2◦. Let us derive the boundary conditions for equation (3). We assume that the limits of integration satisfy the conditions –∞ <a<b<∞. By setting x=aandx=bin (1), we obtain two consequences y(a)+⎝integraldisplayb asin[λ(t–a)]g(t)y(t)dt=f(a), y(b)+⎝integraldisplayb asin[λ(b–t)]g(t)y(t)dt=f(b).(4) Let us express g(x)yfrom (3) via y/prime/prime xxandf/prime/prime xxand substitute the result into (4). Integrating by parts yields the desired boundary conditions for y(x), sin[λ(b–a)]ϕ/prime x(b)–λcos[λ (b–a)]ϕ(b)=λϕ(a), sin[λ(b–a)]ϕ/prime x(a)+λcos[λ (b–a)]ϕ(a)=–λϕ(b);ϕ(x)=y(x)–f(x).(5) Equation (3) under the boundary conditions (5) determines the solution of the original integral equation. Conditions (5) make it possible to calculate the constants of integration that occur in solving the differential equation (3). 380 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 40. y(x)+⎝integraldisplay ⎝integraldisplay∞ –∞⎝bracketleftbig⎝bracketleftbig λe–|x–t|+ϕ(x)ψ(t)]y(t)dt=f(x). The solutions can be obtained by the methods described in Subsection 13.2-3; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.2.14. Solution: y(x)=Yf(x)+AYϕ(x), where Yf(x)=f(x)–λ √ 1+2λ⎝integraldisplay∞ –∞exp⎝parenleftbig –√ 1+2λ|x–t|⎝parenrightbig f(t)dt, Yϕ(x)=ϕ(x)–λ √ 1+2λ⎝integraldisplay∞ –∞exp⎝parenleftbig –√ 1+2λ|x–t|⎝parenrightbig ϕ(t)dt, A=–⎝integraldisplay∞ 0ψ(t)Yf(t)dt 1+⎝integraldisplay∞ 0ψ(t)Yϕ(t)dt,λ>–1 2. 41. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0[λsin(xt)+ϕ(x)ψ(t)]y(t)dt=f(x). The solution can be obtained by the methods described in Subsection 13.2-3; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.5.20. Solution: y(x)=Yf(x)+AYϕ(x), where Yf(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0sin(xt)f(t)dt, Yϕ(x)=ϕ(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0sin(xt)ϕ(t)dt, A=⎝integraldisplay∞ 0ψ(t)Yf(t)dt 1–⎝integraldisplay∞ 0ψ(t)Yϕ(t)dt,λ≠±⎝radicalbigg 2 π. 42. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0[λcos(xt )+ϕ(x)ψ(t)]y(t)dt=f(x). The solution can be obtained by the methods described in Subsection 13.2-3; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.5.6. Solution: y(x)=Yf(x)+AYϕ(x), where Yf(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0cos(xt )f(t)dt, Yϕ(x)=ϕ(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0cos(xt )ϕ(t)dt, A=⎝integraldisplay∞ 0ψ(t)Yf(t)dt 1–⎝integraldisplay∞ 0ψ(t)Yϕ(t)dt,λ≠±⎝radicalbigg 2 π. 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 381 43. y(x)+⎝integraldisplay ⎝integraldisplay∞ 0[λtJν(xt)+ϕ(x)ψ(t)]y(t)dt=f(x), ν> –1. HereJν(z) is the Bessel function of the ﬁrst kind. The solution can be obtained by the methods described in Subsection 13.2-3; it must be taken into account that the truncated equation, withϕ(x) = 0, coincides with equation 4.8.4. Solution: y(x)=Y f(x)+AYϕ(x), where Yf(x)=f(x) 1–λ2–λ 1–λ2⎝integraldisplay∞ 0tJν(xt)f(t)dt, Yϕ(x)=ϕ(x) 1–λ2–λ 1–λ2⎝integraldisplay∞ 0tJν(xt)ϕ(t)dt, A=–⎝integraldisplay∞ 0ψ(t)Yf(t)dt 1+⎝integraldisplay∞ 0ψ(t)Yϕ(t)dt,λ≠±1. 4.9-4. Equations of the Form y(x)+⎝integraltextb aK(x,t)y(···)dt=F(x). 44. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=0 . Eigenfunctions of this integral equation* are determined by the roots of the following char- acteristic (transcendental or algebraic) equation for µ: ⎝integraldisplayb af(t)e x p ( – µt)dt= –1. (1) 1◦. For a real (simple) root µkof equation (1), there is a corresponding eigenfunction yk(x)=e x p ( µkx). 2◦. For a real root µkof multiplicity r, there are corresponding reigenfunctions yk1(x)=e x p ( µkx),yk2(x)=xexp(µkx),...,ykr(x)=xr–1exp(µkx). 3◦. For a complex (simple) root µk=αk+iβkof equation (1), there is a corresponding pair of eigenfunctions y(1) k(x)=e x p ( αkx)c o s (βkx),y(2) k(x)=e x p ( αkx)s i n (βkx). 4◦. For a complex root µk=αk+iβkof multiplicity r, there are corresponding rpairs of eigenfunctions y(1) k1(x)=e x p ( αkx)c o s (βkx), y(1) k2(x)=xexp(αkx)c o s (βkx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(1) kr(x)=xr–1exp(αkx)c o s (βkx),y(2) k1(x)=e x p ( αkx)s i n (βkx), y(2) k2(x)=xexp(αkx)s i n (βkx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(2) kr(x)=xr–1exp(αkx)s i n (βkx). The general solution is the linear combination (with arbitrary constants) of the eigenfunc- tions of the homogeneous integral equation. * In the equations below that contain y(x–t) in the integrand, the arguments can have, for example, the domain (a) –∞<x<∞,–∞<t<∞fora=–∞andb=∞or (b)a≤t≤b,–∞≤x<∞,f o raandbsuch that – ∞<a<b<∞. Case (b) is a special case of (a) if f(t) is nonzero only on the interval a≤t≤b. 382 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION /trianglerightsldF or equations 4.9.45–4.9.50, only particular solutions are given. T o obtain the general solution, one must add the particular solution to the general solution of the corresponding homogeneousequation 4.9.44. 45. y(x)+⎝integraldisplay ⎝integraldisplay b af(t)y(x–t)dt=Ax +B. A solution: y(x)=px+q, where the coefﬁcients pandqare given by p=A 1+I0,q=AI1 (1 +I0)2+B 1+I0,I0=⎝integraldisplayb af(t)dt,I1=⎝integraldisplayb atf(t)dt. 46. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=Aeλx. A solution: y(x)=A Beλx,B=1+⎝integraldisplayb af(t)e x p ( – λt)dt. The general solution of the integral equation is the sum of the speciﬁed particular solution and the general solution of the homogeneous equation 4.9.44. 47. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=Asin(λx). A solution: y(x)=AIc I2c+I2ssin(λx)+AIs I2c+I2scos(λx), where the coefﬁcients IcandIsare given by Ic=1+⎝integraldisplayb af(t)c o s ( λt)dt,Is=⎝integraldisplayb af(t)s i n (λt)dt. 48. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=Acos(λx ). A solution: y(x)=–AIs I2c+I2ssin(λx)+AIc I2c+I2scos(λx), where the coefﬁcients IcandIsare given by Ic=1+⎝integraldisplayb af(t)c o s ( λt)dt,Is=⎝integraldisplayb af(t)s i n (λt)dt. 49. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=eµx(Asinλx +Bcosλx). A solution: y(x)=eµx(psinλx+qcosλx), where the coefﬁcients pandqare given by p=AIc–BIs I2c+I2s,q=AIs+BIc I2c+I2s, Ic=1+⎝integraldisplayb af(t)e–µtcos(λt )dt,Is=⎝integraldisplayb af(t)e–µtsin(λt)dt. 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 383 50. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)dt=g(x). 1◦.F o rg(x)=n⎝summationtext k=1Akexp(λkx), the equation has a solution y(x)=n⎝summationdisplay k=1Ak Bkexp(λkx), Bk=1+⎝integraldisplayb af(t)e x p ( – λkt)dt. 2◦. For polynomial right-hand side of the equation, g(x)=n⎝summationtext k=0Akxk, a solution has the form y(x)=n⎝summationdisplay k=0Bkxk, where the constants Bkcan be found by the method of undetermined coefﬁcients. 3◦.F o rg(x)=eλxn⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=eλxn⎝summationdisplay k=0Bkxk, where the constants Bkcan be found by the method of undetermined coefﬁcients. 4◦.F o rg(x)=n⎝summationtext k=1Akcos(λ kx), a solution of the equation has the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 5◦.F o rg(x)=n⎝summationtext k=1Aksin(λkx), a solution of the equation has the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 6◦.F o rg(x)=c o s ( λx)n⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 7◦.F o rg(x)=s i n ( λx)n⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 384 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 8◦.F o rg(x)=eµxn⎝summationtext k=1Akcos(λ kx), a solution of the equation has the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 9◦.F o rg(x)=eµxn⎝summationtext k=1Aksin(λkx), a solution of the equation has the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 10◦.F o rg(x)=c o s ( λx)n⎝summationtext k=1Akexp(µkx), a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Bkexp(µkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 11◦.F o rg(x)=s i n ( λx)n⎝summationtext k=1Akexp(µkx), a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Bkexp(µkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 51. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x+βt)dt=Ax +B. A solution:* y(x)=px+q, where p=A 1+I0,q=B 1+I0–AI1β (1 +I0)2,I0=⎝integraldisplayb af(t)dt,I1=⎝integraldisplayb atf(t)dt. 52. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x+βt)dt=Aeλx. A solution: y(x)=A Beλx,B=1+⎝integraldisplayb af(t)e x p ( λβt)dt. * In the equations below that contain y(x+βt),β> 0, in the integrand, the arguments can have, for example, the domain (a) 0 ≤x<∞,0≤t<∞fora=0a n d b=∞or (b)a≤t≤b,0≤x<∞foraandbsuch that 0 ≤a<b<∞.C a s e( b )i s a special case of (a) if f(t) is nonzero only on the interval a≤t≤b. 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 385 53. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x+βt)dt=Asinλx +Bcosλx. A solution: y(x)=psinλx+qcosλx, where the coefﬁcients pandqare given by p=AIc+BIs I2c+I2s,q=BIc–AIs I2c+I2s, Ic=1+⎝integraldisplayb af(t)c o s ( λβt)dt,Is=⎝integraldisplayb af(t)s i n (λβt)dt. 54. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x+βt)dt=g(x). 1◦.F o rg(x)=n⎝summationtext k=1Akexp(λkx), a solution of the equation has the form y(x)=n⎝summationdisplay k=1Ak Bkexp(λkx), Bk=1+⎝integraldisplayb af(t)e x p ( βλkt)dt. 2◦. For polynomial right-hand side of the equation, g(x)=n⎝summationtext k=0Akxk, a solution has the form y(x)=n⎝summationdisplay k=0Bkxk, where the constants Bkcan be found by the method of undetermined coefﬁcients. 3◦.F o rg(x)=eλxn⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=eλxn⎝summationdisplay k=0Bkxk, where the constants Bkcan be found by the method of undetermined coefﬁcients. 4◦.F o rg(x)=n⎝summationtext k=1Akcos(λ kx), a solution of the equation has the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 5◦.F o rg(x)=n⎝summationtext k=1Aksin(λkx), a solution of the equation has the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 386 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 6◦.F o rg(x)=c o s ( λx)n⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 7◦.F o rg(x)=s i n ( λx)n⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 8◦.F o rg(x)=eµxn⎝summationtext k=1Akcos(λ kx), a solution of the equation has the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 9◦.F o rg(x)=eµxn⎝summationtext k=1Aksin(λkx), a solution of the equation has the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 10◦.F o rg(x)=c o s ( λx)n⎝summationtext k=1Akexp(µkx), a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Bkexp(µkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 11◦.F o rg(x)=s i n ( λx)n⎝summationtext k=1Akexp(µkx), a solution of the equation has the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Bkexp(µkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 387 55. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=0 . Eigenfunctions of this integral equation* are determined by the roots of the following tran- scendental (or algebraic) equation for λ: ⎝integraldisplayb af(t)tλdt= –1. (1) 1◦. For a real (simple) root λkof equation (1), there is a corresponding eigenfunction yk(x)=xλk. 2◦. For a real root λkof multiplicity r, there are corresponding reigenfunctions yk1(x)=xλk,yk2(x)=xλklnx,...,ykr(x)=xλklnr–1x. 3◦. For a complex (simple) root λk=αk+iβkof equation (1), there is a corresponding pair of eigenfunctions y(1) k(x)=xαkcos(β klnx),y(2) k(x)=xαksin(βklnx). 4◦. For a complex root λk=αk+iβkof multiplicity r, there are corresponding rpairs of eigenfunctions y(1) k1(x)=xαkcos(β klnx), y(1) k2(x)=xαklnxcos(β klnx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(1) kr(x)=xαklnr–1xcos(β klnx),y(2) k1(x)=xαksin(βklnx), y(2) k2(x)=xαklnxsin(βklnx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(2) kr(x)=xαklnr–1xsin(βklnx). The general solution is the linear combination (with arbitrary constants) of the eigenfunc- tions of the homogeneous integral equation. /trianglerightsldF or equations 4.9.56–4.9.62, only particular solutions are given. T o obtain the general solution, one must add the particular solution to the general solution of the corresponding homogeneousequation 4.9.55. 56. y(x)+⎝integraldisplay ⎝integraldisplay b af(t)y(xt)dt=Ax +B. A solution: y(x)=A 1+I1x+B 1+I0,I0=⎝integraldisplayb af(t)dt,I1=⎝integraldisplayb atf(t)dt. 57. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Axβ. A solution: y(x)=A Bxβ,B=1+⎝integraldisplayb af(t)tβdt. * In the equations below that contain y(xt) in the integrand, the arguments can have, for example, the domain (a) 0 ≤x≤1, 0≤t≤1f o ra=0a n d b=1 ,( b )1 ≤x<∞,1≤t<∞fora=1a n d b=∞,( c )0 ≤x<∞,0≤t<∞fora=0a n d b=∞, or (d) a≤t≤b,0≤x<∞foraandbsuch that 0 ≤a<b≤∞. Case (d) is a special case of (c) if f(t) is nonzero only on the interval a≤t≤b. 388 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 58. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Alnx+B. A solution: y(x)=plnx+q, where p=A 1+I0,q=B 1+I0–AIl (1 +I0)2,I0=⎝integraldisplayb af(t)dt,Il=⎝integraldisplayb af(t)l ntd t. 59. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Axβlnx. A solution: y(x)=pxβlnx+qxβ, where p=A 1+I1,q=–AI2 (1 +I1)2,I1=⎝integraldisplayb af(t)tβdt,I2=⎝integraldisplayb af(t)tβlntd t. 60. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Acos(ln x). A solution: y(x)=AIc I2c+I2scos(ln x)+AIs I2c+I2ssin(lnx), Ic=1+⎝integraldisplayb af(t)c o s ( l n t)dt,Is=⎝integraldisplayb af(t)s i n ( l n t)dt. 61. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Asin(ln x). A solution: y(x)=–AIs I2c+I2scos(ln x)+AIc I2c+I2ssin(lnx), Ic=1+⎝integraldisplayb af(t)c o s ( l n t)dt,Is=⎝integraldisplayb af(t)s i n ( l n t)dt. 62. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(xt)dt=Axβcos(λ lnx)+Bxβsin(λlnx). A solution: y(x)=pxβcos(λ lnx)+qxβsin(λlnx), where p=AIc–BIs I2c+I2s,q=AIs+BIc I2c+I2s, Ic=1+⎝integraldisplayb af(t)tβcos(λ lnt)dt,Is=⎝integraldisplayb af(t)tβsin(λlnt)dt. 4.9. E QUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS 389 63. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(ξ)dt=0 , ξ=xϕ(t). Eigenfunctions of this integral equation are determined by the roots of the following tran- scendental (or algebraic) equation for λ: ⎝integraldisplayb af(t)[ϕ(t)]λdt= –1. (1) 1◦. For a real (simple) root λkof equation (1), there is a corresponding eigenfunction yk(x)=xλk. 2◦. For a real root λkof multiplicity r, there are corresponding reigenfunctions yk1(x)=xλk,yk2(x)=xλklnx,...,ykr(x)=xλklnr–1x. 3◦. For a complex (simple) root λk=αk+iβkof equation (1), there is a corresponding pair of eigenfunctions y(1) k(x)=xαkcos(β klnx),y(2) k(x)=xαksin(βklnx). 4◦. For a complex root λk=αk+iβkof multiplicity r, there are corresponding rpairs of eigenfunctions y(1) k1(x)=xαkcos(β klnx), y(1) k2(x)=xαklnxcos(β klnx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(1) kr(x)=xαklnr–1xcos(β klnx),y(2) k1(x)=xαksin(βklnx), y(2) k2(x)=xαklnxsin(βklnx), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y(2) kr(x)=xαklnr–1xsin(βklnx). The general solution is the linear combination (with arbitrary constants) of the eigenfunc- tions of the homogeneous integral equation. 64. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(ξ)dt=Axβ,ξ=xϕ(t). A solution: y(x)=A Bxβ,B=1+⎝integraldisplayb af(t)[ϕ(t)]βdt. It is assumed that B≠0. A linear combination of eigenfunctions of the corresponding homogeneous equation (see 4.9.63) can be added to this solution. 65. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(ξ)dt=g(x), ξ=xϕ(t). 1◦.F o rg(x)=n⎝summationtext k=0Akxk, a solution of the equation has the form y(x)=n⎝summationdisplay k=0Ak Bkxk,Bk=1+⎝integraldisplayb af(t)[ϕ(t)]kdt.( 1) 2◦.F o rg(x)=l nxn⎝summationtext k=0Akxk, a solution has the form y(x)=l nxn⎝summationdisplay k=0Bkxk+n⎝summationdisplay k=0Ckxk,( 2) where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 390 LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 3◦.F o rg(x)=n⎝summationtext k=0Ak(lnx)k, a solution of the equation has the form y(x)=n⎝summationdisplay k=0Bk(lnx)k,( 3) where the constants Bkcan be found by the method of undetermined coefﬁcients. 4◦.F o rg(x)=n⎝summationtext k=1Akcos(λ klnx), a solution of the equation has the form y(x)=n⎝summationdisplay k=1Bkcos(λ klnx)+n⎝summationdisplay k=1Cksin(λklnx), (4) where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 5◦.F o rg(x)=n⎝summationtext k=1Aksin(λklnx), a solution of the equation has the form y(x)=n⎝summationdisplay k=1Bkcos(λ klnx)+n⎝summationdisplay k=1Cksin(λklnx), (5) where the constants BkandCkcan be found by the method of undetermined coefﬁcients. Remark. A linear combination of eigenfunctions of the corresponding homogeneous equation (see 4.9.63) can be added to solutions (1)–(5). 4.10. Some Formulas and Transformations Let the solution of the integral equation y(x)+⎝integraldisplayb aK(x,t)y(t)dt=f(x)( 1 ) have the form y(x)=f(x)+⎝integraldisplayb aR(x,t)f(t)dt.( 2) Then the solution of the more complicated integral equation y(x)+⎝integraldisplayb aK(x,t)g(x) g(t)y(t)dt=f(x)( 3) has the form y(x)=f(x)+⎝integraldisplayb aR(x,t)g(x) g(t)f(t)dt.( 4) Below are formulas for the solutions of integral equations of the form (3) for some speciﬁc func- tionsg(x). In all cases, it is assumed that the solution of equation (1) is known and is given by (2). 4.10. S OME FORMULAS AND TRANSFORMATIONS 391 1◦. The solution of the equation y(x)+⎝integraldisplayb aK(x,t)(x/t )λy(t)dt=f(x) has the form y(x)=f(x)+⎝integraldisplayb aR(x,t)(x/t )λf(t)dt. 2◦. The solution of the equation y(x)+⎝integraldisplayb aK(x,t)eλ(x–t)y(t)dt=f(x) has the form y(x)=f(x)+⎝integraldisplayb aR(x,t)eλ(x–t)f(t)dt. Chapter 5 Nonlinear Equations of the First Kind with Variable Limit of Integration /trianglerightsld Notation: f,g,h, andKare arbitrary functions of an argument speciﬁed in the parentheses (the argument can depend on t,x, andy);A,B,a,b,k,β,λ, andµare arbitrary parameters. 5.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters 5.1-1. Equations of the Form⎝integraltextx 0y(t)y(x–t)dt=f(x). 1.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=Ax +B,A,B>0 . Solutions: y(x)=±√ B⎝bracketleftbigg1 √ πxexp⎝parenleftBig –A Bx⎝parenrightBig +⎝radicalbigg A Berf⎝parenleftbigg⎝radicalbigg A Bx⎝parenrightbigg⎝bracketrightbigg , where erf z=2 √ π⎝integraldisplayz 0exp⎝parenleftbig –t2⎝parenrightbig dtis the error function. 2.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=A2xλ. Solutions: y(x)=±A√ Γ(λ+1 ) Γ⎝parenleftbigλ+1 2⎝parenrightbigxλ–1 2, whereΓ(z) is the gamma function. 3.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=Axλ–1+Bxλ,λ>0 . Solutions: y(x)=±√ AΓ(λ) Γ(λ/2)xλ–2 2exp⎝parenleftBig –λB Ax⎝parenrightBig Φ⎝parenleftBigλ+1 2,λ 2;λB Ax⎝parenrightBig , whereΦ(a,c;x) is the degenerate hypergeometric function (Kummer’s function). 4.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=A2eλx. Solutions: y(x)=±A √ πxeλx. 393 394 NONLINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 5.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=(Ax +B)eλx,A,B>0 . Solutions: y(x)=±√ Beλx⎝bracketleftbigg1 √ πxexp⎝parenleftBig –A Bx⎝parenrightBig +⎝radicalbigg A Berf⎝parenleftbigg⎝radicalbigg A Bx⎝parenrightbigg⎝bracketrightbigg , where erf z=2 √ π⎝integraldisplayz 0exp⎝parenleftbig –t2⎝parenrightbig dtis the error function. 6.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=A2xµeλx. Solutions: y(x)=±A√ Γ(µ+1 ) Γ⎝parenleftbigµ+1 2⎝parenrightbigxµ–1 2eλx. 7.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=⎝parenleftbig⎝parenleftbig Axµ–1+Bxµ⎝parenrightbig⎝parenrightbig eλx. Solutions: y(x)=±√ AΓ(µ) Γ(µ/2)xµ–2 2exp⎝bracketleftBig⎝parenleftBig λ–µB A⎝parenrightBig x⎝bracketrightBig Φ⎝parenleftBigµ+1 2,µ 2;µB Ax⎝parenrightBig , whereΦ(a,c;x) is the degenerate hypergeometric function (Kummer’s function). 8.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=A2cosh(λx ). Solutions: y(x)=±A √ πd dx⎝integraldisplayx 0I0(λt)dt √ x–t,w h e r e I0(z) is the modiﬁed Bessel function. 9.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=Asinh(λx). Solutions: y=±√ AλI 0(λx), where I0(z) is the modiﬁed Bessel function. 10.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=Asinh(λ√ x). Solutions: y=±√ Aπ1/42–7/8λ3/4x–1/8I–1/4⎝parenleftBig λ⎝radicalBig 1 2x⎝parenrightBig ,w h e r e I–1/4(z) is the modiﬁed Bessel function. 11.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=A2cos(λx ). Solutions: y(x)=±A √ πd dx⎝integraldisplayx 0J0(λt)dt √ x–t,w h e r e J0(z) is the Bessel function. 12.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=Asin(λx). Solutions: y=±√ AλJ 0(λx), where J0(z) is the Bessel function. 5.1. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY PARAMETERS 395 13.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=Asin(λ√ x). Solutions: y=±√ Aπ1/42–7/8λ3/4x–1/8J–1/4⎝parenleftBig λ⎝radicalBig 1 2x⎝parenrightBig ,w h e r e J–1/4(z) is the Bessel func- tion. 14.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=A2eµxcosh(λx ). Solutions: y(x)=±A √ πeµxd dx⎝integraldisplayx 0I0(λt)dt √ x–t,w h e r e I0(z) is the modiﬁed Bessel function. 15.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=Aeµxsinh(λx). Solutions: y=±√ AλeµxI0(λx), where I0(z) is the modiﬁed Bessel function. 16.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=A2eµxcos(λx ). Solutions: y(x)=±A √ πeµxd dx⎝integraldisplayx 0J0(λt)dt √ x–t,w h e r e J0(z) is the Bessel function. 17.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=Aeµxsin(λx). Solutions: y=±√ AλeµxJ0(λx), where J0(z) is the Bessel function. 5.1-2. Equations of the Form⎝integraltextx 0K(x,t)y(t)y(x–t)dt=f(x). 18.⎝integraldisplay ⎝integraldisplayx 0tky(t)y(x–t)dt=Axλ,A>0 . Solutions: y(x)=±⎝bracketleftbiggAΓ(λ+1 ) Γ⎝parenleftbigλ+1+k 2⎝parenrightbig Γ⎝parenleftbigλ+1–k 2⎝parenrightbig⎝bracketrightbigg1/2 xλ–k–1 2, whereΓ(z) is the gamma function. 19.⎝integraldisplay ⎝integraldisplayx 0tky(t)y(x–t)dt=Aeλx. Solutions: y(x)=±⎝bracketleftbiggA Γ⎝parenleftbigk+1 2⎝parenrightbig Γ⎝parenleftbig1–k 2⎝parenrightbig⎝bracketrightbigg1/2 x–k+1 2eλx, whereΓ(z) is the gamma function. 20.⎝integraldisplay ⎝integraldisplayx 0tky(t)y(x–t)dt=Axµeλx. Solutions: y(x)=±⎝bracketleftbiggAΓ(µ+1 ) Γ⎝parenleftbigµ+k+1 2⎝parenrightbig Γ⎝parenleftbigµ–k+1 2⎝parenrightbig⎝bracketrightbigg1/2 xµ–k–1 2eλx, whereΓ(z) is the gamma function. 396 NONLINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 21.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t) ax+btdt=Axλ. Solutions: y(x)=±⎝radicalbigg A Ixλ/2,I=⎝integraldisplay1 0zλ/2(1 –z)λ/2dz a+bz. 22.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t) ax+btdt=Aeλx. Solutions: y(x)=±⎝radicalbigg A Ieλx,I=1 bln⎝parenleftBig 1+b a⎝parenrightBig . 23.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t) ax+btdt=Axµeλx. Solutions: y(x)=±⎝radicalbigg A Ixµ/2eλx,I=⎝integraldisplay1 0zµ/2(1 –z)µ/2dz a+bz. 24.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t) √ ax2+bt2dt=Axλ. Solutions: y(x)=±⎝radicalbigg A Ixλ/2,I=⎝integraldisplay1 0zλ/2(1 –z)λ/2dz √ a+bz2. 25.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t) √ ax2+bt2dt=Aeλx. Solutions: y(x)=±⎝radicalbigg A Ieλx,I=⎝integraldisplay1 0dz √ a+bz2. 26.⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t) √ ax2+bt2dt=Axµeλx. Solutions: y(x)=±⎝radicalbigg A Ixµ/2eλx,I=⎝integraldisplay1 0zµ/2(1 –z)µ/2dz √ a+bz2. 5.1-3. Equations of the Form⎝integraltextx 0y(t)y(···)dt=f(x). 27.⎝integraldisplay ⎝integraldisplayx 0y(t)y(ax+bt)dt=Axλ. Solutions: y(x)=±⎝radicalbigg A Ixλ–1 2,I=⎝integraldisplay1 0zλ–1 2(a+bz)λ–1 2dz. 5.2. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 397 28.⎝integraldisplay ⎝integraldisplayx 0y(t)y(ax–t)dt=Aeλx,a≥1. Solutions: y(x)=±⎝radicalbigg A Iexp(λx/a ) √ x,I=⎝integraldisplay1 0dz √ z(a–z). 29.⎝integraldisplay ⎝integraldisplayx 0y(t)y(ax–t)dt=Axµeλx,a≥1. Solutions: y(x)=±⎝radicalbigg A Ixµ–1 2exp(λx/a ), I=⎝integraldisplay1 0zµ–1 2(a–z)µ–1 2dz. 30.⎝integraldisplay ⎝integraldisplayx 0y(t)y(xt)dt=Axµ. Solutions: y(x)=±⎝radicalbigg 1 3A(2µ+1 )xµ–1 3 (A>0 ,µ≥0). 5.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions 5.2-1. Equations of the Form⎝integraltextx aK(x,t)[Ay (t)+By2(t)]dt=f(x). 1.⎝integraldisplay ⎝integraldisplayx a(x–t)[Ay (t)+By2(t)]dt=f(x), f(a)=f/prime(a)=0 . Solution in implicit form: Ay+By2–f/prime/prime xx(x)=0 . 2.⎝integraldisplay ⎝integraldisplayx a(x–t)n[Ay(t)+By2(t)]dt=f(x), f(a)=f/prime x(a)=··· =f(n) x(a)=0 . Heren=1 ,2 , ...Solution in implicit form: n!(Ay+By2)–f(n+1) x(x)=0 . 3.⎝integraldisplay ⎝integraldisplayx aeλ(x–t)[Ay(t)+By2(t)]dt=f(x), f(a)=0 . Solution in implicit form: Ay+By2+λf(x)–f/prime x(x)=0 . 4.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)][Ay (t)+By2(t)]dt=f(x), f(a)=f/prime(a)=0 . Solution in implicit form: λ(Ay+By2)+λ2f(x)–f/prime/prime xx(x)=0 . 398 NONLINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 5.⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)][Ay (t)+By2(t)]dt=f(x), f(a)=0 . Solution in implicit form: Ay+By2+λ2⎝integraldisplayx af(t)dt–f/prime x(x)=0 . 6.⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)][Ay (t)+By2(t)]dt=f(x), f(a)=f/prime(a)=0 . Solution in implicit form: λ(Ay+By2)–λ2f(x)–f/prime/prime xx(x)=0 . 7.⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)][Ay (t)+By2(t)]dt=f(x). Solution in implicit form: Ay+By2–λ2⎝integraldisplayx af(t)dt–f/prime x(x)=0 . 8.⎝integraldisplay ⎝integraldisplayx a[g(x)–g(t)][Ay (t)+By2(t)]dt=f(x). It is assumed that f(a)=f/prime x(a)=0a n d f/prime x/g/prime x≠const. Solution in implicit form: Ay+By2=d dx⎝bracketleftbiggf/prime x(x) g/primex(x)⎝bracketrightbigg . 9.⎝integraldisplay ⎝integraldisplayx aK(x,t)[Ay (t)+By2(t)]dt=f(x). The substitution w(t)=Ay(t)+By2(t) leads to the linear integral equation of the ﬁrst kind ⎝integraldisplayx aK(x,t)w(t)dt=f(x). For the exact solutions of the equation with various K(x,t)a n d f(x), see Chapter 1. 5.2-2. Equations of the Form⎝integraltextx aK(x,t)y(t)y(ax+bt)dt=f(x). 10.⎝integraldisplay ⎝integraldisplayx aK(t)y(x)y(t)dt=f(x). Solutions: y(x)=±f(x)⎝bracketleftbigg 2⎝integraldisplayx aK(t)f(t)dt⎝bracketrightbigg–1/2 . 11.⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)y(x–t)dt=Axλ. Solutions: y(x)=±⎝radicalbigg A Ixλ–1 2,I=⎝integraldisplay1 0f(z)zλ–1 2(1 –z)λ–1 2dz. 5.3. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 399 12.⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)y(x–t)dt=Aeλx. Solutions: y(x)=±⎝radicalbigg A Ieλx √ x,I=⎝integraldisplay1 0f(z)dz √ z(1 –z). 13.⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)y(x–t)dt=Axµeλx. Solutions: y(x)=±⎝radicalbigg A Ixµ–1 2eλx,I=⎝integraldisplay1 0f(z)zµ–1 2(1 –z)µ–1 2dz. 14.⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)y(ax+bt)dt=Axλ. Solutions: y(x)=±⎝radicalbigg A Ixλ–1 2,I=⎝integraldisplay1 0f(z)zλ–1 2(a+bz)λ–1 2dz. 15.⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)y(ax–t)dt=Aeλx,a≥1. Solutions: y(x)=±⎝radicalbigg A Iexp(λx/a ) √ x,I=⎝integraldisplay1 0f(z)dz √ z(a–z). 16.⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)y(ax–t)dt=Axµeλx,a≥1. Solutions: y(x)=±⎝radicalbigg A Ixµ–1 2exp(λx/a ), I=⎝integraldisplay1 0f(z)zµ–1 2(a–z)µ–1 2dz. 5.3. Equations with Nonlinearity of General Form 5.3-1. Equations of the Form⎝integraltextx aK(x,t)f(t,y(t))dt=g(x). 1.⎝integraldisplay ⎝integraldisplayx af⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), g(a)=0 . Solution in implicit form: f(x,y)–g/prime x(x)=0 . 2.⎝integraldisplay ⎝integraldisplayx a(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), g(a)=g/prime(a)=0 . Solution in implicit form: f(x,y)–g/prime/prime xx(x)=0 . 400 NONLINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRA TION 3.⎝integraldisplay ⎝integraldisplayx a(x–t)nf⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), g(a)=g/prime x(a)=··· =g(n) x(a)=0 . Heren=1 ,2 , ...Solution in implicit form: n!f(x,y)–g(n+1) x(x)=0 . 4.⎝integraldisplay ⎝integraldisplayx aeλ(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), g(a)=0 . Solution in implicit form: f(x,y)+λg(x)–g/prime x(x)=0 . 5.⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), g(a)=g/prime(a)=0 . Solution in implicit form: λf(x,y)+λ2g(x)–g/prime/prime xx(x)=0 . 6.⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)]f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), g(a)=0 . Solution in implicit form: f(x,y)+λ2⎝integraldisplayx ag(t)dt–g/prime x(x)=0 . 7.⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)]f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), g(a)=g/prime(a)=0 . Solution in implicit form: λf(x,y)–λ2g(x)–g/prime/prime xx(x)=0 . 8.⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)]f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). Solution in implicit form: f(x,y)–λ2⎝integraldisplayx ag(t)dt–g/prime x(x)=0 . 9.⎝integraldisplay ⎝integraldisplayx a[h(x)–h(t)]f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). It is assumed that g(a)=g/prime x(a)=0a n d g/prime x/h/primex≠const. Solution in implicit form: f(x,y)=d dx⎝bracketleftbiggg/prime x(x) h/primex(x)⎝bracketrightbigg . 10.⎝integraldisplay ⎝integraldisplayx aK(x,t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). The substitution w(t)=f⎝parenleftbig t,y(t)⎝parenrightbig leads to the linear integral equation of the ﬁrst kind ⎝integraldisplayx aK(x,t)w(t)dt=g(x). For the exact solutions of the equation with various K(x,t)a n d g(x), see Chapter 1. 5.3. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 401 5.3-2. Other Equations. 11.⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x,y(t),y(x)⎝parenrightbigg ⎝parenrightbigg dt=Ax. A solution: y(x)=λ,w h e r e λis a root of the algebraic (or transcendental) equation F(λ)–A=0 , F(λ)=⎝integraldisplay1 0f(z,λ,λ)dz. 12.⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x,y(t) y(x)⎝parenrightbigg ⎝parenrightbigg dt=Ax. A solution: y(x)=Cxλ,w h e r e Cis an arbitrary constant and λis a root of the algebraic (or transcendental) equation F(λ)–A=0 , F(λ)=⎝integraldisplay1 0f(z,zλ)dz. 13.⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x,y(t) y(x)⎝parenrightbigg ⎝parenrightbigg yα(t)dt=Axβ,α≠0. A solution: y(x)=A1/αxλ,λ=β–1 α, where λis a root of the algebraic (or transcendental) equation F(λ)–1=0 , F(λ)=⎝integraldisplay1 0f(z,zλ)zαλdz. 14.⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x,y(t) t,y(x) x⎝parenrightbigg ⎝parenrightbigg dt=Ax. A solution: y(x)=λx,w h e r e λis a root of the algebraic (or transcendental) equation F(λ)–A=0 , F(λ)=⎝integraldisplay1 0f(z,λ,λ)dz. 15.⎝integraldisplay ⎝integraldisplay∞ xf⎝parenleftbig⎝parenleftbig t–x,y(t–x)⎝parenrightbig⎝parenrightbig y(t)dt=Ae–λx. Solutions: y(x)=bke–λx,w h e r e bkare roots of the algebraic (or transcendental) equation bI(b)=A,I(b)=⎝integraldisplay∞ 0f(z,be–λz)e–λzdz. Chapter 6 Nonlinear Equations of the Second Kind with Variable Limit of Integration /trianglerightsld Notation: f,g, andhare arbitrary functions of an argument speciﬁed in the parentheses (the argument can depend on t,x, andy);A,B,C,a,b,k,β,λ, andµare arbitrary parameters. 6.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters 6.1-1. Equations of the Form y(x)+⎝integraltextx aK(x,t)y2(t)dt=F(x). 1. y(x)+A⎝integraldisplay ⎝integraldisplayx ay2(t)dt=Bx +C. By differentiation, this integral equation can be reduced to a separable ordinary differential equation. 1◦. Solution with AB>0 : y(x)=k(k+ya)e x p [ 2 Ak(x–a)] +ya–k (k+ya)e x p [ 2 Ak(x–a)] –ya+k,k=⎝radicalbigg B A,ya=aB+C. 2◦. Solution with AB<0 : y(x)=ktan⎝bracketleftBig Ak(a–x) + arctanya k⎝bracketrightBig ,k=⎝radicalbigg –B A,ya=aB+C. 3◦. Solution with B=0 : y(x)=C AC(x–a)+1. 2. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)y2(t)dt=Ax2+Bx +C. This is a special case of equation 6.8.3 with f(y)=ky2. Solution in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig 4Au–2kF(u)+B2–4AC⎝bracketrightbig–1/2du=±(x–a), F(u)=1 3⎝parenleftbig u3–y3 0⎝parenrightbig ,y0=Aa2+Ba+C. 403 404 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 3. y(x)+A⎝integraldisplay ⎝integraldisplayx atλy2(t)dt=Bxλ+1+C. This is a special case of equation 6.8.4 with f(y)=Ay2. By differentiation, this integral equation can be reduced to a separable ordinary differential equation. Solution in an implicit form: (λ+1 )⎝integraldisplayy yadu Au2–B(λ+1 )+xλ+1–aλ+1=0 , ya=Baλ+1+C. 4. y(x)+A⎝integraldisplay ⎝integraldisplayx 0x–λ–1y2(t)dt=Bxλ,λ>–1 2. Solutions: y1(x)=β1xλandy2(x)=β2xλ,w h e r e β1,2are the roots of the quadratic equation Aβ2+( 2λ+1 )β–B(2λ+1 )=0 . 5. y(x)+⎝integraldisplay ⎝integraldisplayx 0y2(t)dt ax+bt=A. Solutions: y1(x)=λ1andy2(x)=λ2,w h e r e λ1,2are the roots of the quadratic equation ln⎝parenleftBig 1+b a⎝parenrightBig λ2+bλ–Ab=0 . 6. y(x)+A⎝integraldisplay ⎝integraldisplayx 0y2(t)dt x2+t2=Bx . Solutions: y1(x)=λ1xandy2(x)=λ2x,w h e r e λ1,2are the roots of the quadratic equation⎝parenleftbig 1–1 4π⎝parenrightbig Aλ2+λ–B=0 . 7. y(x)+⎝integraldisplay ⎝integraldisplayx 0y2(t)dt √ ax2+bt2=A. Solutions: y1(x)=λ1andy2(x)=λ2,w h e r e λ1,2are the roots of the quadratic equation Iλ2+λ–A=0 , I=⎝integraldisplay1 0dz √ a+bz2. 8. y(x)+A⎝integraldisplay ⎝integraldisplayx 0⎝parenleftbig⎝parenleftbig axn+btn⎝parenrightbig⎝parenrightbig–λ+1 ny2(t)dt=Bxλ. Solutions: y1(x)=β1xλandy2(x)=β2xλ,w h e r e β1,2are the roots of the quadratic equation AIβ2+β–B=0 , I=⎝integraldisplay1 0z2λ⎝parenleftbig a+bzn⎝parenrightbig–λ+1 ndz. 9. y(x)+A⎝integraldisplay ⎝integraldisplayx aeλty2(t)dt=Beλx+C. This is a special case of equation 6.8.9 with f(y)=Ay2. By differentiation, this integral equation can be reduced to a separable ordinary differential equation. Solution in an implicit form: λ⎝integraldisplayy y0du Au2–Bλ+eλx–eλa=0 , y0=Beλa+C. 6.1. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY PARAMETERS 405 10. y(x)+A⎝integraldisplay ⎝integraldisplayx aeλ(x–t)y2(t)dt=B. This is a special case of equation 6.8.10. By differentiation, this integral equation can be reduced to the separable ordinary differential equation y/prime x+Ay2–λy+λB=0 , y(a)=B. Solution in an implicit form:⎝integraldisplayy Bdu Au2–λu+λB+x–a=0 . 11. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)y2(t)dt=Aeλx+B. Solution in an implicit form:⎝integraldisplayy y0du λu–ku2–λB=x–a,y0=Aeλa+B. 12. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]y2(t)dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=ky2. Solution in an implicit form:⎝integraldisplayy y0⎝bracketleftbig λ2u2–2λ2Cu–2kλF (u)+λ2(C2–4AB)⎝bracketrightbig–1/2du=±(x–a), F(u)=1 3⎝parenleftbig u3–y3 0⎝parenrightbig ,y0=Aeλa+Be–λa+C. 13. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]y2(t)dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=ky2. Solution in an implicit form:⎝integraldisplayy y0⎝bracketleftbig λ2u2–2λ2Bu–2kλF (u)+λ2(B2–A2)⎝bracketrightbig–1/2du=±(x–a), F(u)=1 3⎝parenleftbig u3–y3 0⎝parenrightbig ,y0=Acosh(λa )+B. 14. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]y2(t)dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=ky2. Solution in an implicit form:⎝integraldisplayy y0⎝bracketleftbig λ2u2–2λ2Bu–2kλF (u)+λ2(A2+B2)⎝bracketrightbig–1/2du=±(x–a), F(u)=1 3⎝parenleftbig u3–y3 0⎝parenrightbig ,y0=Asinh(λa)+B. 15. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)]y2(t)dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=ky2. Solution in an implicit form:⎝integraldisplayy y0⎝bracketleftbig λ2D–λ2u2+2λ2Cu–2kλF (u)⎝bracketrightbig–1/2du=±(x–a), y0=Asin(λa)+Bcos(λa )+C,D=A2+B2–C2,F(u)=1 3⎝parenleftbig u3–y3 0⎝parenrightbig . 406 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 6.1-2. Equations of the Form y(x)+⎝integraltextx aK(x,t)y(t)y(x–t)dt=F(x). 16. y(x)+A⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=AB2x+B. A solution: y(x)=B. 17. y(x)+A⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=(AB2x+B)eλx. A solution: y(x)=Beλx. 18. y(x)+λ 2β⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=1 2βsinh(λx). A solution: y(x)=βI1(λx), where I1(x) is the modiﬁed Bessel function. 19. y(x)–λ 2β⎝integraldisplay ⎝integraldisplayx 0y(t)y(x–t)dt=1 2βsin(λx). A solution: y(x)=βJ1(λx), where J1(x) is the Bessel function. 20. y(x)+A⎝integraldisplay ⎝integraldisplayx 0x–λ–1y(t)y(x–t)dt=Bxλ. Solutions: y1(x)=β1xλandy2(x)=β2xλ,w h e r e β1,2are the roots of the quadratic equation AIβ2+β–B=0 , I=⎝integraldisplay1 0zλ(1 –z)λdz=Γ2(λ+1 ) Γ(2λ+2 ). 6.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions 6.2-1. Equations of the Form y(x)+⎝integraltextx aK(x,t)y2(t)dt=F(x). 1. y(x)+⎝integraldisplay ⎝integraldisplayx af(t)y2(t)dt=A. Solution: y(x)=A⎝bracketleftbigg 1+A⎝integraldisplayx af(t)dt⎝bracketrightbigg–1 . 2. y(x)+⎝integraldisplay ⎝integraldisplayx aeλ(x–t)g(t)y2(t)dt=f(x). Differentiating the equation with respect to xyields y/prime x+g(x)y2+λ⎝integraldisplayx aeλ(x–t)g(t)y2(t)dt=f/prime x(x). (1) Eliminating the integral term from (1) with the aid of the original equation, we arrive at a Riccati ordinary differential equation, y/prime x+g(x)y2–λy+λf(x)–f/prime x(x)=0 , ( 2 ) under the initial condition y(a)=f(a). Equation (2) can be reduced to a second-order linear ordinary differential equation. For the exact solutions of equation (2) with various speciﬁcfunctions fandg, see, for example, E. Kamke (1977) and A. D. Polyanin and V . F. Zaitsev (2003). 6.2. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 407 3. y(x)+⎝integraldisplay ⎝integraldisplayx ag(x)h(t)y2(t)dt=f(x). Differentiating the equation with respect to xyields y/prime x+g(x)h(x)y2+g/prime x(x)⎝integraldisplayx ah(t)y2(t)dt=f/prime x(x). (1) Eliminating the integral term from (1) with the aid of the original equation, we arrive at a Riccati ordinary differential equation, y/prime x+g(x)h(x)y2–g/prime x(x) g(x)y=f/prime x(x)–g/prime x(x) g(x)f(x), (2) under the initial condition y(a)=f(a). Equation (2) can be reduced to a second-order linear ordinary differential equation. For the exact solutions of equation (2) with various speciﬁc functions f,g,a n dh, see, for example, E. Kamke (1977) and A. D. Polyanin and V . F. Zaitsev (2003). 4. y(x)+⎝integraldisplay ⎝integraldisplayx 0x–λ–1f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y2(t)dt=Axλ. Solutions: y1(x)=β1xλandy2(x)=β2xλ,w h e r e β1,2are the roots of the quadratic equation Iβ2+β–A=0 , I=⎝integraldisplay1 0f(z)z2λdz. 5. y(x)–⎝integraldisplay ⎝integraldisplayx –∞eλt+βxf(x–t)y2(t)dt=0 . This is a special case of equation 6.3.19 with k=2 . 6. y(x)–⎝integraldisplay ⎝integraldisplay∞ xeλt+βxf(x–t)y2(t)dt=0 . A solution: y(x)=1 Ae–(λ+β)x,A=⎝integraldisplay∞ 0e–(λ+2β)zf(–z)dz. 6.2-2. Other Equations. 7. y(x)+⎝integraldisplay ⎝integraldisplayx 01 xf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)y(x–t)dt=Aeλx. Solutions: y1(x)=B1eλx,y2(x)=B2eλx, where B1andB2are the roots of the quadratic equation IB2+B–A=0 , I=⎝integraldisplay1 0f(z)dz. 408 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 8. y(x)+A⎝integraldisplay ⎝integraldisplayx 0x–λ–1f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg y(t)y(x–t)dt=Bxλ. Solutions: y1(x)=β1xλandy2(x)=β2xλ,w h e r e β1,2are the roots of the quadratic equation AIβ2+β–B=0 , I=⎝integraldisplay1 0f(z)zλ(1 –z)λdz. 9. y(x)+⎝integraldisplay ⎝integraldisplay∞ xf(t–x)y(t–x)y(t)dt=ae–λx. Solutions: y(x)=bke–λx,w h e r e bk(k= 1, 2) are the roots of the quadratic equation b2I+b–a=0 , I=⎝integraldisplay∞ 0f(z)e–2λzdz. To calculate the integral I, it is convenient to use tables of Laplace transforms (with parameter p=2λ). 6.3. Equations with Power-Law Nonlinearity 6.3-1. Equations Containing Arbitrary Parameters. 1. y(x)=a⎝integraldisplay ⎝integraldisplayx 0yk(t)dt+b,a>0 ,b>0 ,k>0 . Solution: y(x)=⎧ ⎪⎨ ⎪⎩[b1–k+a(1 –k)x]1 1–k if 0 < k<1 , beaxifk=1 , [b1–k–a(k–1 )x]1 1–k ifk>1 . If 0 <k≤1, the solution exists for all x≥0. Ifk> 1, the continuous solution exists only in a limited interval of argument variation 0≤x<x∗=b1–k a(k–1 ). 2. y(x)+A⎝integraldisplay ⎝integraldisplayx atλyk(t)dt=Bxλ+1+C. By differentiation, this integral equation can be reduced to a separable ordinary differential equation. Solution in an implicit form: (λ+1 )⎝integraldisplayy y0du Auk–B(λ+1 )+xλ+1–aλ+1=0 , y0=Baλ+1+C. 3. y(x)+⎝integraldisplay ⎝integraldisplayx 0yk(t) ax +btdt=A. A solution: y(x)=λ,w h e r e λis a root of the algebraic (or transcendental) equation ln⎝parenleftBig 1+b a⎝parenrightBig λk+bλ–Ab=0 . 6.3. E QUATIONS WITH POWER -LAWNONLINEARITY 409 4. y(x)+Ax⎝integraldisplay ⎝integraldisplayx 0yk(t)dt x2+t2=B. A solution: y(x)=λ,w h e r e λis a root of the algebraic (or transcendental) equation λ+1 4Aπλk=B. 5. y(x)+⎝integraldisplay ⎝integraldisplayx 0yk(t)dt √ ax2+bt2=A. A solution: y(x)=λ,w h e r e λis a root of the algebraic (or transcendental) equation Iλk+λ–A=0 , I=⎝integraldisplay1 0dz √ a+bz2. 6. y(x)+A⎝integraldisplay ⎝integraldisplayx a⎝parenleftbig⎝parenleftbig axn+btn⎝parenrightbig⎝parenrightbigλ–kλ–1 nyk(t)dt=Bxλ. A solution: y=βxλ,w h e r e βis a root of the algebraic (or transcendental) equation AIβk+β–B=0 , I=⎝integraldisplay1 0zkλ⎝parenleftbig a+bzn⎝parenrightbigλ–kλ–1 ndz. 7. y(x)+A⎝integraldisplay ⎝integraldisplayx aeλtyµ(t)dt=Beλx+C. By differentiation, this integral equation can be reduced to a separable ordinary differential equation. Solution in an implicit form: λ⎝integraldisplayy y0du Auµ–Bλ+eλx–eλa=0 , y0=Beλa+C. 8. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)yµ(t)dt=Aeλx+B. Solution in an implicit form: ⎝integraldisplayy y0dt λt–ktµ–λB=x–a,y0=Aeλa+B. 9. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]yµ(t)dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=kyµ. Solution in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig λ2u2–2λ2Cu–2kλF (u)+λ2(C2–4AB)⎝bracketrightbig–1/2du=±(x–a), F(u)=1 µ+1⎝parenleftbig uµ+1–yµ+1 0⎝parenrightbig ,y0=Aeλa+Be–λa+C. 410 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 10. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]yµ(t)dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=kyµ. Solution in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig λ2u2–2λ2Bu–2kλF (u)+λ2(B2–A2)⎝bracketrightbig–1/2du=±(x–a), F(u)=1 µ+1⎝parenleftbig uµ+1–yµ+1 0⎝parenrightbig ,y0=Acosh(λa )+B. 11. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]yµ(t)dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=kyµ. Solution in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig λ2u2–2λ2Bu–2kλF (u)+λ2(A2+B2)⎝bracketrightbig–1/2du=±(x–a), F(u)=1 µ+1⎝parenleftbig uµ+1–yµ+1 0⎝parenrightbig ,y0=Asinh(λa)+B. 12. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)]yµ(t)dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=kyµ. Solution in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig λ2D–λ2u2+2λ2Cu–2kλF (u)⎝bracketrightbig–1/2du=±(x–a), y0=Asin(λa)+Bcos(λa )+C,D=A2+B2–C2,F(u)=1 µ+1⎝parenleftbig uµ+1–yµ+1 0⎝parenrightbig . 6.3-2. Equations Containing Arbitrary Functions. 13. y(x)+⎝integraldisplay ⎝integraldisplayx af(t)yk(t)dt=A. Solution: y(x)=⎝bracketleftbigg A1–k+(k–1 )⎝integraldisplayx af(t)dt⎝bracketrightbigg1 1–k . 14. y(x)–⎝integraldisplay ⎝integraldisplayx af(x)g(t)yk(t)dt=0 . 1◦. Differentiating the equation with respect to xand eliminating the integral term (using the original equation), we obtain the Bernoulli ordinary differential equation y/prime x–f(x)g(x)yk–f/prime x(x) f(x)y=0 , y(a)=0 . 2◦. Solution with k<1 : y(x)=f(x)⎝bracketleftbigg (1 –k)⎝integraldisplayx afk(t)g(t)dt⎝bracketrightbigg1 1–k . Additionally, for k> 0, there is the trivial solution y(x)≡0. 6.4. E QUATIONS WITH EXPONENTIAL NONLINEARITY 411 15. y(x)+⎝integraldisplay ⎝integraldisplayx 0xλ–kλ–1f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg yk(t)dt=Axλ. A solution: y(x)=βxλ,w h e r e βis a root of the algebraic equation Iβk+β–A=0 , I=⎝integraldisplay1 0f(z)zkλdz. 16. y(x)+⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg⎝radicalbig y(t)dt=Ax2. Solutions: yk(x)=B2 kx2,w h e r e Bk(k= 1, 2) are the roots of the quadratic equations B2±IB–A=0 , I=⎝integraldisplay1 0zf(z)dz. 17. y(x)–⎝integraldisplay ⎝integraldisplayx 0taf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg yk(t)dt=0 , k≠1. A solution: y(x)=Ax1+a 1–k,A1–k=⎝integraldisplay1 0za+k 1–kf(z)dz. 18. y(x)–⎝integraldisplay ⎝integraldisplay∞ xeλt+βxf(x–t)yk(t)dt=0 , k≠1. A solution: y(x)=Aexp⎝parenleftBigλ+β 1–kx⎝parenrightBig ,A1–k=⎝integraldisplay∞ 0exp⎝parenleftBigλ+βk 1–kz⎝parenrightBig f(–z)dz. 19. y(x)–⎝integraldisplay ⎝integraldisplayx –∞eλt+βxf(x–t)yk(t)dt=0 , k≠1. A solution: y(x)=Aexp⎝parenleftBigλ+β 1–kx⎝parenrightBig ,A1–k=⎝integraldisplay∞ 0exp⎝parenleftBig –λ+βk 1–kz⎝parenrightBig f(z)dz. 6.4. Equations with Exponential Nonlinearity 6.4-1. Equations Containing Arbitrary Parameters. 1. y(x)+A⎝integraldisplay ⎝integraldisplayx aexp[λy(t)]dt=B. Solution: y(x)=–1 λln⎝bracketleftbig Aλ(x–a)+e–Bλ⎝bracketrightbig . 2. y(x)+A⎝integraldisplay ⎝integraldisplayx aexp[λy(t)]dt=Bx +C. ForB= 0, see equation 6.4.1. Solution with B≠0: y(x)=–1 λln⎝bracketleftbiggA B+⎝parenleftBig e–λy0–A B⎝parenrightBig eλB(a–x)⎝bracketrightbigg ,y0=aB+C. 412 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 3. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)e x p [λy(t)]dt=Ax2+Bx +C. 1◦. This is a special case of equation 6.8.3 with f(y)=keλy. The solution of this integral equation is determined by the solution of the second-order autonomous ordinary differential equation y/prime/prime xx+keλy–2A=0 under the initial conditions y(a)=Aa2+Ba+C,y/prime x(a)=2Aa+B. 2◦. Solution in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig 4Au–2F(u)+B2–4AC⎝bracketrightbig–1/2du=±(x–a), F(u)=k λ⎝parenleftbig eλu–eλy0⎝parenrightbig ,y0=Aa2+Ba+C. 4. y(x)+A⎝integraldisplay ⎝integraldisplayx atλexp[βy(t)]dt=Bxλ+1+C. By differentiation, this integral equation can be reduced to a separable ordinary differential equation. Solution in an implicit form: (λ+1 )⎝integraldisplayy y0du Aeβu–B(λ+1 )+xλ+1–aλ+1=0 , y0=Baλ+1+C. 5. y(x)+⎝integraldisplay ⎝integraldisplayx 0exp[λy(t)] ax +btdt=A. A solution: y(x)=β,w h e r e βis a root of the transcendental equation ln⎝parenleftBig 1+b a⎝parenrightBig eλβ+bβ–Ab=0 . 6. y(x)+⎝integraldisplay ⎝integraldisplayx 0exp[λy(t)] √ ax2+bt2dt=A. A solution: y(x)=β,w h e r e βis a root of the transcendental equation keλβ+β–A=0 , k=⎝integraldisplay1 0dz √ a+bz2. 7. y(x)+A⎝integraldisplay ⎝integraldisplayx aexp⎝bracketleftbig⎝bracketleftbig λt+βy(t)⎝bracketrightbig⎝bracketrightbig dt=Beλx+C. By differentiation, this integral equation can be reduced to a separable ordinary differential equation. Solution in an implicit form: λ⎝integraldisplayy y0du Aeβu–Bλ+eλx–eλa=0 , y0=Beλa+C. 6.4. E QUATIONS WITH EXPONENTIAL NONLINEARITY 413 8. y(x)+k⎝integraldisplay ⎝integraldisplayx aexp⎝bracketleftbig⎝bracketleftbig λ(x–t)+βy(t)⎝bracketrightbig⎝bracketrightbig dt=A. Solution in an implicit form: ⎝integraldisplayy Adt λt–keβt–λA=x–a. 9. y(x)+k⎝integraldisplay ⎝integraldisplayx aexp⎝bracketleftbig⎝bracketleftbig λ(x–t)+βy(t)⎝bracketrightbig⎝bracketrightbig dt=Aeλx+B. Solution in an implicit form: ⎝integraldisplayy y0dt λt–keβt–λB=x–a,y0=Aeλa+B. 10. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] exp[ βy(t)]dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=keβy. 11. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] exp[ βy(t)]dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=keβy. 12. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] exp[ βy(t)]dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=keβy. 13. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] exp[ βy(t)]dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=keβy. 6.4-2. Equations Containing Arbitrary Functions. 14. y(x)+⎝integraldisplay ⎝integraldisplayx af(t)e x p [λy(t)]dt=A. Solution: y(x)=–1 λln⎝bracketleftbigg λ⎝integraldisplayx af(t)dt+e–Aλ⎝bracketrightbigg . 15. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t)e x p [λy(t)]dt=f(x). 1◦. By differentiation, this integral equation can be reduced to the ﬁrst-order ordinary differ- ential equation y/prime x+g(x)eλy=f/prime x(x)( 1) under the initial condition y(a)=f(a). The substitution w=e–λyreduces (1) to the linear equation w/prime x+λf/prime x(x)w–λg(x)=0 , w(a)=e x p⎝bracketleftbig –λf(a)⎝bracketrightbig . 2◦. Solution: y(x)=f(x)–1 λln⎝braceleftbigg 1+λ⎝integraldisplayx ag(t)e x p⎝bracketleftbig λf(t)⎝bracketrightbig dt⎝bracerightbigg . 414 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 16. y(x)+1 x⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg exp[λy(t)]dt=A. A solution: y(x)=β,w h e r e βis a root of the transcendental equation β+Ieλβ–A=0 , I=⎝integraldisplay1 0f(z)dz. 6.5. Equations with Hyperbolic Nonlinearity 6.5-1. Integrands with Nonlinearity of the Form cosh[βy (t)]. 1. y(x)+k⎝integraldisplay ⎝integraldisplayx acosh[βy (t)]dt=A. This is a special case of equation 6.8.1 with f(y)=kcosh(βy). 2. y(x)+k⎝integraldisplay ⎝integraldisplayx acosh[βy (t)]dt=Ax +B. This is a special case of equation 6.8.2 with f(y)=kcosh(βy). 3. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)c o s h [ βy(t)]dt=Ax2+Bx +C. This is a special case of equation 6.8.3 with f(y)=kcosh(βy). 4. y(x)+k⎝integraldisplay ⎝integraldisplayx atλcosh[βy (t)]dt=Bxλ+1+C. This is a special case of equation 6.8.4 with f(y)=kcosh(βy). 5. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t)c o s h [ βy(t)]dt=A. This is a special case of equation 6.8.5 with f(y)=c o s h ( βy). 6. y(x)+⎝integraldisplay ⎝integraldisplayx 0cosh[βy (t)] ax +btdt=A. This is a special case of equation 6.8.6 with f(y)=c o s h ( βy). 7. y(x)+⎝integraldisplay ⎝integraldisplayx 0cosh[βy (t)] √ ax2+bt2dt=A. This is a special case of equation 6.8.7 with f(y)=c o s h ( βy). 8. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλtcosh[βy (t)]dt=Beλx+C. This is a special case of equation 6.8.9 with f(y)=kcosh(βy). 9. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)cosh[βy (t)]dt=A. This is a special case of equation 6.8.10 with f(y)=kcosh(βy ). 6.5. E QUATIONS WITH HYPERBOLIC NONLINEARITY 415 10. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)cosh[βy (t)]dt=Aeλx+B. This is a special case of equation 6.8.11 with f(y)=kcosh(βy ). 11. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cosh[βy (t)]dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=kcosh(βy ). 12. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cosh[βy (t)]dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=kcosh(βy ). 13. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cosh[βy (t)]dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=kcosh(βy ). 14. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] cosh[βy (t)]dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=kcosh(βy ). 6.5-2. Integrands with Nonlinearity of the Form sinh[ βy(t)]. 15. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[βy(t)]dt=A. This is a special case of equation 6.8.1 with f(y)=ksinh(βy). 16. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[βy(t)]dt=Ax +B. This is a special case of equation 6.8.2 with f(y)=ksinh(βy). 17. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)s i n h [ βy(t)]dt=Ax2+Bx +C. This is a special case of equation 6.8.3 with f(y)=ksinh(βy). 18. y(x)+k⎝integraldisplay ⎝integraldisplayx atλsinh[βy(t)]dt=Bxλ+1+C. This is a special case of equation 6.8.4 with f(y)=ksinh(βy). 19. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t) sinh[ βy(t)]dt=A. This is a special case of equation 6.8.5 with f(y)=s i n h ( βy). 20. y(x)+⎝integraldisplay ⎝integraldisplayx 0sinh[βy(t)] ax+btdt=A. This is a special case of equation 6.8.6 with f(y)=s i n h ( βy). 416 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 21. y(x)+⎝integraldisplay ⎝integraldisplayx 0sinh[βy(t)] √ ax2+bt2dt=A. This is a special case of equation 6.8.7 with f(y)=s i n h ( βy). 22. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλtsinh[βy(t)]dt=Beλx+C. This is a special case of equation 6.8.9 with f(y)=ksinh(βy). 23. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)sinh[βy(t)]dt=A. This is a special case of equation 6.8.10 with f(y)=ksinh(βy). 24. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)sinh[βy(t)]dt=Aeλx+B. This is a special case of equation 6.8.11 with f(y)=ksinh(βy). 25. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] sinh[ βy(t)]dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=ksinh(βy). 26. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] sinh[ βy(t)]dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=ksinh(βy). 27. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] sinh[ βy(t)]dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=ksinh(βy). 28. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] sinh[ βy(t)]dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=ksinh(βy). 6.5-3. Integrands with Nonlinearity of the Form tanh[ βy(t)]. 29. y(x)+k⎝integraldisplay ⎝integraldisplayx atanh[βy(t)]dt=A. This is a special case of equation 6.8.1 with f(y)=ktanh(βy). 30. y(x)+k⎝integraldisplay ⎝integraldisplayx atanh[βy(t)]dt=Ax +B. This is a special case of equation 6.8.2 with f(y)=ktanh(βy). 31. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)t a n h [ βy(t)]dt=Ax2+Bx +C. This is a special case of equation 6.8.3 with f(y)=ktanh(βy). 6.5. E QUATIONS WITH HYPERBOLIC NONLINEARITY 417 32. y(x)+k⎝integraldisplay ⎝integraldisplayx atλtanh[βy(t)]dt=Bxλ+1+C. This is a special case of equation 6.8.4 with f(y)=ktanh(βy). 33. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t)t a n h [ βy(t)]dt=A. This is a special case of equation 6.8.5 with f(y)=t a n h ( βy). 34. y(x)+⎝integraldisplay ⎝integraldisplayx 0tanh[βy(t)] ax+btdt=A. This is a special case of equation 6.8.6 with f(y)=t a n h ( βy). 35. y(x)+⎝integraldisplay ⎝integraldisplayx 0tanh[βy(t)] √ ax2+bt2dt=A. This is a special case of equation 6.8.7 with f(y)=t a n h ( βy). 36. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλttanh[βy(t)]dt=Beλx+C. This is a special case of equation 6.8.9 with f(y)=ktanh(βy). 37. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)tanh[βy(t)]dt=A. This is a special case of equation 6.8.10 with f(y)=ktanh(βy). 38. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)tanh[βy(t)]dt=Aeλx+B. This is a special case of equation 6.8.11 with f(y)=ktanh(βy). 39. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] tanh[ βy(t)]dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=ktanh(βy). 40. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] tanh[ βy(t)]dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=ktanh(βy). 41. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] tanh[ βy(t)]dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=ktanh(βy). 42. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] tanh[ βy(t)]dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=ktanh(βy). 418 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 6.5-4. Integrands with Nonlinearity of the Form coth[βy (t)]. 43. y(x)+k⎝integraldisplay ⎝integraldisplayx acoth[βy(t)]dt=A. This is a special case of equation 6.8.1 with f(y)=kcoth(βy). 44. y(x)+k⎝integraldisplay ⎝integraldisplayx acoth[βy(t)]dt=Ax +B. This is a special case of equation 6.8.2 with f(y)=kcoth(βy). 45. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)c o t h [ βy(t)]dt=Ax2+Bx +C. This is a special case of equation 6.8.3 with f(y)=kcoth(βy). 46. y(x)+k⎝integraldisplay ⎝integraldisplayx atλcoth[βy (t)]dt=Bxλ+1+C. This is a special case of equation 6.8.4 with f(y)=kcoth(βy). 47. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t)c o t h [ βy(t)]dt=A. This is a special case of equation 6.8.5 with f(y)=c o t h ( βy). 48. y(x)+⎝integraldisplay ⎝integraldisplayx 0coth[βy (t)] ax +btdt=A. This is a special case of equation 6.8.6 with f(y)=c o t h ( βy). 49. y(x)+⎝integraldisplay ⎝integraldisplayx 0coth[βy (t)] √ ax2+bt2dt=A. This is a special case of equation 6.8.7 with f(y)=c o t h ( βy). 50. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλtcoth[βy (t)]dt=Beλx+C. This is a special case of equation 6.8.9 with f(y)=kcoth(βy). 51. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)coth[βy (t)]dt=A. This is a special case of equation 6.8.10 with f(y)=kcoth(βy). 52. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)coth[βy (t)]dt=Aeλx+B. This is a special case of equation 6.8.11 with f(y)=kcoth(βy). 53. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] coth[βy (t)]dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=kcoth(βy). 6.6. E QUATIONS WITH LOGARITHMIC NONLINEARITY 419 54. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] coth[βy (t)]dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=kcoth(βy). 55. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] coth[βy (t)]dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=kcoth(βy). 56. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] coth[βy (t)]dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=kcoth(βy). 6.6. Equations with Logarithmic Nonlinearity 6.6-1. Integrands Containing Power-Law Functions of xandt. 1. y(x)+k⎝integraldisplay ⎝integraldisplayx aln[λy(t)]dt=A. This is a special case of equation 6.8.1 with f(y)=kln(λy). 2. y(x)+k⎝integraldisplay ⎝integraldisplayx aln[λy(t)]dt=Ax +B. This is a special case of equation 6.8.2 with f(y)=kln(λy). 3. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)l n [λy(t)]dt=Ax2+Bx +C. This is a special case of equation 6.8.3 with f(y)=kln(λy). 4. y(x)+k⎝integraldisplay ⎝integraldisplayx atλln[µy(t)]dt=Bxλ+1+C. This is a special case of equation 6.8.4 with f(y)=kln(µy). 5. y(x)+⎝integraldisplay ⎝integraldisplayx 0ln[λy(t)] ax+btdt=A. This is a special case of equation 6.8.6 with f(y)=l n ( λy). 6. y(x)+⎝integraldisplay ⎝integraldisplayx 0ln[λy(t)] √ ax2+bt2dt=A. This is a special case of equation 6.8.7 with f(y)=l n ( λy). 6.6-2. Integrands Containing Exponential Functions of xandt. 7. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλtln[µy(t)]dt=Beλx+C. This is a special case of equation 6.8.9 with f(y)=kln(µy). 420 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 8. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)ln[µy(t)]dt=A. This is a special case of equation 6.8.10 with f(y)=kln(µy). 9. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)ln[µy(t)]dt=Aeλx+B. This is a special case of equation 6.8.11 with f(y)=kln(µy). 6.6-3. Other Integrands. 10. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t)l n [λy(t)]dt=A. This is a special case of equation 6.8.5 with f(y)=l n ( λy). 11. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] ln[µy(t)]dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=kln(µy). 12. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] ln[µy(t)]dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=kln(µy). 13. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] ln[µy(t)]dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=kln(µy). 14. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] ln[µy(t)]dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=kln(µy). 6.7. Equations with Trigonometric Nonlinearity 6.7-1. Integrands with Nonlinearity of the Form cos[ βy(t)]. 1. y(x)+k⎝integraldisplay ⎝integraldisplayx acos[βy (t)]dt=A. This is a special case of equation 6.8.1 with f(y)=kcos(βy ). 2. y(x)+k⎝integraldisplay ⎝integraldisplayx acos[βy (t)]dt=Ax +B. This is a special case of equation 6.8.2 with f(y)=kcos(βy ). 3. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)c o s [βy(t)]dt=Ax2+Bx +C. This is a special case of equation 6.8.3 with f(y)=kcos(βy ). 6.7. E QUATIONS WITH TRIGONOMETRIC NONLINEARITY 421 4. y(x)+k⎝integraldisplay ⎝integraldisplayx atλcos[βy (t)]dt=Bxλ+1+C. This is a special case of equation 6.8.4 with f(y)=kcos(βy ). 5. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t)c o s [βy(t)]dt=A. This is a special case of equation 6.8.5 with f(y)=c o s ( βy). 6. y(x)+⎝integraldisplay ⎝integraldisplayx 0cos[βy (t)] ax +btdt=A. This is a special case of equation 6.8.6 with f(y)=c o s ( βy). 7. y(x)+⎝integraldisplay ⎝integraldisplayx 0cos[βy (t)] √ ax2+bt2dt=A. This is a special case of equation 6.8.7 with f(y)=c o s ( βy). 8. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλtcos[βy (t)]dt=Beλx+C. This is a special case of equation 6.8.9 with f(y)=kcos(βy ). 9. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)cos[βy (t)]dt=A. This is a special case of equation 6.8.10 with f(y)=kcos(βy ). 10. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)cos[βy (t)]dt=Aeλx+B. This is a special case of equation 6.8.11 with f(y)=kcos(βy ). 11. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cos[βy (t)]dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=kcos(βy ). 12. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cos[βy (t)]dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=kcos(βy ). 13. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cos[βy (t)]dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=kcos(βy ). 14. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] cos[βy (t)]dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=kcos(βy ). 422 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 6.7-2. Integrands with Nonlinearity of the Form sin[ βy(t)]. 15. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[βy(t)]dt=A. This is a special case of equation 6.8.1 with f(y)=ksin(βy). 16. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[βy(t)]dt=Ax +B. This is a special case of equation 6.8.2 with f(y)=ksin(βy). 17. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)s i n [βy(t)]dt=Ax2+Bx +C. This is a special case of equation 6.8.3 with f(y)=ksin(βy). 18. y(x)+k⎝integraldisplay ⎝integraldisplayx atλsin[βy(t)]dt=Bxλ+1+C. This is a special case of equation 6.8.4 with f(y)=ksin(βy). 19. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t)s i n [βy(t)]dt=A. This is a special case of equation 6.8.5 with f(y)=s i n ( βy). 20. y(x)+⎝integraldisplay ⎝integraldisplayx 0sin[βy(t)] ax+btdt=A. This is a special case of equation 6.8.6 with f(y)=s i n ( βy). 21. y(x)+⎝integraldisplay ⎝integraldisplayx 0sin[βy(t)] √ ax2+bt2dt=A. This is a special case of equation 6.8.7 with f(y)=s i n ( βy). 22. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλtsin[βy(t)]dt=Beλx+C. This is a special case of equation 6.8.9 with f(y)=ksin(βy). 23. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)sin[βy(t)]dt=A. This is a special case of equation 6.8.10 with f(y)=ksin(βy). 24. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)sin[βy(t)]dt=Aeλx+B. This is a special case of equation 6.8.11 with f(y)=ksin(βy). 25. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] sin[βy(t)]dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=ksin(βy). 6.7. E QUATIONS WITH TRIGONOMETRIC NONLINEARITY 423 26. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] sin[βy(t)]dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=ksin(βy). 27. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] sin[βy(t)]dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=ksin(βy). 28. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] sin[βy(t)]dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=ksin(βy). 6.7-3. Integrands with Nonlinearity of the Form tan[ βy(t)]. 29. y(x)+k⎝integraldisplay ⎝integraldisplayx atan[βy(t)]dt=A. This is a special case of equation 6.8.1 with f(y)=ktan(βy). 30. y(x)+k⎝integraldisplay ⎝integraldisplayx atan[βy(t)]dt=Ax +B. This is a special case of equation 6.8.2 with f(y)=ktan(βy). 31. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)t a n [βy(t)]dt=Ax2+Bx +C. This is a special case of equation 6.8.3 with f(y)=ktan(βy). 32. y(x)+k⎝integraldisplay ⎝integraldisplayx atλtan[βy(t)]dt=Bxλ+1+C. This is a special case of equation 6.8.4 with f(y)=ktan(βy). 33. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t)t a n [βy(t)]dt=A. This is a special case of equation 6.8.5 with f(y)=t a n ( βy). 34. y(x)+⎝integraldisplay ⎝integraldisplayx 0tan[βy(t)] ax+btdt=A. This is a special case of equation 6.8.6 with f(y)=t a n ( βy). 35. y(x)+⎝integraldisplay ⎝integraldisplayx 0tan[βy(t)] √ ax2+bt2dt=A. This is a special case of equation 6.8.7 with f(y)=t a n ( βy). 36. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλttan[βy(t)]dt=Beλx+C. This is a special case of equation 6.8.9 with f(y)=ktan(βy). 424 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 37. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)tan[βy(t)]dt=A. This is a special case of equation 6.8.10 with f(y)=ktan(βy). 38. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)tan[βy(t)]dt=Aeλx+B. This is a special case of equation 6.8.11 with f(y)=ktan(βy). 39. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] tan[βy (t)]dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=ktan(βy). 40. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] tan[βy (t)]dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=ktan(βy). 41. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] tan[βy (t)]dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=ktan(βy). 42. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] tan[βy (t)]dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=ktan(βy). 6.7-4. Integrands with Nonlinearity of the Form cot[ βy(t)]. 43. y(x)+k⎝integraldisplay ⎝integraldisplayx acot[βy(t)]dt=A. This is a special case of equation 6.8.1 with f(y)=kcot(βy). 44. y(x)+k⎝integraldisplay ⎝integraldisplayx acot[βy(t)]dt=Ax +B. This is a special case of equation 6.8.2 with f(y)=kcot(βy). 45. y(x)+k⎝integraldisplay ⎝integraldisplayx a(x–t)c o t [βy(t)]dt=Ax2+Bx +C. This is a special case of equation 6.8.3 with f(y)=kcot(βy). 46. y(x)+k⎝integraldisplay ⎝integraldisplayx atλcot[βy(t)]dt=Bxλ+1+C. This is a special case of equation 6.8.4 with f(y)=kcot(βy). 47. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t)c o t [βy(t)]dt=A. This is a special case of equation 6.8.5 with f(y)=c o t ( βy). 6.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 425 48. y(x)+⎝integraldisplay ⎝integraldisplayx 0cot[βy(t)] ax +btdt=A. This is a special case of equation 6.8.6 with f(y)=c o t ( βy). 49. y(x)+⎝integraldisplay ⎝integraldisplayx 0cot[βy(t)] √ ax2+bt2dt=A. This is a special case of equation 6.8.7 with f(y)=c o t ( βy). 50. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλtcot[βy(t)]dt=Beλx+C. This is a special case of equation 6.8.9 with f(y)=kcot(βy). 51. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)cot[βy(t)]dt=A. This is a special case of equation 6.8.10 with f(y)=kcot(βy). 52. y(x)+k⎝integraldisplay ⎝integraldisplayx aeλ(x–t)cot[βy(t)]dt=Aeλx+B. This is a special case of equation 6.8.11 with f(y)=kcot(βy). 53. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cot[βy (t)]dt=Aeλx+Be–λx+C. This is a special case of equation 6.8.12 with f(y)=kcot(βy). 54. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cot[βy (t)]dt=Acosh(λx )+B. This is a special case of equation 6.8.13 with f(y)=kcot(βy). 55. y(x)+k⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)] cot[βy (t)]dt=Asinh(λx)+B. This is a special case of equation 6.8.14 with f(y)=kcot(βy). 56. y(x)+k⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)] cot[βy (t)]dt=Asin(λx)+Bcos(λx )+C. This is a special case of equation 6.8.15 with f(y)=kcot(βy). 6.8. Equations with Nonlinearity of General Form 6.8-1. Equations of the Form y(x)+⎝integraltextx aK(x,t)G⎝parenleftbig y(t)⎝parenrightbig dt=F(x). 1. y(x)+⎝integraldisplay ⎝integraldisplayx af⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=A. This is a special case of equation 6.8.16. Solution in an implicit form: ⎝integraldisplayy Adu f(u)+x–a=0 . 426 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 2. y(x)+⎝integraldisplay ⎝integraldisplayx af⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=Ax +B. Solution in an implicit form: ⎝integraldisplayy y0du A–f(u)=x–a,y0=Aa+B. 3. y(x)+⎝integraldisplay ⎝integraldisplayx a(x–t)f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=Ax2+Bx +C. 1◦. This is a special case of equation 6.8.17. The solution of this integral equation is determined by the solution of the second-order autonomous ordinary differential equation y/prime/prime xx+f(y)–2A=0 under the initial conditions y(a)=Aa2+Ba+C,y/prime x(a)=2Aa+B. 2◦. Solutions in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig 4Au–2F(u)+B2–4AC⎝bracketrightbig–1/2du=±(x–a), F(u)=⎝integraldisplayu y0f(t)dt,y0=Aa2+Ba+C. 4. y(x)+⎝integraldisplay ⎝integraldisplayx atλf⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=Bxλ+1+C. By differentiation, this integral equation can be reduced to a separable ordinary differential equation. Solution in an implicit form: (λ+1 )⎝integraldisplayy yadu f(u)–B(λ+1 )+xλ+1–aλ+1=0 , ya=Baλ+1+C. 5. y(x)+⎝integraldisplay ⎝integraldisplayx ag(t)f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=A. Solution in an implicit form: ⎝integraldisplayy Adu f(u)+⎝integraldisplayx ag(t)dt=0 . 6. y(x)+⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig ax+btdt=A. A solution: y(x)=λ,w h e r e λis a root of the algebraic (or transcendental) equation ln⎝parenleftBig 1+b a⎝parenrightBig f(λ)+bλ–Ab=0 . 6.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 427 7. y(x)+⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig √ ax2+bt2dt=A. A solution: y(x)=λ,w h e r e λis a root of the algebraic (or transcendental) equation kf(λ)+λ–A=0 , k=⎝integraldisplay1 0dz √ a+bz2. 8. y(x)+x⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbigdt x2+t2=A. A solution: y(x)=λ,w h e r e λis a root of the algebraic (or transcendental) equation λ+1 4πf(λ)=A. 9. y(x)+⎝integraldisplay ⎝integraldisplayx aeλtf⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=Beλx+C. By differentiation, this integral equation can be reduced to a separable ordinary differential equation. Solution in an implicit form: λ⎝integraldisplayy y0du f(u)–Bλ+eλx–eλa=0 , y0=Beλa+C. 10. y(x)+⎝integraldisplay ⎝integraldisplayx aeλ(x–t)f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=A. This is a special case of equation 6.8.19. Solution in an implicit form: ⎝integraldisplayy Adu λu–f(u)–λA=x–a. 11. y(x)+⎝integraldisplay ⎝integraldisplayx aeλ(x–t)f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=Aeλx+B. This is a special case of equation 6.8.19. Solution in an implicit form: ⎝integraldisplayy y0du λu–f(u)–λB=x–a,y0=Aeλa+B. 12. y(x)+⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=Aeλx+Be–λx+C. 1◦. This is a special case of equation 6.8.21. The solution of this integral equation is determined by the solution of the second-order autonomous ordinary differential equation y/prime/prime xx+λf(y)–λ2y+λ2C=0 under the initial conditions y(a)=Aeλa+Be–λa+C,y/prime x(a)=Aλeλa–Bλe–λa. 2◦. Solution in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig λ2u2–2λ2Cu–2λF(u)+λ2(C2–4AB)⎝bracketrightbig–1/2du=±(x–a), F(u)=⎝integraldisplayu y0f(t)dt,y0=Aeλa+Be–λa+C. 428 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 13. y(x)+⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=Acosh(λx )+B. This is a special case of equation 6.8.12. Solution in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig λ2u2–2λ2Bu–2λF(u)+λ2(B2–A2)⎝bracketrightbig–1/2du=±(x–a), F(u)=⎝integraldisplayu y0f(t)dt,y0=Acosh(λa)+B. 14. y(x)+⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=Asinh(λx)+B. This is a special case of equation 6.8.21. Solution in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig λ2u2–2λ2Bu–2λF(u)+λ2(A2+B2)⎝bracketrightbig–1/2du=±(x–a), F(u)=⎝integraldisplayu y0f(t)dt,y0=Asinh(λa)+B. 15. y(x)+⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)]f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=Asin(λx)+Bcos(λx )+C. 1◦. This is a special case of equation 6.8.23. The solution of this integral equation is determined by the solution of the second-order autonomous ordinary differential equation y/prime/prime xx+λf(y)+λ2y–λ2C=0 under the initial conditions y(a)=Asin(λa)+Bcos(λa )+C,y/prime x(a)=Aλcos(λa)–Bλsin(λa). 2◦. Solution in an implicit form: ⎝integraldisplayy y0⎝bracketleftbig λ2D–λ2u2+2λ2Cu–2λF(u)⎝bracketrightbig–1/2du=±(x–a), y0=Asin(λa)+Bcos(λa )+C,D=A2+B2–C2,F(u)=⎝integraldisplayu y0f(t)dt. 6.8-2. Equations of the Form y(x)+⎝integraltextx aK(x–t)G⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x). 16. y(x)+⎝integraldisplay ⎝integraldisplayx af⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). The solution of this integral equation is determined by the solution of the ﬁrst-order ordinary differential equation y/prime x+f(x,y)–g/prime x(x)=0 under the initial condition y(a)=g(a). For the exact solutions of the ﬁrst-order differential equations with various f(x,y)a n d g(x), see E. Kamke (1977) and A. D. Polyanin and V . F. Zaitsev (2003). 6.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 429 17. y(x)+⎝integraldisplay ⎝integraldisplayx a(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). Differentiating the equation with respect to xyields y/prime x+⎝integraldisplayx af⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime x(x). (1) In turn, differentiating this equation with respect to xyields the second-order nonlinear ordinary differential equation y/prime/prime xx+f(x,y)–g/prime/prime xx(x)=0 . ( 2 ) By setting x=ain the original equation and equation (1), we obtain the initial conditions for y=y(x): y(a)=g(a), y/prime x(a)=g/prime x(a). (3) Equation (2) under conditions (3) deﬁnes the solution of the original integral equation. For the exact solutions of the second-order differential equation (2) with various f(x,y)a n dg(x), see A. D. Polyanin and V . F. Zaitsev (2003). 18. y(x)+⎝integraldisplay ⎝integraldisplayx a(x–t)nf⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), n=1 , 2 , ... Differentiating the equation n+1 times with respect to x, we obtain an ( n+1)st-order nonlinear ordinary differential equation for y=y(x): y(n+1) x +n!f(x,y)–g(n+1) x(x)=0 . This equation under the initial conditions y(a)=g(a),y/prime x(a)=g/prime x(a),...,y(n) x(a)=g(n) x(a), deﬁnes the solution of the original integral equation. 19. y(x)+⎝integraldisplay ⎝integraldisplayx aeλ(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). Differentiating the equation with respect to xyields y/prime x+f⎝parenleftbig x,y(x)⎝parenrightbig +λ⎝integraldisplayx aeλ(x–t)f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime x(x). Eliminating the integral term with the aid of the original equation, we obtain the ﬁrst-order nonlinear ordinary differential equation y/prime x+f(x,y)–λy+λg(x)–g/prime x(x)=0 . The unknown function y=y(x) must satisfy the initial condition y(a)=g(a). For the exact solutions of the ﬁrst-order differential equations with various f(x,y)a n dg(x), see E. Kamke (1977) and A. D. Polyanin and V . F. Zaitsev (2003). 430 NONLINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRA TION 20. y(x)+⎝integraldisplay ⎝integraldisplayx acosh[λ (x–t)]f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). Differentiating the equation with respect to xtwice yields y/prime x(x)+f⎝parenleftbig x,y(x)⎝parenrightbig +λ⎝integraldisplayx asinh[λ(x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime x(x), (1) y/prime/prime xx(x)+⎝bracketleftbig f⎝parenleftbig x,y(x)⎝parenrightbig⎝bracketrightbig/prime x+λ2⎝integraldisplayx acosh[λ (x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order nonlinear ordinary differential equation y/prime/prime xx+⎝bracketleftbig f(x,y)⎝bracketrightbig/prime x–λ2y+λ2g(x)–g/prime/prime xx(x)=0 . ( 3 ) By setting x=ain the original equation and in (1), we obtain the initial conditions for y=y(x): y(a)=g(a), y/prime x(a)=g/prime x(a)–f⎝parenleftbig a,g(a)⎝parenrightbig .( 4 ) Equation (3) under conditions (4) deﬁnes the solution of the original integral equation. 21. y(x)+⎝integraldisplay ⎝integraldisplayx asinh[λ(x–t)]f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). Differentiating the equation with respect to xtwice yields y/prime x(x)+λ⎝integraldisplayx acosh[λ(x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime x(x), (1) y/prime/prime xx(x)+λf⎝parenleftbig x,y(x)⎝parenrightbig +λ2⎝integraldisplayx asinh[λ(x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order nonlinear ordinary differential equation y/prime/prime xx+λf(x,y)–λ2y+λ2g(x)–g/prime/prime xx(x)=0 . ( 3 ) By setting x=ain the original equation and in (1), we obtain the initial conditions for y=y(x): y(a)=g(a), y/prime x(a)=g/prime x(a). (4) Equation (3) under conditions (4) deﬁnes the solution of the original integral equation. For the exact solutions of the second-order differential equation (3) with various f(x,y)a n dg(x), see A. D. Polyanin and V . F. Zaitsev (2003). 22. y(x)+⎝integraldisplay ⎝integraldisplayx acos[λ (x–t)]f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). Differentiating the equation with respect to xtwice yields y/prime x(x)+f⎝parenleftbig x,y(x)⎝parenrightbig –λ⎝integraldisplayx asin[λ(x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime x(x), (1) y/prime/prime xx(x)+⎝bracketleftbig f⎝parenleftbig x,y(x)⎝parenrightbig⎝bracketrightbig/prime x–λ2⎝integraldisplayx acos[λ (x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order nonlinear ordinary differential equation y/prime/prime xx+⎝bracketleftbig f(x,y)⎝bracketrightbig/prime x+λ2y–λ2g(x)–g/prime/prime xx(x)=0 . ( 3 ) By setting x=ain the original equation and in (1), we obtain the initial conditions for y=y(x): y(a)=g(a), y/prime x(a)=g/prime x(a)–f⎝parenleftbig a,g(a)⎝parenrightbig .( 4 ) Equation (3) under conditions (4) deﬁnes the solution of the original integral equation. 6.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 431 23. y(x)+⎝integraldisplay ⎝integraldisplayx asin[λ(x–t)]f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). Differentiating the equation with respect to xtwice yields y/prime x(x)+λ⎝integraldisplayx acos[λ (x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime x(x), (1) y/prime/prime xx(x)+λf⎝parenleftbig x,y(x)⎝parenrightbig –λ2⎝integraldisplayx asin[λ(x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime/prime xx(x). (2) Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order nonlinear ordinary differential equation y/prime/prime xx+λf(x,y)+λ2y–λ2g(x)–g/prime/prime xx(x)=0 . ( 3 ) By setting x=ain the original equation and in (1), we obtain the initial conditions for y=y(x): y(a)=g(a), y/prime x(a)=g/prime x(a). (4) Equation (3) under conditions (4) deﬁnes the solution of the original integral equation. For the exact solutions of the second-order differential equation (3) with various f(x,y)a n dg(x), see A. D. Polyanin and V . F. Zaitsev (2003). 6.8-3. Other Equations. 24. y(x)+1 x⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x,y(t),y(x)⎝parenrightbigg ⎝parenrightbigg dt=A. A solution: y(x)=λ,w h e r e λis a root of the algebraic (or transcendental) equation λ+F(λ)–A=0 , F(λ)=⎝integraldisplay1 0f(z,λ,λ)dz. 25. y(x)+⎝integraldisplay ⎝integraldisplayx 0f⎝parenleftbigg ⎝parenleftbiggt x,y(t) t,y(x) x⎝parenrightbigg ⎝parenrightbigg dt=Ax. A solution: y(x)=λx,w h e r e λis a root of the algebraic (or transcendental) equation λ+F(λ)–A=0 , F(λ)=⎝integraldisplay1 0f(z,λ,λ)dz. 26. y(x)+⎝integraldisplay ⎝integraldisplay∞ xf⎝parenleftbig⎝parenleftbig t–x,y(t–x)⎝parenrightbig⎝parenrightbig y(t)dt=ae–λx. Solutions: y(x)=bke–λx,w h e r e bkare roots of the algebraic (or transcendental) equation b+bI(b)=a,I(b)=⎝integraldisplay∞ 0f(z,be–λz)e–λzdz. Chapter 7 Nonlinear Equations of the First Kind with Constant Limits of Integration /trianglerightsld Notation: f,g,ϕ, andψare arbitrary functions of an argument speciﬁed in the parentheses (the argument can depend on t,x, andy); and A,B,a,b,c,β,γ,λ, andµare arbitrary parameters. 7.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters 7.1-1. Equations of the Form⎝integraltextb aK(t)y(x)y(t)dt=F(x). 1.⎝integraldisplay ⎝integraldisplay1 0y(x)y(t)dt=Axλ,A>0 , λ> –1. This is a special case of equation 7.2.1 with f(t)=1 , g(x)=Axλ,a=0 ,a n d b=1 . Solutions: y(x)=±√ A(λ+1 )xλ. 2.⎝integraldisplay ⎝integraldisplay1 0y(x)y(t)dt=Aeβx,A>0 . This is a special case of equation 7.2.1 with f(t)=1 , g(x)=Aeβx,a=0 ,a n d b=1 . Solutions: y(x)=±⎝radicalbigg Aβ eβ–1eβx. 3.⎝integraldisplay ⎝integraldisplay1 0y(x)y(t)dt=Acosh(βx ), A>0 . This is a special case of equation 7.2.1 with f(t)=1 , g(x)=Acosh(βx),a=0 ,a n d b=1 . Solutions: y(x)=±⎝radicalBigg Aβ sinhβcosh(βx ). 4.⎝integraldisplay ⎝integraldisplay1 0y(x)y(t)dt=Asinh(βx), Aβ >0 . This is a special case of equation 7.2.1 with f(t)=1 , g(x)=Asinh(βx),a=0 ,a n d b=1 . Solutions: y(x)=±⎝radicalBigg Aβ coshβ–1sinh(βx). 433 434 NONLINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 5.⎝integraldisplay ⎝integraldisplay1 0y(x)y(t)dt=Atanh(βx), Aβ >0 . This is a special case of equation 7.2.1 with f(t)=1 , g(x)=Atanh(βx),a=0 ,a n d b=1 . Solutions: y(x)=±⎝radicalBigg Aβ ln cosh βtanh(βx). 6.⎝integraldisplay ⎝integraldisplay1 0y(x)y(t)dt=Aln(βx), A(lnβ–1 )>0 . This is a special case of equation 7.2.1 with f(t)=1 , g(x)=Aln(βx),a=0 ,a n d b=1 . Solutions: y(x)=±⎝radicalBigg A lnβ–1ln(βx). 7.⎝integraldisplay ⎝integraldisplay1 0y(x)y(t)dt=Acos(βx ), A>0 . This is a special case of equation 7.2.1 with f(t)=1 , g(x)=Acos(βx ),a=0 ,a n d b=1 . Solutions: y(x)=±⎝radicalBigg Aβ sinβcos(βx ). 8.⎝integraldisplay ⎝integraldisplay1 0y(x)y(t)dt=Asin(βx), Aβ >0 . This is a special case of equation 7.2.1 with f(t)=1 , g(x)=Asin(βx),a=0 ,a n d b=1 . Solutions: y(x)=±⎝radicalBigg Aβ 1–c o s βsin(βx). 9.⎝integraldisplay ⎝integraldisplay1 0y(x)y(t)dt=Atan(βx), Aβ >0 . This is a special case of equation 7.2.1 with f(t)=1 , g(x)=Atan(βx),a=0 ,a n d b=1 . Solutions: y(x)=±⎝radicalBigg –Aβ ln|cosβ|tan(βx). 10.⎝integraldisplay ⎝integraldisplay1 0tµy(x)y(t)dt=Axλ,A>0 , µ+λ> –1. This is a special case of equation 7.2.1 with f(t)=tµ,g(x)=Axλ,a=0 ,a n d b=1 . Solutions: y(x)=±√ A(µ+λ+1 )xλ. 11.⎝integraldisplay ⎝integraldisplay1 0eµty(x)y(t)dt=Aeβx,A>0 . This is a special case of equation 7.2.1 with f(t)=eµt,g(x)=Aeβx,a=0 ,a n d b=1 . Solutions: y(x)=±⎝radicalbigg A(µ+β) eµ+β–1eβx. 7.1. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY PARAMETERS 435 7.1-2. Equations of the Form⎝integraltextb aK(t)y(t)y(xt)dt=F(x). 12.⎝integraldisplay ⎝integraldisplay1 0y(t)y(xt)dt=A,0 ≤x≤1. This is a special case of equation 7.2.2 with f(t)=1 , a=0 ,a n d b=1 . 1◦. Solutions: y1(x)=√ A, y3(x)=√ A(3x–2 ) , y5(x)=√ A(10x2–1 2x+3 ) ,y2(x)=–√ A, y4(x)=–√ A(3x–2 ) , y6(x)=–√ A(10x2–1 2x+3 ) . 2◦. The integral equation has some other solutions; for example, y7(x)=√ A C⎝bracketleftbig (2C+1 )xC–C–1⎝bracketrightbig , y9(x)=√ A(lnx+1 ) ,y8(x)=–√ A C⎝bracketleftbig (2C+1 )xC–C–1⎝bracketrightbig , y10(x)=–√ A(lnx+1 ) , where Cis an arbitrary constant. 3◦. See 7.2.2 for some other solutions. 13.⎝integraldisplay ⎝integraldisplay1 0y(t)y(xtβ)dt=A,β>0 . 1◦. Solutions: y1(x)=√ A, y3(x)=√ B⎝bracketleftbig (β+2 )x–β–1⎝bracketrightbig ,y2(x)=–√ A, y4(x)=–√ B⎝bracketleftbig (β+2 )x–β–1⎝bracketrightbig , where B=⎝radicalBigg 2A β(β+1 ). 2◦. The integral equation has some other (more complicated solutions) of the polynomial formy(x)=n⎝summationtext k=0Bkxk, where the constants Bkcan be found from the corresponding system of algebraic equations. 14.⎝integraldisplay ⎝integraldisplay∞ 1y(t)y(xt)dt=Ax–λ,λ>0 , 1 ≤x<∞. This is a special case of equation 7.2.3 with f(t)=1 , a=1 ,a n d b=∞. 1◦. Solutions: y1(x)=Bx–λ, y2(x)=–Bx–λ, λ>1 2; y3(x)=B⎝bracketleftbig (2λ–3 )x–2λ+2⎝bracketrightbig x–λ,y4(x)=–B⎝bracketleftbig (2λ–3 )x–2λ+2⎝bracketrightbig x–λ,λ>3 2; where B=√ A(2λ–1 ) . 2◦. For sufﬁciently large λ, the integral equation has some other (more complicated) solutions of the polynomial form y(x)=n⎝summationtext k=0Bkxk, where the constants Bkcan be found from the corresponding system of algebraic equations. 436 NONLINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 15.⎝integraldisplay ⎝integraldisplay∞ 0e–λty(t)y(xt)dt=A,λ>0 , 0 ≤x<∞. This is a special case of equation 7.2.2 with f(t)=e–λt,a=0 ,a n d b=∞. 1◦. Solutions: y1(x)=√ Aλ, y3(x)=⎝radicalBig 1 2Aλ(λx–2 ) ,y2(x)=–√ Aλ, y4(x)=–⎝radicalBig 1 2Aλ(λx–2 ) . 2◦. The integral equation has some other (more complicated) solutions of the polynomial formy(x)=n⎝summationtext k=0Bkxk, where the constants Bkcan be found from the corresponding system of algebraic equations. See 7.2.2 for some other solutions. 7.1-3. Other Equations. 16.⎝integraldisplay ⎝integraldisplay1 0y(t)y(x+λt)dt=A,0 ≤x<∞. This is a special case of equation 7.2.7 with f(t)≡1,a=0 ,a n d b=1 . Solutions: y1(x)=√ A, y3(x)=⎝radicalbig 3A/λ ( 1–2x),y2(x)=–√ A, y4(x)=–⎝radicalbig 3A/λ (1 – 2x). 17.⎝integraldisplay ⎝integraldisplay∞ 0y(t)y(x+λt)dt=Ae–βx,A,λ,β>0 , 0 ≤x<∞. This is a special case of equation 7.2.9 with f(t)≡1,a=0 ,a n d b=∞. Solutions: y1(x)=⎝radicalbig Aβ(λ+1 )e–βx, y3(x)=B⎝bracketleftbig β(λ+1 )x–1⎝bracketrightbig e–βx,y2(x)=–⎝radicalbig Aβ(λ+1 )e–βx, y4(x)=–B⎝bracketleftbig β(λ+1 )x–1⎝bracketrightbig e–βx, where B=⎝radicalbig Aβ(λ+1 )/λ. 18.⎝integraldisplay ⎝integraldisplay1 0y(t)y(x–t)dt=A,– ∞<x<∞. This is a special case of equation 7.2.10 with f(t)≡1,a=0 ,a n d b=1 . 1◦. Solutions with A>0 : y1(x)=√ A, y3(x)=√ 5A(6x2–6x+1 ) ,y2(x)=–√ A, y4(x)=–√ 5A(6x2–6x+1 ) . 2◦. Solutions with A<0 : y1(x)=√ –3A ( 1–2x),y2(x)=–√ –3A (1 – 2x). The integral equation has some other (more complicated) solutions of the polynomial formy(x)=n⎝summationtext k=0Bkxk, where the constants Bkcan be found from the corresponding system of algebraic equations. 19.⎝integraldisplay ⎝integraldisplay∞ 0e–λty⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig y(t)dt=Axb,λ>0 . Solutions: y(x)=±√ Aλxb. 7.2. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 437 7.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions 7.2-1. Equations of the Form⎝integraltextb aK(t)y(t)y(···)dt=F(x). 1.⎝integraldisplay ⎝integraldisplayb af(t)y(x)y(t)dt=g(x). Solutions: y(x)=±λg(x), λ=⎝bracketleftbigg⎝integraldisplayb af(t)g(t)dt⎝bracketrightbigg–1/2 . 2.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(xt)dt=A. 1◦. Solutions:* y1(x)=⎝radicalbig A/I 0, y3(x)=q(I1x–I2),y2(x)=–⎝radicalbig A/I 0, y4(x)=–q(I1x–I2), where Im=⎝integraldisplayb atmf(t)dt,q=⎝parenleftbiggA I0I2 2–I2 1I2⎝parenrightbigg1/2 ,m=0 ,1 ,2 . The integral equation has some other (more complicated) solutions of the polynomial formy(x)=n⎝summationtext k=0Bkxk, where the constants Bkcan be found from the corresponding system of algebraic equations. 2◦. Solutions: y5(x)=q(I1xC–I2), y6(x)=–q(I1xC–I2), q=⎝parenleftbiggA I0I2 2–I2 1I2⎝parenrightbigg1/2 ,Im=⎝integraldisplayb atmCf(t)dt,m=0 ,1 ,2 , where Cis an arbitrary constant. The equation has more complicated solutions of the form y(x)=n⎝summationtext k=0BkxkC,w h e r e Cis an arbitrary constant and the coefﬁcients Bkcan be found from the corresponding system of algebraic equations. 3◦. Solutions: y7(x)=p(J0lnx–J1), y8(x)=–p(J0lnx–J1), p=⎝parenleftbiggA J2 0J2–J0J2 1⎝parenrightbigg1/2 ,Jm=⎝integraldisplayb a(lnt)mf(t)dt. The equation has more complicated solutions of the form y(x)=n⎝summationtext k=0Ek(lnx)k,w h e r et h e constants Ekcan be found from the corresponding system of algebraic equations. * The arguments of the equations containing y(xt) in the integrand can vary, for example, within the following intervals: (a) 0 ≤t≤1, 0≤x≤1f o ra=0a n d b=1 ;( b )1 ≤t<∞,1≤x<∞fora=1a n d b=∞;( c )0 ≤t<∞,0≤x<∞for a=0a n d b=∞;o r( d ) a≤t≤b,0≤x<∞for arbitrary aandbsuch that 0 ≤a<b≤∞. Case (d) is a special case of (c) iff(t) is nonzero only on the interval a≤t≤b. 438 NONLINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 3.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(xt)dt=Axβ. 1◦. Solutions: y1(x)=⎝radicalbig A/I 0xβ, y3(x)=q(I1x–I2)xβ,y2(x)=–⎝radicalbig A/I 0xβ, y4(x)=–q(I1x–I2)xβ, where Im=⎝integraldisplayb at2β+mf(t)dt,q=⎝radicalBigg A I2(I0I2–I2 1),m=0 ,1 ,2 . 2◦. The substitution y(x)=xβw(x) leads to an equation of the form 7.2.2: ⎝integraldisplayb ag(t)w(t)w(xt)dt=A,g(x)=f(x)x2β. Therefore, the integral equation in question has more complicated solutions. 4.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(xt)dt=Alnx+B. This equation has solutions of the form y(x)=plnx+q. The constants pandqare determined from the following system of two second-order algebraic equations: I1p2+I0pq=A,I2p2+2I1pq+I0q2=B, where Im=⎝integraldisplayb af(t)(lnt)mdt,m=0 ,1 ,2 . 5.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(xt)dt=Axλlnx+Bxλ. The substitution y(x)=xλw(x) leads to an equation of the form 7.2.4: ⎝integraldisplayb ag(t)w(t)w(xt)dt=Alnx+B,g(t)=f(t)t2λ. 6.⎝integraldisplay ⎝integraldisplay∞ 0f(t)y(t)y⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig dt=Axλ. Solutions: y1(x)=⎝radicalbigg A Ixλ,y2(x)=–⎝radicalbigg A Ixλ,I=⎝integraldisplay∞ 0f(t)dt. 7.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x+λt)dt=A,λ>0 . 1◦. Solutions* y1(x)=⎝radicalbig A/I 0, y3(x)=q(I0x–I1),y2(x)=–⎝radicalbig A/I 0, y4(x)=–q(I0x–I1), * The arguments of the equations containing y(x+λt) in the integrand can vary within the following intervals: (a) 0 ≤t<∞, 0≤x<∞fora=0a n d b=∞or (b) a≤t≤b,0≤x<∞for arbitrary aandbsuch that 0 ≤a<b<∞.C a s e ( b ) i s a special case of (a) if f(t) is nonzero only on the interval a≤t≤b. 7.2. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 439 where Im=⎝integraldisplayb atmf(t)dt,q=⎝radicalBigg A λ(I2 0I2–I0I2 1),m=0 ,1 ,2 . 2◦. The integral equation has some other (more complicated) solutions of the polynomial formy(x)=n⎝summationtext k=0Bkxk, where the constants Bkcan be found from the corresponding system of algebraic equations. 8.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x+λt)dt=Ax +B,λ>0 . A solution: y(x)=βx+µ, where the constants βandµare determined from the following system of two second-order algebraic equations: I0βµ+I1β2=A,I0µ2+(λ+1 )I1βµ+λI2β2=B,Im=⎝integraldisplayb atmf(t)dt.( 1 ) Multiplying the ﬁrst equation by Band the second by – Aand adding the resulting equations, we obtain the quadratic equation AI0z2+⎝bracketleftbig (λ+1 )AI1–BI0⎝bracketrightbig z+λAI 2–BI1=0 , z=µ/β.( 2 ) In general, to each root of equation (2) two solutions of system (1) correspond. Therefore, the original integral equation can have at most four solutions of this form. If the discriminant of equation (2) is negative, then the integral equation has no such solutions. The integral equation has some other (more complicated) solutions of the polynomial formy(x)=n⎝summationtext k=0βkxk, where the constants βkcan be found from the corresponding system of algebraic equations. 9.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x+λt)dt=Ae–βx,λ>0 . 1◦. Solutions: y1(x)=⎝radicalbig A/I 0e–βx, y3(x)=q(I0x–I1)e–βx,y2(x)=–⎝radicalbig A/I 0e–βx, y4(x)=–q(I0x–I1)e–βx, where Im=⎝integraldisplayb atme–β(λ+1)tf(t)dt,q=⎝radicalBigg A λ(I2 0I2–I0I2 1),m=0 ,1 ,2 . 2◦. The equation has more complicated solutions of the form y(x)=e–βxn⎝summationtext k=0Bkxk,w h e r e the constants Bkcan be found from the corresponding system of algebraic equations. 3◦. The substitution y(x)=e–βxw(x) leads to an equation of the form 7.2.7: ⎝integraldisplayb ae–β(λ+1)tf(t)w(t)w(x+λt)dt=A. 440 NONLINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 10.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=A. 1◦. Solutions* y1(x)=⎝radicalbig A/I 0, y3(x)=q(I0x–I1),y2(x)=–⎝radicalbig A/I 0, y4(x)=–q(I0x–I1), where Im=⎝integraldisplayb atmf(t)dt,q=⎝radicalBigg A I0I2 1–I2 0I2,m=0 ,1 ,2 . 2◦. The integral equation has some other (more complicated) solutions of the polynomial formy(x)=n⎝summationtext k=0λkxk, where the constants λkcan be found from the corresponding system of algebraic equations. For n= 3, such a solution is presented in 7.1.18. 11.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=Ax +B. A solution: y(x)=λx+µ, where the constants λandµare determined from the following system of two second-order algebraic equations: I0λµ+I1λ2=A,I0µ2–I2λ2=B,Im=⎝integraldisplayb atmf(t)dt,m=0 ,1 ,2 . ( 1 ) Multiplying the ﬁrst equation by Band the second by – Aand adding the results, we obtain the quadratic equation AI0z2–BI0z–AI2–BI1=0 , z=µ/λ.( 2) In general, to each root of equation (2) two solutions of system (1) correspond. Therefore, the original integral equation can have at most four solutions of this form. If the discriminant of equation (2) is negative, then the integral equation has no such solutions. The integral equation has some other (more complicated) solutions of the polynomial formy(x)=n⎝summationtext k=0λkxk, where the constants λkcan be found from the corresponding system of algebraic equations. 12.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=n⎝summationdisplay k=0Akxk. This equation has solutions of the form y(x)=n⎝summationdisplay k=0λkxk,( 1) where the constants λkare determined from the system o f algebraic equations obtained by substituting solution (1) into the original integral equation and matching the coefﬁcients of like powers of x. * The arguments of the equations containing y(x–t) in the integrand can vary within the following intervals: (a) – ∞<t<∞, –∞<x<∞fora=–∞andb=∞or (b) a≤t≤b,–∞≤x<∞, for arbitrary aandbsuch that – ∞<a<b<∞. Case (b) is a special case of (a) if f(t) is nonzero only on the interval a≤t≤b. 7.2. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 441 13.⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)y(t)dt=Aeλx. Solutions: y1(x)=⎝radicalbig A/I 0eλx, y3(x)=q(I0x–I1)eλx,y2(x)=–⎝radicalbig A/I 0eλx, y4(x)=–q(I0x–I1)eλx, where Im=⎝integraldisplayb atmf(t)dt,q=⎝radicalBigg A I0I2 1–I2 0I2,m=0 ,1 ,2 . The integral equation has more complicated solutions of the form y(x)=eλxn⎝summationtext k=0Bkxk,w h e r e the constants Bkcan be found from the corresponding system of algebraic equations. 14.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=Asinhλx. A solution: y(x)=psinhλx+qcoshλx.( 1) Herepandqare roots of the algebraic system I0pq+Ics(p2–q2)=A,Iccq2–Issp2=0 , ( 2 ) where the notation I0=⎝integraldisplayb af(t)dt,Ics=⎝integraldisplayb af(t)c o s h ( λt)s i n h ( λt)dt, Icc=⎝integraldisplayb af(t)c o s h2(λt)dt,Iss=⎝integraldisplayb af(t)s i n h2(λt)dt is used. Different solutions of system (2) generate different solutions (1) of the integral equation. It follows from the second equation of (2) that q=±⎝radicalbig Iss/Iccp. Using this expression to eliminate qfrom the ﬁrst equation of (2), we obtain the following four solutions: y1,2(x)=p⎝parenleftbig sinhλx±kcoshλx⎝parenrightbig ,y3,4(x)=–p⎝parenleftbig sinhλx±kcoshλx⎝parenrightbig , k=⎝radicalbigg Iss Icc,p=⎝radicalBigg A (1 –k2)Ics±kI0. 15.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=Acoshλx. A solution: y(x)=psinhλx+qcoshλx.( 1) Herepandqare roots of the algebraic system I0pq+Ics(p2–q2)=0 , Iccq2–Issp2=A,( 2 ) where we use the notation introduced in 7.2.14. Different solutions of system (2) generate different solutions (1) of the integral equation. 442 NONLINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 16.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=Asinλx. A solution: y(x)=psinλx+qcosλx.( 1) Herepandqare roots of the algebraic system I0pq+Ics(p2+q2)=A,Iccq2–Issp2=0 , ( 2 ) where I0=⎝integraldisplayb af(t)dt,Ics=⎝integraldisplayb af(t)c o s ( λt)s i n (λt)dt, Icc=⎝integraldisplayb af(t)c o s2(λt)dt,Iss=⎝integraldisplayb af(t)s i n2(λt)dt. It follows from the second equation of (2) that q=±⎝radicalbig Iss/Iccp. Using this expression to eliminate qfrom the ﬁrst equation of (2), we obtain the following four solutions: y1,2(x)=p⎝parenleftbig sinλx±kcosλx⎝parenrightbig ,y3,4(x)=–p⎝parenleftbig sinλx±kcosλx⎝parenrightbig , k=⎝radicalbigg Iss Icc,p=⎝radicalBigg A (1 +k2)Ics±kI0. 17.⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=Acosλx. A solution: y(x)=psinλx+qcosλx.( 1) Herepandqare roots of the algebraic system I0pq+Ics(p2+q2)=0 , Iccq2–Issp2=A,( 2) where we use the notation introduced in 7.2.16. Different solutions of system (2) generate different solutions (1) of the integral equation. 18.⎝integraldisplay ⎝integraldisplay1 0y(t)y(ξ)dt=A,ξ=f(x)t. 1◦. Solutions: y1(t)=√ A, y3(t)=√ A(3t–2 ) , y5(t)=√ A(10t2–1 2t+3 ) ,y2(t)=–√ A, y4(t)=–√ A(3t–2 ) , y6(t)=–√ A(10t2–1 2t+3 ) . 2◦. The integral equation has some other (more complicated) solutions of the polynomial formy(t)=n⎝summationtext k=0Bktk, where the constants Bkcan be found from the corresponding system of algebraic equations. 3◦. The substitution z=f(x) leads to an equation of the form 7.1.12. 7.2. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 443 7.2-2. Equations of the Form⎝integraltextb a[K(x,t)y(t)+M(x,t)y2(t)]dt=F(x). 19.⎝integraldisplay ⎝integraldisplaya 0⎝bracketleftbigg ⎝bracketleftbigg1 |x–t|ky(t)+ϕ(x)ψ(t)y2(t)⎝bracketrightbigg ⎝bracketrightbigg dt=f(x), 0 < k<1 . The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.1.30. 20.⎝integraldisplay ⎝integraldisplayb a⎝bracketleftBig⎝bracketleftBig ln|x–t|y(t)+ϕ(x)ψ(t)y2(t)⎝bracketrightBig⎝bracketrightBig dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.4.2. 21.⎝integraldisplay ⎝integraldisplay∞ 0[sin(xt )y(t)+ϕ(x)ψ(t)y2(t)]dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.5.8. Solutions: y1,2(t)=Yf(t)+A1,2Yϕ(t), where Yf(t)=2 π⎝integraldisplay∞ 0sin(xt)f(x)dx,Yϕ(t)=2 π⎝integraldisplay∞ 0sin(xt)ϕ(x)dx, andA1,2are roots of the quadratic equation pA2+qA+r=0 , p=⎝integraldisplay∞ 0ψ(t)Y2 ϕ(t)dt,q=1+2⎝integraldisplay∞ 0ψ(t)Yf(t)Yϕ(t)dt,r=⎝integraldisplay∞ 0ψ(t)Y2 f(t)dt. Reference: A. D. Polyanin and A. I. Zhurov (2007). 22.⎝integraldisplay ⎝integraldisplay∞ 0[cos(xt)y(t)+ϕ(x)ψ(t)y2(t)]dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.5.1. Solutions: y1,2(t)=Yf(t)+A1,2Yϕ(t), where Yf(t)=2 π⎝integraldisplay∞ 0cos(xt )f(x)dx,Yϕ(t)=2 π⎝integraldisplay∞ 0cos(xt )ϕ(x)dx, andA1,2are roots of the quadratic equation pA2+qA+r=0 , p=⎝integraldisplay∞ 0ψ(t)Y2 ϕ(t)dt,q=1+2⎝integraldisplay∞ 0ψ(t)Yf(t)Yϕ(t)dt,r=⎝integraldisplay∞ 0ψ(t)Y2 f(t)dt. 444 NONLINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 23.⎝integraldisplay ⎝integraldisplay∞ 0[tJν(xt)y(t)+ϕ(x)ψ(t)y2(t)]dt=f(x), ν> –1. HereJν(z) is the Bessel function of the ﬁrst kind. The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.7.17. Solutions: y1,2(t)=Yf(t)+A1,2Yϕ(t), where Yf(t)=⎝integraldisplay∞ 0xJν(xt)f(x)dx,Yϕ(t)=⎝integraldisplay∞ 0xJν(xt)ϕ(x)dx, andA1,2are roots of the quadratic equation pA2+qA+r=0 , p=⎝integraldisplay∞ 0ψ(t)Y2 ϕ(t)dt,q=1+2⎝integraldisplay∞ 0ψ(t)Yf(t)Yϕ(t)dt,r=⎝integraldisplay∞ 0ψ(t)Y2 f(t)dt. 7.3. Equations with Power-Law Nonlinearity That Contain Arbitrary Functions 7.3-1. Equations of the Form⎝integraltextb aK(t)yµ(x)yγ(t)dt=F(x). 1.⎝integraldisplay ⎝integraldisplayb atλyµ(x)yγ(t)dt=g(x). A solution: y(x)=A⎝bracketleftbig g(x)⎝bracketrightbig1 µ,A=⎝braceleftbigg⎝integraldisplayb atλ⎝bracketleftbig g(t)⎝bracketrightbigγ µdt⎝bracerightbigg–1 µ+γ . 2.⎝integraldisplay ⎝integraldisplayb aeλtyµ(x)yγ(t)dt=g(x). A solution: y(x)=A⎝bracketleftbig g(x)⎝bracketrightbig1 µ,A=⎝braceleftbigg⎝integraldisplayb aeλt⎝bracketleftbig g(t)⎝bracketrightbigγ µdt⎝bracerightbigg–1 µ+γ . 7.3-2. Equations of the Form⎝integraltextb aK(t)yγ(t)y(xt)dt=F(x). 3.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(xt)dt=A. This is a special case of equation 7.4.4. 4.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(xt)dt=Ax +B. This is a special case of equation 7.4.5. 7.3. E QUATIONS WITH POWER -LAWNONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 445 5.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(xt)dt=Axβ. This equation has solutions of the form y(x)=kxβ,w h e r e kis a constant. 6.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(xt)dt=Alnx+B. This equation has solutions of the form y(x)=plnx+q,w h e r e pandqare some constants. 7.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(xt)dt=Axβlnx. This equation has solutions of the form y(x)=pxβlnx+qxβ,w h e r e pandqare some constants. 8.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(xt)dt=Acos(β lnx). This equation has solutions of the form y(x)=pcos(β lnx)+qsin(βlnx), where pandqare some constants. 9.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(xt)dt=Asin(βlnx). This equation has solutions of the form y(x)=pcos(β lnx)+qsin(βlnx), where pandqare some constants. 7.3-3. Equations of the Form⎝integraltextb aK(t)yγ(t)y(x+βt)dt=F(x). 10.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(x+βt)dt=Ax +B. This is a special case of equation 7.4.16. 11.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(x+βt)dt=Ae–λx. This is a special case of equation 7.4.17. 12.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(x+βt)dt=Acosλx. This equation has solutions of the form y(x)=psinλx+qcosλx,w h e r e pandqare some constants. 13.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(x+βt)dt=Asinλx. This equation has solutions of the form y(x)=psinλx+qcosλx,w h e r e pandqare some constants. 14.⎝integraldisplay ⎝integraldisplayb af(t)yγ(t)y(x+βt)dt=e–µx(Acosλx +Bsinλx). This equation has solutions of the form y(x)=e–µx(psinλx+qcosλx), where pandqare some constants. 446 NONLINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 7.3-4. Equations of the Form⎝integraltextb a[K(x,t)y(t)+M(x,t)yγ(t)]dt=f(x). 15.⎝integraldisplay ⎝integraldisplaya 0⎝bracketleftbigg ⎝bracketleftbigg1 √ |x–t|y(t)+ϕ(x)ψ(t)yγ(t)⎝bracketrightbigg ⎝bracketrightbigg dt=f(x), 0 < a≤∞. The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.1.22. 16.⎝integraldisplay ⎝integraldisplaya 0⎝bracketleftbigg ⎝bracketleftbigg1 |x–t|ky(t)+ϕ(x)ψ(t)yγ(t)⎝bracketrightbigg ⎝bracketrightbigg dt=f(x), 0 < k<1 . The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.1.30. 17.⎝integraldisplay ⎝integraldisplayb a⎝bracketleftBig⎝bracketleftBig ln|x–t|y(t)+ϕ(x)ψ(t)yγ(t)⎝bracketrightBig⎝bracketrightBig dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.4.2. 18.⎝integraldisplay ⎝integraldisplay∞ 0[sin(xt )y(t)+ϕ(x)ψ(t)yγ(t)]dt=f(x). This is a special case of equation 7.4.24. 19.⎝integraldisplay ⎝integraldisplay∞ 0[cos(xt)y(t)+ϕ(x)ψ(t)yγ(t)]dt=f(x). This is a special case of equation 7.4.25. 7.3-5. Other Equations. 20.⎝integraldisplay ⎝integraldisplay∞ 0f(xat)tbyγ(t)y⎝parenleftbig⎝parenleftbig xkt⎝parenrightbig⎝parenrightbig dt=Axc. A solution: y(x)=⎝parenleftBigA I⎝parenrightBig1 γ+1xλ,λ=a+c+ab k–a–aγ, I=⎝integraldisplay∞ 0f(t)tβdt,β=a+c+aγ+bk+cγ k–a–aγ. 21.⎝integraldisplay ⎝integraldisplay1 0[y(xt)+ϕ(x)ψ(t)yγ(t)]dt=f(x). This is a special case of equation 7.4.27. 22.⎝integraldisplay ⎝integraldisplayπ/2 0[y(xsint)+ϕ(x)ψ(t)yγ(t)]dt=f(x). This is a special case of equation 7.4.28. 7.4. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 447 7.4. Equations with Nonlinearity of General Form 7.4-1. Equations of the Form⎝integraltextb aϕ⎝parenleftbig y(x)⎝parenrightbig K⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x). 1.⎝integraldisplay ⎝integraldisplayb ay(x)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). A solution: y(x)=λg(x), where λis determined by the algebraic (or transcendental) equation λ⎝integraldisplayb af⎝parenleftbig t,λg(t)⎝parenrightbig dt=1 . 2.⎝integraldisplay ⎝integraldisplayb ayk(x)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). A solution: y(x)=λ[g(x)]1/k,w h e r e λis determined from the algebraic (or transcendental) equation λk⎝integraldisplayb af⎝parenleftbig t,λg1/k(t)⎝parenrightbig dt=1 . 3.⎝integraldisplay ⎝integraldisplayb aϕ⎝parenleftbig⎝parenleftbig y(x)⎝parenrightbig⎝parenrightbig f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). A solution in an implicit form: λϕ⎝parenleftbig y(x)⎝parenrightbig –g(x)=0 , ( 1 ) where λis determined by the algebraic (or transcendental) equation λ–F(λ)=0 , F(λ)=⎝integraldisplayb af⎝parenleftbig t,y(t)⎝parenrightbig dt.( 2) Here the function y(x)=y(x,λ) obtained by solving (1) must be substituted into (2). The number of solutions of the integral equation is determined by the number of the solutions obtained from (1) and (2). 7.4-2. Equations of the Form⎝integraltextb ay(xt)K⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x). 4.⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=A. 1◦. Solutions: y(x)=λk,w h e r e λkare roots of the algebraic (or transcendental) equation λ⎝integraldisplayb af(t,λ)dt=A. 2◦. Solutions: y(x)=px+q,w h e r e pandqare roots of the following system of algebraic (or transcendental) equations: ⎝integraldisplayb atf(t,pt+q)dt=0 , q⎝integraldisplayb af(t,pt+q)dt=A. In the case f⎝parenleftbig t,y(t)⎝parenrightbig =¯f(t)y(t), see 7.2.2 for solutions of this system. 2◦. The integral equation has some other (more complicated) solutions of the polynomial formy(x)=n⎝summationtext k=0Bkxk, where the constants Bkcan be found from the corresponding system of algebraic (or transcendental) equations. 4◦. The integral equation can have logarithmic sol utions similar to those presented in item 3◦ of equation 7.2.2. 448 NONLINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 5.⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Ax +B. 1◦. A solution: y(x)=px+q,( 1 ) where pandqare roots of the following system of algebraic (or transcendental) equations: p⎝integraldisplayb atf(t,pt+q)dt–A=0 , q⎝integraldisplayb af(t,pt+q)dt–B=0 . ( 2 ) Different solutions of system (2) generate different solutions (1) of the integral equation. 2◦. The integral equation has some other (more complicated) solutions of the polynomial formy(x)=n⎝summationtext k=0Bkxk, where the constants Bkcan be found from the corresponding system of algebraic (or transcendental) equations. 6.⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Axβ. A solution: y(x)=kxβ,( 1) where kis a root of the algebraic (or transcendental) equation kF(k)–A=0 , F(k)=⎝integraldisplayb atβf⎝parenleftbig t,ktβ⎝parenrightbig dt.( 2) Each root of equation (2) generates a solution of the integral equation which has the form (1). 7.⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Alnx+B. A solution: y(x)=plnx+q,( 1) where pandqare roots of the following system of algebraic (or transcendental) equations: p⎝integraldisplayb af(t,plnt+q)dt–A=0 ,⎝integraldisplayb a(plnt+q)f(t,plnt+q)dt–B=0 . ( 2 ) Different solutions of system (2) generate different solutions (1) of the integral equation. 8.⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Axβlnx. This equation has solutions of the form y(x)=pxβlnx+qxβ,w h e r e pandqare some constants. 9.⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Acos(β lnx). This equation has solutions of the form y(x)=pcos(β lnx)+qsin(βlnx), where pandqare some constants. 7.4. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 449 10.⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Asin(βlnx). This equation has solutions of the form y(x)=pcos(β lnx)+qsin(βlnx), where pandqare some constants. 11.⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Axβcos(β lnx)+Bxβsin(βlnx). This equation has solutions of the form y(x)=pxβcos(β lnx)+qxβsin(βlnx), where pandq are some constants. 7.4-3. Equations of the Form⎝integraltextb ay(x+βt)K⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x). 12.⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Ax +B. This equation has solutions of the form y(x)=px+q,w h e r e pandqare some constants. 13.⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Aeλx. This equation has solutions of the form y(x)=peλx,w h e r e pis some constant. 14.⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Acosλx. This equation has solutions of the form y(x)=psinλx+qcosλx,w h e r e pandqare some constants. 15.⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=e–µx(Acosλx +Bsinλx). This equation has solutions of the form y(x)=e–µx(psinλx+qcosλx), where pandqare some constants. 16.⎝integraldisplay ⎝integraldisplayb ay(x+βt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Ax +B,β>0 . A solution: y(x)=px+q,( 1) where pandqare roots of the following system of algebraic (or transcendental) equations: p⎝integraldisplayb af(t,pt+q)dt–A=0 ,⎝integraldisplayb a(βpt+q)f(t,pt+q)dt–B=0 . ( 2 ) Different solutions of system (2) generate different solutions (1) of the integral equation. 17.⎝integraldisplay ⎝integraldisplayb ay(x+βt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Ae–λx,β>0 . Solutions: y(x)=kne–λx, where knare roots of the algebraic (or transcendental) equation kF(k)–A=0 , F(k)=⎝integraldisplayb af⎝parenleftbig t,ke–λt⎝parenrightbig e–βλtdt. 450 NONLINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 18.⎝integraldisplay ⎝integraldisplayb ay(x+βt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Acosλx,β>0 . This equation has solutions of the form y(x)=psinλx+qcosλx,w h e r e pandqare some constants. 19.⎝integraldisplay ⎝integraldisplayb ay(x+βt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Asinλx,β>0 . This equation has solutions of the form y(x)=psinλx+qcosλx,w h e r e pandqare some constants. 20.⎝integraldisplay ⎝integraldisplayb ay(x+βt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=e–µx(Acosλx +Bsinλx), β>0 . This equation has solutions of the form y(x)=e–µx(psinλx+qcosλx), where pandqare some constants. 7.4-4. Equations of the Form⎝integraltextb a[K(x,t)y(t)+ϕ(x)Ψ(t,y(t))]dt=F(x). 21.⎝integraldisplay ⎝integraldisplaya 0⎝bracketleftbigg ⎝bracketleftbigg1 √ |x–t|y(t)+ϕ(x)Ψ(t,y(t))⎝bracketrightbigg ⎝bracketrightbigg dt=f(x), 0 < a≤∞. The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.1.22. 22.⎝integraldisplay ⎝integraldisplaya 0⎝bracketleftbigg ⎝bracketleftbigg1 |x–t|ky(t)+ϕ(x)Ψ(t,y(t))⎝bracketrightbigg ⎝bracketrightbigg dt=f(x), 0 < k<1 . The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.1.30. 23.⎝integraldisplay ⎝integraldisplayb a⎝bracketleftBig⎝bracketleftBig ln|x–t|y(t)+ϕ(x)Ψ(t,y(t))⎝bracketrightBig⎝bracketrightBig dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.4.2. 24.⎝integraldisplay ⎝integraldisplay∞ 0[sin(xt )y(t)+ϕ(x)Ψ(t,y(t))]dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.5.8. Solutions: ym(t)=Yf(t)+AmYϕ(t), where Yf(t)=2 π⎝integraldisplay∞ 0sin(xt)f(x)dx,Yϕ(t)=2 π⎝integraldisplay∞ 0sin(xt)ϕ(x)dx, andAmare roots of the algebraic (transcendental) equation A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt=0 . 7.4. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 451 25.⎝integraldisplay ⎝integraldisplay∞ 0[cos(xt)y(t)+ϕ(x)Ψ(t,y(t))]dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.5.1. Solutions: ym(t)=Yf(t)+AmYϕ(t), where Yf(t)=2 π⎝integraldisplay∞ 0cos(xt )f(x)dx,Yϕ(t)=2 π⎝integraldisplay∞ 0cos(xt )ϕ(x)dx, andAmare roots of the algebraic (transcendental) equation A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt=0 . 26.⎝integraldisplay ⎝integraldisplay∞ 0[tJν(xt)y(t)+ϕ(x)Ψ(t,y(t))]dt=f(x), ν> –1. HereJν(z) is the Bessel function of the ﬁrst kind. The solutions can be obtained by the methods described in Subsection 16.4-4; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.7.17. Solutions: ym(t)=Yf(t)+AmYϕ(t), where Yf(t)=⎝integraldisplay∞ 0xJν(xt)f(x)dx,Yϕ(t)=⎝integraldisplay∞ 0xJν(xt)ϕ(x)dx, andAmare roots of the algebraic (transcendental) equation A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt=0 . 7.4-5. Other Equations. 27.⎝integraldisplay ⎝integraldisplay1 0[y(xt)+ϕ(x)Ψ(t,y(t))]dt=f(x). Solutions: ym(t)=Yf(t)+AmYϕ(t), where Yf(t)=tf/prime t(t)+f(t),Yϕ(t)=tϕ/prime t(t)+ϕ(t), andAmare roots of the algebraic (transcendental) equation A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt=0 . The functions f(x)a n dϕ(x) are assumed to satisfy the conditions⎝bracketleftbig xf(x)⎝bracketrightbig x=0=⎝bracketleftbig xϕ(x)⎝bracketrightbig x=0=0. 452 NONLINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRA TION 28.⎝integraldisplay ⎝integraldisplayπ/2 0[y(xsint)+ϕ(x)Ψ(t,y(t))]dt=f(x). Forϕ(x) = 0, it is the Schl ¨omilch equation, see Eq. 3.5.40. Solutions: ym(z)=Yf(z)+AmYϕ(z), where Yf(z)=2 π⎝bracketleftbigg f(0) +z⎝integraldisplayπ/2 0f/prime ξ(ξ)dτ⎝bracketrightbigg ,Yϕ(z)=2 π⎝bracketleftbigg ϕ(0) +z⎝integraldisplayπ/2 0ϕ/prime ξ(ξ)dτ⎝bracketrightbigg ,ξ=zsinτ, andAmare roots of the algebraic (transcendental) equation A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt=0 . Reference: A. D. Polyanin and A. I. Zhurov (2007). Chapter 8 Nonlinear Equations of the Second Kind with Constant Limits of Integration /trianglerightsld Notation: f,g,h,ϕ,Ψ, andψare arbitrary functions of an argument speciﬁed in the parentheses (the argument can depend on t,x, andy); and A,B,C,a,b,β,γ,λ, andµare arbitrary parameters. 8.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters 8.1-1. Equations of the Form y(x)+⎝integraltextb aK(x,t)y2(t)dt=F(x). 1. y(x)+A⎝integraldisplay ⎝integraldisplayb axλy2(t)dt=0 . Solutions: y1(x)=0 , y2(x)=–2λ+1 A(b2λ+1–a2λ+1)xλ. 2. y(x)+A⎝integraldisplay ⎝integraldisplayb axλtµy2(t)dt=0 . Solutions: y1(x)=0 , y2(x)=–2λ+µ+1 A(b2λ+µ+1–a2λ+µ+1)xλ. 3. y(x)+A⎝integraldisplay ⎝integraldisplayb ae–λxy2(t)dt=0 . Solutions: y1(x)=0 , y2(x)=2λ A(e–2λb–e–2λa)e–λx. 4. y(x)+A⎝integraldisplay ⎝integraldisplayb ae–λx–µty2(t)dt=0 . Solutions: y1(x)=0 , y2(x)=2λ+µ A[e–(2λ +µ)b–e–(2λ +µ)a]e–λx. 453 454 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 5. y(x)+A⎝integraldisplay ⎝integraldisplayb axλe–µty2(t)dt=0 . This is a special case of equation 8.2.2 with f(x)=Axλandg(t)=e–µt. 6. y(x)+A⎝integraldisplay ⎝integraldisplayb ae–µxtλy2(t)dt=0 . This is a special case of equation 8.2.2 with f(x)=Ae–µxandg(t)=tλ. 7. y(x)+A⎝integraldisplay ⎝integraldisplay1 0y2(t)dt=Bxµ,µ> –1. This is a special case of equation 8.2.4 with g(t)=A,f(x)=Bxµ,a=0 ,a n d b=1 . A solution: y(x)=Bxµ+λ,w h e r e λis determined by the quadratic equation λ2+1 A⎝parenleftbigg 1+2AB µ+1⎝parenrightbigg λ+B2 2µ+1=0 . 8. y(x)+A⎝integraldisplay ⎝integraldisplayb atβy2(t)dt=Bxµ. This is a special case of equation 8.2.4 with g(t)=Atβandf(x)=Bxµ. 9. y(x)+A⎝integraldisplay ⎝integraldisplayb aeβty2(t)dt=Beµx. This is a special case of equation 8.2.4 with g(t)=Aeβtandf(x)=Beµx. 10. y(x)+A⎝integraldisplay ⎝integraldisplayb axβy2(t)dt=Bxµ. This is a special case of equation 8.2.5 with g(x)=Axβandf(x)=Bxµ. 11. y(x)+A⎝integraldisplay ⎝integraldisplayb aeβxy2(t)dt=Beµx. This is a special case of equation 8.2.5 with g(x)=Aeβxandf(x)=Beµx. 8.1-2. Equations of the Form y(x)+⎝integraltextb aK(x,t)y(x)y(t)dt=F(x). 12. y(x)+A⎝integraldisplay ⎝integraldisplayb atβy(x)y(t)dt=Bxµ. This is a special case of equation 8.2.7 with g(t)=Atβandf(x)=Bxµ. 13. y(x)+A⎝integraldisplay ⎝integraldisplayb aeβty(x)y(t)dt=Beµx. This is a special case of equation 8.2.7 with g(t)=Aeβtandf(x)=Beµx. 14. y(x)+A⎝integraldisplay ⎝integraldisplayb axβy(x)y(t)dt=Bxµ. This is a special case of equation 8.2.8 with g(x)=Axβandf(x)=Bxµ. 15. y(x)+A⎝integraldisplay ⎝integraldisplayb aeβxy(x)y(t)dt=Beµx. This is a special case of equation 8.2.8 with g(x)=Aeβxandf(x)=Beµx. 8.1. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY PARAMETERS 455 8.1-3. Equations of the Form y(x)+⎝integraltextb aK(t)y(t)y(···)dt=F(x). 16. y(x)+A⎝integraldisplay ⎝integraldisplay1 0y(t)y(xt)dt=0 . This is a special case of equation 8.2.16 with f(t)=A,a=0 ,a n d b=1 . 1◦. Solutions: y1(x)=–1 A(2C+1 )xC,y2(x)=(I1–I0)x+I1–I2 I0I2–I2 1xC, Im=A 2C+m+1,m=0 ,1 ,2 , where Cis an arbitrary nonnegative constant. There are more complicated solutions of the form y(x)=xCn⎝summationtext k=0Bkxk,w h e r e Cis an arbitrary constant and the coefﬁcients Bkcan be found from the corresponding system of algebraic equations. 2◦. A solution: y3(x)=(I1–I0)xβ+I1–I2 I0I2–I2 1xC,Im=A 2C+mβ+1,m=0 ,1 ,2 , where Candβare arbitrary constants. There are more complicated solutions of the form y(x)=xCn⎝summationtext k=0Dkxkβ,w h e r e Candβ are arbitrary constants and the coefﬁcients Dkcan be found from the corresponding system of algebraic equations. 3◦. A solution: y4(x)=xC(J1lnx–J2) J0J2–J2 1,Jm=⎝integraldisplay1 0t2C(lnt)mdt,m=0 ,1 ,2 , where Cis an arbitrary constant. There are more complicated solutions of the form y(x)=xCn⎝summationtext k=0Ek(lnx)k,w h e r e Cis an arbitrary constant and the coefﬁcients Ekcan be found from the corresponding system of algebraic equations. 17. y(x)+A⎝integraldisplay ⎝integraldisplay∞ 1y(t)y(xt)dt=0 . This is a special case of equation 8.2.16 with f(t)=A,a=1 ,a n d b=∞. 18. y(x)+λ⎝integraldisplay ⎝integraldisplay∞ 1y(t)y(xt)dt=Axβ. This is a special case of equation 8.2.17 with f(t)=λ,a=0 ,a n d b=1 . 456 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 19. y(x)+A⎝integraldisplay ⎝integraldisplay1 0y(t)y(x+λt)dt=0 . This is a special case of equation 8.2.21 with f(t)≡A,a=0 ,a n d b=1 . 1◦. A solution: y(x)=C(λ+1 ) A[1 –eC(λ+1)]eCx, where Cis an arbitrary constant. 2◦. There are more complicated solutions of the form y(x)=eCxn⎝summationtext m=0Bmxm,w h e r e Cis an arbitrary constant and the coefﬁcients Bmcan be found from the corresponding system of algebraic equations. 20. y(x)+A⎝integraldisplay ⎝integraldisplay∞ 0y(t)y(x+λt)dt=0 , λ>0 , 0 ≤x<∞. This is a special case of equation 8.2.21 with f(t)≡A,a=0 ,a n d b=∞. A solution: y(x)=–C(λ+1 ) Ae–Cx, where Cis an arbitrary positive constant. 21. y(x)+A⎝integraldisplay ⎝integraldisplay∞ 0e–λty⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig y(t)dt=0 , λ>0 . A solution: y(x)=–λ AxC,w h e r e Cis an arbitrary constant. 22. y(x)+A⎝integraldisplay ⎝integraldisplay∞ 0e–λty⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig y(t)dt=Bxb,λ>0 . Solutions: y1(x)=β1xb,y2(x)=β2xb, where β1andβ2are the roots of the quadratic equation Aβ2+λβ–Bλ=0 . 8.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions 8.2-1. Equations of the Form y(x)+⎝integraltextb aK(x,t)y2(t)dt=F(x). 1. y(x)+⎝integraldisplay ⎝integraldisplayb af(x)y2(t)dt=0 . Solutions: y1(x)=0a n d y2(x)=λf(x), where λ=–⎝bracketleftBig⎝integraldisplayb af2(t)dt⎝bracketrightBig–1 . 2. y(x)+⎝integraldisplay ⎝integraldisplayb af(x)g(t)y2(t)dt=0 . This is a special case of equation 8.8.9. Solutions: y1(x)=0a n d y2(x)=λf(x), where λ=–⎝bracketleftBig⎝integraldisplayb af2(t)g(t)dt⎝bracketrightBig–1 . 8.2. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 457 3. y(x)+A⎝integraldisplay ⎝integraldisplayb ay2(t)dt=f(x). This is a special case of equation 8.8.7. A solution: y(x)=f(x)+λ,w h e r e λis determined by the quadratic equation A(b–a)λ2+( 1+2 AI1)λ+AI2=0 , w h e r e I1=⎝integraldisplayb af(t)dt,I2=⎝integraldisplayb af2(t)dt. 4. y(x)+⎝integraldisplay ⎝integraldisplayb ag(t)y2(t)dt=f(x). This is a special case of equation 8.8.9. A solution: y(x)=f(x)+λ,w h e r e λis determined by the quadratic equation I0λ2+( 1+2 I1)λ+I2=0 , w h e r e Im=⎝integraldisplayb afm(t)g(t)dt,m=0 ,1 ,2 . 5. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)y2(t)dt=f(x). Solution: y(x)=λg(x)+f(x), where λis determined by the quadratic equation Iggλ2+( 1+2 Ifg)λ+Iff=0 , Igg=⎝integraldisplayb ag2(t)dt,Ifg=⎝integraldisplayb af(t)g(t)dt,Iff=⎝integraldisplayb af2(t)dt. 6. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig g1(x)h1(t)+g2(x)h2(t)⎝bracketrightbig⎝bracketrightbig y2(t)dt=f(x). A solution: y(x)=λ1g1(x)+λ2g2(x)+f(x), where the constants λ1andλ2can be found from a system of two second-order algebraic equations (this system can be obtained from themore general system presented in 8.8.19). 8.2-2. Equations of the Form y(x)+⎝integraltextb a⎝summationtextKnm(x,t)yn(x)ym(t)dt=F(x),n+m≤2. 7. y(x)+⎝integraldisplay ⎝integraldisplayb ag(t)y(x)y(t)dt=f(x). Solutions: y1(x)=λ1f(x), y2(x)=λ2f(x), where λ1andλ2are the roots of the quadratic equation Iλ2+λ–1=0 , I=⎝integraldisplayb af(t)g(t)dt. 8. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)y(x)y(t)dt=f(x). A solution: y(x)=f(x) 1+λg(x), where λis a root of the algebraic (or transcendental) equation λ–⎝integraldisplayb af(t)dt 1+λg(t)=0 . Different roots generate different solutions of the integral equation. 458 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 9. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig g1(t)y2(x)+g2(x)y(t)⎝bracketrightbig⎝bracketrightbig dt=f(x). Solution in an implicit form: y(x)+Iy2(x)+λg2(x)–f(x)=0 , I=⎝integraldisplayb ag1(t)dt,( 1) where λis determined by the algebraic equation λ=⎝integraldisplayb ay(t)dt.( 2) Here the function y(x)=y(x,λ) obtained by solving the quadratic equation (1) must be substituted in the integrand of (2). 10. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig g1(t)y2(x)+g2(x)y2(t)⎝bracketrightbig⎝bracketrightbig dt=f(x). Solution in an implicit form: y(x)+Iy2(x)+λg2(x)–f(x)=0 , I=⎝integraldisplayb ag1(t)dt,( 1) where λis determined by the algebraic equation λ=⎝integraldisplayb ay2(t)dt.( 2) Here the function y(x)=y(x,λ) obtained by solving the quadratic equation (1) must be substituted into the integrand of (2). 11. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig g11(x)h11(t)y2(x)+g12(x)h12(t)y(x)y(t)+g22(x)h22(t)y2(t) +g1(x)h1(t)y(x)+g2(x)h2(t)y(t)⎝bracketrightbig⎝bracketrightbig dt=f(x). This is a special case of equation 8.8.49. 12. y(x)+⎝integraldisplay ⎝integraldisplay∞ –∞⎝bracketleftbig⎝bracketleftbig λe–|x–t|y(t)+ϕ(x)ψ(t)y2(t)]dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.2.14. Solutions for λ>–1 2: y1,2(x)=Yf(x)+A1,2Yϕ(x), where Yf(x)=f(x)–λ √ 1+2λ⎝integraldisplay∞ –∞exp⎝parenleftbig –√ 1+2λ|x–t|⎝parenrightbig f(t)dt, Yϕ(x)=ϕ(x)–λ √ 1+2λ⎝integraldisplay∞ –∞exp⎝parenleftbig –√ 1+2λ|x–t|⎝parenrightbig ϕ(t)dt, andA1,2are roots of the quadratic equation pA2+qA+r=0 , p=⎝integraldisplay∞ 0ψ(t)Y2 ϕ(t)dt,q=1+2⎝integraldisplay∞ 0ψ(t)Yf(t)Yϕ(t)dt,r=⎝integraldisplay∞ 0ψ(t)Y2 f(t)dt. 8.2. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 459 13. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0[λsin(xt)y(t)+ϕ(x)ψ(t)y2(t)]dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.5.20. Solutions for λ≠±⎝radicalBig 2 π: y1,2(x)=Yf(x)+A1,2Yϕ(x), where Yf(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0sin(xt)f(t)dt, Yϕ(x)=ϕ(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0sin(xt)ϕ(t)dt, andA1,2are roots of the quadratic equation pA2+qA+r=0 , p=⎝integraldisplay∞ 0ψ(t)Y2 ϕ(t)dt,q=2⎝integraldisplay∞ 0ψ(t)Yf(t)Yϕ(t)dt–1 ,r=⎝integraldisplay∞ 0ψ(t)Y2 f(t)dt. 14. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0[λcos(xt )y(t)+ϕ(x)ψ(t)y2(t)]dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.5.6. Solutions for λ≠±⎝radicalBig 2 π: y1,2(x)=Yf(x)+A1,2Yϕ(x), where Yf(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0cos(xt )f(t)dt, Yϕ(x)=ϕ(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0cos(xt )ϕ(t)dt, andA1,2are roots of the quadratic equation pA2+qA+r=0 , p=⎝integraldisplay∞ 0ψ(t)Y2 ϕ(t)dt,q=2⎝integraldisplay∞ 0ψ(t)Yf(t)Yϕ(t)dt–1 ,r=⎝integraldisplay∞ 0ψ(t)Y2 f(t)dt. 15. y(x)+⎝integraldisplay ⎝integraldisplay∞ 0[λtJν(xt)y(t)+ϕ(x)ψ(t)y2(t)]dt=f(x), ν> –1. HereJν(z) is the Bessel function of the ﬁrst kind. The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.8.4. Solutions for λ≠±1: y1,2(x)=Yf(x)+A1,2Yϕ(x), where Yf(x)=f(x) 1–λ2–λ 1–λ2⎝integraldisplay∞ 0tJν(xt)f(t)dt, Yϕ(x)=ϕ(x) 1–λ2–λ 1–λ2⎝integraldisplay∞ 0tJν(xt)ϕ(t)dt, 460 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION andA1,2are roots of the quadratic equation pA2+qA+r=0 , p=⎝integraldisplay∞ 0ψ(t)Y2 ϕ(t)dt,q=1+2⎝integraldisplay∞ 0ψ(t)Yf(t)Yϕ(t)dt,r=⎝integraldisplay∞ 0ψ(t)Y2 f(t)dt. 8.2-3. Equations of the Form y(x)+⎝integraltextb aK(t)y(t)y(···)dt=F(x). 16. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(xt)dt=0 . 1◦. Solutions: y1(x)=–1 I0xC,y2(x)=(I1–I0)x+I1–I2 I0I2–I2 1xC, Im=⎝integraldisplayb af(t)t2C+mdt,m=0 ,1 ,2 , where Cis an arbitrary constant. There are more complicated solutions of the form y(x)=xCn⎝summationtext k=0Bkxk,w h e r e Cis an arbitrary constant and the coefﬁcients Bkcan be found from the corresponding system of algebraic equations. 2◦. A solution: y3(x)=(I1–I0)xβ+I1–I2 I0I2–I2 1xC, Im=⎝integraldisplayb af(t)t2C+mβdt,m=0 ,1 ,2 , where Candβare arbitrary constants. There are more complicated solutions of the form y(x)=xCn⎝summationtext k=0Dkxkβ,w h e r e Candβ are arbitrary constants and the coefﬁcients Dkcan be found from the corresponding system of algebraic equations. 3◦. A solution: y4(x)=xC(J1lnx–J2) J0J2–J2 1, Jm=⎝integraldisplayb af(t)t2C(lnt)mdt,m=0 ,1 ,2 , where Cis an arbitrary constant. There are more complicated solutions of the form y(x)=xCn⎝summationtext k=0Ek(lnx)k,w h e r e Cis an arbitrary constant and the coefﬁcients Ekcan be found from the corresponding system of algebraic equations. 4◦. The equation also has the trivial solution y(x)≡0. 5◦. The substitution y(x)=xβw(x) leads to an equation of the same form, w(x)+⎝integraldisplayb ag(t)w(t)w(xt)dt=0 , g(x)=f(x)x2β. 8.2. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 461 17. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(xt)dt=Axβ. 1◦. Solutions: y1(x)=k1xβ,y2(x)=k2xβ, where k1andk2are the roots of the quadratic equation Ik2+k–A=0 , I=⎝integraldisplayb af(t)t2βdt. 2◦. Solutions: y(x)=xβ(λx+µ), where λandµare determined from the following system of two algebraic equations (this system can be reduced to a quadratic equation): I2λ+I1µ+1=0 , I1λµ+I0µ2+µ–A=0 where Im=⎝integraldisplayb af(t)t2β+mdt,m=0 ,1 ,2 . 3◦. There are more complicated solutions of the form y(x)=xβn⎝summationtext m=0Bmxm,w h e r et h e Bm can be found from the corresponding system of algebraic equations. 18. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(xt)dt=Alnx+B. This equation has solutions of the form y(x)=plnx+q, where the constants pandqcan be found from a system of two second-order algebraic equations. 19. y(x)+⎝integraldisplay ⎝integraldisplay∞ 0f(t)y(t)y⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig dt=0 . 1◦. A solution: y(x)=–kxC,k=⎝bracketleftbigg⎝integraldisplay∞ 0f(t)dt⎝bracketrightbigg–1 , where Cis an arbitrary constant. 2◦. The equation has the trivial solution y(x)≡0. 3◦. The substitution y(x)=xβw(x) leads to an equation of the same form, w(x)+⎝integraldisplay∞ 0f(t)w(t)w⎝parenleftBigx t⎝parenrightBig dt=0 . 20. y(x)+⎝integraldisplay ⎝integraldisplay∞ 0f(t)y(t)y⎝parenleftBig ⎝parenleftBigx t⎝parenrightBig ⎝parenrightBig dt=Axb. Solutions: y1(x)=λ1xb,y2(x)=λ2xb, where λ1andλ2are the roots of the quadratic equation Iλ2+λ–A=0 , I=⎝integraldisplay∞ 0f(t)dt. 462 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 21. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x+λt)dt=0 , λ>0 . 1◦. Solutions: y1(x)=–1 I0exp(–Cx),y2(x)=I2–I1x I2 1–I0I2exp(–Cx), Im=⎝integraldisplayb atmexp⎝bracketleftbig –C(λ+1 )t⎝bracketrightbig f(t)dt,m=0 ,1 ,2 , where Cis an arbitrary constant. 2◦. There are more complicated solutions of the form y(x)=e x p ( – Cx)n⎝summationtext k=0Akxk,w h e r e C is an arbitrary constant and the coefﬁcients Akcan be found from the corresponding system of algebraic equations. 3◦. The equation also has the trivial solution y(x)≡0. 4◦. The substitution y(x)=eβxw(x) leads to a similar equation: w(x)+⎝integraldisplayb ag(t)w(t)w(x+λt)dt=0 , g(t)=eβ(λ+1)tf(t). 22. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x+λt)y(t)dt=Ae–µx,λ>0 . 1◦. Solutions: y1(x)=k1e–µx,y2(x)=k2e–µx, where k1andk2are the roots of the quadratic equation Ik2+k–A=0 , I=⎝integraldisplayb ae–µ(λ+1)tf(t)dt. 2◦. There are more complicated solutions of the form y(x)=e–µxn⎝summationtext m=0Bmxm,w h e r et h e Bm can be found from the corresponding system of algebraic equations. 3◦. The substitution y(x)=eβxw(x) leads to an equation of the same form, w(x)+⎝integraldisplayb ag(t)w(t)w(x–t)dt=Ae(λ–β)x,g(t)=f(t)eβ(λ+1)t. 23. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=0 . 1◦. Solutions: y1(x)=–1 I0exp(Cx),y2(x)=I2–I1x I2 1–I0I2exp(Cx), Im=⎝integraldisplayb atmf(t)dt, where Cis an arbitrary constant and m=0 ,1 ,2 . 2◦. There are more complicated solutions of the form y(x)=e x p ( Cx)n⎝summationtext k=0Akxk,w h e r e Cis an arbitrary constant and the coefﬁcients Akcan be found from the corresponding system of algebraic equations. 3◦. The equation also has the trivial solution y(x)≡0. 4◦. The substitution y(x)=e x p ( Cx)w(x) leads to an equation of the same form: w(x)+⎝integraldisplayb af(t)w(t)w(x–t)dt=0 . 8.2. E QUATIONS WITH QUADRATIC NONLINEARITY THATCONTAIN ARBITRARY FUNCTIONS 463 24. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(x–t)y(t)dt=Aeλx. 1◦. Solutions: y1(x)=k1eλx,y2(x)=k2eλx, where k1andk2are the roots of the quadratic equation Ik2+k–A=0 , I=⎝integraldisplayb af(t)dt. 2◦. The substitution y(x)=eβxw(x) leads to an equation of the same form, w(x)+⎝integraldisplayb af(t)w(t)w(x–t)dt=Ae(λ–β)x. 25. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=Asinhλx. A solution: y(x)=psinhλx+qcoshλx.( 1) Herepandqare roots of the algebraic system p+I0pq+Ics(p2–q2)=A,q+Iccq2–Issp2=0 , ( 2 ) where I0=⎝integraldisplayb af(t)dt,Ics=⎝integraldisplayb af(t)c o s h ( λt)s i n h ( λt)dt, Icc=⎝integraldisplayb af(t)c o s h2(λt)dt,Iss=⎝integraldisplayb af(t)s i n h2(λt)dt. Different solutions of system (2) generate different solutions (1) of the integral equation. 26. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=Acoshλx. A solution: y(x)=psinhλx+qcoshλx.( 1) Herepandqare roots of the algebraic system p+I0pq+Ics(p2–q2)=0 , q+Iccq2–Issp2=A,( 2) where we use the notation introduced in 8.2.25. Different solutions of system (2) generate different solutions (1) of the integral equation. 27. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=Asinλx. A solution: y(x)=psinλx+qcosλx.( 1) Herepandqare roots of the algebraic system p+I0pq+Ics(p2+q2)=A,q+Iccq2–Issp2=0 , ( 2 ) where I0=⎝integraldisplayb af(t)dt,Ics=⎝integraldisplayb af(t)c o s ( λt)s i n (λt)dt, Icc=⎝integraldisplayb af(t)c o s2(λt)dt,Iss=⎝integraldisplayb af(t)s i n2(λt)dt. Different solutions of system (2) generate different solutions (1) of the integral equation. 464 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 28. y(x)+⎝integraldisplay ⎝integraldisplayb af(t)y(t)y(x–t)dt=Acosλx. A solution: y(x)=psinλx+qcosλx.( 1) Herepandqare roots of the algebraic system p+I0pq+Ics(p2+q2)=0 , q+Iccq2–Issp2=A,( 2) where we use the notation introduced in 8.2.27. Different solutions of system (2) generate different solutions (1) of the integral equation. 8.3. Equations with Power-Law Nonlinearity 8.3-1. Equations of the Form y(x)+⎝integraltextb aK(x,t)yβ(t)dt=F(x). 1. y(x)+A⎝integraldisplay ⎝integraldisplayb atλyβ(t)dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atλyβ. 2. y(x)+A⎝integraldisplay ⎝integraldisplayb aeµtyβ(t)dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Aeµtyβ. 3. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ(x–t)yβ(t)dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Ayβ. 4. y(x)–⎝integraldisplay ⎝integraldisplayb ag(x)yβ(t)dt=0 . A solution: y(x)=λg(x), λ=⎝bracketleftbigg⎝integraldisplayb agβ(t)dt⎝bracketrightbigg1 1–β . Forβ> 0, the equation also has the trivial solution y(x)≡0. 5. y(x)–⎝integraldisplay ⎝integraldisplayb ag(x)yβ(t)dt=h(x). This is a special case of equation 8.8.9 with f(t,y)=–yβ. 6. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)yβ(t)dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Ayβ. 7. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)yβ(t)dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Ayβ. 8.3. E QUATIONS WITH POWER -LAWNONLINEARITY 465 8. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt)yβ(t)dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Ayβ. 9. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt)yβ(t)dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Ayβ. 10. y(x)+⎝integraldisplay ⎝integraldisplay∞ 0f⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg⎝radicalbig y(t)dt=Ax2. Solutions: yk(x)=β2 kx2,w h e r e βk(k= 1, 2) are the roots of the quadratic equations β2±Iβ–A=0 , I=⎝integraldisplay∞ 0zf(z)dz. 11. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0tλf⎝parenleftbigg ⎝parenleftbiggt x⎝parenrightbigg ⎝parenrightbigg yβ(t)dt=0 , β≠1. A solution: y(x)=Ax1+λ 1–β,A1–β=⎝integraldisplay∞ 0zλ+β 1–βf(z)dz. 12. y(x)–⎝integraldisplay ⎝integraldisplay∞ –∞eλtf(ax +bt)yβ(t)dt=0 , b≠0,aβ≠–b. A solution: y(x)=Aexp⎝parenleftBig –aλ aβ+bx⎝parenrightBig ,A1–β=⎝integraldisplay∞ –∞exp⎝parenleftBigλb aβ+bz⎝parenrightBig f(bz)dz. 8.3-2. Other Equations. 13. y(x)+A⎝integraldisplay ⎝integraldisplayb ayβ(x)yµ(t)dt=f(x). Solution in an implicit form: y(x)+Aλyβ(x)–f(x)=0 , ( 1 ) where λis determined by the algebraic (or transcendental) equation λ=⎝integraldisplayb ayµ(t)dt.( 2) Here the function y(x)=y(x,λ) obtained by solving the quadratic equation (1) must be substituted in the integrand of (2). 14. y(x)+⎝integraldisplay ⎝integraldisplayb ag(t)y(x)yµ(t)dt=f(x). A solution: y(x)=λf(x), where λis determined from the algebraic (or transcendental) equation Iλµ+1+λ–1=0 , I=⎝integraldisplayb ag(t)fµ(t)dt. 466 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 15. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)y(x)yµ(t)dt=f(x). A solution: y(x)=f(x) 1+λg(x), where λis a root of the algebraic (or transcendental) equation λ–⎝integraldisplayb afµ(t)dt [1 +λg(t)]µ=0 . Different roots generate different solutions of the integral equation. 16. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig g1(t)y2(x)+g2(x)yµ(t)⎝bracketrightbig⎝bracketrightbig dt=f(x). Solution in an implicit form: y(x)+Iy2(x)+λg2(x)–f(x)=0 , I=⎝integraldisplayb ag1(t)dt,( 1) where λis determined by the algebraic (or transcendental) equation λ=⎝integraldisplayb ayµ(t)dt.( 2) Here the function y(x)=y(x,λ) obtained by solving the quadratic equation (1) must be substituted in the integrand of (2). 17. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig g1(x)h1(t)yk(x)ys(t)+g2(x)h2(t)yp(x)yq(t)⎝bracketrightbig⎝bracketrightbig dt=f(x). This is a special case of equation 8.8.49. 18. y(x)+A⎝integraldisplay ⎝integraldisplayb af(t)y(xt)yβ(t)dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Af(t)yβ. 19. y(x)+⎝integraldisplay ⎝integraldisplay∞ –∞⎝bracketleftbig⎝bracketleftbig λe–|x–t|y(t)+ϕ(x)ψ(t)yβ(t)]dt=f(x). This is a special case of equation 8.8.21. The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.2.14. 20. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0[λsin(xt)y(t)+ϕ(x)ψ(t)yβ(t)]dt=f(x). This is a special case of equation 8.8.22. The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.5.20. 8.4. E QUATIONS WITH EXPONENTIAL NONLINEARITY 467 21. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0[λcos(xt )y(t)+ϕ(x)ψ(t)yβ(t)]dt=f(x). This is a special case of equation 8.8.23. The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.5.6. 22. y(x)+⎝integraldisplay ⎝integraldisplay∞ 0[λtJν(xt)y(t)+ϕ(x)ψ(t)yβ(t)]dt=f(x). HereJν(z) is the Bessel function of the ﬁrst kind. This is a special case of equation 8.8.24. 8.4. Equations with Exponential Nonlinearity 8.4-1. Integrands with Nonlinearity of the Form exp[ βy(t)]. 1. y(x)+A⎝integraldisplay ⎝integraldisplayb aexp[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Aexp(βy). 2. y(x)+A⎝integraldisplay ⎝integraldisplayb atµexp[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atµexp(βy). 3. y(x)+A⎝integraldisplay ⎝integraldisplayb aexp⎝bracketleftbig⎝bracketleftbig µt+βy(t)⎝bracketrightbig⎝bracketrightbig dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Aexp(µt)e x p ( βy). 4. y(x)+A⎝integraldisplay ⎝integraldisplayb aexp⎝bracketleftbig⎝bracketleftbig λ(x–t)+βy(t)⎝bracketrightbig⎝bracketrightbig dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Aexp(βy). 5. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)e x p [βy(t)]dt=h(x). This is a special case of equation 8.8.9 with f(t,y)=e x p ( βy). 6. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)e x p [βy(t)]dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Aexp(βy). 7. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)e x p [βy(t)]dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Aexp(βy). 8. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt)e x p [βy(t)]dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Aexp(βy). 9. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt)e x p [βy(t)]dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Aexp(βy). 468 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 8.4-2. Other Integrands. 10. y(x)+A⎝integraldisplay ⎝integraldisplayb aexp⎝bracketleftbig⎝bracketleftbig βy(x)+γy(t)⎝bracketrightbig⎝bracketrightbig dt=h(x). This is a special case of equation 8.8.48 with g(x,y)=Aexp(βy)a n df(t,y)=e x p ( γy). 11. y(x)+A⎝integraldisplay ⎝integraldisplayb ay(xt)e x p [βy(t)]dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Aexp(βy). 8.5. Equations with Hyperbolic Nonlinearity 8.5-1. Integrands with Nonlinearity of the Form cosh[βy (t)]. 1. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh[βy (t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Acosh(βy). 2. y(x)+A⎝integraldisplay ⎝integraldisplayb atµcoshk[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atµcoshk(βy). 3. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(µt )c o s h [ βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Acosh(µt)c o s h ( βy). 4. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ(x–t)cosh[βy (t)]dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Acosh(βy). 5. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)c o s h [ βy(t)]dt=h(x). This is a special case of equation 8.8.9 with f(t,y)=c o s h ( βy). 6. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)c o s h [ βy(t)]dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Acosh(βy ). 7. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)c o s h [ βy(t)]dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Acosh(βy ). 8. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt)c o s h [ βy(t)]dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Acosh(βy ). 9. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt)c o s h [ βy(t)]dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Acosh(βy ). 8.5. E QUATIONS WITH HYPERBOLIC NONLINEARITY 469 8.5-2. Integrands with Nonlinearity of the Form sinh[ βy(t)]. 10. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Asinh(βy). 11. y(x)+A⎝integraldisplay ⎝integraldisplayb atµsinhk[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atµsinhk(βy). 12. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(µt)s i n h [ βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Asinh(µt)s i n h ( βy). 13. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ(x–t)sinh[βy(t)]dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Asinh(βy). 14. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)s i n h [ βy(t)]dt=h(x). This is a special case of equation 8.8.9 with f(t,y) = sinh(βy ). 15. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt) sinh[ βy(t)]dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Asinh(βy). 16. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)s i n h [ βy(t)]dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Asinh(βy). 17. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt)s i n h [ βy(t)]dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Asinh(βy). 18. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt)s i n h [ βy(t)]dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Asinh(βy). 8.5-3. Integrands with Nonlinearity of the Form tanh[ βy(t)]. 19. y(x)+A⎝integraldisplay ⎝integraldisplayb atanh[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atanh(βy). 470 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 20. y(x)+A⎝integraldisplay ⎝integraldisplayb atµtanhk[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atµtanhk(βy). 21. y(x)+A⎝integraldisplay ⎝integraldisplayb atanh(µt) tanh[ βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atanh(µt)t a n h ( βy). 22. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ(x–t)tanh[βy(t)]dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Atanh(βy). 23. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x) tanh[ βy(t)]dt=h(x). This is a special case of equation 8.8.9 with f(t,y)=t a n h ( βy). 24. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)t a n h [ βy(t)]dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Atanh(βy). 25. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt) tanh[ βy(t)]dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Atanh(βy). 26. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt) tanh[ βy(t)]dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Atanh(βy). 27. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt) tanh[ βy(t)]dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Atanh(βy). 8.5-4. Integrands with Nonlinearity of the Form coth[βy (t)]. 28. y(x)+A⎝integraldisplay ⎝integraldisplayb acoth[βy (t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Acoth(βy). 29. y(x)+A⎝integraldisplay ⎝integraldisplayb atµcothk[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atµcothk(βy). 30. y(x)+A⎝integraldisplay ⎝integraldisplayb acoth(µt )c o t h [ βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Acoth(µt)c o t h ( βy). 8.5. E QUATIONS WITH HYPERBOLIC NONLINEARITY 471 31. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ(x–t)coth[βy(t)]dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Acoth(βy). 32. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)c o t h [ βy(t)]dt=h(x). This is a special case of equation 8.8.9 with f(t,y)=c o t h ( βy). 33. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)c o t h [ βy(t)]dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Acoth(βy). 34. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)c o t h [ βy(t)]dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Acoth(βy). 35. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt)c o t h [ βy(t)]dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Acoth(βy). 36. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt)c o t h [ βy(t)]dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Acoth(βy). 8.5-5. Other Integrands. 37. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh[βy (x)] cosh[γy (t)]dt=h(x). This is a special case of equation 8.8.48 with g(x,y)=Acosh(βy)a n df(t,y)=c o s h ( γy). 38. y(x)+A⎝integraldisplay ⎝integraldisplayb ay(xt)c o s h [ βy(t)]dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Acosh(βy ). 39. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh[βy(x)] sinh[ γy(t)]dt=h(x). This is a special case of equation 8.8.48 with g(x,y)=Asinh(βy)a n df(t,y)=s i n h ( γy). 40. y(x)+A⎝integraldisplay ⎝integraldisplayb ay(xt) sinh[ βy(t)]dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Asinh(βy). 41. y(x)+A⎝integraldisplay ⎝integraldisplayb atanh[βy(x)] tanh[ γy(t)]dt=h(x). This is a special case of equation 8.8.48 with g(x,y)=Atanh(βy)a n df(t,y)=t a n h ( γy). 472 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 42. y(x)+A⎝integraldisplay ⎝integraldisplayb ay(xt)t a n h [ βy(t)]dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Atanh(βy). 43. y(x)+A⎝integraldisplay ⎝integraldisplayb acoth[βy (x)] coth[γy (t)]dt=h(x). This is a special case of equation 8.8.48 with g(x,y)=Acoth(βy)a n df(t,y)=c o t h ( γy). 44. y(x)+A⎝integraldisplay ⎝integraldisplayb ay(xt)c o t h [ βy(t)]dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Acoth(βy). 8.6. Equations with Logarithmic Nonlinearity 8.6-1. Integrands with Nonlinearity of the Form ln[ βy(t)]. 1. y(x)+A⎝integraldisplay ⎝integraldisplayb aln[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Aln(βy). 2. y(x)+A⎝integraldisplay ⎝integraldisplayb atµlnk[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atµlnk(βy). 3. y(x)+A⎝integraldisplay ⎝integraldisplayb aln(µt)l n [βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Aln(µt)l n ( βy). 4. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ(x–t)ln[βy(t)]dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Aln(βy). 5. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)l n [βy(t)]dt=h(x). This is a special case of equation 8.8.9 with f(t,y)=l n ( βy). 6. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)l n [βy(t)]dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Aln(βy). 7. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)l n [βy(t)]dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Aln(βy). 8.7. E QUATIONS WITH TRIGONOMETRIC NONLINEARITY 473 8. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt)l n [βy(t)]dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Aln(βy). 9. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt)l n [βy(t)]dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Aln(βy). 8.6-2. Other Integrands. 10. y(x)+A⎝integraldisplay ⎝integraldisplayb aln[βy(x)] ln[γy(t)]dt=h(x). This is a special case of equation 8.8.48 with g(x,y)=Aln(βy)a n df(t,y)=l n ( γy). 11. y(x)+A⎝integraldisplay ⎝integraldisplayb ay(xt)l n [βy(t)]dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Aln(βy). 8.7. Equations with Trigonometric Nonlinearity 8.7-1. Integrands with Nonlinearity of the Form cos[ βy(t)]. 1. y(x)+A⎝integraldisplay ⎝integraldisplayb acos[βy (t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Acos(βy ). 2. y(x)+A⎝integraldisplay ⎝integraldisplayb atµcosk[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atµcosk(βy). 3. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(µt )c o s [βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Acos(µt)c o s ( βy). 4. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ(x–t)cos[βy (t)]dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Acos(βy ). 5. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)c o s [βy(t)]dt=h(x). This is a special case of equation 8.8.9 with f(t,y)=c o s ( βy). 6. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)c o s [βy(t)]dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Acos(βy ). 474 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 7. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)c o s [βy(t)]dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Acos(βy ). 8. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt)c o s [βy(t)]dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Acos(βy ). 9. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt)c o s [βy(t)]dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Acos(βy ). 8.7-2. Integrands with Nonlinearity of the Form sin[ βy(t)]. 10. y(x)+A⎝integraldisplay ⎝integraldisplayb asin[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Asin(βy). 11. y(x)+A⎝integraldisplay ⎝integraldisplayb atµsink[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atµsink(βy). 12. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(µt)s i n [βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Asin(µt)s i n ( βy). 13. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ(x–t)sin[βy(t)]dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Asin(βy). 14. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)s i n [βy(t)]dt=h(x). This is a special case of equation 8.8.9 with f(t,y)=s i n ( βy). 15. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)s i n [βy(t)]dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Asin(βy). 16. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)s i n [βy(t)]dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Asin(βy). 17. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt)s i n [βy(t)]dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Asin(βy). 18. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt)s i n [βy(t)]dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Asin(βy). 8.7. E QUATIONS WITH TRIGONOMETRIC NONLINEARITY 475 8.7-3. Integrands with Nonlinearity of the Form tan[ βy(t)]. 19. y(x)+A⎝integraldisplay ⎝integraldisplayb atan[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atan(βy). 20. y(x)+A⎝integraldisplay ⎝integraldisplayb atµtank[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atµtank(βy). 21. y(x)+A⎝integraldisplay ⎝integraldisplayb atan(µt)t a n [βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atan(µt)t a n ( βy). 22. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ(x–t)tan[βy(t)]dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Atan(βy). 23. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)t a n [βy(t)]dt=h(x). This is a special case of equation 8.8.9 with f(t,y)=t a n ( βy). 24. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)t a n [βy(t)]dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Atan(βy). 25. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)t a n [βy(t)]dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Atan(βy). 26. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt)t a n [βy(t)]dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Atan(βy). 27. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt)t a n [βy(t)]dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Atan(βy). 8.7-4. Integrands with Nonlinearity of the Form cot[ βy(t)]. 28. y(x)+A⎝integraldisplay ⎝integraldisplayb acot[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Acot(βy). 476 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 29. y(x)+A⎝integraldisplay ⎝integraldisplayb atµcotk[βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Atµcotk(βy). 30. y(x)+A⎝integraldisplay ⎝integraldisplayb acot(µt)c o t [βy(t)]dt=g(x). This is a special case of equation 8.8.7 with f(t,y)=Acot(µt)c o t ( βy). 31. y(x)+A⎝integraldisplay ⎝integraldisplayb aeλ(x–t)cot[βy(t)]dt=g(x). This is a special case of equation 8.8.8 with f(t,y)=Acot(βy). 32. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)c o t [βy(t)]dt=h(x). This is a special case of equation 8.8.9 with f(t,y)=c o t ( βy). 33. y(x)+A⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)c o t [βy(t)]dt=h(x). This is a special case of equation 8.8.11 with f(t,y)=Acot(βy). 34. y(x)+A⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)c o t [βy(t)]dt=h(x). This is a special case of equation 8.8.12 with f(t,y)=Acot(βy). 35. y(x)+A⎝integraldisplay ⎝integraldisplayb acos(λx +µt)c o t [βy(t)]dt=h(x). This is a special case of equation 8.8.13 with f(t,y)=Acot(βy). 36. y(x)+A⎝integraldisplay ⎝integraldisplayb asin(λx +µt)c o t [βy(t)]dt=h(x). This is a special case of equation 8.8.14 with f(t,y)=Acot(βy). 8.7-5. Other Integrands. 37. y(x)+A⎝integraldisplay ⎝integraldisplayb acos[βy (x)] cos[γy (t)]dt=h(x). This is a special case of equation 8.8.48 with g(x,y)=Acos(βy )a n df(t,y)=c o s ( γy). 38. y(x)+A⎝integraldisplay ⎝integraldisplayb ay(xt)c o s [βy(t)]dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Acos(βy ). 39. y(x)+A⎝integraldisplay ⎝integraldisplayb asin[βy(x)] sin[γy(t)]dt=h(x). This is a special case of equation 8.8.48 with g(x,y)=Asin(βy)a n df(t,y)=s i n ( γy). 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 477 40. y(x)+A⎝integraldisplay ⎝integraldisplayb ay(xt)s i n [βy(t)]dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Asin(βy). 41. y(x)=λ⎝integraldisplay1 0f(x)g(t)s i n⎝parenleftbigg ⎝parenleftbiggy(t) f(t)⎝parenrightbigg ⎝parenrightbigg y(t)dt. Solutions are sought in the form y(x)=Af(x), where the constant Ais determined from the transcendental equation (the trivial solution corresponding to A= 0 is not taken into account): 1=λσsinA,σ=⎝integraldisplay1 0f(t)g(t)dt. For|λ|<1/|σ|, the integral equation has no real solutions (the case σ= 0 is included). For any λsatisfying the inequality |λ|>1/|σ|, the integral equation has inﬁnitely many real solutions. 42. y(x)+A⎝integraldisplay ⎝integraldisplayb atan[βy(x)] tan[γy (t)]dt=h(x). This is a special case of equation 8.8.48 with g(x,y)=Atan(βy)a n df(t,y)=t a n ( γy). 43. y(x)+A⎝integraldisplay ⎝integraldisplayb ay(xt)t a n [βy(t)]dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Atan(βy). 44. y(x)+A⎝integraldisplay ⎝integraldisplayb acot[βy(x)] cot[γy (t)]dt=h(x). This is a special case of equation 8.8.48 with g(x,y)=Acot(βy)a n df(t,y)=c o t ( γy). 45. y(x)+A⎝integraldisplay ⎝integraldisplayb ay(xt)c o t [βy(t)]dt=0 . This is a special case of equation 8.8.25 with f(t,y)=Acot(βy). 8.8. Equations with Nonlinearity of General Form 8.8-1. Equations of the Form y(x)+⎝integraltextb aK(|x–t|)G⎝parenleftbig y(t)⎝parenrightbig dt=F(x). 1. y(x)+⎝integraldisplay ⎝integraldisplayb a|x–t|f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=Ax2+Bx +C. This is a special case of equation 8.8.15 with f(t,y)=f(y)a n dg(x)=Ax2+Bx+C. The function y=y(x) obeys the second-order autonomous differential equation y/prime/prime xx+2f(y)=2A, whose solution can be represented in an implicit form: ⎝integraldisplayy yadu ⎝radicalbig w2a+4A(u–ya)–4F(u,ya)=±(x–a), F(u,v)=⎝integraldisplayu vf(t)dt,( 1 ) 478 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION where ya=y(a)a n dwa=y/prime x(a) are constants of integration. These constants, as well as the unknowns yb=y(b)a n dwb=y/prime x(b), are determined by the algebraic (or transcendental) system ya+yb–(a–b)wa=(b2+2ab–a2)A+2bB+2C, wa+wb=2 (a+b)A+2B, w2 b=w2 a+4A(yb–ya)–4F(yb,ya),⎝integraldisplayyb yadu ⎝radicalbig w2a+4A(u–ya)–4F(u,ya)=±(b–a).(2) Here the ﬁrst equation is obtained from the second condition of (5) in 8.8.15, the second equation is obtained from condition (6) in 8.8.15, and the third and fourth equations are consequences of (1). Each solution of system (2) generates a solution of the integral equation. 2. y(x)+⎝integraldisplay ⎝integraldisplayb aeλ|x–t|f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=A+Beλx+Ce–λx. This is a special case of equation 8.8.16 with f(t,y)=f(y)a n dg(x)=A+Beλx+Ce–λx. The function y=y(x) satisﬁes the second-order autonomous differential equation y/prime/prime xx+2λf(y)–λ2y=–λ2A,( 1) whose solution can be written in an implicit form: ⎝integraldisplayy yadu ⎝radicalbig w2a+λ2(u2–y2a)–2Aλ2(u–ya)–4λF(u,ya)=±(x–a),F(u,v)=⎝integraldisplayu vf(t)dt,( 2 ) where ya=y(a)a n dwa=y/prime x(a) are constants of integration. These constants, as well as the unknowns yb=y(b)a n dwb=y/prime x(b), are determined by the algebraic (or transcendental) system wa+λya=Aλ+2Bλeλa, wb–λyb=–Aλ–2Cλe–λb, w2 b=w2 a+λ2(y2 b–y2 a)–2Aλ2(yb–ya)–4λF(yb,ya), ⎝integraldisplayyb yadu ⎝radicalbig w2a+λ2(u2–y2a)–2Aλ2(u–ya)–4λF(u,ya)=±(b–a).(3) Here the ﬁrst and second equations are obtained from conditions (5) in 8.8.16, and the third and fourth equations are consequences of (2). Each solution of system (3) generates a solution of the integral equation. 3. y(x)+⎝integraldisplay ⎝integraldisplayb aeλ|x–t|f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=βcosh(λx ). This is a special case of equation 8.8.2 with A=0a n d B=C=1 2β. 4. y(x)+⎝integraldisplay ⎝integraldisplayb aeλ|x–t|f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=βsinh(λx). This is a special case of equation 8.8.2 with A=0 ,B=1 2β,a n dC=–1 2β. 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 479 5. y(x)+⎝integraldisplay ⎝integraldisplayb asinh⎝parenleftbig⎝parenleftbig λ|x–t|⎝parenrightbig⎝parenrightbig f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=A+Bcosh(λx )+Csinh(λx). This is a special case of equation 8.8.17 with f(t,y)=f(y)a n dg(x)=A+Bcosh(λx)+ Csinh(λx). The function y=y(x) satisﬁes the second-order autonomous differential equation y/prime/prime xx+2λf(y)–λ2y=–λ2A, whose solution can be represented in an implicit form: ⎝integraldisplayy yadu ⎝radicalbig w2a+λ2(u2–y2a)–2Aλ2(u–ya)–4λF(u,ya)=±(x–a), F(u,v)=⎝integraldisplayu vf(t)dt, where ya=y(a)a n dwa=y/prime x(a) are constants of integration, which can be determined from the boundary conditions (5) in 8.8.17. 6. y(x)+⎝integraldisplay ⎝integraldisplayb asin⎝parenleftbig⎝parenleftbig λ|x–t|⎝parenrightbig⎝parenrightbig f⎝parenleftbig⎝parenleftbig y(t)⎝parenrightbig⎝parenrightbig dt=A+Bcos(λx )+Csin(λx). This is a special case of equation 8.8.18 with f(t,y)=f(y)a n dg(x)=A+Bcos(λx)+ Csin(λx). The function y=y(x) satisﬁes the second-order autonomous differential equation y/prime/prime xx+2λf(y)+λ2y=λ2A, whose solution can be represented in an implicit form: ⎝integraldisplayy yadu ⎝radicalbig w2a–λ2(u2–y2a)+2Aλ2(u–ya)–4λF(u,ya)=±(x–a), F(u,v)=⎝integraldisplayu vf(t)dt, where ya=y(a)a n dwa=y/prime x(a) are constants of integration, which can be determined from the boundary conditions (5) in 8.8.18. 8.8-2. Equations of the Form y(x)+⎝integraltextb aK(x,t)G⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x). 7. y(x)+⎝integraldisplay ⎝integraldisplayb af⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). A solution: y(x)=g(x)+λ,w h e r e λis determined by the algebraic (or transcendental) equation λ+F(λ)=0 , F(λ)=⎝integraldisplayb af⎝parenleftbig t,g(t)+λ⎝parenrightbig dt. 8. y(x)+⎝integraldisplay ⎝integraldisplayb aeλ(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). A solution: y(x)=βeλx+g(x), where λis determined by the algebraic (or transcendental) equation β+F(β)=0 , F(β)=⎝integraldisplayb ae–λtf⎝parenleftbig t,βeλt+g(t)⎝parenrightbig dt. 480 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 9. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=h(x). A solution: y(x)=λg(x)+h(x), where λis determined by the algebraic (or transcendental) equation λ+F(λ)=0 , F(λ)=⎝integraldisplayb af⎝parenleftbig t,λg(t)+h(t)⎝parenrightbig dt. 10. y(x)+⎝integraldisplay ⎝integraldisplayb a(Ax +Bt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). A solution: y(x)=g(x)+λx+µ, where the constants λandµare determined from the algebraic (or transcendental) system λ+A⎝integraldisplayb af⎝parenleftbig t,g(t)+λt+µ⎝parenrightbig dt=0 , µ+B⎝integraldisplayb atf⎝parenleftbig t,g(t)+λt+µ⎝parenrightbig dt=0 . 11. y(x)+⎝integraldisplay ⎝integraldisplayb acosh(λx +µt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=h(x). Using the formula cosh( λx+µt)=c o s h ( λx) cosh( µt)+s i n h ( µt)s i n h ( λx), we arrive at an equation of the form 8.8.19: y(x)+⎝integraldisplayb a⎝bracketleftbig cosh(λx) f1⎝parenleftbig t,y(t)⎝parenrightbig +s i n h ( λx)f2⎝parenleftbig t,y(t)⎝parenrightbig⎝bracketrightbig dt=h(x), f1⎝parenleftbig t,y(t)⎝parenrightbig = cosh( µt)f⎝parenleftbig t,y(t)⎝parenrightbig ,f2⎝parenleftbig t,y(t)⎝parenrightbig = sinh(µt) f⎝parenleftbig t,y(t)⎝parenrightbig . 12. y(x)+⎝integraldisplay ⎝integraldisplayb asinh(λx +µt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=h(x). Using the formula sinh( λx+µt)=c o s h ( λx)s i n h ( µt)+c o s h ( µt)s i n h ( λx), we arrive at an equation of the form 8.8.19: y(x)+⎝integraldisplayb a⎝bracketleftbig cosh(λx) f1⎝parenleftbig t,y(t)⎝parenrightbig +s i n h ( λx)f2⎝parenleftbig t,y(t)⎝parenrightbig⎝bracketrightbig dt=h(x), f1⎝parenleftbig t,y(t)⎝parenrightbig = sinh(µt) f⎝parenleftbig t,y(t)⎝parenrightbig ,f2⎝parenleftbig t,y(t)⎝parenrightbig = cosh( µt)f⎝parenleftbig t,y(t)⎝parenrightbig . 13. y(x)+⎝integraldisplay ⎝integraldisplayb acos(λx +µt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=h(x). Using the formula cos( λx+µt)=c o s ( λx)c o s ( µt)–s i n ( µt)s i n ( λx), we arrive at an equation of the form 8.8.19: y(x)+⎝integraldisplayb a⎝bracketleftbig cos(λx) f1⎝parenleftbig t,y(t)⎝parenrightbig +s i n (λx)f2⎝parenleftbig t,y(t)⎝parenrightbig⎝bracketrightbig dt=h(x), f1⎝parenleftbig t,y(t)⎝parenrightbig =c o s ( µt)f⎝parenleftbig t,y(t)⎝parenrightbig ,f2⎝parenleftbig t,y(t)⎝parenrightbig =–s i n ( µt)f⎝parenleftbig t,y(t)⎝parenrightbig . 14. y(x)+⎝integraldisplay ⎝integraldisplayb asin(λx +µt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=h(x). Using the formula sin( λx+µt)=c o s ( λx)s i n ( µt)+c o s ( µt)s i n ( λx), we arrive at an equation of the form 8.8.19: y(x)+⎝integraldisplayb a⎝bracketleftbig cos(λx) f1⎝parenleftbig t,y(t)⎝parenrightbig +s i n (λx)f2⎝parenleftbig t,y(t)⎝parenrightbig⎝bracketrightbig dt=h(x), f1⎝parenleftbig t,y(t)⎝parenrightbig =s i n (µt)f⎝parenleftbig t,y(t)⎝parenrightbig ,f2⎝parenleftbig t,y(t)⎝parenrightbig =c o s ( µt)f⎝parenleftbig t,y(t)⎝parenrightbig . 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 481 15. y(x)+⎝integraldisplay ⎝integraldisplayb a|x–t|f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), a≤x≤b. 1◦. Let us remove the modulus in the integrand: y(x)+⎝integraldisplayx a(x–t)f⎝parenleftbig t,y(t)⎝parenrightbig dt+⎝integraldisplayb x(t–x)f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x). (1) Differentiating (1) with respect to xyields y/prime x(x)+⎝integraldisplayx af⎝parenleftbig t,y(t)⎝parenrightbig dt–⎝integraldisplayb xf⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime x(x). (2) Differentiating (2), we arrive at a second-order ordinary differential equation for y=y(x): y/prime/prime xx+2f(x,y)=g/prime/prime xx(x). (3) 2◦. Let us derive the boundary conditions for equation (3). We assume that –∞ <a<b<∞. By setting x=aandx=bin (1), we obtain the relations y(a)+⎝integraldisplayb a(t–a)f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(a), y(b)+⎝integraldisplayb a(b–t)f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(b).(4) Let us solve equation (3) for f(x,y) and substitute the result into (4). Integrating by parts yields the desired boundary conditions for y(x): y(a)+y(b)+(b–a)⎝bracketleftbig g/prime x(b)–y/prime x(b)⎝bracketrightbig =g(a)+g(b), y(a)+y(b)+(a–b)⎝bracketleftbig g/prime x(a)–y/prime x(a)⎝bracketrightbig =g(a)+g(b).(5) Let us point out a useful consequence of (5): y/prime x(a)+y/prime x(b)=g/prime x(a)+g/prime x(b), (6) which can be used together with one of conditions (5). Equation (3) under the boundary conditions (5) determines the solution of the original integral equation (there may be several solutions). Conditions (5) make it possible to calculate the constants of integration that occur in solving the differential equation (3). 16. y(x)+⎝integraldisplay ⎝integraldisplayb aeλ|x–t|f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), a≤x≤b. 1◦. Let us remove the modulus in the integrand: y(x)+⎝integraldisplayx aeλ(x–t)f⎝parenleftbig t,y(t)⎝parenrightbig dt+⎝integraldisplayb xeλ(t–x)f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x). (1) Differentiating (1) with respect to xtwice yields y/prime/prime xx(x)+2λf⎝parenleftbig x,y(x)⎝parenrightbig +λ2⎝integraldisplayx aeλ(x–t)f⎝parenleftbig t,y(t)⎝parenrightbig dt+λ2⎝integraldisplayb xeλ(t–x)f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime/prime xx(x). (2) 482 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary differential equation for y=y(x): y/prime/prime xx+2λf(x,y)–λ2y=g/prime/prime xx(x)–λ2g(x). (3) 2◦. Let us derive the boundary conditions for equation (3). We assume that –∞ <a<b<∞. By setting x=aandx=bin (1), we obtain the relations y(a)+e–λa⎝integraldisplayb aeλtf⎝parenleftbig t,y(t)⎝parenrightbig dt=g(a), y(b)+eλb⎝integraldisplayb ae–λtf⎝parenleftbig t,y(t)⎝parenrightbig dt=g(b).(4) Let us solve equation (3) for f(x,y) and substitute the result into (4). Integrating by parts yields eλbϕ/prime x(b)–eλaϕ/prime x(a)=λeλaϕ(a)+λeλbϕ(b),ϕ(x)=y(x)–g(x); e–λbϕ/prime x(b)–e–λaϕ/prime x(a)=λe–λaϕ(a)+λe–λbϕ(b). Hence, we obtain the boundary conditions for y(x): ϕ/prime x(a)+λϕ(a)=0 , ϕ/prime x(b)–λϕ(b)=0 ; ϕ(x)=y(x)–g(x). (5) Equation (3) under the boundary conditions (5) determines the solution of the original integral equation (there may be several solutions). Conditions (5) make it possible to calculatethe constants of integration that occur in solving the differential equation (3). 17. y(x)+⎝integraldisplay ⎝integraldisplay b asinh⎝parenleftbig⎝parenleftbig λ|x–t|⎝parenrightbig⎝parenrightbig f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), a≤x≤b. 1◦. Let us remove the modulus in the integrand: y(x)+⎝integraldisplayx asinh[λ(x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt+⎝integraldisplayb xsinh[λ(t–x)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x). (1) Differentiating (1) with respect to xtwice yields y/prime/prime xx(x)+2λf⎝parenleftbig x,y(x)⎝parenrightbig +λ2⎝integraldisplayx asinh[λ(x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt +λ2⎝integraldisplayb xsinh[λ(t–x)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime/prime xx(x). (2) Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary differential equation for y=y(x): y/prime/prime xx+2λf(x,y)–λ2y=g/prime/prime xx(x)–λ2g(x). (3) 2◦. Let us derive the boundary conditions for equation (3). We assume that –∞ <a<b<∞. By setting x=aandx=bin (1), we obtain the relations y(a)+⎝integraldisplayb asinh[λ(t–a)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(a), y(b)+⎝integraldisplayb asinh[λ(b–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(b).(4) Let us solve equation (3) for f(x,y) and substitute the result into (4). Integrating by parts yields sinh[λ(b–a)]ϕ/prime x(b)–λcosh[λ (b–a)]ϕ(b)=λϕ(a),ϕ(x)=y(x)–g(x); sinh[λ(b–a)]ϕ/prime x(a)+λcosh[λ(b–a)]ϕ(a)=–λϕ(b).(5) Equation (3) under the boundary conditions (5) determines the solution of the original integral equation (there may be several solutions). Conditions (5) make it possible to calculate the constants of integration that occur in solving the differential equation (3). 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 483 18. y(x)+⎝integraldisplay ⎝integraldisplayb asin(λ|x–t|)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), a≤x≤b. 1◦. Let us remove the modulus in the integrand: y(x)+⎝integraldisplayx asin[λ(x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt+⎝integraldisplayb xsin[λ(t–x)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x). (1) Differentiating (1) with respect to xtwice yields y/prime/prime xx(x)+2λf⎝parenleftbig x,y(x)⎝parenrightbig –λ2⎝integraldisplayx asin[λ(x–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt –λ2⎝integraldisplayb xsin[λ(t–x)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime/prime xx(x). (2) Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary differential equation for y=y(x): y/prime/prime xx+2λf(x,y)+λ2y=g/prime/prime xx(x)+λ2g(x). (3) 2◦. Let us derive the boundary conditions for equation (3). We assume that –∞ <a<b<∞. By setting x=aandx=bin (1), we obtain the relations y(a)+⎝integraldisplayb asin[λ(t–a)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(a), y(b)+⎝integraldisplayb asin[λ(b–t)]f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(b).(4) Let us solve equation (3) for f(x,y) and substitute the result into (4). Integrating by parts yields sin[λ(b–a)]ϕ/prime x(b)–λcos[λ (b–a)]ϕ(b)=λϕ(a),ϕ(x)=y(x)–g(x); sin[λ(b–a)]ϕ/prime x(a)+λcos[λ (b–a)]ϕ(a)=–λϕ(b).(5) Equation (3) under the boundary conditions (5) determines the solution of the original integral equation (there may be several solutions). Conditions (5) make it possible to calculate the constants of integration that occur in solving the differential equation (3). 8.8-3. Equations of the Form y(x)+⎝integraltextb aG⎝parenleftbig x,t,y(t)⎝parenrightbig dt=F(x). 19. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbig⎝bracketleftbig g1(x)f1⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig +g2(x)f2⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig⎝bracketrightbig⎝bracketrightbig dt=h(x). A solution: y(x)=h(x)+λ1g1(x)+λ2g2(x), where the constants λ1andλ2are determined from the algebrai c (or transcendental) system λ1+⎝integraldisplayb af1⎝parenleftbig t,h(t)+λ1g1(t)+λ2g2(t)⎝parenrightbig dt=0 , λ2+⎝integraldisplayb af2⎝parenleftbig t,h(t)+λ1g1(t)+λ2g2(t)⎝parenrightbig dt=0 . 484 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 20. y(x)+⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1gk(x)fk⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig⎝bracketrightbigg ⎝bracketrightbigg dt=h(x). A solution: y(x)=h(x)+n⎝summationdisplay k=1λkgk(x), where the coefﬁcients λkare determined from the algebrai c (or transcendental) system λm+⎝integraldisplayb afm⎝parenleftBig t,h(t)+n⎝summationdisplay k=1λkgk(t)⎝parenrightBig dt=0 ; m=1 ,...,n. Different roots of this system generate different solutions of the integral equation. Reference: A. F. Verlan’ and V . S. Sizikov (1986). 21. y(x)+⎝integraldisplay ⎝integraldisplay∞ –∞⎝bracketleftbig⎝bracketleftbig λe–|x–t|y(t)+ϕ(x)Ψ(t,y(t))]dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.2.14. Solutions for λ>–1 2: ym(x)=Yf(x)+AmYϕ(x), where Yf(x)=f(x)–λ √ 1+2λ⎝integraldisplay∞ –∞exp⎝parenleftbig –√ 1+2λ|x–t|⎝parenrightbig f(t)dt, Yϕ(x)=ϕ(x)–λ √ 1+2λ⎝integraldisplay∞ –∞exp⎝parenleftbig –√ 1+2λ|x–t|⎝parenrightbig ϕ(t)dt, andAmare roots of the algebraic (transcendental) equation A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt=0 . Reference: A. D. Polyanin and A. I. Zhurov (2007). 22. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0[λsin(xt)y(t)+ϕ(x)Ψ(t,y(t))]dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.5.20. Solutions for λ≠±⎝radicalBig 2 π: ym(x)=Yf(x)+AmYϕ(x), where Yf(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0sin(xt)f(t)dt, Yϕ(x)=ϕ(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0sin(xt)ϕ(t)dt, andAmare roots of the algebraic (transcendental) equation A–⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt=0 . 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 485 23. y(x)–⎝integraldisplay ⎝integraldisplay∞ 0[λcos(xt )y(t)+ϕ(x)Ψ(t,y(t))]dt=f(x). The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.5.6. Solutions for λ≠±⎝radicalBig 2 π: ym(x)=Yf(x)+AmYϕ(x), where Yf(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0cos(xt )f(t)dt, Yϕ(x)=ϕ(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0cos(xt )ϕ(t)dt, andAmare roots of the algebraic (transcendental) equation A–⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt=0 . 24. y(x)+⎝integraldisplay ⎝integraldisplay∞ 0[λtJν(xt)y(t)+ϕ(x)Ψ(t,y(t))]dt=f(x), ν> –1. HereJν(z) is the Bessel function of the ﬁrst kind. The solutions can be obtained by the methods described in Subsection 16.4-5; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.8.4. Solutions for λ≠±1: ym(x)=Yf(x)+AmYϕ(x), where Yf(x)=f(x) 1–λ2–λ 1–λ2⎝integraldisplay∞ 0tJν(xt)f(t)dt, Yϕ(x)=ϕ(x) 1–λ2–λ 1–λ2⎝integraldisplay∞ 0tJν(xt)ϕ(t)dt, andAmare roots of the algebraic (transcendental) equation A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt=0 . 8.8-4. Equations of the Form y(x)+⎝integraltextb ay(xt)G⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x). 25. y(x)+⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=0 . 1◦. A solution: y(x)=kxC,( 1) where Cis an arbitrary constant and the dependence k=k(C) is determined by the algebraic (or transcendental) equation 1+⎝integraldisplayb atCf⎝parenleftbig t,ktC⎝parenrightbig dt=0 . ( 2 ) Each root of equation (2) generates a solution of the integral equation which has the form (1). 2◦. The integral equation can have some other solutions similar to those indicated in items 1◦–3◦of equation 8.2.16. 486 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 26. y(x)+⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Ax +B. A solution: y(x)=px+q,( 1) where pandqare roots of the following system of algebraic (or transcendental) equations: p+p⎝integraldisplayb atf(t,pt+q)dt–A=0 , q+q⎝integraldisplayb af(t,pt+q)dt–B=0 .(2) Different solutions of system (2) generate different solutions (1) of the integral equation. 27. y(x)+⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Axβ. A solution: y(x)=kxβ,( 1) where kis a root of the algebraic (or transcendental) equation k+kF(k)–A=0 , F(k)=⎝integraldisplayb atβf⎝parenleftbig t,ktβ⎝parenrightbig dt.( 2) Each root of equation (2) generates a solution of the integral equation which has the form (1). 28. y(x)+⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Alnx+B. A solution: y(x)=plnx+q,( 1) where pandqare roots of the following system of algebraic (or transcendental) equations: p+p⎝integraldisplayb af(t,plnt+q)dt–A=0 , q+⎝integraldisplayb a(plnt+q)f(t,plnt+q)dt–B=0 .(2) Different solutions of system (2) generate different solutions (1) of the integral equation. 29. y(x)+⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Axβlnx. A solution: y(x)=pxβlnx+qxβ,( 1) where pandqare roots of the following system of algebraic (or transcendental) equations: p+p⎝integraldisplayb atβf(t,ptβlnt+qtβ)dt=A, q+⎝integraldisplayb a(ptβlnt+qtβ)f(t,ptβlnt+qtβ)dt=0 .(2) Different solutions of system (2) generate different solutions (1) of the integral equation. 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 487 30. y(x)+⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Acos(ln x). A solution: y(x)=pcos(ln x)+qsin(lnx), where pandqare roots of the following system of algebraic (or transcendental) equations: p+⎝integraldisplayb a⎝bracketleftbig pcos(ln t)+qsin(lnt)⎝bracketrightbig f⎝parenleftbig t,pcos(ln t)+qsin(lnt)⎝parenrightbig dt=A, q+⎝integraldisplayb a⎝bracketleftbig qcos(ln t)–psin(lnt)⎝bracketrightbig f⎝parenleftbig t,pcos(ln t)+qsin(lnt)⎝parenrightbig dt=0 . 31. y(x)+⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Asin(ln x). A solution: y(x)=pcos(ln x)+qsin(lnx), where pandqare roots of the following system of algebraic (or transcendental) equations: p+⎝integraldisplayb a⎝bracketleftbig pcos(ln t)+qsin(lnt)⎝bracketrightbig f⎝parenleftbig t,pcos(ln t)+qsin(lnt)⎝parenrightbig dt=0 , q+⎝integraldisplayb a⎝bracketleftbig qcos(ln t)–psin(lnt)⎝bracketrightbig f⎝parenleftbig t,pcos(ln t)+qsin(lnt)⎝parenrightbig dt=A. 32. y(x)+⎝integraldisplay ⎝integraldisplayb ay(xt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Axβcos(ln x)+Bxβsin(ln x). A solution: y(x)=pxβcos(ln x)+qxβsin(lnx), (1) where pandqare roots of the following system of algebraic (or transcendental) equations: p+⎝integraldisplayb atβ⎝bracketleftbig pcos(ln t)+qsin(lnt)⎝bracketrightbig f⎝parenleftbig t,ptβcos(ln t)+qtβsin(lnt)⎝parenrightbig dt=A, q+⎝integraldisplayb atβ⎝bracketleftbig qcos(ln t)–psin(lnt)⎝bracketrightbig f⎝parenleftbig t,ptβcos(ln t)+qtβsin(lnt)⎝parenrightbig dt=B.(2) Different solutions of system (2) generate different solutions (1) of the integral equation. 8.8-5. Equations of the Form y(x)+⎝integraltextb ay(x+βt)G⎝parenleftbig t,y(t)⎝parenrightbig dt=F(x). 33. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=0 . 1◦. A solution: y(x)=keCx,( 1) where Cis an arbitrary constant and the dependence k=k(C) is determined by the algebraic (or transcendental) equation 1+⎝integraldisplayb af⎝parenleftbig t,keCt⎝parenrightbig e–Ctdt=0 . ( 2 ) Each root of equation (2) generates a solution of the integral equation which has the form (1). 2◦. The equation has solutions of the form y(x)=n⎝summationtext m=0Emxm, where the constants Emcan be found by the method of undetermined coefﬁcients. 488 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 34. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Ax +B. A solution: y(x)=px+q,( 1) where pandqare roots of the following system of algebraic (or transcendental) equations: p+p⎝integraldisplayb af(t,pt+q)dt–A=0 , q+⎝integraldisplayb a(q–pt)f(t,pt+q)dt–B=0 .(2) Different solutions of system (2) generate different solutions (1) of the integral equation. 35. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Aeλx. Solutions: y(x)=kneλx, where knare roots of the algebraic (or transcendental) equation k+kF(k)–A=0 , F(k)=⎝integraldisplayb af⎝parenleftbig t,keλt⎝parenrightbig e–λtdt. 36. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Asinhλx. A solution: y(x)=psinhλx+qcoshλx,( 1) where pandqare roots of the following system of algebraic (or transcendental) equations: p+⎝integraldisplayb a(pcoshλt–qsinhλt)f⎝parenleftbig t,psinhλt+qcoshλt⎝parenrightbig dt=A, q+⎝integraldisplayb a(qcoshλt–psinhλt)f⎝parenleftbig t,psinhλt+qcoshλt⎝parenrightbig dt=0 .(2) Different solutions of system (2) generate different solutions (1) of the integral equation. 37. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Acoshλx. A solution: y(x)=psinhλx+qcoshλx, where pandqare roots of the following system of algebraic (or transcendental) equations: p+⎝integraldisplayb a(pcoshλt–qsinhλt)f⎝parenleftbig t,psinhλt+qcoshλt⎝parenrightbig dt=0 , q+⎝integraldisplayb a(qcoshλt–psinhλt)f⎝parenleftbig t,psinhλt+qcoshλt⎝parenrightbig dt=A. 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 489 38. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Asinλx. A solution: y(x)=psinλx+qcosλx,( 1) where pandqare roots of the following system of algebraic (or transcendental) equations: p+⎝integraldisplayb a(pcosλt+qsinλt)f⎝parenleftbig t,psinλt+qcosλt⎝parenrightbig dt=A, q+⎝integraldisplayb a(qcosλt–psinλt)f⎝parenleftbig t,psinλt+qcosλt⎝parenrightbig dt=0 .(2) Different solutions of system (2) generate different solutions (1) of the integral equation. 39. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Acosλx. A solution: y(x)=psinλx+qcosλx, where pandqare roots of the following system of algebraic (or transcendental) equations: p+⎝integraldisplayb a(pcosλt+qsinλt)f⎝parenleftbig t,psinλt+qcosλt⎝parenrightbig dt=0 , q+⎝integraldisplayb a(qcosλt–psinλt)f⎝parenleftbig t,psinλt+qcosλt⎝parenrightbig dt=A. 40. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=eµx(Asinλx +Bcosλx). A solution: y(x)=eµx(psinλx+qcosλx), (1) where pandqare roots of the following system of algebraic (or transcendental) equations: p+⎝integraldisplayb a(pcosλt+qsinλt)e–µtf⎝parenleftbig t,peµtsinλt+qeµtcosλt⎝parenrightbig dt=A, q+⎝integraldisplayb a(qcosλt–psinλt)e–µtf⎝parenleftbig t,peµtsinλt+qeµtcosλt⎝parenrightbig dt=B.(2) Different solutions of system (2) generate different solutions (1) of the integral equation. 41. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x–t)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). 1◦.F o rg(x)=n⎝summationtext k=1Akexp(λkx), the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkexp(λkx), where the constants Bkare determined from the nonlinear algebraic (or transcendental) system Bk+BkFk(/vectorB)–Ak=0 , k=1 ,...,n, /vectorB={B1,...,Bn},Fk(/vectorB)=⎝integraldisplayb af⎝parenleftbigg t,n⎝summationdisplay m=1Bmexp(λmt)⎝parenrightbigg exp(–λkt)dt. Different solutions of this system generate different solutions of the integral equation. 490 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 2◦. For a polynomial right-hand side, g(x)=n⎝summationtext k=0Akxk, the equation has a solution of the form y(x)=n⎝summationdisplay k=0Bkxk, where the constants Bkcan be found by the method of undetermined coefﬁcients. 3◦.F o rg(x)=eλxn⎝summationtext k=0Akxk, the equation has a solution of the form y(x)=eλxn⎝summationdisplay k=0Bkxk, where the constants Bkcan be found by the method of undetermined coefﬁcients. 4◦.F o rg(x)=n⎝summationtext k=1Akcos(λ kx), the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 5◦.F o rg(x)=n⎝summationtext k=1Aksin(λkx), the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 6◦.F o rg(x)=c o s ( λx)n⎝summationtext k=0Akxk, the equation has a solution of the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 7◦.F o rg(x)=s i n ( λx)n⎝summationtext k=0Akxk, the equation has a solution of the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 8◦.F o rg(x)=eµxn⎝summationtext k=1Akcos(λ kx), the equation has a solution of the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 491 9◦.F o rg(x)=eµxn⎝summationtext k=1Aksin(λkx), the equation has a solution of the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 10◦.F o rg(x)=c o s ( λx)n⎝summationtext k=1Akexp(µkx), the equation has a solution of the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Bkexp(µkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 11◦.F o rg(x)=s i n ( λx)n⎝summationtext k=1Akexp(µkx), the equation has a solution of the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Bkexp(µkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 42. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x+βt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Ax +B. A solution: y(x)=px+q,( 1) where pandqare roots of the following system of algebraic (or transcendental) equations: p+p⎝integraldisplayb af(t,pt+q)dt–A=0 , q+⎝integraldisplayb a(βpt+q)f(t,pt+q)dt–B=0 .(2) Different solutions of system (2) generate different solutions (1) of the integral equation. 43. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x+βt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Aeλx. Solutions: y(x)=kneλx, where knare roots of the algebraic (or transcendental) equation k+kF(k)–A=0 , F(k)=⎝integraldisplayb af⎝parenleftbig t,keλt⎝parenrightbig eβλtdt. 492 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 44. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x+βt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=Asinλx +Bcosλx. A solution: y(x)=psinλx+qcosλx,( 1) where pandqare roots of the following system of algebraic (or transcendental) equations: p+⎝integraldisplayb a⎝bracketleftbig pcos(λβt )–qsin(λβt)⎝bracketrightbig f⎝parenleftbig t,psinλt+qcosλt⎝parenrightbig dt=A, q+⎝integraldisplayb a⎝bracketleftbig qcos(λβt )+psin(λβt)⎝bracketrightbig f⎝parenleftbig t,psinλt+qcosλt⎝parenrightbig dt=B.(2) Different solutions of system (2) generate different solutions (1) of the integral equation. 45. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x+βt)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). 1◦.F o rg(x)=n⎝summationtext k=1Akexp(λkx), the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkexp(λkx), where the constants Bkare determined from the nonlinear algebraic (or transcendental) system Bk+BkFk(/vectorB)–Ak=0 , k=1 ,...,n, /vectorB={B1,...,Bn},Fk(/vectorB)=⎝integraldisplayb af⎝parenleftbigg t,n⎝summationdisplay m=1Bmexp(λmt)⎝parenrightbigg exp(λkβt)dt. Different solutions of this system generate different solutions of the integral equation. 2◦. For a polynomial right-hand side, g(x)=n⎝summationtext k=0Akxk, the equation has a solution of the form y(x)=n⎝summationdisplay k=0Bkxk, where the constants Bkcan be found by the method of undetermined coefﬁcients. 3◦.F o rg(x)=eλxn⎝summationtext k=0Akxk, the equation has a solution of the form y(x)=eλxn⎝summationdisplay k=0Bkxk, where the constants Bkcan be found by the method of undetermined coefﬁcients. 4◦.F o rg(x)=n⎝summationtext k=1Akcos(λ kx), the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 493 5◦.F o rg(x)=n⎝summationtext k=1Aksin(λkx), the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkcos(λ kx)+n⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 6◦.F o rg(x)=c o s ( λx)n⎝summationtext k=0Akxk, the equation has a solution of the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 7◦.F o rg(x)=s i n ( λx)n⎝summationtext k=0Akxk, the equation has a solution of the form y(x)=c o s ( λx)n⎝summationdisplay k=0Bkxk+s i n (λx)n⎝summationdisplay k=0Ckxk, where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 8◦.F o rg(x)=eµxn⎝summationtext k=1Akcos(λ kx), the equation has a solution of the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 9◦.F o rg(x)=eµxn⎝summationtext k=1Aksin(λkx), the equation has a solution of the form y(x)=eµxn⎝summationdisplay k=1Bkcos(λ kx)+eµxn⎝summationdisplay k=1Cksin(λkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 10◦.F o rg(x)=c o s ( λx)n⎝summationtext k=1Akexp(µkx), the equation has a solution of the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Ckexp(µkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 11◦.F o rg(x)=s i n ( λx)n⎝summationtext k=1Akexp(µkx), the equation has a solution of the form y(x)=c o s ( λx)n⎝summationdisplay k=1Bkexp(µkx)+s i n ( λx)n⎝summationdisplay k=1Ckexp(µkx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 494 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 8.8-6. Other Equations. 46. y(x)+⎝integraldisplay ⎝integraldisplayb ay(x)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x). A solution: y(x)=λg(x), where λis determined by the algebraic (or transcendental) equation λ+λF(λ)–1=0 , F(λ)=⎝integraldisplayb af⎝parenleftbig t,λg(t)⎝parenrightbig dt. 47. y(x)+⎝integraldisplay ⎝integraldisplayb ag(x)y(x)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=h(x). A solution: y(x)=h(x) 1+λg(x),w h e r e λis determined from the algebraic (or transcendental) equation λ–F(λ)=0 , F(λ)=⎝integraldisplayb af⎝parenleftbigg t,h(t) 1+λg(t)⎝parenrightbigg dt. 48. y(x)+⎝integraldisplay ⎝integraldisplayb ag⎝parenleftbig⎝parenleftbig x,y(x)⎝parenrightbig⎝parenrightbig f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=h(x). Solution in an implicit form: y(x)+λg⎝parenleftbig x,y(x)⎝parenrightbig –h(x)=0 , ( 1 ) where λis determined from the algebraic (or transcendental) equation λ–F(λ)=0 , F(λ)=⎝integraldisplayb af⎝parenleftbig t,y(t)⎝parenrightbig dt.( 2) Here the function y(x)=y(x,λ) obtained by solving (1) must be substituted into (2). The number of solutions of the integral equation is determined by the number of the solutions obtained from (1) and (2). 49. f⎝parenleftbig⎝parenleftbig x,y(x)⎝parenrightbig⎝parenrightbig +⎝integraldisplay ⎝integraldisplayb a⎝bracketleftbigg ⎝bracketleftbiggn⎝summationdisplay k=1gk⎝parenleftbig⎝parenleftbig x,y(x)⎝parenrightbig⎝parenrightbig hk⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig⎝bracketrightbigg ⎝bracketrightbigg dt=0 . Solution in an implicit form: f⎝parenleftbig x,y(x)⎝parenrightbig +n⎝summationdisplay k=1λkgk⎝parenleftbig x,y(x)⎝parenrightbig =0 , ( 1 ) where the λkare determined from the algebrai c (or transcendental) system λk–Hk(/vectorλ)=0 , k=1 ,...,n; Hk(/vectorλ)=⎝integraldisplayb ahk⎝parenleftbig t,y(t)⎝parenrightbig dt,/vectorλ={λ1,...,λn}.(2) Here the function y(x)=y(x,/vectorλ) obtained by solving (1) must be substituted into (2). The number of solutions of the integral equation is determined by the number of the solutions obtained from (1) and (2). 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 495 50. y(x)+⎝integraldisplay ⎝integraldisplayb ay⎝parenleftbig⎝parenleftbig xtβ⎝parenrightbig⎝parenrightbig f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), β>0 . 1◦.F o rg(x)=n⎝summationtext k=1Akxk, the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkxk, where Bkare roots of the algebraic (or transcendental) equations Bk+BkFk(/vectorB)–Ak=0 , Fk(/vectorB)=⎝integraldisplayb atkβf⎝parenleftbigg t,n⎝summationdisplay m=1Bmtm⎝parenrightbigg dt. Different roots of this system generate different solutions of the integral equation. 2◦.F o rg(x)=l nxn⎝summationtext k=0Akxk, the equation has a solution of the form y(x)=l nxn⎝summationdisplay k=0Bkxk+n⎝summationdisplay k=0Ckxk, where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 3◦.F o rg(x)=n⎝summationtext k=0Ak⎝parenleftbig lnx)k, the equation has a solution of the form y(x)=n⎝summationdisplay k=0Bk⎝parenleftbig lnx)k, where the constants Bkcan be found by the method of undetermined coefﬁcients. 4◦.F o rg(x)=n⎝summationtext k=1Akcos(λ klnx), the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkcos(λ klnx)+n⎝summationdisplay k=1Cksin(λklnx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 5◦.F o rg(x)=n⎝summationtext k=1Aksin(λklnx), the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkcos(λ klnx)+n⎝summationdisplay k=1Cksin(λklnx), where the constants BkandCkcan be found by the method of undetermined coefﬁcients. 496 NONLINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRA TION 51. y(x)+⎝integraldisplay ⎝integraldisplayb ay(ξ)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=0 , ξ=xϕ(t). 1◦. A solution: y(x)=kxC,( 1) where Cis an arbitrary constant and the dependence k=k(C) is determined by the algebraic (or transcendental) equation 1+⎝integraldisplayb a⎝bracketleftbig ϕ(t)⎝bracketrightbigCf⎝parenleftbig t,ktC⎝parenrightbig dt=0 . ( 2 ) Each root of equation (2) generates a solution of the integral equation which has the form (1). 2◦. The equation has solutions of the form y(x)=n⎝summationtext m=0Emxm, where the constants Emcan be found by the method of undetermined coefﬁcients. 52. y(x)+⎝integraldisplay ⎝integraldisplayb ay(ξ)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), ξ=xϕ(t). 1◦.F o rg(x)=n⎝summationtext k=1Akxk, the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkxk, where Bkare roots of the algebraic (or transcendental) equations Bk+BkFk(/vectorB)–Ak=0 , k=1 ,...,n, /vectorB={B1,...,Bn},Fk(/vectorB)=⎝integraldisplayb a⎝bracketleftbig ϕ(t)⎝bracketrightbigkf⎝parenleftbigg t,n⎝summationdisplay m=1Bmtm⎝parenrightbigg dt. Different roots generate different solutions of the integral equation. 2◦. A form of solutions for some other functions g(x) can be found in items 2◦–5◦of equation 8.8.50. 53. y(x)+⎝integraldisplay ⎝integraldisplayb ay(ξ)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=0 , ξ=x+ϕ(t). 1◦. A solution: y(x)=keCx,( 1) where Cis an arbitrary constant and the dependence k=k(C) is determined by the algebraic (or transcendental) equation 1+⎝integraldisplayb aeCϕ(t)f⎝parenleftbig t,keCt⎝parenrightbig dt=0 . ( 2 ) Each root of equation (2) generates a solution of the integral equation which has the form (1). 2◦. The equation has a solution of the form y(x)=n⎝summationtext m=0Emxm, where the constants Emcan be found by the method of undetermined coefﬁcients. 8.8. E QUATIONS WITH NONLINEARITY OF GENERAL FORM 497 54. y(x)+⎝integraldisplay ⎝integraldisplayb ay(ξ)f⎝parenleftbig⎝parenleftbig t,y(t)⎝parenrightbig⎝parenrightbig dt=g(x), ξ=x+ϕ(t). 1◦.F o rg(x)=n⎝summationtext k=1Akexp(λkx) the equation has a solution of the form y(x)=n⎝summationdisplay k=1Bkexp(λkx), where the constants Bkare determined from the nonlinear algebraic (or transcendental) system Bk+BkFk(/vectorB)–Ak=0 , k=1 ,...,n, /vectorB={B1,...,Bn},Fk(/vectorB)=⎝integraldisplayb af⎝parenleftbigg t,n⎝summationdisplay m=1Bmexp(λmt)⎝parenrightbigg exp⎝bracketleftbig λkϕ(t)⎝bracketrightbig dt. 2◦. A form of solutions for some other functions g(x) can be found in items 2◦–11◦of equation 8.8.45. Part II Methods for Solving Integral Equations Chapter 9 Main Deﬁnitions and Formulas. Integral Transforms 9.1. Some Deﬁnitions, Remarks, and Formulas 9.1-1. Some Deﬁnitions. A function f(x)i ss a i dt ob e square integrable on an interval [ a,b]i ff2(x) is integrable on [ a,b]. The set of all square integrable functions is denoted by L2(a,b)o r ,b r i e ﬂ y , L2.* Likewise, the set of all integrable functions on [ a,b] is denoted by L1(a,b)o r ,b r i e ﬂ y , L1. Let us list the main properties of functions from L2. 1◦. The sum of two square integrable functions is a square integrable function. 2◦. The product of a square integrable function by a constant is a square integrable function. 3◦. The product of two square integrable functions is an integrable function. 4◦.I ff(x)∈L2andg(x)∈L2, then the following Cauchy–Schwarz–Bunyakovsky inequality holds: (f,g)2≤/bardblf/bardbl2/bardblg/bardbl2, (f,g)=⎝integraldisplayb af(x)g(x)dx,/bardblf/bardbl2=(f,f)=⎝integraldisplayb af2(x)dx. The number ( f,g) is called the inner product of the functions f(x)a n dg(x) and the number /bardblf/bardblis called the L2-norm off(x). 5◦.F o rf(x)∈L2andg(x)∈L2, the following triangle inequality holds: /bardblf+g/bardbl≤/bardblf/bardbl+/bardblg/bardbl. 6◦. Let functions f(x)a n df1(x),f2(x),...,fn(x),...be square integrable on an interval [ a,b]. If lim n→∞⎝integraldisplayb a⎝bracketleftbig fn(x)–f(x)⎝bracketrightbig2dx=0 , then the sequence f1(x),f2(x),...is said to be mean-square convergent tof(x). Note that if a sequence of functions {fn(x)}fromL2converges uniformly to f(x), then f(x)∈L2 and{fn(x)}is mean-square convergent to f(x). * In the most general case the integral is understood as the Lebesgue integral of measurable functions (see Supplement 12.3). As usual, two equivalent functions (i.e., equal everywhere, or distinct on a negligible set (of zero measure)) are regarded as one and the same element of L2. 501 502 MAINDEFINITIONS AND FORMULAS .INTEGRAL TRANSFORMS The notion of an integrable function of several variables is similar. For instance, a function f(x,t) is said to be square integrable in a domain S={a≤x≤b,a≤t≤b}iff(x,t) is measurable and /bardblf/bardbl2≡⎝integraldisplayb a⎝integraldisplayb af2(x,t)dx dt <∞. Here/bardblf/bardbldenotes the norm of the function f(x,t), as above. 9.1-2. Structure of Solutions to Linear Integral Equations. A linear integral equation with variable integration limit has the form βy(x)+⎝integraldisplayx aK(x,t)y(t)dt=f(x), (1) where y(x) is the unknown function. A linear integral equation with constant integration limits has the form βy(x)+⎝integraldisplayb aK(x,t)y(t)dt=f(x). (2) Forβ= 0, Eqs. (1) and (2) are called linear integral equations of the ﬁrst kind ,a n df o r β≠0, linear integral equations of the second kind .* Equations of the form (1) and (2) with speciﬁc conditions imposed on the kernels and the right-hand sides form various classes of integral equations (V olterra equations, Fredholm equations, convolution equations, etc.), which are considered in detail in Chapters 10–14. For brevity, we shall sometimes represent t he linear equations (1) and (2) in the operator form L[y]=f(x). (3) A linear operator Lpossesses the properties L[y1+y2]=L[y1]+L[y2], L[σy]=σL[y],σ= const . A linear equation is called homogeneous iff(x)≡0a n dnonhomogeneous otherwise. An arbitrary homogeneous linear integral equation has the trivial solution y≡0. Ify1=y1(x)a n dy2=y2(x) are particular solutions of a linear homogeneous integral equation, then the linear combination C1y1+C2y2with arbitrary constants C1andC2is also a solution (in physical problems, this property is called the linear superposition principle ). The general solution of a linear nonhomogeneous integral equation (3) is the sum of the general solution Y=Y(x) of the corresponding homogeneous equation L[Y] = 0 and an arbitrary particular solution ¯ y=¯y(x) of the nonhomogeneous equation L[¯y]=f(x), that is, y=Y+¯y.( 4) If the homogeneous integral equation has only the trivial solution Y≡0, then the solution of the corresponding nonhomogeneous equation is unique (if it exists). Let ¯y1and ¯y2be solutions of nonhomogeneous linear integral equations with the same left-hand sides and different right-hand sides, L[¯y1]=f1(x)a n d L[¯y2]=f2(x). Then the function ¯ y=¯y1+¯y2 is a solution of the equation L[¯y]=f1(x)+f2(x). The transformation x=g(z),t=g(τ), y(x)=ϕ(z)w(z)+ψ(z), (5) where g(z),ϕ(z), and ψ(z) are arbitrary continuous functions ( g/prime z≠0), reduces Eqs. (1) and (2) to linear equations of the same form for the unknown function w=w(z). Such transformations are frequently used for constructing exact solutions of linear integral equations. * In Chapters 1–4, which deal with equations with variable and constant limits of integration, we sometimes consider more general equations in which the integrand contains the unknown function y(z), where z=z(x,t), instead of y(t). 9.1. S OME DEFINITIONS ,REMARKS ,AND FORMULAS 503 9.1-3. Integral Transforms. Integral transforms have the form ˜f(λ)=⎝integraldisplayb aϕ(x,λ)f(x)dx. The function ˜f(λ) is called the transform of the function f(x)a n dϕ(x,λ) is called the kernel of the integral transform. The function f(x) is called the inverse transform of˜f(λ). The limits of integration aandbare real numbers (usually, a=0 ,b=∞ora=–∞,b=∞). In Subsections 9.2–9.6, the most popular (Laplace, Mellin, Fourier, etc.) integral transforms, applied in this book to the solution of speciﬁc integral equations, are described. These subsections also describe the corresponding inversion formulas, which have the form f(x)=⎝integraldisplay Lψ(x,λ)˜f(λ)dλ and make it possible to recover f(x)i f˜f(λ) is given. The integration path Lcan lie either on the real axis or in the complex plane. Integral transforms are used in the solution of various differential and integral equations (see, for example, Sections 10.4, 11.3, 11.6, 12.5, and 13.9). Figure 1 outlines the overall scheme of solving some special classes of linear integral equations by means of integral transforms (by applying appropriate integral transforms to this sort of integral equations, one obtains ﬁrst-orderlinear algebraic equations for ˜f(λ)). Solution of the equation for the transformOriginal integral equation for a function ( ) yy x/c61 Application of an integral transform Algebraic equation for the transform yy/c61/c108() Derivation of an explicit form of the function yy/c61/c108() Application of the inverse integral transform Derivation of a solution to the original integral equation Figure 1. Principal scheme of applying integral transforms for solving integral equations. In many cases, to calculate deﬁnite integrals, in particular, to ﬁnd the inverse Laplace, Mellin, and Fourier transforms, methods of the theory of functions of a complex variable can be applied,including the residue theorem and the Jordan lemma, which are presented below in Subsections 9.1-4 and 9.1-5. 504 MAINDEFINITIONS AND FORMULAS .INTEGRAL TRANSFORMS 9.1-4. Residues. Calculation Formulas. Cauchy’s Residue Theorem. 1◦.T h e residue of a function f(z) holomorphic in a deleted neighborhood of a point z=a(thus, ais an isolated singularity of f)o ft h ec o m p l e xp l a n e zis the number res z=af(z)=1 2πi⎝integraldisplay cεf(z)dz,i2= –1, where cεis a circle of sufﬁciently small radius εdescribed by the equation |z–a|=ε. If the point z=ais a pole of order n*o ft h ef u n c t i o n f(z), then we have res z=af(z)=1 (n–1 ) !lim z→adn–1 dxn–1⎝bracketleftbig (z–a)nf(z)⎝bracketrightbig . For a simple pole, which corresponds to n= 1, this implies res z=af(z) = lim z→a⎝bracketleftbig (z–a)f(z)⎝bracketrightbig . Iff(z)=ϕ(z) ψ(z),w h e r e ϕ(a)≠0a n dψ(z) has a simple zero at the point z=a, i.e.,ψ(a)=0a n d ψ/prime z(a)≠0, then res z=af(z)=ϕ(a) ψ/primez(a). 2◦. A function f(z) is said to be continuous on the boundary Cof the domain Dif for each boundary pointz0there exists a limit lim z→z0f(z)=f(z0)a sz→z0,z∈D. CAUCHY ’S RESIDUE THEOREM .Letf(z)be a function continuous on the boundary Cof a domain Dand analytic in the interior of Deverywhere except for ﬁnitely many points a1,...,an.T h e n⎝integraldisplay Cf(z)dz=2πin⎝summationdisplay k=1resf(ak), where the integral is taken in the positive sense of C. The residue of a function f(z) at inﬁnity is deﬁned as resf(∞)=1 2πi⎝contintegraldisplay Γf(z)dz, whereΓis a circle of sufﬁciently large radius |z|=ρand the integral is taken in the clockwise sense (so that the neighborhood of the point z=∞remains to the left of the contour, just as in the case of a ﬁnite point). Note that resf(∞) = lim z→∞[–zf(z)], provided that this limit exists. THEOREM .If a function f(z)has ﬁnitely many singular points a1,...,anin the extended complex plane, then the sum of all its residues, inc luding the residue at inﬁnity, is zero: resf(∞)+n⎝summationdisplay k=1resf(ak)=0 . * In a neighborhood of this point we have f(z)≈const ( z–a)–n. 9.2. L APLACE TRANSFORM 505 9.1-5. Jordan Lemma. JORDAN LEMMA .If a function f(z)is continuous in the domain |z|≥R0,Imz≥α,w h e r e αis a chosen real number, and if lim z→∞f(z)=0,t h e n lim R→∞⎝integraldisplay CReiλzf(z)dz=0 for any λ>0,w h e r e CRis the arc of the circle |z|=Rthat lies in this domain. If a function f(z) is analytic for |z|>R0andzf(z)→0a s|z|→∞ fory≥0( o rx≥0), then lim R→∞⎝integraldisplay CRf(z)dz=0 , where CRis the arc of the circle |z|=Rin the upper half-plane (or right half-plane). References for Section 9.1: A. G. Sveshnikov and A. N. Tikhonov (1970), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), W. R. LePage (1980), A. D. Polyanin and A. V . Manzhirov (1998), A. N. Kolmogorov andS. V . Fomin (1999), S. G. Krantz (1999). 9.2. Laplace Transform 9.2-1. Deﬁnition. Inversion Formula. The Laplace transform of an arbitrary (complex-valued) function f(x) of a real variable x(x≥0) is deﬁned by ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx,( 1) where p=s+iσis a complex variable. The Laplace transform exists for any continuous or piecewise-continuous function satisfying the condition |f(x)|<Meσ0xwith some M>0a n d σ0≥0. In the following, σ0often means the greatest lower bound of the possible values of σ0in this estimate; this value is called the growth exponent of the function f(x). For any f(x), the transform ˜f(p) is deﬁned in the half-plane Re p>σ0and is analytic there. For brevity, we shall write formula (1) as follows: ˜f(p)=L⎝braceleftbig f(x)⎝bracerightbig ,o r ˜f(p)=L⎝braceleftbig f(x),p⎝bracerightbig . Given the transform ˜f(p), the function can be found by means of the inverse Laplace transform f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞˜f(p)epxdp,i2= –1, (2) where the integration path is parallel to the imaginary axis and lies to the right of all singularities of˜f(p), which corresponds to c>σ0. The integral in (2) is understood in the sense of the Cauchy principal value: ⎝integraldisplayc+i∞ c–i∞˜f(p)epxdp= lim ω→∞⎝integraldisplayc+iω c–iω˜f(p)epxdp. In the domain x<0 ,f o r m u l a( 2 )g i v e s f(x)≡0. Formula (2) holds for continuous functions. If f(x) has a (ﬁnite) jump discontinuity at a point x=x0> 0, then the left-hand side of (2) is equal to1 2[f(x0–0 )+f(x0+ 0)] at this point (for x0=0 , the ﬁrst term in the square brackets must be omitted). For brevity, we write the Laplace inversion formula (2) as follows: f(x)=L–1⎝braceleftbig˜f(p)⎝bracerightbig ,o r f(x)=L–1⎝braceleftbig˜f(p),x⎝bracerightbig . 506 MAINDEFINITIONS AND FORMULAS .INTEGRAL TRANSFORMS 9.2-2. Inverse Transforms of Rational Functions. Consider the important case in which the transform is a rational function of the form ˜f(p)=R(p) Q(p),( 3) where Q(p)a n dR(p) are polynomials in the variable pand the degree of Q(p) exceeds that of R(p). Assume that the zeros of the denominator are simple, i.e., Q(p)≡const ( p–λ1)(p–λ2)...(p–λn). Then the inverse transform can be determined by the formula f(x)=n⎝summationdisplay k=1R(λk) Q/prime(λk)exp(λkx), (4) where the primes denote the derivatives. IfQ(p) has multiple zeros, i.e., Q(p)≡const ( p–λ1)s1(p–λ2)s2...(p–λm)sm, then f(x)=m⎝summationdisplay k=11 (sk–1 ) !lim p→skdsk–1 dpsk–1⎝bracketleftbig (p–λk)sk˜f(p)epx⎝bracketrightbig .( 5) Example 1. The transform ˜f(p)=b p2–a2(a,breal numbers) can be represented as the fraction (3) with R(p)=bandQ(p)=(p–a)(p+a). The denominator Q(p) has two simple roots, λ1=aandλ2=–a. Using formula (4) with n=2a n d Q/prime(p)=2p, we obtain the inverse transform in the form f(x)=b 2aeax–b 2ae–ax=b asinh(ax ). Example 2. The transform ˜f(p)=b p2+a2(a,breal numbers) can be written as the fraction (3) with R(p)=bandQ(p)=(p–ia)(p+ia),i2= –1. The denominator Q(p) has two simple pure imaginary roots, λ1=iaandλ2=–ia. Using formula (4) with n= 2, we ﬁnd the inverse transform: f(x)=b 2iaeiax–b 2iae–iax=–bi 2a⎝bracketleftbig cos(ax)+isin(ax)⎝bracketrightbig +bi 2a⎝bracketleftbig cos(ax)–isin(ax)⎝bracketrightbig =b asin(ax). Example 3. The transform ˜f(p)=ap–n, where nis a positive integer, can be written as the fraction (3) with R(p)=aandQ(p)=pn. The denominator Q(p) has one root of multiplicity n,λ1= 0. By formula (5) with m=1a n d s1=n, we ﬁnd the inverse transform: f(x)=a (n–1 ) !xn–1. /trianglerightsldFairly detailed tables of inverse Laplace transforms can be found in Supplement 6. 9.2. L APLACE TRANSFORM 507 9.2-3. Inversion of Functions with Finitely Many Singular Points. If the function ˜f(p) has ﬁnitely many singular points, p1,p2,...,pn, and tends to zero as p→∞ , then the integral in the Laplace inversion formula (2) may be evaluated using the residue theory byapplying the Jordan lemma (see Subsection 9.1-5). In this case f(x)= n⎝summationdisplay k=1res p=pk[˜f(p)epx]. (6) Formula (6) can be extended to the case where ˜f(p) has inﬁnitely many singular points. In this case, f(x) is represented as an inﬁnite series. 9.2-4. Convolution Theorem. Main Properties of the Laplace Transform. 1◦.T h e convolution of two functions f(x)a n dg(x)i sd e ﬁ n e da sa ni n t e g r a l⎝integraldisplayx 0f(t)g(x–t)dt,a n d is usually denoted by f(x)∗g(x), f(x)∗g(x)=⎝integraldisplayx 0f(t)g(x–t)dt. By performing substitution x–t=u, we see that the convolution is symmetric with respect to the convolved functions: f(x)∗g(x)=g(x)∗f(x). The convolution theorem states that L⎝braceleftbig f(x)∗g(x)⎝bracerightbig =L⎝braceleftbig f(x)⎝bracerightbig L⎝braceleftbig g(x)⎝bracerightbig and is frequently applied to solve V olterra equations with kernels depending on the difference of the arguments. 2◦. The main properties of the correspondence between functions and their Laplace transforms are gathered in Table 1. 3◦. The Laplace transforms of some functions are listed in Table 2; for more detailed tables of direct and inverse Laplace transforms, see Supplements 5–6 and the list of references at the end of this section. 9.2-5. Limit Theorems. THEOREM 1.Let0≤x<∞and ˜f(p)=L⎝braceleftbig f(x)⎝bracerightbig be the Laplace transform of f(x). If a limit off(x)asx→0exists, then lim x→0f(x) = lim p→∞⎝bracketleftbig p˜f(p)⎝bracketrightbig . THEOREM 2.If a limit of f(x)asx→∞ exists, then lim x→∞f(x) = lim p→0⎝bracketleftbig p˜f(p)⎝bracketrightbig . 508 MAINDEFINITIONS AND FORMULAS .INTEGRAL TRANSFORMS TABLE 1 Main properties of the Laplace transform No. Function Laplace transform Operation 1 af1(x)+bf2(x) a˜f1(p)+b˜f2(p) Linearity 2 f(x/a),a>0 a˜f(ap) Scaling 3 f(x–a), f(ξ)≡0f o r ξ<0 e–ap˜f(p) Shift of the argument 4 xnf(x);n=1 , 2 , ... (–1)n˜f(n) p(p) Differentiation of the transform 5 1 xf(x) ⎝integraldisplay∞ p˜f(q)dq Integration of the transform 6 eaxf(x) ˜f(p–a) Shift in the complex plane 7 f/prime x(x) p˜f(p)–f(+0) Differentiation 8 f(n) x(x) pn˜f(p)–n⎝summationtext k=1pn–kf(k–1) x(+0) Differentiation 9 xmf(n) x(x),m=1 ,2 , ... (–1)mdm dpm⎝bracketleftBig pn˜f(p)–n⎝summationtext k=1pn–kf(k–1) x(+0)⎝bracketrightBig Differentiation 10 dn dxn⎝bracketleftbig xmf(x)⎝bracketrightbig ,m≥n (–1)mpndm dpm˜f(p) Differentiation 11 ⎝integraldisplayx 0f(t)dt ˜f(p) p Integration 12 ⎝integraldisplayx 0f1(t)f2(x–t)dt ˜f1(p)˜f2(p) Convolution TABLE 2 The Laplace transforms of some functions No. Function, f(x) Laplace transform, ˜f(p) Remarks 1 1 1/p 2 xn n! pn+1 n=1 ,2 , ... 3 xa Γ(a+1 )p–a–1 a>– 1 4 e–ax (p+a)–1 5 xae–bx Γ(a+1 ) (p+b)–a–1 a>– 1 6 sinh(ax) a p2–a2 7 cosh(ax) p p2–a2 8 lnx –1 p(lnp+C) C= 0.5772 ... is the Euler constant 9 sin(ax) a p2+a2 10 cos(ax) p p2+a2 11 erfc⎝parenleftBiga 2√ x⎝parenrightBig 1 pexp⎝parenleftbig –a√ p⎝parenrightbig a≥0 12 J0(ax) 1 ⎝radicalbig p2+a2 J0(x) is the Bessel function 9.2. L APLACE TRANSFORM 509 9.2-6. Representation of Inverse Transforms as Convergent Series. THEOREM 1.Suppose the transform ˜f(p)can be expanded into series in negative powers of p, ˜f(p)=∞⎝summationdisplay n=1an pn, convergent for |p|>R,w h e r e Ris an arbitrary positive number; note that the transform tends to zero as|p|→∞ . Then the inverse transform can be obtained by the formula f(x)=∞⎝summationdisplay n=1an (n–1 ) !xn–1, where the series on the right-hand side is convergent for all x. THEOREM 2.Suppose the transform ˜f(p),|p|>R, is represented by an absolutely convergent series, ˜f(p)=∞⎝summationdisplay n=0an pλn,( 7) where {λn}is any positive increasing sequence, 0<λ0<λ1<···→∞ . Then it is possible to proceed termwise from series (7) to the following inverse transform series: f(x)=∞⎝summationdisplay n=0an Γ(λn)xλn–1,( 8) where Γ(λ)is the Gamma function. Series (8) is convergent for all real and complex values of x other than zero (if λ0≥1, the series is convergent for all x). 9.2-7. Representation of Inverse Transforms as Asymptotic Expansions as x→∞ . 1◦.L e tp=p0be a singular point of the Laplace transform ˜f(p) with the greatest real part (it is assumed there is only one such point). If ˜f(p) can be expanded near p=p0into an absolutely convergent series, ˜f(p)=∞⎝summationdisplay n=0cn(p–p0)λn(λ0<λ1<···→∞ )( 9 ) with arbitrary λn, then the inverse transform f(x) can be expressed in the form of the asymptotic expansion f(x)∼ep0x∞⎝summationdisplay n=0cn Γ(–λn)x–λn–1asx→∞ . The terms corresponding to nonnegative integer λnmust be omitted from the summation, since Γ(0) =Γ(–1) = Γ(–2) =···=∞. 2◦. If the transform ˜f(p) has several singular points, p1,...,pm, with the same greatest real part, Rep1=···=R epm, then expansions of the form (9) should be obtained for each of these points and the resulting expressions must be added together. 510 MAINDEFINITIONS AND FORMULAS .INTEGRAL TRANSFORMS 9.2-8. Post–Widder Formula. In applications, one can ﬁnd f(x) if the Laplace transform ˜f(t) on the real semiaxis is known for t=p≥0. To this end, one uses the Post–Widder formula f(x) = lim n→∞⎝bracketleftbigg(–1)n n!⎝parenleftBign x⎝parenrightBign+1˜f(n) t⎝parenleftBign x⎝parenrightBig⎝bracketrightbigg . (10) Approximate inversion formulas are obtained by taking sufﬁciently large positive integer nin (10) instead of passing to the limit. References for Section 9.2: G. Doetsch (1950, 1956, 1958, 1974), H. Bateman and A. Erd ´elyi (1954), I. I. Hirschman and D. V . Widder (1955), V . A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), F. Oberhettinger (1973), B. Davis (1978), W. R. LePage (1980), R. Bellman and R. Roth (1984), Yu. A. Brychkov and A. P. Prudnikov (1989), W. H. Beyer (1991), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, V ols 4 and 5), R. J. Beerends, H. G. ter Morschem, andJ. C. van den Berg (2003). 9.3. Mellin Transform 9.3-1. Deﬁnition. Inversion Formula. Suppose that a function f(x) is deﬁned for positive xand satisﬁes the conditions ⎝integraldisplay1 0|f(x)|xσ1–1dx<∞,⎝integraldisplay∞ 1|f(x)|xσ2–1dx<∞ for some real numbers σ1andσ2,σ1<σ2. The Mellin transform off(x)i sd e ﬁ n e db y ˆf(s)=⎝integraldisplay∞ 0f(x)xs–1dx,( 1 ) where s=σ+iτis a complex variable ( σ1<σ<σ2). For brevity, we rewrite formula (1) as follows: ˆf(s)=M{f(x)},o r ˆf(s)=M{f(x),s}. Given ˆf(s), the function can be found by means of the inverse Mellin transform f(x)=1 2πi⎝integraldisplayσ+i∞ σ–i∞ˆf(s)x–sds,( σ1<σ<σ2)( 2) where the integration path is parallel to the imaginary axis of the complex plane sand the integral is understood in the sense of the Cauchy principal value. Formula (2) holds for continuous functions. If f(x) has a (ﬁnite) jump discontinuity at a point x=x0> 0, then the left-hand side of (2) is equal to1 2⎝bracketleftbig f(x0–0 )+f(x0+0 )⎝bracketrightbig at this point (for x0=0 , the ﬁrst term in the square brackets must be omitted). For brevity, we rewrite formula (2) in the form f(x)=M–1{ˆf(s)},o r f(x)=M–1{ˆf(s),x}. 9.3. M ELLIN TRANSFORM 511 9.3-2. Main Properties of the Mellin Transform. 1◦. The integral relations ⎝integraldisplay∞ 0f(xt)g(t)dt=M–1{ˆf(s)ˆg(1 –s)},( 3) ⎝integraldisplay∞ 0f⎝parenleftBigx t⎝parenrightBig g(t)dt t=M–1{ˆf(s)ˆg(s)} (4) hold for fairly general assumptions about the integrability of the functions involved (see Ditkin and Prudnikov, 1965). 2◦. The main properties of the correspondence between the functions and their Mellin transforms are gathered in Table 3. TABLE 3 Main properties of the Mellin transform No Function Mellin Transform Operation 1 af1(x)+bf2(x) aˆf1(s)+bˆf2(s) Linearity 2 f(ax),a>0 a–sˆf(s) Scaling 3 xaf(x) ˆf(s+a) Shift of the argument of the transform 4 f(x2) 1 2ˆf⎝parenleftbig1 2s⎝parenrightbig Squared argument 5 f(1/x) ˆf(–s) Inversion of the argument of the transform 6 xλf⎝parenleftbig axβ⎝parenrightbig ,a>0 ,β≠0 1 βa–s+λ βˆf⎝parenleftBigs+λ β⎝parenrightBig Power law transform 7 f/prime x(x) –(s–1 )ˆf(s–1 ) Differentiation 8 xf/prime x(x) –sˆf(s) Differentiation 9 f(n) x(x) (–1)nΓ(s) Γ(s–n)ˆf(s–n) Multiple differentiation 10 ⎝parenleftBig xd dx⎝parenrightBign f(x) (–1)nsnˆf(s) Multiple differentiation 11 xα⎝integraldisplay∞ 0tβf1(xt)f2(t)dt ˆf1(s+α)ˆf2(1 – s–α+β) Complicated integration 12 xα⎝integraldisplay∞ 0tβf1⎝parenleftBigx t⎝parenrightBig f2(t)dt ˆf1(s+α)ˆf2(s+α+β+1 ) Complicated integration 9.3-3. Relation Among the Mellin, Laplace, and Fourier Transforms. There are tables of direct and inverse Mellin transforms (see Supplements 9 and 10), which are useful in solving speciﬁc integral and differential equations. The Mellin transform is related to the Laplace and Fourier transforms as follows: M{f(x),s}=L{f(ex), –s}+L{f(e–x),s}=F{f(ex),is}, which makes it possible to apply much more common tables of direct and inverse Laplace and Fourier transforms. References for Section 9.3: V . A. Ditkin and A. P. Prudnikov (1965), F. Oberhettinger (1974), Yu. A. Brychkov and A. P. Prudnikov (1989). 512 MAINDEFINITIONS AND FORMULAS .INTEGRAL TRANSFORMS 9.4. Fourier Transform 9.4-1. Deﬁnition. Inversion Formula. The F ourier transform is deﬁned as follows: ˜f(u)=1 √ 2π⎝integraldisplay∞ –∞f(x)e–iuxdx.( 1 ) For brevity, we rewrite formula (1) as follows: ˜f(u)=F{f(x)},o r ˜f(u)=F{f(x),u}. Given ˜f(u), the function f(x) can be found by means of the inverse F ourier transform f(x)=1 √ 2π⎝integraldisplay∞ –∞˜f(u)eiuxdu.( 2 ) Formula (2) holds for continuous functions. If f(x) has a (ﬁnite) jump discontinuity at a point x=x0, then the left-hand side of (2) is equal to1 2⎝bracketleftbig f(x0–0 )+ f(x0+0 )⎝bracketrightbig at this point. For brevity, we rewrite formula (2) as follows: f(x)=F–1{˜f(u)},o r f(x)=F–1{˜f(u),x}. 9.4-2. Asymmetric Form of the Transform. Sometimes it is more convenient to deﬁne the Fourier transform by ˇf(u)=⎝integraldisplay∞ –∞f(x)e–iuxdx.( 3 ) For brevity, we rewrite formula (3) as follows: ˇf(u)=F{f(x)}orˇf(u)=F{f(x),u}. In this case, the F ourier inversion formula reads f(x)=1 2π⎝integraldisplay∞ –∞ˇf(u)eiuxdu,( 4 ) and we use the following symbolic notation for relation (4): f(x)=F–1{ˇf(u)},o r f(x)= F–1{ˇf(u),x}. 9.4-3. Alternative Fourier Transform. Sometimes, for instance, in the theory of boundary value problems, the alternative Fourier transform is used (and called merely the F ourier transform ) in the form F(u)=1 √ 2π⎝integraldisplay∞ –∞f(x)eiuxdx.( 5 ) For brevity, we rewrite formula (5) as follows: F(u)=F{f(x)},o r F(u)=F{f(x),u}. For given F(u), the function f(x) can be found by means of the inverse transform f(x)=1 √ 2π⎝integraldisplay∞ –∞F(u)e–iuxdu.( 6 ) 9.4. F OURIER TRANSFORM 513 TABLE 4 Main properties of the Fourier transform No. Function Fourier transform Operation 1 af1(x)+bf2(x) a˜f1(u)+b˜f2(u) Linearity 2 f(x/a),a>0 a˜f(au) Scaling 3 xnf(x);n=1 ,2 , ... in˜f(n) u(u) Differentiation of the transform 4 f/prime/prime xx(x) –u2˜f(u) Differentiation 5 f(n) x(x) (iu)n˜f(u) Differentiation 6 ⎝integraldisplay∞ –∞f1(ξ)f2(x–ξ)dξ ˜f1(u)˜f2(u) Convolution For brevity, we rewrite formula (6) as follows: f(x)=F–1{F(u)},o r f(x)=F–1{F(u),x}. The function F(u) is also called the F ourier integral off(x). We can introduce an asymmetric form for the alternative Fourier transform similarly to that of the Fourier transform: ˇF(u)=⎝integraldisplay∞ –∞f(x)eiuxdx,f(x)=1 2π⎝integraldisplay∞ –∞ˇF(u)e–iuxdu,( 7 ) where the direct and the inverse transforms (7) are brieﬂy denoted by ˇF(u)=ˇF⎝braceleftbig f(x)⎝bracerightbig andf(x)= ˇF–1⎝braceleftbigˇF(u)⎝bracerightbig ,o rb y ˇF(u)=ˇF⎝braceleftbig f(x),u⎝bracerightbig andf(x)=ˇF–1⎝braceleftbigˇF(u)x⎝bracerightbig . 9.4-4. Convolution Theorem. Main Properties of the Fourier Transforms. 1◦.T h e convolution of two functions f(x)a n dg(x)i sd e ﬁ n e da s f(x)∗g(x)≡1 √ 2π⎝integraldisplay∞ –∞f(x–t)g(t)dt. By performing substitution x–t=u, we see that the convolution is symmetric with respect to the convolved functions: f(x)∗g(x)=g(x)∗f(x). The convolution theorem states that F⎝braceleftbig f(x)∗g(x)⎝bracerightbig =F⎝braceleftbig f(x)⎝bracerightbig F⎝braceleftbig g(x)⎝bracerightbig .( 8 ) For the alternative Fourier transform, the convolution theorem reads F⎝braceleftbig f(x)∗g(x)⎝bracerightbig =F⎝braceleftbig f(x)⎝bracerightbig F⎝braceleftbig g(x)⎝bracerightbig .( 9 ) Formulas (8) and (9) will be used in Chapters 12 and 13 for solving linear integral equations with difference kernel. 2◦. The main properties of the correspondence between functions and their Fourier transforms are gathered in Table 4. References for Section 9.4: V . A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), B. Davis (1978), F. Oberhettinger (1980), Yu. A. Brychkov and A. P. Prudnikov (1989), W. H. Beyer (1991), I. Sneddon (1995), A. Pinkus and S. Zafrany(1997), R. Bracewell (1999), A. D. Poularikas (2000), R. J. Beerends, H. G. ter Morschem, J. C. van den Berg (2003), L. Debnath and D. Bhatta (2007). 514 MAINDEFINITIONS AND FORMULAS .INTEGRAL TRANSFORMS 9.5. Fourier Cosine and Sine Transforms 9.5-1. Fourier Cosine Transform. 1◦. Let a function f(x) be integrable on the semiaxis 0 ≤x<∞.T h e F ourier cosine transform is deﬁned by ˜fc(u)=⎝radicalbigg 2 π⎝integraldisplay∞ 0f(x)c o s (xu)dx,0 < u<∞.( 1) For given ˜fc(u), the function can be found by means of the Fourier cosine inversion formula f(x)=⎝radicalbigg 2 π⎝integraldisplay∞ 0˜fc(u)c o s (xu)du,0 < x<∞.( 2) The Fourier cosine transform (1) is denoted for brevity by ˜fc(u)=Fc⎝braceleftbig f(x)⎝bracerightbig . 2◦. It follows from formula (2) that the Fourier cosine transform has the property F2 c=1 . Some other properties of the Fourier cosine transform: Fc⎝braceleftbig x2nf(x)⎝bracerightbig = (–1)nd2n du2nFc⎝braceleftbig f(x)⎝bracerightbig ,n=1 ,2 , ...; Fc⎝braceleftbig f/prime/prime(x)⎝bracerightbig =–u2Fc⎝braceleftbig f(x)⎝bracerightbig . Heref(x) is assumed to vanish sufﬁciently rapidly (exponentially) as x→∞ . For the second formula, the condition f/prime(0) = 0 is assumed to hold. Parseval’s relation for the F ourier cosine transform : ⎝integraldisplay∞ 0Fc⎝braceleftbig f(x)⎝bracerightbig Fc⎝braceleftbig g(x)⎝bracerightbig du=⎝integraldisplay∞ 0f(x)g(x)dx. There are tables of the Fourier cosine transform (see Supplement 7 and the references listed at the end of the current section) which prove useful in the solution of speciﬁc integral equations. 3◦. Sometimes the asymmetric form of the Fourier cosine transform is applied, which is given by the pair of formulas ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (xu)dx,f(x)=2 π⎝integraldisplay∞ 0ˇfc(u)c o s (xu)du.( 3 ) The direct and inverse Fourier cosine transforms (3) are denoted by ˇfc(u)=Fc⎝braceleftbig f(x)⎝bracerightbig andf(x)= F–1 c⎝braceleftbigˇfc(u)⎝bracerightbig , respectively. 9.5-2. Fourier Sine Transform. 1◦. Let a function f(x) be integrable on the semiaxis 0 ≤x<∞.T h e F ourier sine transform is deﬁned by ˜fs(u)=⎝radicalbigg 2 π⎝integraldisplay∞ 0f(x)s i n (xu)dx,0 < u<∞.( 4) For given ˜fs(u), the function f(x) can be found by means of the inverse Fourier sine transform f(x)=⎝radicalbigg 2 π⎝integraldisplay∞ 0˜fs(u)s i n (xu)du,0 < x<∞.( 5) The Fourier sine transform (4) is brieﬂy denoted by ˜fs(u)=Fs⎝braceleftbig f(x)⎝bracerightbig . 9.6. O THER INTEGRAL TRANSFORMS 515 2◦. It follows from formula (5) that the Fourier sine transform has the property F2 s=1 . Some other properties of the Fourier sine transform: Fs⎝braceleftbig x2nf(x)⎝bracerightbig = (–1)nd2n du2nFs⎝braceleftbig f(x)⎝bracerightbig ,n=1 ,2 , ...; Fs⎝braceleftbig f/prime/prime(x)⎝bracerightbig =–u2Fs⎝braceleftbig f(x)⎝bracerightbig . Heref(x) is assumed to vanish sufﬁciently rapidly (exponentially) as x→∞ . For the second formula, the condition f(0) = 0 is assumed to hold. Parseval’s relation for the F ourier sine transform: ⎝integraldisplay∞ 0Fs⎝braceleftbig f(x)⎝bracerightbig Fs⎝braceleftbig g(x)⎝bracerightbig du=⎝integraldisplay∞ 0f(x)g(x)dx. There are tables of the Fourier sine transform (see Supplement 8 and the references listed at the end of the current section), which are useful in solving speciﬁc integral equations. 3◦. Sometimes it is more convenient to apply the asymmetric form of the Fourier sine transform deﬁned by the following two formulas: ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (xu)dx,f(x)=2 π⎝integraldisplay∞ 0ˇfs(u)s i n (xu)du.( 6 ) The direct and inverse Fourier sine transforms (6) are denoted by ˇfs(u)=Fs⎝braceleftbig f(x)⎝bracerightbig andf(x)= F–1 s⎝braceleftbigˇfs(u)⎝bracerightbig , respectively. References for Section 9.5: E. A. C. Paley and N. Wiener (1934), S. Bochner and K. C. Chandrasekharan (1949), G. N. Watson (1952), H. Bateman and A. Erd ´elyi (V ol. 1, 1954), S. Bochner (1959), V . A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), B. Davis (1978), F. Oberhettinger (1980), E. C. Titchmarsh (1986), Ya. A. Brychkov and A. P. Prudnikov(1989), W. H. Beyer (1991), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 440), I. Sneddon (1995), A. D. Poularikas (2000). 9.6. Other Integral Transforms 9.6-1. Hankel Transform. The Hankel transform is deﬁned as follows: ˜fν(u)=⎝integraldisplay∞ 0xJν(ux)f(x)dx,0 < u<∞,( 1) where ν>– 1a n d Jν(x) is the Bessel function of the ﬁrst kind of order ν(see Supplement 11.6). For given ˜fν(u), the function f(x) can be found by means of the Hankel inversion formula f(x)=⎝integraldisplay∞ 0uJν(ux)˜fν(u)du,0 < x<∞.( 2) Note that if f(x)=O(xα)a sx→0, where α+ν+2>0 ,a n d f(x)=O(xβ)a sx→∞ ,w h e r e β+3 2< 0, then the integral (1) is convergent. The inversion formula (2) holds for continuous functions. If f(x) has a (ﬁnite) jump discontinuity at a point x=x0, then the left-hand side of (2) is equal to1 2[f(x0–0 )+ f(x0+ 0)] at this point. For brevity, we denote the Hankel transform (1) by ˜fν(u)=Hν⎝braceleftbig f(x)⎝bracerightbig . It follows from formula (2) that the Hankel transform has the property H2 ν=1 . Parseval’s relation for the Hankel transform : ⎝integraldisplay∞ 0uHν⎝braceleftbig f(x)⎝bracerightbig Hν⎝braceleftbig g(x)⎝bracerightbig du=⎝integraldisplay∞ 0xf(x)g(x)dx,ν>–1 2. 516 MAINDEFINITIONS AND FORMULAS .INTEGRAL TRANSFORMS 9.6-2. Meijer Transform. The Meijer transform is deﬁned as follows: ˆfµ(s)=⎝radicalbigg 2 π⎝integraldisplay∞ 0√ sxKµ(sx)f(x)dx,0 < s<∞, where Kµ(x) is the modiﬁed Bessel function of the second kind (the Macdonald function) of order µ (see Supplement 11.7). For given ˜fµ(s), the function f(x) can be found by means of the Meijer inversion formula f(x)=1 i√ 2π⎝integraldisplayc+i∞ c–i∞√ sxIµ(sx)ˆfµ(s)ds,0 < x<∞, where Iµ(x) is the modiﬁed Bessel function of the ﬁrst kind of order µ(see Supplement 11.7). For the Meijer transform, a convolution is deﬁned and an operational calculus is developed. 9.6-3. Kontorovich–Lebedev Transform. The Kontorovich–Lebedev transform is introduced as follows: F(τ)=⎝integraldisplay∞ 0Kiτ(x)f(x)dx,0 < τ<∞, where Kµ(x) is the modiﬁed Bessel function of the second kind (the Macdonald function) of order µ (see Supplement 11.7) and i=√ –1. For given F(τ), the function can be found by means of the Kontorovich–Lebedev inversion formula f(x)=2 π2x⎝integraldisplay∞ 0τsinh(πτ)Kiτ(x)F(τ)dτ,0 < x<∞. Parseval’s relation for the Kontorovich–Lebedev transform : ⎝integraldisplay∞ 0F1(τ)F2(τ)dτ=⎝integraldisplay∞ 0f1(x)f2(x)dx. 9.6-4. Y-transform. TheY-transform is deﬁned by Fν(u)=⎝integraldisplay∞ 0√ uxY ν(ux)f(x)dx, where Yν(x) is the Bessel function of the second kind of order ν. Given a transform Fν(u), the inverse Y-transform f(x) is found by the inversion formula f(x)=⎝integraldisplay∞ 0√ uxHν(ux)Fν(u)du, where Hν(x) is the Struve function, which is deﬁned as Hν(x)=∞⎝summationdisplay j=0(–1)j(x/2)ν+2j+1 Γ⎝parenleftbig j+3 2⎝parenrightbig Γ⎝parenleftbig ν+j+3 2⎝parenrightbig. 9.6. O THER INTEGRAL TRANSFORMS 517 9.6-5. Summary Table of Integral Transforms. Table 5 summarizes the integral transforms considered above and also lists some other integral transforms; for the constraints imposed on the functions and parameters occurring in the integrand, see the references given at the end of this section. TABLE 5 Main integral transforms Integral transform Deﬁnition Inversion formula Laplacetransform ⎝tildewidef(p)=⎝integraldisplay∞ 0e–pxf(x)dx f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx⎝tildewidef(p)dp Laplace- Carlsontransform ⎝tildewidef(p)=p⎝integraldisplay∞ 0e–pxf(x)dx f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx⎝tildewidestf(p) pdp Two-sided Laplacetransform ⎝tildewidef∗(p)=⎝integraldisplay∞ –∞e–pxf(x)dx f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx⎝tildewidef∗(p)dp Fourier transform ⎝tildewidef(u)=1 √ 2π⎝integraldisplay∞ –∞e–iuxf(x)dx f(x)=1 √ 2π⎝integraldisplay∞ –∞eiux⎝tildewidef(u)du Fourier sine transform ⎝tildewidefs(u)=⎝radicalBig 2 π⎝integraldisplay∞ 0sin(xu)f(x)dx f(x)=⎝radicalBig 2 π⎝integraldisplay∞ 0sin(xu)⎝tildewidefs(u)du Fourier cosine transform ⎝tildewidefc(u)=⎝radicalBig 2 π⎝integraldisplay∞ 0cos(xu)f(x)dx f(x)=⎝radicalBig 2 π⎝integraldisplay∞ 0cos(xu)⎝tildewidefc(u)du Hartley transform ⎝tildewidefh(u)=1 √ 2π⎝integraldisplay∞ –∞(cosxu+s i nxu)f(x)dx f(x)=1 √ 2π⎝integraldisplay∞ –∞(cosxu+s i nxu)⎝tildewidefh(u)du Mellin transform ⎝hatwidef(s)=⎝integraldisplay∞ 0xs–1f(x)dx f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞x–s⎝hatwidef(s)ds Hankel transform ⎝hatwidefν(w)=⎝integraldisplay∞ 0xJν(xw)f(x)dx f(x)=⎝integraldisplay∞ 0wJν(xw)⎝hatwidefν(w)dw Y-transform Fν(u)=⎝integraldisplay∞ 0√ ux Y ν(ux)f(x)dx f(x)=⎝integraldisplay∞ 0√ uxHν(ux)Fν(u)du Meijer transform(K-transform) ⎝hatwidef(s)=⎝radicalBig 2 π⎝integraldisplay∞ 0√ sxKν(sx)f(x)dx f(x)=1 i√ 2π⎝integraldisplayc+i∞ c–i∞√ sxIν(sx)⎝hatwidef(s)ds Bochner transform ⎝tildewidef(r)=⎝integraldisplay∞ 0Jn/2–1(2πxr)G(x,r)f(x)dx, G(x,r)=2πr(x/r)n/2,n=1,2, ... f(x)=⎝integraldisplay∞ 0Jn/2–1(2πrx)G(r,x)⎝tildewidef(r)dr Weber transform Fa(u)=⎝integraldisplay∞ aWν(xu,au)xf(x)dx, Wν(β,µ)≡Jν(β)Yν(µ)–Jν(µ)Yν(β) f(x)=⎝integraldisplay∞ 0Wν(xu,au) J2ν(au)+Y2ν(au)uFa(u)du Hardy transform F(u)=⎝integraldisplay∞ 0Cν(xu)xf(x)dx, Cν(z)≡cos(πp)Jν(z)+s i n ( πp)Yν(z) f(x)=⎝integraldisplay∞ 0Φ(xu)uF(u)du, Φ(z)=∞⎝summationtext n=0(–1)n(z/2)ν+2p+2n Γ(p+n+1 )Γ(ν+p+n+1 ) Kontorovich– Lebedevtransform F(τ)=⎝integraldisplay∞ 0Kiτ(x)f(x)dx f(x)=2 π2x⎝integraldisplay∞ 0τsinh(πτ)Kiτ(x)F(τ)dτ 518 MAINDEFINITIONS AND FORMULAS .INTEGRAL TRANSFORMS TABLE 5 ( continued ) Main integral transforms Integral transform Deﬁnition Inversion formula Mehler–Focktransform F(x)=⎝integraldisplay∞ 0P–1 2+iτ(x)f(τ)dτ,1 ≤x<∞ f(τ)=τtanh(πτ)⎝integraldisplay∞ 1P–1 2+iτ(x)F(x)dx Euler transform ofthe 1st kind F(x)=1 Γ(µ)⎝integraldisplayx af(t)dt (x–t)1–µ, 0<µ<1,x>a f(x)=1 Γ(1 –µ)d dx⎝integraldisplayx aF(t)dt (x–t)µ Euler transform ofthe 2nd kind F(x)=1 Γ(µ)⎝integraldisplaya xf(t)dt (t–x)1–µ, 0<µ<1,x<a f(x)=–1 Γ(1 –µ)d dx⎝integraldisplaya xF(t)dt (t–x)µ Gauss transform F(x)=1 √ πa⎝integraldisplay∞ –∞exp⎝bracketleftBig –(x–t)2 a⎝bracketrightBig f(t)dt f(x)=e x p⎝parenleftBig –a 4d2 dx2⎝parenrightBig F(x) Hilbert transform ⎝hatwideF(s)=1 π⎝integraldisplay∞ –∞f(x) x–sdx f(x)=–1 π⎝integraldisplay∞ –∞⎝hatwideF(s) s–xds Notations for Table 5 :i=√ –1,Jµ(x)a n dYµ(x) are the Bessel functions of the ﬁrst and the second kind, respectively; Iµ(x)a n dKµ(x) are the modiﬁed Bessel functions of the ﬁrst and the second kind, respectively; Pµ(x) is the Legendre spherical function of the ﬁrst kind; and Hµ(x)i st h eS t r u v e function (see Subsection 9.6-4). Remark 1. The Euler transform of the ﬁrst kind is also known as Riemann–Liouville integral (the left fractional integral of order µor, for short, the fractional integral), see Section 10.5. The Euler transform of the second kind is also called the right fractional integral of order µ. Remark 2. Ifa= 4, the Gauss transform is called the Weierstrass transform. In the inversion formula, the exponential is represented by an operator series: exp⎝parenleftBig kd2 dx2⎝parenrightBig ≡1+∞⎝summationtext n=1kn n!d2n dx2n. Remark 3. In the direct and inverse Hilbert transforms, the integrals are understood in the sense of the Cauchy principal value. Remark 4. Some other integral transforms are described in Chapter 3. References for Section 9.6: H. Bateman and A. Erd ´elyi (1954, V ols 1 and 2), J. L. Grifﬁth (1958), V . A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), F. Oberhettinger (1972), I. Sneddon (1972), H. M. Srivastava and R. G. Buschman (1977), B. Davis (1978), D. Zwillinger (1989), Yu. A. Brychkov and A. P. Prudnikov (1989), W. H. Beyer (1991), M. Ya. An- timirov (1993), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993), A. D. Polyanin and A. V . Manzhirov (1998, 2007),A. D. Poularikas (2000), L. Debnath and D. Bhatta (2007). Chapter 10 Methods for Solving Linear Equations of the Form⎝integraldisplay ⎝integraldisplayx aK(x,t)y(t)dt=f(x) 10.1. Volterra Equations of the First Kind 10.1-1. Equations of the First Kind. Function and Kernel Classes. In this chapter we present methods for solving V olterra linear equations of the ﬁrst kind. These equations have the form⎝integraldisplayx aK(x,t)y(t)dt=f(x), (1) where y(x) is the unknown function ( a≤x≤b),K(x,t) is the kernel of the integral equation, and f(x) is a given function, the right-hand side of Eq. (1). The functions y(x)a n df(x) are usually assumed to be continuous or square integrable on [ a,b]. The kernel K(x,t) is usually assumed either to be continuous on the square S={a≤x≤b,a≤t≤b}or to satisfy the condition ⎝integraldisplayb a⎝integraldisplayb aK2(x,t)dx dt =B2<∞,( 2) where Bis a constant, that is, to be square integrable on this square. It is assumed in (2) that K(x,t)≡0f o rt>x. The kernel K(x,t)i ss a i dt ob e degenerate if it can be represented in the form K(x,t)=g1(x)h1(t)+···+gn(x)hn(t). The kernel K(x,t) of an integral equation is called difference kernel if it depends only on the difference of the arguments, K(x,t)=K(x–t). Polar kernels K(x,t)=L(x,t) (x–t)β+M(x,t), 0 < β<1 , ( 3 ) and logarithmic kernels (kernels with logarithmic singularity) K(x,t)=L(x,t)l n (x–t)+M(x,t), (4) where L(x,t)a n d M(x,t) are continuous on SandL(x,x)/ ≡0, are often considered as well. Polar and logarithmic kernels form a class of kernels with weak singularity. Equations containing such kernels are called equations with weak singularity . The following generalized Abel equation is a special case of Eq. (1) with the kernel of the form (3):⎝integraldisplayx ay(t) (x–t)βdt=f(x), 0 < β<1 . 519 520 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextx aK(x,t)y(t)dt=f(x) In case the functions K(x,t)a n d f(x) are continuous, the right-hand side of Eq. (1) must satisfy the following conditions: 1◦.I fK(a,a)≠0, then f(x) must be constrained by f(a)=0 . 2◦.I fK(a,a)=K/prime x(a,a)=···=K(n–1) x(a,a)=0 , 0<⎝vextendsingle⎝vextendsingleK(n) x(a,a)⎝vextendsingle⎝vextendsingle<∞, then the right-hand side of the equation must s atisfy the conditions f(a)=f/prime x(a)=···=f(n) x(a)=0 . 3◦.I fK(a,a)=K/prime x(a,a)=···=K(n–1) x(a,a)=0 , K(n) x(a,a)=∞, then the right-hand side of the equation must satisfy the conditions f(a)=f/prime x(a)=···=f(n–1) x(a)=0 . For polar kernels of the form (3) or (4) and continuous f(x), no additional conditions are imposed on the right-hand side of the integral equation. Remark 1. Generally, the case in which the integration limit ais inﬁnite is not excluded. 10.1-2. Existence and Uniqueness of a Solution. Assume that in Eq. (1) the functions f(x)a n d K(x,t) are continuous together with their ﬁrst derivatives on [ a,b] and on S, respectively. If K(x,x)≠0(x∈[a,b]) and f(a) = 0, then there exists a unique continuous solution y(x)o fE q .( 1 ) . Remark 2. The problem of existence and uniqueness of a solution to a V olterra equation of the ﬁrst kind is closely related to conditions under which this equation can be reduced to V olterra equations of the second kind (see Section 10.3). Remark 3. A V olterra equation of the ﬁrst kind can be treated as a Fredholm equation of the ﬁrst kind whose kernel K(x,t) vanishes for t>x(see Chapter 12). 10.1-3. Some Problems Leading to V olterra Integral Equations of the First Kind. 1◦.Abel problem (generalization of the tautochrone problem*). Statement of the problem . Suppose a point mass (a bead) can move along a curve in the vertical plane ( ξ,η) under the gravitational force. Determine the curve if the bead, initially having an ordinate xand zero velocity, must reach the Oξaxis in a time t=f1(x), where f1(x)i sag i v e n function. Derivation of the integral equation . The absolute value of the bead velocity is expressed as v=⎝radicalbig 2g(x–η). Letβ=β(η) denote the angle between the tangent to the curve and the Oξaxis, as shown in Fig. 2. Then the η-component of the velocity is found as dη dt=–⎝radicalbig 2g(x–η)s i nβ. It follows that dt=–dη √ 2g(x–η)s i nβ. * Find the curve down which a heavy bead having zero initial velocity and placed anywhere will fall to the bottom in the same amount of time. 10.1. V OLTERRA EQUATIONS OF THE FIRST KIND 521 Integrating over ηfrom 0 to xand setting1 sinβ=y(η), one arrives at the Abel equation ⎝integraldisplayx 0y(η) √ x–ηdη=–⎝radicalbig 2gf1(x). Denoting –√ 2gf1(x)=f(x) yields ⎝integraldisplayx 0y(η) √ x–ηdη=f(x). Herey(x) is the unknown function and f(x) is a given function. /c98 /c120/c104 /c104/c32/c61 x O Figure 2. Curve along which the bead moves in the Abel problem. Having found y(η), one readily obtains the equation of the desired curve. Indeed, since y(η)= 1/sinβ,w eh a v e η=Φ(β). Further, dξ=dη tanβ=Φ/prime(β)dβ tanβ, and therefore ξ=⎝integraldisplayΦ/prime(β) tanβdβ=Ψ(β). Hence, the desired curve is determined parametrically by the equations ξ=Ψ(β),η=Φ(β). In particular, if f(x)=C= const, the desired curve is a cycloid. 2◦.A model problem on buying and selling goods. Statement of the problem . There is a shop that buys and sells various types of goods. It is assumed that: 1) buying and selling are continuous processes and the goods bought are put on sale immediately; 2) any type of goods is purchased by the shop in consignments, the quantity of goods in each consignment equal to the quantity sold by the shop for a time T, the same for all types of goods; 3) each new consignment is sold uniformly over the time T. The shop starts selling a new consignment the cost of which is equal to unity. Find the law y(t) according to which the goods should be bought, in order that the cost of the goods present in the shop remains constant. 522 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextx aK(x,t)y(t)dt=f(x) Derivation of the integral equation . The cost of the initially bought goods remaining in the shop by an instant tis equal to K(t)=⎝braceleftBigg 1–t Tift≤T, 0i f t>T. Suppose that on the time interval from τtoτ+dτthe cost of the goods bought is equal to y(τ)dτ. This stock of the goods is decreased through selling, so that by the instant t>τthe cost of the remainder is K(t–τ)y(τ)dτ. Therefore, by the time t, the cost of the unsold portion of the goods purchased by the shop will be equal to ⎝integraldisplayt 0K(t–τ)y(τ)dτ. On the other hand, the cost of the unsold portion of the goods bought by the shop is equal to 1– K(t). Equating these two expressions gives 1–K (t)=⎝integraldisplayt 0K(t–τ)y(τ)dτ. This is a convolution integral equation of the ﬁrst kind for the unknown function y(t). References for Section 10.1: E. Goursat (1923), H. M. M ¨untz (1934), F. G. Tricomi (1957), V . V olterra (1959), S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), C. Corduneanu (1973), V . I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), A. J. Jerry (1985), A. F. Verlan’ and V . S. Sizikov (1986), P. Linz (1987). 10.2. Equations with Degenerate Kernel: K(x,t)=g1(x)h1(t)+··· +gn(x)hn(t) 10.2-1. Equations with Kernel of the Form K(x,t)=g1(x)h1(t)+g2(x)h2(t). Any equation of this type can be rewritten in the form g1(x)⎝integraldisplayx ah1(t)y(t)dt+g2(x)⎝integraldisplayx ah2(t)y(t)dt=f(x). (1) It is assumed that g1(x)≠constg2(x),h1(t)≠consth2(t), 0 < g2 1(a)+g2 2(a)<∞,a n d f(a)=0 . The change of variables u(x)=⎝integraldisplayx ah1(t)y(t)dt (2) followed by the integration by parts in the second integral in (1) with regard to the relation u(a)=0 yields the following V olterra equation of the second kind: [g1(x)h1(x)+g2(x)h2(x)]u(x)–g2(x)h1(x)⎝integraldisplayx a⎝bracketleftbiggh2(t) h1(t)⎝bracketrightbigg/prime tu(t)dt=h1(x)f(x). (3) The substitution w(x)=⎝integraldisplayx a⎝bracketleftbiggh2(t) h1(t)⎝bracketrightbigg/prime tu(t)dt (4) reduces Eq. (3) to the ﬁrst-order linear ordinary differential equation [g1(x)h1(x)+g2(x)h2(x)]w/prime x–g2(x)h1(x)⎝bracketleftbiggh2(x) h1(x)⎝bracketrightbigg/prime xw=f(x)h1(x)⎝bracketleftbiggh2(x) h1(x)⎝bracketrightbigg/prime x.( 5 ) 10.2. E QUATIONS WITH DEGENERATE KERNEL :K(x,t)=g1(x)h1(t)+···+gn(x)hn(t) 523 1◦. In the case g1(x)h1(x)+g2(x)h2(x)/ ≡0, the solution of equatio n (5) satisfying the condition w(a) = 0 (this condition is a consequence of the substitution (4)) has the form w(x)=Φ(x)⎝integraldisplayx a⎝bracketleftbiggh2(t) h1(t)⎝bracketrightbigg/prime tf(t)h1(t)dt Φ(t)[g 1(t)h1(t)+g2(t)h2(t)],( 6) Φ(x)=e x p⎝braceleftbigg⎝integraldisplayx a⎝bracketleftbiggh2(t) h1(t)⎝bracketrightbigg/prime tg2(t)h1(t)dt g1(t)h1(t)+g2(t)h2(t)⎝bracerightbigg .( 7) Let us differentiate relation (4) and substitute the function (6) into the resulting expression. After integrating by parts with regard to the relations f(a)=0a n d w(a)=0 ,f o r f/ ≡constg2we obtain u(x)=g2(x)h1(x)Φ(x) g1(x)h1(x)+g2(x)h2(x)⎝integraldisplayx a⎝bracketleftbiggf(t) g2(t)⎝bracketrightbigg/prime tdt Φ(t). Using formula (2), we ﬁnd a solution of the original equation in the form y(x)=1 h1(x)d dx⎝braceleftbiggg2(x)h1(x)Φ(x) g1(x)h1(x)+g2(x)h2(x)⎝integraldisplayx a⎝bracketleftbiggf(t) g2(t)⎝bracketrightbigg/prime tdt Φ(t)⎝bracerightbigg ,( 8) where the function Φ(x) is given by (7). Iff(x)≡constg2(x), the solution is given by formulas (8) and (7) in which the subscript 1 must be changed by 2 and vice versa. 2◦. In the case g1(x)h1(x)+g2(x)h2(x)≡0, the solution has the form y(x)=1 h1d dx⎝bracketleftbigg(f/g 2)/prime x (g1/g2)/primex⎝bracketrightbigg =–1 h1d dx⎝bracketleftbigg(f/g 2)/prime x (h2/h1)/primex⎝bracketrightbigg . 10.2-2. Equations with General Degenerate Kernel. A V olterra equation of the ﬁrst kind with general degenerate kernel has the form n⎝summationdisplay m=1gm(x)⎝integraldisplayx ahm(t)y(t)dt=f(x). (9) Using the notation wm(x)=⎝integraldisplayx ahm(t)y(t)dt,m=1 ,...,n, (10) we can rewrite Eq. (9) as follows: n⎝summationdisplay m=1gm(x)wm(x)=f(x). (11) On differentiating formulas (10) and eliminating y(x) from the resulting equations, we arrive at the following linear differential equations for the functions wm=wm(x): h1(x)w/prime m=hm(x)w/prime 1, m=2 ,...,n, (12) (the prime stands for the derivative with respect to x) with the initial conditions wm(a)=0 , m=1 ,...,n. Any solution of system (11), (12) determines a solution of the original integral equation (9) by each of the expressions y(x)=w/prime m(x) hm(x),m=1 ,...,n, which can be obtained by differentiating formula (10). System (11), (12) can be reduced to a linear differential equation of order n–1f o ra n y function wm(x)(m=1 ,...,n) by multiple differentiation of Eq. (11) with regard to (12). References for Section 10.2: E. Goursat (1923), A. F. Verlan’ and V . S. Sizikov (1986). 524 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextx aK(x,t)y(t)dt=f(x) 10.3. Reduction of Volterra Equations of the First Kind to Volterra Equations of the Second Kind 10.3-1. First Method. Suppose that the kernel and the right-hand side of the equation ⎝integraldisplayx aK(x,t)y(t)dt=f(x), (1) have continuous derivatives with respect to xand that the condition K(x,x)/ ≡0 holds. In this case, after differentiating relation (1) and dividing the resulting expression by K(x,x) we arrive at the following V olterra equation of the second kind: y(x)+⎝integraldisplayx aK/prime x(x,t) K(x,x)y(t)dt=f/prime x(x) K(x,x).( 2) Equations of this type are considered in Chapter 11. If K(x,x)≡0, then, on differentiating Eq. (1) with respect to xtwice and assuming that K/prime x(x,t)|t=x/ ≡0, we obtain the V olterra equation of the second kind y(x)+⎝integraldisplayx aK/prime/prime xx(x,t) K/primex(x,t)|t=xy(t)dt=f/prime/prime xx(x) K/primex(x,t)|t=x. IfK/prime x(x,x)≡0, we can again apply differentiation, and so on. If the ﬁrst m– 2 partial derivatives of the kernel with respect to xare identically zero and the ( m– 1)st derivative is nonzero, then the m-fold differentiation of the original equation gives the following V olterra equation of the second kind: y(x)+⎝integraldisplayx aK(m) x(x,t) K(m–1) x (x,t)|t=xy(t)dt=f(m) x(x) K(m–1) x (x,t)|t=x. 10.3-2. Second Method. Let us introduce the new variable Y(x)=⎝integraldisplayx ay(t)dt and integrate the right-hand side of Eq. (1) by parts taking into account the relation f(a)=0 .A f t e r dividing the resulting expression by K(x,x), we arrive at the V olterra equation of the second kind Y(x)–⎝integraldisplayx aK/prime t(x,t) K(x,x)Y(t)dt=f(x) K(x,x), for which the condition K(x,x)/ ≡0 must hold. References for Section 10.3: E. Goursat (1923), V . V olterra (1959). 10.4. Equations with Difference Kernel: K(x,t)=K(x–t) 10.4-1. Solution Method Based on the Laplace Transform. V olterra equations of the ﬁrst kind with kernel depending on the difference of the arguments have the form ⎝integraldisplayx 0K(x–t)y(t)dt=f(x). (1) 10.4. E QUATIONS WITH DIFFERENCE KERNEL :K(x,t)=K(x–t) 525 To solve these equations, the Laplace transform can be used (see Section 9.2). In what follows we need the transforms of the kernel and the right-hand side; they are given by the formulas ˜K(p)=⎝integraldisplay∞ 0K(x)e–pxdx,˜f(p)=⎝integraldisplay∞ 0f(x)e–pxdx.( 2 ) Applying the Laplace transform Lto Eq. (1) and taking into account the fact that an integral with kernel depending on the difference of the arguments is transformed to the product by the rule (see Subsection 9.2-4) L⎝braceleftbigg⎝integraldisplayx 0K(x–t)y(t)dt⎝bracerightbigg =˜K(p)˜y(p), we obtain the following equation for the transform ˜ y(p): ˜K(p)˜y(p)=˜f(p). (3) The solution of Eq. (3) is given by the formula ˜y(p)=˜f(p) ˜K(p).( 4) On applying the Laplace inversion formula (if it is applicable) to (4), we obtain a solution of Eq. (1) in the form y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞˜f(p) ˜K(p)epxdp.( 5) When applying formula (5) in practice, the following two technical problems occur: 1◦. Finding the transform ˜K(p)=⎝integraldisplay∞ 0K(x)e–pxdxfor a given kernel K(x). 2◦. Finding the resolvent (5) whose transform ˜R(p) is given by formula (4). To calculate the corresponding in tegrals, tables of direct and inverse Laplace transforms can be applied (see Supplements 5 and 6), and, in many cases, to ﬁnd the inverse transform, methods of thetheory of functions of a complex variable are applied, including the Cauchy residue theorem (see Subsection 9.1-4). Remark. If the lower limit in the integral of a V olterra equation with difference kernel is a,t h e n this equation can be reduced to Eq. (1) by means of the change of variables x=¯x–a,t=¯t–a. 10.4-2. Case in Which the Transform of the Solution is a Rational Function. Consider the important special case in which th e transform (4) of the solution is a rational function of the form ˜y(p)=˜f(p) ˜K(p)≡R(p) Q(p), where Q(p)a n dR(p) are polynomials in the variable pand the degree of Q(p) exceeds that of R(p). 526 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextx aK(x,t)y(t)dt=f(x) If the zeros of the denominator Q(p) are simple, i.e., Q(p)≡const ( p–λ1)(p–λ2)...(p–λn), andλi≠λjfori≠j, then the solution has the form y(x)=n⎝summationdisplay k=1R(λk) Q/prime(λk)exp(λkx), where the prime stands for the derivatives. Example 1. Consider the V olterra integral equation of the ﬁrst kind ⎝integraldisplayx 0e–a(x–t)y(t)dt=Asinh(bx ). We apply the Laplace transform to this equation and obtain (see Supplement 5) 1 p+a˜y(p)=Ab p2–b2. This implies ˜y(p)=Ab(p+a) p2–b2=Ab(p+a) (p–b)(p+b). We have Q(p)=(p–b)(p+b),R(p)=Ab(p+a),λ1=b,a n dλ2=–b. Therefore, the solution of the integral equation has the form y(x)=1 2A(b+a)ebx+1 2A(b–a)e–bx=Aasinh(bx )+Abcosh(bx). 10.4-3. Convolution Representation of a Solution. In solving V olterra integral equations of the ﬁrst kind with difference kernel K(x–t) by means of the Laplace transform, it is sometimes useful to apply the following approach. Let us represent the transform (4) of a solution in the form ˜y(p)=˜N(p)˜M(p)˜f(p), ˜N(p)≡1 ˜K(p)˜M(p).( 6) If we can ﬁnd a function ˜M(p) for which the inverse transforms L–1⎝braceleftbig˜M(p)⎝bracerightbig =M(x), L–1⎝braceleftbig˜N(p)⎝bracerightbig =N(x)( 7 ) exist and can be found in a closed form, then the solution can be written as the convolution y(x)=⎝integraldisplayx 0N(x–t)F(t)dt,F(t)=⎝integraldisplayt 0M(t–s)f(s)ds.( 8) Example 2. Consider the equation ⎝integraldisplayx 0sin⎝parenleftbig λ√ x–t⎝parenrightbig y(t)dt=f(x), f(0) = 0. (9) Applying the Laplace transform, we obtain (see Supplement 5) ˜y(p)=2 √ πλp3/2exp(α/p)˜f(p), α=1 4λ2. (10) Let us rewrite the right-hand side of (10) in the equivalent form ˜y(p)=2 √ πλp2⎝bracketleftbigp–1/2exp(α/p)⎝bracketrightbig˜f(p), α=1 4λ2, (11) where the factor in the square brackets corresponds to ˜M(p)i nf o r m u l a( 6 )a n d ˜N(p) = const p2. By applying the Laplace inversion formula according to the above scheme to formula (11) with regard to the relation (see Supplement 6) L–1⎝braceleftbigp2˜ϕ(p)⎝bracerightbig=d2 dx2ϕ(x), L–1⎝braceleftbigp–1/2exp(α/p)⎝bracerightbig=1 √ πxcosh⎝parenleftbigλ√ x⎝parenrightbig, we ﬁnd the solution y(x)=2 πλd2 dx2⎝integraldisplayx 0cosh⎝parenleftbigλ√ x–t⎝parenrightbig √ x–tf(t)dt. 10.4. E QUATIONS WITH DIFFERENCE KERNEL :K(x,t)=K(x–t) 527 10.4-4. Application of an Auxiliary Equation. Consider the equation⎝integraldisplayx aK(x–t)y(t)dt=f(x), (12) where the kernel K(x) has an integrable singularity at x=0 . Letw=w(x) be the solution of the simpler auxiliary equation with f(x)≡1a n da=0 , ⎝integraldisplayx 0K(x–t)w(t)dt= 1. (13) Then the solution of the original equation (12) with arbitrary right-hand side can be expressed as follows via the solution of the auxiliary equation (13): y(x)=d dx⎝integraldisplayx aw(x–t)f(t)dt=f(a)w(x–a)+⎝integraldisplayx aw(x–t)f/prime t(t)dt. (14) Example 3. Consider the generalized Abel equation ⎝integraldisplayx ay(t)dt (x–t)µ=f(x), 0 < µ<1 . (15) We seek a solution of the corresponding auxiliary equation ⎝integraldisplayx 0w(t)dt (x–t)µ=1 , 0< µ<1 , (16) by the method of indeterminate coefﬁcients in the form w(x)=Axβ. (17) Let us substitute (17) into (16) and then perform the change of variable t=xξin the integral. Taking into account the relationship B(p,q)=⎝integraldisplay1 0ξp–1(1 –ξ)1–qdξ=Γ(p)Γ(q) Γ(p+q) between the beta and gamma functions, we obtain AΓ(β+1 )Γ(1 –µ) Γ(2 +β–µ)xβ+1–µ=1 . From this relation we ﬁnd the coefﬁcients Aandβ: β=µ–1 , A=1 Γ(µ)Γ(1 –µ)=sin(πµ) π. (18) Formulas (17) and (18) deﬁne the solution of the auxiliary equation (16) and make it possible to ﬁnd the solution of the generalized Abel equation (15) by means of formula (14) as follows: y(x)=sin(πµ) πd dx⎝integraldisplayx af(t)dt (x–t)1–µ=sin(πµ) π⎝bracketleftbiggf(a) (x–a)1–µ+⎝integraldisplayx af/prime t(t)dt (x–t)1–µ⎝bracketrightbigg . (19) 10.4-5. Reduction to Ordinary Differential Equations. Consider the special case in which the transform of the kernel of the integral equation (1) can be represented in the form ˜K(p)=M(p) N(p), (20) 528 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextx aK(x,t)y(t)dt=f(x) where M(p)a n dN(p) are some polynomials of degrees mandn, respectively: M(p)=m⎝summationdisplay k=0Akpk,N(p)=n⎝summationdisplay k=0Bkpk. (21) In this case, the solution of the integral equation (1) (if it exists) satisﬁes the following linear nonhomogeneous ordinary differential equation of order mwith constant coefﬁcients: m⎝summationdisplay k=0Aky(k) x(x)=n⎝summationdisplay k=0Bkf(k) x(x). (22) We can rewrite Eq. (22) in the operator form M(D)y(x)=N(D)f(x), D≡d dx. The initial data for the differential equation (22), as well as the conditions that must be imposed on the right-hand side of the integral equa tion (1), can be obtained from the relation m⎝summationdisplay k=0Akk–1⎝summationdisplay s=0pk–1–sy(s) x(0) –n⎝summationdisplay k=0Bkk–1⎝summationdisplay s=0pk–1–sf(s) x(0) = 0 (23) by matching the coefﬁcients of like powers of the parameter p. The proof of this assertion can be given by applying the Laplace transform to the differential equation (22) followed by comparing the resulting expression with Eq. (3) with regard to (20). 10.4-6. Reduction of a V olterra Equation to a Wiener–Hopf Equation. A V olterra equation of the ﬁrst kind with difference kernel of the form ⎝integraldisplayx 0K(x–t)y(t)dt=f(x), 0 < x<∞, (24) can be reduced to the following Wiener–Hopf equation of the ﬁrst kind: ⎝integraldisplay∞ 0K+(x–t)y(t)dt=f(x), 0 < x<∞, (25) where the kernel K+(x–t)i sg i v e nb y K+(s)=⎝braceleftBigK(s)f o r s>0 , 0f o r s<0 . Methods for solving Eq. (25) are presented in Section 12.8. References for Section 10.4: G. Doetsch (1956), V . A. Ditkin and A. P. Prudnikov (1965), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), V . I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. D. Gakhov and Yu. I. Cherskii (1978). 10.5. M ETHOD OF FRACTIONAL DIFFERENTIA TION 529 10.5. Method of Fractional Differentiation 10.5-1. Deﬁnition of Fractional Integrals. A function f(x)i ss a i dt ob e absolutely continuous on a closed interval [ a,b] if for each ε>0t h e r e exists a δ> 0 such that for any ﬁnite system of disjoint intervals [ ak,bk]⊂[a,b],k=1 ,...,n,s u c h thatn⎝summationtext k=1(bk–ak)<δthe inequalityn⎝summationtext k=1|f(bk)–f(ak)|<εholds. The class of all these functions is denoted by AC. LetACn,n=1 ,2 ,... , be the class of functions f(x) that are continuously differentiable on [ a,b] up to the order n–1a n df o rw h i c h f(n–1)(x)∈AC. Letϕ(x)∈L1(a,b). The integrals Iµ a+ϕ(x)≡1 Γ(µ)⎝integraldisplayx aϕ(t) (x–t)1–µdt,x>a,( 1) Iµ b–ϕ(x)≡1 Γ(µ)⎝integraldisplayb xϕ(t) (t–x)1–µdt,x<b,( 2) where µ> 0, are called the integrals of fractional order µ. Sometimes the integral (1) is called left-sided and the integral (2) is called right-sided . The operators Iµ a+and Iµ b–are called the operators of fractional integration . The integrals (1) and (2) are usually called the Riemann–Liouville fractional integrals . The following formula holds: ⎝integraldisplayb aϕ(x)Iµ a+ψ(x)dx=⎝integraldisplayb aψ(x)Iµ b–ϕ(x)dx,( 3 ) which is sometimes called the formula of fractional integration by parts . Fractional integration has the property Iµ a+Iβa+ϕ(x)=Iµ+β a+ϕ(x), Iµ b–Iβ b–ϕ(x)=Iµ+β b–ϕ(x), µ>0 , β>0 . ( 4 ) Property (4) is called the semigroup property of fractional integration. 10.5-2. Deﬁnition of Fractional Derivatives. It is natural to introduce fractional differentiation as the operation inverse to fractional integration. For a function f(x) deﬁned on a closed interval [ a,b], the expressions Dµ a+f(x)=1 Γ(1 –µ)d dx⎝integraldisplayx af(t) (x–t)µdt,( 5) Dµ b–f(x)=–1 Γ(1 –µ)d dx⎝integraldisplayb xf(t) (t–x)µdt (6) are called the leftand the right fractional derivative of order µ, respectively. It is assumed here that 0<µ<1 . The fractional derivatives (5) and (6) are usually called the Riemann–Liouville derivatives. Note that the fractional integrals are deﬁned for any order µ> 0, but the fractional derivatives are so far deﬁned only for 0 < µ<1 . 530 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextx aK(x,t)y(t)dt=f(x) Iff(x)∈AC, then the derivatives Dµ a+f(x)a n d Dµ b–f(x), 0 < µ< 1, exist almost everywhere, and we have Dµ a+f(x)∈Lr(a,b)a n d Dµ b–f(x)∈Lr(a,b), 1≤r<1/µ. These derivatives have the representations Dµ a+f(x)=1 Γ(1 –µ)⎝bracketleftbiggf(a) (x–a)µ+⎝integraldisplayx af/prime t(t) (x–t)µdt⎝bracketrightbigg ,( 7 ) Dµ b–f(x)=1 Γ(1 –µ)⎝bracketleftbiggf(b) (b–x)µ–⎝integraldisplayb xf/prime t(t) (t–x)µdt⎝bracketrightbigg .( 8 ) Finally, let us pass to the fractional derivatives of order µ≥1. We shall use the following notation: [ µ] stands for the integral part of a real number µand{µ}is the fractional part of µ, 0≤{µ}<1 ,s ot h a t µ=[µ]+{µ}.( 9) Ifµis an integer, then by the fractional derivative of order µwe mean the ordinary derivative Dµ a+=⎝parenleftbiggd dx⎝parenrightbiggµ ,Dµ b–=⎝parenleftbigg –d dx⎝parenrightbiggµ ,µ=1 ,2 , ... (10) However, if µis not integral, then Dµ a+fand Dµ b–fare introduced by the formulas Dµ a+f(x)≡⎝parenleftbiggd dx⎝parenrightbigg[µ] D{µ} a+f(x)=⎝parenleftbiggd dx⎝parenrightbigg[µ]+1 I1–{µ} a+f(x), (11) Dµ b–f(x)≡⎝parenleftbigg –d dx⎝parenrightbigg[µ] D{µ} b–f(x)=⎝parenleftbigg –d dx⎝parenrightbigg[µ]+1 I1–{µ} b–f(x). (12) Thus, Dµ a+f(x)=1 Γ(n–µ)⎝parenleftbiggd dx⎝parenrightbiggn⎝integraldisplayx af(t) (x–t)µ–n+1dt,n=[µ] + 1, (13) Dµ b–f(x)=(–1)n Γ(n–µ)⎝parenleftbiggd dx⎝parenrightbiggn⎝integraldisplayb xf(t) (t–x)µ–n+1dt,n=[µ] + 1. (14) A sufﬁcient condition for the existence of the derivatives (13) and (14) is as follows: ⎝integraldisplayx af(t)dt (x–t){µ}∈AC[µ]. This sufﬁcient conditio n holds whenever f(x)∈AC[µ]. Remark. The deﬁnitions of the fractional integrals and fractional derivatives can be extended to the case of complex µ(e.g., see S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993)). 10.5-3. Main Properties. LetIµ a+(L1),µ> 0, be the class of functions f(x) that can be represented by the left fractional integral of order µof an integrable function: f(x)=Iµ a+ϕ(x),ϕ(x)∈L1(a,b), 1≤p<∞. For the relation f(x)∈Iµ a+(L1),µ> 0, to hold, it is necessary and sufﬁcient that fn–µ(x)≡In–µ a+f∈ACn, (15) 10.5. M ETHOD OF FRACTIONAL DIFFERENTIA TION 531 where n=[µ] + 1, and* f(k) n–µ(a)=0 , k=0 ,1 , ...,n– 1. (16) Letµ> 0. We say that a function f(x)∈L1has an integrable fractional derivative Dµ a+fif In–µ a+f(x)∈ACn,w h e r e n=[µ]+1 . In other words, this deﬁnition introduces a notion involving only the ﬁrst of the two condi- tions (15) and (16) describing the class Iµ a+(L1). Letµ> 0. In this case the relation Dµ a+Iµa+ϕ(x)=ϕ(x) (17) holds for any integrable function ϕ(x), and the relation Iµ a+Dµa+f(x)=f(x) (18) holds for any function f(x)s u c ht h a t f(x)∈Iµ a+(L1). (19) If we replace (19) by the condition that the function f(x)∈L1(a,b) has an integrable deriva- tive Dµ a+f(x), then relation (18) fails in general and must be replaced by the formula Iµ a+Dµa+f(x)=f(x)–n–1⎝summationdisplay k=0(x–a)µ–k–1 Γ(µ–k)f(n–k–1) n–µ(a), (20) where n=[µ]+1a n d fn–µ(x)=In–µ a+f(x). In particular, for 0 < µ<1w eh a v e Iµ a+Dµa+f(x)=f(x)–f1–µ(a) Γ(µ)(x–a)µ–1. (21) 10.5-4. Solution of the Generalized Abel Equation. Consider the Abel integral equation⎝integraldisplayx ay(t) (x–t)µdt=f(x), (22) where 0 < µ< 1. Suppose that x∈[a,b],f(x)∈AC,a n dy(t)∈L1, and apply the technique of fractional differentiation. We divide Eq. (22) by Γ(1 –µ), and, by virtue of (1), rewrite this equation as follows: I1–µ a+y(x)=f(x) Γ(1 –µ),x>a. (23) Let us apply the operator of fractional differentiation D1–µ a+to (23). Using the properties of the operators of fractional integration and differentiation, we obtain y(x)=D1–µ a+f(x) Γ(1 –µ), (24) or, in the detailed notation, y(x)=1 Γ(µ)Γ(1 –µ)⎝bracketleftbiggf(a) (x–a)1–µ+⎝integraldisplayx af/prime t(t) (x–t)1–µdt⎝bracketrightbigg . (25) Taking into account the relation 1 Γ(µ)Γ(1 –µ)=sin(πµ) π, we now arrive at the solution of the generalized Abel equation in the form y(x)=sin(πµ) π⎝bracketleftbiggf(a) (x–a)1–µ+⎝integraldisplayx af/prime t(t)dt (x–t)1–µ⎝bracketrightbigg , (26) which coincides with that obtained above in Subsection 10.4-4. * From now on in Section 10.5, by f(n)(x) we mean the nth derivative of f(x) with respect to xandf(n)(a)≡f(n)(x)⎝vextendsingle⎝vextendsingle x=a. 532 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextx aK(x,t)y(t)dt=f(x) 10.5-5. Erd ´elyi–Kober Operators. Generalized Erd ´elyi–Kober operators are deﬁned by the relations Iβ,α[f]≡2 Γ(α)x–2(α+β)⎝integraldisplayx at2β+1(x2–t2)α–1f(t)dtif 0 < α<∞ (27) and Kβ,α[f]≡2 Γ(α)x2β⎝integraldisplayb xt1–2β –2α(t2–x2)α–1f(t)dtif 0 < α<∞. The following identities hold: Iβ,αIα+β,γ=Iβ,α+γ, Iβ,α[t2γf(t)] =x2γIβ+γ,α[f(t)], Kβ,αIα+β,γ=Kβ,α+γ, Kβ,α[t2γf(t)] =x2γKβ–γ,α[f(t)]. Deﬁning the inverse operators, one can show that I–1 β,α=Iα+β,–α, K–1 β,α=Kα+β,–α.(28) Generalized Erd ´elyi–Kober operators (27) and inversion formulas (28) are used for solving some dual integral equations. References for Section 10.5: K. B. Oldham and J. Spanier (1974), C. Nasim and B. D. Aggarwala (1984), Yu. I. Babenko (1986), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 10.6. Equations with Weakly Singular Kernel 10.6-1. Method of Transformation of the Kernel. Consider the V olterra integral equation of the ﬁrst kind with polar kernel K(x,t)=L(x,t) (x–t)α,0 < α<1 . ( 1 ) The integral equation in question can be represented in the form⎝integraldisplayx 0L(x,t) (x–t)αy(t)dt=f(x), (2) where we assume that the functions L(x,t)a n d ∂L(x,t)/∂x are continuous and bounded. To solve Eq. (2), we multiply it by dx/(ξ–x)1–αand integrate from 0 to ξ, thus obtaining ⎝integraldisplayξ 0⎝bracketleftbigg⎝integraldisplayx 0L(x,t) (x–t)αy(t)dt⎝bracketrightbiggdx (ξ–x)1–α=⎝integraldisplayξ 0f(x)dx (ξ–x)1–α. By setting K∗(ξ,t)=⎝integraldisplayξ tL(x,t)dx (ξ–x)1–α(x–t)α, ϕ(ξ)=⎝integraldisplayξ 0f(x)dx (ξ–x)1–α,ϕ(0) = 0, we obtain another integral equation of the ﬁrst kind with the unknown function y(t): ⎝integraldisplayξ 0K∗(ξ,t)y(t)dt=ϕ(ξ), (3) in which the kernel K∗(ξ,t) has no singularities. It can be shown that any solution of Eq. (3) is a solution of Eq. (2). Thus, after transforming Eq. (2) to the form (3), we can apply any methods available for continuous kernels to the latter equation. 10.6. E QUATIONS WITH WEAKLY SINGULAR KERNEL 533 10.6-2. Kernel with Logarithmic Singularity. Consider the equation⎝integraldisplayx 0ln(x–t)y(t)dt=f(x), f(0) = 0. (4) Let us apply the Laplace transform to solve this equation. Note that L⎝braceleftbig xν⎝bracerightbig =⎝integraldisplay∞ 0e–pxxνdx=Γ(ν+1 ) pν+1,ν> –1. (5) Let us differentiate relation (5) with respect to ν. We obtain L⎝braceleftbig xνlnx⎝bracerightbig =Γ(ν+1 ) pν+1⎝bracketleftbiggΓ/prime z(ν+1 ) Γ(ν+1 )+l n1 p⎝bracketrightbigg .( 6) From Supplement 11.4-2, it follows that Γ/prime z(1) Γ(1)=–C, whereC= 0.5772... is the Euler constant. With regard to the last relation, formula (6) with ν=0 becomes L⎝braceleftbig lnx⎝bracerightbig =–lnp+C p.( 7) Applying the Laplace transform to Eq. (4) and taking into account (7), we obtain –lnp+C p˜y(p)=˜f(p), and hence ˜y(p)=–p˜f(p) lnp+C.( 8) Now let us express ˜ y(p) in the form ˜y(p)=–p2˜f(p)–f/prime x(0) p(lnp+C)–f/prime x(0) p(lnp+C).( 9) Sincef(0) = 0, it follows that L⎝braceleftbig f/prime/prime xx(x)⎝bracerightbig =p2˜f(p)–f/prime x(0). (10) Let us rewrite formula (5) as L⎝braceleftbiggxν Γ(ν+1 )⎝bracerightbigg =1 pν+1(11) and integrate (11) with respect to νfrom 0 to ∞. We obtain L⎝braceleftbigg⎝integraldisplay∞ 0xν Γ(ν+1 )dν⎝bracerightbigg =⎝integraldisplay∞ 0dν pν+1=1 plnp. Applying the scaling formula for the Laplace transform (see Table 1 in Subsection 9.2-5) we see that L⎝braceleftbigg⎝integraldisplay∞ 0(x/a)ν Γ(ν+1 )dν⎝bracerightbigg =1 plnap=1 p(lnp+l na). 534 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextx aK(x,t)y(t)dt=f(x) We set a=eCand obtain L⎝braceleftbigg⎝integraldisplay∞ 0xνe–Cν Γ(ν+1 )dν⎝bracerightbigg =1 p(lnp+C). (12) Let us proceed with relation (9). By (12), we have f/prime x(0) p(lnp+C)=L⎝braceleftbigg f/prime x(0)⎝integraldisplay∞ 0xνe–Cν Γ(ν+1 )dν⎝bracerightbigg . (13) Taking into account (10) and (12), we can regard the ﬁrst summand on the right-hand side in (9) as a product of transforms. To ﬁnd this summand itself we apply the convolution theorem: p2˜f(p)–f/prime x(0) p(lnp+C)=L⎝braceleftbigg⎝integraldisplayx 0f/prime/prime tt(t)⎝integraldisplay∞ 0(x–t)νe–Cν Γ(ν+1 )dν dt⎝bracerightbigg . (14) On the basis of relations (9), (13), and (14) we obt ain the solution of the integral equation (4) in the form y(x)=–⎝integraldisplayx 0f/prime/prime tt(t)⎝integraldisplay∞ 0(x–t)νe–Cν Γ(ν+1 )dν dt –f/prime x(0)⎝integraldisplay∞ 0xνe–Cν Γ(ν+1 )dν. (15) References for Section 10.6: V . V olterra (1959), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971). 10.7. Method of Quadratures 10.7-1. Quadrature Formulas. The method of quadratures is a method for constructing an approximate solution of an integral equation based on the replacement of integrals by ﬁnite sums according to some formula. Such formulas are called quadrature formulas and, in general, have the form ⎝integraldisplayb aψ(x)dx=n⎝summationdisplay i=1Aiψ(xi)+εn[ψ], (1) where xi(i=1 ,...,n) are the abscissas of the partition points of the integration interval [ a,b], or quadrature (interpolation )nodes ,Ai(i=1 ,...,n) are numerical coefﬁcients independent of the choice of the function ψ(x), and εn[ψ] is the remainder (the truncation error) of formula (1). As a rule,Ai≥0a n dn⎝summationtext i=1Ai=b–a. There are quite a few quadrature formulas of the form (1). The following formulas are the simplest and most frequently used in practice. Rectangle rule : A1=A2=···=An–1=h,An=0 , h=b–a n–1,xi=a+h(i–1 ) ( i=1 ,...,n).(2) Trapezoidal rule : A1=An=1 2h,A2=A3=···=An–1=h, h=b–a n–1,xi=a+h(i–1 ) ( i=1 ,...,n).(3) Simpson’s rule (orprismoidal formula ): A1=A2m+1=1 3h,A2=···=A2m=4 3h,A3=···=A2m–1=2 3h, h=b–a n–1,xi=a+h(i–1 ) ( n=2m+1 ,i=1 ,...,n),(4) where mis a positive integer. In formulas (2)–(4), his a constant integration step. The quadrature formulas due to Chebyshev and Gauss with various numbers of interpolation nodes are also widely applied. Let us illustrate these formulas by an example. 10.7. M ETHOD OF QUADRATURES 535 Example. For the interval [–1, 1], the parameters in formula (1) acquire the following values: Chebyshev’s formula (n=6 ) : A1=A2=···=2 n=1 3, x2=–x5= –0.4225186538,x1=–x6= –0.8662468181, x3=–x4= –0.2666354015.(5) Gauss’s formula (n=7 ) : A1=A7= 0.1294849662, A3=A5= 0.3818300505, x1=–x7= –0.9491079123, x3=–x5= –0.4058451514,A2=A6= 0.2797053915, A4= 0.4179591837, x2=–x6= –0.7415311856, x4=0 .(6) Note that a vast literature is devoted to quadrature formulas, and the reader can ﬁnd books of interest (e.g., see G. A. Korn and T. M. Korn (1968), N. S. Bakhvalov (1973), S. M. Nikol’skii(1979)). 10.7-2. General Scheme of the Method. Let us solve the V olterra integral equation of the ﬁrst kind ⎝integraldisplayx aK(x,t)y(t)dt=f(x), f(a)=0 , ( 7 ) on an interval a≤x≤bby the method of quadratures. The procedure of constructing the solution involves two stages: 1◦. First, we determine the initial value y(a). To this end, we differentiate Eq. (7) with respect to x, thus obtaining K(x,x)y(x)+⎝integraldisplayx aK/prime x(x,t)y(t)dt=f/prime x(x). By setting x=a,w eﬁ n dt h a t y1=y(a)=f/prime x(a) K(a,a)=f/prime x(a) K11. 2◦. Let us choose a constant integration step hand consider the discrete set of points xi=a+h(i–1), i=1 ,...,n.F o rx=xi, Eq. (7) acquires the form ⎝integraldisplayxi aK(xi,t)y(t)dt=f(xi), i=2 ,...,n,( 8) Applying the quadrature formula (1) to the integral in (8) and choosing xj(j=1 ,...,i)t ob et h e nodes in t, we arrive at the system of equations i⎝summationdisplay j=1AijK(xi,xj)y(xj)=f(xi)+εi[y], i=2 ,...,n,( 9) where the Aijare the coefﬁcients of the quadrature formula on the interval [a ,xi]a n dεi[y]i st h e truncation error. Assume that the εi[y] are small and neglect them; then we obtain a system of linear algebraic equations in the form i⎝summationdisplay j=1AijKijyj=fi,i=2 ,...,n, (10) 536 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextx aK(x,t)y(t)dt=f(x) where Kij=K(xi,xj)(j=1 ,...,i),fi=f(xi), and yjare approximate values of the unknown function at the nodes xi. Now system (10) permits one, provided that AiiKii≠0(i=2 ,...,n), to successively ﬁnd the desired approximate values by the formulas y1=f/prime x(a) K11,y2=f2–A21K21y1 A22K22,...,yn=fn–n–1⎝summationtext j=1AnjKnjyj AnnKnn, whose speciﬁc form depends on the choice of the quadrature formula. 10.7-3. Algorithm Based on the Trapezoidal Rule. According to the trapezoidal rule (3), we have Ai1=Aii=1 2h,Ai2=···=Ai,i–1=h,i=2 ,...,n. The application of the trapezoidal rule in the general scheme leads to the following step algorithm: y1=f/prime x(a) K11,f/prime x(a)=–3f 1+4f2–f3 2h, yi=2 Kii⎝parenleftbiggfi h–i–1⎝summationdisplay j=1βjKijyj⎝parenrightbigg ,βj=⎝braceleftbigg1 2forj=1 , 1f o r j>1 ,i=2 ,...,n, where the notation coincides with th at introduced in Subsection 10.7-2. The trapezoid al rule is quite simple and effective and frequently used in practice for solving integral equations with variable limit of integration. On the basis of Subsections 10.7-1 and 10.7-2, one can write out similar expressions for other quadrature formulas. However, they must be used with care. For example, the application of Simpson’s rule must be alternated, for odd nodes, with some other rule, e.g., the rectangle rule or the trapezoidal rule. For equations with variable integration limit, the use of Chebyshev’s formulaor Gauss’s formula also has some difﬁculties as well. 10.7-4. Algorithm for an Equation with Degenerate Kernel. A general property of the algorithms of the method of quadratures in the solution of the V olterra equations of the ﬁrst kind with arbitrary kernel is that the amount of computational work at each step is proportional to the number of the step: all operations of the previous step are repeated withnew data and another term in the sum is added. However, if the kernel in Eq. (7) is degenerate, i.e., K(x,t)= m⎝summationdisplay k=1pk(x)qk(t), (11) or if the kernel under consideration can be approximated by a degenerate kernel,then an algorithm can be constructed for which the number of operations does not depend on the index of the digitalization node. With regard to (11), Eq. (7) becomes m⎝summationdisplay k=1pk(x)⎝integraldisplayx aqk(t)y(t)dt=f(x). (12) 10.8. E QUATIONS WITH INFINITE INTEGRATION LIMIT 537 By applying the trapezoidal rule to (12), we obtain recurrent expressions for the solution of the equation (see formulas in Subsection 10.7-3): y(a)=f/prime x(a) m⎝summationtext k=1pk(a)qk(a),yi=2 m⎝summationtext k=1pkiqki⎝bracketleftbiggfi h–m⎝summationdisplay k=1pkii–1⎝summationdisplay j=1βjqkjyj⎝bracketrightbigg , where yiare approximate values of y(x)a txi,fi=f(xi),pki=pk(xi), and qki=qk(xi). References for Section 10.7: G. A. Korn and T. M. Korn (1968), N. S. Bakhvalov (1973), V . I. Krylov, V . V . Bobkov, and P. I. Monastyrnyi (1984), A. F. Verlan’ and V . S. Sizikov (1986). 10.8. Equations with Inﬁnite Integration Limit Integral equations of the ﬁrst kind with difference ke rnel in which one of the limits of integration is variable and the other is inﬁnite are of interest. Sometimes the kernels and the functions of these equations do not belong to the classes described in the beginning of the chapter. The investigation of these equations can be performed by the method of model solutions (see Section 11.6) or by the method of reducing to equations of the convolution type. Let us consider these methods for anexample of an equation of the ﬁrst kind with variable lower limit of integration. 10.8-1. Equation of the First Kind with Variable Lower Limit of Integration. Consider the equation of the ﬁrst kind with difference kernel⎝integraldisplay∞ xK(x–t)y(t)dt=f(x). (1) Equation (1) cannot be solved by direct application of the Laplace transform, because the convolution theorem cannot be used here. According to the method of model solutions whose detailed expositioncan be found in Section 11.6, we consider the auxiliary equation with exponential right-hand side⎝integraldisplay ∞ xK(x–t)y(t)dt=epx.( 2) The solution of (2) has the form Y(x,p)=1 ˜K(–p)epx, ˜K(–p)=⎝integraldisplay∞ 0K(–z)epzdz.( 3) On the basis of these formulas and formula (11) from Section 11.6, we obtain the solution of Eq. (1) for an arbitrary right-hand side f(x) in the form y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞˜f(p) ˜K(–p)epxdp,( 4) where ˜f(p) is the Laplace transform of the function f(x). Example. Consider the following integral equation of the ﬁrst kind with variable lower limit of integration: ⎝integraldisplay∞ xea(x–t)y(t)dt=Asin(bx), a>0 . (5) According to (3) and (4), we can write out the expressions for ˜f(p) (see Supplement 5) and ˜K(–p), ˜f(p)=Ab p2+b2, ˜K(–p)=⎝integraldisplay∞ 0e(p–a)zdz=1 a–p, (6) and the solution of Eq. (5) in the form y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞Ab(a–p) p2+b2epxdp. (7) Now using the tables of inverse Laplace transforms (see Supplement 6), we obtain the exact solution y(x)=Aasin(bx)–Abcos(bx), a>0 , (8) which can readily be veriﬁed by substituting (8) into (5) and using the tables of integrals in Supplement 3. 538 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextx aK(x,t)y(t)dt=f(x) 10.8-2. Reduction to a Wiener–Hopf Equation of the First Kind. Equation (1) can be reduced to a ﬁrst-kind one-sided equation ⎝integraldisplay∞ 0K–(x–t)y(t)dt=–f(x), 0 < x<∞,( 9) where the kernel K–(x–t) has the following form: K–(s)=⎝braceleftbigg 0f o r s>0 , –K(s)f o r s<0 . Methods for studying Eq. (9) are described in Chapter 12. References for Section 10.8: F. D. Gakhov and Yu. I. Cherskii (1978), A. D. Polyanin and A. V . Manzhirov (1997). Chapter 11 Methods for Solving Linear Equations of the Form y(x)–⎝integraldisplay ⎝integraldisplayx aK(x,t)y(t)dt=f(x) 11.1. Volterra Integral Equations of the Second Kind 11.1-1. Preliminary Remarks. Equations for the Resolvent. In this chapter we present methods for solving V olterra integral equations of the second kind, which have the form y(x)–⎝integraldisplayx aK(x,t)y(t)dt=f(x), (1) where y(x) is the unknown function ( a≤x≤b),K(x,t) is the kernel of the integral equation, and f(x)i st h e right-hand side of the integral equation. The function classes to which y(x),f(x), and K(x,t) can belong are deﬁned in Subsection 10.1-1. In these function classes, there exists a unique solution of the V olterra integral equation of the second kind. Equation (1) is said to be homogeneous iff(x)≡0a n dnonhomogeneous otherwise. The kernel K(x,t)i ss a i dt ob e degenerate if it can be represented in the form K(x,t)=g1(x)h1(t)+···+gn(x)hn(t). The kernel K(x,t) of an integral equation is called difference kernel if it depends only on the difference of the arguments, K(x,t)=K(x–t). Remark 1. A homogeneous V olterra integral equation of the second kind has only the trivial solution. Remark 2. The existence and uniqueness of the solution of a V olterra integral equation of the second kind hold for a much wider class of kernels and functions. Remark 3. A V olterra equation of the second kind can be regarded as a Fredholm equation of the second kind whose kernel K(x,t) vanishes for t>x(see Chapter 13). Remark 4. The case in which a=–∞and/or b=∞is not excluded, but in this case the square integrability of the kernel K(x,t) on the square S={a≤x≤b,a≤t≤b}is especially signiﬁcant. The solution of Eq. (1) can be presented in the form y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt,( 2) where the resolvent R(x,t) is independent of f(x) and the lower limit of integration aand is determined by the kernel of the integral equation alone. 539 540 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) The resolvent of the V olterra equation (1) satisﬁes the following two integral equations: R(x,t)=K(x,t)+⎝integraldisplayx tK(x,s)R(s,t)ds,( 3) R(x,t)=K(x,t)+⎝integraldisplayx tK(s,t)R(x,s)ds,( 4 ) in which the integration is p erformed with respect to different pairs of variables of the kernel and the resolvent. 11.1-2. Relationship Between Solutions of Some Integral Equations. Let us present two useful formulas that express the solution of one integral equation via the solutions of other integral equations. 1◦. Assume that the V olterra equation of the second kind with kernel K(x,t) has a resolvent R(x,t). Then the V olterra equation of the second kind with kernel K∗(x,t)=–K(t,x) has the resolvent R∗(x,t)=–R(t,x). 2◦. Assume that two V olterra equations of the second kind with kernels K1(x,t)a n d K2(x,t)a r e given and that resolvents R1(x,t)a n dR2(x,t) correspond to these equations. In this case the V olterra equation with kernel K(x,t)=K1(x,t)+K2(x,t)–⎝integraldisplayx tK1(x,s)K2(s,t)ds (5) has the resolvent R(x,t)=R1(x,t)+R2(x,t)+⎝integraldisplayx tR1(s,t)R2(x,s)ds.( 6 ) Note that in formulas (5) and (6), the integration is performed with respect to different pairs of variables. References for Section 11.1: E. Goursat (1923), H. M. M ¨untz (1934), V . V olterra (1959), S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), V . I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), A. J. Jerry (1985), F. G. Tricomi (1985), A. F. Verlan’ and V . S. Sizikov (1986), P. Linz (1987), G. Gripenberg, S.-O. Londen, and O. Staffans (1990), C. Corduneanu (1991), R. Gorenﬂo and S. Vessella (1991), A. C. Pipkin(1991). 11.2. Equations with Degenerate Kernel: K(x,t)=g1(x)h1(t)+··· +gn(x)hn(t) 11.2-1. Equations with Kernel of the Form K(x,t)=ϕ(x)+ψ(x)(x–t). The solution of a V olterra equation (see Subsection 11.1-1) with kernel of this type can be expressed by the formula y=w/prime/prime xx,( 1) where w=w(x) is the solution of the second-order linear nonhomogeneous ordinary differential equation w/prime/prime xx–ϕ(x)w/prime x–ψ(x)w=f(x), (2) with the initial conditions w(a)=w/prime x(a)=0 . ( 3 ) 11.2. E QUATIONS WITH DEGENERATE KERNEL :K(x,t)=g1(x)h1(t)+···+gn(x)hn(t) 541 Letw1=w1(x) be a nontrivial particular solution of the corresponding homogeneous linear differ- ential equation (2) for f(x)≡0. Assume that w1(a)≠0. In this case, the other nontrivial particular solution w2=w2(x) of this homogeneous linear differential equation has the form w2(x)=w1(x)⎝integraldisplayx aΦ(t) [w1(t)]2dt,Φ(x)=e x p⎝bracketleftBig⎝integraldisplayx aϕ(s)ds⎝bracketrightBig . The solution of the nonhomogeneous equation (2) with the initial conditions (3) is given by the formula w(x)=w2(x)⎝integraldisplayx aw1(t) Φ(t)f(t)dt–w1(x)⎝integraldisplayx aw2(t) Φ(t)f(t)dt.( 4) On substituting expression (4) into formula (1) we obtain the solution of the original integral equation in the form y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt, where R(x,t)=[w/prime/prime 2(x)w1(t)–w/prime/prime 1(x)w2(t)]1 Φ(t) =ϕ(x)Φ(x) w1(x)w1(t) Φ(t)+[ϕ(x)w/prime 1(x)+ψ(x)w1(x)]w1(t) Φ(t)⎝integraldisplayx tΦ(s) [w1(s)]2ds. HereΦ(x)=e x p⎝bracketleftBig⎝integraldisplayx aϕ(s)ds⎝bracketrightBig and the primes stand for x-derivatives. For a degenerate kernel of the above form, the resolvent can be deﬁned by the formula R(x,t)=u/prime/prime xx, where the auxiliary function uis the solution of the homogeneous linear second-order ordinary differential equation u/prime/prime xx–ϕ(x)u/prime x–ψ(x)u=0 ( 5 ) with the following initial conditions at x=t: u⎝vextendsingle⎝vextendsingle x=t=0 , u/prime x⎝vextendsingle⎝vextendsingle x=t=1 . ( 6 ) The parameter toccurs only in the initial conditions (6), and Eq. (5) itself is independent of t. Remark 1. The kernel of the integral equation in question can be rewritten in the form K(x,t)= G1(x)+tG2(x), where G1(x)=ϕ(x)+xψ(x)a n dG2(x)=–ϕ(x). 11.2-2. Equations with Kernel of the Form K(x,t)=ϕ(t)+ψ(t)(t–x). For a degenerate kernel of the above form, the resolvent is determined by the expression R(x,t)=–v/prime/prime tt,( 7) where the auxiliary function vis the solution of the homogeneous linear second-order ordinary differential equation v/prime/prime tt+ϕ(t)v/prime t+ψ(t)v=0 ( 8 ) with the following initial conditions at t=x: v⎝vextendsingle⎝vextendsingle t=x=0 , v/prime t⎝vextendsingle⎝vextendsingle t=x=1 . ( 9 ) The point xoccurs only in the initial data (9) as a parameter, and Eq. (8) itself is independent of x. 542 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) Assume that v1=v1(t) is a nontrivial particular solution of Eq. (8). In this case, the general solution of this differential equation is given by the formula v(t)=C1v1(t)+C2v1(t)⎝integraldisplayt ads Φ(s)[v1(s)]2,Φ(t)=e x p⎝bracketleftBig⎝integraldisplayt aϕ(s)ds⎝bracketrightBig . Taking into account the initial data (9), we ﬁnd the dependence of the integration constants C1 andC2on the parameter x. As a result, we obtain the solution of problem (8), (9): v=v1(x)Φ(x)⎝integraldisplayt xds Φ(s)[v1(s)]2. (10) On substituting the expression (10) into formula (7) and eliminating the second derivative by means of Eq. (8) we ﬁnd the resolvent: R(x,t)=ϕ(t)v1(x)Φ(x) v1(t)Φ(t)+v1(x)Φ(x)[ϕ(t)v/prime t(t)+ψ(t)v1(t)]⎝integraldisplayt xds Φ(s)[v1(s)]2. Remark 2. The kernel of the integral equation under consideration can be rewritten in the form K(x,t)=G1(t)+xG 2(t), where G1(t)=ϕ(t)+tψ(t)a n d G2(t)=–ϕ(t). 11.2-3. Equations with Kernel of the Form K(x,t)=⎝summationtextn m=1ϕm(x)(x–t)m–1. To ﬁnd the resolvent, we introduce an auxiliary function as follows: u(x,t)=1 (n–1 ) !⎝integraldisplayx tR(s,t)(x–s)n–1ds+(x–t)n–1 (n–1 ) !; atx=t, this function vanishes together with the ﬁrst n– 2 derivatives with respect to x,a n dt h e (n– 1)st derivative at x=tis equal to 1. Moreover, R(x,t)=u(n) x(x,t), u(n) x=dnu(x,t) dxn. (11) On substituting relation (11) into the resolvent equation (3) of Subsection 11.1-1, we see that u(n) x(x,t)=K(x,t)+⎝integraldisplayx tK(x,s)u(n) s(s,t)ds. (12) Integrating by parts the right-hand side in (12), we obtain u(n) x(x,t)=K(x,t)+n–1⎝summationdisplay m=0(–1)mK(m) s(x,s)u(n–m–1) s (s,t)⎝vextendsingle⎝vextendsingles=x s=t. (13) On substituting the expressions for K(x,t)a n d u(x,t) into (13), we arrive at a linear homogeneous ordinary differential equation of order nfor the function u(x,t). Thus, the resolvent R(x,t) of the V olterra integral equation with degenerate kernel of the above form can be obtained by means of (11), where u(x,t) satisﬁes the following differential equation and initial conditions: u(n) x–ϕ1(x)u(n–1) x –ϕ2(x)u(n–2) x –2ϕ3(x)u(n–3) x –···–(n–1 ) !ϕn(x)u=0 , u⎝vextendsingle⎝vextendsingle x=t=u/prime x⎝vextendsingle⎝vextendsingle x=t=···=u(n–2) x⎝vextendsingle⎝vextendsingle x=t=0 , u(n–1) x⎝vextendsingle⎝vextendsingle x=t=1 . The parameter toccurs only in the initial conditions, and the equation itself is independent of t explicitly. Remark 3. A kernel of the form K(x,t)=n⎝summationtext m=1φm(x)tm–1can be reduced to a kernel of the above type by elementary transformations. 11.2. E QUATIONS WITH DEGENERATE KERNEL :K(x,t)=g1(x)h1(t)+···+gn(x)hn(t) 543 11.2-4. Equations with Kernel of the Form K(x,t)=⎝summationtextn m=1ϕm(t)(t–x)m–1. Let us represent the resolvent of this degenerate kernel in the form R(x,t)=–v(n) t(x,t), v(n) t=dnv(x,t) dtn, where the auxiliary function v(x,t)v a n i s h e sa t t=xtogether with n– 2 derivatives with respect to t, and the ( n– 1)st derivative with respect to tatt=xis equal to 1. On substituting the expression for the resolvent into Eq. (3) of Subsection 11.1-1, we obtain v(n) t(x,t)=⎝integraldisplayx tK(s,t)v(n) s(x,s)ds–K(x,t). Let us apply integration by parts to the integral on the right-hand side. Taking into account the properties of the auxiliary function v(x,t), we arrive at the following Cauchy problem for an nth-order ordinary differential equation: v(n) t+ϕ1(t)v(n–1) t +ϕ2(t)v(n–2) t +2ϕ3(t)v(n–3) t +···+(n–1 ) !ϕn(t)v=0 , v⎝vextendsingle⎝vextendsingle t=x=v/prime t⎝vextendsingle⎝vextendsingle t=x=···=v(n–2) t⎝vextendsingle⎝vextendsingle t=x=0 , v(n–1) t⎝vextendsingle⎝vextendsingle t=x=1 . The parameter xoccurs only in the initial conditions, and the equation itself is independent of x explicitly. Remark 4. A kernel of the form K(x,t)=n⎝summationtext m=1φm(t)xm–1can be reduced to a kernel of the above type by elementary transformations. 11.2-5. Equations with Degenerate Kernel of the General Form. In this case, the V olterra equation of the second kind can be represented in the form y(x)–n⎝summationdisplay m=1gm(x)⎝integraldisplayx ahm(t)y(t)dt=f(x). (14) Let us introduce the notation wj(x)=⎝integraldisplayx ahj(t)y(t)dt,j=1 ,...,n, (15) and rewrite Eq. (14) as follows: y(x)=n⎝summationdisplay m=1gm(x)wm(x)+f(x). (16) On differentiating the expressions (15) with regard to formula (16), we arrive at the following system of linear differential equations for the functions wj=wj(x): w/prime j=hj(x)⎝bracketleftBign⎝summationdisplay m=1gm(x)wm+f(x)⎝bracketrightBig ,j=1 ,...,n, with the initial conditions wj(a)=0 , j=1 ,...,n. Once the solution of this system is found, the solution of the original integral equation (14) is deﬁned by formula (16) or any of the expressions y(x)=w/prime j(x) hj(x),j=1 ,...,n, which can be obtained from formul a (15) by differentiation. References for Section 11.2: E. Goursat (1923), H. M. M ¨untz (1934), A. F. Verlan’ and V . S. Sizikov (1986), A. D. Polyanin and A. V . Manzhirov (1998). 544 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) 11.3. Equations with Difference Kernel: K(x,t)=K(x–t) 11.3-1. Solution Method Based on the Laplace Transform. V olterra equations of the second kind with kernel depending on the difference of the arguments have the form y(x)–⎝integraldisplayx 0K(x–t)y(t)dt=f(x). (1) Applying the Laplace transform Lto Eq. (1) and taking into account the fact that by the convolution theorem (see Subsection 9.2-4) the integral with kernel depending on the difference of the arguments is transformed into the product ˜K(p)˜y(p), we arrive at the following equation for the transform of the unknown function: ˜y(p)–˜K(p)˜y(p)=˜f(p). (2) The solution of Eq. (2) is given by the formula ˜y(p)=˜f(p) 1–˜K(p),( 3) which can be written equivalently in the form ˜y(p)=˜f(p)+˜R(p)˜f(p), ˜R(p)=˜K(p) 1–˜K(p).( 4) On applying the Laplace inversion formula to (4), we obtain the solution of Eq. (1) in the form y(x)=f(x)+⎝integraldisplayx 0R(x–t)f(t)dt, R(x)=1 2πi⎝integraldisplayc+i∞ c–i∞˜R(p)epxdp.(5) When applying formula (5) in practice, the following two technical problems occur: 1◦. Finding the transform ˜K(p)=⎝integraldisplay∞ 0K(x)e–pxdxfor a given kernel K(x). 2◦. Finding the resolvent (5) whose transform ˜R(p) is given by formula (4). To calculate the corresponding in tegrals, tables of direct and inverse Laplace transforms can be applied (see Supplements 5 and 6), and, in many cases, to ﬁnd the inverse transform, methods of the theory of functions of a complex variable are applied, including the Cauchy residue theorem (seeSubsection 9.1-4). Remark. If the lower limit of the integral in the V olterra equation with kernel depending on the difference of the arguments is equal to a, then this equation can be reduced to Eq. (1) by the change of variables x=¯x–a,t=¯t–a. Figure 3 depicts the principal scheme of solving V olterra integral equations of the second kind with difference kernel by means of the Laplace integral transform. 11.3. E QUATIONS WITH DIFFERENCE KERNEL :K(x,t)=K(x–t) 545 Solution of the equation for the transform Figure 3. Scheme of solving V olterra integral equations of the second kind with difference kernel by means of the Laplace integral transform, R(x) is the inverse transform of the function ˜R(p)=˜K(p) 1–˜K(p). Example 1. Consider the equation y(x)+A⎝integraldisplayx 0sin⎝bracketleftbigλ(x–t)⎝bracketrightbigy(t)dt=f(x), (6) which is a special case of Eq. (1) for K(x)=–Asin(λx). We ﬁrst apply the table of Laplace transforms (see Supplement 5) and obtain the transform of the kernel of the integral equation in the form ˜K(p)=–Aλ p2+λ2. Next, by formula (4) we ﬁnd the transform of the resolvent: ˜R(p)=–Aλ p2+λ(A+λ). Furthermore, applying the table of inverse Laplace transforms (see Supplement 6) we obtain the resolvent: R(x)=⎧ ⎪⎨ ⎪⎩–Aλ ksin(kx)f o r λ(A+λ)>0 , –Aλ ksinh(kx)f o r λ(A+λ)<0 ,where k=|λ(A+λ)|1/2. Moreover, in the special case λ=–A,w eh a v e R(x)=A2x. On substituting the expressions for the resolvent into formula (5), we ﬁnd the solution of the integral equation (6). In particular, for λ(A+λ) > 0, this solution has the form y(x)=f(x)–Aλ k⎝integraldisplayx 0sin⎝bracketleftbigk(x–t)⎝bracketrightbigf(t)dt,k=⎝radicalbig λ(A+λ). (7) The Laplace transformation can also be used for ﬁnding solutions of integro-differential equa- tions with difference kernel. 546 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) Example 2. Consider the Cauchy problem for the integro-differential equation dy dx+⎝integraldisplayx 0K(x–t)y(t)dt=f(x)( 0 ≤x<∞) (8) with the initial condition y=aatx=0 . (9) Let us multiply equation (8) by e–pxand integrate the result with respect to xfrom zero to inﬁnity. Using properties 7 and 12 of the Laplace transform (Table 1, Subsection 9.2-4) and taking into account the initial condition (9), we obtain a linear algebraic equation for the transform ˜ y(p): p˜y(p)–a+˜K(p)˜y(p)=˜f(p). It follows that ˜y(p)=˜f(p)+a p+˜K(p). By the inversion formula (see formula (2) of Subsection 9.2-1), the solution to the original problem (8)–(9) is found in the form y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞˜f(p)+a p+˜K(p)epxdp,i2= –1. (10) Consider the special case of a=0a n d K(x)=c o s ( bx). From row 10 of Table 2 it follows that ˜K(p)=p p2+b2. Rearranging the integrand in (10), we get ˜f(p) p+˜K(p)=p2+b2 p(p2+b2+1 )˜f(p)=⎝parenleftbigg1 p–1 p(p2+b2+1 )⎝parenrightbigg ˜f(p). In order to invert this expression, let us use the convolution theorem (see row 12 of Table 1) as well as formulas 1 and 28 for the inversion of rational functions, Supplement 6.2. As a result, we arrive at the solution in the form y(x)=⎝integraldisplayx 0b2+c o s⎝parenleftbigt√ b2+1⎝parenrightbig b2+1f(x–t)dt. 11.3-2. Method Based on the Solution of an Auxiliary Equation. Consider the integral equation Ay(x)+B⎝integraldisplayx aK(x–t)y(t)dt=f(x). (11) Letw=w(x) be a solution of the simpler auxiliary equation with f(x)≡1a n da =0 , Aw(x)+B⎝integraldisplayx 0K(x–t)w(t)dt= 1. (12) In this case, the solution of the original equation (11) with an arbitrary right-hand side can be expressed via the solution of the auxiliary equation (12) by the formula y(x)=d dx⎝integraldisplayx aw(x–t)f(t)dt=f(a)w(x–a)+⎝integraldisplayx aw(x–t)f/prime t(t)dt. (13) Let us prove this assertion. We rewrite expression (13) (in which we ﬁrst redenote the integration parameter tbys)i n the form y(x)=d dxI(x), I(x)=⎝integraldisplayx aw(x–s)f(s)ds (14) and substitute it into the left-hand side of Eq. (11). After some algebraic manipulations and after changing the order of integration in the double integral with regard to (12), we obtain d dxAI(x)+B⎝integraldisplayx aK(x–t)d dtI(t)dt=d dxAI(x)+d dxB⎝integraldisplayx aK(x–t)I(t)dt =d dx⎝bracketleftBig A⎝integraldisplayx aw(x–s)f(s)ds+B⎝integraldisplayx a⎝integraldisplayt aK(x–t)w(t–s)f(s)dsdt⎝bracketrightBig =d dx⎝braceleftBig⎝integraldisplayx af(s)⎝bracketleftBig Aw(x–s)+B⎝integraldisplayx sK(x–t)w(t–s)dt⎝bracketrightBig ds⎝bracerightBig =d dx⎝braceleftBig⎝integraldisplayx af(s)⎝bracketleftBig Aw(x–s)+B⎝integraldisplayx–s 0K(x–s–λ)w(λ)dλ⎝bracketrightBig ds⎝bracerightBig =d dx⎝integraldisplayx af(s)ds=f(x), which proves the desired assertion. 11.3. E QUATIONS WITH DIFFERENCE KERNEL :K(x,t)=K(x–t) 547 11.3-3. Reduction to Ordinary Differential Equations. Consider the special case in which the transform of the kernel of the integral equation (1) can be expressed in the form 1–˜K(p)=Q(p) R(p), (15) where Q(p)a n dR(p) are polynomials of degree n: Q(p)=n⎝summationdisplay k=0Akpk,R(p)=n⎝summationdisplay k=0Bkpk. (16) In this case, the solution of the integral equation (1) satisﬁes the following linear nonhomogeneous ordinary differential equation of order nwith constant coefﬁcients: n⎝summationdisplay k=0Aky(k) x(x)=n⎝summationdisplay k=0Bkf(k) x(x). (17) Equation (17) can be rewritten in the operator form Q(D)y(x)=R(D)f(x), D≡d dx. The initial conditions for Eq. (17) can be found from the relation n⎝summationdisplay k=0Akk–1⎝summationdisplay s=0pk–1–sy(s) x(0) –n⎝summationdisplay k=0Bkk–1⎝summationdisplay s=0pk–1–sf(s) x(0) = 0 (18) by matching the coefﬁcients of like powers of the parameter p. The proof of this assertion can be performed by applying the Laplace transform to the differential equation (17) and by the subsequent comparison of the resulting expression with Eq. (2) with regard to (15). Another method of reducing an integral equation to an ordinary differential equation is described in Section 11.7. 11.3-4. Reduction to a Wiener–Hopf Equation of the Second Kind. A V olterra equation of the second kind with the difference kernel of the form y(x)+⎝integraldisplayx 0K(x–t)y(t)dt=f(x), 0 < x<∞, (19) can be reduced to the Wiener–Hopf equation y(x)+⎝integraldisplay∞ 0K+(x–t)y(t)dt=f(x), 0 < x<∞, (20) where the kernel K+(x–t)i sg i v e nb y K+(s)=⎝braceleftBigK(s)f o r s>0 , 0f o r s<0 . Methods for studying Eq. (20) are described in Chapter 13, where an example of constructing a solution of a V olterra equation of the second kind with difference kernel by means of con-structing a solution of the corresponding Wiener–Hopf equation of the second kind is presented (see Subsection 13.10-3). 548 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) 11.3-5. Method of Fractional Integration for the Generalized Abel Equation. Consider the generalized Abel equation of the second kind y(x)–λ⎝integraldisplayx ay(t) (x–t)µdt=f(x), x>a, (21) where 0 < µ< 1. Let us assume that x∈[a,b],f(x)∈AC,a n dy(t)∈L1, and apply the technique of the fractional integration (see Section 10.5). We set µ=1–β,0 < β<1 , λ=ν Γ(β), (22) and use formula (8) from Subsection 10.5-1 to rewrite Eq. (21) in the form ⎝parenleftbig 1–ν Iβ a+⎝parenrightbig y(x)=f(x), x>a. (23) Now the solution of the generalized Abel equation of the second kind can be symbolically written as follows: y(x)=⎝parenleftbig 1–νIβ a+⎝parenrightbig–1f(x), x>a. (24) On expanding the operator expression in the parentheses in a series in powers of the operator by means of the formula for a geometric progression, we obtain y(x)=⎝bracketleftbigg 1+∞⎝summationdisplay n=1⎝parenleftbig νIβ a+⎝parenrightbign⎝bracketrightbigg f(x), x>a. (25) Taking into account the relation ( Iβ a+)n=Iβn a+, we can rewrite formula (25) in the expanded form y(x)=f(x)+∞⎝summationdisplay n=1νn Γ(βn)⎝integraldisplayx a(x–t)βn–1f(t)dt,x>a. (26) Let us transpose the integration and summation in the expression (26). Note that ∞⎝summationdisplay n=1νn(x–t)βn–1 Γ(βn)=d dx∞⎝summationdisplay n=1νn(x–t)βn Γ(1 +βn). In this case, taking into account the change of variables (22), we see that a solution of the generalized Abel equation of the second kind becomes y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt,x>a, (27) where the resolvent R(x–t)i sg i v e nb yt h ef o r m u l a R(x–t)=d dx∞⎝summationdisplay n=1⎝bracketleftbig λΓ(1 –µ)(x–t)(1–µ)⎝bracketrightbign Γ[1 + (1 – µ)n]. (28) In some cases, the sum of the series in the representation (28) of the resolvent can be found, and a closed-form expression for this sum can be obtained. 11.4. O PERA TOR METHODS FOR SOLVING LINEAR INTEGRAL EQUATIONS 549 Example 3. Consider the Abel equation of the second kind (we set µ=1 2in Eq. (21)) y(x)–λ⎝integraldisplayx ay(t) √ x–tdt=f(x), x>a. (29) By virtue of formula (28), the resolvent for Eq. (29) is given by the expression R(x–t)=d dx∞⎝summationdisplay n=1⎝bracketleftbig λ√ π(x–t)⎝bracketrightbign Γ⎝parenleftbig1+1 2n⎝parenrightbig. (30) We have∞⎝summationdisplay n=1xn/2 Γ⎝parenleftbig1+1 2n⎝parenrightbig=exerf√ x,e r f x≡2 √ π⎝integraldisplayx 0e–t2dt, (31) where erf xis the error function. By (30) and (31), in this case the expression for the resolvent can be rewritten in the form R(x–t)=d dx⎝braceleftBig exp[λ2π(x–t)] erf⎝bracketleftbigλ⎝radicalbig π(x–t)⎝bracketrightbig⎝bracerightBig . (32) Applying relations (27) and (32), we obtain the solution of the Abel integral equation of the second kind (29) in the form y(x)=f(x)+d dx⎝integraldisplayx a⎝braceleftBig exp[λ2π(x–t)] erf⎝bracketleftbig λ⎝radicalbig π(x–t)⎝bracketrightbig⎝bracerightBig f(t)dt,x>a. (33) Note that in the case under consideration, the solution is constructed in the closed form. 11.3-6. Systems of V olterra Integral Equations. The Laplace transform can be applied to solve systems of V olterra integral equations of the form ym(x)–n⎝summationdisplay k=1⎝integraldisplayx 0Kmk(x–t)yk(t)dt=fm(x), m=1 ,...,n. (34) Let us apply the Laplace transform to system (34). We obtain the relations ˜ym(p)–n⎝summationdisplay k=1˜Kmk(p)˜yk(p)=˜fm(p), m=1 ,...,n. (35) On solving this system of linear algebraic equations, we ﬁnd ˜ ym(p), and the solution of the system under consideration becomes ym(x)=1 2πi⎝integraldisplayc+i∞ c–i∞˜ym(p)epxdp. (36) The Laplace transform can be applied to construct a solution of systems of V olterra equations of the ﬁrst kind and of integro-differential equations as well. References for Section 11.3: V . A. Ditkin and A. P. Prudnikov (1965), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), V . I. Smirnov (1974), K. B. Oldham and J. Spanier (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. D. Gakhov and Yu. I. Cherskii (1978), Yu. I. Babenko (1986), R. Gorenﬂo and S. Vessella (1991), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993). 11.4. Operator Methods for Solving Linear Integral Equations 11.4-1. Application of a Solution of a “Truncated” Equation of the First Kind. Consider the linear equation of the second kind y(x)+L[y]=f(x), (1) where Lis a linear (integral) operator. 550 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) Assume that the solution of the auxiliary “truncated” equation of the ﬁrst kind, L[u]=g(x), (2) can be represented in the form u(x)=M⎝bracketleftbig L[g]⎝bracketrightbig ,( 3) where Mis a known linear operator. Formula (3) means that L–1=ML. Let us apply the operator L–1to Eq. (1). The resulting relation has the form M⎝bracketleftbig L[y]⎝bracketrightbig +y(x)=M⎝bracketleftbig L[f]⎝bracketrightbig .( 4) On eliminating y(x) from (1) and (4) we obtain the equation M[w]–w(x)=F(x), (5) in which the following notation is used: w=L[y],F(x)=M⎝bracketleftbig L[f]⎝bracketrightbig –f(x). In some cases, Eq. (5) is simpler than the original equation (1). For example, this is the case if the operator Mis a constant (see Section 13.8) or a differential operator: M=anDn+an–1Dn–1+···+a1D+a0,D≡d dx. In the latter case, Eq. (5) is an ordinary linear differential equation for the function w. If a solution w=w(x) of Eq. (5) is obtained, then a solution of Eq. (1) is given by the formula y(x)=M⎝bracketleftbig L[w]⎝bracketrightbig . Example 1. Consider the Abel equation of the second kind y(x)+λ⎝integraldisplayx ay(t)dt √ x–t=f(x). (6) To solve this equation, we apply a slight modiﬁcation of the above scheme, which corresponds to the case M≡constd dx. Let us rewrite Eq. (6) as follows:⎝integraldisplayx ay(t)dt √ x–t=f(x)–y(x) λ. (7) Let us assume that the right-hand side of Eq. (7) is known and treat Eq. (7) as an Abel equation of the ﬁrst kind. Its solution can be written in the following form (see Example 3 in Subsection 10.4-4): y(x)=1 πd dx⎝integraldisplayx af(t)–y(t) λ√ x–tdt or y(x)+1 πλd dx⎝integraldisplayx ay(t)dt √ x–tdt=1 πλd dx⎝integraldisplayx af(t)dt √ x–t. (8) Let us differentiate both sides of Eq. (6) with respect to x, multiply Eq. (8) by – πλ2, and add the resulting expressions term by term. We eventually arrive at the following ﬁrst-order linear ordinary differential equation for the function y=y(x): y/prime x–πλ2y=F/prime x(x), (9) where F(x)=f(x)–λ⎝integraldisplayx af(t)dt √ x–t. (10) We must supplement Eq. (9) with initial condition y(a)=f(a), (11) which is a consequence of (6). The solution of problem (9)–(11) has the form y(x)=F(x)+πλ2⎝integraldisplayx aexp[πλ2(x–t)]F(t)dt, (12) and deﬁnes the solution of the Abel equation of the second kind (6). 11.4. O PERA TOR METHODS FOR SOLVING LINEAR INTEGRAL EQUATIONS 551 11.4-2. Application of the Auxiliary Equation of the Second Kind. The solution of the Abel equation of the second kind (6) can also be obtained by another method, presented below. Consider the linear equation y(x)–L[y]=f(x), (13) where Lis a linear operator. Assume that the solution of the auxiliary equation w(x)–Ln[w]=Φ(x), Ln[w]≡L⎝bracketleftbig Ln–1[w]⎝bracketrightbig , (14) which involves the nth power of the operator L, is known and is deﬁned by the formula w(x)=M[Φ(x)]. (15) In this case, the solution of the original equation (13) has the form y(x)=M[Φ(x)], Φ(x)=Ln–1[f]+Ln–2[f]+···+L[f]+f(x). (16) This assertion can be proved by applying the operator Ln–1+Ln–2+···+L+ 1 to Eq. (13), with regard to the operator relation ⎝parenleftbig 1–L⎝parenrightbig⎝parenleftbig Ln–1+Ln–2+···+L+1⎝parenrightbig =1– Ln together with formula (16) for Φ(x). In Eq. (14) we may write y(x) instead of w(x). Example 2. Let us apply the operator method (for n= 2) to solve the generalized Abel equation with exponent 3 /4: y(x)–b⎝integraldisplayx 0y(t)dt (x–t)3/4=f(x). (17) We ﬁrst consider the integral operator with difference kernel L[y(x)]≡⎝integraldisplayx 0K(x–t)y(t)dt. Let us ﬁnd L2: L2[y]≡L⎝bracketleftbig L[y]⎝bracketrightbig =⎝integraldisplayx 0⎝integraldisplayt 0K(x–t)K(t–s)y(s)dsdt =⎝integraldisplayx 0y(s)ds⎝integraldisplayx sK(x–t)K(t–s)dt=⎝integraldisplayx 0K2(x–s)y(s)ds, K2(z)=⎝integraldisplayz 0K(ξ)K(z–ξ)dξ.(18) In the proof of this formula, we have reversed the order of integration and performed the change of variables ξ=t–s. For the power-law kernel K(ξ)=bξµ, we have K2(z)=b2Γ2(1 +µ) Γ(2 + 2µ)z1+2µ. (19) For Eq. (17) we obtain µ=–3 4,K2(z)=A1 √ z,A=b2 √ πΓ2(1 4). Therefore, the auxiliary equation (14) corresponding to n= 2 has the form y(x)–A⎝integraldisplayx 0y(t)dt √ x–t=Φ(x), (20) where Φ(x)=f(x)+b⎝integraldisplayx 0f(t)dt (x–t)3/4. After the substitution A→–λandΦ→f, relation (20) coincides with Eq. (6), and the solution of Eq. (20) can be obtained by formula (12). 552 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) Remark. It follows from (19) that the solution of the generalized Abel equation with exponent β y(x)+λ⎝integraldisplayx 0y(t)dt (x–t)β=f(x) can be reduced to the solution of a similar equation with the different exponent β1=2β–1 . I n particular, the Abel equation (6), which corresponds to β=1 2, is reduced to the solution of an equation with degenerate kernel for β1=0 . 11.4-3. Method for Solving “Quadratic” Operator Equations. Suppose that the solution of the linear (integral, differential, etc.) equation y(x)–λL[y]=f(x) (21) is known for an arbitrary right-hand side f(x)a n df o ra n y λfrom the interval ( λmin,λmax). We denote this solution by y=Y(f,λ). (22) Let us construct the solution of the more complicated equation y(x)–aL[y]–bL2[y]=f(x), (23) where aandbare some numbers and f(x) is an arbitrary function. To this end, we represent the left-hand side of Eq. (23) by the product of operators ⎝parenleftbig 1–a L–bL2⎝parenrightbig [y]≡⎝parenleftbig 1–λ 1L⎝parenrightbig⎝parenleftbig 1–λ 2L⎝parenrightbig [y], (24) where λ1andλ2are the roots of the quadratic equation λ2–aλ–b= 0. (25) We assume that λmin<λ1,λ2<λmax. Let us solve the auxiliary equation w(x)–λ2L[w]=f(x), (26) which is the special case of Eq. (21) for λ=λ2. The solution of this equation is given by the formula w(x)=Y(f,λ2). (27) Taking into account (24) and (26), we can rewrite Eq. (23) in the form ⎝parenleftbig 1–λ 1L⎝parenrightbig⎝parenleftbig 1–λ2L⎝parenrightbig [y]=⎝parenleftbig 1–λ 2L⎝parenrightbig [w], or, in view of the identity (1 – λ1L)(1 –λ2L)≡(1 –λ2L)(1 –λ1L), in the form ⎝parenleftbig 1–λ 2L⎝parenrightbig⎝braceleftBig⎝parenleftbig 1–λ 1L⎝parenrightbig [y]–w(x)⎝bracerightBig =0 . This relation holds if the unknown function y(x) satisﬁes the equation y(x)–λ1L[y]=w(x). (28) The solution of this equation is given by the formula y(x)=Y(w,λ1), where w=Y(f,λ2). (29) 11.4. O PERA TOR METHODS FOR SOLVING LINEAR INTEGRAL EQUATIONS 553 If the homogeneous equation y(x)–λ2L[y] = 0 has only the trivial* solution y≡0, then formula (29) deﬁnes the unique solution of the original equation (23). Example 3. Consider the integral equation y(x)–⎝integraldisplayx 0⎝parenleftBigA √ x–t+B⎝parenrightBig y(t)dt=f(x). It follows from the results of Example 2 that this equation can be written in the form of Eq. (23): y(x)–AL[y]–1 πBL2[y]=f(x), L[y]≡⎝integraldisplayx 0y(t)dt √ x–t. Therefore, the solution (in the form of antiderivatives) of the integral equation can be given by the formulas y(x)=Y(w,λ1),w=Y(f,λ2), Y(f,λ)=F(x)+πλ2⎝integraldisplayx 0exp⎝bracketleftbigπλ2(x–t)⎝bracketrightbigF(t)dt,F(x)=f(x)+λ⎝integraldisplayx 0f(t)dt √ x–t, where λ1andλ2are the roots of the quadratic equation λ2–Aλ–1 πB=0 . This method can also be applied to solve (in the form of antiderivatives) more general equations of the form y(x)–⎝integraldisplayx 0⎝bracketleftBigA (x–t)β+B (x–t)2β–1⎝bracketrightBig y(t)dt=f(x), where βis a rational number satisfying the condition 0 < β< 1 (see Example 2 and Eq. 2.1.60 from the ﬁrst part of the book). 11.4-4. Solution of Operator Equations of Polynomial Form. The method described in Subsection 11.4-3 can be generalized to the case of operator equations of polynomial form. Suppose that the solution of the linear nonhomogeneous equation (21) is given by formula (22) and that the corresponding homogeneous equation has only the trivial solution. Let us construct the solution of the more complicated equation with polynomial left-hand side with respect to the operator L: y(x)–n⎝summationdisplay k=1AkLk[y]=f(x), Lk≡L⎝parenleftbig Lk–1⎝parenrightbig , (30) where Akare some numbers and f(x)i sa na r b i t r a r yf u n c t i o n . We denote by λ1,...,λnthe roots of the characteristic equation λn–n⎝summationdisplay k=1Akλn–k= 0. (31) The left-hand side of Eq. (30) can be expressed in the form of a product of operators: y(x)–n⎝summationdisplay k=1AkLk[y]≡n⎝productdisplay k=1⎝parenleftbig 1–λ kL⎝parenrightbig [y]. (32) The solution of the auxiliary equation (26), in which we use the substitution w→yn–1andλ2→λn, is given by the formula yn–1(x)=Y(f,λn). Reasoning similar to that in Subsection 11.4-3 shows that the solution of Eq. (30) is reduced to the solution of the simpler equation n–1⎝productdisplay k=1⎝parenleftbig 1–λ kL⎝parenrightbig [y]=yn–1(x), (33) * If the homogeneous equation y(x)–λ2L[y] = 0 has nontrivial solutions, then the right-hand side of Eq. (28) must contain the function w(x)+y0(x) instead of w(x), where y0is the general solution of the homogeneous equation. 554 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) whose degree is less by one than that of the original equation with respect to the operator L. We can show in a similar way that Eq. (33) can be reduced to the solution of the simpler equation n–2⎝productdisplay k=1⎝parenleftbig 1–λkL⎝parenrightbig [y]=yn–2(x), yn–2(x)=Y(yn–1,λn–1). Successively reducing the order of the equation, we eventually arrive at an equation of the form (28) whose right-hand side contains the function y1(x)=Y(y2,λ2). The solution of this equation is given by the formula y(x)=Y(y1,λ1). The solution of the original equation (30) is deﬁned recursively by the following formulas: yk–1(x)=Y(yk,λk);k=n,...,1 , w h e r e yn(x)≡f(x),y0(x)≡y(x). Note that here the decreasing sequence k=n,...,1i su s e d . 11.4-5. Some Generalizations. Suppose that the left-hand side of a linear (integral) equation y(x)–Q[y]=f(x) (34) can be represented in the form of a product y(x)–Q[y]≡n⎝productdisplay k=1⎝parenleftbig 1–L k⎝parenrightbig [y], (35) where the Lkare linear operators. Suppose that the solutions of the auxiliary equations y(x)–Lk[y]=f(x), k=1 ,...,n (36) are known and are given by the formulas y(x)=Yk⎝bracketleftbig f(x)⎝bracketrightbig ,k=1 ,...,n. (37) The solution of the auxiliary equation (36) for k=n, in which we apply the substitution y→yn–1, is given by the formula yn–1(x)=Yn⎝bracketleftbig f(x)⎝bracketrightbig . Reasoning similar to that used in Subsection 11.4-3 shows that the solution of Eq. (34) can be reduced to the solution of the simpler equation n–1⎝productdisplay k=1⎝parenleftbig 1–L k⎝parenrightbig [y]=yn–1(x). Successively reducing the order of the equation, we eventually arrive at an equation of the form (36) fork= 1, whose right-hand side contains the function y1(x)=Y2⎝bracketleftbig y2(x)⎝bracketrightbig . The solution of this equation is given by the formula y(x)=Y1⎝bracketleftbig y1(x)⎝bracketrightbig . The solution of the original equation (35) can be deﬁned recursively by the following formulas: yk–1(x)=Yk⎝bracketleftbig yk(x)⎝bracketrightbig ;k=n,...,1 , w h e r e yn(x)≡f(x),y0(x)≡y(x). Note that here the decreasing sequence k=n,...,1i su s e d . Reference for Section 11.4: A. D. Polyanin and A. V . Manzhirov (1998). 11.5. C ONSTRUCTION OF SOLUTIONS OF INTEGRAL EQUATIONS WITH SPECIAL RIGHT-HAND SIDE 555 11.5. Construction of Solutions of Integral Equations with Special Right-Hand Side In this section we describe some approaches to the construction of solutions of integral equations with special right-hand side. These approaches are based on the application of auxiliary solutions that depend on a free parameter. 11.5-1. General Scheme. Consider a linear equation, which we shall write in the following brief form: L[y]=fg(x,λ), (1) where Lis a linear operator (integral, differential, etc.) that acts with respect to the variable xand is independent of the parameter λ,a n dfg(x,λ) is a given function that depends on the variable xand the parameter λ. Suppose that the solution of Eq. (1) is known: y=y(x,λ). (2) Let Mbe a linear operator (integral, differential, etc.) that acts with respect to the parameter λ and is independent of the variable x. Consider the (usual) case in which Mcommutes with L.W e apply the operator Mto Eq. (1) and ﬁnd that the equation L[w]=fM(x), fM(x)=M⎝bracketleftbig fg(x,λ)⎝bracketrightbig ,( 3 ) has the solution w=M⎝bracketleftbig y(x,λ)⎝bracketrightbig .( 4) By choosing the operator Min a different way, we can obtain solutions for other right-hand sides of Eq. (1). The original function fg(x,λ) is called the generating function for the operator L. 11.5-2. Generating Function of Exponential Form. Consider a linear equation with exponential right-hand side L[y]=eλx.( 5) Suppose that the solution is known and is given by formula (2). In Table 6 we present solutions of the equation L[y]=f(x) with various right-hand sides; these solutions are expressed via the solution of Eq. (5). Remark 1. When applying the formulas indicated in the table, we need not know the left-hand side of the linear equation (5) (the equation can be integral,differential, etc.) provided that a particular solution of this equation for exponential right-hand side is known. It is only of importance that the left-hand side of the equation is independent of the parameter λ. Remark 2. When applying formulas indicated in the table, the convergence of the integrals occurring in the resulting solution must be veriﬁed. 556 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) TABLE 6 Solutions of the equation L[y]=f(x) with generating function of the exponential form No Right-Hand Side f(x) Solution y Solution Method 1 eλx y(x,λ) Original Equation 2 A1eλ1x+···+Aneλnx A1y(x,λ1)+···+Any(x,λn) Follows from linearity 3 Ax+B A∂ ∂λ⎝bracketleftBig y(x,λ)⎝bracketrightBig λ=0+By(x,0 ) Follows from linearity and the results of row No 4 4 Axn, n=0 ,1 ,2 , ... A⎝braceleftBig∂n ∂λn⎝bracketleftBig y(x,λ)⎝bracketrightBig⎝bracerightBig λ=0 Follows from the results of row No 6 for λ=0 5 A x+a,a>0 A⎝integraldisplay∞ 0e–aλy(x,–λ)dλ Integration with respect to the parameter λ 6 Axneλx, n=0 ,1 ,2 , ... A∂n ∂λn⎝bracketleftBig y(x,λ)⎝bracketrightBig Differentiation with respect to the parameter λ 7 ax y(x,l na) Follows from row No 1 8 Acosh(λx) 1 2A[y(x,λ)+y(x,–λ)⎝bracketrightbig Linearity and relations to the exponential 9 Asinh(λx) 1 2A[y(x,λ)–y(x,–λ)⎝bracketrightbig Linearity and relations to the exponential 10 Axmcosh(λx), m=1 ,3 ,5 , ... 1 2A∂m ∂λm[y(x,λ)–y(x,–λ)⎝bracketrightbig Differentiation with respect toλand relation to the exponential 11 Axmcosh(λx), m=2 ,4 ,6 , ... 1 2A∂m ∂λm[y(x,λ)+y(x,–λ)⎝bracketrightbig Differentiation with respect toλand relation to the exponential 12 Axmsinh(λx), m=1 ,3 ,5 , ... 1 2A∂m ∂λm[y(x,λ)+y(x,–λ)⎝bracketrightbig Differentiation with respect toλand relation to the exponential 13 Axmsinh(λx), m=2 ,4 ,6 , ... 1 2A∂m ∂λm[y(x,λ)–y(x,–λ)⎝bracketrightbig Differentiation with respect toλand relation to the exponential 14 Acos(βx) ARe⎝bracketleftbig y(x,iβ)⎝bracketrightbig Selection of the real part for λ=iβ 15 Asin(βx) AIm⎝bracketleftbig y(x,iβ)⎝bracketrightbig Selection of the imaginary part for λ=iβ 16 Axncos(βx), n=1 ,2 ,3 , ... ARe⎝braceleftBig∂n ∂λn⎝bracketleftBig y(x,λ)⎝bracketrightBig⎝bracerightBig λ=iβ Differentiation with respect toλand selection of the real part for λ=iβ 17 Axnsin(βx), n=1 ,2 ,3 , ... AIm⎝braceleftBig∂n ∂λn⎝bracketleftBig y(x,λ)⎝bracketrightBig⎝bracerightBig λ=iβ Differentiation with respect toλand selection of the imaginary part for λ=iβ 18 Aeµxcos(βx) ARe⎝bracketleftbig y(x,µ+iβ)⎝bracketrightbig Selection of the real part for λ=µ+iβ 19 Aeµxsin(βx) AIm⎝bracketleftbig y(x,µ+iβ)⎝bracketrightbig Selection of the imaginary part for λ=µ+iβ 20 Axneµxcos(βx), n=1 ,2 ,3 , ... ARe⎝braceleftBig∂n ∂λn⎝bracketleftBig y(x,λ)⎝bracketrightBig⎝bracerightBig λ=µ+iβ Differentiation with respect toλand selection of the real part for λ=µ+iβ 21 Axneµxsin(βx), n=1 ,2 ,3 , ... AIm⎝braceleftBig∂n ∂λn⎝bracketleftBig y(x,λ)⎝bracketrightBig⎝bracerightBig λ=µ+iβ Differentiation with respect toλand selection of the imaginary part for λ=µ+iβ 11.5. C ONSTRUCTION OF SOLUTIONS OF INTEGRAL EQUATIONS WITH SPECIAL RIGHT-HAND SIDE 557 Example 1. We seek a solution of the equation with exponential right-hand side y(x)+⎝integraldisplay∞ xK(x–t)y(t)dt=eλx(6) in the form y(x,λ)=keλxby the method of indeterminate coefﬁcients. Then we obtain y(x,λ)=1 B(λ)eλx,B(λ)=1+⎝integraldisplay∞ 0K(–z)eλzdz. (7) It follows from row 3 of Table 6 that the solution of the equation y(x)+⎝integraldisplay∞ xK(x–t)y(t)dt=Ax (8) has the form y(x)=A Dx–AC D2, where D=1+⎝integraldisplay∞ 0K(–z)dz,C=⎝integraldisplay∞ 0zK(–z)dz. For such a solution to exist, it is necessary that the improper integrals of the functions K(–z)a n dzK(–z) exist. This holds if the function K(–z) decreases more rapidly than z–2asz→∞ . Otherwise a solution can be nonexistent. It is of interest that for functions K(–z) with power-law growth as z→∞ in the case λ< 0, the solution of Eq. (6) exists and is given by formula (7), whereas Eq. (8) does not have a solution. Therefore, we must be careful when using formulas fromTable 6 and verify the convergence of the integrals occurring in the solution. It follows from row 15 of Table 6 that the solution of the equation y(x)+⎝integraldisplay ∞ xK(x–t)y(t)dt=Asin(λx) (9) is given by the formula y(x)=A B2c+B2s⎝bracketleftbigBcsin(λx)–Bscos(λx)⎝bracketrightbig, where Bc=1+⎝integraldisplay∞ 0K(–z)c o s (λz)dz,Bs=⎝integraldisplay∞ 0K(–z)s i n (λz)dz. 11.5-3. Power-Law Generating Function. Consider the linear equation with power-law right-hand side L[y]=xλ. (10) Suppose that the solution is known and is given by formula (2). In Table 7, solutions of the equation L[y]=f(x) with various right-hand sides are presented which can be expressed via the solution of Eq. (10). Example 2. We seek a solution of the equation with power-law right-hand side y(x)+⎝integraldisplayx 01 xK⎝parenleftBigt x⎝parenrightBig y(t)dt=xλ in the form y(x,λ)=kxλby the method of indeterminate coefﬁcients. We ﬁnally obtain y(x,λ)=1 1+B(λ)xλ,B(λ)=⎝integraldisplay1 0K(t)tλdt. 558 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) It follows from row 3 of Table 7 that the solu tion of the equation with logarithmic right-hand side y(x)+⎝integraldisplayx 01 xK⎝parenleftBigt x⎝parenrightBig y(t)dt=Alnx has the form y(x)=A 1+I0lnx–AI1 (1 +I0)2, where I0=⎝integraldisplay1 0K(t)dt,I1=⎝integraldisplay1 0K(t)l ntd t. TABLE 7 Solutions of the equation L[y]=f(x) with generating function of power-law form No Right-Hand Side f(x) Solution y Solution Method 1 xλ y(x,λ) Original Equation 2 n⎝summationtext k=0Akxk n⎝summationtext k=0Aky(x,k) Follows from linearity 3 Alnx+B A∂ ∂λ⎝bracketleftBig y(x,λ)⎝bracketrightBig λ=0+By(x,0 ) Follows from linearity and from the results of row No 4 4 Alnnx, n=0 ,1 ,2 , ... A⎝braceleftBig∂n ∂λn⎝bracketleftBig y(x,λ)⎝bracketrightBig⎝bracerightBig λ=0 Follows from the results of row No 5 for λ=0 5 Axλlnnx, n=0 ,1 ,2 , ... A∂n ∂λn⎝bracketleftBig y(x,λ)⎝bracketrightBig Differentiation with respect to the parameter λ 6 Acos(βlnx) ARe⎝bracketleftbig y(x,iβ)⎝bracketrightbig Selection of the real part for λ=iβ 7 Asin(βlnx) AIm⎝bracketleftbig y(x,iβ)⎝bracketrightbig Selection of the imaginary part for λ=iβ 8 Axµcos(βlnx) ARe⎝bracketleftbig y(x,µ+iβ)⎝bracketrightbig Selection of the real part for λ=µ+iβ 9 Axµsin(βlnx) AIm⎝bracketleftbig y(x,µ+iβ)⎝bracketrightbig Selection of the imaginary part for λ=µ+iβ 11.5-4. Generating Function Containing Sines and Cosines. Consider the linear equation L[y]=s i n ( λx). (11) We assume that the solution of this equation is known and is given by formula (2). In Table 8, solutions of the equation L[y]=f(x) with various right-hand sides are given, which are expressed via the solution of Eq. (11). Consider the linear equation L[y]=c o s ( λx). (12) We assume that the solution of this equation is known and is given by formula (2). In Table 9, solutions of the equation L[y]=f(x) with various right-hand sides are given, which are expressed via the solution of Eq. (12). 11.6. M ETHOD OF MODEL SOLUTIONS 559 TABLE 8 Solutions of the equation L[y]=f(x) with sine-shaped generating function No Right-Hand Side f(x) Solution y Solution Method 1 sin(λx) y(x,λ) Original Equation 2 n⎝summationtext k=1Aksin(λkx) n⎝summationtext k=1Aky(x,λk) Follows from linearity 3 Axm, m=1 ,3 ,5 , ... A(–1)m–1 2⎝bracketleftBig∂m ∂λmy(x,λ)⎝bracketrightBig λ=0 Follows from the results of row 5 for λ=0 4 Axmsin(λx), m=2 ,4 ,6 , ... A(–1)m 2∂m ∂λmy(x,λ) Differentiation with respect to the parameter λ 5 Axmcos(λx), m=1 ,3 ,5 , ... A(–1)m–1 2∂m ∂λmy(x,λ) Differentiation with respect to the parameter λ 6 sinh(βx) –iy(x,iβ) Relation to the hyperbolic sine,λ=iβ 7 xmsinh(βx), m=2 ,4 ,6 , ... i(–1)m+2 2⎝bracketleftBig∂m ∂λmy(x,λ)⎝bracketrightBig λ=iβ Differentiation with respect toλand relation to the hyperbolic sine, λ=iβ TABLE 9 Solutions of the equation L[y]=f(x) with cosine-shaped generating function No Right-Hand Side f(x) Solution y Solution Method 1 cos(λx) y(x,λ) Original Equation 2 n⎝summationtext k=1Akcos(λkx) n⎝summationtext k=1Aky(x,λk) Follows from linearity 3 Axm, m=0 ,2 ,4 , ... A(–1)m 2⎝bracketleftBig∂m ∂λmy(x,λ)⎝bracketrightBig λ=0 Follows from the results of row 4 for λ=0 4 Axmcos(λx), m=2 ,4 ,6 , ... A(–1)m 2∂m ∂λmy(x,λ) Differentiation with respect to the parameter λ 5 Axmsin(λx), m=1 ,3 ,5 , ... A(–1)m+1 2∂m ∂λmy(x,λ) Differentiation with respect to the parameter λ 6 cosh(βx) y(x,iβ) Relation to the hyperbolic cosine, λ=iβ 7 xmcosh(βx), m=2 ,4 ,6 , ... (–1)m 2⎝bracketleftBig∂m ∂λmy(x,λ)⎝bracketrightBig λ=iβ Differentiation with respect toλand relation to the hyperbolic cosine, λ=iβ 11.6. Method of Model Solutions 11.6-1. Preliminary Remarks∗. Consider a linear equation, which we brieﬂy write out in the form L[y(x)] =f(x), (1) where Lis a linear (integral) operator, y(x) is an unknown function, and f(x) is a known function. We ﬁrst deﬁne arbitrarily a test solution y0=y0(x,λ), (2) * Before reading this section, it is useful to look over Section 11.5. 560 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) which depends on an auxiliary parameter λ(it is assumed that the operator Lis independent of λ andy0/ ≡const). By means of Eq. (1) we deﬁne the right-hand side that corresponds to the test solution (2): f0(x,λ)=L[y0(x,λ)]. Let us multiply Eq. (1), for y=y0andf=f0, by some function ϕ(λ) and integrate the resulting relation with respect to λover an interval [ a,b]. We ﬁnally obtain L[yϕ(x)] =fϕ(x), (3) where yϕ(x)=⎝integraldisplayb ay0(x,λ)ϕ(λ)dλ,fϕ(x)=⎝integraldisplayb af0(x,λ)ϕ(λ)dλ.( 4 ) It follows from formulas (3) and (4) that, for the right-hand side f=fϕ(x), the function y=yϕ(x) is a solution of the original equation (1). Since the choice of the function ϕ(λ) (as well as of the integration interval) is arbitrary, the function fϕ(x) can be arbitrary in principle. Here the main problem is how to choose a function ϕ(λ) to obtain a given function fϕ(x). This problem can be solved if we can ﬁnd a test solution such that the right-hand side of Eq. (1) is the kernel of a known inverse integral transform (we denote such a test solution by Y(x,λ) and call it a model solution ). 11.6-2. Description of the Method. Indeed, let Pbe an invertible integral transform that takes each function f(x) to the corresponding transform F(λ)b yt h er u l e F(λ)=P{f(x)}.( 5) Assume that the inverse transform P–1has the kernel ψ(x,λ) and acts as follows: P–1{F(λ)}=f(x), P–1{F(λ)}≡⎝integraldisplayb aF(λ)ψ(x,λ)dλ.( 6) The limits of integration aandband the integration path in (6) may well lie in the complex plane. Suppose that we succeeded in ﬁnding a model solution Y(x,λ) of the auxiliary problem for Eq. (1) whose right-hand side is the kernel of the inverse transform P–1: L[Y(x,λ)] =ψ(x,λ). (7) Let us multiply Eq. (7) by F(λ) and integrate with respect to λwithin the same limits that stand in the inverse transform (6). Taking into account the fact that the operator Lis independent of λand applying the relation P–1{F(λ)}=f(x), we obtain L⎝bracketleftBig⎝integraldisplayb aY(x,λ)F(λ)dλ⎝bracketrightBig =f(x). Therefore, the solution of Eq. (1) for an arbitrary function f(x) on the right-hand side is expressed via a solution of the simpler auxiliary equation (7) by the formula y(x)=⎝integraldisplayb aY(x,λ)F(λ)dλ,( 8 ) where F(λ) is the transform (5) of the function f(x). For the right-hand side of the auxiliary equation (7) we can take, for instance, exponential, power- law, and trigonometric function, which are the kernels of the Laplace, Mellin, and sine and cosine Fourier transforms (up to a constant factor). Sometimes it is rather easy to ﬁnd a model solution by means of the method of indeterminate coefﬁcients (by prescribing its structure). Afterwards, toconstruct a solution of the equation with arbitrary right-hand side, we can apply formulas written out below in Subsections 11.6-3–11.6-6. 11.6. M ETHOD OF MODEL SOLUTIONS 561 11.6-3. Model Solution in the Case of an Exponential Right-Hand Side. Assume that we have found a model solution Y=Y(x,λ) that corresponds to the exponential right-hand side: L[Y(x,λ)] =eλx.( 9) Consider two cases: 1◦.Equations on the semiaxis, 0≤x<∞.Let ˜f(p) be the Laplace transform of the function f(x): ˜f(p)=L{f(x)}, L{f(x)}≡⎝integraldisplay∞ 0f(x)e–pxdx. (10) The solution of Eq. (1) for an arbitrary right-hand side f(x) can be expressed via the solution of the simpler auxiliary equation with exponential right-hand side (9) for λ=pby the formula y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞Y(x,p)˜f(p)dp. (11) 2◦.Equations on the entire axis, –∞<x<∞. Let ˜f(u) denote the Fourier transform of the function f(x): ˜f(u)=F{f(x)}, F{f(x)}≡1 √ 2π⎝integraldisplay∞ –∞f(x)e–iuxdx. (12) The solution of Eq. (1) for an arbitrary right-hand side f(x) can be expressed via the solution of the simpler auxiliary equation with exponential right-hand side (9) for λ=iuby the formula y(x)=1 √ 2π⎝integraldisplay∞ –∞Y(x,iu)˜f(u)du. (13) In the calculation of the integrals on the right-hand sides in (11) and (13), methods of the theory of functions of a complex variable are applied, including the Cauchy residue theorem and the Jordan lemma (see Subsections 9.1-4 and 9.1-5). Remark 1. The structure of a model solution Y(x,λ) can differ from that of the kernel of the Laplace or Fourier inversion formula. Remark 2. When applying the method under consideration, the left-hand side of Eq. (1) need not be known (the equation can be integral, differential, functional, etc.) if a particular solution of this equation is known for the exponential right-hand side. Here only the most general information isimportant, namely, that the equation is linear, and its left-hand side is independent of the parameter λ. Remark 3. The above method can be used in the solution of linear integral (differential, integro- differential, and functional) equations with composed argument of the unknown function. Example 1. Consider the following V olterra equation of the second kind with difference kernel: y(x)+⎝integraldisplay∞ xK(x–t)y(t)dt=f(x). (14) This equation cannot be solved by direct application of the Laplace transform because the convolution theorem cannot be used here. In accordance with the method of model solutions, we consider the auxiliary equation with exponential right-hand side y(x)+⎝integraldisplay∞ xK(x–t)y(t)dt=epx. (15) Its solution has the form (see Example 1 of Section 11.5) Y(x,p)=1 1+˜K(–p)epx, ˜K(–p)=⎝integraldisplay∞ 0K(–z)epzdz. (16) This, by means of formula (11), yields a solution of Eq. (14) for an arbitrary right-hand side, y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞˜f(p) 1+˜K(–p)epxdp, (17) where ˜f(p) is the Laplace transform (10) of the function f(x) (see also Section 11.11). Note that a solution to Eq. (12) was obtained in the book of M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971) in a more complicated way. 562 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) 11.6-4. Model Solution in the Case of a Power-Law Right-Hand Side. Suppose that we have succeeded in ﬁnding a model solution Y=Y(x,s) that corresponds to a power-law right-hand side of the equation: L[Y(x,s)] =x–s, λ=–s. (18) Let ˆf(s) be the Mellin transform of the function f(x): ˆf(s)=M{f(x)}, M{f(x)}≡⎝integraldisplay∞ 0f(x)xs–1dx. (19) The solution of Eq. (1) for an arbitrary right-hand side f(x) can be expressed via the solution of the simpler auxiliary equation with power-law right-hand side (18) by the formula y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞Y(x,s)ˆf(s)ds. (20) In the calculation of the corresponding integrals on the right-hand side of formula (20), one can use tables of inverse Mellin transforms (e.g., see Supplement 10), as well as methods of the theory of functions of a complex variable, including the Cauchy residue theorem and the Jordan lemma (see Subsections 9.1-4 and 9.1-5). Example 2. Consider the equation y(x)+⎝integraldisplayx 01 xK⎝parenleftBigt x⎝parenrightBig y(t)dt=f(x). (21) In accordance with the method of model solutions, we consider the following auxiliary equation with power-law right-hand side: y(x)+⎝integraldisplayx 01 xK⎝parenleftBigt x⎝parenrightBig y(t)dt=x–s. (22) Its solution has the form (see Example 2 for λ=–sin Section 11.5) Y(x,s)=1 1+B(s)x–s,B(s)=⎝integraldisplay1 0K(t)t–sdt. (23) This, by means of formula (20), yields the solution of Eq. (21) for an arbitrary right-hand side: y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞ˆf(s) 1+B(s)x–sds, (24) where ˆf(s) is the Mellin transform (19) of the function f(x). 11.6-5. Model Solution in the Case of a Sine-Shaped Right-Hand Side. Suppose that we have succeeded in ﬁnding a model solution Y=Y(x,u) that corresponds to the sine on the right-hand side: L[Y(x,u)] = sin( ux), λ=u. (25) Let ˇfs(u) be the asymmetric sine Fourier transform of the function f(x): ˇfs(u)=Fs{f(x)},Fs{f(x)}≡⎝integraldisplay∞ 0f(x)s i n (ux)dx. (26) The solution of Eq. (1) for an arbitrary right-hand side f(x) can be expressed via the solution of the simpler auxiliary equation with sine-shape right-hand side (25) by the formula y(x)=2 π⎝integraldisplay∞ 0Y(x,u)ˇfs(u)du. (27) 11.6. M ETHOD OF MODEL SOLUTIONS 563 11.6-6. Model Solution in the Case of a Cosine-Shaped Right-Hand Side. Suppose that we have succeeded in ﬁnding a model solution Y=Y(x,u) that corresponds to the cosine on the right-hand side: L[Y(x,u)] = cos( ux), λ=u. (28) Let ˇfc(u) be the asymmetric Fourier cosine transform of the function f(x): ˇfc(u)=Fc{f(x)},Fc{f(x)}≡⎝integraldisplay∞ 0f(x)c o s (ux)dx. (29) The solution of Eq. (1) for an arbitrary right-hand side f(x) can be expressed via the solution of the simpler auxiliary equation with cosine right-hand side (28) by the formula y(x)=2 π⎝integraldisplay∞ 0Y(x,u)ˇfc(u)du. (30) 11.6-7. Some Generalizations. Just as above we assume that Pis an invertible transform taking each function f(x) to the corre- sponding transform F(λ) by the rule (5) and that the inverse transform is deﬁned by formula (6). Suppose that we have succeeded in ﬁnding a model solution Y(x,λ) of the following auxiliary problem for Eq. (1): Lx[Y(x,λ)] = Hλ[ψ(x,λ)]. (31) The right-hand side of Eq. (31) contains an invertible linear operator (which is integral, differential, or functional) that is independent of the variable xand acts with respect to the parameter λon the kernel ψ(x,λ) of the inverse transform, see formula (6). For clarity, the operator on the left-hand side of Eq. (31) is labeled by the subscript x(it acts with respect to the variable xand is independent ofλ). Let us apply the inverse operator H–1 λto Eq. (31). As a result, we obtain the kernel ψ(x,λ)o n the right-hand side. On the left-hand side we intertwine the operators by the rule H–1 λLx=LxH–1 λ (this is as a rule possible because the operators act with respect to different variables). Furthermore, let us multiply the resulting relation by F(λ) and integrate with respect to λwithin the limits that stand in the inverse transform (6). Taking into account the relation P–1{F(λ)}=f(x), we ﬁnally obtain Lx⎝bracketleftBig⎝integraldisplayb aF(λ)H–1 λ[Y(x,λ)]dλ⎝bracketrightBig =f(x). (32) Hence, a solution of Eq. (1) with an arbitrary function f(x) on the right-hand side can be expressed via the solution of the simpler auxiliary equation (31) by the formula y(x)=⎝integraldisplayb aF(λ)H–1 λ[Y(x,λ)]dλ, (33) where F(λ) is the transform of the function f(x) obtained by means of the transform P(5). Since the choice of the operator Hλis arbitrary, this approach extends the abilities of the method of model solutions. References for Section 11.6: A. D. Polyanin and A. V . Manzhirov (1997, 1998). 564 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) 11.7. Method of Differentiation for Integral Equations In some cases, the differentiation of integral equations (once, twice, and so on) with the subsequent elimination of integral terms by means of the original equation makes it possible to reduce a given equation to an ordinary differential equation. Sometimes by differentiating we can reduce a given equation to a simpler integral equation whose solution is known. Below we list some classes ofintegral equations that can be reduced to ordinary differential equations with constant coefﬁcients. 11.7-1. Equations with Kernel Containing a Sum of Exponential Functions. Consider the equation y(x)+⎝integraldisplayx a⎝bracketleftbiggn⎝summationdisplay k=1Akeλk(x–t)⎝bracketrightbigg y(t)dt=f(x). (1) In the general case, this equation can be reduced to a linear nonhomogeneous ordinary differential equation of nth order with constant coefﬁcients (see equation 2.2.19 of the ﬁrst part of the book). In a wide range of the parameters Akandλk, the solution can be represented as follows: y(x)=f(x)+⎝integraldisplayx a⎝bracketleftbiggn⎝summationdisplay k=1Bkeµk(x–t)⎝bracketrightbigg f(t)dt,( 2) where the parameters Bkandµkof the solution are related to the parameters Akandλkof the equation by algebraic relations. For the solution of Eq. (1) with n= 2, see Section 2.2 of the ﬁrst part of the book (equation 2.2.10). 11.7-2. Equations with Kernel Containing a Sum of Hyperbolic Functions. By means of the formulas cosh β=1 2(eβ+e–β) and sinh β=1 2(eβ–e–β), any equation with difference kernel of the form y(x)+⎝integraldisplayx aK(x–t)y(t)dt=f(x), K(x)=m⎝summationdisplay k=1Akcosh(λ kx)+s⎝summationdisplay k=1Bksinh(µkx),(3) can be represented in the form of Eq. (1) with n=2m+2s, and hence these equations can be reduced to linear nonhomogeneous ordinary differential equations with constant coefﬁcients. 11.7-3. Equations with Kernel Containing a Sum of Trigonometric Functions. Equations with difference kernel of the form y(x)+⎝integraldisplayx aK(x–t)y(t)dt=f(x), K(x)=m⎝summationdisplay k=1Akcos(λ kx), (4) y(x)+⎝integraldisplayx aK(x–t)y(t)dt=f(x), K(x)=m⎝summationdisplay k=1Aksin(λkx), (5) can also be reduced to linear nonhomogeneous ordinary differential equations of order 2m with constant coefﬁcients (see equations 2.5.4 and 2.5.19 in the ﬁrst part of the book). 11.8. R EDUCTION OF VOLTERRA EQUATIONS OF THE SECOND KIND TO VOLTERRA EQUATIONS OF THE FIRST KIND 565 In a wide range of the parameters Akandλk, the solution of Eq. (5) can be represented in the form y(x)=f(x)+⎝integraldisplayx aR(x–t)f(t)dt,R(x)=m⎝summationdisplay k=1Bksin(µkx), (6) where the parameters Bkandµkof the solution are related to the parameters Akandλkof the equation by algebraic relations. Equations with difference kernel s containing both cosines and sines can also be reduced to linear nonhomogeneous ordinary differential equations with constant coefﬁcients. 11.7-4. Equations Whose Kernels Contain Combinations of Various Functions. Any equation with difference kernel that contains a linear combination of summands of the form (x–t)m(m=0 ,1 ,2 , ...), exp⎝bracketleftbig α(x–t)⎝bracketrightbig , cosh⎝bracketleftbig β(x–t)⎝bracketrightbig ,s i n h⎝bracketleftbig γ(x–t)⎝bracketrightbig ,c o s⎝bracketleftbig λ(x–t)⎝bracketrightbig ,s i n⎝bracketleftbig µ(x–t)⎝bracketrightbig ,(7) can also be reduced by differentiation to a linear nonhomogeneous ordinary differential equation with constant coefﬁcients, where exponential, hyperbolic, and trigonometric functions can also bemultiplied by ( x–t) n(n=1 ,2 , ...). Remark. The method of differentiation can be successfu lly used to solve more complicated equations with nondifference kernel to which the Laplace transform cannot be applied (see, for instance, Eqs. 2.9.5, 2.9.28, 2.9.30, 2.9.34, and 2.9.36 in the ﬁrst part of the book). 11.8. Reduction of Volterra Equations of the Second Kind to Volterra Equations of the First Kind The V olterra equation of the second kind y(x)–⎝integraldisplayx aK(x,t)y(t)dt=f(x)( 1 ) can be reduced to a V olterra equation of the ﬁrst kind in two ways. 11.8-1. First Method. We integrate Eq. (1) with respect to xfromatoxand then reverse the order of integration in the double integral. We ﬁnally obtain the V olterra equation of the ﬁrst kind ⎝integraldisplayx aM(x,t)y(t)dt=F(x), (2) where M(x,t)a n d F(x) are deﬁned as follows: M(x,t)=1–⎝integraldisplayx tK(s,t)ds,F(x)=⎝integraldisplayx af(t)dt.( 3) 566 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) 11.8-2. Second Method. Assume that the condition f(a) = 0 is satisﬁed. In this case Eq. (1) can be reduced to a V olterra equation of the ﬁrst kind for the derivative of the unknown function, ⎝integraldisplayx aN(x,t)y/prime t(t)dt=f(x), y(a)=0 , ( 4 ) where N(x,t)=1–⎝integraldisplayx tK(x,s)ds.( 5) Indeed, on integrating by parts the right-hand side of formula (4) with regard to formula (5), we arrive at Eq. (1). Remark. Forf(a)≠0, Eq. (1) implies the relation y(a)=f(a). In this case the substitution z(x)=y(x)–f(a) yields the V olterra equation of the second kind z(x)–⎝integraldisplayx aK(x,t)z(t)dt=Φ(x), Φ(x)=f(x)–f(a)+f(a)⎝integraldisplayx aK(x,t)dt, whose right-hand s ide satisﬁes the condition Φ(a) = 0, and hence this equation can be reduced by the second method to a V olterra equation of the ﬁrst kind. References for Section 11.8: V . V olterra (1959), A. F. Verlan’ and V . S. Sizikov (1986). 11.9. Successive Approximation Method 11.9-1. General Scheme. 1◦. Consider a V olterra integral equation of the second kind y(x)–⎝integraldisplayx aK(x,t)y(t)dt=f(x). (1) Assume that f(x) is continuous on the interval [ a,b] and the kernel K(x,t) is continuous for a≤x≤b anda≤t≤x. Let us seek the solution by the successive approximation method. To this end, we set y(x)=f(x)+∞⎝summationdisplay n=1ϕn(x), (2) where the ϕn(x) are determined by the formulas ϕ1(x)=⎝integraldisplayx aK(x,t)f(t)dt, ϕ2(x)=⎝integraldisplayx aK(x,t)ϕ1(t)dt=⎝integraldisplayx aK2(x,t)f(t)dt, ϕ3(x)=⎝integraldisplayx aK(x,t)ϕ2(t)dt=⎝integraldisplayx aK3(x,t)f(t)dt,e t c . 11.9. S UCCESSIVE APPROXIMA TION METHOD 567 Here Kn(x,t)=⎝integraldisplayx aK(x,z)Kn–1(z,t)dz,( 3) where n=2 , 3 , ..., and we have the relations K1(x,t)≡K(x,t)a n d Kn(x,t)=0f o r t>x. The functions Kn(x,t) given by formulas (3) are called iterated kernels . These kernels satisfy the relation Kn(x,t)=⎝integraldisplayx aKm(x,s)Kn–m(s,t)ds,( 4) where mis an arbitrary positive integer less than n. 2◦. The successive approximations can be implemented in a more general scheme: yn(x)=f(x)+⎝integraldisplayx aK(x,t)yn–1(t)dt,n=1 ,2 , ...,( 5 ) where the function y0(x) is continuous on the interval [ a,b]. The functions y1(x),y2(x),...which are obtained from (5) are also continuous on [ a,b]. Under the assumptions adopted in item 1◦forf(x)a n dK(x,t), the sequence {yn(x)}converges, asn→∞ , to the continuous solution y(x) of the integral equation. A successful choice of the “zeroth” approximation y0(x) can result in a rapid convergence of the procedure. Note that in the special case y0(x)=f(x), this method becomes that described in item 1◦. Remark 1. If the kernel K(x,t) is square integrable on the square S={a≤x≤b,a≤t≤b} andf(x)∈L2(a,b), then the successive approximations are mean-square convergent to the solution y(x)∈L2(a,b) of the integral equation (1) for any initial approximation y0(x)∈L2(a,b). Example. Consider the integral equation y(x)+⎝integraldisplayx 0(x–t)y(t)dt=1 and use the method of successive approximations for ﬁnding its solution. To that end, we take f(x)=1 , K(x,t)=– (x–t) in (5) and choose the initial function y0(x) = 0. As a result, we get y1(x)=1 , y2(x)=1–x2 2!, .............................. , yn(x)=1–x2 2!+···+ (–1)n–1x2n–2 (2n–2 ) !. It follows that y(x) = lim n→∞yn(x)=1–x2 2!+x4 4!–x6 6!+···=c o sx. It is easy to check that y(x)=c o s xis an exact solution of the integral equation under consideration. 11.9-2. Formula for the Resolvent. The resolvent of the integral equation (1) is determined via the iterated kernels by the formula R(x,t)=∞⎝summationdisplay n=1Kn(x,t), (6) where the convergent series on the right-hand side is called the Neumann series of the kernel K(x,t). Now the solution of the V olterra equation of the second kind (1) can be rewritten in the traditional form y(x)=f(x)+⎝integraldisplayx aR(x,t)f(t)dt.( 7 ) 568 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) Remark 2. In the case of a kernel with weak singularity, the solution of Eq. (1) can be obtained by the successive approximation method. In this case the kernels Kn(x,t) are continuous starting from some n.F o rα<1 2, even the kernel K2(x,t) is continuous. References for Section 11.9: W. V . Lovitt (1950), V . V olterra (1959), S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), V . I. Smirnov (1974). 11.10. Method of Quadratures 11.10-1. General Scheme of the Method. Let us consider the linear V olterra integral equation of the second kind y(x)–⎝integraldisplayx aK(x,t)y(t)dt=f(x), (1) on an interval a≤x≤b. Assume that the kernel and the right-hand side of the equation are continuous functions. From Eq. (1) we ﬁnd that y(a)=f(a). Let us choose a constant integration step hand consider the discrete set of points xi=a+h(i–1 ) ,i=1 ,...,n.F o rx=xi, Eq. (1) acquires the form y(xi)–⎝integraldisplayxi aK(xi,t)y(t)dt=f(xi), i=1 ,...,n.( 2) Applying the quadrature formula (see Subsection 10.7-1) to the integral in (2) and choosing xj (j=1 ,...,i) to be the nodes in t, we arrive at the system of equations y(xi)–i⎝summationdisplay j=1AijK(xi,xj)y(xj)=f(xi)+εi[y], i=2 ,...,n,( 3) where εi[y] is the truncation error and Aijare the coefﬁcients of the quadrature formula on the interval [ a,xi] (see Subsection 10.7-1). Suppose that εi[y] are small and neglect them; then we obtain a system of linear algebraic equations in the form y1=f1,yi–i⎝summationdisplay j=1AijKijyj=fi,i=2 ,...,n,( 4) where Kij=K(xi,xj),fi=f(xi), and yiare approximate values of the unknown function y(x)a t the nodes xi. From (4) we obtain the recurrent formula y1=f1,yi=fi+i–1⎝summationtext j=1AijKijyj 1–A iiKii,i=2 ,...,n,( 5 ) valid under the condition 1–A iiKii≠0, (6) which can always be ensured by an appropriate choice of the nodes and by guaranteeing that the coefﬁcients Aiiare sufﬁciently small. 11.11. E QUATIONS WITH INFINITE INTEGRATION LIMIT 569 11.10-2. Application of the Trapezoidal Rule. According to the trapezoidal rule (see Subsection 10.7-1), we have Ai1=Aii=1 2h,Ai2=···=Ai,i–1=h,i=2 ,...,n. The application of the trapezoidal rule in the general scheme leads to the following step algorithm: y1=f1,yi=fi+hi–1⎝summationtext j=1βjKijyj 1–1 2hKii,i=2 ,...,n, xi=a+(i–1 )h,n=b–a h+1 , βj=⎝braceleftbigg1 2forj=1 , 1f o r j>1 , where the notation coincides with that in troduced in Subsection 11.10-1. The trapezoidal rule is quite simple and effective, and frequently used in practice. Some peculiarities of using the quadrature method for solving integral equations with variable limits of integration are indicated in Subsection 10.7-3. 11.10-3. Case of a Degenerate Kernel. When solving a V olterra integral equation of the second kind with arbitrary kernel, the amount of calculations increases as the index of the integration step increases. However, if the kernel is degenerate, then it is possible to construct algorithms with a constant amount of calculations at each step. Indeed, for a degenerate kernel K(x,t)=m⎝summationdisplay k=1pk(x)qk(t), we can rewrite Eq. (1) in the form y(x)=m⎝summationdisplay k=1pk(x)⎝integraldisplayx aqk(t)y(t)dt+f(x). The application of the trapezoidal rule makes it possible to obtain the following recurrent expression (see Subsection 11.10-2): y1=f1,yi=fi+hm⎝summationtext k=1pkii–1⎝summationtext j=1βjqkjyj 1–1 2hm⎝summationtext k=1pkiqki, where yiare approximate values of the unknown function y(x) at the nodes xi,fi=f(xi),pki=pk(xi), andqki=qk(xi), and this expression shows that the amount of calculations is the same at each step. References for Section 11.10: S. G. Mikhlin and K. L. Smolitskiy (1967), G. A. Korn and T. M. Korn (1968), V . I. Krylov, V . V . Bobkov, and P. I. Monastyrnyi (1984), A. F. Verlan’ and V . S. Sizikov (1986), H. Brunner (2004). 11.11. Equations with Inﬁnite Integration Limit Integral equations of the second kind with difference kernel and with a variable limit of integration for which the other limit is inﬁnite are also of interest. Kernels and functions in such equations need not belong to the classes described in the beginning of the chapter. In this case their investigation can be performed by the method of model solutions (see Section 11.6) or by the reduction to equationsof convolution type. We consider the latter method by an example of an equation of the second kind with variable lower limit. 570 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextx aK(x,t)y(t)dt=f(x) 11.11-1. Equation of the Second Kind with Variable Lower Integration Limit. Integral equations of the second kind with variable lower limit, in the case of a difference kernel, have the form y(x)+⎝integraldisplay∞ xK(x–t)y(t)dt=f(x), 0 < x<∞.( 1) This equation substantially differs from V olterra equations of the second kind studied above for which a solution exists and is unique. A solution of the corresponding homogeneous equation y(x)+⎝integraldisplay∞ xK(x–t)y(t)dt=0 ( 2 ) can be nontrivial. The eigenfunctions of the integral equation (2) are determined by the roots of the following transcendental (or algebraic) equation for the parameter λ: ⎝integraldisplay∞ 0K(–z)e–λzdz= –1. (3) The left-hand side of this equation is the Laplace transform of the function K(–z) with parameter λ. To a real simple root λkof Eq. (3) there corresponds an eigenfunction yk(x)=e x p ( – λkx). The general solution is the linear combination (with arbitrary constants) of the eigenfunctions of the homogeneous integral equation (2). For solutions of Eq. (2) in the case of multiple or complex roots, see equation 52 in Section 2.9 (see also Example 1 below). The general solution of the integral equation (1) is the sum of the general solution of the homogeneous equation (2) and a particular solution of the nonhomogeneous equation (1). Example 1. Consider the homogeneous Picard–Goursat equation y(x)+A⎝integraldisplay∞ x(t–x)ny(t)dt=0 , n=0 ,1 ,2 , ..., (4) which is a special case of Eq. (1) with K(z)=A(–z)n. The general solution of the homogeneous equation has the form y(x)=m⎝summationdisplay k=1Ckexp(–λkx), (5) where Ckare arbitrary constants and λkare the roots of the algebraic equation λn+1+An!=0 (6) that satisfy the condition Re λk>0(mis the number of the roots of Eq. (6) that satisfy this condition). Equation (6) is a special case of Eq. (3) with K(z)=A(–z)n. The roots of Eq. (6) such that Re λk≤0 must be dropped out, since for them the integral in (3) is divergent. Equation (6) has complex roots. Consider two cases that correspond to different signs of A. 1◦.L e t A< 0. A solution of the Eq. (4) is y(x)=Ce–λx,λ=⎝parenleftbig–An!⎝parenrightbig1 n+1, (7) where Cis an arbitrary constant. This solution is unique for n=0 ,1 ,2 ,3 . 11.11. E QUATIONS WITH INFINITE INTEGRATION LIMIT 571 Forn≥4, taking the real and the imaginary part in (5), one arrives at the general solution of the homogeneous Picard–Goursat equation in the form y(x)=Ce–λx+[n/4]⎝summationdisplay k=1exp(–αkx)⎝bracketleftbig C(1) kcos(βkx)+C(2) ksin(βkx)⎝bracketrightbig , (8) where C(1) kandC(2) kare arbitrary constants, [ a] stands for the integral part of a number a,λis deﬁned in (7), and the coefﬁcients αkandβkare given by αk=|An!|1 n+1cos⎝parenleftBig2πk n+1⎝parenrightBig ,βk=|An!|1 n+1sin⎝parenleftBig2πk n+1⎝parenrightBig . Note that Eq. (8) contains an odd number of terms. 2◦.L e t A> 0. By taking the real and the imaginary part in (5), one obtains the general solution of the homogeneous Picard–Goursat equation in the form y(x)=⎝bracketleftbig n+2 4⎝bracketrightbig ⎝summationdisplay k=0exp(–αkx)⎝bracketleftbig C(1) kcos(βkx)+C(2) ksin(βkx)⎝bracketrightbig , (9) where C(1) kandC(2) kare arbitrary constants, and the coefﬁcients αkandβkare given by αk=(An!)1 n+1cos⎝parenleftBig2πk+π n+1⎝parenrightBig ,βk=(An!)1 n+1sin⎝parenleftBig2πk+π n+1⎝parenrightBig . Note that Eq. (9) contains an even number of terms. In the special cases of n=0a n d n= 1, Eq. (9) gives the trivial solution y(x)≡0. Example 2. Consider the nonhomogeneous Picard–Goursat equation y(x)+A⎝integraldisplay∞ x(t–x)ny(t)dt=Be–µx,n=0 , 1 ,2 , ..., (10) which is a special case of Eq. (1) with K(z)=A(–z)nandf(x)=Be–µx. Letµ> 0. Consider two cases. 1◦.L e t µn+1+An! ≠0. A particular solution of the nonhomogeneous equation is ¯y(x)=De–µx,D=Bµn+1 µn+1+An!. (11) ForA< 0, the general solution of the nonhomogeneous Picard–Goursat equation is the sum of solutions (8) and (11). ForA> 0, the general solution of the Eq. (10) is the sum of solutions (9) and (11). 2◦.L e tµn+1+An! = 0. Since µis positive, it follows that Amust be negative. A particular solution of the nonhomogeneous equation is ¯y(x)=Exe–µx,E=Bµn+2 A(n+1 ) !. (12) The general solution of the nonhomogeneous Picard–Goursat equation is the sum of solutions (8) and (12). 11.11-2. Reduction to a Wiener–Hopf Equation of the Second Kind. Equation (1) can be reduced to a one-sided equation of the second kind of the form y(x)–⎝integraldisplay∞ 0K–(x–t)y(t)dt=f(x), 0 < x<∞, (13) where the kernel K–(x–t)h a st h ef o r m K–(s)=⎝braceleftbigg 0f o r s>0 , –K(s)f o r s<0 . Methods for studying Eq. (13) are described in Chapter 13, where equations of the second kind with constant limits are considered. In the same chapter, in Subsection 13.10-3, an equation of the second kind with difference kernel and variable lower limit is studied by means of reduction to aWiener–Hopf equation of the second kind. Reference for Section 11.11: F. D. Gakhov and Yu. I. Cherskii (1978), A. D. Polyanin and A. V . Manzhirov (1998). Chapter 12 Methods for Solving Linear Equations of the Form⎝integraldisplay ⎝integraldisplayb aK(x,t)y(t)dt=f(x) 12.1. Some Deﬁnition and Remarks 12.1-1. Fredholm Integral Equations of the First Kind. Linear integral equations of the ﬁrst kind with constant limits of integration have the form ⎝integraldisplayb aK(x,t)y(t)dt=f(x), (1) where y(x) is the unknown function ( a≤x≤b),K(x,t)i st h e kernel of the integral equation, and f(x) is a given function, which is called the right-hand side of Eq. (1). The functions y(x)a n df(x) are usually assumed to be continuous or square integrable on [a ,b]. If the kernel of the integral equation (1) is continuous on the square S={a≤x≤b,a≤t≤b}or at least square integrable on this square, i.e.,⎝integraldisplayb a⎝integraldisplayb aK2(x,t)dx dt =B2<∞,( 2) where Bis a constant, then this kernel is called a Fredholm kernel . Equations of the form (1) with constant integration limits and Fredholm kernel are called Fredholm equations of the ﬁrst kind . The kernel K(x,t) of an integral equation is said to be degenerate if it can be represented in the formK(x,t)=g1(x)h1(t)+···+gn(x)hn(t). The kernel K(x,t) of an integral equation is called a difference kernel if it depends only on the difference of the arguments: K(x,t)=K(x–t). The kernel K(x,t) of an integral equation is said to be symmetric if it satisﬁes the condition K(x,t)=K(t,x). The integral equation obtained from (1) by replacing the kernel K(x,t)b yK(t,x)i ss a i dt ob e transposed to (1). Remark 1. The variables tandxin Eq. (1) may vary within different intervals (e.g., a≤t≤b andc≤x≤d). It is important to observe that integral equations of the ﬁrst kind (1), even with very smooth kernels and right-hand sides, may have no solutions at all or have several (inﬁnitely many) solutions. Example 1. The equation⎝integraldisplay1 0y(t)dt=1+t has no solutions. 573 574 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) Example 2. The equation⎝integraldisplay1 0y(t)dt=1 has the solutions y(x)=1a n d y(x)=1+ C(2x–1 ) ,w h e r e Cis an arbitrary constant. Moreover, this equation has the solution y(x)=⎝braceleftbigg ϕ(x)/A ifA≠0, 1+Cϕ(x)i fA=0 ,A=⎝integraldisplay1 0ϕ(x)dx, where ϕ(x) is an arbitrary function. It should also be mentioned that Fredholm integral equations of the ﬁrst kind belong to the class of ill-posed problems (for details see Section 12.12). 12.1-2. Integral Equations of the First Kind with Weak Singularity. If the kernel of the integral equation (1) is polar, i.e., if K(x,t)=L(x,t) |x–t|α+M(x,t), 0 < α<1 , ( 3 ) or logarithmic, i.e., K(x,t)=L(x,t)l n|x–t|+M(x,t), (4) where L(x,t)a n d M(x,t) are continuous on SandL(x,x)/ ≡0, then K(x,t) is called a kernel with weak singularity , and the equation itself is called an equation with weak singularity . Remark 2. Kernels with logarithmic singularity and polar kernels with 0 < α<1 2are Fredholm kernels. Remark 3. In general, the case in which the limits of integration aand/or bcan be inﬁnite is not excluded, but in this case the validity of c ondition (2) must be veriﬁed with special care. 12.1-3. Integral Equations of Convolution Type. The integral equation of the ﬁrst kind with difference kernel on the entire axis (this equation is sometimes called an equation of convolution type of the ﬁrst kind with a single kernel )h a st h ef o r m ⎝integraldisplay∞ –∞K(x–t)y(t)dt=f(x), –∞<x<∞,( 5) where f(x)a n dK(x) are the right-hand side and the kernel of the integral equation and y(x)i st h e unknown function (in what follows we use the above notation). An integral equation of the ﬁrst kind with difference kernel on the semiaxis has the form ⎝integraldisplay∞ 0K(x–t)y(t)dt=f(x), 0 < x<∞.( 6) Equation (6) is also called a one-sided equation of the ﬁrst kind or a Wiener–Hopf integral equation of the ﬁrst kind . An integral equation of convolution type with two kernels of the ﬁrst kind has the form ⎝integraldisplay∞ 0K1(x–t)y(t)dt+⎝integraldisplay0 –∞K2(x–t)y(t)dt=f(x), – ∞<x<∞,( 7 ) where K1(x)a n dK2(x) are the kernels of the integral equation (7). 12.1. S OME DEFINITION AND REMARKS 575 Recall that a function g(x) satisﬁes the H¨older condition on the real axis if for any real x1andx2 we have the inequality |g(x2)–g(x1)|≤A|x2–x1|λ,0 < λ≤1, and for any x1andx2sufﬁciently large in absolute value we have |g(x2)–g(x1)|≤A⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle1 x2–1 x1⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleλ ,0 < λ≤1, where Aandλare positive (the latter inequality is the H ¨older condition in the vicinity of the point at inﬁnity). Assume that the functions y(x)a n df(x) and the kernels K(x),K1(x), and K2(x) are such that their Fourier transforms belong to L2(–∞,∞) and, moreover, satisfy the H ¨older condition. For a function y(x) to belong to the above function class it sufﬁces to require y(x)t ob e l o n gt o L2(–∞,∞)a n d xy(x) to be absolutely integrable on (– ∞,∞). 12.1-4. Dual Integral Equations of the First Kind. Adual integral equation of the ﬁrst kind with difference kernels (of convolution type )h a st h ef o r m ⎝integraldisplay∞ –∞K1(x–t)y(t)dt=f(x), 0 < x<∞, ⎝integraldisplay∞ –∞K2(x–t)y(t)dt=f(x), – ∞<x<0 ,(8) where the notation and the classes of functions and kernels coincide with those introduced above for equations of convolution type. In the general case, a dual integral equation of the ﬁrst kind has the form ⎝integraldisplay∞ aK1(x,t)y(t)dt=f1(x), a<x<b, ⎝integraldisplay∞ aK2(x,t)y(t)dt=f2(x), b<x<∞, where f1(x)a n df2(x) are the right-hand sides, K1(x,t)a n d K2(x,t) are the kernels of Eq. (8), and y(x) is the unknown function. Various forms of this equation are considered in Subsections 12.9-3 and 12.9-4. The integral equations obtained from (5)–(8) by replacing the kernel K(x–t) with K(t–x)a r e called transposed equations. Remark 3. Some equations whose kernels contain the product or the ratio of the variables x andtcan be reduced to equations of the form (5)–(8). Remark 4. Equations (5)–(8) of the convolution type are sometimes written in the form in which the integrals are multiplied by the coefﬁcient 1 /√ 2π. 12.1-5. Some Problems Leading to Integral Equations of the First Kind. 1◦. Historically, one of the ﬁrst problems that can be associated with integral equations was that of inverting the integral g(t)=1 √ 2π⎝integraldisplay∞ –∞f(x)eixtdx, 576 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) i.e., ﬁnding a function f(x) from a given g(t). This problem was solved in 1811 by Fourier, who obtained its solution in the form f(x)=1 √ 2π⎝integraldisplay∞ –∞g(t)e–ixtdt. 2◦. Making an elastic string acquire a given shape under the action of a distributed force. Suppose there is a weightless elastic string of length lthat resists tension but does not resist changing its shape. Assume that the string obeys Hooke’s law in tension, so that the force required to extend the string by ∆lis equal to γ∆l,w h e r e γis some constant. Let the ends of the string be ﬁxed at points AandB(Fig. 4) and let the string position coincide with the segment ABof the Oxaxis when acted upon by only a horizontal tensile force T0,v e r y large compared to any other force under consideration. PCAB yxA1 C0 C1x/c61/c32/c120 /c100/c98 /c97 x Figure 4. Shape of an elastic string ﬁxed at points AandBand acted upon by a force Pat a point C0. Suppose that a force Pis applied to the string at a point C0withx=ξ. Then the string will take the shape of a broken line ACB . Assume that the displacement CC 0=δis small compared to AC 0 andC0B, which results from the assumption that Pis small compared to T0. Also assume that the tension of the string remains equal to T0. Projecting the tensile forces at Cand the force Ponto the vertical, we write down the equilibrium condition to obtain T0sinα+T0sinβ=P. Sinceδis considered to be small, we have sinα≈δ ξ,s i n β≈δ l–ξ. Then the equilibrium condition can be rewritten as T0δ ξ+T0δ l–ξ=P. It follows that δ(ξ)=P(l–ξ)ξ T0l. Lety(x) denote the amount of sag of the string at the point with abscissa x.T h e n y(x)=PG(x,ξ), where G(x,ξ)=⎧ ⎪⎪⎨ ⎪⎪⎩x(l–ξ) T0lif 0 ≤x≤ξ, (l–ξ)ξ T0lifξ≤x≤l. 12.2. I NTEGRAL EQUATIONS OF THE FIRST KIND WITH SYMMETRIC KERNEL 577 Indeed, for x<ξ, from the similarity of the triangles AC 0CandAA 1C1(Fig. 4) it follows that y(x) δ(ξ)=x ξ,o rPG(x,ξ) δ(ξ)=x ξ. Hence, G(x,ξ)=xδ(ξ) Pξ=x(l–ξ) T0l. The case of ξ<xis treated similarly. It is apparent that G(x,ξ)=G(ξ,x). If the string is acted upon by a continuously distributed force with line density p(ξ), then the small segment between ξandξ+∆ξis subjected to the force approximately equal to p(ξ)∆ξand is displaced by the distance G(x,ξ)p(ξ)∆ξ. Since the displacements caused by the elementary forces p(ξ)∆ξare summed (according to the principle of superposition), the total amount of sag y(x)i s approximately equal to⎝summationdisplay (ξ)G(x,ξ)p(ξ)∆ξ. On passing to the limit as ∆ξ→0, one arrives at a Fredholm integral equation of the ﬁrst kind: y(x)=⎝integraldisplayl 0G(x,ξ)p(ξ)dξ. This equation serves to determine the force density p(x) under the action of which the string will take the given shape y=y(x). The function G(x,ξ) is called an inﬂuence function . References for Section 12.1: B. Noble (1958), S. G. Mikhlin (1960), I. C. Gohberg and M. G. Krein (1967), L. Ya. Tslaf (1970), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), P. P. Zabreyko, A. I. Koshelev, et al. (1975), Ya. S. Uﬂyand (1977), F. D. Gakhov and Yu. I. Cherskii (1978), A. J. Jerry (1985), A. F. Verlan’ and V . S. Sizikov (1986),I. Sneddon (1995), A. V . Bitsadze (1995), L. A. Sakhnovich (1996). 12.2. Integral Equations of the First Kind with Symmetric Kernel 12.2-1. Solution of an Integral Equation in Terms of Series in Eigenfunctions of Its Kernel. Suppose K(x,t) is a real symmetric kernel deﬁned on a segment [ a,b]. Let us write out the system of characteristic values and eigenfunctions of this kernel* as the sequences λ1,λ2, ...,λn, ...; y1(x),y2(x),...,yn(x),...,(1) where yn(x)–λn⎝integraldisplayb aK(x,t)yn(t)dt=0 . It is assumed that the following conditions hold: 1) The values λnare ordered so that their moduli form a nondecreasing sequence, i.e., |λn–1|≤ |λn|. 2) Each characteristic value appears as many times as its multiplicity (rank), so that one and the same value λin (1) may occur several times, each corresponding to only one eigenfunction. 3) Eigenfunctions yn(x) are normalized and mutually orthogonal in L2[a,b] (for details, see Subsection 13.6-1). * For deﬁnitions of characteristic values and eigenfunctions of a kernel K(x,t), see Subsection 13.1-1. 578 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) A symmetric kernel K(x,t)d e ﬁ n e do n[ a,b] is called complete (orclosed ), if the system of the corresponding eigenfunctions is complete in L2[a,b]); otherwise, the kernel is called incomplete. Consider a nonhomogeneous integral equation of the ﬁrst kind ⎝integraldisplayb aK(x,t)y(t)dt=f(x)( 2 ) with a real symmetric kernel and f∈L2[a,b]. PICARD THEOREM .Equation (2) has a solution if and only if f(x)can be expanded into a mean-square convergent series with respect to eigenfunctions of the kernel K(x,t): f(x)=∞⎝summationdisplay k=1fkyk(x),fk=⎝integraldisplayb af(x)yk(x)dx,( 3 ) and the series∞⎝summationdisplay k=1λ2 k|fk|2(4) is convergent. In this case, the general solution of equation (2) has the form y(x)=y0(x)+∞⎝summationdisplay k=1λkfkyk(x), where y0(x)is an arbitrary solution of the homogeneous equation (2) for f(x)≡0.I f t h e kernel K(x,t)is complete, then y0(x)≡0, and equation (2) has only one solution, y(x)=∞⎝summationdisplay k=1λkfkyk(x). (5) Example. Consider the integral equation ⎝integraldisplay1 0K(x,t)y(t)dt=s i n3(πx) (6) with the real symmetric kernel K(x,t)=⎝braceleftBig(1 –x)tif 0 ≤t≤x, (1 –t)xifx≤t≤1.(7) Let us use the Picard theorem to ﬁnd its solution. First, we ﬁnd the characteristic values and the corresponding normalized eigenfunctions of the kernel (7): λ1=π2, λ2=( 2π)2, ...,λn=(nπ)2, ...; y1(x)=√ 2s i n (πx),y2(x)=√ 2s i n ( 2 πx),...,yn(x)=√ 2s i n (nπx ),...(8) Then we express the right-hand side of (6) in terms of the eigenfunctions: f(x)≡sin3(πx)=3 4sin(πx)–1 4sin(3πx)=3 4√ 2y1(x)–1 4√ 2y3(x) and write out the corresponding coefﬁcients in the expansion of f(x): f1=3 4√ 2,f2=0 , f3=–1 4√ 2,fm=0 f o r m=4 ,5 , ... The series (4) in this case reduces to the ﬁnite sum ∞⎝summationdisplay k=1λ2 k|fk|2=(π2)2⎝parenleftBig3 4√ 2⎝parenrightBig2 +( 9π2)2⎝parenleftBig –1 4√ 2⎝parenrightBig2 =45 16π4 and is therefore convergent. The system of eigenfunctions (8) is a complete orthonormal system on [0, 1], i.e., the kernel is complete. By the Picard theorem, equation (6)–(7) has the unique solution y(x)=λ1f1y1(x)+λ3f3y3(x), which can be written in the form y(x)=3 4π2[sin(πx)–3s i n ( 3 πx)]. 12.2. I NTEGRAL EQUATIONS OF THE FIRST KIND WITH SYMMETRIC KERNEL 579 12.2-2. Method of Successive Approximations. THEOREM .LetK(x,t)be a symmetric positive kernel and suppose that the equation ⎝integraldisplayb aK(x,t)y(t)dt=f(x), f(x)∈L2[a,b], (9) admits one and only one solution. Then the sequence of functions {yn(x)}deﬁned by the recurrent relation yn(x)=yn–1(x)+λ⎝bracketleftbigg f(x)–⎝integraldisplayb aK(x,t)yn–1(t)dt⎝bracketrightbigg ,n=1 ,2 , ..., (10) where y0(x)∈L2[a,b], 0 < λ<2λ1, (11) λ1is the smallest characteristic value of the kernel K(x,t), is mean-square convergent to the solution of equation (9). Remark. If there is no information about the solution of equation (9), one can take y0(x)=0a s the zero approximation. If the smallest characteristic value λ1is unknown, then λshould be chosen sufﬁciently small and one should check (control) the convergence of the process (10). Example. Consider the integral equation ⎝integraldisplay1 0K(x,t)y(t)dt=s i n (πx), (12) where K(x,t)=⎝braceleftBig(1 –x)tif 0 ≤t≤x, (1 –t)xifx≤t≤1.(13) Let us construct successive approximations by formulas (10), taking y0(x) = 0 and imposing no constraints on λso far. We have y1(x)=λsin(πx), y2(x)=λsin(πx)+λ⎝parenleftbigg 1–λ π2⎝parenrightbigg sin(πx), y3(x)=λsin(πx)+λ⎝parenleftbigg 1–λ π2⎝parenrightbigg sin(πx)+λ⎝parenleftbigg 1–λ π2⎝parenrightbigg2 sin(πx), ·································································· yn(x)=λ⎝bracketleftbigg 1+⎝parenleftbigg 1–λ π2⎝parenrightbigg +⎝parenleftbigg 1–λ π2⎝parenrightbigg2 +···+⎝parenleftbigg 1–λ π2⎝parenrightbiggn–1⎝bracketrightbigg sin(πx),(14) The square brackets contain the ﬁnite sum of a geometrical progression with ratio q=1–λ π2. This sum is calculated by the formula n–1⎝summationdisplay m=0qm=1–qn 1–q,q=1–λ π2. Forn→∞ , this sum has a ﬁnite limit equal to1 1–q, provided that |q|< 1, which yields the following constraint on the coefﬁcient λ: 0<λ<2π2. (15) Passing to the limit in (14) as n→∞ , under the condition (15), we ﬁnd that lim n→∞yn(x)=λ 1–qsin(πx)=π2sin(πx). (16) It is easy to check that the limit solution (16) coincides with the exact solution of the integral equation (12)–(13). It can be shown that the smallest characteristic value of the kernel (13) is λ1=π2. Therefore, condition (11), in this case, turns into (15). References for Section 12.2: M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), M. L. Krasnov (1975), P. P. Zabreyko, A. I. Koshelev et al. (1975). 580 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) 12.3. Integral Equations of the First Kind with Nonsymmetric Kernel 12.3-1. Representation of a Solution in the Form of Series. General Description. Consider an integral equation⎝integraldisplayb aK(x,t)y(t)dt=f(x)( 1 ) with an arbitrary (symmetric or nonsymmetric) kernel. Let us seek its solution in the form of a sum y(t)=N⎝summationdisplay n=1Anϕn(x), (2) where ϕn(x) is a (complete) system of functions on the interval ( a,b), the upper limit of the sum, N, can be either ﬁnite or inﬁnite. It is important to mention that in some cases it is possible to obtainan exact solution of the integral equation (1) in the form of series (2) for N=∞(see Examples 1 and 2 in Subsection 12.3-2). Substituting (2) into (1), we get f(x)= N⎝summationdisplay n=1Angn(x), (3) where gn(x) are known functions, gn(x)=⎝integraldisplayb aK(x,t)ϕn(t)dt.( 4) In order to ﬁnd the coefﬁcients Anin the right-had side of (3), different methods can be used, depending on the structure of the functions gn(x). Some basic methods are described below. 12.3-2. Special Case of a Kernel That is a Generating Function. For power-type functions gn(x)=bnxn, the right hand side of (3) is a polynomial or a power series (forN=∞). The coefﬁcients Anof that series can be found by way of comparison with the corresponding coefﬁcients in the expansion of f(x)i np o w e r so f x. This case takes place if the kernel of the integral equation is a generating function for a system of orthogonal polynomials.Recall that G(x,t) is called a generating function for a system of functions h 0(t),h1(t),...,hm(t),... ifG(x,t) admits the following expansion in powers of x: G(x,t)=∞⎝summationdisplay m=0cmhm(t)xm(cm≠0). Example 1. Consider the equation⎝integraldisplay1 –1y(t)dt √ 1+x2–2xt=f(x). (5) Its kernel is a generating function for the Legendre polynomials (see Supplement 11.11-1): 1 √ 1+x2–2xt=∞⎝summationdisplay m=0Pm(t)xm,Pm(x)=1 m!2mdm dxm(x2–1 )m. (6) 12.3. I NTEGRAL EQUATIONS OF THE FIRST KIND WITH NONSYMMETRIC KERNEL 581 Let us seek a solution of equation (5) in the form y(x)=∞⎝summationdisplay n=0AnPn(x). (7) Substituting (6) and (7) into equation (5) and taking into account the orthogonality conditions for the Legendre polynomials, ⎝integraldisplay1 –1Pn(x)Pm(x)dx=⎝braceleftBigg0i f n≠m, 2 2n+1ifn=m, we ﬁnd that 2∞⎝summationdisplay n=0An 2n+1xn=f(x). Expanding the right-hand side into a Maclaurin series and equating the coefﬁcients of equal powers of x, we obtain An=2n+1 2n!f(n) x(0). Inserting these coefﬁcients into (7), we obtain a solution of the integral equation (5) in the form y(x)=1 2∞⎝summationdisplay n=02n+1 n!f(n) x(0)Pn(x). (8) It is easy to see that if the right-hand side of equation (5) is a polynomial, then its solution (8) is a polynomial of the same degree. Example 2. Consider the equation⎝integraldisplay∞ –∞e–(x–t)2y(t)dt=f(x) (9) whose kernel is a generating function for the Hermitian polynomials (see Supplement 11.17-3) e–(x–t)2=∞⎝summationdisplay m=01 m!e–t2Hm(t)xm,Hm(x) = (–1)mexp⎝parenleftbig x2⎝parenrightbigdm dxmexp⎝parenleftbig –x2⎝parenrightbig . (10) Let us seek a solution of equation (9) in the form of expansion y(x)=∞⎝summationdisplay n=0AnHn(x). (11) Substituting (10) and (11) into (9) and taking into account the orthogonality of the Hermitian polynomials, together with the relations⎝integraldisplay∞ –∞e–t2H2 n(t)dt=2nn!√ π, we obtain f(x)=√ π∞⎝summationdisplay n=0An2nxn. Hence, we ﬁnd the coefﬁcients An: An=f(n) x(0) 2nn!√ π. Substituting these into (11), we obtain a solution of the original integral equation (9): y(x)=1 √ π∞⎝summationdisplay n=0f(n) x(0) 2nn!Hn(x). 582 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) 12.3-3. Special Case of the Right-Hand Side Represented in Terms of Orthogonal Functions. Suppose that the functions gn(x) in the right-hand side of (3) form an orthogonal, with some weight ρ(x), system on the interval ( a,b): ⎝integraldisplayb aρ(x)gn(x)gm(x)dx=0if n ≠m. Then the coefﬁcients Anare obtained by multiplying (3) by ρ(x)gm(x) with subsequent integration inxover the segment [ a,b]. As a result, we get An=1 βn⎝integraldisplayb af(x)ρ(x)gn(x)dx,βn=⎝integraldisplayb aρ(x)g2 n(x)dx. 12.3-4. General Case. Galerkin’s Method. In the general case, one chooses a sequence of functions ψm(x),m=1 ,...,n, multiplies the relation (3) by these in successive order, and then integrates in xover the segment [ a,b]. The result is a system of linear algebraic equations for the coefﬁcients An: N⎝summationdisplay n=1σnmAn=Bm,m=1 ,...,n; σnm=⎝integraldisplayb agn(x)ψm(x)dx,Bm=⎝integraldisplayb af(x)ψm(x)dx. Finding the coefﬁcients Anfrom this system and substituting these in (3), one obtains an approximate solution of the integral equation (1). Remark. The Galerkin method and its modiﬁcations, when applied to the solution of integral equations of the second kind, may result in large errors (connected with the instability of solutions with respect to small perturbations of the right -hand side of the equation; see Section 12.12). For this reason the said methods are rarely used in practice. 12.3-5. Utilization of the Schmidt Kernels for the Construction of Solutions of Equations. LetK(x,t) be a real (or complex) nonsymmetric kernel, K(x,t)≠K(t,x), such that ⎝integraldisplayb a⎝integraldisplayb a|K(x,t)|2dx dt <∞. The kernels K(x,t)a n d K∗(x,t)= K(t,s) are called conjugate . Consider auxiliary functions K1(x,t)=⎝integraldisplayb aK∗(x,s)K(s,t)ds=⎝integraldisplayb a K(s,x)K(s,t)ds, (12) K2(x,t)=⎝integraldisplayb aK(x,s)K∗(s,t)ds=⎝integraldisplayb aK(x,s) K(t,s)ds, (13) representing symmetric positive kernels called the Schmidt kernels corresponding to K(x,t). It can be shown that the system of characteristic values of the kernels (12) and (13) coincide. 12.4. M ETHOD OF DIFFERENTIA TION FOR INTEGRAL EQUATIONS 583 Denote by µn(n=1 ,2 , ...) the characteristic values of the Schmidt kernels, by un(x)t h e orthonormalized eigenfunctions corresponding to K2(x,t), and by vn(x) orthonormalized eigen- functions corresponding to K1(x,t). Each un(x)a n d vn(x) can be multiplied by an arbitrary constant coefﬁcient whose absolute value is equal to unity. These coefﬁcients can be chosen such that the following formulas hold: K(x,t)=∞⎝summationdisplay n=1un(x) vn(t) √ µn,K∗(x,t)=∞⎝summationdisplay n=1vn(x) un(t) √ µn. (14) These series are mean-square convergent on [ a,b] (with respect to the variables xandtjointly). The following inequality holds: ∞⎝summationdisplay n=11 µn≤⎝integraldisplayb a⎝integraldisplayb a|K(x,t)|2dx dt . For a nonhomogeneous equation of the ﬁrst kind ⎝integraldisplayb aK(x,t)y(t)dt=f(x), f∈L2[a,b], (15) with nonsymmetric kernel to have a solution it is necessary and sufﬁcient that the free term f(x) could be expanded into a mean-square convergent series in terms of eigenfunctions un: f(x)=∞⎝summationdisplay n=1fnun(x),fn=⎝integraldisplayb af(x)un(x)dx, and that the series∞⎝summationdisplay n=1µn|fn|2 be convergent. Under these conditions, the general solution of equation (15) has the form y(x)=y0(x)+∞⎝summationdisplay n=1√ µnfnvn(x), where y0(x) is any solution of the homogeneous equation (15) with f(x)≡0. If the Schmidt kernel K1(x,t) is complete, then y0(x)≡0, and equation (15) has only one solution, y(x)=∞⎝summationdisplay n=1√ µnfnvn(x). References for Section 12.3: P. M. Morse and H. Feshbach (1953), L. Ya. Tslaf (1970), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), V . I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev et al. (1975). 12.4. Method of Differentiation for Integral Equations 12.4-1. Equations with Modulus. In some cases, differentiation of integral equations ( once, twice, etc.) with subsequent elimination of integral terms by means of the original equation m akes it possible to ﬁnd s olutions of the latter. 584 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) The class of integral equations whose solutions can be obtained by the method of differentiation includes integral equations of the ﬁrst kind with difference kernel, ⎝integraldisplayb aK(|x–t|)y(t)dt=f(x), (1) for the following types of K(z): K(z)=n⎝summationdisplay m=1Amzm,( 2) K(z)=n⎝summationdisplay m=1Amexp(λmz), (3) K(z)=n⎝summationdisplay m=1Amsinh(λmz), (4) K(z)=n⎝summationdisplay m=1Amsin(λmz). (5) Example 1. Consider the equation⎝integraldisplay1 0|x–t|y(t)dt=f(x), (6) which is a special case of (1), (2) for n=1 . 1◦. Let us remove the modulus in the integrand: ⎝integraldisplayx 0(x–t)y(t)dt+⎝integraldisplay1 x(t–x)y(t)dt=f(x). (7) Differentiating (7) in xyields⎝integraldisplayx 0y(t)dt–⎝integraldisplay1 xy(t)dt=f/prime x(x). (8) Differentiating (8) in xyields the solution y(x)=1 2f/prime/prime xx(x). (9) 2◦. The right-hand side f(x) of the integral equation (6) must satisfy certain additional relations. In order to obtain these, let us substitute the solution (9) into the transformed original equation (7). Integrating by parts, we get ⎝integraldisplayx 0(x–t)y(t)dt=1 2⎝integraldisplayx 0(x–t)f/prime/prime tt(t)dt=1 2⎝bracketleftbig –xf/prime(0) +f(x)–f(0)], ⎝integraldisplay1 x(t–x)y(t)dt=1 2⎝integraldisplay1 x(t–x)f/prime/prime tt(t)dt=1 2⎝bracketleftbig–xf/prime(1) +f(x)+f/prime(1) –f(1)]. Substituting these integrals into the left-hand side of equation (7) and reducing the result by f(x), we obtain –1 2x[f/prime(0) +f/prime(1)] +1 2[f/prime(1) –f(1) –f(0)] = 0. Since this relation must hold for all x, we obtain the following two conditions: f/prime(0) +f/prime(1) = 0, f/prime(1) –f(1) –f(0) = 0, which should be satisﬁed by the right-hand side of the integral equation (6). Example 2. Consider the equation⎝integraldisplayb aeλ|x–t|y(t)dt=f(x), (10) which is a special case of (1) with kernel (3) for n=1 . 12.4. M ETHOD OF DIFFERENTIA TION FOR INTEGRAL EQUATIONS 585 Let us remove the modulus in the integrand: ⎝integraldisplayx aeλ(x–t)y(t)dt+⎝integraldisplayb xeλ(t–x)y(t)dt=f(x). (11) Differentiating (11) with respect to xtwice yields 2λy(x)+λ2⎝integraldisplayx aeλ(x–t)y(t)dt+λ2⎝integraldisplayb xeλ(t–x)y(t)dt=f/prime/prime xx(x). (12) Eliminating the integral terms from (11) and (12), we obtain the solution y(x)=1 2λ⎝bracketleftbig f/prime/prime xx(x)–λ2f(x)⎝bracketrightbig . (13) The right-hand side f(x) of the integral equation (10) must satisfy certain additional relations. In order to obtain these, one should substitute the solution (13) into the original equation (10) or its corollary (11). Another method of ﬁndingadditional conditions on f(x) is described in Section 3.2 (see Eq. 3, Item 2 ◦). Other examples of solutions of such equations can be found in Section 3.1 (equations 2, 8, 11, and 16), Section 3.2 (equations 3, 4, and 6), Section 3.3 (equations 5, 6, and 10), Section 3.5 (equations 10, 11, 12, and 16). In a similar way, one can ﬁnd solutions of the equation ⎝integraldisplayb a|g(x)–h(t)|y(t)dt=f(x). Some equations of this type are considered in 3.8 (see equations 4–6). 12.4-2. Other Equations. Some Generalizations. 1◦. Sometimes, differentiation helps to reduce a given equation ⎝integraldisplayb aK(x,t)y(t)dt=f(x) (14) to a simpler integral equation⎝integraldisplayb aK/prime x(x,t)y(t)dt=f/prime x(x) (15) whose solution is known. Note that equations (14) and (15) may be nonequivalent. Thus, if y(t) is a solution of (14), it is also a solution of (15) (provided that the integral on the left-hand side of the equation exists). On the other hand, if y(t) is a solution of (15), it will satisfy equation (14) only under the additional condition⎝integraldisplayb aK(c,t)y(t)dt=f(c)( a<c<b), (16) which is obtained by taking x=cin the original equation (14). Example 3. Consider the equation ⎝integraldisplay1 –1ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingletanhλ(t–x) 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=–f(x), –1 ≤x≤1.(17) Differentiating this equation in x, we obtain the following singular equation: λ⎝integraldisplay1 –1y(t)dt sinh[λ(t–x)]=f/prime x(x). (18) 586 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) Equations (18) and (17) are equivalent under the additional condition ⎝integraldisplay1 –1ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingletanhλt 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley(t)dt=–f(0). (19) Let us rewrite (18) as follows: λ⎝integraldisplay1 –1y(t)dt cosh(λt)[tanh(λt )–t a n h ( λx)]=c o s h ( λx)f/prime x(x), and make the transformation z=t a n h ( λx),τ=t a n h ( λt),Y(τ)=c o s h ( λt)y(t),h(z)=c o s h ( λx)f/prime x(x). (20) As a result, we obtain a much simpler equation, ⎝integraldisplaya –aY(τ)dτ τ–z=h(z)( a=t a n h λ,|z|≤a), whose solution is given in Section 3.1 (see equation 51). Using this solution and going back to the original variables by formulas (20), one can ﬁnd a solution of the original equation (17). 2◦. As a preliminary step, one can multiply equation (14) by a function ϕ(x), and then integrate the result in x. Thus, one obtains the equation ⎝integraldisplayb a∂ ∂x⎝bracketleftbig ϕ(x)K(x,t)]y(t)dt=[ϕ(x)f(x)]/prime x. If one can ﬁnd a solution of the last equation, its solution should be inserted into the original equation (14) or into (16) for veriﬁcation. Remark 1. Instead of differentiation,one can multiply equation (14) by a function ϕ(x),integrate the result in xfromatox, and try to ﬁnd a solution of the equation thus obtained. Remark 2. If equation (14) does not depend on the parameter µ, it can be multiplied by a function ϕ(x,µ) and then integrated with respect to µfromαtoβ. References for Section 12.4: I. I. V orovich, V . M. Aleksandrov, and V . A. Babeshko (1974), A. D. Polyanin and A. V . Manzhirov (1998). 12.5. Method of Integral Transforms The method of integral transforms enables one to reduce some integral equations on the entire axis and on the semiaxis to algebraic equations for transforms. These algebraic equations can readily be solved for the transform of the desired function. The solution of the original integral equation is then obtained by applying the inverse integral transform. 12.5-1. Equation with Difference Kernel on the Entire Axis. Consider the integral equation ⎝integraldisplay∞ –∞K(x–t)y(t)dt=f(x), –∞<x<∞,( 1) where f(x),y(x)∈L2(–∞,∞)a n d K(x)∈L1(–∞,∞). Let us apply the Fourier transform to Eq. (1). In this case, taking into account the convolution theorem (see Subsection 9.4-4), we obtain √ 2π˜K(u)˜y(u)=˜f(u). (2) 12.5. M ETHOD OF INTEGRAL TRANSFORMS 587 Thus, by means of the Fourier transform we have reduced the solution of the original integral equation (1) to the solution of the algebraic equation (2) for the Fourier transform of the desiredsolution. The solution of the latter equation has the form ˜y(u)=1 √ 2π˜f(u) ˜K(u),( 3) where the function ˜f(u)/˜K(u) must belong to the space L2(–∞,∞). Thus, the Fourier transform of the solution of the original integral equation is expressed via the Fourier transforms of known functions, namely, the kernel and the right-hand side of the equation. The solution itself can be expressed via its Fourier transform by means of the Fourier inversion formula: y(x)=1 √ 2π⎝integraldisplay∞ –∞˜y(u)eiuxdu=1 2π⎝integraldisplay∞ –∞˜f(u) ˜K(u)eiuxdu.( 4 ) 12.5-2. Equations with Kernel K(x,t)=K(x/t)o nt h eS e m i a x i s . The integral equation of the ﬁrst kind ⎝integraldisplay∞ 0K(x/t)y(t)dt=f(x), 0 ≤x<∞,( 5) can be reduced to the form (1) by the change of variables x=eξ,t=eτ,w(τ)=ty(t). The solution to this equation can also be obtained by straightforward application of the Mellin transform, and this method is applied in a similar situation in the next section. 12.5-3. Equation with Kernel K(x,t)=K(xt) and Some Generalizations. 1◦. We ﬁrst consider the equation ⎝integraldisplay∞ 0K(xt)y(t)dt=f(x), 0 ≤x<∞.( 6) By changing variables x=eξandt=e–τthis equation can be reduced to the form (1), but it is more convenient here to apply the Mellin transform (see Section 9.3). On multiplying Eq. (6) by xs–1and integrating with respect to xfrom 0 to ∞, we obtain ⎝integraldisplay∞ 0y(t)dt⎝integraldisplay∞ 0K(xt)xs–1dx=⎝integraldisplay∞ 0f(x)xs–1dx. We make the change of variables z=xtin the inner integral of the double integral. This implies the relation ˆK(s)⎝integraldisplay∞ 0y(t)t–sdt=ˆf(s). (7) Taking into account the formula ⎝integraldisplay∞ 0y(t)t–sdt=ˆy(1 –s), we can rewrite Eq. (7) in the form ˆK(s)ˆy(1 –s)=ˆf(s). (8) 588 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) Replacing 1 – sbysin (8) and solving the resulting relation for ˆ y(s), we obtain the transform ˆy(s)=ˆf(1 –s) ˆK(1 –s)(9) of the desired solution. Applying the Mellin inversion formula, we obtain the solution of the integral equation (6) in the form y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞ˆf(1 –s) ˆK(1 –s)x–sds. 2◦. Now we consider the more complicated equation ⎝integraldisplay∞ 0K⎝parenleftbig ϕ(x)ψ(t)⎝parenrightbig g(t)y(t)dt=f(x). (10) Assume that the conditions ϕ(0) = 0, ϕ(∞)=∞,ϕ/prime x>0 ,ψ(0) = 0, ψ(∞)=∞,a n d ψ/prime x>0a r e satisﬁed. The transform z=ϕ(x), τ=ψ(t), y(t)=g(t) ψ/prime t(t)w(τ) takes (10) to the following equation of the form (6): ⎝integraldisplay∞ 0K(zτ)w(τ)dτ=F(z), where the function F(z) is deﬁned parametrically by F=f(x),z=ϕ(x). In many cases, on eliminating xfrom these relations, we obtain the dependence F=F(z) in an explicit form. References for Section 12.5: V . A. Ditkin and A. P. Prudnikov (1965), M. L. Krasnov, A. I. Kiselev, and G. I. Maka- renko (1971). 12.6. Krein’s Method and Some Other Exact Methods for Integral Equations of Special Types 12.6-1. Krein’s Method for an Equation with Difference Kernel with a Weak Singularity. 1◦. Here we describe a method for constructing exact closed-form solutions of linear integral equations of the ﬁrst kind with weak singularity and with arbitrary right-hand side. The method is based on the construction of the auxiliary solution of the simpler equation whose right-hand side isequal to one. The auxiliary solution is then used to construct the solution of the original equation for an arbitrary right-hand side. Consider the equation ⎝integraldisplay a –aK(x–t)y(t)dt=f(x), – a≤x≤a.( 1 ) Suppose that the kernel of the integral equation (1) is polar or logarithmic and that K(x)i sa ne v e n positive deﬁnite function that can be expressed in the form K(x)=β|x|–µ+M(x), 0 < µ<1 , K(x)=βln1 |x|+M(x), respectively, where β>0 ,– 2 a≤x≤2a,a n dM(x) is a sufﬁciently smooth function. 12.6. K REIN’SMETHOD AND SOME OTHER EXACT METHODS FOR INTEGRAL EQUATIONS OF SPECIAL TYPES 589 Along with (1), we consider the following auxiliary equation containing a parameter ξ(0≤ξ≤a): ⎝integraldisplayξ –ξK(x–t)w(t,ξ)dt=1 , – ξ≤x≤ξ.( 2) 2◦. For any continuous function f(x), the solution of the original equation (1) can be expressed via the solution of the auxiliary equation (2) by the formula y(x)=1 2M/prime(a)⎝bracketleftBigd da⎝integraldisplaya –aw(t,a)f(t)dt⎝bracketrightBig w(x,a) –1 2⎝integraldisplaya |x|w(x,ξ)d dξ⎝bracketleftBig1 M/prime(ξ)d dξ⎝integraldisplayξ –ξw(t,ξ)f(t)dt⎝bracketrightBig dξ –1 2d dx⎝integraldisplaya |x|w(x,ξ) M/prime(ξ)⎝bracketleftBig⎝integraldisplayξ –ξw(t,ξ)df(t)⎝bracketrightBig dξ,(3) where M(ξ)=⎝integraltextξ 0w(x,ξ)dx, the prime stands for the derivative, and the last inner integral is treated as a Stieltjes integral. Formula (3) permits one to obtain some exact solutions of integral equations of the form (1) with arbitrary right-hand side, see Section 3.6 of the ﬁrst part of the book. Example 1. The solution of the integral equation ⎝integraldisplaya –aln⎝parenleftbiggA |x–t|⎝parenrightbigg y(t)dt=f(x), which arises in elasticity, is given by formula (3), where M(ξ)=⎝parenleftBig ln2A ξ⎝parenrightBig–1 ,w(t,ξ)=M(ξ) π⎝radicalbig ξ2–t2. Example 2. Consider the integral equation ⎝integraldisplaya –ay(t)dt |x–t|µ=f(x), 0 < µ<1 , which arises in the theory of elasticity. The solution is given by formula (3), where M(ξ)=2√ π µΓ⎝parenleftBigµ 2⎝parenrightBig Γ⎝parenleftBig1–µ 2⎝parenrightBigξµ,w(t,ξ)=1 πcos⎝parenleftBigπµ 2⎝parenrightBig⎝parenleftbig ξ2–t2⎝parenrightbigµ–1 2. 12.6-2. Kernel is the Sum of a Nondegenerate Kernel and an Arbitrary Degenerate Kernel. 1◦. Consider the Fredholm equation of the ﬁrst kind ⎝integraldisplayb aK(x,t)y(t)dt=f(x). (4) Suppose equation (4) can be solved for any f(x) from some class of functions LF.L e tyf(x) denote the corresponding solution. Now consider the more complex integral equation ⎝integraldisplayb a[K(x,t)+ϕ(x)ψ(t)]u(t)dt=f(x)( 5 ) 590 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) with its kernel containing an additional term ϕ(x)ψ(t). A solution to equation (5) will be sought in the form u(x)=yf(x)+Ayϕ(x), (6) where yϕ(x) is the solution to equation (4) in which f(x) must be replaced with ϕ(x). Substituting (6) into (5) results in the coefﬁcient A: A=–⎝integraldisplayb aψ(t)yf(t)dt 1+⎝integraldisplayb aψ(t)yϕ(t)dt.( 7) Formulas (6)–(7) deﬁne a solution to equation (5), provided the integrals in the numerator and denominator exist, with⎝integraltextb aψ(t)yϕ(t)dt≠–1. In addition, the condition ϕ(x)∈LFmust be satisﬁed. Example 3. The solution of Carleman’s equation ⎝integraldisplay1 0ln|x–t|y(t)dt=f(x) (8) is expressed as yf(x)=1 π2√ x(1 –x)⎝bracketleftbigg⎝integraldisplay1 0√ t(1 –t)f/prime t(t)dt t–x–1 ln 4⎝integraldisplay1 0f(t)dt √ t(1 –t)⎝bracketrightbigg . (9) Now consider the more complex integral equation ⎝integraldisplay1 0⎝bracketleftbigln|x–t|+ψ(t)]u(t)dt=f(x) (10) with its kernel containing an arbitrary additive function ψ(t). In terms of equation (5), we have ϕ(x) = 1 in (10). The corresponding solution (9) to equation (8) with f(x) = 1 is written as y1(x)=–1 πln 4√ x(1 –x). (11) Hence, equation (10) has the solution u(x)=yf(x)+Ay1(x),A=–⎝integraltext1 0ψ(t)yf(t)dt 1+⎝integraltext1 0ψ(t)y1(t)dt. Example 4. Consider the integral equation ⎝integraldisplay∞ 0[cos(xt)+ϕ(x)ψ(t)]y(t)dt=f(x). Its solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.5.1 from Section 3.5. Therefore the solution is y(t)=yf(t)+Ayϕ(t), where yf(t)=2 π⎝integraldisplay∞ 0cos(xt)f(x)dx,yϕ(t)=2 π⎝integraldisplay∞ 0cos(xt)ϕ(x)dx,A=–⎝integraltext∞ 0ψ(t)yf(t)dt 1+⎝integraltext∞ 0ψ(t)yϕ(t)dt. 2◦. The integral equation ⎝integraldisplayb a⎝bracketleftbigg K(x,t)+n⎝summationdisplay m=1ϕm(x)ψm(t)⎝bracketrightbigg u(t)dt=f(x) (12) whose kernel is the sum of the kernel of equation (4) and an arbitrary degenerate kernel can be solved in a similar manner. The solution is sought in the additive form u(x)=yf(x)+n⎝summationdisplay m=1Amyϕm(x), (13) 12.6. K REIN’SMETHOD AND SOME OTHER EXACT METHODS FOR INTEGRAL EQUATIONS OF SPECIAL TYPES 591 where yϕm(x) is the solution to equation (4) in which f(x) must be replaced with ϕm(x). Substituting (13) into (12) results in the following linear algebraic system of equations for the coefﬁcients Am: Am+n⎝summationdisplay j=1Ajσmj=–σm0,m=1 ,...,n; σmj=⎝integraldisplayb aψm(t)yϕj(t)dt,σm0=⎝integraldisplayb aψm(t)yf(t)dt.(14) Corollary . Given a solution to the integral equation (4) with a difference kernel K(x,t)=K(x–t), one can obtain a solution to the integral equation with a difference kernel of the form K(x–t)+Pn(x–t), where Pn(x) is an arbitrary polynomial of any (ﬁnite) degree n. 3◦. Let a function y(x) solve equation (4) and let the condition ⎝integraldisplayb aK(x,t)dt=0 be satisﬁed. Then the function y(x)+C, withCan arbitrary constant, also solves equation (4). 12.6-3. Reduction of Integral Equations of the First Kind to Equations of the Second Kind. In some cases it is possible to reduce integral equations of the ﬁrst kind with constant limits of integration to integral equations of the second kind. In order to be deﬁnite, let us consider an integral equation of the ﬁrst kind on semiaxis ⎝integraldisplay∞ 0[K(x,t)g(t)+L(x,t)]y(t)dt=f(x). (15) Suppose the truncated linear equation ⎝integraldisplay∞ 0K(x,t)u(t)dt=f(x) (16) obtained from (15) by setting L(x,t)=0and g(t)=1,to be an integral transform (see Subsections 9.1-3 and 9.6-5) with the following inverse formula: u(x)=⎝integraldisplay∞ 0M(x,s)f(s)ds. (17) Let us rewrite (15) in such a way, that its left-hand side coincides with (16): ⎝integraldisplay∞ 0K(x,t)u(t)dt=f(x)–⎝integraldisplay∞ 0L(x,t)y(t)dt,u(t)=g(t)y(t). (18) Applying the inverse formula (17) to (18), provided that the function f(x) must be substituted for f(x)–⎝integraltext∞ 0L(x,t)y(t)dt, and changing the integration order, we obtain an integral equation of the second kind with constant limits of integration y(x)+⎝integraldisplay∞ 0N(x,t)y(t)dt=F(x), (19) where N(x,t)=1 g(x)⎝integraldisplay∞ 0M(x,s)L(s,t)ds,F(x)=1 g(x)⎝integraldisplay∞ 0M(x,s)f(s)ds. Here, all integrals are supposed to converge. 592 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) Example 5. Consider the integral equation of the ﬁrst kind ⎝integraldisplay∞ 0[sin(xt)+L(x,t)]y(t)dt=f(x). (20) The solution of the truncated linear integral equation ⎝integraldisplay∞ 0sin(xt)u(t)dt=f(x) (21) is expressed as (see equation 3.5.8 in Section 3.5) u(x)=2 π⎝integraldisplay∞ 0sin(xs)f(s)ds. (22) Up to constant factors, the function f(x) and the solution u(x) in (21)–(22) are the Fourier sine transform pair. In accordance with the method described, the integral equation of the ﬁrst kind with constant limits of integration (20) can be reduced to the integral equation of the second kind with constant limits of integration (19) where N(x,t)=2 π⎝integraldisplay∞ 0sin(xs)L(s,t)ds,F(x)=2 π⎝integraldisplay∞ 0sin(xs)f(s)ds. References for Section 12.6: N. Kh. Arutyunyan (1959), I. C. Gohberg and M. G. Krein (1967), F. D. Gakhov (1977, 1990), S. Feny ¨o and H. W. Stolle (1984, pp. 236–237), A. D. Polyanin and A. I. Zhurov (2007). 12.7. Riemann Problem for the Real Axis The Riemann boundary value problem is one of the main tools for constructing solutions of integral equations provided that various integral transforms can be applied to a given equation and thecorresponding convolution-type theorems can be applied. This problem is investigated by an example of the Fourier integral transform. 12.7-1. Relationships Between the Fourier Integral and the Cauchy Type Integral. LetY(τ) be a function integrable on a closed or nonclosed contour Lon the complex plane of the variable z=u+iv(τis the complex coordinate of the contour points). Consider the integral of the Cauchy type (see Section 14.2): 1 2πi⎝integraldisplay LY(τ) τ–zdτ. This integral deﬁnes a function that is analytic on the complex plane with a cut along the contour L. IfLis a closed curve, then the integral is a function that is analytic on each of the connected parts of the plane bounded by L. If the contour Lis the real axis, then we have 1 2πi⎝integraldisplay∞ –∞Y(τ) τ–zdτ=⎝braceleftbigg Y+(z)i f I m z>0 , Y–(z)i f I m z<0 .(1) Moreover, there exist limit values of the functions Y±(z) on the real axis, and these values are related to the density Yof the integral by the Sokhotski–Plemelj formulas Y+(u)=1 2Y(u)+1 2πi⎝integraldisplay∞ –∞Y(τ) τ–udτ, Y–(u)=–1 2Y(u)+1 2πi⎝integraldisplay∞ –∞Y(τ) τ–udτ,(2) or Y+(u)–Y–(u)=Y(u),Y+(u)+Y–(u)=1 πi⎝integraldisplay∞ –∞Y(τ) τ–udτ.( 3) 12.7. R IEMANN PROBLEM FOR THE REALAXIS 593 In the latter formulas, the integral is understood as a singular integral in the sense of the Cauchy principal value. In the Fourier integral* Y(u)=1 √ 2π⎝integraldisplay∞ –∞y(x)eiuxdx, the real parameter uoccurs in an analytic function, and therefore we can replace uin this integral by a complex variable z. The function Y(z) deﬁned by the integral Y(z)=1 √ 2π⎝integraldisplay∞ –∞y(x)eizxdx (4) is analytic in the part of the complex plane of the variable z=u+ivin which the integral (4) is absolutely convergent. If thi s is a domain indeed, i.e., if it is not reduced to the real axis, then the integral (4) gives an analytic continuation of the Fourier integral into the complex plane. The integral (4) will also be called the F ourier integral . Let us establish a relationship between this integral and the integral of the Cauchy type with density Y(u) taken along the entire axis. We have 1 2πi⎝integraldisplay∞ –∞Y(τ) τ–zdτ=1 √ 2π⎝integraldisplay∞ 0y(x)eizxdx, 1 2πi⎝integraldisplay∞ –∞Y(τ) τ–zdτ=–1 √ 2π⎝integraldisplay0 –∞y(x)eizxdx,Imz>0 , Imz<0 .(5) (6) 12.7-2. One-Sided Fourier Integrals. IfY(z)=Y+(z) is an analytic function in the upper half-plane whose limit value on the real axis is given by the function Y(u)=Y+(u)∈L2(–∞,∞), then the function Y+(z) can be expressed by means of the Cauchy integral. Hence, by virtue of (5) we have Y+(z)=1 √ 2π⎝integraldisplay∞ 0y(x)eizxdx, and, since the integral deﬁnes a continuous function, the limit values on the axis can be obtained from the last relation merely by setting z=u: Y+(u)=1 √ 2π⎝integraldisplay∞ 0y(x)eiuxdx, where, according to (5), y(x) the inverse transform of Y(u). The right-hand side can be regarded as the Fourier integral of a function that is identically zero for negative x. Hence, by the uniqueness of the representation of the function Y+(u) by a Fourier integral, it follows that y(x)≡0 on the negative semiaxis. Conversely, if y≡0f o rx< 0, then the Fourier integral of this function becomes Y(u)=1 √ 2π⎝integraldisplay∞ 0y(x)eiuxdx. * In Sections 12.7–12.9, the alternative Fourier transform is used (see Subsection 9.4-3). 594 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) If we replace the parameter uby a complex number zbelonging to the upper half-plane, then the integral will converge even better. This implies the analyticity of the function Y(z)=1 √ 2π⎝integraldisplay∞ 0y(x)eizxdx in the upper half-plane. The case of the lower half-plane can be treated in a similar way. The integrals Y+(z)=1 √ 2π⎝integraldisplay∞ 0y(x)eizxdx,Y–(z)=–1 √ 2π⎝integraldisplay0 –∞y(x)eizxdx (7) are called one-sided F ourier integrals , namely, the right and the leftFourier integral, respectively. As well as in formula (1), the symbols ±over symbols of functions mean that the corresponding function is analytic in the upper or lower half-plane, respectively. Let us introduce the functions y+(x)=⎝braceleftbigg y(x)f o r x>0 , 0f o r x<0 ,y–(x)=⎝braceleftbigg0f o r x>0 , –y(x)f o r x<0 .(8) These functions are said to be one-sided functions fory(x), namely, the right function and the left function , respectively. Obviously, the following relation holds: y(x)=y+(x)–y–(x). (9) Applying the well-known function sign xdeﬁned by signx=⎝braceleftbigg 1f o r x>0 , –1 for x<0 ,(10) we can express y±in terms of yas follows: y±(x)=1 2(±1+s i g n x)y(x). (11) The symbols ±on symbols of one-sided functions will be always subscripts. The Fourier integrals of the right and left one-sided functions are the boundary values of functions that are analytic on the upper and lower half-planes, respectively. Let us indicate the following analogs of the Sokhotski–Plemelj formulas (3) in the Fourier integrals: Y(u)=1 √ 2π⎝integraldisplay∞ –∞y(x)eiuxdx =1 √ 2π⎝integraldisplay∞ 0y(x)eiuxdx+1 √ 2π⎝integraldisplay0 –∞y(x)eiuxdx=Y+(u)–Y–(u), 1 πi⎝integraldisplay∞ –∞Y(τ) τ–udτ=Y+(u)+Y–(u) =1 √ 2π⎝integraldisplay∞ 0y(x)eiuxdx–1 √ 2π⎝integraldisplay0 –∞y(x)eiuxdx=1 √ 2π⎝integraldisplay∞ –∞y(x)s i g nxeiuxdx.(12) Thus, in this setting, the ﬁrst Sokhotski–Plemelj f ormula (a representation of an arbitrary function in the form of the difference of boundary values of analytic functions) is an obvious consequence of the decomposition of a Fourier integral into the right and the left integral. The second formula canalso be rewritten as follows: F{y(x)s i g nx}=1 πi⎝integraldisplay∞ –∞Y(τ) τ–udτ,F–1⎝braceleftbigg1 πi⎝integraldisplay∞ –∞Y(τ) τ–udτ⎝bracerightbigg =y(x)s i g nx. (13) 12.7. R IEMANN PROBLEM FOR THE REALAXIS 595 12.7-3. Analytic Continuation Theorem and the Generalized Liouville Theorem. Below is the analytic continuation theorem and the generalized Liouville theorem combined into a single statement, which will be used in Chapters 12 and 13. Let functions Y1(z)a n dY2(z) be analytic in the upper and lower half-planes, respectively, possibly except for a point z∗≠∞, at which these functions have a pole. If Y1(z)a n dY2(z)a r e bounded at inﬁnity, the principal parts of their expansions in a neighborhood of z∗have the form c1 z–z∗+c2 (z–z∗)2+···+cm (z–z∗)m≡Pm–1(z) (z–z∗)m, and if the functions themselves coincide on the real axis, then these functions represent a single rational function on the entire plane: Y(z)=c0+Pm–1(z) (z–z∗)m, where c0is a constant. The pole z∗can belong either to the open half-planes or to the real axis. Let us also give a more general version of the above statement. If functions Y1(z)a n dY2(z) are analytic in the upper and lower half-planes, respectively, possibly except for ﬁnitely many points z0=∞,zk(k=1 ,...,n), at which these functions can have poles if the principal parts of the expansions of these functions in a neighborhood of a pole have the form c0 1z+c0 2z2+···+c0 m0zm0≡P0(z) ck 1 z–zk+ck 2 (z–zk)2+···+ck mk (z–zk)mk≡Pmk–1(z) (z–zk)mkat the point z0, at the points zk, and if the functions themselves coincide on the real axis, then these functions represent a single rational function on the entire plane: Y(z)=C+P0(z)+n⎝summationdisplay k=1Pmk–1(z) (z–zk)mk where Cis a constant. The poles zkcan belong either to the open half-planes or to the real axis. 12.7-4. Riemann Boundary Value Problem. The solution of the Riemann problem in this section differs from the traditional one, because it is expressed not by means of integrals of the Cauchy type (see Subsection 14.3-8) but by means of Fourier integrals. To solve equations of convolution type under consideration, the Fourier integraltechnique is more convenient. By the index of a continuous complex-valued nonvanishing function M(u)(M(u)=M 1(u)+ iM2(u), –∞<u<∞,M(–∞)=M(∞)) we mean the variation of the argument of this function on the real axis expressed in the number of full rotations: IndM(u)=1 2π⎝bracketleftbig argM(u)⎝bracketrightbig∞ –∞=1 2πi⎝bracketleftbig lnM(u)⎝bracketrightbig∞ –∞=1 2πi⎝integraldisplay∞ –∞dlnM(u). IfM(u) is not differentiable but is of bounded variation, then the last integral must be understood as the Stieltjes integral. If an analytic function Y(z) has a representation of the form Y(z)=(z–z0)mY1(z) in a neighborhood of some point z0,w h e r e Y1(z) is analytic and Y1(z0)≠0, then the integer m (which can be positive, negative, or zero) is called the order of the function Y(z) at the point z0. 596 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) Ifm> 0, then the order of the function is the order of its zero, and if m< 0, then the order of the function is minus the order of its pole. If the order of the function at z0is zero, then at this point the function takes a ﬁnite nonzero value. When considering the point at inﬁnity we must replace the difference z–z0by 1/z . Let us pose the Riemann problem. Let two functions be given on the real axis, namely, D(u), the coefﬁcient of the problem ,a n dH(u), the right-hand side , and let the following normality condition hold:D(u)≠0. The functions H(u)a n dD(u) – 1 belong to L2(–∞,∞) and simultaneously satisfy the H ¨older condition. The problem is to ﬁnd two functions Y±(z) that are analytic in the upper and the lower half-plane, respectively,* whose limit values on the real axis satisfy the following boundary condition: Y+(u)=D(u)Y–(u)+H(u). (14) It follows from the representation of D(u)t h a tD(∞) = 1. The last condition implies no loss of generality of subsequent reasoning because by dividing the boundary condition (14) by D(∞)w e can always obtain the necessary form of the problem.** IfD(u)≡1, then the Riemann problem is called the jump problem .F o rH(u)≡0, the Riemann problem is said to be homogeneous . The index νof the coefﬁcient D(u) of the boundary value problem is called the index of the Riemann problem. Consider the jump problem , i.e., the problem of ﬁnding Y±(z) from the boundary condition Y+(u)–Y–(u)=H(u). (15) The solution of this problem is given by the ﬁrst formula in (12): Y+(z)=1 √ 2π⎝integraldisplay∞ 0H(x)eizxdx,Y–(z)=–1 √ 2π⎝integraldisplay0 –∞H(x)eizxdx, (16) where H(x)=1 √ 2π⎝integraldisplay∞ –∞H(u)e–iuxdu. (17) Let us construct a particular solution X(z) of the homogeneous Riemann problem (14), which we need in what follows: X+(u)=D(u)X–(u),D(∞) = 1, (18) whereX(z) is assumed to be nonzero on the real axis with the additional condition X±(∞)=1 . Denote by N+andN–the numbers of zeros of the functions X+(z)a n dX–(z) in the upper and lower half-planes, respectively. On calculating the index of both sides of the boundary condition (18) and applying the properties of the index, we obtain N++N–=I n dD(u)=ν. (19) We ﬁrst assume that ν= 0. In this case, ln D(u) is a single-valued function. It follows from relation (19) that N+=N–= 0, i.e., the solution has no zeros on the entire plane. Therefore, the functions ln X+(z)a n dl n X–(z) are analytic in the corresponding half-planes, and hence are single-valued together with their boundary values ln X+(u)a n dl n X–(u). Taking the logarithm of the boundary condition (18), we obtain lnX+(u)–l nX–(u)=l nD(u). (20) * A couple of functions Y±(z) can be treated as a single function Y(z) piecewise analytic in the entire complex plane. In some cases, we use the latter notation. ** Since the boundary condition is the main analytic expression of the Riemann problem, in references to the corresponding problem we shall often indicate its boundary condition only and write, for instance, “Riemann problem (14).” 12.7. R IEMANN PROBLEM FOR THE REALAXIS 597 On choosing a branch of ln D(u) such that ln D(∞) = 0 (it can be shown that the ﬁnal result does not depend on the choice of the branch) we arrive at a jump problem. In this case, on the basisof (15)–(17) and (20), the solution of problem (18) can be represented in the form X +(z)=eG+(z),X–(z)=eG–(z), G+(z)=1 √ 2π⎝integraldisplay∞ 0g(x)eizxdx,G–(z)=–1 √ 2π⎝integraldisplay0 –∞g(x)eizxdx, g(x)=1 √ 2π⎝integraldisplay∞ –∞lnD(u)e–iuxdu.(21) Relations (21) imply the followi ng important fact: a function D(u) of zero index that is nonvanishing on the real axis and s atisﬁes the condition D(∞) = 1 can be represented as the ratio of functions that are the boundary values of nonzero analytic functions in the upper and the lower half-plane, respectively. Let us pass to the case in which the index of the homogeneousRiemann problem (18) is arbitrary. By a canonical function X(z) (of the homogeneous Riemann problem) we mean a function that satisﬁes the boundary condition (18) and the condition X±(∞) = 1 and has zero order everywhere possibly except for the point – i, at which the order of X(z) is equal to the index νof the Riemann problem. Such a function can be constructed by reducing the homogeneous Riemann problem to the above case of zero index. Indeed, let us write out the boundary condition of the homogeneous Riemann problem (18) in the form X+(u)=⎝bracketleftbigg⎝parenleftbiggu–i u+i⎝parenrightbigg–ν D(u)⎝bracketrightbigg⎝bracketleftbigg⎝parenleftbiggu–i u+i⎝parenrightbiggν X–(u)⎝bracketrightbigg . (22) In this case, the function in the ﬁrst square brackets has zero index and can be represented as the ratio of the boundary values of functions that are analytic in the upper and the lower half-plane. This, together with the boundary condition (22), gives the following expression for the canonical function: X+(z)=eG+(z),X–(z)=⎝parenleftbiggz–i z+i⎝parenrightbigg–ν eG–(z), G+(z)=1 √ 2π⎝integraldisplay∞ 0g(x)eizxdx,G–(z)=–1 √ 2π⎝integraldisplay0 –∞g(x)eizxdx, g(x)=1 √ 2π⎝integraldisplay∞ –∞ln⎝bracketleftbigg⎝parenleftbiggu–i u+i⎝parenrightbigg–ν D(u)⎝bracketrightbigg e–iuxdu,(23) where, at the point – i,X–(z) has a zero of order νforν> 0 and a pole of order |ν|for the case ν<0 . The coefﬁcient D(u) of the Riemann boundary value problem can be represented as the ratio of the boundary values of the canonical function (see (22) and (23)): D(u)=X+(u) X–(u). (24) Such a representation of D(u) in the form of the ratio of boundary values of the canonical function is often called a factorization . Now we consider the homogeneous Riemann problem with the boundary condition Y+(u)=D(u)Y–(u),D(∞) = 1. (25) On substituting the expression (24) for D(u) into (25) we reduce the boundary condition to the form Y+(u) X+(u)=Y–(u) X–(u). (26) 598 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) According to formulas (23) for X(z), the left- and the right-hand sides of Eq. (26) contain the boundary values of functions that are analytic on the upper and lower half-planes, respectively,possibly except for the point – iat which the order is equal to ν. In the chosen function class, each function vanishes at inﬁnity. In this case, it follows from the analytic continuation theorem and the generalized Liouville theorem (see Subsection 12.7-3) that for ν>0w eh a v e Y +(z) X+(z)=Y–(z) X–(z)=Pν–1(z) (z+i)ν, (27) wherePν–1(z) is an arbitrary polynomial of degree ν– 1 (the degree of the numerator is less than that of the denominator because Y(∞) = 0). Hence, Y(z)=X(z)Pν–1(z) (z+i)ν. (28) Forν≤0, it follows from Y(∞)=0t h a t Y(z)≡0 by the generalized Liouville theorem. Hence, for ν> 0, the homogeneous Riemann boundary value problem has precisely νlinearly independent solutions of the form zk–1X(z) (z+i)ν,k=1 ,2 , ...,ν, and for ν≤0, there are no nontrivial solutions. The right-hand side of Eq. (28) has exactly νzeros on the entire plane, including the zero at inﬁnity. These zeros can lie at arbitrary points of the upper and lower half-plane or on the real axis.Denote the number of zeros on the real axis by N 0. In the general case (without the requirement that there are no zeros on the real axis), formula (19) is replaced by the relation N++N–+N0=I n dD(u)=ν. (29) Let us pass to the solution of the nonhomogeneous Riemann problem with the boundary condi- tion (14). We apply relation (24) and reduce the boundary condition to the form Y+(u) X+(u)=Y–(u) X–(u)+H(u) X+(u). (30) Let us express the last summand as the difference of the boundary values of functions that are analytic in the upper and the lower half-plane (see the jump problem), that is, W+(u)–W–(u)=H(u) X+(u), (31) where W+(z)=1 √ 2π⎝integraldisplay∞ 0w(x)eizxdx,W–(z)=–1 √ 2π⎝integraldisplay0 –∞w(x)eizxdx, w(x)=1 √ 2π⎝integraldisplay∞ –∞H(u) X+(u)e–iuxdu.(32) On substituting (31) into (30), we obtain Y+(u) X+(u)–W+(u)=Y–(u) X–(u)–W–(u). (33) 12.7. R IEMANN PROBLEM FOR THE REALAXIS 599 Forν> 0, it follows from the analytic continuation theorem and the generalized Liouville theorem that Y+(z) X+(z)–W+(z)=Y–(z) X–(z)–W–(z)=Pν–1(z) (z+i)ν. Hence, for ν>0w eh a v e Y(z)=X(z)⎝bracketleftbigg W(z)+Pν–1(z) (z+i)ν⎝bracketrightbigg . (34) The right-hand side of formula (34) contains the general solution (28) of the homogeneous problem as a summand, and hence the general solution of the nonhomogeneous problem is obtained. Forν≤0w em u s ts e t Pν–1(z)≡0, and the desired solution becomes Y(z)=X(z)W(z). (35) However, formula (35) gives a so lution that satisﬁes all conditions for ν= 0 only. For ν<0 ,t h e function X(z) has a pole of order |ν|at the point – i. In this case, for the existence of a solution in the chosen class of functions it is necessary that the second factor have a zero of the corresponding order at the point – i. On the basis of relations (6) and (32), we represent the function W–(z)i nt h e form W–(z)=1 2πi⎝integraldisplay∞ –∞H(τ) X+(τ)dτ τ–z. On expanding the last integral in series in powers of z+iand equating the coefﬁcients of ( z+i)k–1 (k=1 ,2 , ...,|ν|) with zero, we obtain the solvability conditions for the problem in the form ⎝integraldisplay∞ –∞H(u) X+(u)du (u+i)k=0 , k=1 ,2 , ...,|ν|. (36) Figure 5 depicts a scheme of the above method for solving the Riemann problem on the real axis. Let us state the results concerning the solution of the Riemann problem in the ﬁnal form. If the indexνof the problem satisﬁes the condition ν> 0, then the homogeneous and the nonhomogeneous Riemann problems are unconditionally solvable, and their solutions Y±(z)=X±(z)Pν–1(z) (z+i)ν(the homogeneous problem), (37) Y±(z)=X±(z)⎝bracketleftbigg W±(z)+Pν–1(z) (z+i)ν⎝bracketrightbigg (the nonhomogeneous problem) (38) depend on νarbitrary complex constants, where Pν–1(z) is a polynomial of degree ν–1 . I f ν≤0, then the homogeneous problem has only the trivial zero solution, and the nonhomogeneous problem has the unique solution Y±(z)=X±(z)W±(z) (39) provided that |ν|conditions (36) hold. Here we have g(x)=1 √ 2π⎝integraldisplay∞ –∞ln⎝bracketleftbigg⎝parenleftbiggu–i u+i⎝parenrightbigg–ν D(u)⎝bracketrightbigg e–iuxdu, (40) G+(z)=1 √ 2π⎝integraldisplay∞ 0g(x)eizxdx,G–(z)=–1 √ 2π⎝integraldisplay0 –∞g(x)eizxdx, (41) X+(z)=eG+(z),X–(z)=⎝parenleftbiggz–i z+i⎝parenrightbigg–ν eG–(z), (42) w(x)=1 √ 2π⎝integraldisplay∞ –∞H(u) X+(u)e–iuxdu, (43) W+(z)=1 √ 2π⎝integraldisplay∞ 0w(x)eizxdx,W–(z)=–1 √ 2π⎝integraldisplay0 –∞w(x)eizxdx. (44) 600 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) Application of the theorem on analytical continuation and the Liouville theorem Figure 5. Scheme of solving the Riemann boundary value problem for the functions Y+(z)a n dY–(z) that are analytic, respectively, in the upper and the lower half-plane of the complex plane z=u+iv. It is assumed that D(u)≠0a n d Pν–1(z)≡0f o rν≤0. The sequence of operations to construct a solution can be described as follows. 1◦. By virtue of formula (40) we ﬁnd g(x), and then, with the help of (41), for the given g(x)w e ﬁndG±(z). 2◦. By formulas (42) the canonical function X±(z) is determined. 3◦. By formula (43) we determine w(x), and then apply formula (44) to ﬁnd W±(z). 12.7. R IEMANN PROBLEM FOR THE REALAXIS 601 After this, solutions of the homogeneous and nonhomogeneous problems can be found by formulas (37)–(39) and (42). For the case ν< 0, it is also necessary to verify the solvability conditions (36). 12.7-5. Problems with Rational Coefﬁcients. The solution of the Riemann problem thus obtained requires evaluation of several Fourier integrals.This can also be readily expressed by means of integrals of the Cauchy type. As a rule, the integrals cannot be evaluated in the closed form and are calculated by various approximate methods. This process is rather cumbersome, and therefore it is of interest to select cases in which the solution canbe obtained directly from the boundary condition by applying the method of analytic continuation without using the antiderivatives. Assume that in the boundary condition (14) we have D(u)=R +(u) Q+(u)R–(u) Q–(u). HereR+(u)a n dQ+(u)(R–(u)a n dQ–(u)) are polynomials whose zeros belong to the upper (lower) half-plane (we must avoid confusing these polynomials with the one-sided functions introduced above, which have similar notation). Denote the degrees of the polynomials P+,R–,Q+,a n dQ– bym+,m–,n+,a n dn–, respectively. Since, by the assumption of the problem, the value D(∞) can be neither zero nor inﬁnity, i t follows that the relation m++m–=n++n–holds. The index of the problem can be expressed by the formula ν=I n dD(u)=m+–n+=– (m––n–). On multiplying the boundary condition by Q–(u)/P–(u) we obtain Q–(u) R–(u)Y+(u)–R+(u) Q+(u)Y–(u)=Q–(u) R–(u)H(u). IfH(u) is a rational function as well, then the jump problem can readily be solved: W+(u)–W–(u)=Q–(u) R–(u)H(u). (45) To this end, it sufﬁces to decompose the right-hand side into the sum of partial fractions. Then W+(u) andW–(u) are the sums of the partial fractions with poles in the lower and the upper half-planes, respectively. We can directly apply the continuity principle (the analytic continuation theorem) andthe generalized Liouville theorem to the resulting relation Q –(u) R–(u)Y+(u)–W+(u)=R+(u) Q+(u)Y–(u)–W–(u). The only exceptional point at which the analytic function, which is the same on the entire complex plane, can have a nonzero order is the point at inﬁnity, at which the order of the function is equal to ν–1=m+–n+–1=n––m––1 . Forν> 0, the solution can be written in the form Y+(z)=R–(z) Q–(z)[W+(z)+Pν–1(z)],Y–(z)=Q+(z) R+(z)[W–(z)+Pν–1(z)]. Forν≤0w em u s ts e t Pν–1≡0; moreover, for ν< 0 we must also write out the solvability conditions that can be obtained by equating with zero the ﬁrst |ν|terms of the expansion of the rational function W(z) in a series (in powers of 1/z ) in a neighborhood of the point at inﬁnity. 602 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) The solution of the jump problem (45) can be obtained either by applying the method of indeterminate coefﬁcients, as is usually performed in the integration of rational functions, or usingthe theory of residuals of analytic functions. Let z kbe a pole, of multiplicity m, of the function⎝bracketleftbig Q–(z)/R–(z)⎝bracketrightbig H(z). Then the coefﬁcients of the principal part of the decomposition of this function in a neighborhood of the point zk, which has the form ck 1 z–zk+···+ck m (z–zk)m, can be found by the formula ck j=1 (j–1 ) !dj–1 dzj–1⎝bracketleftbiggQ–(z) R–(z)H(z)⎝bracketrightbigg z=zk. The above case is not only of independent interest, as it frequently occurs in practice, but also of importance as a possible way of solving the problem under general assumptions. The approximation of arbitrary coefﬁcients of the class under consideration by rational functions is a widespread methodof approximate solution of the Riemann boundary value problem. 12.7-6. Exceptional Cases. The Homogeneous Problem. Assume that the coefﬁcient D(u) of a Riemann boundary value problem has zeros of orders α1,...,αr at points a1,...,ar, respectively, and poles* of the orders β1,...,βsat points b1,...,bs(α1,...,αr andβ1,...,βsare positive integers). Thus, the coefﬁcient can be represented in the form D(u)=r⎝productdisplay i=1(u–ai)αi s⎝productdisplay j=1(u–bj)βjD1(u),D1(u)≠0, –∞ <u<∞,r⎝summationdisplay i=1αi=m,s⎝summationdisplay j=1βj=n. (46) In turn, we represent the function D1(u) (see Subsection 12.7-5) in the form D1(u)=R+(u)R–(u) Q+(u)Q–(u)D2(u), (47) where, as above, R+(u)a n dQ+(u)(R–(u)a n dQ–(u)) are polynomials of degrees m+andn+(m– andn–) whose zeros belong to the upper (lower) half-plane. The function D2(u) satisﬁes the H ¨older condition, has zero index, and nowhere vanishes on the real axis. Moreover, this function can be subjected to some differentiability conditions in neighborhoods of the points aiandbjand possibly in a neighborhood of the point at inﬁnity. The boundary condition of the homogeneous Riemann problem can be rewritten in the form Y+(u)=r⎝productdisplay i=1(u–ai)αiR+(u)R–(u) s⎝productdisplay j=1(u–bj)βjQ+(u)Q–(u)D2(u)Y–(u). (48) * For the case in which the function D(u) is not analytic, the term “pole” will be used for points at which the function tends to inﬁnity with integer order. 12.7. R IEMANN PROBLEM FOR THE REALAXIS 603 We seek a solution in the class of functions that are bounded on the real axis and vanish at inﬁnity: Y(∞) = 0. (49) The coefﬁcient D(u) has the order η=n+n++n––m–m+–m– (50) at inﬁnity. The number ν=m+–n+ (51) is called the index of the problem. Let us introduce the notation h=n––m–. (52) Then the order at inﬁnity is expressed by the formula η=h–ν+n–m. (53) Now let us proceed with the solution of problem (48). Applying general methods, we set D2(u)=eG+(u) eG–(u),g(x)=1 √ 2π⎝integraldisplay∞ –∞lnD2(u)e–iuxdu, G+(z)=1 √ 2π⎝integraldisplay∞ 0g(x)eizxdx,G–(z)=–1 √ 2π⎝integraldisplay0 –∞g(x)eizxdx(54) and rewrite the boundary condition in the form Q–(u)Y+(u) r⎝productdisplay i=1(u–ai)αiR–(u)eG+(u)=R+(u)Y–(u) s⎝productdisplay j=1(u–bj)βjQ+(u)eG–(u). (55) As above, we can apply the analytic continua tion and the generalized Liouville theorem and obtain a pole at inﬁnity as the only possible singularity. Two cases are possible: 1◦. Let the order ηof the coefﬁcient of the boundary value problem at inﬁnity satisfy the condition η≥0, i.e., let D(u) have a zero of order ηat inﬁnity. It follows from (53) that n–ν≥m–h.O n equating the left- and right-hand sides of relation (55) with a polynomial Pν–n–1(z), we obtain the solution of the boundary value problem in the form Y+(z)=r⎝productdisplay i=1(z–ai)αiR–(z) Q–(z)eG+(z)Pν–n–1(z), Y–(z)=s⎝productdisplay j=1(z–bj)βjQ+(z) R+(z)eG–(z)Pν–n–1(z).(56) This problem has ν–nlinearly independent solutions for ν–n> 0 and only the trivial zero solution for ν–n≤0. 2◦.L e tη< 0, i.e., let D(u) have a pole of order – ηat inﬁnity. In this case, m–h>n–ν,a n dw e can obtain the general solution from (56) by replacing Pν–n–1(z)b yPh–m–1(z) in this expression. In this case, the problem has h–msolutions for h–m> 0 and only the trivial zero solution for h–m≤0. According to (53), we have h–m=ν–n+η. (57) Thus, in both cases under consideration, the number of linearly independent solutions is equal to the index minus the total number of the poles (including the pole at inﬁnity) of the coefﬁcient D(u). Hence, we have the following law: the number of linearly independent solutions of a homogeneousRiemann problem is not affected by the number of zeros of the coefﬁcient and is reduced by the total number of its poles. 604 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) 12.7-7. Exceptional Cases. The Nonhomogeneous Problem. Assume that the right-hand side has the same poles as the coefﬁcient. The boundary condition can be rewritten as follows: Y+(u)=r⎝productdisplay i=1(u–ai)αiR+(u)R–(u) s⎝productdisplay j=1(u–bj)βjQ+(u)Q–(u)D2(u)Y–(u)+H1(u) s⎝productdisplay j=1(u–bj)βj, (58) whereD2(u)a n dH1(u) satisfy the H ¨older condition and some additional differentiability conditions near the points ai,bj,a n d∞. 1◦. Assume that the order ηat inﬁnity of the coefﬁcient of the boundary value problem satisﬁes the condition η≥0. Since the ﬁrst two terms of relation (58) vanish at inﬁnity, it follows that the minimal possible order of H1(u) at inﬁnity is equal to 1 – n. Just as in the homogeneous problem, we replace D2(u) by the ratio of two functions (54) and write out the boundary condition in the following form (under the braces, the orders of the functions at inﬁnity are indicated): s⎝productdisplay j=1(u–bj)βjQ–(u)Y+(u) R–(u)eG+(u) ⎝bracehtipupleft ⎝bracehtipdownright⎝bracehtipdownleft ⎝bracehtipupright 1–n–h=r⎝productdisplay i=1(u–ai)αiR+(u)Y–(u) Q+(u)eG–(u) ⎝bracehtipupleft ⎝bracehtipdownright⎝bracehtipdownleft ⎝bracehtipupright 1–m –ν+H1(u)Q–(u) R–(u)eG+(u) ⎝bracehtipupleft ⎝bracehtipdownright⎝bracehtipdownleft ⎝bracehtipupright 1–n–h. Assume that a polynomial S(u) of the degree n+h– 1 represents the principal part of the decompo- sition of the last term in a neighborhood of the point at inﬁnity (for the case in which n+h–1≥0): H1(u)Q–(u) R–(u)eG+(u)=S(u)+W(u),W(∞)=0 . On replacing the function W(u) by the difference of boundary values of analytic functions W(u)=W+(u)–W–(u), (59) where w(x)=1 √ 2π⎝integraldisplay∞ –∞W(u)e–iuxdu, W+(u)=1 √ 2π⎝integraldisplay∞ 0w(x)eiuxdx,W–(u)=–1 √ 2π⎝integraldisplay0 –∞w(x)eiuxdx,(60) we reduce the boundary condition to the form s⎝productdisplay j=1(u–bj)βjQ–(u)Y+(u) R–(u)eG+(u)–S(u)–W+(u)=r⎝productdisplay i=1(u–ai)αiR+(u)Y–(u) Q+(u)eG–(u)–W–(u). On applying the analytic continuation theorem and the generalized Liouville theorem and taking into account the fact that the only possible singular point of the function under consideration is thepoint at inﬁnity, while we have the relation – n–h≤–m–ν(η≥0), we obtain the expressions Y +(z)=R–(z)eG+(z) s⎝productdisplay j=1(z–bj)βjQ–(z)[W+(z)+S(z)+Pν+m–1(z)], Y–(z)=Q+(z)eG–(z) r⎝productdisplay i=1(z–ai)αiR+(z)[W–(z)+Pν+m–1(z)].(61) 12.7. R IEMANN PROBLEM FOR THE REALAXIS 605 The last formulas deﬁne a solution that has pole singularities at the points aiandbj. To obtain a bounded solution, we apply the canonical function of the nonhomogeneous problem. By a canonical function V(z)of the nonhomogeneous Riemann problem in the exceptional case we mean a piecewise analytic function that satisﬁes the boundary condition (58), has the zero orderon the entire ﬁnite part of the complex plane, including the points a iandbj, and has the least possible order at inﬁnity. LetUp(z) be the Hermite interpolation polynomial with interpolation nodes of orders αiandβj at the points aiandbj, respectively. Such a polynomial of degree p=m+n– 1 exists and is determined uniquely (see Subsection 14.3-2). The functions D1(u)a n dH1(u) must be subjected to the additional condition that in neighborhoodsof the points aiandbjthese functions have derivatives of the orders αiandβj, respectively, and these derivatives satisfy the H ¨older condition. Then the canonical function of the nonhomogeneous problem can be represented in the form V+(z)=R–(z)eG+(z) s⎝productdisplay j=1(z–bj)βjQ–(z)[W+(z)+S(z)–Up(z)], V–(z)=Q+(z)eG–(z) r⎝productdisplay i=1(z–ai)αiR+(z)[W–(z)–Up(z)].(62) Adding V(z) to the above general solution of the homogeneous problem, we ﬁnd the general solution of the nonhomogeneous problem under consideration: Y+(z)=V+(z)+r⎝productdisplay i=1(z–ai)αiR–(z) Q–(z)eG+(z)Pν–n–1(z), Y–(z)=V–(z)+s⎝productdisplay j=1(z–bj)βjQ+(z) R+(z)eG–(z)Pν–n–1(z).(63) Forν–n> 0, the problem has ν–nlinearly independent solutions. In the case ν–n≤0w em u s ts e t Pν–n–1(z)≡0. For ν–n< 0, the canonical function V(z) has the order ν–n< 0 at inﬁnity and hence is no longer a solution of the nonhomogeneous problem. However, on subjecting the right-hand side ton–νconditions we can increase the order of the function V(u) at inﬁnity by n–νand thus make the canonical function V(z) be a solution of the nonhomogeneous problem again. To make the above operations possible, it sufﬁces to require that the functions ukH1(u)a n d D2(u) have derivatives of order ≤n–νat inﬁnity, and these derivatives satisfy the H ¨older condition. 2◦.L e tη< 0. The least possible order at inﬁnity of H1(u)i sh–ν–m+ 1. In this case, the function [H1(u)Q–(u)]/[R–(u)eG+(u)] in the boundary condition (58) has the order 1 – m–νat inﬁnity. After selecting the princip al part of the expansion of [ H1(u)Q–(u)]/[R–(u)eG+(u)] in a neighborhood of the point at inﬁnity for m+ν– 1 > 0, the boundary condition can be rewritten in the form s⎝productdisplay j=1(u–bj)βjQ–(u)Y+(u) R–(u)eG+(u)–W+(u)=r⎝productdisplay i=1(u–ai)αiR+(u)Y–(u) Q+(u)eG–(u)–W–(u)+S(u). The canonical function of the nonhomogeneous problem can be expressed via the interpolation 606 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) polynomial as follows: V+ 1(z)=R–(z)eG+(z) s⎝productdisplay j=1(z–bj)βjQ–(z)[W+(z)–Up(z)], V– 1(z)=Q+(z)eG–(z) r⎝productdisplay i=1(z–ai)αiR+(z)[W–(z)–S(z)–Up(z)].(64) The general solution of problem (58) becomes Y+(z)=V+ 1(z)+r⎝productdisplay i=1(z–ai)αiR–(z) Q–(z)eG+(z)Ph–m–1(z), Y–(z)=V– 1(z)+s⎝productdisplay j=1(z–bj)βjQ+(z) R+(z)eG–(z)Ph–m–1(z).(65) Forh–m> 0, the problem has h–mlinearly independent solutions. In the case h–m≤0, we must set the polynomial Ph–m–1(z) to be identically zero and, for the case in which h–m<0 , impose m–hconditions of the same type as in the previ ous case on the right-hand side. Under these conditions, the nonhomogeneous problem (58) has a unique solution. Remark. In Section 12.8 we consider equations that can be reduced to the problem by applying the convolution theorem for the Fourier transform. Equations to which the convolution theorems for other integral transforms can be applied, for instance, for the Mellin transform, can be investigatedin a similar way. References for Section 12.7: F. D. Gakhov and Yu. I. Cherskii (1978), S. G. Mikhlin and S. Pr ¨ossdorf (1986), A. V . Bit- sadze (1995), N. I. Muskhelishvili (1992). 12.8. Carleman Method for Equations of the Convolution Type of the First Kind By the Carleman method we mean the method of reducing an integral equation to a boundary value problem of the theory of analytic functions, in particular, to the Riemann problem. For equations of convolution type, this reduction can be performed by means of the integral transforms. Aftersolving the boundary value problem, the desired function can be obtained by applying the inverse integral transform. 12.8-1. Wiener–Hopf Equation of the First Kind. Consider the Wiener–Hopf equation of the ﬁrst kind 1 √ 2π⎝integraldisplay∞ 0K(x–t)y(t)dt=f(x), 0 < x<∞,( 1) which is frequently encountered in applications. Let us extend its domain to the negative semiaxis by introducing one-sided functions, y+(x)=⎝braceleftbigg y(x)f o r x>0 , 0f o r x<0 ,f+(x)=⎝braceleftbigg f(x)f o r x>0 , 0f o r x<0 ,y–(x)=0 f o r x>0 . 12.8. C ARLEMAN METHOD FOR EQUATIONS OF THE CONVOLUTION TYPE OF THE FIRST KIND 607 Using these one-sided functions, we can rewrite Eq. (1) in the form 1 √ 2π⎝integraldisplay∞ –∞K(x–t)y+(t)dt=f+(x)+y–(x), – ∞<x<∞.( 2) The auxiliary function y–(x) is introduced to compensate for the left-hand side of Eq. (2) for x<0 . Note that y–(x) is unknown in the domain x< 0 and is to be found in solving the problem. Let us now apply the alternative Fourier transform to Eq. (2). Then we obtain the boundary value problem Y+(u)=1 K(u)Y–(u)+F+(u) K(u).( 3) Ifσis the order of K(u) at inﬁnity, then the order of the coefﬁcient of the boundary value problem at inﬁnity is η=–σ< 0. The general solution of problem (3) can be obtained on the basis of relations (65) from Subsection 12.7-7 by replacing Ph–m–1(z) withPν–n+η–1(z) there. The solution of the original equation (1) can be obtained from the solution of problem (3) by means of the inversion formula y(x)=y+(x)=1 √ 2π⎝integraldisplay∞ –∞Y+(u)e–iuxdu,x>0 . ( 4 ) Note that in formula (4), only the function Y+(u) occurs explicitly, which is related to the function Y–(u)b y( 3 ) . 12.8-2. Integral Equations of the First Kind with Two Kernels. Consider the integral equation of the ﬁrst kind 1 √ 2π⎝integraldisplay∞ 0K1(x–t)y(t)dt+1 √ 2π⎝integraldisplay0 –∞K2(x–t)y(t)dt=f(x), – ∞<x<∞.( 5 ) The Fourier transform of Eq. (5) results in the following boundary value problem: Y+(u)=K2(u) K1(u)Y–(u)+F(u) K1(u),– ∞<u<∞.( 6) The coefﬁcient of this problem is the ratio of functions that vanish at inﬁnity, and hence, in contrast to the preceding case, it can have a zero or a pole of some order at inﬁnity. LetK1(u)=T1(u)/uλandK2(u)=T2(u)/uµ, where the functions T1(u)a n dT2(u) have zero order at inﬁnity. In the dependence of the sign of the difference η=µ–λ, two cases can occur. For generality, we assume that there are exceptional points at ﬁnite distances as well. Let the functions K1(u)a n dK2(u) have the representations K1(u)=s⎝productdisplay j=1(u–bj)βjp⎝productdisplay k=1(u–ck)γkK11(u), K2(u)=r⎝productdisplay i=1(u–ai)αip⎝productdisplay k=1(u–ck)γkK12(u). Along with the common zeros at points ckof multiplicity γk, the functions K1(u)a n dK2(u) have a common zero of order min( λ,µ) at inﬁnity. 608 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) The coefﬁcient of the Riemann problem can be represented in the form D(u)=r⎝productdisplay i=1(u–ai)αiR+(u)R–(u) s⎝productdisplay j=1(u–bj)βjQ+(u)Q–(u)D2(u). It follows from (6) that this problem and the integral equation (5) are solvable if at any point ckthat is a common zero of the functions K1(u)a n dK2(u), the function F(u) has zero of order γk,i . e . , F(u)h a st h ef o r m F(u)=p⎝productdisplay k=1(u–ck)γkF1(u). To this end, the following γ1+···+γp=lconditions must hold: ⎝bracketleftbig F(jk) u(u)⎝bracketrightbig u=ck=0 , jk=0 ,1 , ...,γk–1 , ( 7 ) or, which is the same, the conditions ⎝integraldisplay∞ –∞f(x)xjkeickxdx=0 . ( 8 ) For the case under consideration in which the equation is of the ﬁrst kind, we must add other dconditions, where d=m i n ( λ,µ)+1 , ( 9 ) that are imposed on the behavior of F(u) at inﬁnity because the functions K1(u)a n dK2(u)h a v ea common zero of order min( λ,µ) at inﬁnity. Hence, F(u) must satisfy the conditions (8) and have at least the order dat inﬁnity. If these conditions are satisﬁed, then the boundary value problem (6) becomes Y+(u)=r⎝productdisplay i=1(u–ai)αiR+(u)R–(u) s⎝productdisplay j=1(u–bj)βjQ+(u)Q–(u)D2(u)Y–(u)+H1(u) s⎝productdisplay j=1(u–bj)βj. The solution was given above in Subsection 12.7-7. For the case in which η≥0(µ≥λ), this solution can be rewritten in the form Y+(z)=V+(z)+r⎝productdisplay i=1(z–ai)αiR–(z) Q–(z)eG+(z)Pν–n+1(z), Y–(z)=V–(z)+s⎝productdisplay j=1(z–bj)βjQ+(z) R+(z)eG–(z)Pν–n+1(z).(10) For the case in which η<0(µ<λ), this solution becomes Y+(z)=V+ 1(z)+r⎝productdisplay i=1(z–ai)αiR–(z) Q–(z)eG+(z)Ph–m–1(z), Y–(z)=V– 1(z)+s⎝productdisplay j=1(z–bj)βjQ+(z) R+(z)eG–(z)Ph–m–1(z).(11) 12.8. C ARLEMAN METHOD FOR EQUATIONS OF THE CONVOLUTION TYPE OF THE FIRST KIND 609 In both cases, the solution of the original integral equation can be obtained by substituting the expressions (10) and (11) into the formula y(x)=1 √ 2π⎝integraldisplay∞ –∞[Y+(u)–Y–(u)]e–iuxdu. (12) Example. Consider the following equation of the ﬁrst kind: 1 √ 2π⎝integraldisplay∞ 0K1(x–t)y(t)dt+1 √ 2π⎝integraldisplay0 –∞K2(x–t)y(t)dt=f(x), where K1(x)=⎝braceleftBig0f orx>0 , √ 2π(e3x–e2x)f o rx<0 ,K2(x)=⎝braceleftbigg –√ 2πi e–2xforx>0 , 0f o r x<0 ,f(x)=⎝braceleftBig0f orx>0 , √ 2π(e3x–e2x)f o rx<0 .(13) Applying the Fourier transform to the functions in (13), we obtain K1(u)=1 (u–2i)(u–3i),K2(u)=1 u+2i,F(u)=1 (u–2i)(u–3i). Here the boundary value problem (6) becomes Y+(u)=(u–2i)(u–3i) u+2iY–(u)+1 . The coefﬁcient D(u) has a ﬁrst-order pole at inﬁnity ( ν= –1). In this case m+=2 , n+=0 , ν=m+–n+=2 , m i n ( λ,µ)=1 , d=2 . The function F(u) has second-order zero at inﬁnity, and hence the necessary condition for the solvability is satisﬁed. In the class of functions that vanish at inﬁnity, the homogeneous problem Y+(u)=(u–2i)(u–3i) u+2iY–(u) has the following solution: Y+(z)=C z+2i,Y–(z)=C (z–2i)(z–3i), where Cis an arbitrary constant. The number of linearly independent solutions of problem (13) is less by one than the index, because D(u) has a ﬁrst-order pole at inﬁnity. The solution of the nonhomogeneous problem in the class of functions vanishing at inﬁnity has the form Y+(z)=C z+2i,Y–(z)=C–2i–z (z–2i)(z–3i), y(x)=⎝braceleftbigg –√ 2πi Ce–2xforx>0 ,√ 2πC(e2x–e3x)–4i√ 2πe2x+5i√ 2πe3xforx<0 . For the chosen right-hand side, the equation turns out to be solvable. However, if we take, for instance, f(x)=⎝braceleftbigg0f o r x>0 ,√ 2πi(5e3x–4e2x)f o r x<0 ,(14) then we have F(u)=(u+2i)/[(u–2i)(u–3i)]. The corresponding Riemann boundary value problem has the form Y+(u)=(u–2i)(u–3i) u+2iY–(u)+u+2i. In the class of functions bounded at inﬁnity, its solution can be represented in the form Y+(z)=C–z z+2i,Y–(z)=C–z–(z+2i)2 (z–2i)(z–3i). (15) For no choice of the constant Cthe solution vanishes at inﬁnity, and hence the equation with the right-hand side deﬁned by (14) has no solutions integrable on the real axis. References for Section 12.8: F. D. Gakhov and Yu. I. Cherskii (1978), S. G. Mikhlin and S. Pr ¨ossdorf (1986), N. I. Muskhe- lishvili (1992). 610 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) 12.9. Dual Integral Equations of the First Kind 12.9-1. Carleman Method for Equations with Difference Kernels. Consider the following dual integral equation of convolution type: 1 √ 2π⎝integraldisplay∞ –∞K1(x–t)y(t)dt=f(x), 0 < x<∞, 1 √ 2π⎝integraldisplay∞ –∞K2(x–t)y(t)dt=f(x), – ∞<x<0 ,(1) in which the function y(x) is to be found. In order to apply the Fourier transform technique (see Subsections 9.4-3, 12.7-1, and 12.7-2), we extend the domain of both conditions in Eq. (1) by formally rewriting them for all real values ofx. This can be achieved by introducing new unknown functions into the right-hand sides. These functions must be chosen so that the conditions give n on the semiaxis are not violated. Hence, the ﬁrst condition in (1) must be complemented by a summand that vanishes on the positive semiaxis and the second by a summand that vanishes on the negative semiaxis. Thus, the dual equation can be written in the form 1 √ 2π⎝integraldisplay∞ –∞K1(x–t)y(t)dt=f(x)+ξ–(x), 1 √ 2π⎝integraldisplay∞ –∞K2(x–t)y(t)dt=f(x)+ξ+(x),–∞<x<∞, where the ξ±(x) are some right and left one-sided functions so far unknown. On applying the Fourier integral transform, we have K1(u)Y(u)=F(u)+Ξ–(u),K2(u)Y(u)=F(u)+Ξ+(u). (2) Here the three functions Y(u),Ξ+(u), andΞ–(u) are unknown. Let us eliminate Y(u) from relations (2). We obtain the Riemann boundary value problem in the form Ξ+(u)=K2(u) K1(u)Ξ–(u)+K2(u)–K1(u) K1(u)F(u), – ∞<u<∞. In the present case, the coefﬁcient of the boundary condition is the ratio of functions that vanish at inﬁnity, and hence this coefﬁcient can have a zero or a pole of some order at inﬁnity. The solution of the Riemann boundary value problem can be constructed on the basis of Subsections 12.7-6 and 12.7-7, and the solution of the integral equation (1) can be deﬁned by the formula y(x)=1 √ 2π⎝integraldisplay∞ –∞Ξ+(u)+F(u) K2(u)e–iuxdu=1 √ 2π⎝integraldisplay∞ –∞Ξ–(u)+F(u) K1(u)e–iuxdu.( 3 ) Example 1. Let us solve the dual equation (1), where K1(x)=⎝braceleftbigg√ 2π(e3x–e2x)f o r x<0 , 0f orx>0 ,K2(x)=⎝braceleftBig0f o r x<0 , –√ 2πi e–2xforx>0 ,f(x)=⎝braceleftBigg1 4√ 2πe2xforx<0 , –1 4√ 2πe–2xforx>0 . We ﬁnd the Fourier integrals K1(u)=1 (u–2i)(u–3i),K2(u)=1 u+2i,F(u)=1 u2+4. 12.9. D UALINTEGRAL EQUATIONS OF THE FIRST KIND 611 In this case, the boundary value problem (2) corresponding to this equation becomes Ξ+(u)=(u–2i)(u–3i) u+2iΞ–(u)+u–3i (u+2i)2–1 u2+4. (4) The coefﬁcient D(u) has a ﬁrst-order pole at inﬁnity (with index ν= –1). The functions K1(u)a n dK2(u)h a v ea common zero of the ﬁrst order at inﬁnity. We ﬁnd that m+=2 , n+=0 , ν=m+–n+=2 . On representing the boundary condition in the form (u+2i)Ξ+(u)–u–3i u+2i=(u–2i)(u–3i)Ξ–(u)–1 u–2i and applying the analytic continuation and the generalized Liouville theorem, we see that the general solution of problem (4) in the class of functions vanishing at inﬁnity is given by Ξ+(z)=1 z+2i⎝parenleftbiggz–3i z+2i+C⎝parenrightbigg ,Ξ–(z)=1 (z–2i)(z–3i)⎝parenleftbigg1 z–2i+C⎝parenrightbigg , (5) where Cis an arbitrary constant. The solution of the integral equation in question is given by the expression y(x)=1 √ 2π⎝integraldisplay∞ –∞Ξ+(u)+F(u) K2(u)e–iuxdu. Since the function K2(u) has a ﬁrst-order zero at inﬁnity, it follows that the function Ξ+(u)+F(u) must have a zero at inﬁnity whose order is at least two. This condition implies the relation C= –1. ForC= –1, formulas (5) become Ξ+(z)=–5i (z+2i)2,Ξ–(z)=1+2i–z (z–2i)2(z–3i),y(x)=⎝braceleftbigg i√ 2πe2xforx<0 , 5√ 2πe–2xforx>0 . Thus, we have succeeded in satisfying the solvability condition, which follows from the existence of a common zero of the functions K1(u)a n dK2(u), by choosing an appropriate constant that enters the general solution, and the integral equation turns out to be unconditionally and uniquely solvable. 12.9-2. General Scheme of Finding Solutions of Dual Integral Equations. In applications (for example, in elasticity, thermal conduction, and electrostatics), one encounters dual integral equations of the form ⎝integraldisplay∞ 0K(x,t)y(t)dt=f(x)i f0 ≤x≤a, ⎝integraldisplay∞ 0M(x,t)y(t)dt=g(x)i f a<x<∞,(6) where K(x,t),M(x,t),f(x), and g(x) are known functions and y(x) is the function to be found. Methods for solving various types of these equations are described, for instance, in the books mentioned in the references at the end of this section. Below we outline the general scheme ofﬁnding solutions of such equations. A solution of equation (6) can be represented as the sum y(x)=y 1(x)+y2(x), where y1(x)a n dy2(x) are solutions of simpler auxiliary dual equations ⎝integraldisplay∞ 0K(x,t)y1(t)dt=f(x)i f0 ≤x≤a, ⎝integraldisplay∞ 0M(x,t)y1(t)dt=0 i f a<x<∞(7) 612 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) and ⎝integraldisplay∞ 0K(x,t)y2(t)dt=0 i f 0 ≤x≤a, ⎝integraldisplay∞ 0M(x,t)y2(t)dt=g(x)i f a<x<∞.(8) For instance, consider dual equation (7) [dual equation (8) can be considered in a similar manner]. Let us supplement the second equation by the relation ⎝integraldisplay∞ 0M(x,t)y1(t)dt=ϕ(x)i f0 ≤x≤a,( 9 ) where ϕ(x) is an auxiliary function to be determined. Suppose that the integral transformation ⎝integraldisplay∞ 0M(x,t)y(t)dt=z(x)( 0 ≤x<∞) can be inverted in the form ⎝integraldisplay∞ 0M1(x,t)z(t)dt=y(x)( 0 ≤x<∞). (10) Then, from (7), (9), and (10), using the relation z(x)=⎝braceleftbigg ϕ(x)i f 0 ≤x≤a, 0i f a<x<∞,we obtain y1(x)=⎝integraldisplaya 0M1(x,t)ϕ(t)dt (0≤x<∞). (11) Substituting this expression into the ﬁrst equation in (7) and changing the integration order, we obtain Fredholm integral equations of the ﬁrst kind for the auxiliary function ϕ(x): ⎝integraldisplaya 0N(x,s)ϕ(s)ds=f(x), N(x,s)=⎝integraldisplay∞ 0K(x,t)M1(t,s)dt. (12) After ﬁnding a solution of equation (12), one can use formula (11) to obtain a solution of the dual integral equation. In some cases, it is possible to ﬁnd a solution of equation (12) in closed form (see Example 2, Subsection 12.9-3, and Section 3.9). In a number of cases the kernel of integral equation (12) can be presented in the form of sum of the kernel of integral transform and some function. Then, using method described in Subsection 12.6-3, one can reduce integral equation of the ﬁrst kind with constant limits of integration (12) to an integralequation of the second kind (see Subsection 12.9-4). Example 2. Consider the dual integral equations ⎝integraldisplay∞ 0cos(xt)y(t)dt=f(x)i f0 < x<1 , ⎝integraldisplay∞ 0sin(xt)y(t)dt=0 i f 1< x<∞,(13) which arises in crack problems in the classical theory of elasticity. The second equation in (13) can be written as ⎝radicalbigg 2 π⎝integraldisplay∞ 0sin(xt)y(t)dt=⎝braceleftBigϕ(x)i f 0 < x<1 , 0i f 1 < x<∞,(14) where ϕ(x) is an auxiliary function. The right-hand side of equation (14) is the Fourier sine transform. Applying the Fourier sine inversion formula (see Subsection 9.5-2) to (14), we get y(t)=⎝radicalbigg 2 π⎝integraldisplay1 0sin(xt)ϕ(x)dx. (15) 12.9. D UALINTEGRAL EQUATIONS OF THE FIRST KIND 613 Using the integral representation (see Supplement 4.6-2, integral 3) sin(xt)=t⎝integraldisplayx 0uJ0(ut) √ x2–u2du, one can transform equation (15) (after changing the integration order) to y(t)=⎝radicalbigg 2 πt⎝integraldisplay1 0ϕ(x)⎝bracketleftbigg⎝integraldisplayx 0uJ0(ut) √ x2–u2du⎝bracketrightbigg dx =⎝radicalbigg 2 πt⎝integraldisplay1 0uJ0(ut)⎝bracketleftbigg⎝integraldisplay1 uϕ(x)dx √ x2–u2⎝bracketrightbigg du=t⎝integraldisplay1 0J0(ut)uψ(u)du,(16) where ψ(u)=⎝radicalbigg 2 π⎝integraldisplay1 uϕ(x)dx √ x2–u2. (17) Let us rewrite the ﬁrst equation in (13) as follows: f(x)=d dx⎝integraldisplay∞ 01 tsin(xt)y(t)dt (0 <x<1 ) . (18) Substituting y(t) from (16) into (18), we have f(x)=d dx⎝integraldisplay∞ 0sin(xt)⎝bracketleftbigg⎝integraldisplay1 0J0(ut)uψ(u)du⎝bracketrightbigg dt =d dx⎝integraldisplay1 0uψ(u)⎝bracketleftbigg⎝integraldisplay∞ 0sin(xt)J0(ut)dt⎝bracketrightbigg du.(19) The last integral in square brackets can be calculated (see Supplement 4.6-1, integral 3). As a result, equation (19) becomes f(x)=d dx⎝integraldisplayx 0uψ(u)du √ x2–u2(0 <x<1 ) . (20) To within obvious changes of notation, the right-hand side of equation (20) coincides with the inverse Abel-type integral equation 41 from Subsection 1.1-6. Therefore, the solution of equation (20) has the form ψ(u)=2 π⎝integraldisplayu 0f(x)dx √ u2–x2. Substituting this function into (16), we ﬁnd a solution of the original dual integral equation (13): y(t)=2 πt⎝integraldisplay1 0uJ0(ut)⎝bracketleftbigg⎝integraldisplayu 0f(x)dx √ u2–x2⎝bracketrightbigg du. (21) Below we give solutions for some classes of dual integral equations that occur most frequently in applications. 12.9-3. Exact Solutions of Some Dual Equations of the First Kind. Below we present solutions of some classes of dual integral equations that occur most frequently in applications. 1◦. Consider the following dual integral equation: ⎝integraldisplay∞ 0J0(xt)y(t)dt=f(x) ⎝integraldisplay∞ 0tJ0(xt)y(t)dt=0for 0 < x<a, fora<x<∞,(22) where J0(x) is the Bessel function of zero order. We can obtain the solution of Eqs. (22) by applying the Hankel transform. This solution is given by y(x)=2 π⎝integraldisplaya 0cos(xt )⎝bracketleftbiggd dt⎝integraldisplayt 0sf(s)ds √ t2–s2⎝bracketrightbigg dt. (23) 614 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) 2◦. The exact solution of the dual integral equation ⎝integraldisplay∞ 0tJ0(xt)y(t)dt=f(x) ⎝integraldisplay∞ 0J0(xt)y(t)dt=0for 0 < x<a, fora<x<∞,(24) where J0(x) is the Bessel function of zero order, can be constructed by means of the Hankel transform, y(x)=2 π⎝integraldisplaya 0sin(xt)⎝bracketleftbiggd dt⎝integraldisplayt 0sf(s)ds √ t2–s2⎝bracketrightbigg dt. (25) 3◦. The exact solution of the dual integral equation ⎝integraldisplay∞ 0tJµ(xt)y(t)dt=f(x) ⎝integraldisplay∞ 0Jµ(xt)y(t)dt=0for 0 < x<a, fora<x<∞,(26) where Jµ(x) is the Bessel function of order µ, can be deﬁned by the following expression (here the calculation also involves the Hankel transform): y(x)=⎝radicalbigg 2x π⎝integraldisplaya 0t3/2Jµ+1 2(xt)⎝bracketleftbigg⎝integraldisplayπ/2 0sinµ+1θf(tsinθ)dθ⎝bracketrightbigg dt. (27) 4◦. Consider the dual integral equation ⎝integraldisplay∞ 0t2βJµ(xt)y(t)dt=f(x) ⎝integraldisplay∞ 0Jµ(xt)y(t)dt=0for 0 < x<1 , for 1 < x<∞,(28) where Jµ(x) is the Bessel function of order µ. The solution of Eq. (28) can be obtained by applying the Mellin transform. For β>0 ,t h i s solution is deﬁned by the formulas y(x)=(2x)1–β Γ(β)⎝integraldisplay1 0t1+βJµ+β(xt)F(t)dt,F(t)=⎝integraldisplay1 0f(tζ)ζµ+1(1 –ζ2)β–1dζ. (29) Forβ> –1, the solution of the dual equation (28) has the form y(x)=(2x)–β Γ(1 +β)⎝bracketleftbigg x1+βJµ+β(x)⎝integraldisplay1 0tµ+1(1 –t2)βf(t)dt+⎝integraldisplay1 0tµ+1(1 –t2)βΦ(x,t)dt⎝bracketrightbigg , (30) Φ(x,t)=⎝integraldisplay1 0(xξ)2+βJµ+β+1(xξ)f(ξt)dξ. Formula (30) holds for β>– 1a n df o r– µ–1 2<2β<µ+3 2. It can be shown that for β>0t h e solution of Eq. (30) can be reduced to the form (29). 12.9. D UALINTEGRAL EQUATIONS OF THE FIRST KIND 615 5◦. The exact solution of the dual integral equation ⎝integraldisplay∞ 0tP–1 2+it(coshx)y(t)dt=f(x) ⎝integraldisplay∞ 0tanh(πt)P–1 2+it(coshx)y(t)dt=0for 0 < x<a, fora<x<∞,(31) where Pµ(x) is the Legendre spherical function of the ﬁrst kind (see Supplement 11.11) and i2= –1, can be constructed by means of the Meler–Fock integral transform (see Section 9.6) and is given by the formula y(x)=√ 2 π⎝integraldisplaya 0sin(xt)⎝bracketleftbigg⎝integraldisplayt 0f(s)s i n hs √ cosht–c o s h sds⎝bracketrightbigg dt. (32) Note that P–1 2+it(coshx)=√ 2 π⎝integraldisplayx 0cos(ts ) √ coshx–c o s h sds,x>0 , where the integral on the right-hand side is called the Meler integral . 12.9-4. Reduction of Dual Equations to a Fredholm Equation. One of the most effective methods for the approximate solution of dual integral equations of the ﬁrst kind is the method of reducing these equations to Fredholm integral equations of the second kind (see Chapter 13). In what follows, we present some dual equations encountered in problems of mechanics and physics and related Fredholm equations of the second kind. 1◦. The solution of the dual integral equation of the ﬁrst kind ⎝integraldisplay∞ 0g(t)J0(xt)y(t)dt=f(x) ⎝integraldisplay∞ 0tJ0(xt)y(t)dt=0for 0 < x<a, fora<x<∞,(33) where g(x) is a given function and J0(x) is the Bessel function of zero order, has the form y(x)=⎝integraldisplaya 0ϕ(t)c o s ( xt)dt, (34) where the function ϕ(x) to be found from the following Fredholm equation of the second kind: ϕ(x)–1 π⎝integraldisplaya 0K(x,t)ϕ(t)dt=ψ(x), 0 < x<a, (35) where the symmetric kernel K(x,t) and the right-hand side ψ(x)a r eg i v e nb y K(x,t)=2⎝integraldisplay∞ 0[1 –g(s)] cos(x s)c o s (ts)ds,ψ(x)=2 πd dx⎝integraldisplayx 0tf(t) √ x2–t2dt. (36) Methods for the investigation of these equations are presented in Chapter 13. 616 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) 2◦. The solution of the dual integral equation of the ﬁrst kind ⎝integraldisplay∞ 0tg(t)J0(xt)y(t)dt=f(x) ⎝integraldisplay∞ 0J0(xt)y(t)dt=0for 0 < x<a, fora<x<∞,(37) where g(x) is a given function and J0(x) is the Bessel function of zero order, has the form y(x)=⎝integraldisplaya 0ϕ(t)s i n (xt)dt, (38) where the function ϕ(x) is to be found from the Fredholm equation (35) of the second kind with K(x,t)=2⎝integraldisplay∞ 0[1 –g(s)] sin(x s)s i n (ts)ds,ψ(x)=2 π⎝integraldisplayx 0tf(t) √ x2–t2dt. Note that the kernel K(x,t) is symmetric. 3◦. The solution of the dual integral equation of the ﬁrst kind ⎝integraldisplay∞ 0g(t)Jµ(xt)y(t)dt=f(x) ⎝integraldisplay∞ 0tJµ(xt)y(t)dt=0for 0 < x<a, fora<x<∞,(39) where g(x) is a given function and Jµ(x) is the Bessel function of order µ,h a st h ef o r m y(x)=⎝radicalbigg πx 2⎝integraldisplaya 0√ tJµ–1 2(xt)ϕ(t)dt, (40) where the function ϕ(x) is to be found from the Fredholm equation (35) of the second kind with K(x,t)=π√ xt⎝integraldisplay∞ 0[1 –g(s)]sJµ–1 2(xs)Jµ–1 2(ts)ds, ψ(x)=2 π⎝braceleftbigg f(0) +⎝integraldisplayπ/2 0⎝bracketleftbig µ(sinθ)µ–1f(xsinθ)+x(sinθ)µf/prime(xsinθ)⎝bracketrightbig dθ⎝bracerightbigg . Note that f/prime(xsinθ)=f/prime ξ(ξ)⎝vextendsingle⎝vextendsingle ξ=xsinθ, and the kernel K(x,t) is symmetric. 4◦. The solution of the integral equation of the ﬁrst kind ⎝integraldisplay∞ 0tg(t)Jµ(xt)y(t)dt=f(x) ⎝integraldisplay∞ 0Jµ(xt)y(t)dt=0for 0 < x<a, fora<x<∞,(41) where g(x) is a given function and Jµ(x) is the Bessel function of order µ,h a st h ef o r m y(x)=⎝radicalbigg πx 2⎝integraldisplaya 0√ tJµ+1 2(xt)ϕ(t)dt, (42) where the function ϕ(x) is to be found by solving the Fredholm equation (35) of the second kind with K(x,t)=π√ xt⎝integraldisplay∞ 0[1 –g(s)]sJµ+1 2(xs)Jµ+1 2(ts)ds,ψ(x)=2x π⎝integraldisplayπ/2 0f(xsinθ)(sinθ)µ+1dθ, and the kernel K(x,t) is symmetric. 12.9. D UALINTEGRAL EQUATIONS OF THE FIRST KIND 617 5◦. The solution of the dual integral equation of the ﬁrst kind ⎝integraldisplay∞ 0g(t)Jµ(xt)y(t)dt=f(x) ⎝integraldisplay∞ 0Jµ(xt)y(t)dt=0for 0 < x<a, fora<x<∞,(43) where g(x) is a given function and Jµ(x) is the Bessel function of order µ,h a st h ef o r m y(x)=x⎝radicalbigg πx 2⎝integraldisplaya 0√ tJµ–1 2(xt)ϕ(t)dt, (44) and the function ϕ(x) is to be found from the Fredholm equation (35) of the second kind with K(x,t)=xµ√ 2πt⎝integraldisplaya xρ1–µ ⎝radicalbig ρ2–x2⎝integraldisplay∞ 0[1 –g(s)]s3/2Jµ(ρs)Jµ–1 2(ts)dsdρ, ψ(x)=2 πxµ⎝integraldisplaya xρ1–µ ⎝radicalbig ρ2–x2dρ. 6◦. The solution of the dual integral equation of the ﬁrst kind ⎝integraldisplay∞ 0t2βg(t)Jµ(xt)y(t)dt=f(x) ⎝integraldisplay∞ 0Jµ(xt)y(t)dt=0for 0 < x<a, fora<x<∞,(45) where 0 < β<1 ,g(x) is a given function, and Jµ(x) is the Bessel function of order µ,h a st h ef o r m y(x)=⎝radicalbigg π 2x1–β⎝integraldisplaya 0√ tJµ+β(xt)ϕ(t)dt, (46) and the function ϕ(x) is to be found from the Fredholm equation (35) of the second kind with K(x,t)=π√ xt⎝integraldisplay∞ 0[1 –g(s)]sJµ+β(xs)Jµ+β(ts)ds, ψ(x)=21–β Γ(β)⎝radicalbigg 2x πxβ⎝integraldisplayπ/2 0f(xsinθ)(sinθ)µ+1(cosθ)2β–1dθ, and the kernel K(x,t) is symmetric. 7◦. The solution of the dual integral equation of the ﬁrst kind ⎝integraldisplay∞ 0g(t)P–1 2+it(coshx)y(t)dt=f(x) ⎝integraldisplay∞ 0ttanh(πt)P–1 2+it(coshx)y(t)dt=0for 0 < x<a, fora<x<∞,(47) where Pµ(x) is the Legendre spherical function of the ﬁrst kind (see Supplement 11.11), i2= –1, andg(x) is a given function, is determined by the formula y(x)=⎝integraldisplaya 0cos(xt )ϕ(t)dt, (48) and the function ϕ(x) is to be found from the Fredholm equation (35) of the second kind in which K(x,t)=⎝integraldisplay∞ 0[1 –g(s)]{cos[(x +t)s]+c o s [ ( x–t)s]}ds, ψ(x)=√ 2 πd dx⎝integraldisplayx 0f(s)s i n hs √ coshx–c o s h sds. (49) On the basis of relations (49), we can readily see that the kernel K(x,t) is symmetric. 618 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) 8◦. The solution of the dual integral equation of the ﬁrst kind ⎝integraldisplay∞ 0tg(t)P–1 2+it(coshx)y(t)dt=f(x) ⎝integraldisplay∞ 0tanh(πt)P–1 2+it(coshx)y(t)dt=0for 0 < x<a, fora<x<∞,(50) where Pµ(x) is the spherical Legendre function of the ﬁrst kind (see Supplement 11.11), i2= –1, andg(x) is a given function, is determined by the formula y(x)=⎝integraldisplaya 0sin(xt)ϕ(t)dt, (51) and the function ϕ(x) is to be found from the Fredholm equation (35) of the second kind in which K(x,t)=⎝integraldisplay∞ 0[1 –g(s)]{cos[(x –t)s]–c o s [ ( x+t)s]}ds, ψ(x)=√ 2 π⎝integraldisplayx 0f(s)s i n hs √ coshx–c o s h sds. (52) On the basis of relations (52), we can readily see that the kernel K(x,t) is symmetric. References for Section 12.9: Ya. S. Uﬂyand (1977), F. D. Gakhov and Yu. I. Cherskii (1978), C. Nasim and B. D. Ag- garwala (1984), E. C. Titchmarsh (1986), I. Sneddon (1995), B. N. Mandal and N. Mandal (1999, pp. 134–136). 12.10. Asymptotic Methods for Solving Equations with Logarithmic Singularity 12.10-1. Preliminary Remarks. Consider the Fredholm integral equation of the ﬁrst kind of the form ⎝integraldisplay1 –1K⎝parenleftbiggx–t λ⎝parenrightbigg y(t)dt=f(x), –1 ≤x≤1, (1) with parameter λ(0 <λ<∞). We assume that the kernel K=K(x) is an even function continuous for x≠0 which has a logarithmic singularity as x→0 and exponentially decays as x→∞ . Equations with such a kernel arise in solving various problems of continuum mechanics with mixed boundary conditions. Letf(x) belong to the space of functions whose ﬁrst derivatives satisfy the H ¨older condition with exponent α>1 2on [–1, 1]. In this case, the solution of the integral equation (1) in the class of functions satisfying the H ¨older condition exists and is unique for any λ∈(0,∞) and has the structure y(x)=ω(x) √ 1–x2,( 2) where ω(x) is a continuous function that does not vanish at x=±1.* It follows from formula (2) that the solution of Eq. (1) is unbounded as x→± 1. This important circumstance will be taken into account in Subsection 12.10-3 in constructing the asymptotic solutionin the case λ→0. Note that more general equations with difference kernel and arbitrary ﬁnite limits of integration can always be reduced to Eq. (1) by a change of variables. The form (1) is taken here for furtherconvenience. * The situation ω(±1) = 0 is only possible in exceptional cases for special values of λ. 12.10. A SYMPTOTIC METHODS FOR SOLVING EQUATIONS WITH LOGARITHMIC SINGULARITY 619 12.10-2. Solution for Large λ. Let the representation K(x)=l n |x|∞⎝summationdisplay n=0an|x|n+∞⎝summationdisplay n=0bn|x|n,( 3) where a0≠0, be valid for the kernel of the integral equation (1) as x→0. It is obvious from (3) that two different-scale large parameters λand ln λoccur in Eq. (1) as λ→∞ . The latter, “quasiconstant” parameter grows much slower than the former (for instance, for λ= 100 and λ= 1000 we have ln λ≈4.6 and ln λ≈6.9, respectively). Let us drop out all terms decaying as λ→∞ in Eq. (1). In view of (3), for the main (zeroth) approximation we have ⎝integraldisplay1 –1⎝parenleftbig a0ln|x–t|–a0lnλ+b0⎝parenrightbig y0(t)dt=f(x), –1 ≤x≤1. (4) It should be noted that one cannot retain in the integrand only one term proportional to ln λ(since the corresponding “truncated” equation is unsolvable). The constant b0must also be included in (4) for the main-approximation equation to be invariant with respect to the scaling parameter λin Eq. (1). The exact closed-form solution of Eq. (4) is given in Section 3.4 (see equations 3 and 4). To construct an asymptotic solution of Eq. (1) as λ→∞ , it is convenient to do the following. First, we consider the auxiliary integral equation ⎝integraldisplay1 –1K(x–t,β,λ)y(t)dt=f(x), –1 ≤x≤1, K(x,β,λ)=⎝parenleftbig ln|x|–β⎝parenrightbig∞⎝summationdisplay n=0an λn|x|n+∞⎝summationdisplay n=0bn λn|x|n,(5) with two parameters λandβ. We seek its solution in the form of a regular asymptotic expansion in negative powers of λ(for ﬁxed β). That is, we have y(x,β,λ)=N⎝summationdisplay n=0λ–nyn(x,β)+o⎝parenleftbig λ–N⎝parenrightbig .( 6 ) Substituting (6) into (5) yields a recurrent chain of integral equations of the form (4): ⎝integraldisplay1 –1⎝parenleftbig a0ln|x–t|–a0β+b0⎝parenrightbig yn(t,β)dt=gn(x,β), –1 ≤x≤1, (7) from which the functions yn(x,β) can be successively calculated. The right-hand sides gn(x,β) depend only on the previously determined functions y0,y1,...,yn–1. Note that for β=l nλthe auxiliary equation (5) coincides with the original equation (1) into which the expansion (3) is substituted. Therefore, the asymptotic solution of Eq. (1) can be obtained with the aid of (6) and (7) with β=l nλ. Some contact problems of elasticity can be reduced to Eq. (1), in which the kernel can be represented in the form (3) with an=0f o ra l l n>0a n d b2m+1=0f o r m=0 ,1 ,2 , ...In this case, one must set yn(x,β)≡0(n=1 ,3 ,5 , ...) in the solution (6). In practice, it usually sufﬁces to r e t a i nt h et e r m su pt o λ–4. 620 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) 12.10-3. Solution for Small λ. In analyzing the limit case λ→0, we take into account the singularities of the solution at the endpoints of the interval –1 ≤x≤1 (see formula (2)). Consider the following auxiliary system of two integral equations: ⎝integraldisplay∞ –1K⎝parenleftbiggx–t λ⎝parenrightbigg y1(t)dt=f1(x)+⎝integraldisplay–1 –∞K⎝parenleftbiggx–t λ⎝parenrightbigg y2(t)dt,– 1 ≤x<∞, ⎝integraldisplay1 –∞K⎝parenleftbiggx–t λ⎝parenrightbigg y2(t)dt=f2(x)+⎝integraldisplay∞ 1K⎝parenleftbiggx–t λ⎝parenrightbigg y1(t)dt,– ∞<x≤1.(8) The former equation provides for selecting the singularity at x= –1 and the latter for selecting the singularity at x=+ 1 . The functions f1(x)a n df2(x) are such that f1(x)+f2(x)=f(x), –1 ≤x≤1, f1(x)=O⎝parenleftbig e–α1x⎝parenrightbig asx→∞ , f2(x)=O⎝parenleftbig eα2x⎝parenrightbig asx→–∞,(9) where α1>0a n d α2>0 . The ﬁrst condition in (9) makes it possible to seek the solution of the integral equation (1) as the sum of the solutions of the integral equations (8), that is, y(x)=y1(x)+y2(x), –1 ≤x≤1. (10) Note that by virtue of the last two conditions in (9), the relations y1(x)=O⎝parenleftbig e–β1x⎝parenrightbig asx→∞ , y2(x)=O⎝parenleftbig eβ2x⎝parenrightbig asx→–∞,(11) where β1>0a n d β2> 0, are valid. Recall that the kernel K(x) is an even function. Therefore, if f(x)i nE q .( 1 )i sa ne v e no ro d d function, then one must set f1(x)=±f2(–x),y1(x)=±y2(–x) (12) in system (8).* In both cases, system (8) can be reduced by changes of variables to the same integral equation ⎝integraldisplay∞ 0K(z–τ)w(τ)dτ=F(z)±⎝integraldisplay∞ 2/λK(2/λ–z–τ)w(τ)dτ,0 ≤z<∞, (13) in which the following notation is used: z=x+1 λ,τ=t+1 λ,w(τ)=y(t),F(z)=1 λf1(x). (14) In view of the properties of the kernel K(x) (see Subsection 12.10-1) and the ﬁrst relation in (11), the asymptotic estimate I(w)≡⎝integraldisplay∞ 2/λK(2/λ–z–τ)w(τ)dτ=O⎝parenleftbig e–2β 1/λ⎝parenrightbig (15) can be obtained, which is uniform with respect to τ. * In formulas (12), (13), (16), and (17), the plus sign corresponds to even f(x) and the minus sign to odd f(x). 12.11. R EGULARIZATION METHODS 621 According to (15), for small λthe iterative scheme ⎝integraldisplay∞ 0K(z–τ)wn(τ)dτ=F(z)±I⎝parenleftbig wn–1⎝parenrightbig ,n=1 ,2 , ..., (16) can be used to solve the integral equation (13) by the method of successive approximations. In the main approximation, the integral I(w0) can be omitted on the right-hand side. Equations (16) are Wiener–Hopf integral equations of the ﬁrst kind, which can be solved in a closed form (see Subsection 12.8-1). It follows from formulas (10), (12), and (14) that, as λ→0, the leading term of the asymptotic expansion of the solution of the integral equation (1) has the form y(x)=w1⎝parenleftbigg1+x λ⎝parenrightbigg ±w1⎝parenleftbigg1–x λ⎝parenrightbigg , (17) where w1=w1(τ) is the solution of Eq. (16) with n=1a n d w0≡0. For practical purposes, formula (17) is usually sufﬁcient. 12.10-4. Integral Equation of Elasticity. The integral equation (1) whose kernel is given via the Fourier cosine transform, K(x)=⎝integraldisplay∞ 0L(u) ucos(ux )du, (18) frequently occurs in contact problems of elasticity. The function L(u) in (18) is continuous and positive for 0 < u<∞and satisﬁes the asymptotic relations L(u)=Au+O(u3)a su→0, L(u)=N–1⎝summationdisplay n=0Bnu–n+O⎝parenleftbig u–N⎝parenrightbig asu→∞ ,(19) where A>0a n d B0>0 . Formula (18) implies that the kernel is an even function: K(x)=K(–x). It is usually assumed that L(u)u–1andu[L(u)]–1, treated as functions of the complex variable w=u+iv, are regular at the pole |v|≤γ1and the pole |v|≤γ2, respectively. It follows in particular that the kernel K(x) decays at least as exp(–γ 1|t|) at inﬁnity. Formulas (18) and (19) imply that K(x) has a logarithmic singularity at x= 0. Moreover, the representation (3) is valid with an=0f o r n=1 ,3 ,5 , ... Thus, the kernel given by (18) has the same characteristic features as those inherent by assumption in the kernel of the integral equation (1). Therefore, the results of Subsections 12.10-2 and 12.10-3 can be used for the asymptotic analysis of Eq. (1) with kernel (18) as λ→∞ andλ→0. References for Section 12.10: I. I. V orovich, V . M. Aleksandrov, and V . A. Babeshko (1974), V . M. Aleksandrov and E. V . Kovalenko (1986), V . M. Aleksandrov (1993). 12.11. Regularization Methods 12.11-1. Lavrentiev Regularization Method. Consider the Fredholm equation of the ﬁrst kind ⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b,( 1 ) 622 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) where f(x)∈L2(a,b)a n dy(x)∈L2(a,b). The kernel K(x,t) is square integrable, symmetric, and positive deﬁnite (see Subsection 13.6-2), that is, for all ϕ(x)∈L2(a,b), we have ⎝integraldisplayb a⎝integraldisplayb aK(x,t)ϕ(x)ϕ(t)dx dt ≥0, where the equality is attained only for ϕ(x)≡0. In the above classes of functions and kernels, the problem of ﬁnding a solution of Eq. (1) is ill-posed, i.e., unstable with respect to small variations in the right-hand side of the integral equation. Following the Lavrentiev regularization method, along with Eq. (1) we consider the regularized equation εyε(x)+⎝integraldisplayb aK(x,t)yε(t)dt=f(x), a≤x≤b,( 2 ) where ε> 0 is the regularization parameter. This equation is a Fredholm equation of the second kind, so it can be solved by the methods presented in Chapter 13, whence the solution exists and is unique. On taking a sufﬁciently small εin Eq. (2), we ﬁnd a solution yε(x) of the equation and substitute this solution into Eq. (1), thus obtaining ⎝integraldisplayb aK(x,t)yε(t)dt=fε(x), a≤x≤b.( 3) If the function fε(x) thus obtained differs only slightly from f(x), that is, /bardblf(x)–fε(x)/bardbl≤δ,( 4) where δis a prescribed small positive number, then the solution yε(x) is regarded as a sufﬁciently good approximate solution of Eq. (1). The parameter δusually deﬁnes the error of the initial data provided that the right-hand side of Eq. (1) is deﬁned or determined by an experiment with some accuracy. For the case in which, for a given ε, condition (4) fails, we must choose another value of the regularization parameter and repeat the above procedure. The next subsection describes the regularization method suitable for equations of the ﬁrst kind with arbitrary square-integrable kernels. 12.11-2. Tikhonov Regularization Method. Consider the Fredholm integral equation of the ﬁrst kind ⎝integraldisplayb aK(x,t)y(t)dt=f(x), c≤x≤d.( 5) Assume that K(x,t) is any function square-integrable in the domain {a≤t≤b,c≤x≤d}, f(x)∈L2(c,d), andy(x)∈L2(a,b). The problem of ﬁnding the solution of Eq. (5) is also ill-posed in the above sense. Following the Tikhonov (zero-order) regularization method, along with (5) we consider the following Fredholm integral equation of the second kind (see Chapter 13): εyε(x)+⎝integraldisplayb aK∗(x,t)yε(t)dt=f∗(x), a≤x≤b,( 6) 12.12. F REDHOLM INTEGRAL EQUATION OF THE FIRST KIND AS AN ILL-POSED PROBLEM 623 where K∗(x,t)=K∗(t,x)=⎝integraldisplayd cK(s,x)K(s,t)ds,f∗(x)=⎝integraldisplayd cK(s,x)f(s)ds,( 7 ) and the positive number εis the regularization parameter. Equation (6) is said to be a regularized integral equation , and its solution exists and is unique. Taking a sufﬁciently small εin Eq. (6), we ﬁnd a solution yε(x) of the equation and substitute this solution into Eq. (5), thus obtaining ⎝integraldisplayb aK(x,t)yε(t)dt=fε(x), c≤x≤d.( 8) By comparing the right-hand side with the given f(x) using formula (4), we either regard fε(x) as a satisfactory approximate solution obtained in accordance with the above simple algorithm, or continue the procedure for a new value of the regularization parameter. Presented above are the simplest principles of ﬁnding an approximate solution of the Fredholm equation of the ﬁrst kind. More perfect and complex algorithms can be found in the references cited below. References for Section 12.11: M. M. Lavrentiev (1967), A. N. Tikhonov and V . Ya. Arsenin (1979), M. M. Lavrentiev, V . G. Romanov, and S. P. Shishatskii (1980), A. F. Verlan’ and V . S. Sizikov (1986), R. Kress (1999). 12.12. Fredholm Integral Equation of the First Kind as an Ill-Posed Problem 12.12-1. General Notions of Well-Posed and Ill-Posed Problems. To solve a quantitative mathematical problem usually means to ﬁnd an element y, called a “solution of the problem”, from a given element f, called “data of the problem”. Assume that yandfare elements of some metric spaces Y(space of solutions) and F(space of data) with the respective distances between their elements ρY(y1,y2)a n dρF(f1,f2). A solution ycorresponding to fis called stable ,i ff o ra n y ε>0t h e r ei s δ(ε) > 0 such that for anyf1∈Fsuch that ρF(f,f1)≤δ(ε)w eh a v e ρY(y,y1)≤ε,w h e r e y1is a solution corresponding tof1. In other words, if small variations of data cause a small variation of solutions. Such a problem is called well-posed on a pair of metrics spaces ( Y,F), if the following conditions hold: 1) for each f∈F, there is a solution y∈Y; 2) the solution is unique; 3) the solution is stable. Problems that do not satisfy one of these requirements are called ill-posed . Remark. The metrics in the spaces YandFdetermine in what sense small variations of yand fare understood. The choice of these metri cs determines whether a solution yis stable or not under the variation of f, and therefore, a particular problem may be well-posed or ill-posed, depending on the metrics. Now, for deﬁniteness, assume that yandfare elements of the space of continuous functions on an interval [ a,b] with the metrics ρY(y1,y2)= s u p a≤x≤b|y1(x)–y2(x)|,ρF(f1,f2)= s u p a≤x≤b|f1(x)–f2(x)|.( 1 ) 624 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORM⎝integraltextb aK(x,t)y(t)dt=f(x) 12.12-2. Integral Equation of the First Kind is an Ill-Posed Problem. Consider the Fredholm equation of the ﬁrst kind ⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b,( 2 ) with continuous kernel K(x,t), and assume that yandfare elements of the space of continuous functions with the metrics (1). It is easy to see that the continuity of f(x), in general, does not guarantee the existence of a continuous solution. Indeed, suppose that the function f(x) is continuous, but its derivative is discontinuous at some points x∈(a,b), and the kernel is continuously differentiable in x. Then for any continuous y(x) the left-hand side of (2) has a continuous derivative at all points of ( a,b), while the derivative of the right-hand side has discontinuities on ( a,b). Therefore, relation (2) can hold for no continuous function y(x), which means that equation (2) has no solutions. Consider the problem of stability of a solution. Assume that the kernel K(x,t) is continuous, together with its derivative in t.L e t y(x) be a solution of equation (2). Take z(x)=y(x)+c o s ( ωx), where ωis a parameter. We have ⎝integraldisplayb aK(x,t)[z(t)–c o s ( ωt)]dt=f(x). After elementary transformations, we get ⎝integraldisplayb aK(x,t)z(t)dt=g(x)≡f(x)+⎝integraldisplayb aK(x,t)c o s ( ωt)dt =f(x)+K(x,t)sin(ωt) ω⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleb a–⎝integraldisplayb aK/prime t(x,t)c o s ( ωt)sin(ωt) ωdt. This relation, for a ﬁnite interval [ a,b], implies the estimate sup a≤x≤b|f(x)–g(x)|≤C ω, where Cis a constant that does not depend on ω. Therefore, for sufﬁciently large ω,t h ev a l u e ρF(f,g)= s u p a≤x≤b|f(x)–g(x)| becomes arbitrarily small, ρF(f,g)→0. On the other hand, we have ρY(y,z)= s u p a≤x≤b|y(x)–z(x)|=s u p a≤x≤b|cos(ωx )|=1 , and this quantity is not small. Hence, an important conclusion can be made: the Fredholm integral equation of the ﬁrst kind (2) admits a solution which is unstable with respect to perturbations of theright-hand side f(x). Thus, equation (2) belongs to the class of ill-posed problems. This instability of solutions of integral equations of the ﬁrst kind causes great difﬁculties when using such equations for practical purposes,since small errors in input data may cause large variationsof a solution. For this reason, there existed a widespread opinion that Fredholm equations of the ﬁrst kind (as well as other ill-posed problems) are unsuitable for the description of physical processes. At present this view has changed drastically due to the development of the general theory of ill- posed problems and the corresponding regularization methods (see Section 12.11 and the references below). References for Section 12.12: M. M. Lavrentiev (1967), A. N. Tikhonov and V . Ya. Arsenin (1979), M. M. Lavrentiev, V . G. Romanov, and S. P. Shishatskii (1980), A. B. Vasilieva and A. N. Tikhonov (1989), R. Kress (1999). Chapter 13 Methods for Solving Linear Equations of the Form y(x)–⎝integraldisplay ⎝integraldisplayb aK(x,t)y(t)dt=f(x) 13.1. Some Deﬁnition and Remarks 13.1-1. Fredholm Equations and Equations with Weak Singularity of the Second Kind. Linear integral equations of the second kind with constant limits of integration have the form y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt=f(x), (1) where y(x) is the unknown function ( a≤x≤b),K(x,t)i st h e kernel of the integral equation, andf(x) is a given function, which is called the right-hand side of Eq. (1). For convenience of analysis, a number λis traditionally singled out in Eq. (1), which is called the parameter of integral equation . The classes of functions and kernels under consideration were deﬁned above in Subsections 12.1-1 and 12.1-2. Note t hat equations of the form (1) with constant limits of integration and with Fredholm kernels or kernels with weak singularity are called Fredholm equations of the second kind and equations with weak singularity of the second kind, respectively. A number λis called a characteristic value of the integral equation (1) if there exist nontrivial solutions of the corresponding homogeneous equation (with f(x)≡0). The nontrivial solutions themselves are called the eigenfunctions of the integral equation corresponding to the characteristic value λ.I fλis a characteristic value, the number 1 /λis called an eigenvalue of the integral equation (1). A value of the parameter λis said to be regular if for this value the above homogeneous equation has only the trivial solution. Sometimes the characteristic values and the eigenfunctions of a Fredholm integral equation are called the characteristic values and the eigenfunctions of the kernel K(x,t). The kernel K(x,t)of the integral equation (1) is called a degenerate kernel if it has the form K(x,t)=g1(x)h1(t)+···+gn(x)hn(t), a difference kernel if it depends on the difference of the arguments ( K(x,t)=K(x–t)), and a symmetric kernel if it satisﬁes the condition K(x,t)=K(t,x). The transposed integral equation is obtained from (1) by replacing the kernel K(x,t)b yK(t,x). Remark 1. The variables tandxmay vary in different ranges (e.g., a≤t≤bandc≤x≤d). To be speciﬁc, from now on we assume that c=aandd=b(this can be achieved by the linear substitution x=α¯x+βwith the aid of an appropriate choice of the constants αandβ). Remark 2. In general, the case in which the limits of integration aand/or bcan be inﬁnite is not excluded; however, in this case, the v alidity of the condition that the kernel K(x,t) is square integrable on the square S={a≤x≤b,a≤t≤b}is especially signiﬁcant. 625 626 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.1-2. Structure of the Solution. The solution of Eq. (1) can be presented in the form y(x)=f(x)+λ⎝integraldisplayb aR(x,t;λ)f(t)dt, where the resolvent R(x,t;λ) is independent of f(x) and is determined by the kernel of the integral equation. The resolvent of the Fredholm equation (1) satisﬁes the following two integral equations: R(x,t;λ)=K(x,t)+⎝integraldisplayb aK(x,s)R(s,t;λ)ds, R(x,t;λ)=K(x,t)+⎝integraldisplayb aK(s,t)R(x,s;λ)ds, in which the integration is performed with respect to different pairs of arguments of the kernel and the resolvent. 13.1-3. Integral Equations of Convolution Type of the Second Kind. By the integral equations of convolution type (see also Subsection 12.1-3) we mean the integral equations that can be reduced, by applying some integral transform and the convolution theoremfor this transform, to an algebraic equation for the transforms or to boundary value problems of the theory of analytic functions. Consider equations of convolution type of the second kind related to the Fourier transform. An integral equation of the second kind with difference kernel on the entire axis (this equation is sometimes called an equation of convolution type of the second kind with a single kernel )h a st h e form y(x)+⎝integraldisplay ∞ –∞K(x–t)y(t)dt=f(x), – ∞<x<∞,( 2) where f(x)a n dK(x) are the right-hand side and the kernel of the integral equation and y(x)i st h e function to be found. An integral equation of the second kind with difference kernel on the semiaxis has the form y(x)+⎝integraldisplay∞ 0K(x–t)y(t)dt=f(x), 0 < x<∞.( 3) Equation (3) is also called a one-sided equation of the second kind or a Wiener–Hopf integral equation of the second kind . An integral equation of convolution type of the second kind with two kernels has the form y(x)+⎝integraldisplay∞ 0K1(x–t)y(t)dt+⎝integraldisplay0 –∞K2(x–t)y(t)dt=f(x), – ∞<x<∞,( 4 ) where K1(x)a n dK2(x)a r et h e kernels of the integral equation (4). The class of functions and kernels for equations of convolution type was introduced above in Subsection 12.1-3. 13.2. F REDHOLM EQUATIONS OF THE SECOND KIND WITH DEGENERATE KERNEL .SOME GENERALIZATIONS 627 13.1-4. Dual Integral Equations of the Second Kind. Adual integral equation of the second kind with difference kernels (of convolution type) has the form y(x)+⎝integraldisplay∞ –∞K1(x–t)y(t)dt=f(x), 0 < x<∞, y(x)+⎝integraldisplay∞ –∞K2(x–t)y(t)dt=f(x), – ∞<x<0 ,(5) where the notation and the class of the functions and kernels coincide with those introduced for the equations of convolution type in Subsection 12.1-3. In a sufﬁciently general case, a dual integral equation of the second kind has the form y(x)+⎝integraldisplay∞ aK1(x,t)y(t)dt=f1(x), a<x<b, y(x)+⎝integraldisplay∞ aK2(x,t)y(t)dt=f2(x), b<x<∞,(6) where f1(x)a n df2(x)( a n d K1(x,t)a n d K2(x,t)) are the known right-hand sides (and the kernels) of Eq. (6) and y(x) is the function to be found. These equations can be studied by the methods of various integral transforms with reduction to boundary value problems of the theory of analyticfunctions and also by other methods developed for dual integral equations of the ﬁrst kind (e.g., see I. Sneddon (1995) and Ya. S. Uﬂyand (1977)). The integral equations obtained from (2)–(5) by replacing the kernel K(x–t)b yK(t–x)a r e said to be transposed to the original equations. If the right-hand sides of Eqs. (1)–(6) are identically zero, then these equations are said to be homogeneous . For the case in which the right-hand side of an equation of the type (1)–(6) does not vanish on the entire domain, the corresponding equation is said to be nonhomogeneous . Remark 3. Some equations whose kernel contains the product or the ratio of the variables xand tcan be reduced to Eqs. (2)–(5). Remark 4. Sometimes equations of convolution type of the form (2)–(5) are written in the form in which the integrals are multiplied by the coefﬁcient 1 /√ 2π. Remark 5. The cases in which the class of functions and kernels for equations of convolution type (in particular, for Wiener–Hopf equations) differs from those introduced in Subsections 12.1-3are always mentioned explicitly (see Sections 13.11 and 13.12). References for Section 13.1: E. Goursat (1923), F. Riesz and B. Sz.-Nagy (1955), I. G. Petrovskii (1957), B. Noble (1958), M. G. Krein (1958), S. G. Mikhlin (1960), L. V . Kantorovich and G. P. Akilov (1964), A. N. Kolmogorov andS. V . Fomin (1970), L. Ya. Tslaf (1970), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), V . I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. D. Gakhov and Yu. I. Cherskii (1978), A. G. Butkovskii (1979), L. M. Delves and J. L. Mohamed (1985), F. G. Tricomi (1985), A. J. Jerry (1985), A. F. Verlan’ and V . S. Sizikov (1986),A. Golberg (1990), D. Porter and D. S. G. Stirling (1990), C. Corduneanu (1991), J. Kondo (1991), S. Pr ¨ossdorf and B. Silbermann (1991), W. Hackbusch (1995), R. P. Kanwal (1996). 13.2. Fredholm Equations of the Second Kind with Degenerate Kernel. Some Generalizations 13.2-1. Simplest Degenerate Kernel. Consider Fredholm integral equations of the second kind with the simplest degenerate kernel: y(x)–λ⎝integraldisplayb ag(x)h(t)y(t)dt=f(x), a≤x≤b.( 1) 628 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) We seek a solution of Eq. (1) in the form y(x)=f(x)+λAg(x). (2) On substituting the expressions (2) into Eq. (1), after simple algebraic manipulations we obtain A⎝bracketleftbigg 1–λ⎝integraldisplayb ah(t)g(t)dt⎝bracketrightbigg =⎝integraldisplayb af(t)h(t)dt.( 3) Both integrals occurring in Eq. (3) are supposed to exist. On the basis of (1)–(3) and taking into account the fact that the unique characteristic value λ1of Eq. (1) is given by the expression λ1=⎝bracketleftbigg⎝integraldisplayb ah(t)g(t)dt⎝bracketrightbigg–1 ,( 4) we obtain the following results. 1◦.I fλ≠λ1, then for an arbitrary right-hand side there exists a unique solution of Eq. (1), which can be written in the form y(x)=f(x)+λλ1f1 λ1–λg(x), f1=⎝integraldisplayb af(t)h(t)dt.( 5) 2◦.I fλ=λ1andf1= 0, then any solution of Eq. (1) can be represented in the form y=f(x)+Cy1(x), y1(x)=g(x), (6) where Cis an arbitrary constant and y1(x) is an eigenfunction that corresponds to the characteristic valueλ1. 3◦.I fλ=λ1andf1≠0, then there are no solutions. 13.2-2. Degenerate Kernel in the General Case. In the general case, a Fredholm integral equation of the second kind with degenerate kernel has the form y(x)–λ⎝integraldisplayb a⎝bracketleftBiggn⎝summationdisplay k=1gk(x)hk(t)⎝bracketrightBigg y(t)dt=f(x), n=2 ,3 , ... (7) Let us rewrite Eq. (7) in the form y(x)=f(x)+λn⎝summationdisplay k=1gk(x)⎝integraldisplayb ahk(t)y(t)dt,n=2 ,3 , ... (8) We assume that Eq. (8) has a solution and introduce the notation Ak=⎝integraldisplayb ahk(t)y(t)dt.( 9) In this case we have y(x)=f(x)+λn⎝summationdisplay k=1Akgk(x), (10) 13.2. F REDHOLM EQUATIONS OF THE SECOND KIND WITH DEGENERATE KERNEL .SOME GENERALIZATIONS 629 and hence the solution of the integral equation with degenerate kernel is reduced to the deﬁnition of the constants Ak. Let us multiply Eq. (10) by hm(x) and integrate with respect to xfromatob. We obtain the following system of linear algebraic equations for the coefﬁcients Ak: Am–λn⎝summationdisplay k=1smkAk=fm,m=1 ,...,n, (11) where smk=⎝integraldisplayb ahm(x)gk(x)dx,fm=⎝integraldisplayb af(x)hm(x)dx;m,k=1 ,...,n. (12) In the calculation of the coefﬁcients smkandfmfor speciﬁc degenerate kernels, the tables of integrals can be applied; see Supplements 3 and 4, as well as I. S. Gradshtein and I. M. Ryzhik (1980), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986). Once we construct a solution of system (11), we obtain a solution of the integral equation with degenerate kernel (7) as well. The values of the parameter λat which the determinant of system (11) vanishes are characteristic values of the integral equation (7), and it is clear that there are just nsuch values counted according to their multiplicities. Now we can state the main results on the solution of Eq. (7). 1◦.I fλis a regular value, then for an arbitrary right-hand side f(x), there exists a unique solution of the Fredholm integral equation with degenerate kernel and this solution can be represented in the form (10), in which the coefﬁcients Akmake up a solution of system (11). The constants Akcan be determined, for instance, by Cramer’s rule (see equation 4.9.20, Chapter 4). 2◦.I fλis a characteristic value and f(x)≡0, then every solution of the homogeneous equation with degenerate kernel has the form y(x)=p⎝summationdisplay i=1Ciyi(x), (13) where the Ciare arbitrary constants and the yi(x) are linearly independent eigenfunctions of the kernel corresponding to the characteristic value λ: yi(x)=n⎝summationdisplay k=1Ak(i)gk(x). (14) Here the constants Ak(i)formp(p≤n) linearly independent solutions of the following homogeneous system of algebraic equations: Am(i)–λn⎝summationdisplay k=1smkAk(i)=0 ; m=1 ,...,n,i=1 ,...,p. (15) 3◦.I fλis a characteristic value and f(x)≠0, then for the nonhomogeneous integral equation (7) to be solvable, it is necessary and sufﬁcient that the right-hand side f(x)i ss u c ht h a tt h e pconditions n⎝summationdisplay k=1Bk(i)fk=0 , i=1 ,...,p,p≤n, (16) 630 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) are satisﬁed. Here the constants Bk(i)formplinearly independent solutions of the homogeneous system of algebraic equations which is the transpose of system (15). In this case, every solution ofEq. (7) has the form y(x)=y 0(x)+p⎝summationdisplay i=1Ciyi(x), (17) where y0(x) is a particular solution of the nonhomogeneous equation (7) and the sum represents the general solution of the corresponding homogeneous equation (see item 2◦). In particular, if f(x)≠0 but all fkare zero, we have y(x)=f(x)+p⎝summationdisplay i=1Ciyi(x). (18) Remark. When studying Fredholm equations of the second kind with degenerate kernel, it is useful for the reader to be acquainted with equations 4.9.18 and 4.9.20 of the ﬁrst part of the book. Example 1. Let us solve the integral equation y(x)–λ⎝integraldisplayπ –π(xcost+t2sinx+c o sxsint)y(t)dt=x,– π≤x≤π. (19) Let us denote A1=⎝integraldisplayπ –πy(t)c o std t,A2=⎝integraldisplayπ –πt2y(t)dt,A3=⎝integraldisplayπ –πy(t)s i ntd t, (20) where A1,A2,a n dA3are unknown constants. Then Eq. (19) can be rewritten in the form y(x)=A1λx+A2λsinx+A3λcosx+x. (21) On substituting the expression (21) into relations (20), we obtain A1=⎝integraldisplayπ –π(A1λt+A2λsint+A3λcost+t)c o std t, A2=⎝integraldisplayπ –π(A1λt+A2λsint+A3λcost+t)t2dt, A3=⎝integraldisplayπ –π(A1λt+A2λsint+A3λcost+t)s i ntd t. On calculating the integrals occurring in these equations, we obtain the following system of algebraic equations for the unknowns A1,A2,a n dA3: A1–λπA 3=0 , A2+4λπA 3=0 , –2λπA 1–λπA 2+A3=2π.(22) The determinant of this system is ∆(λ)=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle10 – λπ 01 4 λπ –2λπ –λπ 1⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle=1+2 λ 2π2≠0. Thus, system (22) has the unique solution A1=2λπ2 1+2λ2π2,A2=–8λπ2 1+2λ2π2,A3=2π 1+2λ2π2. On substituting the above values of A1,A2,a n dA3into (21), we obtain the solution of the original integral equation: y(x)=2λπ 1+2λ2π2(λπx –4λπsinx+c o sx)+x. 13.2. F REDHOLM EQUATIONS OF THE SECOND KIND WITH DEGENERATE KERNEL .SOME GENERALIZATIONS 631 13.2-3. Kernel is the Sum of a Nondegenerate Kernel and an Arbitrary Degenerate Kernel. 1◦. Consider a linear integral equation of the second kind y(x)+⎝integraldisplayb aK(x,t)y(t)dt=f(x). (23) Suppose equation (23) can be solved for any f(x) from some class of functions LF.L e tyf(x) denote the corresponding solution. Now consider the more complex integral equation u(x)+⎝integraldisplayb a[K(x,t)+ϕ(x)ψ(t)]u(t)dt=f(x), (24) with its kernel containing an additional term ϕ(x)ψ(t). A solution to equation (24) will be sought in the form u(x)=yf(x)+Ayϕ(x), (25) where yϕ(x) is the solution to equation (23) in which f(x) must be replaced with ϕ(x). Substituting (25) into (24) results in the coefﬁcient A: A=–⎝integraldisplayb aψ(t)yf(t)dt 1+⎝integraldisplayb aψ(t)yϕ(t)dt. (26) Formulas (25)–(26) deﬁne a solution to equation (24), provided the integrals in the numerator and denominator exist, with⎝integraltextb aψ(t)yϕ(t)dt≠–1. In addition, the condition ϕ(x)∈LFmust be satisﬁed. Example 2. The solution of the integral equation y(x)–λ⎝integraldisplay∞ 0sin(xt)y(t)dt=f(x) (27) is expressed as (see Eq. 4.5.20) yf(x)=f(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0sin(xt)f(t)dt, (28) where λ≠±⎝radicalbig 2/π. Now consider the more complex integral equation u(x)–λ⎝integraldisplay∞ 0[sin(xt )+ϕ(x)ψ(t)]u(t)dt=f(x) (29) with its kernel containing an arbitrary additive function ϕ(x)ψ(t). The corresponding solution (28) to equation (27) with f(x)=ϕ(x) is written as yϕ(x)=ϕ(x) 1–π 2λ2+λ 1–π 2λ2⎝integraldisplay∞ 0sin(xt)ϕ(t)dt. Hence, equation (29) has the solution u(x)=yf(x)+Ayϕ(x),A=–⎝integraltext1 0ψ(t)yf(t)dt 1+⎝integraltext1 0ψ(t)y1(t)dt. 2◦. The integral equation u(x)+⎝integraldisplayb a⎝bracketleftbigg K(x,t)+n⎝summationdisplay m=1ϕm(x)ψm(t)⎝bracketrightbigg u(t)dt=f(x), (30) 632 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) with its kernel being the sum of the kernel of equation (23) and an arbitrary degenerate kernel, can be solved in a similar manner. The solution is sought in the additive form u(x)=yf(x)+n⎝summationdisplay m=1Amyϕm(x), (31) where yϕm(x) is the solution to equation (23) in which f(x) must be replaced with ϕm(x). Substitut- ing (31) into (30) results in the following linear algebraic system of equations for the coefﬁcients Am: Am+n⎝summationdisplay j=1Ajσmj=–σm0,m=1 ,...,n; σmj=⎝integraldisplayb aψm(t)yϕj(t)dt,σm0=⎝integraldisplayb aψm(t)yf(t)dt. References for Section 13.2: S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), P. P. Zabreyko, A. I. Koshelev, et al. (1975), A. J. Jerry (1985), A. F. Verlan’ and V . S. Sizikov (1986), A. D. Polyanin andA. I. Zhurov (2007). 13.3. Solution as a Power Series in the Parameter. Method of Successive Approximations 13.3-1. Iterated Kernels. Consider the Fredholm integral equation of the second kind: y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b.( 1 ) We seek the solution in the form of a series in powers of the parameter λ: y(x)=f(x)+∞⎝summationdisplay n=1λnψn(x). (2) Substitute series (2) into Eq. (1). On matching the coefﬁcients of like powers of λ, we obtain a recurrent system of equations for the functions ψn(x). The solution of this system yields ψ1(x)=⎝integraldisplayb aK(x,t)f(t)dt, ψ2(x)=⎝integraldisplayb aK(x,t)ψ1(t)dt=⎝integraldisplayb aK2(x,t)f(t)dt, ψ3(x)=⎝integraldisplayb aK(x,t)ψ2(t)dt=⎝integraldisplayb aK3(x,t)f(t)dt,e t c . Here Kn(x,t)=⎝integraldisplayb aK(x,z)Kn–1(z,t)dz,( 3) where n=2 ,3 , ..., and we have K1(x,t)≡K(x,t). The functions Kn(x,t) deﬁned by formulas (3) are called iterated kernels . These kernels satisfy the relation Kn(x,t)=⎝integraldisplayb aKm(x,s)Kn–m(s,t)ds,( 4 ) where mis an arbitrary positive integer less than n. 13.3. S OLUTION AS A POWER SERIES IN THE PARAMETER .M ETHOD OF SUCCESSIVE APPROXIMA TIONS 633 The iterated kernels Kn(x,t) can be directly expressed via K(x,t) by the formula Kn(x,t)=⎝integraldisplayb a⎝integraldisplayb a···⎝integraldisplayb a⎝bracehtipupleft ⎝bracehtipdownright⎝bracehtipdownleft ⎝bracehtipupright n–1K(x,s1)K(s1,s2)...K (sn–1,t)ds1ds2... dsn–1. All iterated kernels Kn(x,t), beginning with K2(x,t), are continuous functions on the square S={a≤x≤b,a≤t≤b}if the original kernel K(x,t) is square integrable on S. IfK(x,t) is symmetric, then all iterated kernels Kn(x,t) are also symmetric. 13.3-2. Method of Successive Approximations. The results of Subsection 13.3-1 can also be obtained by means of the method of successive approximations. To this end, one should use the recurrent formula yn(x)=f(x)+λ⎝integraldisplayb aK(x,t)yn–1(t)dt,n=1 ,2 , ..., with the zeroth approximation y0(x)=f(x). 13.3-3. Construction of the Resolvent. The resolvent of the integral equation (1) is deﬁned via the iterated kernels by the formula R(x,t;λ)=∞⎝summationdisplay n=1λn–1Kn(x,t), (5) where the series on the right-hand side is called the Neumann series of the kernel K(x,t). It converges to a unique square integrable solution of Eq. (1) provided that |λ|<1 B,B=⎝radicalBigg ⎝integraldisplayb a⎝integraldisplayb aK2(x,t)dx dt .( 6 ) If, in addition, we have⎝integraldisplayb aK2(x,t)dt≤A,a≤x≤b, where Ais a constant, then the Neumann series converges absolutely and uniformly on [ a,b]. A solution of a Fredholm equation of the second kind of the form (1) is expressed by the formula y(x)=f(x)+λ⎝integraldisplayb aR(x,t;λ)f(t)dt,a≤x≤b.( 7) Inequality (6) is essential for the convergence of the series (5). However, a solution of Eq. (1) can exist for values |λ|>1/Bas well. Remark 1. A solution of the equation y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b, 634 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) with weak singularity, where the kernel K(x,t)h a st h ef o r m K(x,t)=L(x,t) |x–t|α,0 < α<1 , andL(x,t) is a function continuous on the square S={a≤x≤b,a≤t≤b}, can be obtained by the successive approximation method provided that |λ|<1–α 2B∗(b–a)1–α,B∗=s u p |L(x,t)|. The equation itself can be reduced to a Fredholm equation of the form y(x)–λn⎝integraldisplayb aKn(x,t)y(t)dt=F(x), a≤x≤b, F(x)=f(x)+n–1⎝summationdisplay p=1λp⎝integraldisplayb aKp(x,t)f(t)dt, where Kp(x,t)(p=1 ,...,n)i st h e pth iterated kernel, with Kn(x,t) being a Fredholm kernel for n>1 2(1 –α)–1and bounded for n>( 1– α)–1. Example 1. Let us solve the integral equation y(x)–λ⎝integraldisplay1 0xty(t)dt=f(x), 0 ≤x≤1, by the method of successive approximations. Here we have K(x,t)=xt,a=0 ,a n d b= 1. We successively deﬁne K1(x,t)=xt,K2(x,t)=⎝integraldisplay1 0(xz)(zt)dz=xt 3,K3(x,t)=1 3⎝integraldisplay1 0(xz)(zt)dz=xt 32,...,Kn(x,t)=xt 3n–1. According to formula (5) for the resolvent, we obtain R(x,t;λ)=∞⎝summationdisplay n=1λn–1Kn(x,t)=xt∞⎝summationdisplay n=1⎝parenleftbiggλ 3⎝parenrightbiggn–1 =3xt 3–λ, where |λ|< 3, and it follows from formula (7) that the solution of the integral equation can be rewritten in the form y(x)=f(x)+λ⎝integraldisplay1 03xt 3–λf(t)dt,0 ≤x≤1,λ≠3. In particular, for f(x)=xwe obtain y(x)=3x 3–λ,0 ≤x≤1,λ≠3. 13.3-4. Orthogonal Kernels. For some Fredholm equations, the Neumann series (5) for the resolvent is convergent for all values ofλ. Let us establish this fact. Assume that two kernels K(x,t)a n d L(x,t) are given. These kernels are said to be orthogonal if the following two conditions hold: ⎝integraldisplayb aK(x,z)L(z,t)dz=0 ,⎝integraldisplayb aL(x,z)K(z,t)dz=0 ( 8 ) for all admissible values of xandt. There exist kernels that are orthogonal to themselves. For these kernels we have K2(x,t)≡0, where K2(x,t) is the second iterated kernel. It is clear that in this case all the subsequent iterated kernels also vanish, and the resolvent coincides with the kernel K(x,t). 13.4. M ETHOD OF FREDHOLM DETERMINANTS 635 Example 2. Let us ﬁnd the resolvent of the kernel K(x,t)=s i n ( x–2t), 0≤x≤2π,0≤t≤2π. We have ⎝integraldisplay2π 0sin(x–2z)s i n (z–2t)dz=1 2⎝integraldisplay2π 0[cos(x+2t–3z)–c o s ( x–2t–z)]dz= =1 2⎝bracketleftbig –1 3sin(x+2t–3z)+s i n ( x–2t–z)⎝bracketrightbigz=2π z=0=0 . Thus, in this case the resolvent of the kernel is equal to the kernel itself: R(x,t;λ)≡sin(x–2t), so that the Neumann series (5) consists of a single term and clearly converges for any λ. Example 3. The kernel K(x,t)=∞⎝summationdisplay n=1ansin(nx)c o s (nt), 0 ≤x,t≤2π, with a convergent series∞⎝summationtext n=1|an|is orthogonal to itself. Remark 2. If the kernels M(1)(x,t),...,M(n)(x,t) are pairwise orthogonal, then the resolvent corresponding to the sum K(x,t)=n⎝summationdisplay m=1M(m)(x,t) is equal to the sum of the resolvents corresponding to each of the summands. References for Section 13.3: S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), V . I. Smirnov (1974), A. J. Jerry (1985). 13.4. Method of Fredholm Determinants 13.4-1. Formula for the Resolvent. A solution of the Fredholm equation of the second kind y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b,( 1 ) is given by the formula y(x)=f(x)+λ⎝integraldisplayb aR(x,t;λ)f(t)dt,a≤x≤b,( 2) where the resolvent R(x,t;λ)i sd e ﬁ n e db yt h er e l a t i o n R(x,t;λ)=D(x,t;λ) D(λ),D(λ)≠0. (3) HereD(x,t;λ)a n dD(λ)a r ep o w e rs e r i e si n λ, D(x,t;λ)=∞⎝summationdisplay n=0(–1)n n!An(x,t)λn,D(λ)=∞⎝summationdisplay n=0(–1)n n!Bnλn,( 4) 636 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) with coefﬁcients deﬁned by the formulas A0(x,t)=K(x,t),An(x,t)=⎝integraldisplayb a···⎝integraldisplayb a⎝bracehtipupleft ⎝bracehtipdownright⎝bracehtipdownleft ⎝bracehtipupright n⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleK(x,t)K(x,t 1)···K(x,tn) K(t1,t)K(t1,t1)···K(t1,tn) ............ K(t n,t)K(tn,t1)···K(tn,tn)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle dt1...d t n,( 5 ) B0=1 , Bn=⎝integraldisplayb a···⎝integraldisplayb a⎝bracehtipupleft ⎝bracehtipdownright⎝bracehtipdownleft ⎝bracehtipupright n⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleK(t 1,t1)K(t1,t2)···K(t1,tn) K(t2,t1)K(t2,t2)···K(t2,tn) ............ K(t n,t1)K(tn,t2)···K(tn,tn)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingledt1...d t n;n=0 ,1 ,2 ,... (6) The function D(x,t;λ) is called the Fredholm minor andD(λ)t h e Fredholm determinant.T h e series (4) converge for all values of λand hence deﬁne entire analytic functions of λ.T h er e s o l v e n t R(x,t;λ) is an analytic function of λeverywhere except for the values of λthat are roots of D(λ). These roots coincide with the characteristic values of the equation and are poles of the resolvent R(x,t;λ). Example 1. Consider the integral equation y(x)–λ⎝integraldisplay1 0xety(t)dt=f(x), 0 ≤x≤1,λ≠1. We have A0(x,t)=xet,A1(x,t)=⎝integraldisplay1 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglexe txet1 t1ett1et1⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingledt 1=0 , A2(x,t)=⎝integraldisplay1 0⎝integraldisplay1 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglexe txet1xet2 t1ett1et1t1et2 t2ett2et1t2et2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingledt 1dt2=0 , since the determinants in the integrand are zero. It is clear that the relation An(x,t) = 0 holds for the subsequent coefﬁcients. Let us ﬁnd the coefﬁcients Bn: B1=⎝integraldisplay1 0K(t1,t1)dt1=⎝integraldisplay1 0t1et1dt1=1 , B2=⎝integraldisplay1 0⎝integraldisplay1 0⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglet1et1t1et2 t2et1t2et2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingledt1dt2=0 . It is clear that Bn= 0 for all subsequent coefﬁcients as well. According to formulas (4), we have D(x,t;λ)=K(x,t)=xet;D(λ)=1– λ. Thus, R(x,t;λ)=D(x,t;λ) D(λ)=xet 1–λ, and the solution of the equation can be represented in the form y(x)=f(x)+λ⎝integraldisplay1 0xet 1–λf(t)dt,0 ≤x≤1,λ≠1. In particular, for f(x)=e–xwe obtain y(x)=e–x+λ 1–λx,0 ≤x≤1,λ≠1. 13.4-2. Recurrent Relations. In practice, the calculation of the coefﬁcients An(x,t)a n d Bnof the series (4) by means of formulas (5) and (6) is seldom possible. However, formulas (5) and (6) imply the following recurrent relations: An(x,t)=BnK(x,t)–n⎝integraldisplayb aK(x,s)An–1(s,t)ds,( 7 ) Bn=⎝integraldisplayb aAn–1(s,s)ds.( 8) 13.5. F REDHOLM THEOREMS AND THE FREDHOLM ALTERNATIVE 637 Example 2. Let us use formulas (7) and (8) to ﬁnd the resolvent of the kernel K(x,t)=x–2t,w h e r e0 ≤x≤1a n d 0≤t≤1. Indeed, we have B0=1a n d A0(x,t)=x–2t. Applying formula (8), we see that B1=⎝integraldisplay1 0(–s)ds=–1 2. Formula (7) implies the relation A1(x,t)=–x–2t 2–⎝integraldisplay1 0(x–2s)(s–2t)ds=–x–t+2xt+2 3. Furthermore, we have B2=⎝integraldisplay1 0⎝parenleftbig –2s+2s2+2 3⎝parenrightbig ds=1 3, A2(x,t)=x–2t 3–2⎝integraldisplay1 0(x–2s)⎝parenleftbig–s–t+2st+2 3⎝parenrightbigds=0 , B3=B4=···=0 , A3(x,t)=A4(x,t)=···=0 . Hence, D(λ)=1+1 2λ+1 6λ2;D(x,t;λ)=x–2t+λ⎝parenleftbigx+t–2xt–2 3⎝parenrightbig. The resolvent has the form R(x,t;λ)=x–2t+λ⎝parenleftbig x+t–2xt–2 3⎝parenrightbig 1+1 2λ+1 6λ2. References for Section 13.4: S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), V . I. Smirnov (1974). 13.5. Fredholm Theorems and the Fredholm Alternative 13.5-1. Fredholm Theorems. THEOREM 1.Ifλis a regular value, then both the Fredholm integral equation of the second kind and the transposed equation are solvable for any right-hand side, and both the equations have unique solutions. The corresponding homogeneous equations have only the trivial solutions. THEOREM 2.For the nonhomogeneous integral equation to be solvable, it is necessary and sufﬁcient that the right-hand side f(x)satisﬁes the conditions ⎝integraldisplayb af(x)ψk(x)dx=0 , k=1 ,...,n, where ψk(x)is a complete set of linearly independent solutions of the corresponding transposed homogeneous equation. THEOREM 3.Ifλis a characteristic value, then both the homogeneous integral equation and the transposed homogeneous equation have nontrivial solutions. The number of linearly independent solutions of the homogeneous integral equation is ﬁnite and is equal to the number of linearly independent solutions of the transposed homogeneous equation. THEOREM 4.A Fredholm equation of the second kind has at most countably many characteristic values, whose only possible accumulation point is the point at inﬁnity. Example. To illustrate the Fredholm theorems, consider the degenerate integral equation y(x)–λ⎝integraldisplayπ 0sin(x+t)y(t)dt=f(x). (1) 638 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) Using the trigonometric formula sin(x+t)=s i n xcost+c o sxsint, we transform equation (1) to y(x)=λAsinx+λBcosx+f(x), (2) A=⎝integraldisplayπ 0costy(t)dt,B=⎝integraldisplayπ 0sinty(t)dt. (3) Substituting (2) into (3), we come to the system of linear algebraic equations for the coefﬁcients AandB: A–1 2πλB =f1, –1 2πλA +B=f2,(4) where f1=⎝integraldisplayπ 0f(t)c o std t,f2=⎝integraldisplayπ 0f(t)s i ntd t. The determinant of this system⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle1– 1 2πλ –1 2πλ 1⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle=1– 1 4π2λ2 has the roots λ1=– 2/π,λ2=2/π, (5) which coincide with the characteristic values of equation (1); the other values λ≠±2/πare regular. Ifλdiffers from the characteristic values (5), then the determinant of system (4) differs from zero and the coefﬁcients AandBare uniquely deﬁned by A=f1+1 2πλf 2 1–1 4π2λ2,B=1 2πλf 1+f2 1–1 4π2λ2 and yield the unique solution (2) of the nonhomogeneous integral equation (1). For these values λ≠±2/π, the corresponding homogeneous integral equation (1) with f(x)≡0 has only the trivial solution y(x)≡0. (This illustrates Theorem 1.) Now, suppose that λ=λ1,2is one of the characteristic values (5). In this case, both equations of the homogeneous system (4) with f(x)≡0 are proportional and one can take A=1 2πλB ,w h e r e Bis an arbitrary constant. The corresponding eigenfunctions have the form y1,2(x)=2 πB(sinx∓cosx). (6) The constant Bcan be chosen, for instance, from the following normalization condition for eigenfunctions: /bardbly1,2/bardbl2=⎝integraldisplayπ 0|y1,2(x)|2dx=1 , which yields B=1 2√ π. If we take λ=λ1=– 2/πin the nonhomogeneous equation (1), then the algebraic system (4) takes the form A+B=f1,A+B=f2; and for its solvability it is necessary and sufﬁcient that f1=f2. This condition means that the right-hand side f(x)i s orthogonal to the eigenfunction y1(x). Similarly, for λ=λ2=2/π, system (4) has a solution if and only if f1=–f2, i.e., the functions f(x)a n dy2(x) are orthogonal. The eigenfunctions obtained in this example and the orthogonality conditions illustrate the statements of Theorems 2 and 3 (in the case under consideration, the kernel K(x,t) coincides with its conjugate). 13.5-2. Fredholm Alternative. The Fredholm theorems imply the so-called Fredholm alternative, which is most frequently used in the investigation of integral equations. THEFREDHOLM ALTERNATIVE .Either the nonhomogeneous equation is solvable for any right- hand side or the corresponding homogeneous equation has nontrivial solutions. The ﬁrst part of the alternative holds if the given value of the parameter is regular and the second if it is characteristic. Remark. The Fredholm theory is also valid for integral equations of the second kind with weak singularity. References for Section 13.5: S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), V . I. Smirnov (1974), A. J. Jerry (1985), D. Porter and D. S. G. Stirling (1990), C. Corduneanu (1991), J. Kondo (1991), W. Hackbusch (1995), R. P. Kanwal (1996), R. Kress (1999). 13.6. F REDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND WITH SYMMETRIC KERNEL 639 13.6. Fredholm Integral Equations of the Second Kind with Symmetric Kernel 13.6-1. Characteristic Values and Eigenfunctions. Integral equations whose kernels are symmetric , that is, satisfy the condition K(x,t)=K(t,x), are called symmetric integral equations . Each symmetric kernel that is not iden tically zero has at least one characteristic value. For any n, the set of characteristic values of the nth iterated kernel coincides with the set of nth powers of the characteristic values of the ﬁrst kernel. The eigenfunctions of a symmetric kernel corresponding to distinct characteristic values are orthogonal, i.e., if ϕ1(x)=λ1⎝integraldisplayb aK(x,t)ϕ1(t)dt,ϕ2(x)=λ2⎝integraldisplayb aK(x,t)ϕ2(t)dt,λ1≠λ2, then (ϕ1,ϕ2)=0 , ( ϕ,ψ)≡⎝integraldisplayb aϕ(x)ψ(x)dx. The characteristic values of a symmetric kernel are real. The eigenfunctions can be normalized; namely, we can divide each characteristic function by its norm. If several linearly independenteigenfunctions correspond to the same characteristic value, say, ϕ1(x),...,ϕn(x), then each linear combination of these functions is an eigenfunction as well, and these linear combinations can be chosen so that the corresponding eigenfunctions are orthonormal. Indeed, the function ψ1(x)=ϕ1(x) /bardblϕ1/bardbl,/bardblϕ1/bardbl=⎝radicalbig (ϕ1,ϕ1), has the norm equal to one, i.e., /bardblψ1/bardbl= 1. Let us form a linear combination αψ1+ϕ2and choose α so that (αψ1+ϕ2,ψ1)=0 , i.e., α=–(ϕ2,ψ1) (ψ1,ψ1)=– (ϕ2,ψ1). The function ψ2(x)=αψ1+ϕ2 /bardblαψ1+ϕ2/bardbl is orthogonal to ψ1(x) and has the unit norm. Next, we choose a linear combination αψ1+βψ2+ϕ3, where the constants αandβcan be found from the orthogonality relations (αψ1+βϕ2+ϕ3,ψ1)=0 , ( αψ1+βψ2+ϕ3,ψ2)=0 . For the coefﬁcients αandβthus deﬁned, the function ψ3=αψ1+βψ2+ϕ2 /bardblαψ1+βϕ2+ϕ3/bardbl is orthogonal to ψ1andψ2and has the unit norm, and so on. As was noted above, the eigenfunctions correspondi ng to distinct characteristic values are orthogonal. Hence, the sequence of eigenfunctions of a symmetric kernel can be made orthonormal. 640 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) In what follows we assume that the sequence of eigenfunctions of a symmetric kernel is or- thonormal. We also assume that the characteristic values are always numbered in the increasing order of their absolute values. Thus, if λ1,λ2,...,λn,... (1) is the sequence of characteristic values of a s ymmetric kernel, and if a sequence of eigenfunctions ϕ1,ϕ2,...,ϕn,... (2) corresponds to the sequence (1) so that ϕn(x)–λn⎝integraldisplayb aK(x,t)ϕn(t)dt=0 , ( 3 ) then⎝integraldisplayb aϕi(x)ϕj(x)dx=⎝braceleftbigg1f o r i=j, 0f o r i≠j,(4) and |λ1|≤|λ2|≤···≤|λn|≤···.( 5 ) If there are inﬁnitely many characteristic values, then it follows from the fourth Fredholm theorem that their only accumulation point is the point at inﬁnity, and hence λn→∞ asn→∞ . The set of all characteristic values and the corresponding normalized eigenfunctions of a sym- metric kernel is called the system of characteristic values and eigenfunctions of the kernel. The system of eigenfunctions is said to be incomplete if there exists a nonzero square integrable function that is orthogonal to all functions of the system. Otherwise, the system of eigenfunctions is said to becomplete . 13.6-2. Bilinear Series. Assume that a kernel K(x,t) admits an expansion in a uniformly convergent series with respect to the orthonormal system of its eigenfunctions: K(x,t)=∞⎝summationdisplay k=1ak(x)ϕk(t)( 6) for all xin the case of a continuous kernel or for almost all xin the case of a square integrable kernel. We have ak(x)=⎝integraldisplayb aK(x,t)ϕk(t)dt=ϕk(x) λk,( 7) and hence K(x,t)=∞⎝summationdisplay k=1ϕk(x)ϕk(t) λk.( 8) Conversely, if the series ∞⎝summationdisplay k=1ϕk(x)ϕk(t) λk(9) 13.6. F REDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND WITH SYMMETRIC KERNEL 641 is uniformly convergent, then K(x,t)=∞⎝summationdisplay k=1ϕk(x)ϕk(t) λk. The following assertion holds: the bilinear series (9) converges in mean-square to the ker- nelK(x,t). If a symmetric kernel K(x,t) has ﬁnitely many characteristic values, then it is degenerate, because in this case we have K(x,t)=n⎝summationdisplay k=1ϕk(x)ϕk(t) λk. (10) Ak e r n e l K(x,t)i ss a i dt ob e positive deﬁnite if for all functions ϕ(x) that are not identically zero we have⎝integraldisplayb a⎝integraldisplayb aK(x,t)ϕ(x)ϕ(t)dx dt >0 , and the above quadratic functional vanishes for ϕ(x)=0 only. Such a kernel has positive characteristic values only. A negative deﬁnite kernel is deﬁned similarly. Each symmetric positive deﬁnite (or negative deﬁnite) continuous kernel can be decomposed in a bilinear series in eigenfunctions that is absolutely and uniformly convergent with respect to the variables x,t. The assertion remains valid if we assume that the kernel has ﬁnitely many negative (positive, respectively) characteristic values. If a kernel K(x,t) is symmetric, continuous on the square S={a≤x≤b,a≤t≤b}, and has uniformly bounded partial derivatives on this square, then this kernel can be expanded in a uniformly convergent bilinear series in eigenfunctions. 13.6-3. Hilbert–Schmidt Theorem. If a function f(x) can be represented in the form f(x)=⎝integraldisplayb aK(x,t)g(t)dt, (11) where the symmetric kernel K(x,t) is square integrable and g(t) is a square integrable function, thenf(x) can be represented by its F ourier series with respect to the orthonormal system of eigenfunctions of the kernel K(x,t): f(x)=∞⎝summationdisplay k=1akϕk(x), (12) where ak=⎝integraldisplayb af(x)ϕk(x)dx,k=1 ,2 , ... Moreover, if⎝integraldisplayb aK2(x,t)dt≤A<∞, (13) then the series (12) is absolutely and uniformly convergent for any function f(x) of the form (11). Remark 1. In the Hilbert–Schmidt theorem, the completeness of the system of eigenfunctions is not assumed. 642 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.6-4. Bilinear Series of Iterated Kernels. By the deﬁnition of the iterated kernels, we have Km(x,t)=⎝integraldisplayb aK(x,z)Km–1(z,t)dz,m=2 ,3 , ... (14) The Fourier coefﬁcients ak(t)o ft h ek e r n e l Km(x,t), regarded as a function of the variable x, with respect to the orthonormal system of eigenfunctions of the kernel K(x,t) are equal to ak(t)=⎝integraldisplayb aKm(x,t)ϕk(x)dx=ϕk(t) λm k. (15) On applying the Hilbert–Schmidt theorem to (14), we obtain Km(x,t)=∞⎝summationdisplay k=1ϕk(x)ϕk(t) λm k,m=2 ,3 , ... (16) In formula (16), the sum of the series is understood as the limit in mean-square. If in addition to the above assumptions, inequality (13) is satisﬁed, then the series in (16) is uniformly convergent. 13.6-5. Solution of the Nonhomogeneous Equation. Let us represent an integral equation y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b, (17) where the parameter λis not a characteristic value, in the form y(x)–f(x)=λ⎝integraldisplayb aK(x,t)y(t)dt (18) and apply the Hilbert–Schmidt theorem to the function y(x)–f(x): y(x)–f(x)=∞⎝summationdisplay k=1Akϕk(x), Ak=⎝integraldisplayb a[y(x)–f(x)]ϕk(x)dx=⎝integraldisplayb ay(x)ϕk(x)dx–⎝integraldisplayb af(x)ϕk(x)dx=yk–fk. Taking into account the expansion (8), we obtain λ⎝integraldisplayb aK(x,t)y(t)dt=λ∞⎝summationdisplay k=1yk λkϕk(x), and thus λyk λk=yk–fk,yk=λkfk λk–λ,Ak=λfk λk–λ. (19) Hence, y(x)=f(x)+λ∞⎝summationdisplay k=1fk λk–λϕk(x). (20) 13.6. F REDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND WITH SYMMETRIC KERNEL 643 However, if λis a characteristic value, i.e., λ=λp=λp+1=···=λq, (21) then, for k≠p,p+1 ,...,q, the terms (20) preserve their form. For k=p,p+1 ,...,q, formula (19) implies the relation fk=Ak(λ–λk)/λ, and by (21) we obtain fp=fp+1=···=fq=0 . T h el a s t relation means that⎝integraldisplayb af(x)ϕk(x)dx=0 fork=p,p+1,...,q, i.e., the right-hand side of the equation must be orthogonal to the eigenfunctions that correspond to the characteristic value λ. In this case, the solutions of Eqs. (17) have the form y(x)=f(x)+λ∞⎝summationdisplay k=1fk λk–λϕk(x)+q⎝summationdisplay k=pCkϕk(x), (22) where the terms in the ﬁrst of the sums (22) with indices k=p,p+1 ,...,qmust be omitted (for these indices, fkandλ–λkvanish in this sum simultaneously). The coefﬁcients Ckin the second sum are arbitrary constants. Remark 2. On the basis of the bilinear expansion (8) and the Hilbert–Schmidt theorem, the solution of the symmetric Fredholm integral equation of the ﬁrst kind ⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b, can be constructed in a similar way in the form y(x)=∞⎝summationdisplay k=1fkλkϕk(x), and the necessary and sufﬁcient c ondition for the existence and uniqueness of such a solution inL2(a,b) is the completeness of the system of the eigenfunctions ϕk(x)o ft h ek e r n e l K(x,t) together with the convergenceof the series∞⎝summationtext k=1f2 kλ2k,w h e r et h e λkare the corresponding characteristic values. It should be noted that the veriﬁcation of the last condition for speciﬁc equations is quite complicated. In the solution of Fredholm equations of the ﬁrst kind, the methods presented in Chapter 12 are usually applied. 13.6-6. Fredholm Alternative for Symmetric Equations. The above results can be uniﬁed in the following alternative form. A symmetric integral equation y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b, (23) for a given λ, either has a unique square integrable solution for an arbitrarily given function f(x)∈L2(a,b), in particular, y=0f o r f= 0, or the corresponding homogeneous equation has ﬁnitely many linearly independent solutions Y1(x),...,Yr(x),r>0 . For the second case, the nonhomogeneous equation has a solution if and only if the right-hand sidef(x) is orthogonal to all the functions Y1(x),...,Yr(x) on the interval [ a,b]. Here the solution is deﬁned only up to an arbitrary additive linear combination A1Y1(x)+···+ArYr(x). 644 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.6-7. Resolvent of a Symmetric Kernel. The solution of a Fredholm equation of the second kind (23) can be written in the form y(x)=f(x)+λ⎝integraldisplayb aR(x,t;λ)f(t)dt, (24) where the resolvent R(x,t;λ) is given by the series R(x,t;λ)=∞⎝summationdisplay k=1ϕk(x)ϕk(t) λk–λ. (25) Here the collections ϕk(x)a n dλkform the system of eigenfunctions and characteristic values of Eqs. (23). It follows from formula (25) that the resolvent of a symmetric kernel has only simplepoles. 13.6-8. Extremal Properties of Characteris tic Values and Eigenfunctions. Let us introduce the notation (u,w)=⎝integraldisplayb au(x)w(x)dx,/bardblu/bardbl2=(u,u), (Ku,u)=⎝integraldisplayb a⎝integraldisplayb aK(x,t)u(x)u(t)dx dt , where ( u,w)i st h e inner product of functions u(x)a n dw(x),/bardblu/bardblis the norm of a function u(x), and (Ku ,u)i st h e quadratic form generated by the kernel K(x,t). Letλ1be the characteristic value of the symmetric kernel K(x,t) with minimum absolute value and let y1(x) be the eigenfunction corresponding to this value. Then 1 |λ1|=m a x y/ ≡0|(Ky,y)| /bardbly/bardbl2; (26) in particular, the maximum is attained, and y=y1is a maximum point. Letλ1,...,λnbe the ﬁrst ncharacteristic values of a symmetric kernel K(x,t) (in the ascending order of their absolute values) and let y1(x),...,yn(x) be orthonormal eigenfunctions corresponding toλ1,...,λn, respectively. Then the formula 1 |λn+1|=m a x|(Ky,y)| /bardbly/bardbl2(27) is valid for the characteristic value λn+1following λn. The maximum is taken over the set of functions ywhich are orthogonal to all y1,...,ynand are not identically zero, that is, y≠0 (y,yj)=0 , j=1 ,...,n; (28) in particular, the maximum in (27) is attained, and y=yn+1is a maximum point, where yn+1is any eigenfunction corresponding to the characteristic value λn+1which is orthogonal to y1,...,yn. Remark 3. For a positive deﬁnite kernel K(x,t), the symbol of modulus on the right-hand sides of (27) and (28) can be omitted. 13.6. F REDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND WITH SYMMETRIC KERNEL 645 13.6-9. Kellog’s Method for Finding Characteristic Values in the Case of Symmetric Kernel. LetK(x,t) be a symmetric positive kernel ( a≤x,t≤b). For an arbitrary function ϕ0(x)∈L2(a,b), let us construct a sequence of functions by the recurrent formula ϕn(x)=⎝integraldisplayb aK(x,t)ϕn–1(t)dt,n=1 ,2 , ... and consider the numerical sequence⎝braceleftbigg/bardblϕn–1/bardbl /bardblϕn/bardbl⎝bracerightbigg , (29) where /bardblϕn/bardbl=⎝radicalbigg ⎝integraldisplayb a|ϕn(x)|2dx.L e ty1(x),y2(x),...be orthonormal eigenfunctions of the kernel K(x,t), and λ1≤λ2≤···the corresponding characteristic values. Suppose that the initial function ϕ0(x) has been chosen orthogonal to the functions y1(x),...,yk–1(x), but nonorthogonal to the eigenfunction yk(x). Then the limit of the sequence (29) is equal to the kth characteristic value λk. The sequence⎝braceleftbigg1 n⎝radicalbig /bardblϕn/bardbl⎝bracerightbigg (30) has the same limit as (29). In this case, the sequence of functions ⎝braceleftbiggϕn(x) /bardblϕn/bardbl⎝bracerightbigg converges to a function which is a linear combination of eigenfunctions corresponding to the characteristic value λk. Suppose that the functions y1(x)a n dϕ0(x) are nonorthogonal,⎝integraldisplayb ay1(x)ϕ0(x)dx≠0. Then, from (29) and (30) we obtain the following two approximation formulas for the smallest characteristic value: λ1≈/bardblϕn–1/bardbl//bardblϕn/bardbl, (31) λ1≈(/bardblϕn/bardbl)–1/n. (32) Formula (31) yields an upper bound for λ1. For a suitably chosen initial function ϕ0(x), the Kellog method is relatively simple with regard to calculations. Example 1. Let us apply the Kellog method for the calculation of the smallest characteristic value of the kernel K(x,t)=x2t2,0≤x,t≤1. Taking ϕ0(x)=xas the initial function, we ﬁnd that ϕ1(x)=⎝integraldisplay1 0x2t2td t=⎝integraldisplay1 0x2t3dt=1 4x2, ϕ2(x)=⎝integraldisplay1 0x2t21 4t2dt=1 4⎝integraldisplay1 0x2t4dt=1 4×5x2, ϕ3(x)=⎝integraldisplay1 0x2t21 4×5t2dt=1 4×5⎝integraldisplay1 0x2t4dt=1 4×52x2, .................................................................. ϕn(x)=⎝integraldisplay1 0x2t2 1 4×5n–2t2dt=1 4×5n–2⎝integraldisplay1 0x2t4dt=1 4×5n–1x2. Now, we deﬁne the norm /bardblϕn/bardbl=1 4×5n–1⎝radicalBigg ⎝integraldisplay1 0|x4dx=1 4×5n–1√ 5. 646 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) According to (31), we ﬁnd the ﬁrst characteristic value: λ1≈/bardblϕn–1/bardbl /bardblϕn/bardbl=5 . It is easy to check that λ1= 5 is the exact characteristic value. If the kernel K(x,t) is not positive deﬁnite, then formulas (31) and (32) yield approximations for the smallest absolute value of the corresponding characteristic values. 13.6-10. Trace Method for the Approximation of Characteristic Values. Them-trace of the kernel K(x,t)i sd e ﬁ n e db y Am=⎝integraldisplayb aKm(t,t)dt, where Km(x,t)i st h e mth iterated kernel (see Subsection 13.3-1). Formula (16) for a symmetric kernel implies that Am=∞⎝summationdisplay n=11 λmn(m=2 ,3 , ...). For sufﬁciently large m, the leading term in this expression is 1 /λm 1, and therefore, we obtain the approximate relations A2m≈1 λ2m 1,A2m+2≈1 λ2m+2 1. It follows that for the smallest characteristic value λ1, for large enough m, the following approxi- mation formula holds: |λ1|≈⎝radicalbigg A2m A2m+2, (33) which is an upper bound for |λ1|. In order to calculate the second characteristic value, one can use the approximation formulas |λ2|≈1 |λ1|⎝radicalbigg B2m B2m+2,|λ2|≈1 |λ1|⎝parenleftbigg2 B2m⎝parenrightbigg1/(2m) , where B2m=A2 2m–A4m. Traces of even orders for a symmetric kernel are calculated by the formula A2m=⎝integraldisplayb a⎝integraldisplayb aK2 m(x,t)dx dt =2⎝integraldisplayb a⎝integraldisplayx aK2 m(x,t)dt dx . (34) Example 2. Let us use the trace method to ﬁnd the ﬁrst characteristic value of the kernel K(x,t)=⎝braceleftBigxif 0 ≤x≤t≤1, tif 0 ≤t≤x≤1. SinceK(x,t) is symmetric, it sufﬁces to ﬁnd K2(x,t)f o rt<x.W eh a v e K2(x,t)=⎝integraldisplay1 0K(x,z)K(z,t)dz=⎝integraldisplayt 0z2dz+⎝integraldisplayx tztdz +⎝integraldisplay1 xxt dt =xt–1 2x2t–1 6t3. 13.6. F REDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND WITH SYMMETRIC KERNEL 647 Now, using (34) for m=1a n d m=2 ,w eﬁ n dt h a t A2=2⎝integraldisplay1 0dx⎝integraldisplayx 0K2 1(x,t)dt=2⎝integraldisplay1 0dx⎝integraldisplayx 0t2dt=2⎝integraldisplay1 0x3 3dx=1 6, A4=2⎝integraldisplay1 0dx⎝integraldisplayx 0K2 2(x,t)dt =2⎝integraldisplay1 0dx⎝integraldisplayx 0⎝parenleftbigg x2t2+x4t2 4+t6 36–x3t2–xt4 3+x2t4 6⎝parenrightbigg dt =2⎝integraldisplay1 0⎝parenleftbiggx2t3 3+x4t3 12+t7 7×36–x3t3 3–xt5 15+x2t5 30⎝parenrightbigg⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglet=x t=0dx =2⎝integraldisplay1 0⎝parenleftbiggx5 3+x7 12+x7 7×36–x6 3–x6 15+x7 30⎝parenrightbigg dx=17 630. By (33) we obtain an approximation for the smallest characteristic value, λ1≈⎝radicalBigg A2 A4=⎝radicalBigg 1 6 17 630≈2.485. This is in good agreement with the exact value λ1=1 4π2≈2.467 (the error is less than 1%). 13.6-11. Integral Equations Reducible to Symmetric Equations. An equation of the form y(x)–λ⎝integraldisplayb aK(x,t)ρ(t)y(t)dt=f(x), (35) where K(x,t) is a symmetric kernel and ρ(t) > 0 is a continuous function on [ a,b], can be reduced to a symmetric equation. Indeed, on multiplying Eq. (35) by√ ρ(x) and introducing the new unknown function z(x)=√ ρ(x)y(x), we arrive at the integral equation z(x)–λ⎝integraldisplayb aL(x,t)z(t)dt=f(x)⎝radicalbig ρ(x), L(x,t)=K(x,t)⎝radicalbig ρ(x)ρ(t), (36) where L(x,t) is a symmetric kernel. 13.6-12. Skew-Symmetric Integral Equations. By a skew-symmetric inte gral equation we mean an equation whose kernel is skew-symmetric, i.e., an equation of the form y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt=f(x) (37) whose kernel K(x,t) has the property K(t,x)=–K(x,t). (38) Equation (37) with the skew-symmetric kernel (38) has at least one characteristic value, and all its characteristic values are purely imaginary. 13.6-13. Remark on Nonsymmetric Kernels. An integral equation with a nonsymmetric kernel (i.e., such that K(x,t)≠K(t,x)f o rs o m e x,t)m a y happen to have no characteristic values. 648 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) Example. Consider the homogeneous integral equation with nonsymmetric degenerate kernel y(x)=λ⎝integraldisplayπ 0cosxsinty(t)dt, (1) which can be written in the form y(x)=Acosx,A=λ⎝integraldisplayπ 0sinty(t)dt. (2) Substituting (2) into (1) and dividing the result by cos x,w eg e t A=λ⎝integraldisplayπ 0sintAcostd t=0 . Therefore, equation (1) has only the trivial solution for any λ. References for Section 13.6: E. Goursat (1923), G. Wiarda (1930), R. Courant and D. Hilbert (1931), S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), V . I. Smirnov (1974), P. P. Zabreyko,A. I. Koshelev, et al. (1975), A. J. Jerry (1985), F. G. Tricomi (1985), D. Porter and D. S. G. Stirling (1990), C. Corduneanu (1991), J. Kondo (1991), W. Hackbusch (1995), R. P. Kanwal (1996). 13.7. Integral Equations with Nonnegative Kernels 13.7-1. Positive Principal Eigenvalues. Generalized Jentzch Theorem. In this section, we consider nonnegative kernels K(x,t)≥0 that are either continuous or square- summable in the domain a≤x,t≤b. Such kernels, under minimal additional assumptions, admit nonnegative eigenfunctions. The associated eigenvalues µ(resp., characteristic values λ=1/µ) will be called positive principal eigenvalues (resp., positive principal characteristic values ). Each positive principal eigenvalue is obviously nonzero. Under fairly general conditions, a nonnegative eigenfunction is unique (up to a constant coefﬁcient) and the corresponding positive principal eigenvalue is an upper bound for the modulus of any other eigenvalue. THEOREM 1(GENERALIZED JENTZCH THEOREM ).If a continuous or polar kernel K(x,t)is positive, then its characteristic values λ0with the smallest modulus is positive and simple, and the corresponding eigenfunction y0(x)does not change sign on the interval a≤x≤b. Remark 1. The generalized Jentzch theorem holds for a s ymmetric, as well as a nonsymmetric, polar positive kernel. It is allowed that the kernel may vanish at isolated points (on a set of zero measure) of the domain a≤x,t≤b. THEOREM 2.Suppose a nonnegativekernel K(x,t)has at least one (real or complex) eigenvalue. Then it has a nonnegative eigenvalue µ0. Remark 2. Not every nonnegative kernel has a nonnegative eigenfunction. Example. Any nonnegative kernel K(x,t)≥0(a≤x,t≤b) satisfying the condition K(x,t)≡0f o rt≥xhas no eigenfunctions corresponding to nonzero eigenvalues. THEOREM 3.LetK(x,t)be a nonnegative kernel. Suppose that there is a function u0(x)which is positive on a set of nonzero measure and satisﬁes the inequality ⎝integraldisplayb aKn(x,t)u0(t)dt≥βu0(x)( a≤x≤b), where β>0andKn(x,t)is an iterated kernel of some order n. Then the kernel K(x,t)has at least one positive principal eigenvalue µ0. This eigenvalue satisﬁes the inequality µ0≥β1/n. THEOREM 4.All (real and complex) eigenvalues µof the nonnegative kernel K(x,t)satisfy the inequality |µ|<∆,w h e r e ∆is the largest positive principal eigenvalue. 13.7. I NTEGRAL EQUATIONS WITH NONNEGATIVE KERNELS 649 13.7-2. Positive Solutions of a Nonhomogeneous Integral Equation. Consider a nonhomogeneous integral equation with a parameter µ: µy(x)=⎝integraldisplayb aK(x,t)y(t)dt+f(x)( a≤x≤b), (1) where the kernel K(x,t)≥0 is either continuous or square-summable. The functions y(t)a n d f(x) are also assumed either continuous or square-summable. THEOREM 1.Letµ>∆,w h e r e ∆is the largest positive principal eigenvalue of the kernel K(x,t). Then, for any nonnegative function f(x), equation (1) has one and only one nonnegative solution y(x), which can be obtained by the method of successive approximations based on the formula µyn+1(x)=⎝integraldisplayb aK(x,t)yn(t)dt+f(x)( n=0 ,1 , ...)( 2 ) with any initial approximation y0(x). Fory0(x) = 0, the solution can be represented as the series y(x)=∞⎝summationdisplay n=0K(n)[f(x)] µn+1, K[f(x)] =⎝integraldisplayb aK(x,t)f(t)dt,K(n)[f(x)] = K[K(n–1)[f(x)]]. Under the assumptions of Theorem 1, the rate of convergence of the successive approximations to the solution of equation (1) is characterized by the inequality /bardbly–yn/bardbl≤C(µ)⎝parenleftbigg∆ µ⎝parenrightbiggn (n=1 ,2 , ...), where C(µ) is a constant. If the kernel K(x,t) and the function f(x) are continuous, then the norm is introduced by /bardbly/bardbl=m a x a≤x≤b|y(x)|.I fK(x,t)a n d f(x) are square-summable, then one takes the norm/bardbly/bardbl=⎝bracketleftBig⎝integraldisplayb ay2(x)dx⎝bracketrightBig1/2 . THEOREM 2.If equation (1) admits a positive solu tion for at least one positive function f0(x), thenµ>∆, and therefore, equation (1) has a nonnegative solution for any nonnegative function f(x). 13.7-3. Estimates for the Spectral Radius. 1◦. The greatest among the moduli of the eigenvalues of the kernel K(x,t) is called the spectral radius of the kernel orspectral radius of the integral operator K[y(x)] =⎝integraldisplayb aK(x,t)y(t)dt and is denoted ρ(K). The role of the spectral radius can be characterized, for instance, by the fact that the integral equation (1) with a continuous kernel and a continuous free term has a continuous solution that can be obtained by the method of successive approximations (2) if and only if |µ|>ρ(K). Theorem 4 of Subsection 13.7-1 implies that the spectral radius of the nonnegative kernel K(x,t)≥0 is either equal to zero or coincides with its largest positive principal eigenvalue. Therefore, estimates for the 650 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) spectral radius in the case of a nonnegative kernel coincide with estimates for the largest positive principal eigenvalue. Estimates for spectral radii of nonnegative kernel s can be used for studying kernels of alternating sign, since the spectral radius of a nonnegative kernel K(x,t) is an upper bound for the spectral radius of any kernel M(x,t)s u c ht h a t |M(x,t)|≤K(x,t)( a≤x≤b). In the following statements, it is assumed that all functions are either continuous or square- summable and K(x,t)≥0. 2◦. The simplest upper bounds for the spectral radius have the form ρ(K)≤max a≤x≤b⎝integraldisplayb aK(x,t)dt, ρ(K)≤⎝bracketleftbigg⎝integraldisplayb a⎝integraldisplayb aK2(x,t)dx dt⎝bracketrightbigg1/2 . More precise estimates are obtained in terms of iterations of the kernel: ρ(K)≤⎝bracketleftbigg max a≤x≤b⎝integraldisplayb aKn(x,t)dt⎝bracketrightbigg1/n , ρ(K)≤⎝bracketleftbigg⎝integraldisplayb a⎝integraldisplayb aK2 n(x,t)dx dt⎝bracketrightbigg1/(2n) . Let us give two more estimates. Suppose that for some β1> 0, the following inequality holds: ⎝integraldisplayb aK(x,t)⎝bracketleftbigg β1–⎝integraldisplayb aKn(t,τ)dτ⎝bracketrightbigg dt≥0. Thenρ(K)≤β1/n 1. Suppose that for some β2>0 ,w eh a v e ⎝integraldisplayb aKn(x,t)⎝bracketleftbigg β2–⎝integraldisplayb aK(t,τ)dτ⎝bracketrightbigg dt≥0. Thenρ(K)≤β2. THEOREM 1.Suppose that for some β1>0and some nonnegative function u1(x)taking positive values on a set of nonzero measure, the following inequality holds: ⎝integraldisplayb aK(x,t)u1(t)dt≥β1u1(x)( a≤x≤b). Thenρ(K)≥β1. THEOREM 2.Suppose that for some β2>0and some nonnegative function u2(x)taking zero values only on a set of zero measure (say, at ﬁnitely many points), the following inequality holds: ⎝integraldisplayb aKn(x,t)u2(t)dt≤β2u2(x)( a≤x≤b). 13.7. I NTEGRAL EQUATIONS WITH NONNEGATIVE KERNELS 651 Thenρ(K)≤β1/n 2. 3◦. Consider a continuous kernel K(x,t) deﬁned on the square a≤x,t≤b. Let us split the segment [a,b]i n t onparts: a=x0<x1<...<xn–1<xn=b. Setting mij=m a x xi–1≤x≤xi⎝integraldisplayxj xj–1|K(x,t)|dt (i,j=1 ,...,n), (3) let us construct the matrix S≡⎛ ⎜⎜⎝m11m12···m1n m21m22···m2n ............ m n1mn2···mnn⎞ ⎟⎟⎠.( 4) THEOREM 3.The spectral radius ρ(K)does not exceed the largest eigenvalue of the matrix S. The likewise is true if, instead of (3), the elements of the matrix (4) are deﬁned by mij=⎝parenleftbigg⎝integraldisplayxi xi–1⎝integraldisplayxj xj–1K2(x,t)dx dt⎝parenrightbigg1/2 .( 5) Example. Consider the kernel K(x,t) that coincides with the Green function G(x,t) for the equation of vibrations of a string with ﬁxed ends, K(x,t)=G(x,t)=⎝braceleftBigx(1 –t)i f 0 ≤x≤t≤1, t(1 –x)i f 0 ≤t≤x≤1. Let us construct the matrix (4), taking n=5 ,xi=1 5i(i=0 ,1 , ..., 5). The elements of this matrix are calculated as in (5). The largest eigenvalue of the matrix Sin this case is equal to 0.10216. This gives the estimate ρ(K)≤0.10216. The exact largest eigenvalue is 1 /π2≈0.10132. 13.7-4. Basic Deﬁnition and Theorems for Oscillating Kernels. 1◦. A continuous function of two variables K(x,t)(a≤x,t≤b)i sc a l l e da noscillation kernel,i f the following inequalities hold: (a)K(x,t)>0 , a<x<b,a<t<b; (b) det K(xi,tj)≥0,a<x1<x2<···<xn<b,a<t1<t2<···<tn<b; (c) det K(xi,xj)>0 , a<x1<x2<···<xn<b, where nis an arbitrary positive integer and the points xi,tjthat satisfy the above inequalities are otherwise selected arbitrarily. It can be shown that the product of two (or ﬁnitely many) oscillation kernels is an oscillating kernel. THEOREM 1.Consider an integral equation of the form y(x)=λ⎝integraldisplayb aK(x,t)σ(t)y(t)dt,( 6) where K(x,t)is an oscillation kernel and σ(t)>0is a continuous function. Then the following statements hold: 1. All characteristic values (6) are positive and simple; 0<λ0<λ1<···. 2. The eigenfunction y0(x)corresponding to λ0has no zeros on the interval a<x<b. 652 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 3. The eigenfunction yk(x)corresponding to λkhas precisely knodes ( yk(x)changes sign at each node) and has no other zeros. 4. For arbitrary integers kandm(0≤k≤m)a n dr e a l ck,ck+1,...,cm(m⎝summationtext i=kc2 i>0), the linear combination y(x)=m⎝summationdisplay i=kciyi(x) has at least knodes and at most mzeros. 5. The nodes of neighboring eigenfunctions alternate. In order to apply the theory of integral equations with oscillation kernels to the investigation of ordinary differential equations, one has to reduce the latter to the former with the help of the Green’s function, and it is necessary to answer the question whether the Green’s function represents an oscillation kernel. Next, we give some results that suggest an answer to this question. 2◦. Consider a differential operator L[y]=n⎝summationdisplay s=0γs(x)dsy dxs,n≥2, (7) on the interval a≤x≤bwith positive coefﬁcients γs(x)>0 and the homogeneous boundary conditions n–1⎝summationdisplay m=0αimy(m) x=0 f o r x=a (i=1 ,...,p), n–1⎝summationdisplay m=0βimy(m) x=0 f o r x=b (i=1 ,...,q),(8) where p+q=1 . THEOREM 2.Suppose that the system of boundary conditions (8) corresponds to the Green’s function G(x,t)of the differential operator (7) such that (–1)qG(x,t)is an oscillation kernel. Then the same property holds for the following simpler system of boundary conditions: y(a)=y/prime x(a)=···=y(p–1) x(a)=0 , y(b)=y/prime x(b)=···=y(q–1) x(b)=0 ,(9) where p+q=n,i . e . , (–1)qGp,q(x,t)is an oscillation kernel, where Gp,q(x,t)is the Green’s function of the operator (7) with the boundary conditions (9). THEOREM 3.The system of boundary conditions (9) ( 1≤p<n) of the operator (7) corresponds to an oscillation kernel (–1)qGp,q(x,t)if and only if the following two conditions hold: 1. The diﬀerential equation with the truncated system of boundary conditions L[y]=0 ; y(b)=y/prime x(b)=···=y(q–1) x(b)=0 haspsolutions y1=y1(x),...,yp=yp(x)such that y1>0 , W(y1,y2)>0 , ...,W(y1,...,yp)>0 fora<x<b, 13.7. I NTEGRAL EQUATIONS WITH NONNEGATIVE KERNELS 653 where W(y1,...,yk)is the Wronskian determinant W(y1,...,yk)=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley 1(x)··· yk(x) y/prime 1(x)··· y/prime k(x) ··· ··· ··· y(k–1) 1(x)···y(k–1) k(x)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle. 2. The diﬀerential equation with the truncated system of boundary conditions L[y]=0 ; y(a)=y /prime x(a)=···=y(p–1) x(a)=0 hasq=n–psolutions yp+1=y1(x),...,yn=yn(x)such that W(y1,...,yp,yp+1)>0 , ...,W(y1,...,yp,...,yn)>0 fora<x<b. THEOREM 4.Conditions 1 and 2 of Theorem 3, under the assumption that the Green’s function exists, are equivalent to the condition that the differential operator L[y]admits a representation of the form L[y]=µ0(x)d dxµ1(x)d dxµ2(x)...d dxµn(x)y, (10) where µk(x)are positive weight functions with kcontinuous derivatives on (a,b). If the differential operator L[y] admits the representatio n (10), then the equation L[y]=0h a sa particular solution y= const /µn(x). THEOREM 5( K REIN’S CRITERION ).The condition that for each p(1≤p<n), the differen- tial operator (7) with the boundary conditions (9) admits a Green’s function Gp,q(x,t)such that (–1)qGp,q(x,t)is an oscillation kernel, is equivalent to the condition that the operator L[y]on the in- terval (a,b)admits the representation (10) with strictly positive functions µk(x)having kcontinuous derivatives on (a,b). Remark. Suppose that the operator (10) with the boundary conditions (8) admits a Green’s function G(x,t) (it is assumed that µk(x) > 0 and have kcontinuous derivatives). Then the function (–1)qG(x,t) is an oscillation kernel. Example 1. Consider the second-order linear differential operator L[y]=f(x)y/prime/prime xx+g(x)y/prime x, (11) where f(x)>0a n d g(x)>0f o r x∈[a,b], with the homogeneous boundary conditions of the ﬁrst kind y(a)=0 , y(b)=0 . (12) The differential operator (11) can be represented as an iterated operator (10) with positive weights: L[y]=µ0(x)d dxµ1(x)d dxµ2(x)y, µ0(x)=f(x)e x p⎝bracketleftbigg⎝integraldisplayg(x) f(x)dx⎝bracketrightbigg ,µ1(x)=e x p⎝bracketleftbigg⎝integraldisplayg(x) f(x)dx⎝bracketrightbigg ,µ2(x)=1 . Boundary conditions (12) represent a special case of (9) for p=q= 1. It is not difﬁcult to show that the operator (11) with the conditions (12) has the following Green’s function [constructed with the help of formulas (12) from Subsection 18.3-3]: G(x,t)=–⎧ ⎪⎪⎨ ⎪⎪⎩Y(a,x)Y(t,b) f(t)Φ(t)Y(a,b)ifa≤x≤t, Y(a,t)Y(x,b) f(t)Φ(t)Y(a,b)ift≤x≤b, 654 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) where Φ(t)=e x p⎝bracketleftbigg –⎝integraldisplayt 0g(τ) f(τ)dτ⎝bracketrightbigg ,Y(a,b)=⎝integraldisplayb aΦ(z)dz. Krein’s criterion ensures that the function K(x,t)=–G(x,t) is an oscillation kernel. Example 2. Consider the third-order differential operator L[y]=y/prime/prime/prime xxx (13) with the boundary conditions y/prime x=0 f o r x=0 , y=y/prime/prime xx=0 f o r x=1 .(14) It is easy to check that the operator (13) with the boundary conditions (14) has the Green’s function G(x,t)=⎝braceleftbigg t–1 2(x2+t2)f o r 0 ≤x≤t≤1, t–xt for 0 ≤t≤x≤1.(15) Herep=1a n d q= 2. Therefore, it follows from Theorem 5 (see the remark) that G(x,t) is an oscillation kernel. Now, let us examine the eigenvalue problem for the third-order equation y/prime/prime/prime xxx =λσ(x)y,σ(x)>0 , with boundary conditions (14). With the help of the Green’s function (15), this problem is reduced to the integral equation y(x)=λ⎝integraldisplay1 0G(x,t)σ(t)y(t)dt. (16) SinceG(x,t) is an oscillation kernel and σ(x) > 0, the results of Theorem 1 can be applied to equation (16). 13.7-5. Stochastic Kernels. 1◦. A nonnegative continuous kernel K(x,t) in the domain a≤x,t≤bis called a stochastic kernel , if⎝integraldisplayb aK(x,t)dt≡1( a≤x≤b). Obviously, for any integral operator with a stochastic kernel K(x,t), y0(x)≡1( a≤x≤b) is an eigenfunction corresponding to the characteristic value λ0= 1. The other characteristic values λsatisfy the inequality |λ|≥1. Integral operators with stochastic kernels may have characteristic values λ≠1s u c ht h a t |λ|= 1. The corresponding eigenvalues µ=1/λare called permutators . 2◦. Properties of stochastic kernels: 1. All eigenvalues µof an integral operator with sto chastic kernel such that |µ|= 1 are integer roots of unity. 2. The set of all eigenfunctions of an integral operator with stochastic kernel corresponding to an eigenvalue µ=1/λ= 1 contains a basis that consists of nonnegative functions y1(x),...,ym(x) with the following properties: (a) for every yj(x)(j=1 ,...,m), there is at least one point at which this function is positive and all other functions of the basis are equal to zero; (b) for each x∈[a,b], there is at least one function of the basis that is positive at x. References for Section 13.7: M. G. Krein (1939), F. P. Gantmakher and M. G. Krein (1950), S. Karlin (1968), J. M. Karon (1969), P. P. Zabreyko, A. I. Koshelev, et al. (1975), D. D. Joseph (1976), R. P. Agarwal, D. O’Regan, and P. J. Y . Wong (1998). 13.8. O PERA TOR METHOD FOR SOLVING INTEGRAL EQUATIONS OF THE SECOND KIND 655 13.8. Operator Method for Solving Integral Equations of the Second Kind 13.8-1. Simplest Scheme. Consider a linear equation of the second kind of the special form y(x)–λL[y]=f(x), (1) where Lis a linear (integral) operator such that L2=k,k= const. Let us apply the operator Lto Eq. (1). We obtain L[y]–kλy(x)=L[f(x)]. (2) On eliminating the term L[y] from (1) and (2), we ﬁnd the solution y(x)=1 1–kλ2⎝braceleftbig f(x)+λL[f]⎝bracerightbig .( 3) Remark. In Section 11.4, various generalizations of the above method are described. 13.8-2. Solution of Equations of the Second Kind on the Semiaxis. 1◦. Consider the equation y(x)–λ⎝integraldisplay∞ 0cos(xt )y(t)dt=f(x). (4) In this case, the operator Lcoincides, up to a constant factor, with the Fourier cosine transform: L[y]=⎝integraldisplay∞ 0cos(xt )y(t)dt=⎝radicalbigg π 2Fc[y]( 5) and acts by the rule L2=k,w h e r e k=π 2(see Subsection 9.5-1). We obtain the solution by formula (3) taking into account Eq. (5): y(x)=2 2–πλ2⎝bracketleftbigg f(x)+λ⎝integraldisplay∞ 0cos(xt )f(t)dt⎝bracketrightbigg ,λ≠±⎝radicalbigg 2 π.( 6) 2◦. Consider the equation y(x)–λ⎝integraldisplay∞ 0tJν(xt)y(t)dt=f(x), (7) where Jν(x) is the Bessel function, Re ν> –1. Here the operator Lcoincides, up to a constant factor, with the Hankel transform: L[y]=⎝integraldisplay∞ 0tJν(xt)y(t)dt (8) and acts by the rule L2= 1 (see Subsection 9.6-1). We obtain the solution by formula (3), for k= 1, taking into account Eq. (8): y(x)=1 1–λ2⎝bracketleftbigg f(x)+λ⎝integraldisplay∞ 0tJν(xt)f(t)dt⎝bracketrightbigg ,λ≠±1. (9) Reference for Section 13.8: A. D. Polyanin and A. V . Manzhirov (1998). 656 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.9. Methods of Integral Transforms and Model Solutions 13.9-1. Equation with Difference Kernel on the Entire Axis. Consider an integral equation of convolution type of the second kind with one kernel y(x)+1 √ 2π⎝integraldisplay∞ –∞K(x–t)y(t)dt=f(x), – ∞<x<∞,( 1) where f(x)a n dK(x) are the known right-hand side and the kernel of the integral equation and y(x) is the unknown function. Let us apply the (alternative) Fourier transform to Eq. (1). In this case, taking into account the convolution theorem (see Subsection 9.4-4), we obtain Y(u)[1 +K(u)] =F(u). (2) Thus, on applying the Fourier transform we reduce the solution of the original integral equation (1) to the solution of the algebraic equation (2) for the transform of the unknown function. The solution of Eq. (2) has the form Y(u)=F(u) 1+K(u).( 3) Formula (3) gives the transform of the solution of the original integral equation in terms of the transforms of the known functions, namely, the kernel and the right-hand side of the equation. The solution itself can be obtained by appl ying the Fourier inversion formula: y(x)=1 √ 2π⎝integraldisplay∞ –∞Y(u)e–iuxdu=1 √ 2π⎝integraldisplay∞ –∞F(u) 1+K(u)e–iuxdu.( 4 ) In fact, formula (4) solves the problem; however, sometimes it is not convenient because it requires the calculation of the transform F(u) for each right-hand side f(x). In many cases, the representation of the solution of the nonhomogeneous integral equation via the resolvent of the original equation is more convenient. To obtain the desired representation, we note that formula (3) can be transformed to the expression Y(u)=[ 1– R(u)]F(u),R(u)=K(u) 1+K(u).( 5) On the basis of (5), by applying the Fourier in version formula and the convolution theorem (for transforms) we obtain y(x)=f(x)–1 √ 2π⎝integraldisplay∞ –∞R(x–t)f(t)dt,( 6 ) where the resolvent R(x–t) of the integral equation (1) is given by the relation R(x)=1 √ 2π⎝integraldisplay∞ –∞K(u) 1+K(u)e–iuxdu,( 7 ) Thus, to determine the solution of the original integral equation (1), it sufﬁces to ﬁnd the func- tionR(x)b yf o r m u l a( 7 ) . The function R(x) is a solution of Eq. (1) for a special form of the function f(x). Indeed, it follows from formulas (3) and (5) that for Y(u)=R(u) the function F(u) is equal to K(u). This means that, for f(x)≡K(x), the function y(x)≡R(x) is a solution of Eq. (1), i.e., the resolvent of Eq. (1) satisﬁes the integral equation R(x)+1 √ 2π⎝integraldisplay∞ –∞K(x–t)R(t)dt=K(x), – ∞<x<∞.( 8) Note that to calculate direct and inverse Fourier transforms, one can use the corresponding tables from Supplements 7 and 8 and the books by H. Bateman and A. Erd ´elyi (1954) and by V . A. Ditkin and A. P. Prudnikov (1965). 13.9. M ETHODS OF INTEGRAL TRANSFORMS AND MODEL SOLUTIONS 657 Example. Let us solve the integral equation y(x)–λ⎝integraldisplay∞ –∞exp⎝parenleftbig α|x–t|⎝parenrightbig y(t)dt=f(x), – ∞<x<∞, (9) which is a special case of Eq. (1) with kernel K(x–t) given by the expression K(x)=–√ 2πλ e–α|x|,α>0 . (10) Let us ﬁnd the function R(x). To this end, we calculate the integral K(u)=–⎝integraldisplay∞ –∞λe–α|x|eiuxdx=–2αλ u2+α2. (11) In this case, formula (5) implies R(u)=K(u) 1+K(u)=–2αλ u2+α2–2αλ, (12) and hence R(x)=1 √ 2π⎝integraldisplay∞ –∞R(u)e–iuxdu=–⎝radicalbigg 2 π⎝integraldisplay∞ –∞αλ u2+α2–2αλe–iuxdu. (13) Assume that λ<1 2α. In this case the integral (13) makes sense and can be calculated by means of the theory of residues on applying the Jordan lemma (see Subsections 9.1-4 and 9.1-5). After some algebraic manipulations, we obtain R(x)=–√ 2παλ √ α2–2αλexp⎝parenleftbig –|x|√ α2–2αλ⎝parenrightbig(14) and ﬁnally, in accordance with (6), we obtain y(x)=f(x)+αλ √ α2–2αλ⎝integraldisplay∞ –∞exp⎝parenleftbig –|x–t|√ α2–2αλ⎝parenrightbig f(t)dt,–∞<x<∞. (15) 13.9-2. Equation with the Kernel K(x,t)=t–1Q(x/t)o nt h eS e m i a x i s . Here we consider the following equation on the semiaxis: y(x)–⎝integraldisplay∞ 01 tQ⎝parenleftBigx t⎝parenrightBig y(t)dt=f(x). (16) To solve this equation we apply the Mellin transform which is deﬁned as follows (see also Sec- tion 9.3): ˆf(s)=M{f(x),s}≡⎝integraldisplay∞ 0f(x)xs–1dx, (17) where s=σ+iτis a complex variable ( σ1<σ<σ2)a n d ˆf(s) is the transform of the function f(x). In what follows, we brieﬂy denote the Mellin transform by M{f(x)}≡M{f(x),s}. For known ˆf(s), the original function can be found by means of the Mellin inversion formula f(x)=M–1{ˆf(s)}≡1 2πi⎝integraldisplayc+i∞ c–i∞ˆf(s)x–sds,σ1<c<σ2, (18) where the integration path is parallel to the imaginary axis of the complex plane sand the integral is understood in the sense of the Cauchy principal value. On applying the Mellin transform to Eq. (16) and taking into account the fact that the integral with such a kernel is transformed into the product by the rule (see Subsection 9.3-2) M⎝braceleftbigg⎝integraldisplay∞ 01 tQ⎝parenleftBigx t⎝parenrightBig y(t)dt⎝bracerightbigg =ˆQ(s)ˆy(s), 658 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) we obtain the following equation for the transform ˆ y(s): ˆy(s)–ˆQ(s)ˆy(s)=ˆf(s). The solution of this equation is given by the formula ˆy(s)=ˆf(s) 1–ˆQ(s). (19) On applying the Mellin inversion formula to Eq. (19) we obtain the solution of the original integral equation y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞ˆf(s) 1–ˆQ(s)x–sds. (20) This solution can also be represented via the resolvent in the form y(x)=f(x)+⎝integraldisplay∞ 01 tN⎝parenleftBigx t⎝parenrightBig f(t)dt, (21) where we have used the notation N(x)=M–1{ˆN(s)}, ˆN(s)=ˆQ(s) 1–ˆQ(s). (22) Under the application of this analytical method of solution, the following technical difﬁculties can occur: (a) in the calculation of the transform for a given kernel K(x) and (b) in the calculation of the solution for the known transform ˆ y(s). To ﬁnd the corresponding integrals, tables of direct and inverse Mellin transforms are applied (e.g., see Supplements 9 and 10). In many cases, the relationship between the Mellin transform and the Fourier and Laplace transforms is ﬁrst used: M{f(x),s}=F{f(ex),is}=L{f(ex), –s}+L{f(e–x),s}, (23) and then tables of direct and inverse Fourier transforms and Laplace transforms are applied (see Supplements 5–8). Remark 1. The equation y(x)–⎝integraldisplay∞ 0H⎝parenleftBigx t⎝parenrightBig xαt–α–1y(t)dt=f(x) (24) can be rewritten in the form of Eq. (16) under the notation K(z)=zαH(z). 13.9-3. Equation with the Kernel K(x,t)=tβQ(xt)o nt h eS e m i a x i s . Consider the following equation on the semiaxis: y(x)–⎝integraldisplay∞ 0tβQ(xt)y(t)dt=f(x). (25) To solve this equation, we apply the Mellin transform. On multiplying Eq. (25) by xs–1and integrating with respect to xfrom zero to inﬁnity, we obtain ⎝integraldisplay∞ 0y(x)xs–1dx–⎝integraldisplay∞ 0y(t)tβdt⎝integraldisplay∞ 0Q(xt)xs–1dx=⎝integraldisplay∞ 0f(x)xs–1dx. (26) 13.9. M ETHODS OF INTEGRAL TRANSFORMS AND MODEL SOLUTIONS 659 Let us make the change of variables z=xt. We ﬁnally obtain ˆy(s)–ˆQ(s)⎝integraldisplay∞ 0y(t)tβ–sdt=ˆf(s). (27) Taking into account the relation ⎝integraldisplay∞ 0y(t)tβ–sdt=ˆy(1 +β–s), we rewrite Eq. (27) in the form ˆy(s)–ˆQ(s)ˆy(1 +β–s)=ˆf(s). (28) On replacing sby 1 + β–sin Eq. (28), we obtain ˆy(1 +β–s)–ˆQ(1 +β–s)ˆy(s)=ˆf(1 +β–s). (29) Let us eliminate ˆ y(1 +β–s) and solve the resulting equation for ˆ y(s). We thus ﬁnd the transform of the solution: ˆy(s)=ˆf(s)+ˆQ(s)ˆf(1 +β–s) 1–ˆQ(s)ˆQ(1 +β–s). (30) On applying the Mellin inversion formula, we obtain the solution of the integral equation (25) in the form y(x)=1 2πi⎝integraldisplayc+i∞ c–i∞ˆf(s)+ˆQ(s)ˆf(1 +β–s) 1–ˆQ(s)ˆQ(1 +β–s)x–sds. (31) Remark 2. The equation y(x)–⎝integraldisplay∞ 0H(xt)xptqy(t)dt=f(x) can be rewritten in the form of Eq. (25) under the notation Q(z)=zpH(z), where β=q–p. 13.9-4. Method of Model Solutions for Equations on the Entire Axis. Let us illustrate the capability of a generalized modiﬁcation of the method of model solutions (see Subsection 11.6) by an example of the equation Ay(x)+⎝integraldisplay∞ –∞Q(x+t)eβty(t)dt=f(x), (32) where Q=Q(z)a n df(x) are arbitrary functions and Aandβare arbitrary constants satisfying some constraints. For clarity, instead of the original equation (32) we write L[y(x)] =f(x). (33) For a test solution, we take the exponential function y0=epx. (34) 660 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) On substituting (34) into the left-hand side of Eq. (33), after some algebraic manipulations we obtain L[epx]=Aepx+q(p)e–(p+β)x,w h e r e q(p)=⎝integraldisplay∞ –∞Q(z)e(p+β)zdz. (35) The right-hand side of (35) can be regarded as a functional equation for the kernel epxof the inverse Laplace transform. To solve it, we replace pby –p –βin Eq. (35). We ﬁnally obtain L[e–(p+β)x]=Ae–(p+β)x+q(–p–β)epx. (36) Let us multiply Eq. (35) by Aand Eq. (36) by – q(p) and add the resulting relations. This yields L[Aepx–q(p)e–(p+β)x]=[A2–q(p)q(–p–β)]epx. (37) On dividing Eq. (37) by the constant A2–q(p)q(–p–β), we obtain the original model solution Y(x,p)=Aepx–q(p)e–(p+β)x A2–q(p)q(–p–β), L[Y(x,p)] =epx. (38) Since here – ∞<x<∞, one must set p=iuand use the formulas from Subsection 11.6-3. Then the solution of Eq. (32) for an arbitrary function f(x) can be represented in the form y(x)=1 √ 2π⎝integraldisplay∞ –∞Y(x,iu)˜f(u)du, ˜f(u)=⎝integraldisplay∞ –∞f(x)e–iuxdx. (39) References for Section 13.9: M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), V . I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. D. Gakhov and Yu. I. Cherskii (1978), A. D. Polyanin and A. V . Manzhirov (1997, 1998). 13.10. Carleman Method for Integral Equations of Convolution Type of the Second Kind 13.10-1. Wiener–Hopf Equation of the Second Kind. Equations of convolution type of the second kind of the form* y(x)+1 √ 2π⎝integraldisplay∞ 0K(x–t)y(t)dt=f(x), 0 < x<∞,( 1) frequently occur in applications. Here the domain of the kernel K(x) is the entire real axis. Let us extend the equation domain to the negative semiaxis by introducing one-sided functions, y+(x)=⎝braceleftbigg y(x)f o r x>0 , 0f o r x<0 ,f+(x)=⎝braceleftbigg f(x)f o r x>0 , 0f o r x<0 ,y–(x)=0 f o r x>0 . Then we obtain an equation, y+(x)+1 √ 2π⎝integraldisplay∞ –∞K(x–t)y+(t)dt=y–(x)+f+(x), – ∞<x<∞,( 2 ) which coincides with (1) for x>0 . * Prior to reading this section looking through Sections 12.7 and 12.8 is recommended. 13.10. C ARLEMAN METHOD FOR INTEGRAL EQUATIONS OF CONVOLUTION TYPE OF THE SECOND KIND 661 The auxiliary function y–(x) is introduced to compensate for the left-hand side of Eq. (2) for x< 0. Note that y–(x) is unknown for x< 0 and is to be found in solving the problem. Let us pass to the Fourier integrals in Eq. (2) (see Subsections 9.4-3, 12.7-1, and 12.7-2). We obtain a Riemann problem in the form Y+(u)=Y–(u) 1+K(u)+F+(u) 1+K(u),– ∞<u<∞.( 3) 1◦. Assume that the normality condition is satisﬁed, i.e., 1+K(u)≠0, then we rewrite the Riemann problem in the usual form Y+(u)=D(u)Y–(u)+H(u), – ∞<u<∞,( 4) where D(u)=1 1+K(u),H(u)=F(u) 1+K(u).( 5) The Riemann problem (4) is equivalent to Eq. (1); in particular, these equations are simulta- neously solvable or unsolvable and have an equal number of arbitrary constants in their general solutions. If the indexνof the Riemann problem, which is given by the relation ν=I n d1 1+K(u)(6) (which is also sometimes called the index of the Wiener–Hopf equation of the second kind ), is positive, then the homogeneous equation (1) ( f(x)≡0) has exactly νlinearly independent solutions, and the nonhomogeneousequation is unconditionally solvable and its solution depends on νarbitrary complex constants. In the case ν≤0, the homogeneous equation has no nonzero solutions. For ν= 0, the nonhomo- geneous equation is unconditionally solvable, and the solution is unique. If the index νis negative, then the conditions ⎝integraldisplay∞ –∞F(u)du X+(u)[1 +K(u)](u+i)k=0 , k=1 ,2 , ...,–ν,( 7 ) are necessary and sufﬁcient for the solvability of the nonhomogeneous equation (see Subsec- tion 12.7-4). For all cases in which the solution of Eq. (1) exists, it can be found by the formula y(x)=y+(x)=1 √ 2π⎝integraldisplay∞ –∞Y+(u)e–iuxdu,x>0 , ( 8 ) whereY+(u) is the solution of the Riemann problem (4) and (5) that is constructed by the scheme of Subsection 12.7-4 (see Fig. 5). The last formula shows that the solution does not depend on Y–(u), i.e., is independent of the choice of the extension o f the equation to the negative semiaxis. 2◦. Now let us study the exceptional case of the integral equation (1) in which the normality condition for the Riemann problem (3) (see Subsections 12.7-6 and 12.7-7) is violated. In this case,the coefﬁcient D(u)=[ 1+ K(u)] –1has no zeros, and its order at inﬁnity is η= 0. The general solution to the boundary value problem (3) can be obtained by formulas (63) of Subsection 12.7-7 662 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) Introduction of one-sided functions Extension of the domain of the equation to the negative semiaxis Application of the inverse Fourier transformSolution of the Riemann problem (see Section 12.7 and Fig. 5) Figure 6. Scheme of solving the Wiener–Hopf integral equations. For β= 0, we have the equation of the ﬁrst kind, and for β= 1, we have the equation of the second kind. forαi= 0. The solution of the original integral equation (1) can be determined from the solution of the boundary value problem on applying formula (8). Figure 6 depicts a scheme of solving the Wiener–Hopf equations (see also Subsection 12.8-1). Example. Consider the equation y(x)+⎝integraldisplay∞ 0(a+b|x–t|)e–|x–t|y(t)dt=f(x), x>0 , (9) where the constants aandbare real, and b≠0. The kernel K(x–t) of Eq. (1) is given by the expression K(x)=√ 2π(a+b|x|)e–|x|. Let us ﬁnd the transform of the kernel, K(u)=⎝integraldisplay∞ –∞(a+b|x|)e–|x|+iuxdx=2u2(a–b)+a+b (u2+1 )2. Hence, 1+K(u)=P(u) (u2+1 )2,P(z)=z4+2 (a–b+1 )z2+2a+2b+1 . 13.10. C ARLEMAN METHOD FOR INTEGRAL EQUATIONS OF CONVOLUTION TYPE OF THE SECOND KIND 663 On the basis of the normality condition, we assume that the constants aandbare such that the polynomial P(z) has no real roots. Let α+iβbe a root of the biquadratic equation P(z) = 0 such that α>0a n d β> 0. Since the coefﬁcients of the equation are real, it is clear that ( α–iβ), (–α+iβ), and (– α–iβ) are the other three roots. Since the function 1 + K(u)i s real as well, it follows that it has zero index, and hence Eq. (9) is uniquely solvable. On factorizing, we obtain the relation 1 + K(u)=X–(u)/X+(u), where X+(u)=(u+i)2 (u+α+iβ)(u–α+iβ),X–(u)=(u–α–iβ)(u+α–iβ) (u–i)2. Applying this result, we represent the boundary condition (4), (5) in the form Y+(u) X+(u)–(u–i)2F+(u) (u–α–iβ)(u+α–iβ)=Y–(u) X–(u),–∞<u<∞. (10) It follows from the theorem on the analytic continuation and the generalized Liouville theorem (see Subsection 12.7-3) that both sides of the above relation are equal to C1 u–α–iβ+C2 u+α–iβ, where the constants C1andC2must be deﬁned. Hence, Y+(u)=X+(u)⎝parenleftbigg(u–i)2F+(u) (u–α–iβ)(u+α–iβ)+C1 u–α–iβ+C2 u+α–iβ⎝parenrightbigg . (11) For the poles ( α+iβ)a n d( – α+iβ) to be deleted, it is necessary and sufﬁcient that C1=–(α+iβ–i)2F+(α+iβ) 2α,C2=–(–α+iβ–i)2F+(–α+iβ) –2α. (12) Since the problem is more or less cumbersome, we pass from the transform (11) to the corresponding original function in two stages. We ﬁrst ﬁnd the inverse transform of the summand Y1(u)=X+(u)(u–i)2F+(u) (u–α–iβ)(u+α–iβ)=1 1+K(u)F+(u)=F+(u)+R(u)F+(u). Here R(u)=–2u2(a–b)+2a+2b [u2–(α+iβ)2][u2–(α–iβ)2]=µ u2–(α+iβ)2+¯µ u2–(α–iβ)2,µ=i(α+iβ)2(a–b)+a+b 2αβ. Let us ﬁnd the inverse transform of the ﬁrst fraction: F–1⎝braceleftbiggµ u2–(α+iβ)2⎝bracerightbigg =⎝radicalbigg π 2µ β–iαe–(β–iα)|x|. The inverse transform of the second fraction can be found in the form F–1⎝braceleftbigg¯µ u2–(α–iβ)2⎝bracerightbigg =⎝radicalbigg π 2¯µ β+iαe–(β+iα)|x|. (13) Thus, R(x)=⎝radicalbigg π 2ρ⎝parenleftbig eiθ+iα|x|+e–iθ–iα|x|⎝parenrightbig e–β|x|=√ 2πρ e–β|x|cos(θ+α|x|) and y1(x)=f(x)+ρ⎝integraldisplay∞ 0e–β|x–t|cos(θ+α|x–t|)f(t)dt,x>0 , ρeiθ=µ β–iα. (14) Note that, as a by-product, we have found the resolvent R(x–t) of the following integral equation on the entire axis: y0(x)+⎝integraldisplay∞ –∞(a+b|x–t|)e–|x–t|y0(t)dt=f0(x), – ∞<x<∞. Now consider the remaining part of the transform (11): Y2(u)=X+(u)⎝parenleftbiggC1 u–α–iβ+C2 u+α–iβ⎝parenrightbigg . We can calculate the integrals F–1{Y2(u)}=C1 √ 2π⎝integraldisplay∞ –∞(u+i)2e–iuxdu (u+iβ–α)(u+iβ+α)(u–α–iβ)+C2 √ 2π⎝integraldisplay∞ –∞(u+i)2e–iuxdu (u+iβ–α)(u+iβ+α)(u+α–iβ) 664 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) by means of the residue theory (see Subsections 9.1-4 and 9.1-5) and substitute the values (12) into the constants C1andC2. Forx> 0, we obtain y2(x)=[α+(β–1 )2]2 4α2β⎝integraldisplay∞ 0e–β(x+t)cos[α(x–t)]f(t)dt +ρ∗ 4α2⎝integraldisplay∞ 0e–β(x+t)cos[ψ+α(x+t)]f(t)dt,ρ∗eiψ=(β–1–iα )4 8α2(β–iα).(15) SinceY+(u)=Y1(u)+Y2(u), it follows that the desired solution is the sum of the functions (14) and (15). 13.10-2. Integral Equation of the Second Kind with Two Kernels. Consider an integral equation of convolution type of the second kind with two kernels of the form y(x)+1 √ 2π⎝integraldisplay∞ 0K1(x–t)y(t)dt+1 √ 2π⎝integraldisplay0 –∞K2(x–t)y(t)dt=f(x), – ∞<x<∞. (16) Note that each of the kernels K1(x)a n dK2(x) is deﬁned on the entire real axis. On representing the desired function as the difference of one-sided functions, y(x)=y+(x)–y–(x), (17) we rewrite the equation in the form y+(x)+1 √ 2π⎝integraldisplay∞ –∞K1(x–t)y+(t)dt–y–(x)–1 √ 2π⎝integraldisplay∞ –∞K2(x–t)y–(t)dt=f(x). (18) Applying the Fourier integral transform (see Subsection 9.4-3), we obtain [1 +K1(u)]Y+(u)–[ 1+ K2(u)]Y–(u)=F(u). (19) This implies the relation Y+(u)=1+K2(u) 1+K1(u)Y–(u)+F(u) 1+K1(u). (20) HereK1(u),K2(u), andF(u) stand for the Fourier integrals of known functions. The unknown transforms Y+(u)a n dY–(u) are the boundary values of functions that are analytic on the upper and lower half-planes, respectively. Thus, w e have obtained a Riemann boundary value problem. 1◦. Assume that the normality conditions are satisﬁed, i.e., 1+K1(u)≠0, 1 + K2(u)≠0, then we can rewrite the Riemann problem in the usual form (see Subsection 12.7-4): Y+(u)=D(u)Y–(u)+H(u), – ∞<u<∞, (21) where D(u)=1+K2(u) 1+K1(u),H(u)=F(u) 1+K1(u). (22) The Riemann problem (21), (22) is equivalent to Eq. (16): these problems are solvable or unsolvable simultaneously,and have the same number of arbitrary constants in their general solutions. If the index ν=I n d1+K2(u) 1+K1(u)(23) 13.10. C ARLEMAN METHOD FOR INTEGRAL EQUATIONS OF CONVOLUTION TYPE OF THE SECOND KIND 665 is positive, then the homogeneous equation (16) ( f(x)≡0) has precisely νlinearly independent solutions, and the nonhomogeneous equation is unconditionally solvable; moreover, the solution ofthis equation depends on νarbitrary complex constants. In the case ν≤0, the homogeneous equation has no nonzero solutions. The nonhomogeneous equation is unconditionally solvable for ν= 0, and the solution is unique. For the case in which the indexνis negative, the conditions ⎝integraldisplay ∞ –∞F(u)du X+(u)[1 +K1(u)](u+i)k=0 , k=1 ,2 , ...,–ν, (24) are necessary and sufﬁcient for the solvability of the nonhomogeneous equation. In all cases for which the solution of Eq. (16) exists, this solution can be found by the formula y(x)=1 √ 2π⎝integraldisplay∞ –∞[Y+(u)–Y–(u)]e–iuxdu,– ∞<x<∞, (25) whereY+(u),Y–(u) is the solution of the Riemann problem (21), (22) constructed with respect to the scheme of Subsection 12.7-4 (see Fig. 5). Thus, the solution of Eq. (16) is equivalent to the solution of a Riemann boundary value problem and is reduced to the calculation of ﬁnitely many Fourier integrals. 2◦. Now let us study the exceptional case of an integral equation of the form (16). Assume that the functions 1 + K1(u)a n d1+ K2(u) can have zeros, and these zeros can be both different and coinciding points of the contour. Let us write out the expansion of these functions on selecting the coinciding zeros: 1+K1(u)=s⎝productdisplay j=1(u–bj)βjp⎝productdisplay k=1(u–dk)γkK11(u), 1+K2(u)=r⎝productdisplay i=1(u–ai)αip⎝productdisplay k=1(u–dk)γkK12(u),p⎝summationdisplay k=1γk=l.(26) Hereai≠bj, but it is possible that some points dk(k=1 ,...,p) coincide with either aiorbj.T h i s corresponds to the case in which the functions 1 + K1(u)a n d1+ K2(u) have a common zero of different multiplicity. We do not select these points especially because their presence does not affect the solvability conditions and the number of solutions of the problem. It follows from Eq. (19) and from the condition that a solution must be ﬁnite on the contour that, for the solvability of the problem, and all the more for the solvability of Eq. (16), it is necessary that the function F(u) have zero of order γkat any point dk, i.e.,F(u) must have the form F(u)=p⎝productdisplay k=1(u–dk)γkF1(u). To this end, the following γ1+···+γp=lconditions must be satisﬁed: F(jk) u(dk)=0 , jk=0 ,1 , ...,γk– 1, (27) or, which is the same, ⎝integraldisplay∞ –∞f(x)xjkeidkxdx= 0. (28) Since the functions K1(u)a n dK2(u) vanish at inﬁnity, it follows that the point at inﬁnity is a regular point of D(u). 666 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) Assume that conditions (28) are satisﬁed. In this case the Riemann boundary value problem (20) can be rewritten in the form (see Subsections 12.7-6 and 12.7-7) Y+(u)=r⎝producttext i=1(u–ai)αiR+(u)R–(u) s⎝producttext j=1(u–bj)βjQ+(u)Q–(u)D2(u)Y–(u)+H1(u) s⎝producttext j=1(u–bj)βj. (29) On ﬁnding its general solution in the exceptional case under consideration, we obtain the general solution of the original equation by means of formula (25). Let us state the conclusions on the solvability conditions and on the number of solutions of Eq. (16). For the solvability of Eq. (16), it is necessary that the Fourier transform of the right-hand side of the equation satisﬁes lconditions of the form (27). If these conditions are satisﬁed, then, forν–n> 0, problem (20) and the integral equation (16) have exactly ν–nlinearly independent solutions. For ν–n≤0, we must take the polynomial Pν–n–1(z) to be identically zero, and, for the case in which ν–n< 0, the right-hand side must satisfy another n–νconditions. If the latter conditions are satisﬁed, then the integral equation has a unique solution. Example. Consider Eq. (16) for which K1(x)=⎝braceleftbigg –(1 +α)√ 2πe–xforx>0 , 0f o r x<0 ,K2(x)=⎝braceleftbigg –(1 +β)√ 2πe–xforx>0 , 0f orx<0 ,f(x)=⎝braceleftbigg0f o r x>0 , –√ 2πexforx<0 , where αandβare real constants. In this case, K1(x–t)=0f o r x<tandK2(x–t)=0f o r x<t. Hence, the equation under consideration has the form y(x)–( 1+ α)⎝integraldisplayx 0e–(x–t)y(t)dt–( 1+ β)⎝integraldisplay0 –∞e–(x–t)y(t)dt=0 , y(x)–( 1+ β)⎝integraldisplayx –∞e–(x–t)y(t)dt=–√ 2πex,x>0 , x<0 . Let us calculate the Fourier integrals K1(u)=– ( 1+ α)⎝integraldisplay∞ 0e–xeiuxdx=–i(1 +α) u+i,K2(u)=–i(1 +β) u+i,F(u)=i u–i,D(u)=u–iβ u–iα. The boundary condition can be rewritten in the form Y+(u)=u–iβ u–iαY–(u)+i(u+i) (u–i)(u–iα). (30) The solution of the Riemann problem depends on the signs of αandβ. 1◦.L e t α>0a n d β> 0. In this case we have ν=I n dD(u) = 0. The left-hand side and the right-hand side of the boundary condition contain functions that have analytic continuations to the upper and the lower half-plane, respectively. On applying the theorem on the analytic continuation directly and the gener alized Liouville theorem (Subsection 12.7-3), we see that Y+(z)=0 ,z–iβ z–iαY–(z)+i(z+i) (z–i)(z–iα)=0 . Hence, y+(x)=0 , y(x)=–y–(x)=1 √ 2π⎝integraldisplay∞ –∞i(u+i) (u–i)(u–iβ)e–iuxdu. On calculating the last integral, under the assumption that β≠1, by the Cauchy residue theorem (see Subsections 9.1-4 and 9.1-5) we obtain y(x)=⎧ ⎨ ⎩0f o r x>0 , –√ 2π 1–β[2ex–( 1+ β)eβx]f o r x<0 . In the case β=1 ,w eh a v e y(x)=⎝braceleftBig0f orx>0 , –√ 2πex(1 + 2x)f o r x<0 . 13.10. C ARLEMAN METHOD FOR INTEGRAL EQUATIONS OF CONVOLUTION TYPE OF THE SECOND KIND 667 2◦.L e t α<0a n d β<0 . H e r ew ea g a i nh a v e ν=0 ,X+(z)=(z–iβ)(z–iα)–1,a n dX–(z) = 1. On grouping the terms containing the boundary values of functions that are analytic in each of the half-planes and then applying the analyticcontinuation theorem and the generalized Liouville theorem (Subsection 12.7-3), we see that Y +(z) X+(z)+β+1 i(β–1 )1 z–iβ=Y–(z) X–(z)+2 i(β–1 )1 z–i=0 . Hence, Y+(z)=β+1 β–1i z–iα,Y–(z)=2i β–11 z–i, y(x)=1 √ 2π⎝integraldisplay∞ –∞⎝bracketleftbigY+(u)–Y–(u)⎝bracketrightbige–iuxdu=⎧ ⎪⎪⎨ ⎪⎪⎩√ 2πβ+1 β–1eαxforx>0 , 2√ 2π β–1exforx<0 . 3◦.L e t α<0a n d β> 0. In this case we have ν= 1. Let us rewrite the boundary condition (30) in the form Y+(u)+i(1 +α) 1–α1 u–iα=u–iβ u–iαY–(u)–2i 1–α1 u–i. On applying the analytic continuation theorem and the generalized Liouville theorem (Subsection 12.7-3), we see that Y+(z)+i(1 +α) 1–α1 z–iα=z–iβ z–iαY–(z)–2i 1–α1 z–i=C z–iα. Therefore, Y+(z)=⎝parenleftbigg C–i1+α 1–α⎝parenrightbigg1 z–iα,Y–(z)=C z–iβ–2i 1–αz–iα (z–i)(z–iβ), where Cis an arbitrary constant. Now, by means of the Fourier inversion formula, we obtain the general solution of the integral equation in the form y(x)=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩–√ 2π⎝parenleftbigg iC+1+α 1–α⎝parenrightbigg eαxforx>0 , –√ 2π⎝bracketleftbigg iC+2(α–β) (1 –α)(1 –β)⎝bracketrightbigg eβx–2√ 2π 1–βexforx<0 . 4◦.L e t α>0a n d β< 0. In this case we have ν= –1. By the Liouville theorem (see Subsection 12.7-3), we obtain Y+(z)=z–iβ z–iαY–(z)+i(z+i) (z–i)(z–iα)=0 , and hence Y+(z)=0 , Y–(z)=–i(z+i) (z–i)(z–iβ). It can be seen from the expression for Y–(z) that the singularity of the function Y–(z) at the point iβdisappears if we set β= –1. The last condition is exactly the solvability condition of the Riemann problem. In this case we have the unique solution y(x)=1 √ 2π⎝integraldisplay∞ –∞i u–ie–iuxdu=⎝braceleftbigg0f o r x>0 , –√ 2πexforx<0 . Remark 1. Some equations whose kernels contain not the difference but certain other combina- tions of arguments, namely, the product or, more frequently, the ratio, can be reduced to equations considered in Subsection 13.10-2. For instance, the equation Y(ξ)+⎝integraldisplay1 01 τN1⎝parenleftbiggξ τ⎝parenrightbigg Y(τ)dτ+⎝integraldisplay∞ 11 τN2⎝parenleftbiggξ τ⎝parenrightbigg Y(τ)dτ=g(ξ), ξ> 0, (31) becomes a usual equation with two kernels after the following changes of the functions and their arguments: ξ=ex,τ=et,N1(ξ)=K1(x),N2(ξ)=K2(x),g(ξ)=f(x), and Y(ξ)=y(x). 668 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.10-3. Equations of Convolution Type with Variable Integration Limit. 1◦. Consider the V olterra integral equation of the second kind y(x)+1 √ 2π⎝integraldisplayx 0K(x–t)y(t)dt=f(x), 0 ≤x<T, (32) where the interval [0, T) can be either ﬁnite or inﬁnite. In contrast with Eq. (1), where the kernel is deﬁned on the entire real axis, here the kernel is deﬁned on the positive semiaxis. Equation (32) can be regarded as a special case of the one-sided equation (1) of Subsection 13.10- 1. To see this, we can rewrite Eq. (32) in the form y(x)+1 √ 2π⎝integraldisplay∞ 0K+(x–t)y(t)dt=f(x), 0 < x<∞, which can be reduced to the following boundary value problem: Y+(u)=Y–(u) 1+K+(u)+F+(u) 1+K+(u). Here the coefﬁcient [1 + K+(u)]–1of the problem is a function that has an analytic continuation to the upper half-plane, possibly except for ﬁnitely many poles that are zeros of the function 1 + K+(z) (we assume that 1 + K+(z)≠0 on the real axis). Therefore, the index νof the problem is always nonpositive, ν≤0. On rewriting the problem in the form [1 + K+(u)]Y+(u)=Y–(u)+F+(u), we see that Y–(u)≡0, which implies Y+(u)=F+(u) 1+K+(u). (33) Consider the following cases. 1.1. The function 1 + K+(z) has no zeros on the upper half-plane (this means that ν= 0). In this case, Eq. (32) has a unique solution for an arbitrary right-hand side f(x), and this solution can be expressed via the resolvent: y(x)=f(x)+1 √ 2π⎝integraldisplayx 0R(x–t)f(t)dt,x> 0, (34) where R(x)=–1 √ 2π⎝integraldisplay∞ –∞K+(u) 1+K+(u)e–iuxdu. 1.2. The function 1 + K+(z) has zeros at the points z=a1,...,amof the upper half-plane (in this case we have ν<0 ,a n d νis equal to the minus total order of the zeros). The following two possibilities can occur. (a) The function F+(z) vanishes at the points a1,...,am, and the orders of these zeros are not less than the orders of the corresponding zeros of the function 1 + K+(z). In this case, the function F+(z)[1 +K+(z)]–1has no poles again, and thus the equation has the unique solution (34). The assumption dkF+(aj)/dzk= 0 on the zeros of the function F+(z) is equivalent to the conditions ⎝integraldisplay∞ –∞f(t)e–iajttkdt=0 , k=0 ,...,ηj–1 , j=1 ,...,m, (35) where ηjis the multiplicity of the zero of the function 1 + K+(z) at the point aj. In this case, conditions (35) are imposed directly on the right-hand side of the equation. (b) The function F+(z) does not vanish at the points a1,...,am(or vanishes with less multi- plicity than 1 + K+(z)). In this case, the function F+(z)[1 +K+(z)]–1has poles, and therefore the function (33) does not belong to the class under consideration. Equation (32) has no solutions in the chosen class of functions. In this case, conditions (35) fail. The last result does not contradict the well-known fact that a V olterra equation always has a unique solution. Equation (32) belongs to the class of V olterra type equations, and therefore is also solvable in case (b), but in a broader space of functions with exponential growth. 13.10. C ARLEMAN METHOD FOR INTEGRAL EQUATIONS OF CONVOLUTION TYPE OF THE SECOND KIND 669 2◦. Another simple special case of Eq. (1) in Subsection 13.10-1 is the following equation with variable lower limit: y(x)+1 √ 2π⎝integraldisplay∞ xK(x–t)y(t)dt=f(x), 0 < x<∞. (36) This corresponds to the case in which the function K(x) in Eq. (1) is left one-sided: K(x)=K–(x). Under the assumption 1 + K–(u)≠0, the Riemann problem becomes Y+(u)=Y–(u) 1+K–(u)+F+(u) 1+K–(u). (37) 2.1. The function 1 + K–(z) has no zeros on the lower half-plane. This means that the inverse transform of the function Y–(u)[1+K–(u)]–1is left one-sided, and such a function does not inﬂuence the relation between the inverse transforms of (37) for x> 0. Thus, if we introduce the function R–(u)=–K–(u) 1+K–(u) (for convenience of the ﬁnal formula), then by applying the Fourier inversion formula to Eq. (37) and by setting x> 0 we obtain the unique solution to Eq. (36), y(x)=f(x)+1 √ 2π⎝integraldisplay∞ xR–(x–t)f(t)dt,x>0 . 2.2. The function 1 + K–(z) has zeros in the lower half-plane. Since this function is nonzero both on the entire real axis and at inﬁnity, it follows th at the number of zeros is ﬁnite. The Riemann problem (37) has a positive index which is just equal to the number of zeros in the lower half-plane(the zeros are counted according to their multiplicities): ν=I n d1 1+K–(u)=–I n d [ 1+ K–(u)] =η1+···+ηn>0 . Hereηkare the multiplicities of the zeros zkof the function 1 + K–(z),k=1 ,...,n. LetC1k z–zk+C2k (z–zk)2+···+Cηkk (z–zk)ηk be the principal part of the Laurent series expansion of the function Y–(z)[1 +K–(z)]–1in powers of (z–zk),k=1 ,...,n. In this case, Eq. (37) becomes Y+(u)=F+(u) 1+K–(u)+n⎝summationdisplay k=1ηk⎝summationdisplay j=1Cjk (z–zk)j+···, (38) where the dots denote a function whose inverse transform vanishes for x> 0. Under the passage to the inverse transforms in Eq. (38), for x> 0 we obtain y(x)=f(x)+1 √ 2π⎝integraldisplay∞ xR–(x–t)f(t)dt+n⎝summationdisplay k=1Pk(x)e–izkx,x> 0. (39) Here the Pk(x) are polynomials of degree ηk– 1. We can verify that the function (39) is a solution of Eq. (36) for arbitrary coefﬁcients of the polynomials. Since the number of linearlyindependent solutions of the homogeneous equation (36) is equal to the index, it follows that the above solution (39) is the general solution of the nonhomogeneous equation. 670 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.10-4. Dual Equation of Convolution Type of the Second Kind. Consider the dual integral equation of the second kind y(x)+1 √ 2π⎝integraldisplay∞ –∞K1(x–t)y(t)dt=f(x), 0 < x<∞, y(x)+1 √ 2π⎝integraldisplay∞ –∞K2(x–t)y(t)dt=f(x), – ∞<x<0 ,(40) in which the function y(x) is to be found. In order to apply the Fourier transform technique (see Subsections 11.4-3, 12.7-1, and 12.7-2), we extend the domain of both conditions in Eq. (40) by formally rewriting them for all real valuesofx. This can be achieved by introducing new unknown functions into the right-hand sides. These functions must be chosen so that the conditions give n on the semiaxis are not violated. Hence, the ﬁrst condition in (40) must be complemented by a summand that vanishes on the positive semiaxis and the second by a summand that vanishes on the negative semiaxis. Thus, the dual equation can be written in the form y(x)+1 √ 2π⎝integraldisplay∞ –∞K1(x–t)y(t)dt=f(x)+ξ–(x), y(x)+1 √ 2π⎝integraldisplay∞ –∞K2(x–t)y(t)dt=f(x)+ξ+(x),–∞<x<∞, (41) where the ξ±(x) are some right and left one-sided functions so far unknown. On applying the Fourier integral transform, we arrive at the relations [1 +K1(u)]Y(u)=F(u)+Ξ–(u), [1 + K2(u)]Y(u)=F(u)+Ξ+(u). (42) Here the three functions Y(u),Ξ+(u), andΞ–(u) are unknown. Now on the basis of (42) we can ﬁnd Y(u)=F(u)+Ξ–(u) 1+K1(u)=F(u)+Ξ+(u) 1+K2(u)(43) and eliminate the function Y(u) from relations (42) by applying formula (43). We obtain the Riemann boundary value problem in the form Ξ+(u)=1+K2(u) 1+K1(u)Ξ–(u)+K2(u)–K1(u) 1+K1(u)F(u), – ∞<u<∞. (44) 1◦. Assume that the normality conditions are satisﬁed, i.e., 1+K1(u)≠0, 1 + K2(u)≠0; then we can rewrite the Riemann problem (44) in the usual form (see Subsection 12.7-4) Ξ+(u)=D(u)Ξ–(u)+H(u), – ∞<u<∞, (45) where D(u)=1+K2(u) 1+K1(u),H(u)=K2(u)–K1(u) 1+K1(u)F(u). (46) The Riemann problem (45), (46) is equivalent to Eq. (40); in particular, they are solvable and unsolvable simultaneously and have the same number of arbitrary constants in the general solutions. 13.11. W IENER –HOPFMETHOD 671 If the index ν=I n d1+K2(u) 1+K1(u)(47) is positive, then the homogeneous equation (40) ( f(x)≡0) has exactly νlinearly independent solutions, and the nonhomogeneous equation is unconditionally solvable and the solution depends onνarbitrary complex constants. For the case ν≤0, the homogeneous equation has no nonzero solutions. For ν=0 ,t h e nonhomogeneous equation is unconditionally solvable, and a solution is unique. If the index νis negative, then the conditions ⎝integraldisplay∞ –∞K2(u)–K1(u) X+(u)[1 +K1(u)]F(u)du (u+i)k=0 , k=1 ,2 , ...,–ν (48) are necessary and sufﬁcient for the solvability of the nonhomogeneous equation. For all cases in which a solution of Eq. (40) exists, it can be found by the formula y(x)=1 √ 2π⎝integraldisplay∞ –∞F(u)+Ξ–(u) 1+K1(u)e–iuxdu=1 √ 2π⎝integraldisplay∞ –∞F(u)+Ξ+(u) 1+K2(u)e–iuxdu, (49) whereΞ+(u),Ξ–(u) is a solution of the Riemann problem (45), (46) that is constructed by the scheme of Subsection 12.7-4 (see Fig. 5). 2◦. Let us investigate the exceptional case of the i ntegral equation (40). Assume that the functions 1+K1(u)a n d1+ K2(u) can have zeros that can be either different or coinciding points of the contour. Take the expansions of these functions on selecting the coinciding zeros in the form of (26) and further repeat the reasoning performed for the equations of convolution type of the second kind with two kernels. After ﬁnding the general solution of the Riemann boundary value problem (44) in thisexceptional case (see Subsection 12.7-7), we obtain the general solution of the original equation (40) by formula (49). The conclusions on the solvability conditions and on the number of solutions of Eq. (40) are similar to those made above for the equations with two kernels in Subsection 13.10-2. Remark 2. Equations treated in Section 13.10 are sometimes called characteristic equations of convolution type . References for Section 13.10: F. D. Gakhov and Yu. I. Cherskii (1978), F. D. Gakhov (1990). 13.11. Wiener–Hopf Method 13.11-1. Some Remarks. Suppose that the Fourier transform of the function y(x) exists (see Subsection 9.4-3): Y(z)=1 √ 2π⎝integraldisplay∞ –∞y(x)eizxdx.( 1 ) Assume that the parameter zthat enters the transform (1) can take complex values as well. Let us study the properties of the function Y(z) regarded as a function of the complex variable z.T o t h i s end, we represent the function y(x) in the form* y(x)=y+(x)+y–(x), (2) * Do not confuse the functions y±(x)a n dY±(x) introduced in this section with the functions y±(x)a n dY±(x) introduced in Subsection 12.7-2 and used in solving the Riemann boundary value problem on the real axis. 672 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) where the functions y+(x)a n dy–(x) are given by the relations y+(x)=⎝braceleftbigg y(x)f o r x>0 , 0f o r x<0 ,y–(x)=⎝braceleftbigg0f o r x>0 , y(x)f o r x<0 .(3) In this case the transform Y(z) of the function y(x) is clearly equal to the sum of the transforms Y+(z) andY–(z) of the functions y+(x)a n dy–(x), respectively. Let us clarify the analytic properties of the function Y(z) by establishing the analy tic properties of the functions Y+(z)a n dY–(z). Consider the function y+(x) given by relations (3). Its transform is equal to Y+(z)=1 √ 2π⎝integraldisplay∞ 0y+(x)eizxdx.( 4 ) It can be shown that if the function y+(x) satisﬁes the condition |y+(x)|<Mev–xasx→∞ ,( 5 ) where Mis a constant, then the function Y+(z) given by formula (4) is an analytic function of the complex variable z=u+ivin the domain Im z>v–, and in this domain we have Y+(z)→0a s |z|→∞ . We can also show that the functions y+(x)a n dY+(z) are related as follows: y+(x)=1 √ 2π⎝integraldisplay∞+iv –∞+ivY+(z)e–izxdz,( 6) where the integration is performed over any line Im z=v>v–in the complex plane z,w h i c hi s parallel to the real axis. Forv–< 0 (i.e., for functions y(x) with exponential decay at inﬁnity), the real axis belongs to the domain in which the function Y+(z) is analytic, and we can integrate over the real axis in formula (6). However, if the only possible values of v–are positive (for instance, if the function y+(x) has nontrivial growth at inﬁnity, which does not exceed the exponential growth with linear exponent), then the analyticity domain of the function Y+(z) is strictly above the real axis of the complex plane z (and in this case, the integral (4) can be divergent on the real axis). Similarly, if the function y–(x) in relations (3) satisﬁes the condition |y–(x)|<Mev+xasx→–∞,( 7) then its transform, i.e., the function Y–(z)=1 √ 2π⎝integraldisplay0 –∞y–(x)eizxdx,( 8 ) is an analytic function of the complex variable zin the domain Im z<v+. The function y–(x) can be expressed via Y–(z) by means of the relation y–(x)=1 √ 2π⎝integraldisplay∞+iv –∞+ivY–(z)e–izxdz,I m z=v<v+.( 9) Forv+> 0, the analyticity domain of the function Y–(z) contains the real axis. It is clear that for v–<v+, the function Y(z) deﬁned by formula (1) is an analytic function of the complex variable zin the strip v–<I mz<v+. In this case, the functions y(x)a n dY(z) are related by the Fourier inversion formula y(x)=1 √ 2π⎝integraldisplay∞+iv –∞+ivY(z)e–izxdz, (10) where the integration is performed over an arbitrary line in the complex plane zbelonging to the stripv–0 ,t h ef u n c t i o n Y(z) is analytic in the strip containing the real axis of the complex plane z. Example 1. Forα> 0, the function K(x)=e–α|x|has the transform K(z)=1 √ 2π2α α2+z2, which is an analytic function of the complex variable zin the strip – α<I mz<α, which contains the real axis. 13.11. W IENER –HOPFMETHOD 673 13.11-2. Homogeneous Wiener–Hopf Equation of the Second Kind. Consider a homogeneous integral Wiener–Hopf equation of the second kind in the form y(x)=⎝integraldisplay∞ 0K(x–t)y(t)dt, (11) whose solution can obviously be determined up to an arbitrary constant factor only. Here the domain of the function K(x) is the entire real axis. This factor can be found from additional conditions of the problem, for instance, from normalization conditions. We assume that Eq. (11) deﬁnes a function y(x) for all values of the variable x, positive and negative. Let us introduce the functions y–(x)a n d y+(x) by formulas (3). Obviously, we have y(x)=y+(x)+y–(x), and Eq. (11) can be rewritten in the form y+(x)=⎝integraldisplay∞ 0K(x–t)y+(t)dt,x> 0 (12) y–(x)=⎝integraldisplay∞ 0K(x–t)y+(t)dt,x< 0. (13) That is, the function y+(x) can be determined by the solution of the integral equation (12) and the function y–(x) can be expressed via the functions y+(x)a n dK(x) by means of formulas (13). In this case, we have the relation y+(x)+y–(x)=⎝integraldisplay∞ –∞K(x–t)y+(t)dt, (14) which is equivalent to the original equation (11). Let the function K(x) satisfy the condition |K(x)|<Mev–xasx→∞ , |K(x)|<Mev+xasx→–∞,(15) where v–<0a n d v+> 0. In this case, the function K(z)=1 √ 2π⎝integraldisplay∞ –∞K(x)eizxdx (16) is analytic in the strip v–<I mz<v+. Let us seek the solution of E q. (11) satisf ying the condition |y+(x)|<M1eµxasx→∞ , (17) where µ<v+(such a solution exists). In this case we can readily verify that the integrals on the right-hand sides in (12) and (13) are convergent, and the function y–(x) satisﬁes the estimate |y–(x)|<M2ev+xasx→–∞. (18) It follows from conditions (17) and (18) that the transforms Y+(z)a n dY–(z) of the functions y+(x)a n d y–(x) are analytic functions of the complex variable zfor Im z>µand Im z<v+, respectively. Let us pass to the solution of the integral equation (11) or of Eq. (14), which is equivalent to (11). To this end, we apply the (alternative) Fourier transform. By the convolution theorem (see Subsection 9.4-4), it follows from (14) that Y+(z)+Y–(z)=√ 2πK(z)Y+(z), 674 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) or W(z)Y+(z)+Y–(z) = 0, (19) where W(z)=1–√ 2πK(z)≠0. (20) Thus, by means of the Fourier transform, we succeeded in the passage from the original integral equation to an algebraic equation for the transforms. However, in this case Eq. (19) involves two unknown functions. In general, a single algebraic equation cannot uniquely determine two unknown functions. The Wiener–Hopf method makes it possible to solve this problem for a certain class offunctions. This method is mainly related to the study of the analyticity domains of the functions that enter the equation and to a special representation of this equation. The main idea of the Wiener–Hopf method is as follows. Let Eq. (19) be representable in the form W +(z)Y+(z)=–W–(z)Y–(z), (21) where the left-hand side is analytic in the upper half-plane Im z>µand the right-hand side is analytic in the lower half-plane Im z<v+,w h e r e µ<v+, so that there exists a common analyticity strip of these functions: µ<I mz<v+. Since the analytic continuation is unique, it follows that there exists a unique entire function of the complex variable that coincides with the left-hand side of (21)in the upper half-plane and with the right-hand side of (21) in the lower half-plane, respectively. If, in addition, the functions that enter Eq. (21) have at most power-law growth with respect to z at inﬁnity, then it follows from the generalized Liouville theorem (see Subsection 12.7-3) that the entire function under consideration is a polynomial. In particular, for the case of a function that is bounded at inﬁnity we obtain W +(z)Y+(z)=–W–(z)Y–(z) = const . (22) These relations uniquely determine the functions Y+(z)a n dY–(z). Thus, let us apply the above scheme to the solution of Eq. (19). It follows from the above reasoning that the analyticity domains of the functions Y+(z),Y–(z), andW(z)=1–√ 2πK(z), respectively, are the upper half-plane Im z>µ, the lower half-plane Im z<v+, and the strip v–<I mz<v+. Therefore, this equation holds in the strip* µ<I mz<v+, which is the common analyticity domain for all functions that enter the equation. In order to transform Eq. (19) to the form (21), we assume that it is possible to decompose the function W(z) as follows: W(z)=W+(z) W–(z), (23) where the functions W+(z)a n dW–(z) are analytic for Im z>µand Im z<v+, respectively. Moreover, we assume that, in the corresponding analyticity domains, these functions grow at inﬁnity no faster thanzn,w h e r e nis a positive integer. A representation of an analytic function W(z) in the form (23) is often called a factorization ofW(z). Thus, as the result of factorization, the original equation is reduced to the form (21). It follows from the above reasoning that this equation determines an entire function of the complex variable z. SinceY±(z)→0a s |z|→∞ and the growth of the functions W±(z) does not exceed that of a power function zn, it follows that the entire function under consideration can be only a polynomial Pn–1(z)o fd e g r e ea tm o s t n–1 . If the growth of the functions W±(z) at inﬁnity is only linear with respect to the variable z,t h e n it follows from relations (22), by virtue of the Li ouville theorem (see Subsection 12.7-3), that the * To be deﬁnite, we set µ>v–. Otherwise, the common domain of analyticity is the strip v–<I mz<v+. 13.11. W IENER –HOPFMETHOD 675 corresponding entire function is a constant C. In this case we obtain the following relations for the unknown functions Y+(z)a n dY–(z): Y+(z)=C W+(z),Y–(z)=–C W–(z), (24) which deﬁne the transform of the solution up to a constant factor, which can be found at least from the normalizatio n conditions. In the general case, the expressions Y+(z)=Pn–1(z) W+(z),Y–(z)=–Pn–1(z) W–(z), (25) deﬁne the transform of the desired solution of the integral equation (11) up to indeterminate constants, which can be found from the additional conditions of the problem. The solution itself is deﬁned bymeans of the Fourier inversion formula (6), (9), and (10). Example 2. Consider the equation y(x)=λ⎝integraldisplay∞ 0e–|x–t|y(t)dt,0 < λ<∞, (26) whose kernel has the form K(x)=λe–|x|. Let us ﬁnd the transform of the function K(x): K(z)=λ √ 2π⎝integraldisplay∞ –∞K(x)eizxdx=⎝radicalbigg 2 πλ z2+1. (27) The function K(z) is analytic with respect to the complex variable zin the strip –1 < Im z< 1. Let us represent the expression W(z)=1–√ 2πK(z)=z2–2λ+1 z2+1(28) in the form (23), where W+(z)=z2–2λ+1 z+i,W–(z)=z–i. (29) The function W+(z) in Eq. (29) is analytic with respect to zand nonzero in the domain Im z>I m√ 2λ–1 .F o r0< λ<1 2, this domain is deﬁned by the condition Im z>√ 1–2λ,a n d√ 1–2λ≤µ<1 .F o r λ>1 2, the function W+(z) is analytic and nonzero in the domain Im z> 0. It is clear that the function W–(z) is a nonzero analytic function in the domain Im z<1 . Therefore, for 0 < λ<1 2both functions satisfy the required conditions in the domain µ1 2,t h es t r i p0<I m z< 1 is the common domain of analyticity of the functions W+(z)a n dW–(z). Thus, we have obtained the desired factorization of the function (28). Consider the expressions Y±(z)W±(z). Since Y±(z)→0a s |z|→∞ , and, according to (29), the growth of the functions W±(z) at inﬁnity is linear with respect to z, it follows that the entire function Pn–1(z) that coincides with Y+(z)W+(z)f o rI m z>µand with Y–(z)W–(z)f o rI m z< 1 can be a polynomial of zero degree only. Therefore, Y+(z)W+(z)=C. (30) Hence, Y+(z)=Cz+i z2–2λ+1, (31) and it follows from (6) that y+(x)=C √ 2π⎝integraldisplay∞+iv –∞+ivz+i z2–2λ+1e–izxdz, (32) where µ<v<1 . On closing the integration contour for x> 0 by a semicircle in the lower half-plane and estimating the integral over this semicircle by means of the Jordan lemma (see Subsections 9.1-4 and 9.1-5), after some calculations we obtain y+(x)=C⎝bracketleftbigg cos(√ 2λ–1x)+sin(√ 2λ–1x) √ 2λ–1⎝bracketrightbigg , (33) where Cis a constant. For 0 < λ<1 2, this solution has exponential growth with respect to x,a n df o r1 2<λ<∞,i ti s bounded at inﬁnity. 676 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.11-3. General Scheme of the Method. The Factorization Problem. In the general case, the problem which is solved by the Wiener–Hopf method can be reduced to the following problem. It is required to ﬁnd functions Y+(z)a n dY–(z) of the complex variable zthat are analytic in the half-planes Im z>v–and Im z<v+, respectively ( v–<v+), vanish as |z|→∞ in their analyticity domains, and satisfy the following functional equation in the strip ( v–<I mz<v+): A(z)Y+(z)+B(z)Y–(z)+C(z) = 0. (34) HereA(z),B(z), andC(z) are given functions of the complex variable zthat are analytic in the strip v–<I mz<v+, and the functions A(z)a n dB(z) are nonzero in this strip. The main idea of the solution of this problem is based on the possibility of a factorization of the expression A(z)/B(z), i.e., of a representation in the form A(z) B(z)=W+(z) W–(z), (35) where the functions W+(z)a n dW–(z) are analytic and nonzero in the half-planes Im z>v/prime –and Imz<v/prime +, and the strips v–<I mz<v+andv/prime –<I mz<v/prime +have a nonempty common part. In this case Eq. (34), with regard to Eq. (35), can be rewritten in the form W+(z)Y+(z)+W–(z)Y–(z)+W–(z)C(z) B(z)= 0. (36) If the last summand in Eq. (36) can be represented as the sum W–(z)C(z) B(z)=D+(z)+D–(z), (37) where the functions D+(z)a n dD–(z) are analytic in the half-planes Im z>v/prime/prime –and Im z<v/prime/prime +, respectively, and all three strips v–<I mz<v+,v/prime –<I mz<v/prime +,a n dv/prime/prime –<I mz<v/prime/prime +have a nonempty common part, for a strip v0 –<I mz<v0 +, then, in this common strip, the following functional equation holds: W+(z)Y+(z)+D+(z)=–W–(z)Y–(z)–D–(z). (38) The left-hand side of Eq. (38) is a function analytic in the half-plane v0 –<I mz, and the right-hand side is a function analytic in the domain Im z<v0 +. Since these functions coincide in the strip v0 –<I mz<v0 +, it follows that there exists a unique entire function that coincides with the left-hand side and the right-hand side of (38) in their analyticity domains, respectively. If the growth at inﬁnity of all functions that enter the right-hand sides of Eqs. (35) and (37), in their analyticity domains, is at most that of zn, then it follows from the limit relation Y±(z)→0a s |z|→∞ that this entire function is a polynomial Pn–1(z)o fd e g r e ea tm o s t n– 1. Thus, the relations Y+(z)=Pn–1(z)–D+(z) W+(z),Y–(z)=–Pn–1(z)–D–(z) W–(z)(39) determine the desired functions up to constants. These constants can be found from the additional conditions of the problem. The application of the Wiener–Hopf method is based on the representations (35) and (37). If a function G(z) is analytic in the strip v–<I mz<v+and if in this strip the function G(z) uniformly tends to zero as |z|→∞ , then in this strip the following representation is possible: G(z)=G+(z)+G–(z), (40) 13.11. W IENER –HOPFMETHOD 677 where the function G+(z) is analytic in the half-plane Im z>v–, the function G–(z) is analytic in the half-plane Im z<v+,a n d G+(z)=1 2πi⎝integraldisplay∞+iv/prime – –∞+iv/prime–G(τ) τ–zdτ, v–<v/prime –<I mz<v+, (41) G–(z)=–1 2πi⎝integraldisplay∞+iv/prime + –∞+iv/prime +G(τ) τ–zdτ,v–<I mz<v/prime +<v+. (42) The integrals (41) and (42), being regarded as integrals depending on a parameter, deﬁne analytic functions of the complex variable zunder the assumption that the point zdoes not belong to the integration contour. In particular, G+(z) is an analytic function in the half-plane Im z>v/prime –andG–(z) in the half-plane Imz>v/prime +. Moreover, if a function H(z) is analytic and nonzero in the strip v–<I mz<v+and ifH(z)→1 uniformly in this strip as |z|→∞ , then the following representation holds in the strip: H(z)=H+(z)H–(z), (43) H+(z)=e x p⎝bracketleftbigg1 2πi⎝integraldisplay∞+iv/prime – –∞+iv/prime–lnH(τ) τ–zdτ⎝bracketrightbigg , v–<v/prime –<I mz<v+, (44) H–(z)=e x p⎝bracketleftbigg –1 2πi⎝integraldisplay∞+iv/prime + –∞+iv/prime +lnH(τ) τ–zdτ⎝bracketrightbigg ,v–v–and Imz<v+, respectively. The representation (43) is called a factorization of the function H(z). 13.11-4. Nonhomogeneous Wiener–Hopf Equation of the Second Kind. Consider the Wiener–Hopf equation of the second kind y(x)–⎝integraldisplay∞ 0K(x–t)y(t)dt=f(x). (46) Suppose that the kernel K(x) of the equation and the right-hand side f(x) satisfy conditions (15). Let us seek the solution y+(x) to Eq. (46) for which condition (17) is satisﬁed. In this case, reasoning similar to that in the derivation of the functional equation (19) for a homogeneous integral equation shows that, in the case of Eq. (46), the following functional equation must hold on the strip µ<I mz<v+: Y+(z)+Y–(z)=√ 2πK(z)Y+(z)+F+(z)+F–(z), (47) or W(z)Y+(z)+Y–(z)–F(z) = 0, (48) whereW(z) is subjected to condition (20), as well as in the case of a homogeneous equation. We now note that Eq. (48) is a special case of Eq. (34). In the strip v–<I mz<v+,t h e function W(z) is analytic and uniformly tends to 1 as |z|→∞ because |K(z)|→0a s|z|→∞ .I n this case, this function has the representation (see (43)–(45)) W(z)=W+(z) W–(z), (49) 678 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) where the function W+(z) is analytic in the upper half-plane Im z>v–andW–(z) is analytic in the lower half-plane Im z<v+, and the growth at inﬁnity of the functions W±(z) does not exceed that ofzn. On the basis of the representation (49), Eq. (48) becomes W+(z)Y+(z)+W–(z)Y–(z)–W–(z)F–(z)–W–(z)F+(z) = 0. (50) To reduce Eq. (50) to the form (38), it sufﬁces to decompose the last summand F+(z)W–(z)=D+(z)+D–(z) (51) into the sum of functions D+(z)a n dD–(z) that are analytic in the half-planes Im z>µand Im z<v+, respectively. To establish the possibility of a representation (51), we note that the function F+(z) is analytic in the upper half-plane Im z>v–and uniformly tends to zero as |z|→∞ . The function W–(z)i s analytic in the lower half-plane Im z<v+, and, according to the method of its construction, we can perform the factorization (49) so that the function W–(z) remains bounded in the strip v–<I mz<v+ as|z|→∞ . Hence (see (40)–(42)), the functions F+(z)W–(z)i nt h es t r i p v–<I mz<v+satisfy all conditions that are sufﬁcient for th e validity of the representation (51). The above reasoning makes it possible to take into account the fact that the growth at inﬁnity of the functions W±(z) does not exceed that of zn, and thus to present the transform of the solution of the nonhomogeneous integral equation (46) in the form Y+(z)=Pn–1(z)+D+(z) W+(z),Y–(z)=–Pn–1(z)+W–(z)F–(z)+D–(z) W–(z). (52) The solution itself can be obtained from (52) by means of the Fourier inversion formula (6), (9), and (10). 13.11-5. Exceptional Case of a Wiener–Hopf Equation of the Second Kind. Consider the exceptional case of a Wiener–Hopf equation of the second kind in which the func-tionW(z)=1–√ 2πK(z) has ﬁnitely many zeros N(counted according to their multiplicities) in the strip v–<I mz<v+. In this case, the factorization is also possible. To this end, it sufﬁces to introduce the auxiliary function W1(z)=l n⎝bracketleftbigg (z2+b2)N/2W(z)⎝productdisplay i(z–zi)–αi⎝bracketrightbigg , (53) where αiis the multiplicity of the zero ziand a positive constant b>{|v–|,|v+|}is chosen so that the function in the square brackets has no additional zeros in the strip v–<I mz<v+. However, in the exceptional case, the Wiener–Hopf method gives the answer only if the number of zeros of the function W(z) is even. This restriction is due to the fact that only for the case in which the number of zeros is even is it possible to achieve the necessary behavior at inﬁnity (for the application of the Wiener–Hopf method) of the function ( z2+b2)N/2(see F. D. Gakhov and Yu. I. Cherskii (1978)). The last restriction makes no real obstacle to the broad use of the Wiener– Hopf method in solving applied problems in which the kernel K(x) of the corresponding integral equation is frequently an even function, and thus th e reasoning below can be applied completely. Remark 1. The Wiener–Hopf equation of the second kind for functions vanishing at inﬁnity can be reduced to a Riemann boundary value problem on the real axis (see Subsection 13.10-1).In this case, the assumption that the number of zeros of the function W(z) is even, as well as the assumption that the kernel K(x) is even in the exceptional case, are unessential. 13.12. K REIN’SMETHOD FOR WIENER –HOPFEQUATIONS 679 Remark 2. For functions with nontrivial growth at inﬁnity, the complete solution of Wiener– Hopf equations of the second kind is presented in the cited book by F. D. Gakhov and Yu. I. Cherskii(1978). Remark 3. The Wiener–Hopf method can be applied to solve Wiener–Hopf integral equations of the ﬁrst kind under the assumption that the kernels of these equations are even. References for Section 13.11: B. Noble (1958), A. G. Sveshnikov and A. N. Tikhonov (1970), V . I. Smirnov (1974), F. D. Gakhov (1977, 1990), F. D. Gakhov and Yu. I. Cherskii (1978). 13.12. Krein’s Method for Wiener–Hopf Equations 13.12-1. Some Remarks. The Factorization Problem. Consider the Wiener–Hopf equation of the second kind y(x)–⎝integraldisplay∞ 0K(x–t)y(t)dt=f(x), 0 ≤x<∞,( 1) where f(x),y(x)∈L1(0,∞)a n d K(x)∈L1(–∞,∞). Let us use the classes of functions that can be represented as Fourier transforms (alternative Fourier transform in the asymmetric form, see Subsection 9.4-3), of functions from L1(–∞,∞),L1(0,∞), and L1(–∞, 0). For brevity, instead of these symbols we simply write L,L+,a n dL–. Let functions h(x),h1(x), andh2(x) belong to L,L+, andL–, respectively; in this case, their transforms can be represented in the form ˇH(u)=⎝integraldisplay∞ –∞h(x)eiuxdx,ˇH1(u)=⎝integraldisplay∞ 0h1(x)eiuxdx,ˇH2(u)=⎝integraldisplay0 –∞h2(x)eiuxdx. LetQ,Q+,a n dQ–be the classes of functions representable in the form ˇW(u)=1+ ˇH(u), ˇW1(u)=1+ ˇH1(u), ˇW2(u)=1+ ˇH2(u), (2) respectively, where the functions from the classes Q+andQ–, treated as functions of the complex variable z=u+iv, are analytic for Im z>0a n dI m z< 0, respectively, and are continuous up to the real axis. LetT(x) belong to Land let ˇT(u) be its transform. Assume that 1–ˇT(u)≠0, Ind[1 – ˇT(u)] =1 2π⎝braceleftBig arg[1 – ˇT(u)]⎝bracerightBig∞ –∞=0 , – ∞<u<∞.( 3 ) In this case there exists a q(x)∈Lsuch that ln[1 – ˇT(u)] =⎝integraldisplay∞ –∞q(x)eiuxdx.( 4 ) This formula readily implies the relation ln[1 – ˇT(u)]→0a su→± ∞ . In what follows, we apply the factorization of functions ˇM(u) of the class Qthat are continuous on the interval – ∞≤u≤∞. Here the factorization means a representation of the function ˇM(u)i n the form of a product ˇM(u)= ˇM+(u)⎝parenleftbiggu–i u+i⎝parenrightbiggk ˇM–(u), (5) where ˇM–(z)a n d ˇM+(z) are analytic functions in the corresponding half-planes Im z>0a n d Imz< 0 continuous up to the real axis. Moreover, ˇM+(z)≠0f o r I m z≥0a n d ˇM–(z)≠0f o r I m z≤0. (6) 680 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) Relation (5) implies the formula k=I n d ˇM(u). The factorization (5) is said to be canonical provided that k=0 . In what follows we consider only functions of the form ˇM(u)=1– ˇT(u)( 7) such that ˇM(±∞) = 1. We can also assume that ˇM+(±∞)= ˇM–(±∞)=1 . ( 8 ) Let us state the main results concerning the factorization problem. A function (7) admits a canonical factorization if and only if the following two conditions hold: ˇM(u)≠0, Ind ˇM(u)=0 . ( 9 ) In this case, the canonical factorization is unique. Moreover, if conditions (9) hold, then there exists a function M(x) in the class Lsuch that ˇM(u)=e x p⎝bracketleftbigg⎝integraldisplay∞ –∞M(x)eiuxdx⎝bracketrightbigg , (10) ˇM+(u)=e x p⎝bracketleftbigg⎝integraldisplay∞ 0M(x)eiuxdx⎝bracketrightbigg , ˇM–(u)=e x p⎝bracketleftbigg⎝integraldisplay0 –∞M(x)eiuxdx⎝bracketrightbigg . (11) Hence, we have ˇM(u)∈Qand ˇM±(u)∈Q±. The factors in the canonical factorization are also described by the following formulas: lnˇM+(z)=1 2πi⎝integraldisplay∞ –∞lnˇM(τ) τ–zdτ,I m z> 0, (12) lnˇM–(z)=–1 2πi⎝integraldisplay∞ –∞lnˇM(τ) τ–zdτ,I m z< 0. (13) In the general case of the factorization, the following assertion holds. A function (7) admits a factorization (5) if and only if the following condition is satisﬁed: ˇM(u)≠0, –∞ <u<∞. In this case, relation (5) can be rewritten in the form ⎝parenleftbiggu–i u+i⎝parenrightbigg–k ˇM(u)= ˇM–(u)ˇM+(u), – ∞<u<∞. The last relation implies the canonical factorization for the function ˇM1(u)=⎝parenleftbiggu–i u+i⎝parenrightbigg–k ˇM(u). Hence, the factors ˇM±(u) satisfy formulas (10)–(13) if we replace ˇM(u) in these formulas byˇM1(u). Now we return to Eq. (1) for which ˇK(u)=⎝integraldisplay∞ –∞K(x)eiuxdx. (14) 13.12. K REIN’SMETHOD FOR WIENER –HOPFEQUATIONS 681 13.12-2. Solution of the Wiener–Hopf Equations of the Second Kind. THEOREM 1.For Eq. (1) to have a unique solution of the class L+for an arbitrary f(x)∈L+,i t is necessary and sufﬁcient that the following conditions hold: 1–ˇK(u)≠0, –∞ <u<∞, (15) ν=–I n d [ 1– ˇK(u)] = 0. (16) THEOREM 2.If condition (15) holds, then the inequality ν>0is necessary and sufﬁcient for the existence of nonzero solutions in the class L+of the homogeneous equation y(x)–⎝integraldisplay∞ 0K(x–t)y(t)dt= 0. (17) The set of these solutions has a basis formed by νfunctions ϕk(x)(k=1 ,...,ν) that tend to zero asx→∞ and that are related as follows: ϕk(x)=⎝integraldisplayx 0ϕk+1(t)dt,k=1 ,2 , ...,ν–1 , ϕν(x)=⎝integraldisplayx 0ψ(t)dt+C, (18) where Cis a nonzero constant and the functions ϕk(t)andψ(t)belong to L+. THEOREM 3.If condition (15) holds and if ν>0,t h e nf o ra n y f(x)∈L+Eq. (1) has inﬁnitely many solutions in L+. However, if ν< 0, then, for a given f(x)∈L+, Eq. (1) has either no solutions from L+or a unique solution. For the latter case to hold, it is necessary and sufﬁcient that the following conditions be satisﬁed: ⎝integraldisplay∞ 0f(x)ψk(x)dx=0 , k=1 ,2 , ...,|ν|, (19) where ψk(x) is a basis of the linear space of all solutions of the transposed homogeneous equation ψ(x)–⎝integraldisplay∞ 0K(t–x)ψ(t)dt= 0. (20) 1◦. If conditions (15) and (16) hold, then there exists a unique factorization [1 – ˇK(u)]–1=ˇM+(u)ˇM–(u), (21) and ˇM+(u)=1+⎝integraldisplay∞ 0R+(t)eiutdt, ˇM–(u)=1+⎝integraldisplay∞ 0R–(t)e–iutdt. (22) The resolvent is deﬁned by the formula R(x,t)=R+(x–t)+R–(t–x)+⎝integraldisplay∞ 0R+(x–s)R–(t–s)ds (23) where 0 ≤x<∞,0 ≤t<∞,R+(x)=0 ,a n d R–(x)=0f o r x<0 ,s ot h a t ,f o r f(x) from L+,t h e solution of the equation is determined by the expression y(x)=f(x)+⎝integraldisplay∞ 0R(x,t)f(t)dt. (24) 682 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) Formula (23) can be rewritten as follows: R(x,t)=R(x–t,0 )+ R(0,t–x)+⎝integraldisplay∞ 0R(x–s,0 )R(0,t–s)ds. (25) IfK(x–t)=K(t–x), then formula (25) becomes R(x,t)=R(|x–t|,0 )+⎝integraldisplaymin(x,t) 0R(x–s,0 )R(t–s,0 )ds. (26) Note that R+(x)=R(x,0 )a n d R–(x)=R(0,x) are unique solutions, in the class L+, of the following equations (0 ≤x<∞): R+(x)+⎝integraldisplay∞ 0K(x–t)R+(t)dt=K(x), R–(x)+⎝integraldisplay∞ 0K(t–x)R–(t)dt=K(–x).(27) 2◦. Suppose that condition (15) holds, but ν=–I n d [ 1– ˇK(u)] > 0. In this case, the function [1 – ˇK(u)]–1admits the factorization [1 – ˇK(u)]–1=ˇG–(u)⎝parenleftbiggu–i u+i⎝parenrightbiggν ˇG+(u), – ∞<u<∞. (28) For the functions ˇM–(u)a n d ˇM+(u) deﬁned by the relations ˇM–(u)=ˇG–(u)a n d ˇM+(u)=⎝parenleftbiggu–i u+i⎝parenrightbiggν ˇG+(u), (29) we have the representation (22) and formula (23) for the resolvent. Moreover, for k=1 ,...,ν, the following representations hold: ikˇM+(u) (u–i)k=⎝integraldisplay∞ 0gk(x)eiuxdx, (30) where gk(x) is the solution of the homogeneous equation (17). The solutions ϕk(x) mentioned in Theorem 2 can also naturally be expressed via the functions gk(x). 3◦.I fν=–I n d [ 1– ˇK(u)] < 0, then the transposed equation y(x)–⎝integraldisplay∞ 0K(t–x)y(t)dt=f(x) (31) has the index –ν > 0. If formula (28) deﬁnes a factorization for Eq. (1), then the transposed equation admits a factorization of the form [1 – ˇK(u)]–1=ˇM–(–u)ˇM+(–u), and ˇM–(–u) plays the role of ˇM+(u), and ˇM+(–u) plays the role of ˇM–(u). 13.13. M ETHODS FOR SOLVING EQUATIONS WITH DIFFERENCE KERNELS ON A FINITE INTERV AL 683 13.12-3. Hopf–Fock Formula. Let us give a useful formula that allows one to express the solution of Eq. (1) with an arbitrary right-hand side f(x) via the solution to a simpler auxiliary integral equation with an exponential right-hand side. Assume that in Eq. (1) we have f(x)=eiζx,I m ζ>0 , y(x)=yζ(x), (32) and moreover, conditions (15) and (16) hold. In this case yζ(x)=eiζx+⎝integraldisplay∞ 0R(x,t)eiζtdt, (33) where R(x,t) has the form (25). After some manipulations, we can see that yζ(x)= ˇM–(–ζ)⎝bracketleftbigg 1+⎝integraldisplayx 0R(t,0 )e–iζtdt⎝bracketrightbigg eiζx. (34) On setting x= 0 in (34), we have yζ(0) = ˇM–(–ζ), (35) and if the function K(x) describing the kernel of the integral equation is even, then yζ(0) = ˇM+(ζ). (36) On the basis of formula (34), we can obtain the solution of Eq. (1) for a general f(x)a sw e l l (see also Section 11.6): y(x)=1 2π⎝integraldisplay∞ –∞ˇF+(–ζ)yζ(x)dζ, ˇF+(u)=⎝integraldisplay∞ 0f(x)eiuxdx. (37) Remark 1. All results obtained in Section 13.12 concerning Wiener–Hopf equations of the sec- ond kind remain valid for continuous, square integrable, and some other classes of functions, which are discussed in detail in the paper by M. G. Krein (1958) and in the book by C. Corduneanu (1973). Remark 2. The solution of the Wiener–Hopf equation can be also obtained in other classes of functions for the exceptional case in which 1 – ˇK(u) = 0 (see Subsections 13.10-1 and 13.11-5). References for Section 13.12: V . A. Fock (1942), M. G. Krein (1958), C. Corduneanu (1973), V . I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975). 13.13. Methods for Solving Equations with Difference Kernels on a Finite Interval 13.13-1. Krein’s Method. Consider a method for constructing exact analy tic solutions of linear integral equations with an arbitrary right-hand side. The method is based on the construction of two auxiliary solutions ofsimpler equations with the right-hand side equal to 1. The auxiliary solutions are used to construct a solution of the original equation for an arbitrary right-hand side. 684 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 1◦. Let the equation y(x)–⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b,( 1 ) be given. Along with (1), we consider two auxiliary equations depending on a parameter ξ(a≤ξ≤b): w(x,ξ)–⎝integraldisplayξ aK(x,t)w(t,ξ)dt=1 , w∗(x,ξ)–⎝integraldisplayξ aK(t,x)w∗(t,ξ)dt=1 ,(2) where a≤x≤ξ. Assume that for any ξthe auxiliary equations (2) have unique continuous solutions w(x,ξ)a n dw∗(x,ξ), respectively, which satisfy the condition w(ξ,ξ)w∗(ξ,ξ)≠0(a≤ξ≤b). In this case, for any continuous function f(x), the unique continuous solution of Eq. (1) can be obtained by the formula y(x)=F(b)w(x,b)–⎝integraldisplayb xw(x,ξ)F/prime ξ(ξ)dξ,F(ξ)=1 m(ξ)d dξ⎝integraldisplayξ aw∗(t,ξ)f(t)dt,( 3 ) where m(ξ)=w(ξ,ξ)w∗(ξ,ξ). Formula (3) permits one to construct a solution of Eq. (1) with an arbitrary right-hand side f(x) by means of solutions to the two simpler auxiliary equations (2) (depending on the parameter ξ) with a constant right-hand side equal to 1. 2◦. Consider now an equation with the kernel depending on the difference of the arguments: y(x)+⎝integraldisplayb aK(x–t)y(t)dt=f(x), a≤x≤b.( 4 ) It is assumed that K(x) is an even function integrable on [ a–b,b–a]. Along with (4) we consider the following auxiliary equation depending on a parameter ξ(a≤ξ≤b): w(x,ξ)+⎝integraldisplayξ aK(x–t)w(t,ξ)dt=1 , a≤x≤ξ.( 5 ) Assume that for an arbitrary ξthe auxiliary equation (5) has a unique continuous solution w(x,ξ). In this case, for any continuous function f(x), a solution of Eq. (4) can be obtained from formula (3) by setting w∗(x,t)=w(x,t) in this formula. Now let us indicate another useful formula for equations whose kernel depends on the difference of the arguments: y(x)+⎝integraldisplaya –aK(x–t)y(t)dt=f(x), – a≤x≤a.( 6 ) It is assumed that K(x) is an even function that is integrable on the segment [–2 a,2a]. Along with (6) we consider an auxiliary equation depending on a parameter ξ(0 <ξ≤a): w(x,ξ)+⎝integraldisplayξ –ξK(x–t)w(t,ξ)dt=1 , – ξ≤x≤ξ.( 7 ) 13.13. M ETHODS FOR SOLVING EQUATIONS WITH DIFFERENCE KERNELS ON A FINITE INTERV AL 685 Let the auxiliary equation (7) have a unique continuous solution w(x,ξ)f o ra n y ξ. In this case, for an arbitrary continuous function f(x), the solution of Eq. (6) can be obtained by the following formula: y(x)=1 2M(a)⎝bracketleftbiggd da⎝integraldisplaya –aw(t,a)f(t)dt⎝bracketrightbigg w(x,a) –1 2⎝integraldisplaya |x|w(x,ξ)d dξ⎝bracketleftbigg1 M(ξ)d dξ⎝integraldisplayξ –ξw(t,ξ)f(t)dt⎝bracketrightbigg dξ –1 2d dx⎝integraldisplaya |x|w(x,ξ) M(ξ)⎝bracketleftbigg⎝integraldisplayξ –ξw(t,ξ)df(t)⎝bracketrightbigg dξ,( 8 ) where M(ξ)=w2(ξ,ξ), and the last inner integral is treated as a Stieltjes integral. 13.13-2. Kernels with Rational Fourier Transforms. Consider an equation of the form y(x)–⎝integraldisplayT 0K(x–t)y(t)dt=f(x), (9) where 0 ≤x≤T<∞.I ft h ek e r n e l K(x) is integrable on [– T,T], then the Fredholm theory can be applied to this equation. Since the equation involves the values of the kernel K(x) for the points of [– T,T] only, it follows that we can extend the kernel outside this interval in an arbitrary way. Assume that the kernel isextended to the entire axis so that the extended function is integrable. In the general case, Eq. (9) in the space L 2(0,T) can be reduced to a boundary value problem of the theory of analytic functions (Riemann problem) for two pairs of unknown functions. If the Fourier transform of the kernel ˇK(u)=⎝integraldisplay∞ –∞K(x)eiuxdx is rational, then Eq. (9) can be solved in the closed form. Assume that 1 – ˇK(u)≠0( –∞<u<∞). In this case, the transform of the solution of the integral equation (9) is given by the formula ˇY(u)=1 1–ˇK(u)⎝bracketleftbigˇF(u)–ˇW+(u)–e–iTuˇW–(u)⎝bracketrightbig (10) in which ˇW±(u)=⎝summationdisplay np± n⎝summationdisplay k=1M± nk (u–b±n)k, where the b+ nand the b– nare poles of the functions 1 – ˇK(u) that belong to the upper and lower half-planes, respectively, and the p± nare their multiplicities. The constants M± nkcan be determined from the conditions ds dus⎝bracketleftbigˇW+(u)+e–iTuˇW–(u)–ˇF(u)⎝bracketrightbig u=a+n=0 ; s=0 ,1 , ...,q+ n–1 ; ds dus⎝bracketleftbigˇW+(u)+e–iTu–ˇF(u)⎝bracketrightbig u=a–n=0 ; s=0 ,1 , ...,q– n–1 ; where a+ nanda– nare the zeros of the functions 1 – ˇK(u) that belong to the upper and lower half- planes, respectively, and q± nare their multiplicities. The constants M± nkcan also be determined by substituting the solution into the original equation. The solution of the integral equation (9) can be obtained by inverting formula (10). 686 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.13-3. Reduction to Ordinary Differential Equations. 1◦. Consider the special case in which the Fourier transform of the kernel of the integral equation (9) can be represented in the form ˇK(u)=ˇM(u) ˇN(u), (11) where ˇM(u)a n d ˇN(u) are some polynomials of degrees mandn, respectively: ˇM(u)=m⎝summationdisplay k=0Akuk,ˇN(u)=n⎝summationdisplay k=0Bkuk. (12) In this case, the solution of the integral equation (9) (if it exists) satisﬁes the following linear nonhomogeneous ordinary differential equation of the order mwith constant coefﬁcients: ˇM⎝parenleftBig id dx⎝parenrightBig y(x)=ˇN⎝parenleftBig id dx⎝parenrightBig f(x), 0 < x<T. (13) The solution of Eq. (13) contains marbitrary constants that are deﬁned by substituting the solution into the original equation (9). Here a system o f linear algebraic equations is obtained for these constants. 2◦. Consider the Fredholm equation of the second kind with a difference kernel that contains a sum of the exponential functions: y(x)+⎝integraldisplayb a⎝parenleftbiggn⎝summationdisplay k=1Akeλk|x–t|⎝parenrightbigg y(t)dt=f(x). (14) In the general case, this equation can be reduced to a linear nonhomogeneous ordinary differential equation of order 2 nwith constant coefﬁcients (see equation 4.2.16 in the ﬁrst part of the book). For the solution of Eq. (14) with n= 1, see equation 4.2.15 in the ﬁrst part of the book. 3◦. Equations with a difference kernel that contains a sum of hyperbolic functions, y(x)+⎝integraldisplayb aK(x–t)y(t)dt=f(x), K(x)=n⎝summationdisplay k=1Aksinh⎝parenleftbig λk|x|⎝parenrightbig , (15) can be also reduced by differentiation to linear nonhomogeneous ordinary differential equations of order 2 nwith constant coefﬁcients (see equation 4.3.29 in the ﬁrst part of the book). For the solution of Eq. (15) with n= 1, see equation 4.3.26 in the ﬁrst part of the book. 4◦. Equations with a difference kernel containing a sum of trigonometric functions y(x)+⎝integraldisplayb aK(x–t)y(t)dt=f(x), K(x)=n⎝summationdisplay k=1Aksin⎝parenleftbig λk|x|⎝parenrightbig , (16) can be also reduced to linear nonhomogeneous ordinary differential equations of order 2 nwith constant coefﬁcients (see equations 4.5.29 and 4.5.32 in the ﬁrst part of the book). References for Section 13.13: W. B. Davenport and W. L. Root (1958), I. C. Gohberg and M. G. Krein (1967), P. P. Zabreyko, A. I. Koshelev, et al. (1975), A. D. Polyanin and A. V . Manzhirov (1998). 13.14. M ETHOD OF APPROXIMA TING A KERNEL BY A DEGENERATE ONE 687 13.14. Method of Approximating a Kernel by a Degenerate One 13.14-1. Approximation of the Kernel. For the approximate solution of the Fredholm integral equation of the second kind y(x)–⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b,( 1 ) where, for simplicity, the functions f(x)a n dK(x,t) are assumed to be continuous, it is useful to replace the kernel K(x,t) by a close degenerate kernel K(n)(x,t)=n⎝summationdisplay k=0gk(x)hk(t). (2) Let us indicate several ways to perform such a change. If the kernel K(x,t) is differentiable with respect to xon [a,b] sufﬁciently many times, then, for a degenerate kernel K(n)(x,t), we can take a ﬁnite segment of the Taylor series: K(n)(x,t)=n⎝summationdisplay m=0(x–x0)m m!K(m) x(x0,t), (3) where x0∈[a,b]. A similar trick can be applied for the case in which K(x,t) is differentiable with respect to ton [a,b] sufﬁciently many times. To construct a degenerate kernel, a ﬁnite segment of the double Fourier series can be used: K(n)(x,t)=n⎝summationdisplay p=0n⎝summationdisplay q=0apq(x–x0)p(t–t0)q,( 4) where apq=1 p!q!∂p+q ∂xp∂tqK(x,t)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle x=x0t=t0,a≤x0≤b,a≤t0≤b. A continuous kernel K(x,t) admits an approximation by a trigonometric polynomial of period 2 l, where l=b–a. For instance, we can set K(n)(x,t)=1 2a0(t)+n⎝summationdisplay k=1ak(t)c o s⎝parenleftbiggkπx l⎝parenrightbigg ,( 5) where the ak(t)(k=0 ,1 ,2 ,... ) are the Fourier coefﬁcients ak(t)=2 l⎝integraldisplayb aK(x,t)c o s⎝parenleftbiggpπx l⎝parenrightbigg dx.( 6) A similar decomposition can be obtained by interchanging the roles of the variables xandt.A ﬁnite segment of the double Fourier series can also be applied by setting, for instance, ak(t)≈1 2ak0+n⎝summationdisplay m=1akmcos⎝parenleftbiggmπt l⎝parenrightbigg ,k=0 ,1 , ...,n,( 7) 688 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) and it follows from formulas (5)–(7) that K(n)(x,t)=1 4a00+1 2n⎝summationdisplay k=1ak0cos⎝parenleftbiggkπx l⎝parenrightbigg +1 2n⎝summationdisplay m=1a0mcos⎝parenleftbiggmπt l⎝parenrightbigg +n⎝summationdisplay k=1n⎝summationdisplay m=1akmcos⎝parenleftbiggkπx l⎝parenrightbigg cos⎝parenleftbiggmπt l⎝parenrightbigg , where akm=4 l2⎝integraldisplayb a⎝integraldisplayb aK(x,t)c o s⎝parenleftbiggkπx l⎝parenrightbigg cos⎝parenleftbiggmπt l⎝parenrightbigg dx dt .( 8) One can also use other methods of interpolating and approximating the kernel K(x,t). 13.14-2. Approximate Solution. IfK(n)(x,t) is an approximate degenerate kernel for a given exact kernel K(x,t) and if a func- tionfn(x)i sc l o s et of (x), then the solution yn(x) of the integral equation yn(x)–⎝integraldisplayb aK(n)(x,t)yn(t)dt=fn(x)( 9) can be regarded as an approximation to the solution y(x) of Eq. (1). Assume that the following error estimates hold: ⎝integraldisplayb a|K(x,t)–K(n)(x,t)|dt≤ε, |f(x)–fn(x)|≤δ. Next, let the resolvent Rn(x,t) of Eq. (9) satisfy the relation ⎝integraldisplayb a|Rn(x,t)|dt≤Mn fora≤x≤b. Finally, assume that the following inequality holds: q=ε(1 +Mn)<1 . In this case, Eq. (1) has a unique solution y(x)a n d |y(x)–yn(x)|≤εN(1 +Mn)2 1–q+δ,N=m a x a≤x≤b|f(x)|. (10) Example. Let us ﬁnd an approximate solution of the equation y(x)–⎝integraldisplay1/2 0e–x2t2y(t)dt=1 . (11) Applying the expansion in a double Taylor series, we replace the kernel K(x,t)=e–x2t2 by the degenerate kernel K(2)(x,t)=1– x2t2+1 2x4t4. 13.15. B ATEMAN METHOD 689 Hence, instead of Eq. (11) we obtain y2(x)=1+⎝integraldisplay1/2 0⎝parenleftbig1–x2t2+1 2x4t4⎝parenrightbigy2(t)dt. (12) Therefore, y2(x)=1+ A1+A2x2+A3x4, (13) where A1=⎝integraldisplay1/2 0y2(x)dx,A2=–⎝integraldisplay1/2 0x2y2(x)dx,A3=1 2⎝integraldisplay1/2 0x4y2(x)dx. (14) From (13) and (14) we obtain a system of three equations with three unknowns; to the fourth decimal place, the solution is A1= 0.9930, A2= –0.0833, A3= 0.0007. Hence, y(x)≈y2(x) = 1.9930 – 0.0833 x2+ 0.0007 x4,0 ≤x≤1 2. (15) An error estimate for the approximate solution (15) can be performed by formula (10). References for Section 13.14: L. V . Kantorovich and V . I. Krylov (1958), S. G. Mikhlin (1960), B. P. Demidovich, I. A. Maron, and E. Z. Shuvalova (1963), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), K. E. Atkinson (1997). 13.15. Bateman Method 13.15-1. General Scheme of the Method. In some cases it is useful, instead of replacing a given kernel by a degenerate kernel, to represent the given kernel approximately as the sum of a k ernel whose resolvent is known and a degenerate kernel. For the latter, the resolvent can be written out in a closed form. Consider the Fredholm integral equation of the second kind y(x)–λ⎝integraldisplayb ak(x,t)y(t)dt=f(x)( 1 ) with kernel k(x,t) whose resolvent r(x,t;λ) is known; thus, the solution of (1) can be represented in the form y(x)=f(x)+λ⎝integraldisplayb ar(x,t;λ)f(t)dt.( 2) Then, for the integral equation with kernel K(x,t)=1 ∆(aij)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglek(x,t)g 1(x)···gn(x) h1(t)a11··· a1n ............ h n(t)an1··· ann⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,∆(a ij)=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglea 11a12···a1n a21a22···a2n ............ a n1an2···ann⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,( 3 ) where g k(x)a n dhk(t)(k=1 ,...,n) are arbitrary functions and aij(i,j=1 ,...,n) are arbitrary numbers, the resolvent has the form R(x,t;λ)=1 ∆(aij+λbij)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingler(x,t;λ)ϕ 1(x)··· ϕn(x) ψ1(t) a11+λb11···a1n+λb1n ............ ψ n(t)an1+λbn1···ann+λbnn⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,( 4 ) where ϕ k(x)=gk(x)+λ⎝integraldisplayb ar(x,t;λ)gk(t)dt,ψk(x)=hk(x)+λ⎝integraldisplayb ar(x,t;λ)hk(t)dt, bij=⎝integraldisplayb agj(x)hi(x)dx,k,i,j=1 ,...,n.(5) 690 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.15-2. Some Special Cases. Assume that K(x,t)=k(x,t)–n⎝summationdisplay k=1gk(x)hk(t), (6) i.e., in formula (3) we have aij=0f o r i≠jandaii= 1. For this case, the resolvent is equal to R(x,t;λ)=1 ∆∗⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingler(x,t;λ)ϕ 1(x)···ϕn(x) ψ1(t)1 + λb11···λb1n ............ ψn(t)λbn1···1+λbnn⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,∆ ∗=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle1+λb 11λb12···λb1n λb21 1+λb22···λb2n ............ λbn1λbn2···1+λbnn⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle.( 7 ) Moreover, assume that k(x,t) = 0, i.e., the kernel K(x,t) is degenerate: K(x,t)=– n⎝summationdisplay k=1gk(x)hk(t). (8) In this case it is clear that r(x,t;λ) = 0 and, by virtue of (7), ϕk(x)=gk(x),ψk(x)=hk(x),bij=⎝integraldisplayb agj(x)hi(x)dx. Therefore, the resolvent becomes R(x,t;λ)=1 ∆∗⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle0 g 1(x)··· gn(x) h1(t)1 + λb11··· λb1n ............ h n(t)λbn1··· 1+λbnn⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle.( 9) Now we consider an integral equation with some kernel Q(x,t). On the interval (a ,b)w e arbitrarily choose points x 1,...,xnandt1,...,tn, and in relation (3) we set k(x,t)=0 , gk(x)=Q(x,tk),hk(t)=–Q(xk,t),aij=Q(xi,tj). In this case it is clear that r(x,t;λ) = 0, and the kernel K(x,t) acquires the form K(x,t)=1 D⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle0 Q(x,t 1)···Q(x,tn) Q(x1,t)Q(x1,t1)···Q(x1,tn) ............ Q(x n,t)Q(xn,t1)···Q(xn,tn)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,D=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleQ(x 1,t1)···Q(x1,tn) ......... Q(xn,t1)···Q(xn,tn)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle. It is convenient to rewrite this formula in the form K(x,t)=Q(x,t)–1 D⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleQ(x,t)Q(x,t 1)···Q(x,tn) Q(x1,t)Q(x1,t1)···Q(x1,tn) ............ Q(x n,t)Q(xn,t1)···Q(xn,tn)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle. (10) The kernel K(x,t) is degenerate and, moreover, it coincides with the kernel Q(x,t)o nt h e straight lines x=x i,t=tj(i,j=1 ,...,n). Indeed, if we set x=xiort=tj, then the determinant in the numerator of the second term has two equal rows or columns and hence vanishes, and therefore, K(xi,t)=Q(xi,t),K(x,tj)=Q(x,tj). 13.15. B ATEMAN METHOD 691 This coincidence on 2 nstraight lines permits us to expect that K(x,t) is close to Q(x,t)a n dt h e solution of the equation with kernel K(x,t) is close to the solution of the equation with kernel Q(x,t). It should be noted that if Q(x,t) is degenerate, i.e., has the form Q(x,t)=n⎝summationdisplay k=1gk(x)hk(t), (11) then the determinant in the numerat or is identically zero, and hence in this case we have K(x,t)≡Q(x,t). (12) For the kernel K(x,t), the resolvent can be evaluated on the basis of the following relations: r(x,t;λ)=0 , ϕi(x)=gi(x)=Q(x,ti),ψj(t)=hj(t)=–Q(xj,t), bij=–⎝integraldisplayb aQ(x,tj)Q(xi,x)dx=–Q2(xi,tj), i,j=1 ,...,n,(13) where Q2(x,t) is the second iterated kernel for Q(x,t): Q2(x,y)=⎝integraldisplayb aQ(x,s)Q(s,t)ds, and hence R(x,t;λ)=1 D–λD 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle0 Q(x,t 1) ··· Q(x,tn) Q(x1,t)Q(x1,t1)–λQ 2(x1,t1)···Q(x1,tn)–λQ 2(x1,tn) ............ Q(x n,t)Q(xn,t1)–λQ 2(xn,t1)···Q(xn,tn)–λQ 2(xn,tn)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle, (14) where D 2=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleQ 2(x1,t1)···Q2(x1,tn) ......... Q 2(xn,t1)···Q2(xn,tn)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle. By using the resolvent R(x,t;λ), we can obtain an approximate solution of the equation with kernel Q(x,t). In particular, approximate characteristic values ˜λof this kernel can be found by equating the determinant in the denominator of (14) with zero. Example. Consider the equation y(x)–λ⎝integraldisplay1 0Q(x,t)y(t)dt=0 , 0 ≤x≤1, (15) Q(x,t)=⎝braceleftBigx(t–1 ) f o r x≤t, t(x–1 ) f o r x≥t. Let us ﬁnd its characteristic values. To this end, we apply formula (14), where for the second iterated kernel we have Q2(x,t)=⎝integraldisplay1 0Q(x,s)Q(s,t)ds=⎝braceleftBigg1 6x(1 –t)(2t–x2–t2)f o r x≤t, 1 6t(1 –x)(2x–x2–t2)f o r x≥t. We choose equidistant points xiandtjand take n= 5. This implies x1=t1=1 6,x2=t2=2 6,x3=t3=3 6,x4=t4=4 6,x5=t5=5 6. Let us equate the determinant in the denominator of (14) with zero. After some algebraic manipulations, we obtain the following equation: 130µ5– 441µ4+ 488µ3– 206µ2+3 0µ–1=0 ( ˜λ= 216µ ), 692 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) which can be rewritten in the form (µ– 1)(2µ– 1)(5µ– 1)(13 µ2–2 2µ+1 )=0 . (16) On solving (16), we obtain ˜λ1= 10.02, ˜λ2= 43.2, ˜λ3= 108, ˜λ4= 216, ˜λ5= 355.2. The exact values of the characteristic values of the equation under consideration are known: λ1=π2= 9.869 ..., λ2=( 2π)2= 39.478 ...,λ3=( 3π)2= 88.826 ..., and hence the calculation error is 2% for the ﬁrst characteristic value, 9% for the second characteristic value, and 20% for the third characteristic value. The result can be improved by choosing another collection of points xiandti(i=1 ,..., 5). However, for this number of ordinates we cannot have very high precision, because the kernel Q(x,t) itself has a singularity, namely, its derivative is discontinuous for x=t, and thus the kernels under consideration cannot provide a good approximation of the given kernel. References for Section 13.15: H. Bateman (1922), E. Goursat (1923), L. V . Kantorovich and V . I. Krylov (1958), P. K. Kythe and P. Puri (2002). 13.16. Collocation Method 13.16-1. General Remarks. Let us rewrite the Fredholm integral equation of the second kind in the form ε[y(x)]≡y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt–f(x)=0 . ( 1 ) Let us seek an approximate solution of Eq. (1) in the special form Yn(x)=Φ(x,A1,...,An)( 2) with free parameters A1,...,An(undetermined coefﬁcients). On substituting the expression (2) into Eq. (1), we obtain the residual ε[Yn(x)] =Yn(x)–λ⎝integraldisplayb aK(x,t)Yn(t)dt–f(x). (3) Ify(x) is an exact solution, then, clearly, the residual ε[y(x)] is zero. Therefore, one tries to choose the parameters A1,...,Anso that, in a sense, the residual ε[Yn(x)] is as small as possible. The residual ε[Yn(x)] can be minimized in several ways. Usually, to simplify the calculations, a function Yn(x) linearly depending on the parameters A1,...,Anis taken. On ﬁnding the parameters A1,...,An, we obtain an approximate solution (2). If lim n→∞Yn(x)=y(x), (4) then, by taking a sufﬁciently large number of parameters A1,...,An, we ﬁnd that the solution y(x) can be found with an arbitrary prescribed precision. Now let us go to the description of a concrete method of construction of an approximate solution Yn(x). 13.16. C OLLOCATION METHOD 693 13.16-2. Approximate Solution. We set Yn(x)=ϕ0(x)+n⎝summationdisplay i=1Aiϕi(x), (5) where ϕ0(x),ϕ1(x),...,ϕn(x) are given functions ( coordinate functions )a n d A1,...,Anare indeterminate coefﬁcients,and assume that the functions ϕi(x)(i=1,...,n) are linearly independent. Note that, in particular, we can take ϕ0(x)=f(x)o rϕ0(x)≡0. On substituting the expression (5) into the left-hand side of Eq. (1), we obtain the residual ε[Yn(x)] =ϕ0(x)+n⎝summationdisplay i=1Aiϕi(x)–f(x)–λ⎝integraldisplayb aK(x,t)⎝bracketleftbigg ϕ0(t)+n⎝summationdisplay i=1Aiϕi(t)⎝bracketrightbigg dt, or ε[Yn(x)] =ψ0(x,λ)+n⎝summationdisplay i=1Aiψi(x,λ), (6) where ψ0(x,λ)=ϕ0(x)–f(x)–λ⎝integraldisplayb aK(x,t)ϕ0(t)dt, ψi(x,λ)=ϕi(x)–λ⎝integraldisplayb aK(x,t)ϕi(t)dt,i=1 ,...,n.(7) According to the collocation method, we require that the residual ε[Yn(x)] be zero at the given system of the collocation points x1,...,xnon the interval [ a,b], i.e., we set ε[Yn(xj)] = 0, j=1 ,...,n, where a≤x1<x2<···<xn–1<xn≤b. It is common practice to set x1=aandxn=b. This, together with formula (6), implies the linear algebraic system n⎝summationdisplay i=1Aiψi(xj,λ)=–ψ0(xj,λ), j=1 ,...,n,( 8 ) for the coefﬁcients A1,...,An. If the determinant of system (8) is nonzero, det[ψi(xj,λ)] =⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleψ 1(x1,λ)ψ1(x2,λ)···ψ1(xn,λ) ψ2(x1,λ)ψ2(x2,λ)···ψ2(xn,λ) ............ ψ n(x1,λ)ψn(x2,λ)···ψn(xn,λ)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle≠0, then system (8) uniquely determines the numbers A 1,...,An, and hence makes it possible to ﬁnd the approximate solution Yn(x)b yf o r m u l a( 5 ) . 694 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.16-3. Eigenfunctions of the Equation. On equating the determinant with zero, we obtain the relation det[ψi(xj,λ)] = 0, which, in general, enables us to ﬁnd approximate values ˜λk(k=1 ,...,n) for the characteristic values of the kernel K(x,t). If we set f(x)≡0,ϕ0(x)≡0,λ=˜λk, then, instead of system (8), we obtain the homogeneous system n⎝summationdisplay i=1˜A(k) iψi(xj,˜λk)=0 , j=1 ,...,n.( 9 ) On ﬁnding nonzero solutions ˜A(k) i(i=1 ,...,n) of system (9), we obtain approximate eigen- functions for the kernel K(x,t): ˜Y(k) n(x)=n⎝summationdisplay i=1˜A(k) iϕi(x), that correspond to its characteristic value λk≈˜λk. Example. Let us solve the equation y(x)–⎝integraldisplay1 0t2y(t) x2+t2dt=xarctan1 x(10) by the collocation method. We set Y2(x)=A1+A2x. On substituting this expression into Eq. (10), we obtain the residual ε[Y2(x)] = –A1xarctan1 x+A2⎝bracketleftbigg x–1 2+x2 2ln⎝parenleftbigg 1+1 x2⎝parenrightbigg⎝bracketrightbigg –xarctan1 x. On choosing the collocation points x1=0a n d x2= 1 and taking into account the relations lim x→0xarctan1 x= 0, lim x→0x2ln⎝parenleftbigg 1+1 x2⎝parenrightbigg =0 , we obtain the following system for the coefﬁcients A1andA2: 0×A1–1 2A2=0 , –π 4A1+1 2(1 + ln 2)A 2=π 4. This implies A2=0a n d A1= –1. Thus, Y2(x) = –1. (11) We can readily verify that the approximate solution (11) thus obtained is exact. References for Section 13.16: L. Collatz (1960), B. P. Demidovich, I. A. Maron, and E. Z. Shuvalova (1963), A. F. Verlan’ and V . S. Sizikov (1986), K. E. Atkinson (1997), R. Kress (1998, 1999), P. K. Kythe and P. Puri (2002), H. Brunner (2004). 13.17. M ETHOD OF LEAST SQUARES 695 13.17. Method of Least Squares 13.17-1. Description of the Method. By analogy with the collocation method, for the equation ε[y(x)]≡y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt–f(x)=0 ( 1 ) we set Yn(x)=ϕ0(x)+n⎝summationdisplay i=1Aiϕi(x), (2) where ϕ0(x),ϕ1(x),...,ϕn(x) are given functions, A1,...,Anare indeterminate coefﬁcients, and theϕi(x)(i=1 ,...,n) are linearly independent. On substituting (2) into the left-hand side of Eq. (1), we obtain the residual ε[Yn(x)] =ψ0(x,λ)+n⎝summationdisplay i=1Aiψi(x,λ), (3) where ψ0(x,λ)a n dt h e ψi(x,λ)(i=1 ,...,n) are deﬁned by formulas (7) of Subsection 13.16-2. According to the method of least squares, the coefﬁcients Ai(i=1 ,...,n) can be found from the condition for the minimum of the integral I=⎝integraldisplayb a{ε[Yn(x)]}2dx=⎝integraldisplayb a⎝bracketleftbigg ψ0(x,λ)+n⎝summationdisplay i=1Aiψi(x,λ)⎝bracketrightbigg2 dx.( 4 ) This requirement leads to the algebraic system of equations ∂I ∂Aj=0 , j=1 ,...,n,( 5 ) and hence, on the basis of (4), by differen tiating with respect to the parameters A1,...,Anunder the integral sign, we obtain 1 2∂I ∂Aj=⎝integraldisplayb aψj(x,λ)⎝bracketleftbigg ψ0(x,λ)+n⎝summationdisplay i=1Aiψi(x,λ)⎝bracketrightbigg dx=0 , j=1 ,...,n.( 6 ) Using the notation cij(λ)=⎝integraldisplayb aψi(x,λ)ψj(x,λ)dx,( 7) we can rewrite system (6) in the form of the normal system of the method of least squares : c11(λ)A1+c12(λ)A2+···+c1n(λ)An=–c10(λ), c21(λ)A1+c22(λ)A2+···+c2n(λ)An=–c20(λ), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ cn1(λ)A1+cn2(λ)A2+···+cnn(λ)An=–cn0(λ).(8) Note that if ϕ0(x)≡0, then ψ0(x)=–f(x). Moreover, since cij(λ)=cji(λ), the matrix of system (8) is symmetric. 696 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.17-2. Construction of Eigenfunctions. The method of least squares can also be applied f or the approximate construction of characteristic values and eigenfunctions of the kernel K(x,s), similarly to the way in which it can be done in the collocation method. Namely, by setting f(x)≡0a n d ϕ0(x)≡0, which implies ψ0(x)≡0, we determine approximate v alues of the characteristic values from the algebraic equation det[c ij(λ)] = 0. (9) After this, approximate eigenfunctions can be found from the homogeneous system of the form (8), where, instead of λ, the corresponding approximate value is substituted. Example. Let us ﬁnd an approximate solution of the equation y(x)=x2+⎝integraldisplay1 –1sinh(x+t)y(t)dt (10) by the method of least squares. For the form of an approximate solution we take Y2(x)=x2+A2x+A1. This implies ϕ1(x)=1 , ϕ2(x)=x,ϕ0(x)=x2. Taking into account the relations ⎝integraldisplay1 –1sinh(x+t)dt=asinhx,⎝integraldisplay1 –1tsinh(x+t)dt=bsinhx,⎝integraldisplay1 –1t2sinh(x+t)dt=csinhx, a= 2 sinh 1 = 2.3504, b=2e–1= 0.7358, c= 6 sinh 1 – 4 cosh 1 = 0.8788, on the basis of formulas (7) of Subsection 13.16-2 we have ψ1=1–asinhx,ψ2=x–bcoshx,ψ0=–csinhx. Furthermore, we see that (to the fourth decimal place) c11=2+a2⎝parenleftbig1 2sinh 2 – 1⎝parenrightbig= 6.4935, c22=2 3+b2⎝parenleftbig1 2sinh 2 + 1⎝parenrightbig= 2.1896, c12= –4(ae–1+bsinh 1) = –8e–1sinh 1 = –3.4586, c10=ac⎝parenleftbig1 2s i n h2–1⎝parenrightbig= 1.6800, c20=– 2ce–1= –0.6466, and obtain the following system for the coefﬁcients A1andA2: 6.4935A 1– 3.4586A 2= –1.6800, –3.4586A 1+ 2.1896A 2= 0.6466. Hence, we have A1= –0.5423 and A2= –0.5613. Thus, Y2(x)=x2– 0.5613x – 0.5423. (11) Since the kernel K(x,t)=s i n h ( x+t)=s i n h xcosht+c o s h xsinht of Eq. (10) is degenerate, we can readily obtain the exact solution y(x)=x2+αsinhx+βcoshx, (12) α=6s i n h1–4c o s h1 2–⎝parenleftbig1 2sinh 2⎝parenrightbig2= –0.6821, β=α⎝parenleftbig1 2s i n h2–1⎝parenrightbig= –0.5548. On comparing formulas (11) and (12) we conclude that the approximate solution Y2(x) is close to the exact solution y(x)i f |x|is small. At the endpoints x=±1, the discrepancy |y(x)–Y2(x)|is rather signiﬁcant. References for Section 13.17: L. V . Kantorovich and V . I. Krylov (1958), B. P. Demidovich, I. A. Maron, and E. Z. Shu- valova (1963), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), P. K. Kythe and P. Puri (2002). 13.18. B UBNOV –GALERKIN METHOD 697 13.18. Bubnov–Galerkin Method 13.18-1. Description of the Method. Let ε[y(x)]≡y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt–f(x)=0 . ( 1 ) Similarly to the above reasoning, we seek an approximate solution of Eq. (1) in the form of a ﬁnite sum Yn(x)=f(x)+n⎝summationdisplay i=1Aiϕi(x), i=1 ,...,n,( 2 ) where the ϕi(x)(i=1 ,...,n) are some given linearly independent functions ( coordinate functions ) andA1,...,Anare indeterminate coefﬁcients. On substituting the expression (2) into the left-hand side of Eq. (1), we obtain the residual ε[Yn(x)] =n⎝summationdisplay j=1Aj⎝bracketleftbigg ϕj(x)–λ⎝integraldisplayb aK(x,t)ϕj(t)dt⎝bracketrightbigg –λ⎝integraldisplayb aK(x,t)f(t)dt.( 3 ) According to the Bubnov–Galerkin method, the coefﬁcients Ai(i=1 ,...,n) are deﬁned from the condition that the residual is orthogonal to all coordinate functions ϕ1(x),...,ϕn(x). This gives the system of equations ⎝integraldisplayb aε[Yn(x)]ϕi(x)dx=0 , i=1 ,...,n, or, by virtue of (3), n⎝summationdisplay j=1(αij–λβij)Aj=λγi,i=1 ,...,n,( 4) where αij=⎝integraldisplayb aϕi(x)ϕj(x)dx,βij=⎝integraldisplayb a⎝integraldisplayb aK(x,t)ϕi(x)ϕj(t)dt dx ,γi=⎝integraldisplayb a⎝integraldisplayb aK(x,t)ϕi(x)f(t)dt dx . If the determinant of system (4) D(λ)=d e t [ αij–λβij] is nonzero, then this system uniquely determines the coefﬁcients A1,...,An. In this case, formula (2) gives an approximate solution of the integral equation (1). 13.18-2. Characteristic Values. The equation D(λ) = 0 gives approximate characteristic values ˜λ1,...,˜λnof the integral equation. On ﬁnding nonzero solutions of the homogeneous linear system n⎝summationdisplay j=1(αij–˜λkβij)˜A(k) j=0 , i=1 ,...,n, we can construct approximate eigenfunctions ˜Y(k) n(x) corresponding to characteristic values ˜λk: ˜Y(k) n(x)=n⎝summationdisplay i=1˜A(k) iϕ(x). It can be shown that the Bubnov–Galerkin method is equivalent to the replacement of the kernel K(x,t) by some degenerate kernel K(n)(x,t). Therefore, for the approximate solution Yn(x) we have an error estimate similar to that presented in Subsection 13.14-2. 698 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) Example. Let us ﬁnd the ﬁrst two characteristic values of the integral equation ε[y(x)]≡y(x)–λ⎝integraldisplay1 0K(x,t)y(t)dt=0 , where K(x,t)=⎝braceleftBigtfort≤x, xfort>x.(5) On the basis of (5), we have ε[y(x)] =y(x)–λ⎝braceleftbigg⎝integraldisplayx 0ty(t)dt+⎝integraldisplay1 xxy(t)dt⎝bracerightbigg . We set Y2(x)=A1x+A2x2. In this case ε[Y2(x)] =A1x+A2x2–λ⎝bracketleftbig1 3A1x3+1 4A2x4+x⎝parenleftbig1 2A1+1 3A2⎝parenrightbig–⎝parenleftbig1 2A1x3+1 3A2x4⎝parenrightbig⎝bracketrightbig= =A1⎝bracketleftbig⎝parenleftbig1–1 2λ⎝parenrightbigx+1 6λx3⎝bracketrightbig+A2⎝parenleftbig–1 3λx+x2+1 12λx4⎝parenrightbig. On orthogonalizing the residual ε[Y2(x)], we obtain the system ⎝integraldisplay1 0ε[Y2(x)]xd x =0 , ⎝integraldisplay1 0ε[Y2(x)]x2dx=0 , or the following homogeneous system of two algebraic equations with two unknowns: A1(120 – 48 λ)+A2( 9 0–3 5 λ)=0 A1(630 – 245 λ)+A2(504 – 180 λ)=0 .(6) On equating the determinant of system (6) with zero, we obtain the following equation for the characteristic values: D(λ)≡⎝vextendsingle⎝vextendsingle⎝vextendsingle120 – 48λ 9 0–3 5 λ 630 – 245λ 504 – 180λ⎝vextendsingle⎝vextendsingle⎝vextendsingle=0 . Hence, λ 2– 26.03λ + 58.15 = 0. (7) Equations (7) imply ˜λ1= 2.462 ... and ˜λ2= 23.568 ... For comparison we present the exact characteristic values: λ1=1 4π2= 2.467 ... andλ2=9 4π2= 22.206 ..., which can be obtained from the solution of the following boundary value problem equivalent to the original equation: y/prime/prime xx(x)+λy(x)=0 ; y(0) = 0, y/prime x(1) = 0. Thus, the error of ˜λ1is approximately equal to 0.2% and that of ˜λ2,t o6 % . References for Section 13.18: L. V . Kantorovich and V . I. Krylov (1958), B. P. Demidovich, I. A. Maron, and E. Z. Shu- valova (1963), A. F. Verlan’ and V . S. Sizikov (1986), K. E. Atkinson (1997), R. Kress (1999). 13.19. Quadrature Method 13.19-1. General Scheme for Fredholm Equations of the Second Kind. In the solution of an integral equation, the reduction to the solution of systems of algebraic equations obtained by replacing the integrals with ﬁnite sums is one of the most effective tools. The method of quadratures is related to the approximation methods. It is widespread in practice because it israther universal with respect to the principle of constructing algorithms for solving both linear and nonlinear equations. 13.19. Q UADRATURE METHOD 699 Just as in the case of V olterra equations , the method is based on a quadrature formula (see Subsection 10.7-1):⎝integraldisplayb aϕ(x)dx=n⎝summationdisplay j=1Ajϕ(xj)+εn[ϕ], (1) where the xjare the nodes of the quadrature formula, the Ajare given coefﬁcients that do not depend on the function ϕ(x), and εn[ϕ] is the error of replacement of the integral by the sum (the truncation error). If in the Fredholm integral equation of the second kind, y(x)–λ⎝integraldisplayb aK(x,t)y(t)dt=f(x), a≤x≤b,( 2 ) we set x=xi(i=1 ,...,n), then we obtain the following relation that is the basic formula for the method under consideration: y(xi)–λ⎝integraldisplayb aK(xi,t)y(t)dt=f(xi), i=1 ,...,n.( 3) Applying the quadrature formula (1) to the integral in (3), we arrive at the following system of equations: y(xi)–λn⎝summationdisplay j=1AjK(xi,xj)y(xj)=f(xi)+λεn[y]. (4) By neglecting the small term λεn[y] in this formula, we obtain t he system of linear algebraic equations for approximate values yiof the solution y(x) at the nodes x1,...,xn: yi–λn⎝summationdisplay j=1AjKijyj=fi,i=1 ,...,n,( 5) where Kij=K(xi,xj),fi=f(xi). The solution of system (5) gives the values y1,...,yn, which determine an approximate solution of the integral equation (2) on the entire interval [ a,b] by interpolation. Here for the approximate solution we can take the function obtained by linear interpolation, i.e., the function that coincides withyiat the points xiand is linear on each of the intervals [ xi,xi+1]. Moreover, for an analytic expression of the approximate solution to the equation, a function ˜y(x)=f(x)+λn⎝summationdisplay j=1AjK(x,xj)yj (6) can be chosen, which also takes the values y1,...,ynat the points x1,...,xn. 13.19-2. Construction of the Eigenfunctions. The method of quadratures can also be applied for solutions of homogeneous Fredholm equations of the second kind. In this case, system (5) becomes homogeneous ( fi= 0) and has a nontrivial solution only if its determinant D(λ) is equal to zero. The algebraic equation D(λ) = 0 of degree n forλmakes it possible to ﬁnd the roots ˜λ1,...,˜λn, which are approximate values of ncharacteristic values of the equation. The substitution of each value ˜λk(k=1 ,...,n) into (5) for fi≡0 leads to the system of equations y(k) i–˜λkn⎝summationdisplay j=1AjKijy(k) j=0 , i=1 ,...,n, 700 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) whose nonzero solutions y(k) imake it possible to obtain approximate expressions for the eigenfunc- tions of the integral equation: ˜yk(x)=˜λkn⎝summationdisplay j=1AjK(x,xj)y(k) j. Ifλdiffers from each of the roots ˜λk, then the nonhomogeneous system of linear algebraic equations (5) has a unique solution. In the same case, the homogeneous system of equations (5) hasonly the trivial solution. 13.19-3. Speciﬁc Features of the Application of Quadrature Formulas. The accuracy of the resulting solutions essentially depends on the smoothness of the kernel andthe constant term. When choosing the quadrature formula, it is necessary to take into account that the more accurate an applied formula is, the more serious requirements must be imposed on the smoothness of the kernel, the solution, and the right-hand side. If the right-hand side or the kernel have singularities,then it is reasonable to perform a preliminary transform of the original equation to obtain a more accurate approximate solution. Here the following methods can be applied. If the right-hand side f(x) has singularities and the kernel is smooth, then we can introduce the new unknown function z(x)=y(x)–f(x) instead of y(x), and the substitution of z(x) in the original equation leads to the equation z(x)–λ⎝integraldisplay b aK(x,t)z(t)dt=λ⎝integraldisplayb aK(x,t)f(t)dt, in which the right-hand side is smoothed, and hence a solution z(x) is smoother. From the func- tionz(x) thus obtained we can readily ﬁnd the desired solution y(x). For the cases in which the kernel K(x,t) or its derivatives with respect to thave discontinuities on the diagonal x=t, it is useful to rewrite the equation under consideration in the equivalent form y(x)⎝bracketleftbigg 1–λ⎝integraldisplayb aK(x,t)dt⎝bracketrightbigg –λ⎝integraldisplayb aK(x,t)[y(t)–y(x)]dt=f(x), where the integrand in the second integral has no singularities because the difference y(t)–y(x) vanishes on the diagonal x=t, and the calculation of the integral⎝integraldisplayb aK(x,t)dtis performed without unknown functions and is possible in the explicit form. Example. Consider the equation y(x)–1 2⎝integraldisplay1 0xty(t)dt=5 6x. Let us choose the nodes x1=0 ,x2=1 2,x3= 1 and calculate the values of the right-hand side f(x)=5 6xand of the kernel K(x,t)=xtat these nodes: f(0) = 0, f⎝parenleftbig1 2⎝parenrightbig =5 12,f(1) =5 6, K(0, 0) = 0, K⎝parenleftbig0,1 2⎝parenrightbig=0 , K(0, 1) = 0, K⎝parenleftbig1 2,0⎝parenrightbig=0 , K⎝parenleftbig1 2,1 2⎝parenrightbig=1 4, K⎝parenleftbig1 2,1⎝parenrightbig=1 2,K(1, 0) = 0, K⎝parenleftbig1,1 2⎝parenrightbig=1 2,K(1, 1) = 1. On applying Simpson’s rule (see Subsection 10.7-1) ⎝integraldisplay1 0F(x)dx≈1 6⎝bracketleftbigF(0) + 4F⎝parenleftbig1 2⎝parenrightbig+F(1)⎝bracketrightbig 13.20. S YSTEMS OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND 701 to determine the approximate values yi(i= 1, 2, 3) of the solution y(x) at the nodes xiwe obtain the system y1=0 , 11 12y2–1 24y3=5 12, –2 12y2+11 12y3=5 6, whose solution is y1=0 ,y2=1 2,y3= 1. In accordance with the expression (6), the approximate solution can be presented in the form ˜y(x)=5 6x+1 2×1 6⎝parenleftbig0+4 ×1 2×1 2x+1×1×x⎝parenrightbig=x. We can readily verify that it coincides with the exact solution. References for Section 13.19: N. S. Bakhvalov (1973), V . I. Krylov, V . V . Bobkov, and P. I. Monastyrnyi (1984), A. F. Verlan’ and V . S. Sizikov (1986). 13.20. Systems of Fredholm Integral Equations of the Second Kind 13.20-1. Some Remarks. A system of Fredholm integral equations of the second kind has the form yi(x)–λn⎝summationdisplay j=1⎝integraldisplayb aKij(x,t)yj(t)dt=fi(x), a≤x≤b,i=1 ,...,n.( 1 ) Assume that the kernels Kij(x,t) are continuous or square integrable on the square S={a≤x≤b, a≤t≤b}and the right-hand sides fi(x) are continuous or square integrable on [ a,b]. We also assume that the functions yi(x) to be deﬁned are continuous or square integrable on [ a,b] as well. The theory developed above for Fredholm equations of the second kind can be completely extended to such systems. In particular, it can be shown that for systems (1), the successive approximations converge in mean-square to the solution of the system if λsatisﬁes the inequality |λ|<1 B∗,( 2) where n⎝summationdisplay i=1n⎝summationdisplay j=1⎝integraldisplayb a⎝integraldisplayb a|Kij(x,t)|2dx dt =B2 ∗<∞.( 3) If the kernel Kij(x,t) satisﬁes the additional condition ⎝integraldisplayb aK2 ij(x,t)dt≤Aij,a≤x≤b,( 4 ) where Aijare some constants, then the successive approximations converge absolutely and uni- formly. If all kernels Kij(x,t) are degenerate, then system (1) can be reduced to a linear algebraic system. It can be established that for a system of Fredholm integral equations, all Fredholm theorems are satisﬁed. 702 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) 13.20-2. Method of Reducing a System of Equations to a Single Equation. System (1) can be transformed into a single Fredholm integral equation of the second kind. Indeed, let us introduce the functions Y(x)a n dF(x)o n[a,nb–(n–1 )a] by setting Y(x)=yi⎝parenleftbig x–(i–1 ) (b–a)⎝parenrightbig ,F(x)=fi⎝parenleftbig x–(i–1 ) (b–a)⎝parenrightbig , for (i–1 )b–(i–2 )a≤x≤ib–(i–1 )a. Let us deﬁne a kernel K(x,t) on the square {a≤x≤nb–(n–1 )a,a≤t≤nb–(n–1 )a}as follows: K(x,t)=Kij⎝parenleftbig x–(i–1 ) (b–a),t–(j–1 ) (b–a)⎝parenrightbig for (i–1 )b–(i–2 )a≤x≤ib–(i–1 )a,(j–1 )b–(j–2 )a≤t≤jb–(j–1 )a. Now system (1) can be rewritten as the single Fredholm equation Y(x)–λ⎝integraldisplaynb–(n–1)a aK(x,t)Y(t)dt=F(x), a≤x≤nb–(n–1 )a. If the kernels Kij(x,t) are square integrable on the square S={a≤x≤b,a≤t≤b}and the right-hand sides fi(x) are square integrable on [a ,b], then the kernel K(x,t) is square integrable on the new square Sn={a<x<nb–(n–1 )a,a<t<nb–(n–1 )a}, and the right-hand side F(x) is square integrable on [ a,nb–(n–1 )a]. If condition (4) is satisﬁed, then the kernel K(x,t) satisﬁes the inequality ⎝integraldisplayb aK2(x,t)dt≤A∗,a<x<nb–(n–1 )a, where A∗is a constant. Reference for Section 13.20: S. G. Mikhlin (1960). 13.21. Regularization Method for Equations with Inﬁnite Limits of Integration 13.21-1. Basic Equation and Fredholm Theorems. Consider an integral equation of the second kind in the form y(x)+1 √ 2π⎝integraldisplay∞ 0K1(x–t)y(t)dt+1 √ 2π⎝integraldisplay0 –∞K2(x–t)y(t)dt+⎝integraldisplay∞ –∞M(x,t)y(t)dt=f(x), (1) where – ∞<x<∞. We assume that the functions y(x)a n df(x) and the kernels K1(x)a n dK2(x) are such that their Fourier transforms belong to L2(–∞,∞) and satisfy the H ¨older condition. We also assume that the Fourier transforms of the kernel M(x,t) with respect to each variable belong toL2(–∞,∞) and satisfy the H ¨older condition and, in addition, ⎝integraldisplay∞ –∞⎝integraldisplay∞ –∞|M(x,t)|2dx dt <∞. 13.21. R EGULARIZATION METHOD FOR EQUATIONS WITH INFINITE LIMITS OF INTEGRATION 703 It should be noted that Eq. (1) with M(x,t)≡0 is the convolution-type integral equation with two kernels which was discussed in Subsection 13.10-2. The transposed homogeneous equation has the form ϕ(x)+1 √ 2π⎝integraldisplay∞ 0K1(t–x)ϕ(t)dt+1 √ 2π⎝integraldisplay0 –∞K2(t–x)ϕ(t)dt+⎝integraldisplay∞ –∞M(t,x)ϕ(t)dt=0 , ( 2 ) where – ∞<x<∞. Assume that the normality conditions (see Subsection 13.10-2) hold, that is, 1+K1(u)≠0, 1 + K2(u)≠0, –∞ <u<∞.( 3) THEOREM 1.The number of linearly independent solutions of the homogeneous ( f(x)≡0) equation (1) and that of the transposed homogeneous ( g(x)≡0) equation (2) are ﬁnite. THEOREM 2.For the nonhomogeneous equation (1) to be solvable, it is necessary and sufﬁcient that ⎝integraldisplay∞ –∞f(t)ϕk(t)dt=0 , k=1 ,...,N,( 4 ) where ϕk(x)is a complete ﬁnite set of linearly independent solutions to the transposed homogeneous equation (2). THEOREM 3.The difference between the number of linearly independent solutions to the ho- mogeneous equation (1) and the number of linearly independent solutions to the homogeneous transposed equation (2) is equal to the index ν=I n d1+K2(u) 1+K1(u)=1 2π⎝bracketleftbigg arg1+K2(u) 1+K1(u)⎝bracketrightbigg∞ –∞.( 5) 13.21-2. Regularizing Operators. An important method for the theoretical investigation and practical solution of the integral equations in question is a regularization of these equations, i.e., their reduction to a Fredholm equation of the second kind. Let us denote by Kthe operator determined by the left-hand side of Eq. (1): K[y(x)]≡y(x)+1 √ 2π⎝integraldisplay∞ 0K1(x–t)y(t)dt+1 √ 2π⎝integraldisplay0 –∞K2(x–t)y(t)dt+⎝integraldisplay∞ –∞M(x,t)y(t)dt(6) and introduce the similar operator L[ω(x)]≡ω(x)+1 √ 2π⎝integraldisplay∞ 0L1(x–t)ω(t)dt+1 √ 2π⎝integraldisplay0 –∞L2(x–t)ω(t)dt+⎝integraldisplay∞ –∞Q(x,t)ω(t)dt.( 7 ) Let us ﬁnd an operator Lsuch that the product LKis determined by the left-hand side of a Fredholm equation of the second kind with a kernel K(x,t): LK[y(x)]≡y(x)+⎝integraldisplay∞ –∞K(x,t)y(t)dt,⎝integraldisplay∞ –∞⎝integraldisplay∞ –∞|K(x,t)|2dx dt <∞.( 8 ) The operator Lis called a left regularizer . 704 METHODS FOR SOLVING LINEAR EQUATIONS OF THE FORMy(x)–⎝integraltextb aK(x,t)y(t)dt=f(x) For the operator Kof the integral equation (1) to have a left regularizer Lof the form (7), it is necessary and sufﬁcient that the normality conditions (3) hold. If conditions (3) are satisﬁed, then the left regularizer Lhas the form Lω(x)≡ω(x)–1 √ 2π⎝integraldisplay∞ 0R1(x–t)ω(t)dt–1 √ 2π⎝integraldisplay0 –∞R2(x–t)ω(t)dt+⎝integraldisplay∞ –∞Q(x,t)ω(t)dt,( 9 ) where the resolvents R1(x–t)a n d R2(x–t)o ft h ek e r n e l s K1(x–t)a n d K2(x–t)a r eg i v e nb y( s e e Subsection 13.9-1) Rj(x)=1 √ 2π⎝integraldisplay∞ –∞Kj(u) 1+Kj(u)e–iuxdu,Kj(u)=1 √ 2π⎝integraldisplay∞ –∞Kj(x)eiuxdx,j=1 ,2 , andQ(x,t) is any function such that ⎝integraldisplay∞ –∞⎝integraldisplay∞ –∞|Q(x,t)|2dx dt <∞. If condition (3) is satisﬁed, then the operator Lgiven by formula (9) is simultaneously a right regularizer of the operator K: KL[y(x)]≡y(x)+⎝integraldisplay∞ –∞K∗(x,t)y(t)dt, (10) where the function K∗(x,t) satisﬁes the condition ⎝integraldisplay∞ –∞⎝integraldisplay∞ –∞|K∗(x,t)|2dx dt <∞. (11) 13.21-3. Regularization Method. Consider the equation of the form K[y(x)] =f(x), – ∞<x<∞, (12) where the operator Kis deﬁned by (6). There are several ways of regularizing this equation, i.e., of its reduction to a Fredholm equation. First, this equation can be reduced to an equation with a Cauchy kernel. On regularizing the last equation by a method presented in Section 15.4, we can achieve our aim. This approach can be applied if we can ﬁnd, for given functions K1(x),K2(x),M(x,t), and f(x), simple expressions for their Fourier integrals. Otherwise it is natural to perform the regularization of Eq. (12) directly, without passing to the inverse transforms. A left regularization of Eq. (12) involves the application of the regularizer Lconstructed in the previous subsection to both its sides: LK[y(x)] = L[f(x)]. (13) It follows from (8) that Eq. (13) is a Fredholm equation y(x)+⎝integraldisplay∞ –∞K(x,t)y(t)dt=L[f(x)]. (14) 13.21. R EGULARIZATION METHOD FOR EQUATIONS WITH INFINITE LIMITS OF INTEGRATION 705 Thus, Eq. (12) can be transformed by left regularization to a Fredholm equation with the same unknown function y(x) and the known right-hand side L[f(x)]. Left regularization is known to imply no loss of solutions: all solutions of the original equation (12) are solutions of the regularized equation. However, in the general case, a solution of the regularized equation need not be a solution of the original equation. The right regularization consists in the substitution of the expression y(x)=L[ω(x)] (15) for the desired function into Eq. (12), where ω(x) is a new unknown function. We ﬁnally arrive at the following integral equation: KL[ω(x)] =f(x), (16) which is a Fredholm equation as well by virtue of (10): KL[ω(x)]≡ω(x)+⎝integraldisplay∞ –∞K∗(x,t)ω(t)dt=f(x), – ∞<x<∞. (17) Thus, we have passed from Eq. (12) for the unknown function y(x) to a Fredholm integral equation for a new unknown function ω(x). On solving the Fredholm equation (17), we ﬁnd a solution of the original equation (12) by formula (15). Right regularization can give no irrelevant solutions, but it is known that it can lead to a loss of a solution. A solution of the problem on an equivalent regularization, for which neither the loss of solutions nor the appearance of irrelevant “solutions” occur, is of signiﬁcant theoretical and practical interest. For Eq. (12) with an arbitrary right-hand side f(x) to admit an equivalent left regularization, it is necessary and sufﬁcient that the index νgiven by formula (5) be nonnegative. For an equivalently regularizing operator we can take the operator L◦[ω(x)]≡ω(x)–1 √ 2π⎝integraldisplay∞ 0R1(x–t)ω(t)dt–1 √ 2π⎝integraldisplay0 –∞R2(x–t)ω(t)dt. Thus, the Fredholm equation L◦K[y(x)] = L◦[f(x)], (18) for the case ν≥0, has those and only those solutions that are solutions to Eq. (12). For the case in which the index νis nonpositive, the operator L◦performs an equivalent right regularization of Eq. (12) for an arbitrary right-hand side f(x). In other words, for ν≤0, on ﬁnding the solution to the Fredholm equation KL◦[ω(x)] =f(x), we can obtain all solutions of the original equation (12) by the formula y(x)=L◦[ω(x)]. Another method of regularization is known, the so-called Carleman–Vekua regularization, which is based on the solution of the corresponding characteristic equation. Equation (12) can formally be rewritten as a convolution type equation with two kernels: y(x)+1 √ 2π⎝integraldisplay∞ 0K1(x–t)y(t)dt+1 √ 2π⎝integraldisplay0 –∞K2(x–t)y(t)dt=f1(x), (19) where f1(x)=f(x)–⎝integraldisplay∞ –∞M(x,t)y(t)dt. Next, the function f1(x) is provisionally assumed to be known, and Eq. (19) is solved (see Subsec- tion 13.10-2). The analysis of the resulting formula for the function y(x) shows that, for ν=0 ,t h i s is a Fredholm integral equation with the unknown function y(x). For the case in which ν>0 ,t h e resulting equation contains νarbitrary constants. For a negative index ν, solvability conditions must be added to the equation. Reference for Section 13.21: F. D. Gakhov and Yu. I. Cherskii (1978). Chapter 14 Methods for Solving Singular Integral Equations of the First Kind 14.1. Some Deﬁnitions and Remarks 14.1-1. Integral Equations of the First Kind with Cauchy Kernel. Asingular integral equation of the ﬁrst kind with Cauchy kernel has the form 1 πi⎝integraldisplay Lϕ(τ) τ–tdτ=f(t), i2= –1, (1) where Lis a smooth closed or nonclosed contour in the complex plane of the variable z=x+iy, tandτare the complex coordinates on L,ϕ(t) is the unknown function,1 τ–tis the Cauchy kernel, andf(t) is a given function, which is called the right-hand side of Eq. (1). The integral on the left-hand side only exists in the sense of the Cauchy principal value (see Subsection 14.2-5). A singular integral equation in which Lis a smooth closed contour, as well as an equation of the form 1 π⎝integraldisplay∞ –∞ϕ(t) t–xdt=f(x), –∞<x<∞,( 2) on the real axis and an equation with Cauchy kernel 1 π⎝integraldisplayb aϕ(t) t–xdt=f(x), a≤x≤b,( 3) on a ﬁnite interval, are special cases of Eq. (1). A general singular integral equation of the ﬁrst kind with Cauchy kernel has the form 1 πi⎝integraldisplay LM(t,τ) τ–tϕ(τ)dτ=f(t), (4) where M(t,τ) is a given function. This equation can also be rewritten in a different (equivalent) form, which is given in Subsection 14.4-4. Assume that all functions in Eqs. (1)–(4) satisfy the H ¨older condition (Subsection 14.2-2) and the function M(t,τ) satisﬁes this condition with re spect to both variables. 14.1-2. Integral Equations of the First Kind with Hilbert Kernel. The simplest singular integral equation of the ﬁrst kind with Hilbert kernel has the form 1 2π⎝integraldisplay2π 0cot⎝parenleftbiggξ–x 2⎝parenrightbigg ϕ(ξ)dξ=f(x), (5) 707 708 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND where ϕ(x) is the unknown function (0 ≤x≤2π), cot⎝bracketleftbig1 2(ξ–x)⎝bracketrightbig is the Hilbert kernel, and f(x)i s the given right-hand side of the equation (0 ≤x≤2π). A general singular integral equation of the ﬁrst kind with Hilbert kernel has the form –1 2π⎝integraldisplay2π 0N(x,ξ)c o t⎝parenleftbiggξ–x 2⎝parenrightbigg ϕ(ξ)dξ=f(x), (6) where N(x,ξ) is a given function. Equation (6) can often be rewritten in an equivalent form, which is presented in Subsection 14.4-5. Assume that all functions in Eqs. (5) and (6) also satisfy the H ¨older condition (see Subsec- tion 14.2-2) and the function N(x,ξ) satisﬁes this condition with re spect to both variables. If the right-hand sides of Eqs. (1)–(6) are identically zero, then the equations are said to be homogeneous , otherwise they are said to be nonhomogeneous . References for Section 14.1: F. D. Gakhov (1977), S. G. Mikhlin and S. Pr ¨ossdorf (1986), S. Pr ¨ossdorf and B. Silbermann (1991), A. Dzhuraev (1992), N. I. Muskhelishvili (1992), I. K. Lifanov (1996), R. Estrada and R. P. Kanwal (1999), E. G. Ladopoulos (2000). 14.2. Cauchy Type Integral 14.2-1. Deﬁnition of the Cauchy Type Integral. LetLbe a smooth closed contour* on the plane of a complex variable z=x+iy. The domain inside the contour Lis called the interior domain and is denoted by Ω+, and the complement of Ω+∪L, which contains the point at inﬁnity, is called the exterior domain and is denoted by Ω–. If a function f(z) is analytic in Ω+and continuous in Ω+∪L, then according to the familiar Cauchy formula in the theory of functions of a complex variable we have 1 2πi⎝integraldisplay Lf(τ) τ–zdτ=⎝braceleftBigf(z)f o r z∈Ω+, 0f o r z∈Ω–.(1) If a function f(z) is analytic in Ω–and continuous in Ω–∪L,t h e n 1 2πi⎝integraldisplay Lf(τ) τ–zdτ=⎝braceleftbigg f(∞)f o r z∈Ω+, –f(z)+f(∞)f o r z∈Ω–.(2) As usual, the positive direction on Lis deﬁned as the direction for which the domain Ω+remains to the left of the contour. The Cauchy formula permits one to calculate the values of a function at any point of the domain provided that the values on the boundary of the domain are known, i.e., the Cauchy formula solves the boundary value problem for analytic functions. The integral on the left-hand side in (1) and (2) is called the Cauchy integral . Assume that Lis a smooth closed or nonclosed contour that entirely belongs to the ﬁnite part of the complex plane. Let τbe the complex coordinate on L,a n dl e t ϕ(τ) be a continuous function of a point of the contour. In this case the integral Φ(z)=1 2πi⎝integraldisplay Lϕ(τ) τ–zdτ,( 3) which is constructed in the same way as the Cauchy integral, is called a Cauchy type integral .T h e function ϕ(τ) is called its density and 1/(τ–z) its kernel . *B y a smooth contour we mean a simple curve (i.e., a curve without points of self-intersection) that is either closed or nonclosed, has a continuous tangent, and has no cuspidal points. 14.2. C AUCHY TYPEINTEGRAL 709 For a Cauchy type integral with continuous density ϕ(τ), the only points at which the integrand is not analytic with respect to zare the points of the integration curve L. This curve is singular for the function Φ(z). IfLis a nonclosed contour, then Φ(z) is an analytic function on the entire plane with the singularity curve L. Assume that Lis a closed contour. In this case, Φ(z) splits into two independent functions: a function Φ+(z) deﬁned on the domain Ω+and a function Φ–(z) deﬁned on the domain Ω–. In general, these functions are not analytic continuations of each other. By a piecewise analytic function we mean an analytic function Φ(z) deﬁned by two independent expressions Φ+(z)a n dΦ–(z) on two complementary domains Ω+andΩ–of the complex plane. We note an important property of a Cauchy type integral. The function Φ(z) expressed by a Cauchy type integral of the form (3) vanishes at inﬁnity, i.e., Φ–(∞) = 0. This condition is also sufﬁcient for the representability of a piecewise analytic function by a Cauchy type integral. 14.2-2. H ¨older Condition. LetLbe a smooth curve in the complex plane z=x+iy,a n dl e t ϕ(t)b eaf u n c t i o no nt h i sc u r v e . We say that ϕ(t) satisﬁes the H¨older condition onLif for any two points t1,t2∈Lwe have |ϕ(t2)–ϕ(t1)|<A|t2–t1|λ,( 4) where Aandλare positive constants. The number Ais called the H¨older constant andλis called the H¨older exponent .I fλ> 1, then by condition (4) the derivative ϕ/prime t(t) vanishes everywhere, and ϕ(t) must be constant. Therefore, we assume that 0 < λ≤1. For λ=1 ,t h eH ¨older condition is often called the Lipschitz condition . Sometimes the H ¨older condition is called the Lipschitz condition of orderλ. Ift1andt2are sufﬁciently close to each other and if the H ¨older condition holds for some exponent λ1, then this condition certainly holds for each exponent λ<λ1. In general, the converse assertion fails. The smaller λ, the broader the class of H ¨older continuous functions is. The narrowest class is that of functions satisfying the Lipschitz condition. It follows from the last property that if functions ϕ1(t)a n d ϕ2(t) satisfy the H ¨older condition with exponents λ1andλ2, respectively, then their sum and the product, as well as their ratio provided that the denominator is nonzero, satisfy the H ¨older condition with exponent λ=m i n ( λ1,λ2). Ifϕ(t) is differentiable and has a bounded derivative, then ϕ(t) satisﬁes the Lipschitz condition. In general, the converse assertion fails. 14.2-3. Principal Value of a Singular Integral. Consider the integral⎝integraldisplayb adx x–c,a<c<b. Evaluating this integral as an improper integral, we obtain ⎝integraldisplayb adx x–c= lim ε1→0 ε2→0⎝parenleftbigg –⎝integraldisplayc–ε1 adx c–x+⎝integraldisplayb c+ε2dx x–c⎝parenrightbigg =l nb–c c–a+ lim ε1→0 ε2→0lnε1 ε2.( 5 ) The limit of the last expression obviously depends on the way in which ε1andε2tend to zero. Hence, the improper integral does not exist. This integral is called a singular integral .H o w e v e r , t h i s integral can be assigned a meaning if we assume that there is some relationship between ε1andε2. For example, if the deleted interval is symmetric with respect to the point c, i.e., ε1=ε2=ε,( 6) we arrive at the notion of the Cauchy principal value of a singular integral. 710 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND The Cauchy principal value of the singular integral ⎝integraldisplayb adx x–c,a<c<b is the number lim ε→0⎝parenleftbigg⎝integraldisplayc–ε adx x–c+⎝integraldisplayb c+εdx x–c⎝parenrightbigg . With regard to formula (5), we have ⎝integraldisplayb adx x–c=l nb–c c–a.( 7) Consider the more general integral ⎝integraldisplayb aϕ(x) x–cdx,( 8) where ϕ(x)∈[a,b] is a function satisfying the H ¨older condition. Let us understand this integral in the sense of the Cauchy principal value, which we deﬁne as follows: ⎝integraldisplayb aϕ(x) x–cdx= lim ε→0⎝parenleftbigg⎝integraldisplayc–ε aϕ(x) x–cdx+⎝integraldisplayb c+εϕ(x) x–cdx⎝parenrightbigg . We have the identity ⎝integraldisplayb aϕ(x) x–cdx=⎝integraldisplayb aϕ(x)–ϕ(c) x–cdx+ϕ(c)⎝integraldisplayb adx x–c; moreover, the ﬁrst integral on the right-hand side is convergent as an improper integral, because it follows from the H ¨older condition that ⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleϕ(x)–ϕ(c) x–c⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle<A |x–c|1–λ,0 < λ≤1, and the second integral coincides with (7). Thus, we see that the singular integral (8), where ϕ(x) satisﬁes the H ¨older condition, exists in the sense of the Cauchy principal value and is equal to ⎝integraldisplayb aϕ(x) x–cdx=⎝integraldisplayb aϕ(x)–ϕ(c) x–cdx+ϕ(c)l nb–c c–a. Some authors denote singular integrals by special symbols like v.p.⎝integraltext (valeur principale). How- ever, this is not necessary because, on one hand, if an integral of the form (8) exists as a proper or an improper integral, then it exists in the sense of the Cauchy principal value, and their values coincide;on the other hand, we shall always understand a singular integral in the sense of the Cauchy principal value. For this reason, we denote a singular integral by the usual integral sign. 14.2. C AUCHY TYPEINTEGRAL 711 14.2-4. Multivalued Functions. In the representation z=ρeiθof a complex number, the modulus ρis determined uniquely, whereas the argument θis only deﬁned modulo 2 π. This does not make the representation of a number ambiguous, because the argument enters this representation via the function eiθ,w h i c hi s2 π- periodic. However, if the dependence of an analytic function on the argument θis not 2π -periodic, then this function turns out to be multivalued. Of the elementary functions, the logarithm and the power function with noninteger exponent have this property: ln(z–z0)=l n |z–z0|+iarg(z–z0)=l nρ+iΘ,( 9 ) (z–z0)γ=ργeiγΘ=ρα[cos(βlnρ)+isin(βlnρ)]eiγθei2πkγ,γ=α+iβ. (10) In our reasoning, the logarithm of the modulus of a complex number is always understood as a real number, according to the usual deﬁnition. The general representatio n of the argument Θhas the form Θ=θ+2πk, where kranges over all integers ( k=0 ,±1,±2,...)a n dθis the argument with the least absolute value. To any k, there corresponds a branch of the multivalued function. The logarithmic function has inﬁnitely many branches. The same holds for the power function with an irrational or nonrealexponent. However, if the exponent is rational, γ=p/q, with gcd( p,q) = 1, then the power function hasqbranches. The branches of the logarithm differ by a constant of the form i2πm,a n dt h e branches of a power function differ by a factor of the form e i2πmγ(mis an integer). Obviously, to deﬁne a multivalued function, it is n ecessary to indicate which branch is chosen. However, in contrast to the case of functions of a real variable, this is not sufﬁcient for the complete deﬁnition ofa multivalued function of a complex variable. For the latter functions, there are points on the plane with the following property: as the independent variable goes along a closed contour surrounding this point and returns to the initial value, the chosen branch of the function changes to some otherbranch. Such points are called the branching points of the multivalued function. For the functions (9) and (10), the branching points are z 0and the point at inﬁnity. If the variable is going along a contour surrounding the point z0counterclockwise or clockwise, then the argument Θis changed by 2 πor by –2π , respectively. Accordingly, the logarithm is increased or decreased by i2π, and the power function is multiplied byei2πγore–i2πγ. Hence, the branch corresponding to the value k=npasses to the neighboring branch corresponding to k=n+1ork=n–1. As usual, the study of the point at inﬁnity is performed by the substitution z=1/ζwith the subsequent investigation at the point ζ=0 . We can preserve a chosen branch of a function only if we forbid going around an arbitrary branching point. To this end, we may use cuts joining the branching points. In the above cases of the logarithmic and the power function, we can make a cut along a curve issuing from the point z0 and passing to inﬁnity. A multivalued function is deﬁned uniquely if the branch is chosen and the cut is given. The range of Θis determined by the position of the cut. For example, if the cut passes along the ray that forms an angle θ0with the real axis, then for the principal branch ( k=0 )w eh a v e θ0≤Θ≤θ0+2π. In particular, for the cut that passes along the positive real axis, we have 0≤Θ≤2π; and for cut along the negative real axis, we obtain – π≤Θ≤π. If the cut is curvilinear, then the range of the argument depends on the functions of a point. The initial value of the argument corresponds to the left edge of the cut (with respect to z0) and the ﬁnal value corresponds to the right edge. Let us denote the value of the argument on the left and on the right edge of the cut by Θ+andΘ–, respectively. Then we have Θ––Θ+=2π. 712 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND For the chosen branch, the cut is a curve of discontinuity. On the edges of the cut we have ln(z––z0)=l n ( z+–z0)+i2π, (z––z0)γ=ei2πγ(z+–z0)γ. This discontinuity property of branches of multivalued functions on the edges of a cut is widely used in the solution of boundary value problems with discontinuous boundary conditions. The logarithm is applied for the case in which a discontinuous function enters the boundary condition asa summand, and the power function corresponds to the case of a discontinuous factor in the boundary conditions. 14.2-5. Principal Value of a Singular Curvilinear Integral. LetLbe a smooth contour and let τandtbe complex coordinates of its points. Consider the singular curvilinear integral⎝integraldisplay Lϕ(τ) τ–tdτ. (11) Let us take a circle of some radius ρcentered at the point ton the contour. Let t1andt2be the points of intersection of this circle with the curve. Assume that the radius is so small that the circle has no other points of intersection with L.L e tlbe the part of the contour Lcut out by the circle. Consider the integral over the remaining arc, ⎝integraldisplay L–lϕ(τ) τ–tdτ. (12) The limit of the integral (12) as ρ→0 is called the principal value of the singular integral (11). Using the representation ⎝integraldisplay Lϕ(τ) τ–tdτ=⎝integraldisplay Lϕ(τ)–ϕ(t) τ–tdτ+ϕ(t)⎝integraldisplay Ldτ τ–t and the same reasoning as above, we see that the singular integral (11) exists in the sense of the Cauchy principal value for any function ϕ(τ) satisfying the H ¨older condition. At any point of smoothness, this integral can be presented in two forms: ⎝integraldisplay Lϕ(τ) τ–tdτ=⎝integraldisplay Lϕ(τ)–ϕ(t) τ–tdτ+ϕ(t)⎝parenleftbigg lnb–t a–t+iπ⎝parenrightbigg ⎝integraldisplay Lϕ(τ) τ–tdτ=⎝integraldisplay Lϕ(τ)–ϕ(t) τ–tdτ+ϕ(t)l nb–t t–a, where aandbare the endpoints of L. In particular, if the contour is closed, then by setting a=bwe obtain ⎝integraldisplay Lϕ(τ) τ–tdτ=⎝integraldisplay Lϕ(τ)–ϕ(t) τ–tdτ+iπϕ(t). Throughout the following, any singular integral will be understood in the sense of the Cauchy principal value. LetLbe a smooth contour (closed or nonclosed) and let ϕ(τ)b eaH ¨older function of a point on the contour. Then the Cauchy type integral Φ(z)=1 2πi⎝integraldisplay Lϕ(τ) τ–zdτ (13) 14.2. C AUCHY TYPEINTEGRAL 713 has limit values Φ+(t)a n d Φ–(t) at any point of t∈Lother than the endpoints of the contour, as z→tfrom the left or from the right along any path; and these limit values can be expressed via the density ϕ(t) of the integral and via the singular integral (13) by the Sokhotski–Plemelj formulas Φ+(t)=1 2ϕ(t)+1 2πi⎝integraldisplay Lϕ(τ) τ–tdτ,Φ–(t)=–1 2ϕ(t)+1 2πi⎝integraldisplay Lϕ(τ) τ–tdτ. (14) The sum and the difference of formulas (14) give the equivalent formulas Φ+(t)–Φ–(t)=ϕ(t), (15) Φ+(t)+Φ–(t)=1 πi⎝integraldisplay Lϕ(τ) τ–tdτ, (16) which are often used instead of (14). The Sokhotski–Plemelj formulas for the real axis have the form Φ+(x)=1 2ϕ(x)+1 2πi⎝integraldisplay∞ –∞ϕ(τ) τ–xdτ,Φ–(x)=–1 2ϕ(x)+1 2πi⎝integraldisplay∞ –∞ϕ(τ) τ–xdτ. (17) Moreover, we have Φ+(∞)=1 2ϕ(∞), Φ–(∞)=–1 2ϕ(∞). This, together with (17), implies Φ+(∞)+Φ–(∞) = 0, (18) lim x→∞⎝integraldisplay∞ –∞ϕ(τ) τ–xdτ= 0. (19) Any function representable by a Cauchy type integral on the real axis necessarily satisﬁes condition (18). This condition is also sufﬁcient for the representability of a piecewise analytic function in the upper and the lower half-plane by an integral over the real axis. Consider a Cauchy type integral over the real axis and assume that zis not real: Φ(z)=1 2πi⎝integraldisplay∞ –∞ϕ(x) x–zdx, (20) where ϕ(x) is a complex function of a real variable xsatisfying the H ¨older condition on the real axis. If a function ϕ(z) is analytic in the upper half-plane, is continuous in the closed upper half-plane, and satisﬁes the H ¨older condition on the real axis, then 1 2πi⎝integraldisplay∞ –∞ϕ(x) x–zdx=⎝braceleftBigg ϕ(z)–1 2ϕ(∞)f o r I m z>0 , –1 2ϕ(∞)f o r I m z<0 .(21) We also have the formula 1 2πi⎝integraldisplay∞ –∞ϕ(x)–ϕ(∞) x–zdx=⎝braceleftBigg1 2ϕ(∞)f o r I m z>0 , –ϕ(z)+1 2ϕ(∞)f o r I m z<0(22) provided that ϕ(z) is analytic in the lower half-plane, continuous in the closed lower half-plane, and satisﬁes the H ¨older condition on the real axis. 714 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND 14.2-6. Poincar ´e–Bertrand Formula. Consider the following pair of iterated singular integrals: N(t)=1 πi⎝integraldisplay Ldτ τ–t1 πi⎝integraldisplay LK(τ,τ1) τ1–τdτ1, (23) M(t)=1 πi⎝integraldisplay Ldτ11 πi⎝integraldisplay LK(τ,τ1) (τ–t)(τ 1–τ)dτ, (24) where Lis a smooth contour and the function K(τ,τ1) satisﬁes the H ¨older condition with respect to both variables. Both integrals make sense, and although Ndiffers from Monly by the order of integration, they are not equal, as shown by the following Poincar ´e–Bertrand formula 1 πi⎝integraldisplay Ldτ τ–t1 πi⎝integraldisplay LK(τ,τ1) τ1–τdτ1=K(t,t)+1 πi⎝integraldisplay Ldτ11 πi⎝integraldisplay LK(τ,τ1) (τ–t)(τ 1–τ)dτ, (25) which can also be rewritten in the form ⎝integraldisplay Ldτ τ–t⎝integraldisplay LK(τ,τ1) τ1–τdτ1=–π2K(t,t)+⎝integraldisplay Ldτ1⎝integraldisplay LK(τ,τ1) (τ–t)(τ 1–τ)dτ. (26) Example. Let us evaluate the Cauchy type integral over the unit circle |z|= 1 with density ϕ(τ)=2/[τ(τ– 2)], i.e., Φ(z)=1 2πi⎝integraldisplay L1 τ–2dτ τ–z–1 2πi⎝integraldisplay L1 τdτ τ–z. The function 1 /(z– 2) is analytic in Ω+,a n d1 /zis analytic in Ω–and vanishes at inﬁnity. By formula (1), the ﬁrst integral is equal to 1 /(z–2 )f o r z∈Ω+and is zero for z∈Ω–. By formula (2), the second integral is equal to –1 /zforz∈Ω–and is zero for z∈Ω+. Hence, Φ+(z)=1 z–2,Φ–(z)=1 z. References for Section 14.2: F. D. Gakhov (1977), S. G. Mikhlin and S. Pr ¨ossdorf (1986), N. I. Muskhelishvili (1992). 14.3. Riemann Boundary Value Problem 14.3-1. Principle of Argument. The Generalized Liouville Theorem. THETHEOREM ON THE ANALYTIC CONTINUATION (THE PRINCIPLE OF CONTINUITY ).Assume that a domain Ω1borders a domain Ω2along a smooth curve L. Let analytic functions f1(z)andf2(z) be given in Ω1andΩ2. Assume that, as the point ztends to L, both functions tend to the same continuous limit function on the curve L. Under these assumptions, the functions f1(z)andf2(z) are analytic continuations of each other. Assume that a function f(z) is analytic in a domain Ωbounded by a contour Lexcept for ﬁnitely many points, where it may have poles. Let us write out the power series expansion of f(z) around some point z0: f(z)=cn(z–z0)n+cn+1(z–z0)n+1+···=(z–z0)nf1(z), f1(z0)=cn≠0. The number nis called the order of the function f(z)at the point z0.I fn> 0, then the order of the function is the order of zero; if n< 0, then the order of the function is minus the order of the pole. If the order of a function at z0is zero, then at z0the function has a ﬁnite nonzero value at z0. When considering the point at inﬁnity, we must replace the difference z–z0by 1/z .I fz0∈L,t h e n we deﬁne the order of the function to be equal to1 2n. 14.3. R IEMANN BOUNDARY VALUE PROBLEM 715 LetNΩandPΩ(NLandPL) be the numbers of zeros and poles on the domain (on the contour, respectively), where each zero and pole is taken according to its multiplicity. Let [ δ]Ldenote the increment of the variable δwhen going around the contour in the positive direction. As usual, by the positive direction we mean the direction t he domain under consideration remains to the left of the contour. THEPRINCIPLE OF ARGUMENT .Letf(z)be a single-valued analytic function in a multiply connected domain Ωbounded by a smooth contour L=L0+L1+···+Lmexcept for ﬁnitely many points at which f(z)may have poles, and let f(z)be continuous in the closed domain Ω∪L(except for these poles) and have at most ﬁnitely many zeros of integer order on the contour. In this case, the following formula holds: NΩ–PΩ+1 2(NL–PL)=1 2π[argf(z)]L. THEGENERALIZED LIOUVILLE THEOREM .Assume that a function f(z)is analytic on the entire complex plane except for points a0=∞,ak(k=1 ,...,n), where it has poles, and that the principal parts of the Laurent series expansions of f(z)at the poles have the form Q0(z)=c0 1z+c0 2z2+···+c0 m0zm0 Qk⎝parenleftbigg1 z–ak⎝parenrightbigg =ck 1 z–ak+ck 2 (z–ak)2+···+ck mk (z–ak)mkat the point a0, at the points ak. Thenf(z)is a rational function, and can be represented by the formula f(z)=C+Q0(z)+n⎝summationdisplay k=1Qk⎝parenleftbigg1 z–ak⎝parenrightbigg , where Cis a constant. In particular, if the only singularity of f(z)is a pole of order mat inﬁnity, thenf(z)is a polynomial of degree m, f(z)=c0+c1z+···+cmzm. The following notation is customary: (a) f(z) is the function conjugate to a given function f(z); (b)f(¯z) is the function obtained from f(z) by replacing zby ¯z, i.e.,yby –y inf(z); (c) ¯f(z) is the function deﬁned by the condition ¯f(z)= f(¯z). Ifz=x+iyandf(z)=u(x,y)+iv(x,y), then f(z)=u(x,y)–iv(x,y),f(¯z)=u(x,–y)+iv(x,–y), ¯f(z)=u(x,–y)–iv(x,–y). In particular, if f(z) is given by a series f(z)=n⎝summationtext k=0ckzk,t h e n f(z)=n⎝summationdisplay k=0¯ck¯zk,f(¯z)=n⎝summationdisplay k=0ck¯zk,¯f(z)=n⎝summationdisplay k=0¯ckzk. For a function represented by a Cauchy type integral f(z)=1 2πi⎝integraldisplay Lϕ(τ) τ–zdτ, we have f(z)=–1 2πi⎝integraldisplay L ϕ(τ) ¯τ–¯z dτ,f(¯z)=1 2πi⎝integraldisplay Lϕ(τ) τ–¯zdτ,¯f(z)=–1 2πi⎝integraldisplay L ϕ(τ) ¯τ–z dτ. Note that if a functio n satisﬁes the condition ¯f(z)=f(z), then it takes real values for all real values ofz. The converse assertion also holds. 716 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND 14.3-2. Hermite Interpolation Polynomial. The Hermite interpolation polynomial is used for the construction of the canonical function of the nonhomogeneous Riemann problem in Subsections 12.4-7 and 14.3-9. Let distinct points zk(k=1 ,...,m) be given, and a number ∆(j) k(j=0 ,1 , ...,nk–1 )b e assigned to each point zk,w h e r et h e nkare given positive integers. It is required to construct a polynomial Up(z) of the least possible degree such that U(j) p(zk)=∆(j) k,k=1 ,...,m,j=0 ,1 , ...,nk–1 , where the U(j) p(zk)a r et h ev a l u e so ft h e jth-order derivatives of the polynomial at the points zk.T h e numbers zkare called the interpolation nodes andnkthe interpolation multiplicities at the nodes zk. There exists a unique polynomial with these properties. It has the form (e.g., see V . I. Smirnov and N. A. Lebedev (1964)) Up(z)=m⎝summationdisplay k=1ζ(z) (z–zk)nknk–1⎝summationdisplay r=0Ak,r(z–zk)r,p=m⎝summationdisplay k=1nk–1 , ζ(z)=m⎝productdisplay k=1(z–zk)nk,Ak,r=r⎝summationdisplay j=0∆(j) k j!(r–j)!⎝bracketleftbiggdr–j dzr–j(z–zk)nk ζ(z)⎝bracketrightbigg z=zk, k=1 ,...,m,r=0 ,1 , ...,nk–1 ; and this polynomial is unique. The interpolation polynomial Up(z) constructed for some function f(z) must satisfy the following conditions at the points zk: U(j) p(zk)=∆(j) k=f(j)(zk),k=1 ,...,m,j=0 ,1 , ...,nk–1 , where f(j)(zk)i st h ev a l u eo ft h e jth-order derivative of f(z) at the point zk. 14.3-3. Notion of the Index. LetLbe a smooth closed contour, and let D(t) be a continuous nowhere vanishing function on this contour. The indexνof the function D(t) with respect to the contour Lis the increment of the argument ofD(t) along L(traversed in the positive direction) divided by 2 π: ν=I n dD(t)=1 2π[argD(t)]L.( 1) Since ln D(t)=l n |D(t)|+iargD(t) and since after the traverse the function |D(t)|returns to its original value, it follows that [ln D(t)]L=i[argD(t)]L, and hence ν=1 2πi[lnD(t)]L.( 2) The index can be expressed in the form of an integral as follows: ν=I n dD(t)=1 2πi⎝integraldisplay LdlnD(t)=1 2π⎝integraldisplay LdargD(t). (3) If the function D(t) is not differentiable but has bounded variation, then the integral is regarded as the Stieltjes integral. Since D(t) is continuous, the image ˘Γof the closed contour Lis a closed contour as well, and the increment of the argument D(t) along Lis a multiple of 2 π. Hence, the following assertions hold. 14.3. R IEMANN BOUNDARY VALUE PROBLEM 717 1◦. The index of a function that is continuous on a closed contour and vanishes nowhere is an integer (possibly zero). 2◦. The index of the product of two functions is equal to the sum of the indexes of the factors. The index of a ratio is equal to the difference of the indexes of the numerator and the denominator. We now assume that D(t) is differentiable and is the boundary value of a function analytic in the interior or exterior of L. In this case, the number ν=1 2πi⎝integraldisplay LdlnD(t)=1 2πi⎝integraldisplay LD/prime t(t) D(t)dt (4) is equal to the logarithmic residue of the function D(t). The principle of argument (see Subsec- tion 14.3-1) implies the following properties of the index: 3◦.I fD(t) is the boundary value of a function analytic in the interior or exterior of the contour, then its index is equal to the number of zeros inside the contour or minus the number of zeros outside the contour, respectively. 4◦. If a function D(z) is analytic in the interior of the contour except for ﬁnitely many points at which it may have poles, then the number of zeros must be replaced by the difference of the number of zeros and the number of poles. Here the zeros and the poles are counted according to their multiplicities. We also note that the indexes of complex conjugate functions have opposite signs. Let t=t1(s)+it2(s)( 0 ≤s≤l) be the equation of the contour L. On substituting the expression of the complex coordinate tinto the function D(t), we obtain D(t)=D⎝parenleftbig t1(s)+it2(s)⎝parenrightbig =ξ(s)+iη(s). (5) Let us regard ξandηas Cartesian coordinates. Then ξ=ξ(s),η=η(s) is a parametric equation of some curve Γ. Since the function D(t) is continuous and the contour L is closed, it follows that the curve Γis closed as well. The number of turns of the curve Γaround the origin, i.e., the number of full rotations of the radius vector as the variable svaries from 0 to l, is obviously the index of the function D(t). This number is often called the winding number of the curve Γwith respect to the origin. If the curve Γis successfully constructed, then the winding number can be observed directly. There are many examples for which the index can be found by analyzing the shape of the curve Γ. For instance, if D(t) is a real or a pure imaginary function that does not vanish, then Γis a line segment (traversed an even number of times), and the index D(t) is equal to zero. If the real part ξ(s) or the imaginary part η(s) preserves its sign, then the index is obviously zero, and so on. If the function D(t) can be represented as the product or the ratio of functions that are limit values of functions analytic in the interior or exterior of the contour, then the index can be calculated on thebasis of properties 2 ◦,3◦,a n d4◦. In the general case, the calculation of the index can be performed by formula (3). On the basis of formula (5) we substitute the expression dargD(t)=darctanη(s) ξ(s) 718 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND into (3) and assume that ξandηare differentiable. Then we obtain ν=1 2π⎝integraldisplay Γξd η –ηd ξ ξ2+η2=1 2π⎝integraldisplayl 0ξ(s)η/prime s(s)–η(s)ξ/prime s(s) ξ2(s)+η2(s)ds.( 6) Example 1. Let us calculate the index of D(t)=tnwith respect to an arbitrary contour Lsurrounding the origin. First method. The function tnis the boundary value of the function zn, which has precisely one zero of order ninside the contour. Hence ν=I n dtn=n. Second method . If the argument of tisϕ, then the argument of tnisnϕ. As the point ttraverses the contour Land returns to the original value, the argument ϕobtains the increment 2 π. Hence, Indtn=n. The index can also be found numerically. Since the index is integer-valued, an approximate value whose error is less than1 2can be rounded off to the nearest integer to obtain the exact value. 14.3-4. Statement of the Riemann Problem. LetLbe a simple smooth closed contour which divides the complex plane into the interior domain Ω+ and the exterior domain Ω–, and let two functions of points of the contours D(t)a n d H(t) satisfying the H ¨older condition (see Subsection 14.2-2) be given; moreover, suppose that D(t) does not vanish. The Riemann Problem . Find two functions (or a single piecewise analytic function), namely, a function Φ+(z) analytic in Ω+and a function Φ–(z) analytic in the domain Ω–including z=∞,s o that the following linear relation is satisﬁed on the contour L: Φ+(t)=D(t)Φ–(t) or Φ+(t)=D(t)Φ–(t)+H(t)(the homogeneous problem) (the nonhomogeneous problem).(7) (8) The function D(t) is called the coefﬁcient of the Riemann problem, and the function H(t)i s called the right-hand side . We ﬁrst consider a Riemann problem of special form that is called the jump problem .L e t a function ϕ(t) deﬁned on a closed contour Lsatisfy the H ¨older condition. The problem is to ﬁnd a piecewise analytic function Φ(z)(Φ(z)=Φ+(z)f o rz∈Ω+andΦ(z)=Φ–(z)f o rz ∈Ω–)t h a t vanishes at inﬁnity and has a jump of magnitude ϕ(t)o nL, i.e., such that Φ+(t)–Φ–(t)=ϕ(t). It follows from the Sokhotski–Plemelj formulas (see Subsection 14.2-5) that the function Φ(z)=1 2πi⎝integraldisplay Lϕ(τ) τ–zdτ is the unique solution to the above problem. Thus, an arbitrary function ϕ(t) given on the closed contour and satisfying the H ¨older condition can be uniquely represented as the difference of functions Φ+(t)a n d Φ–(t) that are the boundary values of analytic functions Φ+(z)a n dΦ–(z) under the additional condition Φ–(∞)=0 . If we neglect the additional condition Φ–(∞) = 0, then the solution will be given by the formula Φ(z)=1 2πi⎝integraldisplay Lϕ(τ) τ–zdτ+ const . (9) 14.3. R IEMANN BOUNDARY VALUE PROBLEM 719 Let us seek a particular solution of the homogeneous problem (7) in the class of functions that do not vanish on the contour. Let N+andN–be the numbers of zeros of the desired functions in the domains Ω+andΩ–, respectively. Taking the index of both parts of Eq. (7), on the basis of properties 2◦and 3◦we obtain N++N–=I n dD(t)=ν. (10) We call the index νof the coefﬁcient D(t)t h e index of the Riemann problem. Letν= 0. Under this condition, ln D(t) is a single-valued function. It follows from (10) that N+=N–= 0, i.e., the solution has no zeros on the entire plane. Therefore, the functions ln Φ±(z) are analytic in their domains and hence singl e-valued together with the boundary values ln Φ±(t). Taking the logarithm of the boundary condition (7), we obtain lnΦ+(t)–l nΦ–(t)=l n D(t). (11) We can choose an arbitrary branch of ln D(t) because the ﬁnal result is independent of the choice of this branch. Thus, we must ﬁnd a piecewise analytic function ln Φ(z) with a prescribed jump on L. The solution of this problem under the additional condition ln Φ–(∞) = 0 is given by the formula lnΦ(z)=1 2πi⎝integraldisplay LlnD(τ) τ–zdτ. (12) For brevity, we write 1 2πi⎝integraldisplay LlnD(τ) τ–zdτ=G(z). (13) It readily follows from the Sokhotski–Plemelj formulas that the functions Φ+(z)=eG+(z)andΦ–(z)=eG–(z)(14) are the solution of the boundary value problem (7) with the condition Φ–(∞)=1 . If we neglect the additional condition Φ–(∞) = 1, then in formula (12) we must add an arbitrary constant, and the solution becomes Φ+(z)=CeG+(z),Φ–(z)=CeG–(z), (15) where Cis an arbitrary constant. Since G–(∞) = 0, it follows that Cis the value of Φ–(z) at inﬁnity. Thus, in the case ν= 0 and for arbitrary Φ–(∞)≠0, the solution contains a single arbitrary constant, and hence there is a unique linearly independent solution. If Φ–(∞)=0 ,t h e n C=0 ,a n d the problem has only the trivial solution (which is identically zero), which is natural because N–=0 . This gives an important corollary. An arbitrary function D(t)≠0o nLthat satisﬁes the H ¨older condition and has zero index can be represented as the ratio of the boundary values Φ+(t)a n dΦ–(t) of functions that are analytic in Ω+andΩ–and have no zeros in these domains. These functions are determined modulo an arbitrary constant factor and are given by formulas (15). On passing to the general case, we seek a piecewi se analytic function s atisfying the homogeneous boundary condition (7) and having zero order on the entire plane except for the point at inﬁnity, where the order of the function is equal to the index of the problem. By the canonical function (of the homogeneous Riemann problem) X(z) we mean the function satisfying the boundary condition (7) and piecewise analytic on the entire plane except for the point at inﬁnity, where the order of this function is equal to the index of the problem. This function can be constructed by reducing the problem to the case of zero index. Indeed, let us rewrite the boundary condition (7) in the form Φ+(t)=t–νD(t)tνΦ–(t). 720 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND On representing the function t–νD(t) with zero index as the ratio of boundary values of analytic functions, t–νD(t)=eG+(t) eG–(t),G(z)=1 2πi⎝integraldisplay Lln[τ–νD(τ)] τ–zdτ, (16) we obtain the following expression for the canonical function: X+(z)=eG+(z),X–(z)=z–νeG–(z). (17) SinceX+(t)=D(t)X–(t), it follows that the coefﬁcient of the Riemann problem can be represented as the ratio of canonical functions: D(t)=X+(t) X–(t). (18) The representation (18) is often called a factorization . Forν≥0, the canonical function, which has a zero of order νat inﬁnity, is a particular solution of the boundary value problem (7). For ν< 0, the canonical function has a pole of order |ν|at inﬁnity and is not a solution, but in this case it is still used as an auxiliary function in the solution of thenonhomogeneous problem. 14.3-5. Solution of the Homogeneous Problem. Letν=I n dD(t) be an arbitrary integer. On representing D(t) by formula (18), we reduce the boundary condition (7) to the form Φ+(t) X+(t)=Φ–(t) X–(t). The left-hand side of the last relation contains the boundary value of a function that is analytic inΩ+, and the right-hand side contains the boundary value of a function that has at least the order – ν at inﬁnity. By the principle of continuity (see Subsection 14.3-1), the functions on the left-hand side and on the right-hand side are analytic continuations of each other to the entire plane possibly except for the point at inﬁnity at which, in the case ν> 0, a pole of order ≤νcan occur. Hence, for ν> 0, by the generalized Liouville theorem (see Subsection 14.3-1), this single analytic function is a polynomial of degree ≤νwith arbitrary coefﬁcients. For ν< 0, it follows from the Liouville theorem that this function is constant. However, since this function must vanish at inﬁnity, it follows that it is identically zero. Hence, for ν< 0, the homogeneous problem has only the trivial solution (which is identically zero). A problem that has no nontrivial solutions is said to be unsolvable . Thus, for a negative index, the homogeneous problem (7) is unsolvable. Letν>0 .L e t Pν(z) stand for a polynomial of degree νwith arbitrary coefﬁcients. In this case, w eo b t a i nas o l u t i o ni nt h ef o r m Φ(z)=Pν(z)X(z), or Φ+(z)=Pν(z)eG+(z),Φ–(z)=z–νPν(z)eG–(z), (19) where G(z) is determined by formula (16). Thus, if the index νof the Riemann boundary value problem is nonnegative, then the homoge- neous problem (7) has ν+ 1 linearly independent solutions Φ+ k(z)=zkeG+(z),Φ– k(z)=zk–νeG–(z)(k=0 ,1 , ...,ν). (20) The general solution contains ν+ 1 arbitrary constants and is given by formula (19). For a negative index, problem (7) is unsolvable. 14.3. R IEMANN BOUNDARY VALUE PROBLEM 721 The polynomial Pν(z) has exactly νzeros in the complex plane. It follows from formulas (19) that the number of all zeros of a solution to the homogeneous Riemann boundary value problem isequal to the index ν. Depending on the choice of the coefﬁcients of the polynomial, these zeros can occur in each of the domains Ω ±and also on the contour itself. Just as above, we denote by N±the number of zeros in the domains Ω±and by N0the number of zeros on the contour L. We can see that in the general case (without the condition that there are no zeros on the contour), formula (10) becomes N++N–+N0=ν. (21) 14.3-6. Solution of the Nonhomogeneous Problem. On replacing the coefﬁcient D(t) in the boundary condition (8) by the ratio of the boundary values of the canonical functions by formula (18), we reduce (8) to the form Φ+(t) X+(t)=Φ–(t) X–(t)+H(t) X+(t). (22) The function H(t)/X+(t) satisﬁes the H ¨older condition. Let us replace it by the difference of the boundary values of analytic functions (see the jump problem in Subsection 14.3-4): H(t) X+(t)=Ψ+(t)–Ψ–(t), where Ψ(z)=1 2πi⎝integraldisplay LH(τ) X+(τ)dτ τ–z. (23) Then the boundary condition (22) can be rewritten in the form Φ+(t) X+(t)–Ψ+(t)=Φ–(t) X–(t)–Ψ–(t). Note that for ν≥0 the function Φ–(z)/X–(z) has a pole at inﬁnity, and for ν< 0 it has a zero of orderν. By the same reasoning as in the solution of the homogeneous problem, we obtain the following results. Letν≥0. In this case, Φ+(t) X+(t)–Ψ+(t)=Φ–(t) X–(t)–Ψ–(t)=Pν(t). This gives the solution Φ(z)=X(z)[Ψ(z)+Pν(z)], (24) where the functions X(z)a n dΨ(z) are expressed by formulas (17) and (23) and Pνis a polynomial of degree νwith arbitrary coefﬁcients. We can readily see that formula (24) gives the general solution of the nonhomogeneous problem because it contains the general solution X(z)Pν(z) of the homogeneous problem as a summand. Letν< 0. In this case, Φ–(z)/X–(z) vanishes at inﬁnity and Φ+(t) X+(t)–Ψ+(t)=Φ–(t) X–(t)–Ψ–(t)=0 , so that Φ(z)=X(z)Ψ(z). (25) 722 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND In the expression for the function Φ–(z), the ﬁrst factor has a pole of order – νat inﬁnity by virtue of formula (17), and the second factor is the Cauchy type integral (23) and, in general, has aﬁrst-order zero at inﬁnity. Hence, Φ –(z) has a pole of order ≤–ν– 1 at inﬁnity. Thus, if ν< –1, then the nonhomogeneous problem is unsolvable in general. It is solvable only if the constant term satisﬁes some additional c onditions. To ﬁnd these conditions, we expand the Cauchy type integral (23) in a series in a neighborhood of the point at inﬁnity: Ψ–(z)=∞⎝summationdisplay k=1ckz–k,w h e r e ck=–1 2πi⎝integraldisplay LH(τ) X+(τ)τk–1dτ. ForΦ–(z) to be analytic at the point at inﬁnity, it is necessary that the ﬁrst – ν– 1 coefﬁcients of the expansion of Ψ–(z) be zero. This means that for the solvability of the nonhomogeneous problem in the case of negative index ( ν< –1), it is necessary and sufﬁcient that the following – ν–1 conditions hold: ⎝integraldisplay LH(τ) X+(τ)τk–1dτ=0 , k=1 ,2 , ...,–ν– 1. (26) Thus, in the case ν≥0, the nonhomogeneous Riemann problem is solvable for an arbitrary right-hand side, and the general solution is given by the formula Φ(z)=X(z) 2πi⎝integraldisplay LH(τ) X+(τ)dτ τ–z+X(z)Pν(z), (27) where the canonical function X(z) is given by (17) and Pν(z) is a polynomial of degree νwith arbitrary complex coefﬁcients. If ν= –1, then the nonhomogeneous problem is also solvable and has a unique solution. In the case ν< –1, the nonhomogeneous problem is unsolvable in general. For this problem to be solvable, it is necessary and sufﬁcient that the right-hand side of the problem satisfy – ν–1 conditions (26). If these conditions are satisﬁed, th en the solution of the problem is unique and is given by formula (27), where we must set Pν(z)≡0. The solution with the additional condition of vanishing at inﬁnity has important applications. In this case, instead of a polynomial of degree ν, we must take a polynomial of degree ν–1 . F o r the solvability of the problem in the case of negative index, it is necessary that the coefﬁcient c–νbe zero as well. Hence, under the assumption that Φ–(∞) = 0, the solution is given for ν≥0b yt h ef o r m u l a Φ(z)=X(z)[Ψ(z)+Pν–1(z)], (28) where, for ν=0 ,w em u s ts e t Pν–1(z)≡0. Ifν< 0, then the solution is still given by formula (28) with Pν–1(z)≡0 under the following – ν solvability conditions: ⎝integraldisplay LH(τ) X+(τ)τk–1dτ=0 , k=1 ,2 , ...,–ν. (29) In this case, the assertion on the s olvability of the nonhomogeneous problem acquires a more symmetric form. For ν≥0, the general solution of the nonhomogeneous problem linearly depends onνarbitrary constants. For ν< 0, the number of the solvability conditions is equal to – ν.N o t e that for ν= 0 the nonhomogeneous problem is unconditionally solvable, and the solution is unique. On the basis of the above reasoning, the solution of the Riemann boundary value problem is mainly reduced to the following two operations: 1◦. A representation of an arbitrary function given on the contour in the form of the difference of boundary values of analytic functions in the domains Ω+andΩ–(the jump problem). 14.3. R IEMANN BOUNDARY VALUE PROBLEM 723 2◦. A representation of a nonvanishing function in the form of the ratio of boundary values of analytic functions (factorization). Here the second operation can be reduced to the ﬁrst by taking the logarithm. Some complications related to the case of a nonzero index are due to the multivaluedness of the logarithm only. The ﬁrstoperation for arbitrary functions is equivalent to the calculation of a Cauchy type integral. In this connection, the solution to the problem by formulas (17) and (23)–(25) is explicitly expressed (in the closed form) via Cauchy type integrals. 14.3-7. Riemann Problem with Rational Coefﬁcients. Consider the Riemann boundary value problem with a contour that consists of ﬁnitely many simplecurves and with coefﬁcient D(t) a rational function that has neither zeros nor poles on the contour. Note that an arbitrary continuous function (and all the functions satisfying the H ¨older condition) can be approximated with arbitrary accuracy by rational functions, and the solution of problems with rational coefﬁcients can serve as a basis for t he approximate solution in the general case. Assume that the Riemann problem has the form Φ +(t)=p(t) q(t)Φ–(t)+H(t), (30) and the polynomials p(z)a n dq(z) can be factorized as follows: p(z)=p+(z)p–(z),q(z)=q+(z)q–(z), (31) where p+(z)a n dq+(z) are polynomials whose roots belong to Ω+andp–(z)a n dq–(z) are polynomials with roots in Ω–. It readily follows from property 4◦of the index (Subsection 14.3-3) that ν=m+–n+, where m+andn+are the numbers of zeros of the polynomials p+(z)a n dq+(z). Since the coefﬁcient of the problem is a function that can be analytically continued to the domain Ω±, it follows that in this case it is reasonabl e to avoid using the general formulas and obtain a solution directly by analytic continuation; he re the role of the standard function of the type tνthat is used in the reduction of the index to zero can be played by the productν⎝producttext j=1(t–aj), where a1,...,aν are arbitrary points of the domain Ω+. On representing the boundary condition in the form q–(t) p–(t)Φ+(t)–p+(t) q+(t)Φ–(t)=q–(t) p–(t)H(t), where the canonical function is determined by the expressions X+(z)=p–(z) q–(z),X–(z)=q+(z) p+(z), (32) we obtain the solution by the same reasoning as in Subsection 14.3-6 in the following form: Φ+(z)=p–(z) q–(z)[Ψ(z)+Pν–1(z)],Φ–(z)=q+(z) p+(z)[Ψ(z)+Pν–1(z)], (33) where Ψ(z)=1 2πi⎝integraldisplay Lq–(τ) p–(τ)H(τ)dτ τ–z,Φ–(∞)=0 . If the index is negative, then we must set Pν–1(z)≡0 and add the solvability conditions ⎝integraldisplay Lq–(τ) p–(τ)H(τ)τk–1dτ=0 , k=1 ,2 , ...,–ν, (34) which agree with the general formula (29), because the canonical function has the form (32). 724 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND Note that for the general case, in the practical solution of the Riemann problem, it can also be convenient to express the coefﬁcient in the form D(t)=p+(t)p–(t) q+(t)q–(t)D1(t), where D1(t) is a function with zero index and the polynomials p±(t)a n d q±(t) are chosen for a given coefﬁcient in a special way. For an appropriate choice of such polynomials, the solution canbe obtained in the simplest possible way. Example 2. Consider the Riemann problem Φ+(t)=t t2–1Φ–(t)+t3–t2+1 t3–t under the assumption that Φ–(∞)=0a n d Lis an arbitrary smooth closed contour of one of the following forms: 1◦. The interior of the contour Lcontains the point z1= 0 and does not contain the points z2=1a n d z3= –1. 2◦. The interior of the contour Lcontains the points z1=0a n d z2= 1 and does not contain the point z3= –1. 3◦. The interior of the contour Lcontains the points z1=0 ,z2=1 ,a n d z3= –1. 4◦. The interior of the contour Lcontains the points z2=1a n d z3= –1 and does not contain the point z1=0 . Consider cases 1◦–4◦in order. In the solution we apply the method of Subsection 14.3-7. 1◦.W e h a v e p+(t)=t,p–(t)=1 , q+(t)=1 , q–(t)=t2–1 ; m+=1 , n+=0 , ν=m+–n+=1 . Let us rewrite the boundary condition in the form (t2–1 )Φ+(t)–tΦ–(t)=1 t(t3–t2+1 ) (t+1 ) . Hence, Ψ(z)=1 2πi⎝integraldisplay Lq–(τ) p–(τ)H(τ)dτ τ–z=1 2πi⎝integraldisplay Lτ3–τ+1 τ–zdτ+1 2πi⎝integraldisplay L1/τ τ–zdτ, and the formulas for the Cauchy integral (see Subsection 14.2-1) imply Ψ+(z)=z3–z+1 ,Ψ–(z)=–1 z. The general solution of the problem contains a single (arbitrary) constant. By formula (33), we obtain Φ+(z)=1 z2–1(z3–z+1+C)=z3–z+1 z2–1+C z2–1,Φ–(z)=1 z⎝parenleftbigg –1 z+C⎝parenrightbigg =–1 z2+C z, where Cis an arbitrary constant. On replacing CbyC– 1 we can rewrite the solution in the form Φ+(z)=z+C z2–1,Φ–(z)=–z+1 z2+C z. 2◦.W e h a v e p+(t)=t,p–(t)=1 , q+(t)=t–1 , q–(t)=t+1 , m+=n+=1 , ν=0 , (t+1 )Φ+(t)–t t–1Φ–(t)=(t+1 ) (t3–t2+1 ) t(t–1 ), Ψ(z)=1 2πi⎝integraldisplay Lτ2+τ τ–zdτ+1 2πi⎝integraldisplay L(τ+1 )/[τ(τ–1 ) ] τ–zdτ=⎧ ⎨ ⎩z2+z forz∈Ω+, –z+1 z(z–1 )forz∈Ω–. The problem has the unique solution Φ+(z)=p–(z) q–(z)Φ+(z)=1 z+1(z2+z)=z, Φ–(z)=q+(z) p+(z)Φ–(z)=z–1 z⎝parenleftbigg –z+1 z(z–1 )⎝parenrightbigg =–z+1 z2. 14.3. R IEMANN BOUNDARY VALUE PROBLEM 725 3◦.W e h a v e p+(t)=t,p–(t)=1 , q+(t)=t2–1 , q–(t)=1 , m+=1 , n+=2 , ν= –1, Ψ(z)=1 2πi⎝integraldisplay Lτ τ–zdτ+1 2πi⎝integraldisplay L1/[τ(τ–1 ) ] τ–zdτ=⎧ ⎨ ⎩z forz∈Ω+, –1 z(z–1 )forz∈Ω–. The solution of the problem exists only under the solvability conditions (34) or, for the case in question, under the single condition ⎝integraldisplay Lq–(τ) p–(τ)H(τ)dτ=0 . On calculating this integral, we obtain ⎝integraldisplay Lτ3–τ2+1 τ2–τdτ=⎝integraldisplay Lτd τ +⎝integraldisplay Ldτ τ–1–⎝integraldisplay Ldτ τ=0+2 πi–2πi=0 . Thus, the solvability condition holds, and the unique solution of the problem is Φ+(z)=z,Φ–(z)=–z+1 z2. 4◦.W e h a v e p+(t)=1 , p–(t)=t,q+(t)=t2–1 , q–(t)=1 , ν=m+–n+=– 2<0 . For the solvability of the problem, the following two conditions are necessary: ⎝integraldisplay Lq–(τ) p–(τ)H(τ)τk–1dτ=0 , k=1 ,2 . On calculating the last integral for k= 1, we obtain ⎝integraldisplay Lτ3–τ2+1 τ(τ2–τ)dτ=⎝integraldisplay L⎝parenleftbigg 1–1 τ–1 τ2+1 τ–1⎝parenrightbigg dτ=2πi≠0. Thus, the solvability condition fails, and hence the problem has no solution. Note that if we formally calculate the function Φ(z), then it has a pole at inﬁnity, and hence cannot be a solution of the problem. 14.3-8. Riemann Problem for a Half-Plane. Let the contour Lbe the real axis. Just as above, the Riemann problem is to ﬁnd two bounded analytic functions Φ+(z)a n dΦ–(z) in the upper and the lower half-plane, re spectively (or a single piecewise analytic function Φ(z) on the plane), whose limit values on the contour satisfy the boundary condition Φ+(x)=D(x)Φ–(x)+H(x), – ∞<x<∞. (35) The given functions D(x)a n dH(x) satisfy the H ¨older condition both at the endpoints and in a neighborhood of the point at inﬁnity on the contour. We also assume that D(x)≠0. The main difference from the above case of a ﬁnite curve is that here the point at inﬁnity and the origin belong to the contour itself, and therefore cannot be taken as exceptional points at which the canonical function can have a nonzero order. Instead of the auxiliary function twhich was used in the above discussion (and has the unit index with respect to L), we use the linear-fractional function on the real axis with the same property: x–i x+i. The argument of this function argx–i x+i=a r g(x–i)2 x2+i=2a r g ( x–i) 726 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND increases by 2 πasxranges over the real axis in the positive direction. Thus, Indx–i x+i=1 . If IndD(x)=ν, then the function ⎝parenleftbiggx–i x+i⎝parenrightbigg–ν D(x) has zero index. Its logarithm is single-valued on the real axis. We construct the canonical function for which the point –i is the exceptional point as follows: X+(z)=eG+(z),X–(z)=⎝parenleftbiggz–i z+i⎝parenrightbigg–ν eG–(z), (36) where G(z)=1 2πi⎝integraldisplay∞ –∞ln⎝bracketleftbigg⎝parenleftbiggτ–i τ+i⎝parenrightbigg–ν D(τ)⎝bracketrightbiggdτ τ–z. Using the limit values of this function, we transform the boundary condition (35) to the form Φ+(x) X+(x)=Φ–(x) X–(x)+H(x) X+(x). Next, introducing the analytic function Ψ(z)=1 2πi⎝integraldisplay∞ –∞H(τ) X+(τ)dτ τ–z, (37) we represent the boundary condition in the form Φ+(x) X+(x)–Ψ+(x)=Φ–(x) X–(x)–Ψ–(x). Note that, in contrast with the case of a ﬁnite contour, here we have Ψ–(∞)≠0i ng e n e r a l . O n applying the theorem on analytic continuation and taking into account the fact that the only possible singularity of the function under consideration is a pole at the point z=–iof order ≤ν(forν>0 ) , on the basis of the generalized Liouville theorem we obtain (see Subsection 14.3-1) Φ+(z) X+(z)–Ψ+(z)=Φ–(z) X–(z)–Ψ–(z)=Pν(z) (z+i)ν,ν≥0, where Pν(z) is a polynomial of degree ≤νwith arbitrary coefﬁcients. This gives the general solution of the problem: Φ(z)=X(z)⎝bracketleftbigg Ψ(z)+Pν(z) (z+i)ν⎝bracketrightbigg Φ(z)=X(z)[Ψ(z)+C]forν≥0, forν<0 ,(38) (39) where Cis an arbitrary constant. For ν< 0, the function X(z) has a pole of order – νat the point z=–i, and therefore for the solvability of the problem we must set C=–Ψ–(–i). For ν< –1, the following conditions must additionally hold: ⎝integraldisplay∞ –∞H(x) X+(x)dx (x+i)k=0 , k=2 ,3 , ...,–ν. (40) Thus, we obtained results similar to those for a ﬁnite contour. 14.3. R IEMANN BOUNDARY VALUE PROBLEM 727 Indeed, for ν≥0, the homogeneous and nonhomogeneous Riemann boundary value problems for the half-plane are unconditionally solvable, and their solution linearly depends on ν+ 1 arbitrary constants. For ν< 0, the homogeneous problem is unsolvable. For ν< 0, the nonhomogeneous problem is uniquely solvable; moreover, in the case ν= –1 the problem is unconditionally solvable, and in the case ν< –1, it is solvable under – ν– 1 solvability conditions (40) only. Let us also discuss the case of solutions vanishing at inﬁnity. On substituting the relation Φ+(∞)=Φ–(∞) = 0 into the boundary condition, we obtain H(∞) = 0. Hence, for a Riemann problem to have a solution that vanishes at inﬁnity, the right-hand side of the boundary conditionmust vanish at inﬁnity. Assume that this condition is satisﬁed. To obtain a solution for the case under consideration, we must replace the expression P ν(z) in (38) by Pν–1(z) and equate the constant C in (39) with zero. Thus, Φ(z)=X(z)⎝bracketleftbigg Ψ(z)+Pν–1(z) (z+i)ν⎝bracketrightbigg . (41) Forν≤0, we must set Pν–1(z)≡0 in this formula. We must add another condition to the solvability conditions (40), namely, Ψ(–i) = 0, and ﬁnally we obtain the following solvability conditions: ⎝integraldisplay∞ –∞H(x) X+(x)dx (x+i)k=0 , k=1 ,2 , ...,–ν. (42) Now, for ν> 0 we have a solution that depends on νarbitrary constants. For ν≤0, a solution is unique, and for ν< 0, a solution exists if and only if – νconditions hold. 14.3-9. Exceptional Cases of the Riemann Problem In the statement of the Riemann boundary value problem it was required that the coefﬁcient D(t) satisﬁes the H ¨older condition (this prevents inﬁnite values of this coefﬁcient) and vanishes nowhere. As can be observed from the solution (the use of ln D(t)), these restrictions are essential. Now we assume that D(t) vanishes or tends to inﬁnity, with an integer order, at some points of the contour. We assume that the contour Lconsists of a single closed curve. Consider the homogeneous problem. We rewrite the boundary condition of the homogeneous Riemann problem in the form Φ+(t)=µ⎝productdisplay k=1(t–αk)mk κ⎝productdisplay j=1(t–βj)pjD1(t)Φ–(t). (43) Hereαk(k=1 ,...,µ)a n dβj(j=1 ,...,κ) are some points of the contour, mkandpjare positive integers, and D1(t) is a function that is everywhere nonzero and satisﬁes the H ¨older condition. The points αkare zeros of the function D(t). The points βjwill be called the poles of this function. The use of the term “pole” is not completely rigorous because the function D(t) is not analytic. We shall use this term for brevity for a point at which a function (not analytic) tends to inﬁnity with some integer order. We write IndD1(t)=ν,κ⎝summationdisplay j=1pj=p,µ⎝summationdisplay k=1mk=m. We seek the solution in the class of functions bounded on the contour. 728 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND LetX(z) be the canonical function of the Riemann problem with coefﬁcient D1(t). Let us substitute the expression D1(t)=X+(t)/X–(t) into (43) and rewrite the boundary condition in the form Φ+(t) X+(t)µ⎝productdisplay k=1(t–αk)mk=Φ–(t) X–(t)κ⎝productdisplay j=1(t–βj)pj. (44) To the last relation we apply the theorem on analytic continuation and the generalized Liouville theorem (see Subsection 14.3-1). The points αkandβjcannot be singular points of the same analytic function because this would contradict the assumption that Φ+(t)o rΦ–(t) be bounded. Hence, the only possible singularity is the point at inﬁnity. The order at inﬁnity of X–(z)i sν, and the order of κ⎝producttext j=1(z–βj)pjis equal to – p. Hence, the order at inﬁnity of the function Φ–(z)/⎝bracketleftbig X–(z)κ⎝producttext j=1(z–βj)pj⎝bracketrightbig is –ν+p.F o rν–p≥0 it follows from the generalized Liouville theorem that Φ+(z) X+(z)µ⎝productdisplay k=1(z–αk)mk=Φ–(z) X–(z)κ⎝productdisplay j=1(z–βj)pj=Pν–p(z), and hence Φ+(z)=X+(z)µ⎝productdisplay k=1(z–αk)mkPν–p(z),Φ–(z)=X–(z)κ⎝productdisplay j=1(z–βj)pjPν–p(z). (45) Ifν–p< 0, then we must set Pν–p(z)≡0, and hence the problem has no solutions. The boundary value problem with coefﬁcient D1(t) is called the reduced problem . The index ν of the reduced problem will be called the index of the original problem. Formulas (45) show that the degree of the occurring polynomial is less by pthan the index νof the problem. Hence, the number of solutions of problem (43) in the class of functions bounded on the contour is independent of the number of zeros of the coefﬁcient and is diminished by the total number of all poles. In particular, if the index is less than the total order of the poles, then the problem isunsolvable. If the problem is solvable, then its solution can be expressed by formulas (45) in which the canonical function X(z) of the reduced problem can be found by formulas (16) and (17) after replacing D(t)b yD 1(t) in these formulas. Under the additional condition Φ–(∞) = 0, the number of solutions is diminished by one, and the degree of the polynomial in (45) must be at most ν–p–1 . Now let us extend the class of solutions by assuming that one of the desired functions Φ+(z) andΦ–(z) can tend to inﬁnity with integral order at some points of the contour, and at the same time another function remains bounded at these points. We can readily see that this assumption implies no modiﬁcations at nonexceptional points. Here the boundedness of one of the functionsautomatically implies the boundedness of the other. This is not the case for the exceptional points. Let us rewrite the boundary condition (43) in the form κ⎝productdisplay j=1(t–βj)pjΦ+(t) X+(t)=µ⎝productdisplay k=1(t–αk)mkΦ–(t) X–(t). (46) Applying the above reasoning and taking into account the fact that the right-hand side has a pole of orderν+mat inﬁnity, we obtain the general solution in the form Φ+(z)=X+(z)µ⎝productdisplay k=1(z–αk)–mkPν+m(z),Φ–(z)=X–(z)κ⎝productdisplay j=1(z–βj)–pjPν+m(z). (47) 14.3. R IEMANN BOUNDARY VALUE PROBLEM 729 Formulas (47) show that in the class of solutions with admissible polar singularity for one of the functions, the number of solutions is greater than that in the class of functions bounded on thecontour (for ν> 0) by the total order of all zeros and poles of the coefﬁcient. We now consider the nonhomogeneous problem. Let us write out the boundary condition in the form Φ +(t)=µ⎝productdisplay k=1(t–αk)mk κ⎝productdisplay j=1(t–βj)pjD1(t)Φ–(t)+H(t). (48) We can readily see that the boundary condition cannot be satisﬁed by ﬁnite functions Φ+(t)a n dΦ–(t) if we assume that H(t) has poles at points that differ from βjor if at these points, the orders of the poles of H(t) exceed pj. Hence, we assume that H(t) can have poles at the points βjonly and that their orders do not exceed pj. To perform the subsequent reasoning, we must also assume that the functions D1(t)a n dκ⎝producttext j=1(t–βj)pjH(t) at the exceptional points are differentiable sufﬁciently many times. Just as in the homogeneous problem, we replace D1(t) by the ratio of the canonical functions X+(t)/X–(t) and rewrite the boundary condition (48) in the form κ⎝productdisplay j=1(t–βj)pjΦ+(t) X+(t)=µ⎝productdisplay k=1(t–αk)mkΦ–(t) X–(t)+κ⎝productdisplay j=1(t–βj)pjH(t) X+(t). (49) On replacing the function deﬁned by the second summand on the right-hand side in (49) by the difference of the boundary values of analytic functions κ⎝productdisplay j=1(t–βj)pjH(t) X+(t)=Ψ+(t)–Ψ–(t), where Ψ(z)=1 2πi⎝integraldisplay Lκ⎝productdisplay j=1(τ–βj)pjH(τ) X+(τ)dτ τ–z, (50) we reduce the boundary condition to the form κ⎝productdisplay j=1(t–βj)pjΦ+(t) X+(t)–Ψ+(t)=µ⎝productdisplay k=1(t–αk)mkΦ–(t) X–(t)–Ψ–(t). On applying the theorem on analytic continuation and the generalized Liouville theorem (see Subsection 14.3-1), we obtain Φ+(z)=X+(z) κ⎝productdisplay j=1(z–βj)pj[Ψ+(z)+Pν+m(z)],Φ–(z)=X–(z) µ⎝productdisplay k=1(z–αk)mk[Ψ–(z)+Pν+m(z)]. (51) In general, the last formulas give solutions that can tend to inﬁnity at the points αkandβk.F o r a solution to be bounded it is necessary that the function Ψ+(z)+Pν+m(z) have zeros of orders pjat the points βjand the function Ψ–(z)+Pν+m(z) have zeros of orders mkat the points αk.T h e s e requirements form m+pconditions for the coefﬁc ients of the polynomial Pν+m(z). If the coefﬁcients 730 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND of the polynomial Pν+m(z) are chosen in accordance with the above conditions, then formulas (51) give a solution of the nonhomogeneous problem (48) in the class of bounded functions. Consider another way of constructing a solution, which is more convenient and based on the construction of a special particular solution. By the canonical function Y(z)of the nonhomogeneous problem we mean a piecewise analytic function that satisﬁes the boundary condition (48), has zero order everywhere in the ﬁnite part of the domain (including the points αkandβj), and has the least possible order at inﬁnity. In the construction of the canonical function, we start from the solution given by formulas (51). Let us construct a polynomial Un(z) that satisﬁes the following conditions: U(i) n(βj)=Ψ+(i)(βj), U(l) n(αk)=Ψ–(l)(αk),i=0 ,1 , ...,pj–1 , l=0 ,1 , ...,mk–1 ,j=1 ,...,κ, k=1 ,...,µ, whereΨ+(i)(βj)a n dΨ–(l)(αk)a r et h ev a l u e so ft h e ith and the lth derivatives at the corresponding points. Thus, Un(z) is the Hermite interpolation polynomial for the functions Ψ(z)=⎝braceleftbigg Ψ+(z) at the points βj, Ψ–(z) at the points αk with interpolation nodes βjandαkof multiplicities pjandmk, respectively (see Subsection 14.3-2). Such a polynomial is uniquely determined, and its degree is at most n=m+p–1 . The canonical function of the nonhomogeneous problem can be expressed via the interpolation polynomial as follows: Y+(z)=X+(z)Ψ+(z)–Un(z) κ⎝productdisplay j=1(z–βj)pj,Y–(z)=X–(z)Ψ–(z)–Un(z) µ⎝productdisplay k=1(z–αk)mk. (52) To construct the general solution of the nonhomogeneous problem (48), we use the fact that this general solution is the sum of a particular solution of the nonhomogeneous problem and of thegeneral solution of the homogeneous problem. Applying formulas (47) and (52), we obtain Φ +(z)=Y+(z)+X+(z)µ⎝productdisplay k=1(z–αk)mkPν–p(z), Φ–(z)=Y–(z)+X–(z)κ⎝productdisplay j=1(z–βj)pjPν–p(z).(53) For the case in which ν–p<0 ,w em u s ts e t Pν–p(z)≡0. Applying formula (52), we readily ﬁnd that the order of Y–(z) at inﬁnity is equal to ν–p+1 . I f ν<p–1 ,t h e n Y–(z) has a pole at inﬁnity, and the canonical function is no longer a solution of the nonhomogeneous problem. However, on subjecting the constant term H(t)t op–ν– 1 conditions, we can increase the order of the functions Y(z) at inﬁnity by p–ν– 1 and thus again make the canonical function Y(z)a solution of the nonhomogeneous problem. Obviously, to this end it is necessary and sufﬁcient that in the expansion of the function Ψ(z)–Un(z) in a neighborhood of the point at inﬁnity, the ﬁrst p–ν– 1 coefﬁcients be zero. This gives just p–ν– 1 solvability conditions of the problem for the case under consideration. Let us clarify the character of these conditions. The expansion of Ψ(z)–Un(z) can be represented in the form Ψ(z)–Un(z)=–anzn–an–1zn–1–···–a0+a–1z–1+a–2z–2+···+a–kz–k+···, 14.3. R IEMANN BOUNDARY VALUE PROBLEM 731 where a0,a1,...,anare the coefﬁcients of the polynomial Un(z), and the a–kare the coefﬁcients of the expansion of the function Ψ(z), which are given by the obvious formula a–k=–1 2πi⎝integraldisplay Lκ⎝productdisplay j=1(τ–βj)pjH(τ)τk–1 X+(τ)dτ. The solvability conditions acquire the form an=an–1=···=an–p+ν+2=0 . If a solution must satisfy the additional condition Φ–(∞)=0 ,t h e n ,f o r ν–p>0 ,i nf o r m u l a s( 5 3 ) we must take the polynomial Pν–p–1(z), and for ν–p<0 ,p–νconditions must be satisﬁed. 14.3-10. Riemann Problem for a Multiply Connected Domain. LetL=L0+L1+···+Lmbe a collection of m+ 1 disjoint contours, and let the interior of the contour L0contain the other contours. By Ω+we denote the ( m+ 1)-connected domain interior forL0and exterior for L1,...,Lm.B yΩ–we denote the complement of Ω++Lin the entire complex plane. To be deﬁnite, we assume that the origin lies in Ω+. The positive direction of the contour Lis that for which the domain Ω+remains to the left, i.e., the contour L0must be traversed counterclockwise and the contours L1,...,Lm,c l o c k w i s e . We ﬁrst note that the jump problem Φ+(t)–Φ–(t)=H(t) is solved by the same formula Φ(z)=1 2πi⎝integraldisplay LH(τ)dτ τ–z as in the case of a simply connected domain. This follows from the Sokhotski–Plemelj formulas, which have the same form for a multiply connected domain as for a simply connected domain. The Riemann problem (homogeneous and nonhomogeneous) can be posed in the same way as for a simply connected domain. We write νk=1 2π[argD(t)]Lk(all contours are passed in the positive direction). By the index of the problem we mean the number ν=m⎝summationdisplay k=0νk. (54) Ifνk(k=1 ,...,m) are zero for the inner contours, then the solution of the problem has just the same form as for a simply connected domain. To reduce the general case to the simple one, we introduce the function m⎝productdisplay k=1(t–zk)νk, where the zkare some points inside the contours Lk(k=1 ,...,m). Taking into account the fact that [arg( t–zk)]Lj=0f o r k≠jand [arg( t–zj)]Lj=– 2π, we obtain 1 2π⎝bracketleftbigg argm⎝productdisplay k=1(t–zk)νk⎝bracketrightbigg Lj=1 2π⎝bracketleftbig arg(t–zj)νj⎝bracketrightbig Lj=–νj,j=1 ,...,m. 732 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND Hence,⎝bracketleftbigg arg⎝parenleftbigg D(t)m⎝productdisplay k=1(t–zk)νk⎝parenrightbigg⎝bracketrightbigg Lj=0 , j=1 ,...,m. Let us calculate the increment of the argument of the function D(t)m⎝producttext k=1(t–zk)νkwith respect to the contour L0: 1 2π⎝bracketleftbigg arg⎝parenleftbigg D(t)m⎝productdisplay k=1(t–zk)νk⎝parenrightbigg⎝bracketrightbigg L0=1 2π⎝bracketleftbig argD(t)⎝bracketrightbig L0+1 2πm⎝summationdisplay k=1[νkarg(t–zk)]L0=ν0+m⎝summationdisplay k=1νk=ν. Since the origin belongs to the domain Ω+, it follows that [argt]Lk=0 , k=1 ,...,m,[ a r g t]L0=2π. Therefore,⎝bracketleftbigg arg⎝parenleftbigg t–νm⎝productdisplay k=1(t–zk)νkD(t)⎝parenrightbigg⎝bracketrightbigg L= 0. (55) 1◦.The Homogeneous Problem. Let us rewrite the boundary condition Φ+(t)=D(t)Φ–(t) (56) in the form Φ+(t)=tν m⎝productdisplay k=1(t–zk)νk⎝parenleftbigg t–νm⎝productdisplay k=1(t–zk)νkD(t)⎝parenrightbigg Φ–(t). (57) The function t–νm⎝producttext k=1(t–zk)νkD(t) has zero index on each of the contours Lk(k=1 ,...,m), and hence it can be expressed as the ratio t–νm⎝productdisplay k=1(t–zk)νkD(t)=eG+(t) eG–(t), (58) where G(z)=1 2πi⎝integraldisplay Lln⎝parenleftbigg τ–νm⎝productdisplay k=1(τ–zk)νkD(τ)⎝parenrightbiggdτ τ–z. (59) The canonical function of the problem is given by the formulas X+(z)=m⎝productdisplay k=1(z–zk)–νkeG+(z),X–(z)=z–νeG–(z). (60) Now the boundary condition (57) can be rewritten in the form Φ+(t) X+(t)=Φ–(t) X–(t). 14.3. R IEMANN BOUNDARY VALUE PROBLEM 733 As usual, by applying the theorem on analytic continuation and the generalized Liouville theorem (see Subsection 14.3-1), we obtain Φ+(z)=m⎝productdisplay k=1(z–zk)–νkeG+(z)Pν(z),Φ–(z)=z–νeG–(z)Pν(z). (61) We can see that this solution differs from the above solution of the problem for a simply connected domain only in that the function Φ+(z) has the factorm⎝producttext k=1(z–zk)–νk. Under the additional condition Φ–(∞) = 0, in formulas (61) we must take the polynomial Pν–1(z). Applying the Sokhotski–Plemelj formulas, we obtain G±(t)=±1 2ln[t–νΠ(t)D(t)] +G(t), where G(t) is the Cauchy principal value of the integral (59) and Π(t)=m⎝productdisplay k=1(t–zk)νk. On passing to the limit as z→tin formulas (60) we obtain X+(t)=⎝radicalBigg D(t) tνΠ(t)eG(t),X–(t)=1 √ tνΠ(t)D(t)eG(t). (62) The sign of the root is determined by the (arbitrary) choice of a branch of the function ln[ t–νΠ(t)D(t)]. 2◦.The Nonhomogeneous Problem. By the same reasoning as above, we represent the boundary condition Φ+(t)=D(t)Φ–(t)+H(t) (63) in the form Φ+(t) X+(t)–Ψ+(t)=Φ–(t) X–(t)–Ψ–(t), whereΨ(z) is deﬁned by the formula Ψ(z)=1 2πi⎝integraldisplay LH(τ) X+(τ)dτ τ–z. This gives the general solution Φ(z)=X(z)[Ψ(z)+Pν(z)] (64) or Φ(z)=X(z)[Ψ(z)+Pν–1(z)], (65) if the solution s atisﬁes the condition Φ–(∞)=0 . Forν< 0, the nonhomogeneous problem is solvable if and only if the following conditions are satisﬁed: ⎝integraldisplay LH(t) X+(t)tk–1dt= 0, (66) where kranges from 1 to –ν – 1 if we seek solutions bounded at inﬁnity and from 1 to – νif we assume that Φ–(∞)=0 . Under conditions (66), the solution can also be found from formulas (64) or (65) by setting Pν≡0. If the external contour L0is absent and the domain Ω+is the plane with holes, then the main difference from the preceding case is that here the zero index with respect to all contours Lk (k=1 ,...,m) is attained by the functionm⎝producttext k=1(t–zk)νkD(t) that does not involve the factor t–ν. Therefore, to obtain a solution to the problem, it sufﬁces to repeat the above reasoning on omitting this factor. 734 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND 14.3-11. Riemann Problem for Open Curves. LetLbe a curve that consists of msimple smooth open curves (arcs) Lk=akbk,k=1 , 2 , ...,m, without common interior or endpoints. Let D(t),H(t) be two functions deﬁned on Land satisfying the H ¨older condition on each arc Lk, and suppose that D(t)≠0f o ra l l t. On different arcs Lkthe functions D(t),H(t) can be deﬁned by the analytical formulas: D(t)=Dk(t),H(t)=Hk(t),t∈Lk,k=1 ,2 , ...,m. The arc Lkis directed from the point akto the point bk. For the points ak,bkwe will use a uniﬁed notation cj, so that the set of all c-points consists of 2m ordered points c1,c2,...,c2m, each coinciding with some akorbk, but their order may be different from that of the sequences akandbk. For instance, one can take {c1;c2;c3;c4;...;c2m–1;c2m}={a1;b1;a2;b2;...;am;bm}, or {c1;c2;...;cm;cm+1;cm+2;...;c2m}={a1;a2;...;am;b1;b2;...;bm}, or some other combination of ak,bk. On each arc Lk=akbk, we ﬁx some continuous branch of the function ln D(t)=l n Dk(t),t∈Lk by the condition 0≤Im lnD(ak)<2π⇐⇒ 0≤argD(ak)<2π. (67) Then lnD(bk)=l nD(ak)+i⎝bracketleftbig argD(t)⎝bracketrightbig Lk=l nD(ak)+i⎝integraldisplay LkdargD(t). (68) Note that a branch ln D(t)o nLkmay be ﬁxed by other conditions. For instance, instead of (67), one can take –π <a r gD(ak)≤πor 0≤argD(bk)<2π, or some other condition. For each point ck, we calculate the numbers αk+iβk=∓lnD(ck) 2πi,k=1 ,2 , ...,2m, (69) where the upper minus corresponds to ckcoinciding with some ajand the lower plus corresponds to ckcoinciding with some bj. The points ck, as well as the corresponding endpoints aj,bjof the curve L,f o rw h i c h αkare integer numbers, are called singular , while the other ckand the corresponding endpoints of the curve Lare called nonsingular . Clearly, ckis a singular point if and only if D(ck) is real and positive. Let us renumber the points c1,c2,...,c2mso that the ﬁrst and the second groups of subscripts would respectively designate nonsingular and singular points. Let c1,c2,...,cn(0≤n≤2m)b ea l l nonsingular endpoints of the curve L. From these points, we choose p(0≤p≤n) points and move them to the ﬁrst pplaces; we may assume these to be c1,c2,...,cp(after renumbering, if necessary). THERIEMANN PROBLEM . Find a function Φ(z) which is analytic on the entire plane outside the curveL, on which it has continuous boundary values Φ+(t),Φ–(t),t∈L\{endpoints }, satisfying the boundary condition (7) or (8), bounded near the nonsingular endpoints c1,c2,...,cp, and admitting integrable singularities near the other nonsingular points, i.e., |Φ(z)|≤Mk |z–ck|λk,Mk= const, λk= const, 0 ≤λk< 1 near ck,k=p+1 ,...,n. 14.3. R IEMANN BOUNDARY VALUE PROBLEM 735 In this case, in contrast to the Riemann problem for closed curves, the boundary condition (7) or (8) should hold only at the points other than the endpoints, near which the sought functionshould have a prescribed behavior. Moreover, the behavior of the sought function is prescribed only near nonsingular endpoints and is left unspeciﬁed near singular endpoints, since any solution of the homogeneous problem (7) near a singular endpoint c kis always bounded and a solution of the nonhomogeneous problem for βk≠0 is bounded and for βk= 0 has a logarithmic singularity. If ck is a nonsingular endpoint, then any solution of the homogeneous Riemann problem always vanishes at the point ck, while a solution of the nonhomogeneous problem will only be bounded. A solution of the above Riemann problem is called a solution of class hporclassh(c1,c2,...,cp) if nonsingular endpoints c1,c2,...,cpare ﬁxed a priori. The classh0consists of all solutions of the problem that admit integrable singularities near all nnonsingular points of the line L.T h i s c l a s s contains all other classes hp,1≤p≤n. The class hnbelongs to all other classes hp,0≤p≤n–1,a nd consists of all solutions of the Riemann problem that are bounded near all nonsingular endpoints of the line L. As in the case of one or several closed curves, let us construct a particular solution X(z)o ft h e homogeneous problem (7), which, in addition, does not vani sh on the entire plane including the edges of the cuts Lk=akbk, except at the endpoints of the arcs near which its behavior is determined by the class hp=h(c1,c2,...,cp). Consider the Cauchy integral Γ(z)=1 2πi⎝integraldisplay LlnD(τ)dτ τ–z=m⎝summationdisplay k=11 2πibk⎝integraldisplay aklnDk(τ)dτ τ–z, (70) where ln Dk(t) are the logarithmic branches ﬁxed above. This integral has a discontinuity on the curveLwith the jump Γ+(t)–Γ–(t)=l n D(t),t∈L\{c1,c2,...,c2m}, and near the endpoints ckadmits the representation Γ(z)=(αk+iβk)l n (z–ck)+Γ∗(z). Here, ln( z–ck) is a branch which is single-valued on the plane with the cut joining the points ck and∞and going along the arc Ljwith an endpoint at ck; the function Γ∗(z) is analytic in a small neighborhood of ckwith the cut along Ljand tends to a certain limit as z→ckalong any path. Therefore, the function X(z)=eΓ(z)2m⎝productdisplay k=1(z–ck)–νk, (71) where νkare integers such that 0<αk–νk<1⇐⇒ νk=[αk],k=1 ,2 , ...,p, –1 <αk–νk<0⇐⇒ νk=1+[ αk],k=p+1 ,...,n, αk–νk=0⇐⇒ νk=αk,k=n+1 ,...,2m(72) ([αk] is the integer part of αk), has all the above properties of a particular solution of the homogeneous problem (7): both functions X(z)a n d1 /X(z) are analytic on the plane with the cut along L,o n which X±(t)≠0,X+(t)=D(t)X–(t),t∈L\{ck},a n d X(z)∼Ak(z–ck)λkasz→ck,λk=αk–νk,k=1 ,2 , ...,n, X(z)∼Akasz→ck,k=n+1 ,...,2m, 736 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND where Ak= const ≠0f o ra l l k,a n d0< λk<1f o rk =1 ,2 , ...,p, and –1 < λk<0f o r k=p+1,...,n. At∞, this function is of the order ν=ν1+ν2+···+ν2m, i.e.,X(z)∼Az–ν,A= const ≠0a sz→∞ . The function X(z) is called a canonical function of a problem of class hp=h(c1,c2,...,cp), and the integer νis called the index of a problem of class hp. With the help of the canonical function X(z), the Riemann problem for open curves is solved along the same lines as in the case of a simply-connected domain. 1◦.The jump problem Φ+(t)–Φ–(t)=H(t),t∈L\{endpoints } in the class of functions vanishing at ∞(independently of the class hp) has a unique solution, Φ(z)=1 2πi⎝integraldisplay LH(τ)dτ τ–z=m⎝summationdisplay k=11 2πibk⎝integraldisplay akHk(τ)dτ τ–z. In order to obtain a solution bounded at ∞or with a pole at ∞of an order ≤µ, one should take the sum of Φ(z) and an arbitrary constant Cor an arbitrary polynomial of degree µ, respectively. 2◦.T h e homogeneous problem (7), with the help of the represen tation (18), is reduced to the construction of a function Φ(z)/X(z) which is analytic on the entire plane and has removable singularities at all the endpoints ck. A solution of this problem that vanishes at ∞and belongs to the class hp=h(c1,c2,...,cp), forν≥0, is given by the formula Φ(z)=Pν(z)X(z), where Pν(z) is an arbitrary polynomial of degree ν.F o rν< 0, the homogeneous problem has no nontrivial solutions. 3◦.T h e nonhomogeneous problem (8), with the help of the transform ation (18), is reduced to the jump problem (22), and its solution of class hp=h(c1,c2,...,cp) decaying at ∞forν≥0i sa g a i n given by formulas (23), (28). For ν< 0, a nontrivial solution of class hpexists and is unique, provided that –ν solvability conditions (29) are satisﬁed; the solution has the form Φ(z)=X(z)Ψ(z), where X(z) is a canonical function of class hp,a n dΨ(z) is the integral (23). In order to obtain a solution of the homogeneous or the nonhomogeneous problem bounded at ∞or with a pole at ∞of order < µ, one should replace νbyν+1o r ν+µ, respectively. Remark. IfX0(z) is a canonical function of the widest class h0,t h e n X(z)=(z–c1)(z–c2)...(z–cp)X0(z) is a canonical function of class hp=h(c1,c2,...,cp). A similar relation holds for canonical functions of any two classes hpandhq. Thus, for the construction of a canonical function of class hp, it sufﬁces to construct a canonical function of any other class hq, in particular, h0. In order to obtain the canonical function Xn(z)o ft h en a r r o w e s tc l a s s hn, one should take νk=[αk]f o ra l l k=1 ,2 , ...,2min (71). This function is bounded near all endpoints of the line L, both singular and nonsingular. In terms of Xn(z), the canonical function of class hp=h(c1,c2,...,cp) is found by the formula X(z)=(z–cp+1)–1...(z–cn)–1Xn(z). 14.3. R IEMANN BOUNDARY VALUE PROBLEM 737 Example 3. Let the line Lconsist of a segment L1=[a;b](a> 0) of the real axis and the segment L2=[ 2πi;3πi]o f the imaginary axis, and let D(t)=⎝braceleftbiggit ift∈L1, etift∈L2. Let us ﬁnd possible classes hpof solutions of the Riemann problem and construct the canonical function in these classes. 1) Let us ﬁx the branches lnD(t)=l n ( it)=l n ( t)+πi 2,t∈[a;b], lnD(t)=l n ( et)=t–2πi,t∈[2πi;3πi], so that the values lnD(a)=l n ( a)+πi 2,l nD(2πi)=0 satisfy condition (67). We have lnD(b)=l n ( b)+πi 2,l nD(3πi)=πi. 2) Taking c1=a,c2=b,c3=2πi,c4=3πi, let us calculate the numbers α1+iβ1=–lnD(a) 2πi=–1 4+ilna 2π,α2+iβ2=lnD(b) 2πi=1 4–ilnb 2π, α3+iβ3=–lnD(2πi) 2πi=0 , α4+iβ4=lnD(3πi) 2πi=1 2. Since α3= 0 is integer and all the other αkare noninteger, the endpoint c3=2πiis singular and the rest of the endpoints c1,c2,c4are nonsingular. In this connection, let is renumber the points ckas follows: c1=a,c2=b,c3=3πi,c4=2πi, and for these we have the new α1=–1 4,α2=1 4,α3=1 2,α4=0 . 3) All possible classes of solutions of the Riemann problem (and therefore, the classes of the canonical function) are determined by the points c1,c2,c3. These classes are the following: h0,h(c1),h(c2),h(c3),h(c1,c2),h(c1,c3),h(c2,c3), h3=h(c1,c2,c3), with h0being the widest class and h3=h(a,b,3πi) the narrowest class. 4) Let us construct the canonical function X0(z)o fc l a s s h0. In view of (72), we have ν1=1+[ α1]=0 , ν2=1+[ α2]=1 , ν3=1+[ α3]=1 , ν4=α4=0 , and by (70) and (71), X0(z)=eΓ(z)(z–b)–1(z–3πi)–1, Γ(z)=1 2πib⎝integraldisplay a⎝parenleftbiglnτ+πi 2⎝parenrightbigdτ τ–z+1 2πi3πi⎝integraldisplay 2πiτ–2πi τ–zdτ, and therefore, X0(z)=√ e(τ–2πi) (z–b)(τ–3πi)2⎝parenleftbiggz–b z–a⎝parenrightbigg1 4⎝parenleftbiggz–3πi z–2πi⎝parenrightbiggz 2πi exp⎝parenleftbigg1 2πib⎝integraldisplay alnτ τ–zdτ⎝parenrightbigg , where we have chosen branches (of the multiple-valued functions involved) which are single-valued on the plane with cuts along the segments L1=[a;b]a n dL2=[ 2πi;3πi], respectively, and take the value 1 at ∞. 5) According to the above remark, the canonical function of class h(a) is obtained from X0(z) by its multiplication by z–a, and the canonical function of class h(b) is obtained by multiplying X0(z)b yz–b, etc. These functions can be found directly with the help of formulas (70)–(72). 4◦.The case of a piecewise constant coefﬁcient of the problem. The canonical function of the Riemann problem for open curves can be found explicitly, provided that its coefﬁcient D(t) takes constant values on the arcs Lk=akbk(the values may be different on different arcs). Let D(t)=Dk,Dk= const ≠0,t∈akbk,k=1 ,2 , ...,m. Then, according to (67)–(69), we have lnD(ak)=l nD(bk)=l nDk,αk+iβk=∓(γk–iδk), γk=argDk 2π,δk=ln|Dk| 2π,0 ≤argDk<2π,(73) 738 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND where the upper sign corresponds to the initial points ak, and the lower sign corresponds to the endpoints bkof the arcs Lk. Thus for each speciﬁc arc Lk, its endpoints ak,bkare singular or nonsingular simultaneously, depending on whether γkis integer or noninteger, and if γkis integer, it must be equal to zero. Therefore, the number of nonsingular endpoints, as well as of singular ones, is always even. Suppose that a1,b1,...,an,bn(0≤n≤m) are nonsingular endpoints and an+1,bn+1,...,am,bm are singular endpoints of the line L. For this distribution of nonsingular and singular endpoints, it might be necessary to renumber the arcs Lk. According to (70)–(72), the canonical function X0(z) of class h0(i.e., for this function, all nonsingular endpoints are inﬁnity points of order < 1, and this function is bounded near all singular endpoints) is given by the formula X0(z)=Xn(z)n⎝productdisplay k=11 z–bk,Xn(z)=n⎝productdisplay k=1⎝parenleftBigz–bk z–ak⎝parenrightbiggγk+iδkm⎝productdisplay k=n+1⎝parenleftbiggz–bk z–ak⎝parenrightbiggiδk , (74) where the multiple-valued functions in the last two products are replaced by their branches that take the value 1 at ∞and are single-valued on the plane along the arcs Lk=akbk.S i n c e 0 ≤γk<1 for all k, it can be seen that Xn(z) is a canonical function of class h(b1,b2,...,bn). This function is bounded near all endpoints bk(both singular and nonsingular); the points a1,a2,...,anare its inﬁnity points of order < 1, and near the points an+1,...,am(if these exist) it is also bounded. In order to obtain a canonical function of class h(c1,c2,...,cp), where ckis the general notation foraj,bj, one should multiply X0(z)b y(z–c1)(z–c2)...(z–cp). Taking different systems ck, one obtains, in particular, the following canonical functions: Xn∗(z)=Xn(z)n⎝productdisplay k=1z–ak z–bk= =m⎝productdisplay k=1⎝parenleftbiggz–ak z–bk⎝parenrightbigg1–γk–iδkn⎝productdisplay k=m+1⎝parenleftbiggz–bk z–ak⎝parenrightbiggiδk ,Xn∗(z)∈h(a1,a2,...,an), X2n(z)=Xn(z)n⎝productdisplay k=1(z–ak),X2n(z)∈h2n,h2n=h(a1,b1,...,an,bn)(75) etc. The last function in (75) is bounded near all endpoints of the curve L. 5◦.The case of a constant coefﬁcient of the problem. Let D(t)=D0,D0= const, t∈akbk,k=1 ,2 , ...,m. Then, for nonreal or negative real D0, formulas (73)–(75) yield Xm(z)=⎝parenleftbiggm⎝productdisplay k=1z–bk z–ak⎝parenrightbiggγ+iδ ,γ=argD0 2π,δ=ln|D0| 2π, 0<a r g D0<2π,Xm(z)∈h(b1,b2,...,bn); Xm∗(z)=⎝parenleftbiggm⎝productdisplay k=1z–ak z–bk⎝parenrightbiggγ+iδ ,Xm∗(z)∈h(a1,a2,...,an); X0(z)=Xm(z)m⎝productdisplay k=11 z–bk,X0(z)∈h0; X2m(z)=Xm(z)m⎝productdisplay k=1(z–ak),X2m(z)∈h2m,h2m=h(a1,b1,...,am,bm).(76) 14.3. R IEMANN BOUNDARY VALUE PROBLEM 739 The function Xm(z) is bounded near all points bk,a n dakare its inﬁnity points of an order < 1. Conversely, the function Xm∗(z) is bounded near ak, and at the points bkhas integrable singularities. For the function X0(z), the endpoints ak,bkare inﬁnity points of integrable character. The function X2m(z) is bounded near all endpoints ak,bk. IfD0≠1 is a real positive number, then all endpoints of the curve Lare singular and different classes hpcannot be deﬁned for the Riemann problem. In this case, there is a single (to within a nonzero constant coefﬁcient) canonical function X(z)=⎝parenleftbiggm⎝productdisplay k=1z–bk z–ak⎝parenrightbiggiδ ,δ=1 2πln|D0|, which is bounded near all endpoints of the curve, although for z→akandz→bkit has no limits. In applications, one often encounters the Riemann problem with Φ+(t)–Φ–(t)=H(t),t∈L, the coefﬁcient D(t)≡–1, and γ=1 2,δ= 0 in (76). In this situation, X0(z)=m⎝productdisplay k=11 √ (z–ak)(z–bk),X0(z)∈h0; Xm(z)=m⎝productdisplay k=1⎝radicalbigg z–bk z–ak,Xm(z)∈h(b1,b2,...,bm); Xm∗(z)=m⎝productdisplay k=1⎝radicalbigg z–ak z–bk,Xm∗(z)∈h(a1,a2,...,am); X2m(z)=m⎝productdisplay k=1⎝radicalbig (z–ak)(z–bk),X2m(z)∈h2m. 14.3-12. Riemann Problem with a Discontinuous Coefﬁcient. LetLbe a smooth closed curve and suppose that the coefﬁcient D(t) of the Riemann problem is continuous on Lexcept at ﬁnitely many points t1,t2,...,tmin which it has jumps of the ﬁrst kind. On each arc Lk=tktk+1,k=1 , 2 , ...,m(it is assumed that tm+1=t1), the functions D(t),H(t) satisfy the H ¨older condition and D(t)≠0f o ra l l t. On an arc Lk, we ﬁx a continuous branch of the logarithmic function ln D(t). This can also be done as in the case of an open curve Lby ﬁxing the values of ln D(t) at the initial points of the arcs: lnD(tk+0 )=|lnD(tk+0 )|+iargD(tk+0 ) , 0≤argD(tk+0 )<2 π,k=1 ,2 , ...,m,(77) where D(tk+ 0) = lim t→tk,t∈LkD(t) is the value of the function at the point tkregarded as the initial point of the arc Lk. Then, at the ﬁnite point tk+1of this arc, we have lnD(tk+1–0 )=l n D(tk+0 )+ i⎝bracketleftbig argD(t)⎝bracketrightbig Lk,k=1 ,2 , ...,m. Let us calculate the numbers γk+iδk=1 2πi⎝bracketleftbig lnD(tk–0 )–l n D(tk+0 )⎝bracketrightbig ,k=1 ,2 , ...,m, (78) 740 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND and determine nonsingular discontinuity points tkfor which γkare noninteger, and singular points tk for which γkare integer. Then we denote the points tkbycj, so that all the nonsingular points cj occupy the ﬁrst places and all singular points appear after these. Moreover, for cj=tk, the point cj is associated with the number αj+iβj=γk+iδk. Letc1,c2,...,cn(0≤n≤m) be all the nonsingular discontinuity points of the coefﬁcient D(t). The Riemann problem consists in ﬁnding two functions Φ+(t)a n d Φ–(t) that are analytic in the interior Ω+and the exterior Ω–of the curve L, respectively, have boundary values Φ+(t),Φ–(t) continuous on Lexcept, possibly, the points t1,t2,...,tm, satisfying the boundary condition (8), bounded near the nonsingular points c1,c2,...,cp(0≤p≤n), having inﬁnity of an order < 1 at the other nonsingular points cp+1,...,cn, and possibly, having logarithmic singularities at the points cn+1,...,cm. Often, the condition of logarithmic singularity of the sought functions is replaced by the more general condition of almost boundedness: lim z→cj|z–cj|εΦ±(z)=0 f o r a n y ε>0 . Similarly to the case of open curves, a canonical function X(z)o fc l a s s hp=h(c1,c2,...,cp) for this problem can be constructed in the form X(z)=eΓ(z)m⎝productdisplay k=1(z–cj)–νj,Γ(z)=1 2πi⎝integraldisplay LlnD(τ)dτ τ–z, where νj=[αj] for the nonsingular points cj,j=1 ,2 , ...,p, that determine the class hp;νj=1+[αj] for the other nonsingular points cj,j=p+1 ,...,n;νj=αjfor singular points cj,j=n+1 ,...,m. In particular, in order to obtain a canoni cal function in the narrowest class hn=h(c1,c2,...,cn) of functions bounded near all discontinuity points (both singular and nonsingular), one should takeνj=[αj],j=1 ,...,m. With the help of the canonical function X(z), the Riemann problem with a discontinuous coefﬁcient is solved in the same way as in Subsections 12.3-4, 12.3-10, and 12.3-11. All the results of these subsections are valid for this problem, provided that one takes into account that the index of the problem is equal to ν=ν1+ν2+···+νm. In the case of a piecewise-constant coefﬁcient D(t)=Dk= const, t∈tktk+1,k=1 ,2 , ...,m, the canonical function of the narrowest class hn, which is bounded at the discontinuity points tk, k=1 ,2 , ...,m,h a st h ef o r m Xn(z)=m⎝productdisplay k=1(z–tk){γk}+iδk, (79) where {γk}=γk–[γk] is the fractional part of γk, which is the real part of the complex number γk+iδk=1 2πi⎝parenleftBig lnDk–1–l nDk⎝parenrightBig , lnDk=l n|Dk|+iargDk,0 ≤argDk<2π,k=1 ,2 , ...,m (it is assumed that D0=Dm).(80) Note that in general the difference of logarithms in (78), (80) cannot be replaced by the logarithm of fraction. For instance, if Dk–1=2a n d Dk=2i, then for the logarithmic branch ﬁxed by the condition 0 ≤argDj<2π,w eh a v e lnDk–1–l nDk=l n2–⎝parenleftbig ln 2 +πi 2⎝parenrightbig =–πi 2and lnDk–1 Dk=l n ( –i)=3πi 2. 14.3. R IEMANN BOUNDARY VALUE PROBLEM 741 Example 4. LetLbe the unit circle t=eiϕ,0≤ϕ≤2π, and let the coefﬁcient of the Riemann problem have the form D(t)=D(eiϕ)=⎧ ⎨ ⎩–i,0 < ϕ<π 2, 1+i,π 2<ϕ<π, –1, π<ϕ<2π. The function D(t) is piecewise constant with discontinuities of the ﬁrst kind at the points t1=ei0=1 ,t2=eiπ/2=i, t3=eiπ= –1. By (78), we ﬁnd that lnD1=l n ( –i)=3πi 2,l nD2=l n ( 1+ i)=l n√ 2+πi 4, lnD3= ln(–1) = πi,l nD0=l nD3=πi, γ1+iδ1=–1 4,γ2+iδ2=5 8+iln 2 4π,γ3+iδ3=–3 8–iln 2 4π. Since all γkare noninteger, all three discontinuity points are nonsingular. Then, according to (79), the canonical func- tionX3(z)o fc l a s s h3=h(1,i, –1) (i.e., the function bounded near all discontinuity points) has the form X3(z)=(z–1 )3/4(z–i)5/8+iδ(z+1 )5/8–iδ,δ=ln 2 4π. In order to obtain the canonical function X0(z)o ft h ew i d e s tc l a s s h0, one should divide X3(z)b y(z–1 ) (z–i)(z+1 ) ; and to obtain the functionX (z)∈h(1), one should divide X3(z)b y(z–i)(z+ 1), etc. 14.3-13. Riemann Problem in the General Case. LetLbe the union of ﬁnitely many smooth closed and open oriented curves with ﬁnitely many common points ( Lis a piecewise smooth line), and let D(t),H(t) be two functions on Lthat satisfy the H ¨older condition everywhere except for ﬁnitely many ﬁrst kind discontinuity points, D(t)≠0 everywhere on L. Denote by tkthe endpoints, the nodes, the angular points of the line L,a n dt h e discontinuity points of the function D(t). On the closed curves belonging to L,w ec h o s ea r b i t r a r y points regarded as the initial and the ending points of these curves and include these points into the set oftk. Lett1,t2,...,tmbe all the above-speciﬁed points of the line L, which is split into ﬁnitely many oriented open arcs LJby these points. On each arc, we ﬁx a certain continuous branch of the logarithmic function ln D(t), so that if tk+0 is the initial point of some arc Lj, then the value ln D(t) at that point is found by the formula (77). Then, at the ending point tl– 0 of that arc, we have lnD(tl–0 )=l n D(tk+0 )+i [argD(t)]Lj. Note that one and the same tkmay happen to be the initial point of some arcs Lj(one or more) and the ending point of other arcs. For each tk, we calculate the number γk+iδk=1 2πi⎝bracketleftBig⎝summationdisplay lnD(tk–0 )–⎝summationdisplay lnD(tk+0 )⎝bracketrightBig ,k=1 ,2 , ...,m, where the ﬁrst sum is over all tk– 0 that are the endpoints of the arcs ending at the point tk,a n dt h e second sum is over all tk+ 0 that are the initial points of the arcs issuing from tk. For instance, if tk is the initial point of the arc L1,L2,...,Lm1, then the second sum has the form ⎝summationdisplay lnD(tk+0 )=m1⎝summationdisplay j=1lnD(tk)⎝vextendsingle⎝vextendsingle⎝vextendsingle tk∈Lj=m1⎝summationdisplay j=1lim t→tk,t∈LkD(t). Further, as in the previous subsection, the condition that γkis integer or noninteger determines singular and nonsingular nodes cjof the line L, after which the Riemann problem is formulated and solved as in Subsections 12.3-11 and 12.3-12. 742 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND Remark. One of the crucial steps when solving the Riemann problem is the construction of a canonical function of a given class. This function can be constructed in a simpler way. Suppose that the line Lis split by the points t1,t2,...,tminto (closed and open) curves L1,L2,...,Ln. For each Lj, we can construct a canonical function Xj(z) of the homogeneous Riemann problem X+ j(t)=D(t)X– j(t),t∈Lj, without taking care of its speciﬁc class. Then the function X0(z)=X1(z)X2(z)...X n(z) satisﬁes the homogeneous boundary condition (7) and near the points tkadmits the representation X0(z)∼Ak(z–tk)λk+iµk,Ak= const ≠0,k=1 ,2 , ...,m, where λk+iµkare certain numbers found on the basis of the behavior of the functions Xj(z)i na neighborhood of tk. Knowing λk, it is easy to determine singular and nonsingular points tkand ﬁnd the canonical function X(z) of a given class. This function has the form X(z)=n⎝productdisplay j=1Xj(z)m⎝productdisplay k=1(z–tk)–ωk, where ωkare integers to be chosen such that X(z) should belong to the given class. Example 5. LetLconsist of the segment L1= [–1; 1] on the real axis and the segment L2=[ 0 ;i] on the imaginary axis, and D(t)=⎧ ⎨ ⎩2i,t∈[–1; 0), 2,t∈(0; 1], –1,t∈(0;i]. Let us construct the canonical function of the homogeneous Riemann problem with the coefﬁcient D(t), requiring that this function is bounded near all endpoints of the line Land near the node t= 0. For the points t1= –1 (the initial point of the segment [–1; 0]), t2= 0 (the ending point of the segment [–1; 0] and the initial point of the segments [0; 1], [0; i]),t3=1( t h e ending point of the segment [0; 1]) , t4=i(the ending point of the segment [0; i]), we ﬁnd the numbers γ1+iδ1=–1 2πiln(2i)=–1 4+iln 2 2π,γ2+iδ2=1 2πi[ln(2i)–l n2–l n ( – 1 ) ]=–1 4, γ3+iδ3=1 2πiln 2 = – iln 2 2π,γ4+iδ4=1 2πiln(–1) =1 2. Sinceγ3= 0, the point t3= 1 is singular, while the rest of tkare nonsingular. The canonical function that is bounded near all endpoints is found by (79) and has the form X3(z)=(z+1 )3/4+iδz3/4(z–1 )–iδ(z–i)1/2,δ=ln 2 2π. The canonical functions of the other classes are obtained from X3(z) by its division by z+1 ,z,z–i, all or some of these, depending on the class. 14.3-14. Hilbert Boundary Value Problem. Let a simple smooth closed contour Land real H ¨older functions a(s),b(s), andc(s) of the arc length s on the contour be given. By the Hilbert boundary value problem we mean the following problem. Find a function f(z)=u(x,y)+iv(x,y) that is analytic on the domain Ω+and continuous on the contour for which the limit values of the real and the imaginary part on the contour satisfy the linear relation a(s)u(s)+b(s)v(s)=c(s). (81) Forc(s)≡0 we obtain the homogeneous problem and, for nonzero c(s), a nonhomogeneous . The Hilbert boundary value problem can be reduced to the Riemann boundary value problem. The methods of this reduction can be found in the references cited at the end of the section. References for Section 14.3: F. D. Gakhov (1977), N. I. Muskhelishvili (1992). 14.4. S INGULAR INTEGRAL EQUATIONS OF THE FIRST KIND 743 14.4. Singular Integral Equations of the First Kind 14.4-1. Simplest Equation with Cauchy Kernel. Consider the singular integral equation of the ﬁrst kind 1 πi⎝integraldisplay Lϕ(τ) τ–tdτ=f(t), (1) where Lis a closed contour. Let us construct the solution. In this relation we replace the variable t byτ1, multiply by1 πidτ1 τ1–t, integrate along the contour L, and change the order of integration according to the Poincar ´e–Bertrand formula (see Subsection 14.2-6). Then we obtain 1 πi⎝integraldisplay Lf(τ1) τ1–tdτ1=ϕ(t)+1 πi⎝integraldisplay Lϕ(τ)dτ1 πi⎝integraldisplay Ldτ1 (τ1–t)(τ–τ1).( 2) Let us calculate the second integral on the right-hand side of (2): ⎝integraldisplay Ldτ1 (τ1–t)(τ–τ1)=1 τ–t⎝parenleftbigg⎝integraldisplay Ldτ1 τ1–t–⎝integraldisplay Ldτ1 τ1–τ⎝parenrightbigg =1 τ–t(iπ–iπ)=0 . Thus, ϕ(t)=1 πi⎝integraldisplay Lf(τ) τ–tdτ.( 3) The last formula gives the solution of the singular integral equation of the ﬁrst kind (1) for a closed contour L. 14.4-2. Equation with Cauchy Kernel on the Real Axis. Consider the following singular integral equation of the ﬁrst kind on the real axis: 1 πi⎝integraldisplay∞ –∞ϕ(t) t–xdt=f(x), – ∞<x<∞.( 4) Equation (4) is a special case of the characteristic integral equation on the real axis (see Subsec- tion 15.2-3). In the class of functions vanishing at inﬁnity, Eq. (4) has the solution ϕ(x)=1 πi⎝integraldisplay∞ –∞f(t) t–xdt,– ∞<x<∞.( 5) Denoting f(x)=F(x)i–1, we rewrite Eqs. (4) and (5) in the form 1 π⎝integraldisplay∞ –∞ϕ(t) t–xdt=F(x),ϕ(x)=–1 π⎝integraldisplay∞ –∞F(t) t–xdt,– ∞<x<∞.( 6 ) The two formulas (6) are called the Hilbert transform pair (see Subsection 9.6-5). 744 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND 14.4-3. Equation of the First Kind on a Finite Interval. Consider the singular integral equation of the ﬁrst kind 1 π⎝integraldisplayb aϕ(t) t–xdt=f(x), a≤x≤b,( 7) on a ﬁnite interval. Its solutions can be constructed by using the theory of the Riemann boundary value problem for a nonclosed contour (see Subsection 14.3-11). Let us present the ﬁnal results. 1◦. A solution that is unbounded at both endpoints: ϕ(x)=–1 π1 √ (x–a)(b–x)⎝parenleftbigg⎝integraldisplayb a√ (t–a)(b–t) t–xf(t)dt+C⎝parenrightbigg ,( 8) where Cis an arbitrary constant and⎝integraldisplayb aϕ(t)dt=C.( 9 ) 2◦. A solution bounded at the endpoint aand unbounded at the endpoint b: ϕ(x)=–1 π⎝radicalbigg x–a b–x⎝integraldisplayb a⎝radicalbigg b–t t–af(t) t–xdt. (10) 3◦. A solution bounded at both endpoints: ϕ(x)=–1 π⎝radicalbig (x–a)(b–x)⎝integraldisplayb af(t) √ (t–a)(b–t)dt t–x, (11) under the condition that⎝integraldisplayb af(t)dt √ (t–a)(b–t)= 0. (12) Solutions that have a singularity point sinside the interval [ a,b] can also be constructed. These solutions have the following form: 4◦. A singular solution that is unbounded at both endpoints: ϕ(x)=–1 π1 √ (x–a)(b–x)⎝parenleftbigg⎝integraldisplayb a√ (t–a)(b–t) t–xf(t)dt+C1+C2 x–s⎝parenrightbigg , (13) where C1andC2are arbitrary constants. 5◦. A singular solution bounded at one endpoint: ϕ(x)=–1 π⎝radicalbig (x–a)(b–x)⎝parenleftbigg⎝integraldisplayb a⎝radicalbigg b–t t–af(t) t–xdt+C x–s⎝parenrightbigg , (14) where Cis an arbitrary constant. 6◦. A singular solution bounded at both endpoints: ϕ(x)=–1 π⎝radicalbig (x–a)(b–x)⎝parenleftbigg⎝integraldisplayb af(t) √ (t–a)(b–t)dt t–x+A x–s⎝parenrightbigg ,A=⎝integraldisplay1 –1f(t)dt √ (t–a)(b–t). (15) 14.4. S INGULAR INTEGRAL EQUATIONS OF THE FIRST KIND 745 14.4-4. General Equation of the First Kind with Cauchy Kernel. Consider the general equation of the ﬁrst kind with Cauchy kernel 1 πi⎝integraldisplay LM(t,τ) τ–tϕ(τ)dτ=f(t), (16) where the integral is understood in the sense of the Cauchy principal value and is taken over a closed or nonclosed contour L. As usual, the functions a(t),f(t), and M(t,τ)o nLare assumed to satisfy the H ¨older condition, where the last function satisﬁes th is condition with resp ect to both variables. We perform the following manipulation with the kernel: M(t,τ) τ–t=M(t,τ)–M(t,t) τ–t+M(t,t) τ–t and write M(t,t)=b(t),1 πiM(t,τ)–M(t,t) τ–t=K(t,τ). (17) We can rewrite Eq. (16) in the form b(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ+⎝integraldisplay LK(t,τ)ϕ(τ)dτ=f(t). (18) It follows from formulas (17) that the function b(t) satisﬁes the H ¨older condition on the entire contour LandK(t,τ) satisﬁes this condition everywh ere except for the points with τ=tat which this function satisﬁes the estimate |K(t,τ)|<A |τ–t|λ,0 ≤λ<1 . The general singular integral equation of the ﬁrst kind with Cauchy kernel is frequently written in the form (18). The general singular integral equation of the ﬁrst kind is a special case o f the complete singular integral equation whose theory is treated in Chapter 15. In general, it cannot be solved in a closed form. However, there are some cases in wh ich such a solutio n is possible. Let the function M(t,τ) in Eq. (16), which satisﬁes the H ¨older condition with respect to both variables on the smooth closed contour Lby assumption, have an analytic continuation to the domain Ω+with respect to each of the variables. If M(t,t)≡1, then the solution of Eq. (16) can be obtained by means of the Poincar ´e–Bertrand formula (see Subsection 14.2-6). This solution is given by the relation ϕ(t)=1 πi⎝integraldisplay LM(t,τ) τ–tf(τ)dτ. (19) Equation (16) can be solved without the assumption that the function M(t,τ) satisﬁes the condition M(t,t)≡1. Namely, assume that the function M(t,τ) has the analytic continuation to Ω+ with respect to each of the variables and that M(z,z)≠0f o rz∈ Ω+. In this case, the solution of Eq. (16) has the form ϕ(t)=1 πi1 M(t,t)⎝integraldisplay LM(t,τ) M(τ,τ)f(τ) τ–tdτ. (20) In Section 14.5, a numerical method for solving a special case of the general equation of the ﬁrst kind is given, which is of independent interest from the viewpoint of applications. Remark 1. The solutions of complete singular integral equations that are constructed in Sub- section 14.4-4 can also be applied for the case in which the contour Lis a collection of ﬁnitely many disjoint smooth closed contours. 746 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND 14.4-5. Equations of the First Kind with Hilbert Kernel. 1◦. Consider the simplest singular integral equation of the ﬁrst kind with Hilbert kernel 1 2π⎝integraldisplay2π 0cot⎝parenleftbiggξ–x 2⎝parenrightbigg ϕ(ξ)dξ=f(x), 0 ≤x≤2π, (21) under the additional assumption⎝integraldisplay2π 0ϕ(x)dx= 0. (22) Equation (21) can have a solution only if a solvability condition is satisﬁed. This condition is obtained by integrating Eq. (21) with respect to xfrom zero to 2 πand, with regard for the relation ⎝integraldisplay2π 0cot⎝parenleftbiggξ–x 2⎝parenrightbigg dx=0 , becomes⎝integraldisplay2π 0f(x)dx= 0. (23) To construct a solution of Eq. (21), we apply the solution of the simplest singular integral equation of the ﬁrst kind with Cauchy kernel by assuming that the contour Lis the circle of unit radius centered at the origin (see Subsection 14.4-1). We rewrite the equation with Cauchy kernel and its solution in the form 1 π⎝integraldisplay Lϕ1(τ) τ–tdτ=f1(t), (24) ϕ1(t)=–1 π⎝integraldisplay Lf1(τ) τ–tdτ, (25) which is obtained by substituting the function ϕ1(t) instead of ϕ(t) and the function f1(t)i–1instead off(t) into the relations of 14.4-1. We set t=eixandτ=eiξand ﬁnd the relationship between the Cauchy kernel and the Hilbert kernel: dτ τ–t=1 2cot⎝parenleftbiggξ–x 2⎝parenrightbigg dξ+i 2dξ. (26) On substituting relation (26) into Eq. (24) and into solution (25), with regard to the change of variables ϕ(x)=ϕ1(t)a n d f(x)=f1(t) we obtain 1 2π⎝integraldisplay2π 0cot⎝parenleftbiggξ–x 2⎝parenrightbigg ϕ(ξ)dξ+i 2π⎝integraldisplay2π 0ϕ(ξ)dξ=f(x), (27) ϕ(x)=–1 2π⎝integraldisplay2π 0cot⎝parenleftbiggξ–x 2⎝parenrightbigg f(ξ)dξ–i 2π⎝integraldisplay2π 0f(ξ)dξ. (28) Equation (21), under the additional assumption (22), coincides with Eq. (27), and hence its solution is given by the expression (28). Taking into account the solvability conditions (23), on thebasis of (28) we rewrite a solution of Eq. (21) in the form ϕ(x)=–1 2π⎝integraldisplay2π 0cot⎝parenleftbiggξ–x 2⎝parenrightbigg f(ξ)dξ. (29) Formulas (21) and (29), together with conditions (22) and (23), are called the Hilbert inversion formula . 14.5. M ULTHOPP –KALANDIYA METHOD 747 Remark 2. Equation (21) is a special case of the characteristic singular integral equation with Hilbert kernel (see Subsections 15.1-2 and 15.2-5). 2◦. Consider the general singular integral equation of the ﬁrst kind with Hilbert kernel 1 2π⎝integraldisplay2π 0N(x,ξ)c o t⎝parenleftbiggξ–x 2⎝parenrightbigg ϕ(ξ)dξ=f(x). (30) Let us represent its kernel in the form N(x,ξ)c o tξ–x 2=⎝bracketleftbig N(x,ξ)–N(x,x)⎝bracketrightbig cotξ–x 2+N(x,x)c o tξ–x 2. We introduce the notation N(x,x)=–b(x),1 2π⎝bracketleftbig N(x,ξ)–N(x,x)⎝bracketrightbig cotξ–x 2=K(x,ξ), (31) and rewrite Eq. (30) as follows: –b(x) 2π⎝integraldisplay2π 0cot⎝parenleftbiggξ–x 2⎝parenrightbigg ϕ(ξ)dξ+⎝integraldisplay2π 0K(x,ξ)ϕ(ξ)dξ=f(x). (32) It follows from formulas (31) that the function b(x) satisﬁes the H ¨older condition, whereas the kernel K(x,ξ) satisﬁes the H ¨older condition everywhere except possibly for the points x=ξ,a t which the following estimate holds: |K(x,ξ)|<A |ξ–x|λ,A= const < ∞,0 ≤λ<1 . The general singular integral equation of the ﬁrst kind with Hilbert kernel is frequently written in the form (32). It is a special case of the complet e singular integral equation with Hilbert kernel, which is treated in Subsections 15.1-2 and 15.4-8. References for Section 14.4: F. D. Gakhov (1977), F. D. Gakhov and Yu. I. Cherskii (1978), S. G. Mikhlin and S. Pr ¨ossdorf (1986), N. I. Muskhelishvili (1992), I. K. Lifanov (1996). 14.5. Multhopp–Kalandiya Method Consider a general singular integral equation of the ﬁrst kind with Cauchy kernel on the ﬁnite interval [–1, 1] of the form 1 π⎝integraldisplay1 –1ϕ(t)dt t–x+1 π⎝integraldisplay1 –1K(x,t)ϕ(t)dt=f(x). (1) This equation frequently occurs in applications, especially in aerodynamics and 2D elasticity. We present here a method of approximate solution of Eq. (1) under the assumption that this equation has a solution in the classes indicated below. 14.5-1. Solution That is Unbounded at the Endpoints of the Interval. According to the general theory of singular integral equations (e.g., see N. I. Muskhelishvili (1992)), such a solution can be represented in the form ϕ(x)=ψ(x) √ 1–x2,( 2) 748 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND where ψ(x) is a bounded function on [–1, 1]. Let us substitute the expression (2) into Eq. (1) and introduce new variables θandτby the relations x=c o sθandt=c o sτ,0≤θ≤π,0≤τ≤π.I n this case, Eq. (1) becomes 1 π⎝integraldisplayπ 0ψ(cosτ)dτ cosτ–c o sθ+1 π⎝integraldisplayπ 0K(cosθ,c o sτ)ψ(cosτ)dτ=f(cosx). (3) Let us construct the Lagrange interpolation polynomial for the desired function ψ(x) with the Chebyshev nodes xm=c o sθm,θm=2m–1 2nπ,m=1 ,...,n. This polynomial is known to have the form Ln(ψ;c o sθ)=1 nn⎝summationdisplay l=1(–1)l+1ψ(cosθl)cosnθsinθl cosθ–c o sθl.( 4) Note that for each lthe fraction on the right-hand side in (4) is an even trigonometric polynomial of degree ≤n– 1. We deﬁne the coefﬁcients of this polynomial by means of the known relations 1 π⎝integraldisplayπ 0cosnτ dτ cosτ–c o sθ=sinnθ sinθ,0 ≤θ≤π,n=0 ,1 ,2 ,... (5) and rewrite (4) in the form Ln(ψ;c o sθ)=2 nn⎝summationdisplay l=1ψ(cosθl)n–1⎝summationdisplay m=0cosmθlcosmθ–1 nn⎝summationdisplay l=1ψ(cosθl). (6) On the basis of the above two relations we write out the following quadrature formula for the singular integral: 1 π⎝integraldisplay1 –1ϕ(t)dt t–x=2 nsinθn⎝summationdisplay l=1ψ(cosθl)n–1⎝summationdisplay m=1cosmθlsinmθ.( 7 ) This formula is exact for the case in which ψ(t) is a polynomial of order ≤n–1i nt. To the second integral on the left-hand side of Eq. (1), we apply the formula 1 π⎝integraldisplay1 –1P(x)dx √ 1–x2=1 nn⎝summationdisplay l=1P(cosθl), (8) which holds for any polynomial P(x)o fd e g r e e ≤2n– 1. In this case, by (8) we have 1 π⎝integraldisplay1 –1K(x,t)ϕ(t)dt=1 nn⎝summationdisplay l=1K(cosθ,c o sθl)ψ(cosθl). (9) On substituting relations (7) and (9) into Eq. (1), we obtain 2 nsinθn⎝summationdisplay l=1ψ(cosθl)n–1⎝summationdisplay m=1cosmθlsinmθ+1 nn⎝summationdisplay l=1K(cosθ,c o sθl)ψ(cosθl)=f(cosθ). (10) 14.5. M ULTHOPP –KALANDIYA METHOD 749 By setting θ=θk(k=1 ,...,n) and with regard to the formula n–1⎝summationdisplay m=1cosmθlsinmθk=1 2cotθk±θl 2, (11) where the sign “plus” is taken for the case in which |k–l|is even and “minus” if |k–l|odd, we obtain the following system of linear algebraic equations for the approximate values ψlof the desired function ψ(x) at the nodes: n⎝summationdisplay l=1aklψl=fk,fk=f(cosθk),k=1 ,...,n, akl=1 n⎝bracketleftbigg1 sinθkcotθk±θl 2+K(cosθk,c o sθl)⎝bracketrightbigg .(12) After solving the system (12), the corresponding approximate solution to Eq. (1) can be found by formulas (2) and (4). 14.5-2. Solution Bounded at One Endpoint of the Interval. In this case we set ϕ(x)=⎝radicalbigg 1–x 1+xζ(x), (13) where ζ(x) is a bounded function on [–1, 1]. We take the same interpolation nodes as in Subsection 14.5-1, replace ζ(x) by the polynomial Ln(ζ;c o sθ)=1 nn⎝summationdisplay l=1(–1)l+1ζ(cosθl)cosnθsinθl cosθ–c o sθl, (14) and substitute the result into the singular integral that enters the expression (1). Just as above, we obtain the following quadrature formula: 1 π⎝integraldisplay1 –1ϕ(t)dt t–x=21–c o s θ nsinθn⎝summationdisplay l=1ζ(cosθl)n–1⎝summationdisplay m=1cosmθlsinmθ–1 nn⎝summationdisplay l=1ζ(cosθl). (15) This formula is exact for the case in which ζ(t) is a polynomial of order ≤n–1i n t. The formula for the second summand on the left-hand side of the equation becomes 1 π⎝integraldisplay1 –1K(x,t)ϕ(t)dt=1 nn⎝summationdisplay l=1( 1–c o s θl)K(cosθ,c o sθl)ζ(cosθl). (16) This formula is exact if the integrand is a polynomial in tof degree ≤2n–2 . On substituting relations (15) and (16) into Eq. (1) and on setting θ=θk(k=1 ,...,n), with regard to formula (11), we obtain a system of linear algebraic equations for the approximate values ζl of the desired function ζ(x) at the nodes: n⎝summationdisplay l=1bklζl=fk,fk=f(cosθk),k=1 ,...,n, bkl=1 n⎝bracketleftbigg tanθk 2cotθk±θl 2–1+2s i n2θl 2K(cosθk,c o sθl)⎝bracketrightbigg .(17) After solving system (17), the corresponding approximate solution to Eq. (1) can be found by formulas (13) and (14). 750 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND 14.5-3. Solution Bounded at Both Endpoints of the Interval. A solution of Eq. (1) that is bounded at the endpoints of the interval vanishes at the endpoints, ϕ(1) =ϕ(–1) = 0. (18) Let us approximate the function ϕ(x) by an even trigonometric polynomial of θconstructed for the interpolation nodes that are the roots of the corresponding Chebyshev polynomial of the second kind : xk=c o sθk,θk=kπ n+1,k=1 ,...,n. (19) This polynomial has the form Mn(ϕ;c o sθ)=2 n+1n⎝summationdisplay l=1ϕ(cosθl)n⎝summationdisplay m=1sinmθlsinmθ. (20) We thus obtain the following quadrature formula: 1 π⎝integraldisplay1 –1ϕ(t)dt t–x=–2 n+1n⎝summationdisplay l=1ϕ(cosθl)n⎝summationdisplay m=1sinmθlcosmθ. (21) This formula holds for any odd trigonometric polynomial ϕ(x)o fd e g r e e ≤n. To the regular integral in Eq. (1) we apply the formula ⎝integraldisplay1 –1√ 1–x2P(x)dx=π n+1n⎝summationdisplay l=1sin2θlP(cosθl), (22) whose accuracy coincides with that of formula (8). On the basis of (22), we have 1 π⎝integraldisplay1 –1K(x,t)ϕ(t)dt=1 n+1n⎝summationdisplay l=1sinθlK(cosθ,c o sθl)ϕ(cosθl). (23) On substituting relations (21) and (23) into Eq. (1) and on setting θ=θk(k=1 ,...,n), we obtain a system of linear algebraic equations in the form n⎝summationdisplay l=1cklϕl=fk,k=1 ,...,n, ckl=sinθl n+1⎝bracketleftbigg2εkl cosθl–c o sθk+K(cosθk,c o sθl)⎝bracketrightbigg ,εkl=⎝braceleftbigg0f o r e v e n |k–l|, 1 for odd |k–l|,(24) where fk=f(cosθk)a n dϕlare approximate values of the unknown function ϕ(x) at the nodes. After solving system (24), the corresponding approximate solution is deﬁned by formula (20). When solving a singular integral equation by the Multhopp–Kalandiya method, it is important that the desired solutions have a representation ϕ(x)=( 1– x)α(1 +x)βχ(x), (25) where α=±1 2,β=±1 2,a n dχ(x) is a bounded function on the interval with well-deﬁned values at the endpoints. If the representation (25) holds, then the method can be applied to the complete singular integral equation, which is treated in Chapter 15. In the literature cited below, some other methods of numerical solution of singular integral equations are discussed as well. References for Section 14.5: A. I. Kalandiya (1973), N. I. Muskhelishvili (1992), S. M. Belotserkovskii and I. K. Lifanov (1993), and I. K. Lifanov (1996). 14.6. H YPERSINGULAR INTEGRAL EQUATIONS 751 14.6. Hypersingular Integral Equations 14.6-1. Hypersingular Integral Equations with Cauchy- and Hilbert-Type Kernels. The simplest hypersingular integral equation of the ﬁrst kind with Cauchy-type kernel on a ﬁnite interval has the form 1 π⎝integraldisplayb aϕ(t) (x–t)2dt=f/prime x(x), a≤x≤b,( 1 ) where ϕ(t) is the unknown function,1 (x–t)2isCauchy-type kernel ,f/prime x(x)i saf u n c t i o nc a l l e dt h e free term or the right-hand side of equation (1). The integral on the left-hand side exists only in the sense of Hadamard principal value (see Subsection 14.6-2). The general hypersingular equation of the ﬁrst kind with Cauchy-type kernel on a ﬁnite interval has the form 1 π⎝integraldisplayb aϕ(t) (x–t)2dt+1 π⎝integraldisplayb aK/prime x(x,t)ϕ(t)dt=f/prime x(x), a≤x≤b.( 2) Assume that the functions ϕ(x),f(x) in equations (1), (2) are differentiable and K(x,t)i s differentiable in both variables everywhere except at the points x=t, near which it satisﬁes the estimate |K(x,t)|≤A |x–t|λ,A= const < ∞,0 ≤λ<1 . Remark 1. The notation in (1) and (2) is meant to emphasize the fact that these equations are obtained from equation (3) of Subsection 14.1-1 and equation (1) of Section 14.5 by theirdifferentiation in x. The simplest hypersingular equation of the ﬁrst kind with Hilbert-type kernel has the form 1 4π⎝integraldisplay2π 0⎝bracketleftBig sin⎝parenleftBigξ–x 2⎝parenrightBig⎝bracketrightBig–2 ϕ(ξ)dξ=f/prime x(x), 0 ≤x≤2π,( 3 ) where ϕ(x) is the unknown function, 1 /sin2⎝bracketleftbig1 2(ξ–x)⎝bracketrightbig isHilbert-type kernel ,f(x)i sag i v e n right-hand side of the equation. The general hypersingular equation of the ﬁrst kind with Hilbert-type kernel has the form 1 4π⎝integraldisplay2π 0⎝bracketleftBig sin⎝parenleftBigξ–x 2⎝parenrightBig⎝bracketrightBig–2 ϕ(ξ)dξ+1 4π⎝integraldisplayb aK/prime x(x,ξ)ϕ(ξ)dξ=f/prime x(x), 0 ≤x≤2π,( 4 ) where ϕ(x),f(x), and K(x,t) are functions with the properties speciﬁed above. If the right-hand sides of equations (1)–(4) are identically equal to zero, the equations are called homogeneous ; otherwise, they are called nonhomogeneous . Remark 2. Note that there is a relation between hypersingular integral equations (3), (4) and singular integral equations (5) from Subsection 14.1-2 and (32) from Subsection 14.4-5: the latterare obtained from the former by the integration in ξ. 14.6-2. Deﬁnition of Hypersingular Integrals. Hypersingular integrals in equations (1)–(4) exist neither in the sense of improper integrals nor in the sense of the Cauchy principal value. Takin g as an example hypersingular integrals with the Cauchy-type kernel, consider some deﬁnitions of such integrals. 1◦. Hypersingular integral as the derivative of an inte gral in the sense of the Cauchy principal value: ⎝integraldisplayb aϕ(t) (x–t)2dt=d dx⎝integraldisplayb aϕ(t) t–xdt.( 5) 752 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND 2◦. Hypersingular integral in the sense of Hadamard principal value : ⎝integraldisplayb aϕ(t) (x–t)2dt= lim ε→+0⎝bracketleftbigg⎝parenleftbigg⎝integraldisplayx–ε α+⎝integraldisplayb x+ε⎝parenrightbiggϕ(t)dt (x–t)2–2ϕ(x) ε⎝bracketrightbigg .( 6) 3◦. Hypersingular integral as an analytic continuation of the integral ⎝integraldisplayb a|x–t|αϕ(t)dt (7) understood in the sense of distributions, where α= –2. Example. Let us calculate the values of hypersingular integrals using formulas (5)–(7) for ϕ(x)≡1. 1◦. Using the ﬁrst deﬁnition, we have ⎝integraldisplayb adt (x–t)2=d dx⎝integraldisplayb adt t–x=d dxln⎝parenleftbiggb–x x–a⎝parenrightbigg =a–b (x–a)(b–x). 2◦. For the Hadamard principal value, we have ⎝integraldisplayb adt (x–t)2= lim ε→+0⎝bracketleftBig⎝parenleftBig –1 x–a+1 ε+1 ε+1 x–b⎝parenrightBig –2 ε⎝bracketrightBig =a–b (x–a)(b–x). 3◦. Using formula (3), for Re( α) > –1 we get ⎝integraldisplayb a|x–t|αdt=⎝integraldisplayx a(x–t)αdt+⎝integraldisplayb x(t–x)αdt=(x–a)α+1 α+1+(b–x)α+1 α+1. Analytic continuation of this function from the half-plane Re( α)>– 1f o r α= –2 yields ⎝integraldisplayb adt (x–t)2= lim α→–2⎝bracketleftbigg(x–a)α+1 α+1+(b–x)α+1 α+1⎝bracketrightbigg =a–b (x–a)(b–x). For a differentiable function ϕ(x) on the segment ( a,b) the above three deﬁnitions of hypersin- gular integrals are equivalent. The expression ⎝parenleftbigg⎝integraldisplayx–ε α+⎝integraldisplayb x+ε⎝parenrightbiggϕ(t)–ϕ(x)–ϕ/prime x(x)(t–x) (x–t)2dt+ϕ(x)a–b (x–a)(b–x)+ϕ/prime x(x)l nb–x x–a, which is equivalent to the right-hand side of (2), shows that for a differentiable ϕ(x),x∈(a,b), a ﬁnite value of the hypersingular integral⎝integraltextb aϕ(t)(x–t)–2dtexists always, since this expression has a ﬁnite limit as ε→+0. Remark 3. Hypersingular integrals with the Hilbert -type kernel can be deﬁned by analogy with the above deﬁnitions in the case of integrals with the Cauchy-type kernel. Note also that equation (26) of Section 14.4 establishes a relation between the Cauchy and the Hilbert kernels. Remark 4. From deﬁnition of hypersingular integral in the sense of Hadamard principal value (6) we can see that⎝integraldisplayb aϕ(t) (x–t)2dt=ϕ(a) a–x–ϕ(b) b–x+⎝integraldisplayb aϕ/prime t(t)dt t–x, which means that the right-hand side of this equation can be understood as a result of formal integration by parts. 14.6. H YPERSINGULAR INTEGRAL EQUATIONS 753 14.6-3. Exact Solution of the Simplest Hypersingular Equation with Cauchy-Type Kernel. Consider the simplest hypersingular equation of the ﬁrst kind with Cauchy-type kernel on a ﬁnite interval 1 π⎝integraldisplayb aϕ(t) (x–t)2dt=f/prime x(x), a≤x≤b,( 8 ) where ϕ(a)=ϕ(b) = 0. Let us construct its solution by two methods. 1◦. According to deﬁnition (5) from Subsection 14.6-2, this simplest equation can be written in the form 1 πd dx⎝integraldisplayb aϕ(t)dt t–x=f/prime x(x), a≤x≤b. Integrating the last equation with respect to x, we obtain 1 π⎝integraldisplayb aϕ(t)dt t–x=f(x)+C,a≤x≤b,( 9) where Cis an arbitrary constant. A bounded solution of equation (9) has been obtained in Subsec- tion 14.4-3. This solution has the form ϕ(x)=–1 π⎝radicalbig (x–a)(b–x)⎝integraldisplayb af(t) √ (t–a)(b–t)dt t–x,C=1 π⎝integraldisplayb af(t) √ (t–a)(b–t)dt. (10) 2◦. Integrating by parts equation (8)(see Remark 4), we get 1 π⎝bracketleftBigϕ(a) a–x–ϕ(b) b–x+⎝integraldisplayb aϕ/prime t(t)dt t–x⎝bracketrightBig =f/prime x(x). Using this relation and the conditions ϕ(a)=ϕ(b) = 0, we ﬁnally come to the equation 1 π⎝integraldisplayb aϕ/prime t(t)dt t–x=f/prime x(x). (11) Consider the solution of equation (11) given in Subsection 14.4-3: ϕ/prime x(x)=–1 π1 √ (x–a)(b–x)⎝integraldisplayb a√ (t–a)(b–t) t–xf/prime t(t)dt,⎝integraldisplayb aϕ/prime t(t)dt= 0. (12) Integrating (12) from atox,w eg e t ϕ(x)=–1 π⎝integraldisplayx a1 √ (τ–a)(b–τ)⎝integraldisplayb a√ (t–a)(b–t) t–τf/prime t(t)dt dτ . Hence, changing the order of integration, we obtain ϕ(x)=1 π⎝integraldisplayb a⎝integraldisplayx a1 √ (τ–a)(b–τ)dτ τ–tf/prime t(t)⎝radicalbig (t–a)(b–t)dt. (13) The internal integral in (6) can be calculated by the formula⎝integraldisplay1 √ (τ–a)(b–τ)dτ τ–t =–1 2√ (t–a)(b–t)ln1 2(a+b)(t+τ)–ab–tτ+√ (τ–a)(b–τ)(t–a)(b–t) 1 2(a+b)(t+τ)–ab–tτ–√ (τ–a)(b–τ)(t–a)(b–t) =1 √ (t–a)(b–t)ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle√ (b–t)(τ–a)–√ (b–τ)(t–a) √ (b–t)(τ–a)+√ (b–τ)(t–a)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle. Thus, a solution of the simplest hypersingular equation with Cauchy-type kernel (8) can be obtained in the form ϕ(x)=1 π⎝integraldisplayb aln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle√ (b–t)(x–a)–√ (b–x)(t–a) √ (b–t)(x–a)+√ (b–x)(t–a)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglef /prime t(t)dt, (14) which, in contrast to (10), contains no singular integrals. 754 METHODS FOR SOLVING SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND 14.6-4. Exact Solution of the Simplest Hypersingular Equation with Hilbert-Type Kernel. Consider the simplest hypersingular integral equation of the ﬁrst kind with Hilbert-type kernel on the ﬁnite interval 1 4π⎝integraldisplay2π 0⎝bracketleftBig sin⎝parenleftBigξ–x 2⎝parenrightBig⎝bracketrightBig–2 ϕ(ξ)dξ=f/prime x(x), 0 ≤x≤2π (15) with the periodic conditions ϕ(0) =ϕ(2π). Let us construct its solution by two methods. 1◦. According to deﬁnition (1) from Subsection 14.2-1, this equation can be written in the form 1 2πd dx⎝integraldisplay2π 0cot⎝parenleftBigξ–x 2⎝parenrightBig ϕ(ξ)dξ=f/prime x(x), 0 ≤x≤2π. (16) Integrating (16), we reduce the problem of ﬁnding a solution of the hypersingular equation under consideration to that of ﬁnding a solution of the following singular integral equation with the Hilbert kernel: 1 2π⎝integraldisplay2π 0cot⎝parenleftBigξ–x 2⎝parenrightBig ϕ(ξ)dξ=f(x)+C,0 ≤x≤2π, (17) where Cis an arbitrary constant. This equation is considered in Subsection 14.4-5. 2◦. Integrating (15) by parts, we obtain 1 2π⎝integraldisplay2π 0cot⎝parenleftBigξ–x 2⎝parenrightBig ϕ/prime ξ(ξ)dξ=f/prime x(x), 0 ≤x≤2π. (18) To ﬁnd a solution of equation (18), let us use the result obtained in Subsection 14.4-5 for a singular integral equation of the ﬁrst kind with the Hilbert kernel. We ﬁnally get ϕ(x)=–1 π⎝integraldisplay2π 0f/prime ξ(ξ)l n⎝vextendsingle⎝vextendsingle⎝vextendsinglesin⎝parenleftBigξ–x 2⎝parenrightBig⎝vextendsingle⎝vextendsingle⎝vextendsingledξ+C, where Cis an arbitrary constant. 14.6-5. Numerical Methods for Hypersingular Equations. 1◦. Consider collocation method for the simplest equation (1). Let us partition the interval [ a,b]i n t o nequal segments of length h=(b–a)/nwith endpoints at the nodes a=t0,t1,t2,...,tn–1,tn=b, t=a+jh,j=0 , 1 , ...,n. Denote the midpoints of the segments [ ti–1,ti]b yxi. It is easy to see thatxi=a+(i–l/2)hfori=1 ,...,n. Let us represent an approximate value of the integral from (1) as a ﬁnite sum. Then, for x=xi, we have 1 π⎝integraldisplayb aϕ(t)dt (xi–t)2≈1 πn⎝summationdisplay j=0ϕ(ti)⎝integraldisplaytj tj–1dt (xi–t)2=1 πϕ(ti)⎝integraldisplayh/2 –h/2dt t2 +1 π⎝summationdisplay j≠iϕ(tj)⎝parenleftbigg1 xi–tj–1 xi–tj–1⎝parenrightbigg =1 πn⎝summationdisplay j=1ϕ(tj)⎝parenleftbigg1 xi–tj–1 xi–tj–1⎝parenrightbigg .(19) Now, let us replace the hypersingular integral equation under consideration by an approximate expression in the form of a system of linear algebraic equations: 1 πn⎝summationdisplay j=1ϕ(tj)⎝parenleftbigg1 xi–tj–1 xi–tj–1⎝parenrightbigg =f/prime(xi),i=1 ,...,n. (20) 14.6. H YPERSINGULAR INTEGRAL EQUATIONS 755 It can be shown that for a ﬁxed x=xl∈(a,b), the difference of the solutions ϕ(xl) of system (20) and equation (19) tends to zero as n→∞ , i.e., ϕ(xl)∼–h π⎝radicalbig (xl–a)(b–xl)n⎝summationdisplay m=1f(tm) √ (tm–a)(b–tm)(xl–tm) ∼–√ (xl–a)(b–xl) π⎝integraldisplayb af(t) √ (t–a)(b–t)dt t–xl. 2◦. By analogy with the above considerations, one can obtain an approximate solution of the general hypersingularintegral equation (2) by a collocation method solving the following system of algebraic equations: n⎝summationdisplay j=1⎝bracketleftbigg1 xi–tj–1 xi–tj–1+hK/prime x(xi,tj)⎝bracketrightbigg ϕ(tj)=f/prime x(xi),i=1 ,...,n. 3◦. Consider the general hypersingular integral equation (2) of the ﬁrst kind with the Cauchy-type kernel on a ﬁnite interval and write this equation in the form 1 π⎝integraldisplayb aϕ(t) (x–t)2dt=f/prime x(x)–1 π⎝integraldisplayb aK/prime x(x,t)ϕ(t)dt,a≤x≤b. (21) A bounded solution of equation (21) can be constructed by resolving this equation with respect to the right-hand side, and thereby reducing it to a Fredholm equation of the second kind. Indeed, ϕ(x)–⎝integraldisplayb aN(x,t)ϕ(t)dt=F(x), a≤x≤b, where N(x,t)=–√ (x–a)(b–x) π2⎝integraldisplayb aK(τ,t) √ (τ–a)(b–τ)dτ τ–x, F(x)=–√ (x–a)(b–x) π⎝integraldisplayb af(τ) √ (τ–a)(b–τ)dτ τ–x. The problem of solving Fredholm equations of the second kind is considered in detail in Chapter 13. References for Section 14.6: N. I. Muskhelishvili (1968), A. I. Kalandia (1973), F. D. Gakhov (1977, 1990), F. D. Gakhov and Yu. I. Cherskii (1978), S. G. Mikhlin and S. Pr ¨ossdorf (1986), S. Pr ¨ossdorf and B. Silbermann (1991), I. K. Lifanov (1996), S.G. Samko (2000), G. Iovane, I. K. Lifanov, and M. A. Sumbatyan (2003), M.A. I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004). Chapter 15 Methods for Solving Complete Singular Integral Equations 15.1. Some Deﬁnitions and Remarks 15.1-1. Integral Equations with Cauchy Kernel. A complete singular integral equation with Cauchy kernel has the form a(t)ϕ(t)+1 πi⎝integraldisplay LM(t,τ) τ–tϕ(τ)dτ=f(t), i2= –1, (1) where the integral, which is understood in the sense of the Cauchy principal value, is taken over a closed or nonclosed contour Landtandτare the complex coordinates of points of the contour. It is assumed that the functions a(t),f(t), and M(t,τ)g i v e no nL and the unknown function ϕ(t) satisfy the H ¨older condition (see Subsection 14.2-2), and M(t,τ) satisﬁes this conditio n with respect to both variables. The integral in Eq. (1) can also be written in a frequently used equivalent form. To this end, we consider the following transformation of the kernel: M(t,τ) τ–t=M(t,τ)–M(t,t) τ–t+M(t,t) τ–t,( 2) where we set M(t,t)=b(t),1 πiM(t,τ)–M(t,t) τ–t=K(t,τ). (3) In this case Eq. (1), with regard to (2) and (3), becomes a(t)ϕ(t)+b(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ+⎝integraldisplay LK(t,τ)ϕ(τ)dτ=f(t). (4) It follows from formulas (3) that the function b(t) satisﬁes the H ¨older condition on the entire contour LandK(t,τ) satisﬁes the H ¨older condition everywhere except for the points τ=t,a tw h i c h one has the estimate |K(t,τ)|<A |τ–t|λ,A= const < ∞,0 ≤λ<1 . Naturally, Eq. (4) is also called a complete singular integral equation with Cauchy kernel .T h e functions a(t)a n d b(t) are called the coefﬁcients of Eq. (4),1 τ–tis called the Cauchy kernel ,a n d the known function f(t) is called the right-hand side of the equation. The ﬁrst and the second terms 757 758 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS on the left-hand side of Eq. (4) form the characteristic part or the characteristic of the complete singular equation and the third summand is called the regular part , and the function K(t,τ) is called the kernel of the regular part . It follows from the above estimate for the kernel of the regular part thatK(t,τ) is a Fredholm kernel. For Eqs. (1) and (4) we shall use the operator notation K[ϕ(t)] =f(t), (5) where the operator Kis called a singular operator . The equation K◦[ϕ(t)]≡a(t)ϕ(t)+b(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ=f(t)( 6) is called the characteristic equation corresponding to the complete equation (4), and the operator K◦ is called the characteristic operator . For the regular part of the equation we introduce the notation Kr[ϕ(t)]≡⎝integraldisplay LK(t,τ)ϕ(τ)dτ, where the operator Kris called a regular (Fredholm )operator , and we rewrite the complete singular equation in another operator form: K[ϕ(t)]≡K◦[ϕ(t)] + Kr[ϕ(t)] =f(t), (7) which will be used in what follows. The equation K∗[ψ(t)]≡a(t)ψ(t)–1 πi⎝integraldisplay Lb(τ)ψ(τ) τ–tdτ+⎝integraldisplay LK(τ,t)ψ(τ)dτ=g(t), (8) obtained from Eq. (4) by transposing the variables in the kernel, is said to be transposed to (4). The operator K∗is said to be transposed to the operator K. In particular, the equation K◦∗[ψ(t)]≡a(t)ψ(t)–1 πi⎝integraldisplay Lb(τ) τ–tψ(τ)dτ=g(t)( 9) is the equation transposed to the characteristic equa tion (6). It should be noted that the operator K◦∗ transposed to the characteristic operator K◦differs from the operator K∗◦that is characteristic for the transposed equation (9). The latter is deﬁned by the formula K∗◦[ψ(t)]≡a(t)ψ(t)–b(t) πi⎝integraldisplay Lψ(τ) τ–tdτ. (10) Throughout the following we assume that in the general case the contour Lconsists of m+1 closed smooth curves L=L0+L1+···+Lm. For equations with nonclosed contours, see, for example, the books by F. D. Gakhov (1977, 1990) and N. I. Muskhelishvili (1992). Remark 1. The above relationship between Eqs. (1) and (4) that involves the properties of these equations is violated if we modify the cond ition and assume that in Eq. (1) the function M(t,τ) satisﬁes the H ¨older condition everywhere on the contour except for ﬁnitely many points at which M has jump discontinuities. In this case, the complete singular integral equation must be represented in the form (4) with separated ch aracteristic and regular parts in some way that differs from the transformation (2) and (3) because the above transformation of Eq. (1) does not lead to the desired decomposition. For equations with disconti nuous coefﬁcients, see the cited books. 15.1. S OME DEFINITIONS AND REMARKS 759 15.1-2. Integral Equations with Hilbert Kernel. Acomplete singular integral equation with Hilbert kernel has the form a(x)ϕ(x)+1 2π⎝integraldisplay2π 0N(x,ξ)c o tξ–x 2ϕ(ξ)dξ=f(x), (11) where the real functions a(x),f(x), and N(x,ξ) and the unknown function ϕ(x) satisfy the H ¨older condition (see Subsection 14.2-2), with the function N(x,ξ) satisfying the condition with respect to both variables. The integral equation (11) can also be written in the following equivalent form, which is frequently used. We transform the kernel as follows: N(x,ξ)c o tξ–x 2=⎝bracketleftbig N(x,ξ)–N(x,x)⎝bracketrightbig cotξ–x 2+N(x,x)c o tξ–x 2, (12) where we write N(x,x)=–b(x),1 2π⎝bracketleftbig N(x,ξ)–N(x,x)⎝bracketrightbig cotξ–x 2=K(x,ξ). (13) In this case, Eq. (11) with regard to (12) and (13) becomes a(x)ϕ(x)–b(x) 2π⎝integraldisplay2π 0cotξ–x 2ϕ(ξ)dξ+⎝integraldisplay2π 0K(x,ξ)ϕ(ξ)dξ=f(x). (14) It follows from formulas (13) that the function b(x) satisﬁes the H ¨older condition, and the ker- nelK(x,ξ) satisﬁes the H ¨older condition everywhere except possibly for the points x=ξat which the following estimate holds: |K(x,ξ)|<A |ξ–x|λ,A= const < ∞,0 ≤λ<1 . The equation in the form (14) is also called a complete singular integral equation with Hilbert kernel. The functions a(x)a n db(x) are called the coefﬁcients of Eq. (14), cot⎝bracketleftbig1 2(ξ–x)⎝bracketrightbig is called the Hilbert kernel , and the known function f(x) is called the right-hand side of the equation. The ﬁrst and second summands in Eq. (14) form the so-called characteristic part or the characteristic of the complete singular equation, and the third summand is called its regular part ; the function K(x,ξ)i s called the kernel of the regular part . The equation a(x)ϕ(x)–b(x) 2π⎝integraldisplay2π 0cotξ–x 2ϕ(ξ)dξ=f(x) (15) is called the characteristic equation corresponding to the complete equation (14). As usual, the above and the forthcoming equations whose right-hand sides are zero everywhere on their domains are said to be homogeneous , and otherwise they are said to be nonhomogeneous . 15.1-3. Fredholm Equations of the Second Kind on a Contour. Fredholm theory and methods for solving Fredholm integral equations of the second kind presented in Chapter 13 remain valid if all functions and parameters in the equations are treated as complex ones and an interval of the real axis is replaced by a contour L. Here we present only some information and write the Fredholm integral equation of the second kind in the form that is convenient for the purposes of this chapter. 760 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS Consider the Fredholm integral equation ϕ(t)+λ⎝integraldisplay LK(t,τ)ϕ(τ)dτ=f(t), (16) where Lis a smooth contour, tandτare complex coordinates of its points, ϕ(t) is the desired function, f(t) is the right-hand side of the equation, and K(t,τ)i st h ek e r n e l . If for some λ, the homogeneous Fredholm equation has a nontrivial solution (or nontrivial solutions), then λis called a characteristic value , and the nontrivial solutions themselves are called eigenfunctions of the kernel K(t,τ) or of Eq. (16). The set of characteristic values of Eq. (16) is at most countable. If this set is inﬁnite, then its only limit point is the point at inﬁnity. To each characteristic value, there are corresponding ﬁnitely many linearly independent eigenfunctions. The set of characteristic values of an integral equation is called its spectrum . The spectrum of a Fredholm integral equation is a discrete set. Ifλdoes not coincide with any characteristic value (in this case the value λis said to be regular), i.e., the homogeneous equation has only the trivial solution, then the nonhomogeneous equation (16)is solvable for any right-hand side f(t). The general solution is given by the formula ϕ(t)=f(t)–⎝integraldisplay LR(t,τ;λ)f(τ)dτ, (17) where the function R(t,τ;λ)i sc a l l e dt h e resolvent of the equation or the resolvent of the kernel K(t,τ) and can be expressed via K(t,τ). If a value of the parameter λis characteristic for Eq. (16), then the homogeneous integral equation ϕ(t)+λ⎝integraldisplay LK(t,τ)ϕ(τ)dτ= 0, (18) as well as the transposed homogeneous equation ψ(t)+λ⎝integraldisplay LK(τ,t)ϕ(τ)dτ= 0, (19) has nontrivial solutions, and the number of solutions of Eq. (18) is ﬁnite and is equal to the number of linearly independent solutions of Eq. (19). The general solution of the homogeneous equation can be represented in the form ϕ(t)=n⎝summationdisplay k=1Ckϕk(t), (20) where ϕ1(t),...,ϕn(t) is a (complete) ﬁnite set of linearly independent eigenfunctions that corre- spond to the characteristic value λ,a n dCkare arbitrary constants. If the homogeneous equation (18) is solvable, then the nonhomogeneous equation (16) is, in general, unsolvable. This equatio n is solvable if and only if t he following conditions hold: ⎝integraldisplay Lf(t)ψk(t)dt= 0, (21) where {ψk(t)}(k=1 ,...,n) is a (complete) ﬁnite set of linearly independent eigenfunctions of the transposed equation that correspond to the characteristic value λ. 15.2. C ARLEMAN METHOD FOR CHARACTERISTIC EQUATIONS 761 If conditions (21) are satisﬁed, then the general solution of the nonhomogeneous equation (16) can be given by the formula (e.g., see Subsection 15.6-5) ϕ(t)=f(t)–⎝integraldisplay LRg(t,τ;λ)f(τ)dτ+n⎝summationdisplay k=1Ckϕk(t), (22) where Rg(t,τ;λ) is called the generalized resolvent and the sum on the right-hand side of (22) is the general solution of the corresponding homogeneous equation. Now we consider an equation of the second kind with weak singularity on the contour: ϕ(t)+⎝integraldisplay LM(t,τ) |τ–t|αϕ(τ)dτ=f(t), (23) where M(t,τ) is a continuous function and 0 < α< 1. By iterating we can reduce this equation to a Fredholm integral equation of the second kind (e.g., see Remark 1 in Section 13.3). It has all properties of a Fredholm equation. For the above reasons, in the theory of singular integral equations it is customary to make no difference between Fredholm equations and equations with weak singularity and use for them the same notation ϕ(t)+λ⎝integraldisplay LK(t,τ)ϕ(τ)dτ=0 , K(t,τ)=M(t,τ) |τ–t|α,0 ≤α< 1. (24) The integral equation (24) is called simply a Fredholm equation , and its kernel is called a Fredholm kernel . If in Eq. (24) the known functions satisfy the H ¨older condition, and M(t,τ) satisﬁes this condition with respect to both variables, then each bounded integrable solution of Eq. (24) alsosatisﬁes the H ¨older condition. Remark 2. By the above estimates, the kernels of the regular parts of the above singular integral equations are Fredholm kernels. Remark 3. The complete and characteristic singular in tegral equations are sometimes called singular integral equations of the second kind. References for Section 15.1: F. D. Gakhov (1977, 1990), F. G. Tricomi (1985), S. G. Mikhlin and S. Pr ¨ossdorf (1986), A. Dzhuraev (1992), N. I. Muskhelishvili (1992), I. K. Lifanov (1996), R. Estrada and R. P. Kanwal (1999), E. G. Ladopoulos (2000). 15.2. Carleman Method for Characteristic Equations 15.2-1. Characteristic Equation with Cauchy Kernel. Consider a characteristic equation with Cauchy kernel: K◦[ϕ(t)]≡a(t)ϕ(t)+b(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ=f(t), (1) where the contour Lconsists of m+ 1 closed smooth curves L=L0+L1+···+Lm. Solving Eq. (1) can be reduced to solving a Riemann boundary value problem (see Subsec- tion 14.3-10), and the solution of the equation can be presented in a closed form. Let us introduce the piecewise analytic function given by the Cauchy integral whose density is the desired solution of the characteristic equation: Φ(z)=1 2πi⎝integraldisplay Lϕ(τ) τ–zdτ.( 2) 762 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS According to the Sokhotski–Plemelj formulas (see Subsection 14.2-5), we have ϕ(t)=Φ+(t)–Φ–(t), 1 πi⎝integraldisplay Lϕ(τ) τ–zdτ=Φ+(t)+Φ–(t).(3) On substituting (3) into (1) and solving the resultant equation for Φ+(t), we see that the piecewise analytic function Φ(z) must be a solution of the Riemann boundary value problem Φ+(t)=D(t)Φ–(t)+H(t), (4) where D(t)=a(t)–b(t) a(t)+b(t),H(t)=f(t) a(t)+b(t).( 5) Since the function Φ(z) is represented by a Cauchy type integral, it follows that this function must satisfy the additi onal condition Φ–(∞)=0 . ( 6 ) The index νof the coefﬁcient D(t) of the Riemann problem (4) is called the index of the integral equation (1). On solving the boundary value problem (4), we ﬁnd the solution of Eq. (1) by the ﬁrst formula in (3). Thus, the integral equation (1) is reduced to the Riemann boundary value problem (4). To establish the equivalence of the equation to the boundary value problem we note that, conversely, the function ϕ(t) that is found by the above-mentioned method from the solution of the boundary value problem nece ssarily satisﬁes Eq. (1). We ﬁrst consider the following normal (nonexceptional) case in which the coefﬁcient D(t)o f the Riemann problem (4) admits no zero or inﬁnite values, which amounts to the condition a(t)±b(t)≠0( 7) for Eq. (1). To simplify the subsequent formulas, we assume that the coefﬁcients of Eq. (1) satisfy the condition a2(t)–b2(t)=1 . ( 8 ) This can always be achieved by dividing the equation by⎝radicalbig a2(t)–b2(t). Let us write out the solution of the Riemann boundary value problem (4) under the assumption ν≥0 and then use the Sokhotski–Plemelj formulas to ﬁnd the limit values of the corresponding functions (see Subsections 14.2-5, 14.3-6, and 14.3-10): Φ+(t)=X+(t)⎝bracketleftbigg1 2H(t) X+(t)+Ψ(t)–1 2Pν–1(t)⎝bracketrightbigg ,Φ–(t)=X–(t)⎝bracketleftbigg –1 2H(t) X+(t)+Ψ(t)–1 2Pν–1(t)⎝bracketrightbigg ,( 9 ) where Ψ(t)=1 2πi⎝integraldisplay LH(τ) X+(τ)dτ τ–t. (10) The arbitrary polynomial is taken in the form –1 2Pν–1(t), which is convenient for the subsequent notation. Hence, by formula (3) we have ϕ(t)=1 2⎝bracketleftbigg 1+X–(t) X+(t)⎝bracketrightbigg H(t)+X+(t)⎝bracketleftbigg 1–X–(t) X+(t)⎝bracketrightbigg⎝bracketleftbigg Ψ(t)–1 2Pν–1(t)⎝bracketrightbigg . 15.2. C ARLEMAN METHOD FOR CHARACTERISTIC EQUATIONS 763 Representing the coefﬁcient of the Riemann problem in the form D(t)=X+(t)/X–(t) and replacing the function Ψ(t) by the expression on the right-hand side in (10), we obtain ϕ(t)=1 2⎝bracketleftbigg 1+1 D(t)⎝bracketrightbigg H(t)+X+(t)⎝bracketleftbigg 1–1 D(t)⎝bracketrightbigg⎝bracketleftbigg1 2πi⎝integraldisplay LH(τ) X+(τ)dτ τ–t–1 2Pν–1(t)⎝bracketrightbigg . Finally, on replacing X+(t) by the expression (62) in Subsection 14.3-10 and substituting the expressions for D(t)a n d H(t) given in (5), we obtain ϕ(t)=a(t)f(t)–b(t)Z(t) πi⎝integraldisplay Lf(τ) Z(τ)dτ τ–t+b(t)Z(t)Pν–1(t), (11) where Z(t)=[a(t)+b(t)]X+(t)=[a(t)–b(t)]X–(t)=eG(t) √ tνΠ(t), G(t)=1 2πi⎝integraldisplay Lln⎝bracketleftbigg τ–νΠ(τ)a(τ)–b(τ) a(τ)+b(τ)⎝bracketrightbiggdτ τ–t,Π(t)=m⎝summationdisplay k=1(t–zk)νk,(12) and the coefﬁcients a(t)a n d b(t) satisfy condition (7). Here Π(t)≡1 for the case in which Lis a simple contour enclosing a simply connected domain. Since the functions a(t),b(t), and f(t) satisfy the H ¨older condition, it follows from the properties of the limit values of the Cauchy type integral that the function ϕ(t) also satisﬁes the H ¨older condition. The last term in formula (11) is the general solution of the homogeneous equation ( f(t)≡0), and the ﬁrst two terms form a particular solution of the nonhomogeneous equation. The particular solution of Eq. (1) can be represented in the form R[f(t)], where Ris the operator deﬁned by R[f(t)] =a(t)f(t)–b(t)Z(t) πi⎝integraldisplay Lf(τ) Z(τ)dτ τ–t. In this case, the general solution of Eq. (1) becomes ϕ(t)= R[f(t)] +ν⎝summationdisplay k=1ckϕk(t), (13) where ϕk(t)=b(t)Z(t)tk–1(k=1 , 2 , ...,ν) are the linearly independent eigenfunctions of the characteristic equation. Ifν< 0, then the Riemann problem (4) is in general unsolvable. The solvability conditions ⎝integraldisplay LH(τ) X+(τ)τk–1dτ=0 , k=1 ,2 , ...,–ν, (14) for problem (4) are the solvability conditions for Eq. (1) as well. Replacing H(τ)a n dX+(τ) by their expressions from (5) and (12), we can rewrite the solvability conditions in the form ⎝integraldisplay Lf(τ) Z(τ)τk–1dτ=0 , k=1 ,2 , ...,–ν. (15) If the solvability conditions hold, then the solution of the nonhomogeneous equation (4) is given by formula (11) for Pν–1≡0. 1.◦Ifν> 0, then the homogeneous equation K◦[ϕ(t)] = 0 has νlinearly independent solutions ϕk(t)=b(t)Z(t)tk–1,k=1 ,2 , ...,ν. 2.◦Ifν≤0, then the homogeneous equation is unsolvable (has only the trivial solution). 764 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS 3.◦Ifν≥0, then the nonhomogeneous equation is solvable for an arbitrary right-hand side f(t), and its general solution linearly depends on νarbitrary constants. 4.◦Ifν< 0, then the nonhomogeneous equation is solvable if and only if its right-hand side f satisﬁes the – νconditions, ⎝integraldisplay Lψk(t)f(t)dt=0 , ψk(t)=tk–1 Z(t). (16) The above properties of characteristic singular integral equations are essentially different from the properties of Fredholm integral equations (see Subsection 15.1-3). With Fredholm equations, if the homogeneous equation is solvable, then the nonhomogeneous equation is in general unsolvable, and conversely, if the homogeneous equation is unsolvable, then the nonhomogeneous equationis solvable. However, for a singular equation, if the homogeneous equation is solvable, then the nonhomogeneous equation is unconditionally solvable, and if the homogeneous equation is unsolvable, then the nonhomogeneous equation is in general unsolvable as well. By analogy with the case of Fredholm equations, we introduce a parameter λinto the kernel of the characteristic equation and consider the equation a(t)ϕ(t)+λb(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ=0 . As shown above, the last equation is solvable if ν=I n da(t)–λb(t) a(t)+λb(t)>0 . The index of a continuous function changes by jumps and only for the values of λsuch that a(t)∓λb(t) = 0. If in the complex plane λ=λ1+iλ2we draw the curves λ=±a(t)/b(t), then these curves divide the plane into domains in each of which the index is constant. Thus, the characteristic values of the characteristic integral equation occupy entire domains, and hence the spectrum is continuous, in contrast with the spectrum of a Fredholm equation. 15.2-2. Transposed Equation of a Characteristic Equation. The equation K◦∗[ψ(t)]≡a(t)ψ(t)–1 πi⎝integraldisplay Lb(τ)ψ(τ) τ–tdτ=g(t), (17) which is transposed to the characteristic equation K◦[ϕ(t)] =f(t), is not characteristic. However, the substitution b(t)ψ(t)=ω(t) (18) reduces it to a characteristic equation for the function ω(t): a(t)ω(t)–b(t) πi⎝integraldisplay Lω(τ) τ–tdτ=b(t)g(t). (19) From the last equation we ﬁnd ω(t), by the formula obtained by adding (17) to (18), and determine the desired function ψ(t): ψ(t)=1 a(t)+b(t)⎝bracketleftbigg ω(t)+1 πi⎝integraldisplay Lω(τ) τ–tdτ+g(t)⎝bracketrightbigg . 15.2. C ARLEMAN METHOD FOR CHARACTERISTIC EQUATIONS 765 Introducing the piecewise analytic function Φ∗(z)=1 2πi⎝integraldisplay Lω(τ) τ–zdτ, (20) we arrive at the Riemann boundary value problem Φ+ ∗(t)=a(t)+b(t) a(t)–b(t)Φ– ∗(t)+b(t)g(t) a(t)–b(t). (21) The coefﬁcient of the boundary value problem (21) is the inverse of the coefﬁcient of the Riemann problem (4) corresponding to the equation K◦[ϕ(t)] =f(t). Hence, ν∗=I n da(t)+b(t) a(t)–b(t)=–I n da(t)–b(t) a(t)+b(t)=–ν. (22) Note that it follows from formulas (17) in Subsection 14.3-4 that the canonical function X∗(z)f o r Eq. (21) and the canonical function X(z) for (4) are reciprocal: X∗(z)=1 X(z). By analogy with the reasoning in Subsection 15.2-1, we obtain a solution of the singular integral equation (17) for ν∗=–ν≥0 in the form ψ(t)=a(t)g(t)+1 πiZ(t)⎝integraldisplay Lb(τ)Z(τ)g(τ) τ–tdτ+1 Z(t)Qν∗–1(t), (23) where Z(t) is given by formula (12) and Qν∗–1(t) is a polynomial of degree at most ν∗– 1 with arbitrary coefﬁcients. If ν∗= 0, then we must set Qν∗–1(t)≡0. Ifν∗=–ν< 0, then for the solvability of Eq. (17) it is necessary and sufﬁcient that ⎝integraldisplay Lb(t)Z(t)g(t)tk–1dt=0 , k=1 ,2 , ...,–ν∗, (24) and if these conditions hold, then the solu tion is given by formula (23), where we must set Qν∗–1(t)≡0. The results of simultaneous investigation of a characteristic equation and the transposed equation show another essential difference from the properties of Fredholm equations (see Subsection 15.1-3). Transposed homogeneous characteristic equations cannot be solvable simultaneously. Either theyare both unsolvable ( ν= 0), or, for a nonzero index, only the equation with a positive index is solvable. We point out that the difference between the numbers of solutions of a characteristic homoge- neous equation and the transposed equation is equal to the index ν. Assertions 1 ◦and 2◦and assertions 3◦and 4◦in Subsection 15.2-1 are called, respectively, the ﬁrst Fredholm theorem and the second Fredholm theorem for a characteristic equation, and the relationship between the index of an equation and the number of solutions of the homogeneous equations K◦[ϕ(t)] = 0 and K◦∗[ψ(t)] = 0 is called the third Fredholm theorem . 15.2-3. Characteristic Equation on the Real Axis. The theory of the Cauchy type integral (see Section 14.2) shows that if the density of the Cauchy type integral taken over an inﬁnite curve vanishes at inﬁnity, then the properties of the integral for the cases in which the contour is ﬁnite and inﬁnite are essentially the same. Therefore, the theoryof singular integral equations on an inﬁnite contour in the class of functions that vanish at inﬁnity coincides with the theory of equations on a ﬁnite contour. 766 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS Just as for the case of a ﬁnite contour, the characteristic integral equation a(x)ϕ(x)+b(x) πi⎝integraldisplay∞ –∞ϕ(τ) τ–xdτ=f(x) (25) can be reduced by means of the Cauchy type integral Φ(z)=1 2πi⎝integraldisplay∞ –∞ϕ(τ) τ–zdτ (26) and the Sokhotski–Plemelj formulas (see Subsection 14.2-5), to the following Riemann boundary value problem for the real axis (see Subsection 14.3-8): Φ+(x)=a(x)–b(x) a(x)+b(x)Φ–(x)+f(x) a(x)+b(x),– ∞<x<∞. (27) We assume that a2(x)–b2(x) = 1, (28) because Eq. (25) can always be reduced to case (28) by the division by⎝radicalbig a2(t)–b2(t). Note that the indexνof the integral equation (25) is given by the formula ν=I n da(x)–b(x) a(x)+b(x). (29) In this case for ν≥0 we obtain ϕ(x)=a(x)f(x)–b(x)Z(x) πi⎝integraldisplay∞ –∞f(τ) Z(τ)dτ τ–x+b(x)Z(x)Pν–1(x) (x+i)ν, (30) where Z(x)=[a(x)+b(x)]X+(x)=[a(x)–b(x)]X–(x)=⎝parenleftbiggx–i x+i⎝parenrightbigg–ν/2 eG(x), G(x)=1 2πi⎝integraldisplay∞ –∞ln⎝bracketleftbigg⎝parenleftbiggτ–i τ+i⎝parenrightbigg–νa(τ)–b(τ) a(τ)+b(τ)⎝bracketrightbiggdτ τ–x. For the case in which ν≤0w em u s ts e t Pν–1(x)≡0. For ν< 0, we must also impose the solvability conditions ⎝integraldisplay∞ –∞f(x) Z(x)dx (x+i)k=0 , k=1 ,2 , ...,–ν. (31) For the solution of Eq. (25) in the class of functions bounded at inﬁnity, see F. D. Gakhov (1977, 1990). The analog of the characteristic equation on the real axis is the equation of the form a(x)ϕ(x)+b(x) πi⎝integraldisplay∞ –∞x–z0 τ–z0ϕ(τ) τ–xdτ=f(x), (32) where z0is a point that does not belong to the contour. For this equation, all qualitative results obtained for the characteristic equation with ﬁnite contour are still valid together with the formulas. In particular, the following inversion formulas for the Cauchy type integral hold: ψ(x)=1 πi⎝integraldisplay∞ –∞x–z0 τ–z0ϕ(τ) τ–xdτ,ϕ(x)=1 πi⎝integraldisplay∞ –∞x–z0 τ–z0ψ(τ) τ–xdτ. (33) 15.2. C ARLEMAN METHOD FOR CHARACTERISTIC EQUATIONS 767 15.2-4. Exceptional Case of a Characteristic Equation. In the study of the characteristic equation in Subsection 15.2-1, the case in which the functions a(t)±b(t) can vanish on the contour Lwas excluded. The reason was that the coefﬁcient D(t)o f the Riemann problem to which the characteristic equation can be reduced has in the exceptional case zeros and poles on the contour, and hence this problem is outside the framework of the general theory. Let us perform an investigation of the above exceptional case. We assume that the coefﬁcients of the singular equations under consideration have properties that provide the additional differentiability requirements that were introduced in the consideration of exceptional cases of the Riemann problem (see 14.3-9). Consider a characteristic equation with Cauchy kernel (1) under the assumption that the functions a(t)–b(t)a n da(t)+b(t) have zeros on the contour at the points α1,...,αµandβ1...,βη, respectively, of integral orders, and hence are representable in the form a(t)–b(t)=µ⎝productdisplay k=1(t–αk)mkr(t),a(t)+b(t)=η⎝productdisplay j=1(t–βj)pjs(t), where r(t)a n d s(t) vanish nowhere. We assume that all points αkandβjare different. Assume that the coefﬁcients of Eq. (1) satisfy the relation a2(t)–b2(t)=µ⎝productdisplay k=1(t–αk)mkη⎝productdisplay j=1(t–βj)pj=A0(t). (34) The equation under consideration can be reduced to the above case by dividing it by√ s(t)r(t). In the exceptional case, by analogy with the case studied in Subsection 15.2-1, Eq. (1) can be reduced to the Riemann problem Φ+(t)=µ⎝productdisplay k=1(t–αk)mk η⎝productdisplay j=1(t–βj)pjD1(t)Φ–(t)+f(t) η⎝productdisplay j=1(t–βj)pjs(t), (35) where D1(t)=r(t)/s(t). The solution of this problem in th e class of functi ons that satisfy the condition Φ(∞) = 0 is given by the formulas Φ+(z)=X+(z) η⎝productdisplay j=1(z–βj)pj[Ψ+(z)–Uρ(z)+A0(z)Pν–p–1(z)], Φ–(z)=X–(z) µ⎝productdisplay k=1(z–αk)mk[Ψ–(z)–Uρ(z)+A0(z)Pν–p–1(z)],(36) where Ψ(z)=1 2πi⎝integraldisplay Lf(τ) s(τ)X+(τ)dτ τ–z, (37) andUρ(z) is the Hermite interpolation polynomial (see Subsection 14.3-2) for the function Ψ(z) of degree ρ=m+p– 1 with nodes at the points αkandβj, respectively, and of the multiplicities mkandpj, respectively, where m=⎝summationtextmkandp=⎝summationtextpj. 768 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS We regard the polynomial Uρ(z) as an operator that maps the right-hand side f(t)o fE q .( 1 )t o the polynomial that interpolates the Cauchy type integral (37) as above. Let us denote this operatorby 1 2T[f(t)] =Uρ(z). (38) Here the coefﬁcient1 2is taken for the convenience of the subsequent manipulations. Furthermore, by analogy with the normal case, from (36) we can ﬁnd Φ+(t)=X+(t) η⎝productdisplay j=1(t–βj)pj⎝bracketleftbigg1 2f(t) s(t)X+(t)+1 2πi⎝integraldisplay Lf(τ) s(τ)X+(τ)dτ τ–t–1 2T[f(t)] –1 2A0(t)Pν–p–1(t)⎝bracketrightbigg , Φ–(t)=X–(t) µ⎝productdisplay k=1(t–αk)mk⎝bracketleftbigg –1 2f(t) s(t)X+(t)+1 2πi⎝integraldisplay Lf(τ) s(τ)X+(τ)dτ τ–t–1 2T[f(t)] –1 2A0(t)Pν–p–1(t)⎝bracketrightbigg . We introduced the coefﬁcient –1 2in the last summands of these formulas using the fact that the coefﬁcients of the polynomial Pν–p–1(t) are arbitrary. Hence, ϕ(t)=Φ+(t)–Φ–(t)=∆1(t)f(t) s(t)X+(t)+∆2(t)⎝bracketleftbigg1 πi⎝integraldisplay Lf(τ)dτ s(τ)X+(τ)(τ–t)–T[f(t)]–A0(t)Pν–p–1(t)⎝bracketrightbigg , (39) where ∆1(t)=X+(t) 2η⎝productdisplay j=1(t–βj)pj+X–(t) 2µ⎝productdisplay k=1(t–αk)mk,∆2(t)=X+(t) 2η⎝productdisplay j=1(t–βj)pj–X–(t) 2µ⎝productdisplay k=1(t–αk)mk. We write Z(t)=s(t)X+(t)=r(t)X–(t), (40) and, applying relation (34), represent formula (39) as follows: ϕ(t)=1 A0(t)⎝bracketleftbigg a(t)f(t)–b(t)Z(t) πi⎝integraldisplay Lf(τ) Z(τ)dτ τ–t+b(t)Z(t)T[f(t)]⎝bracketrightbigg +b(t)Z(t)Pν–p–1(t). Let us introduce the operator R1[f(t)] by the formula R1[f(t)]≡1 A0(t)⎝bracketleftbigg a(t)f(t)–b(t)Z(t) πi⎝integraldisplay Lf(τ) Z(τ)dτ τ–t+b(t)Z(t)T[f(t)]⎝bracketrightbigg , (41) and ﬁnally obtain ϕ(t)= R1[f(t)] +b(t)Z(t)Pν–p–1(t). (42) Formula (42) gives a solution of Eq. (1) for the exceptional case in which ν–p>0 . T h i s solution linearly depends on ν–parbitrary constants. If ν–p< 0, then the solution exists only under p–νspecial solvability conditions imposed on f(t), which follow from the solvability conditions for the Riemann problem (35) corresponding to this case. 15.2. C ARLEMAN METHOD FOR CHARACTERISTIC EQUATIONS 769 15.2-5. Characteristic Equation with Hilbert Kernel. Consider the characteristic equation with Hilbert kernel a(x)ϕ(x)–b(x) 2π⎝integraldisplay2π 0cotξ–x 2ϕ(ξ)dξ=f(x). (43) Just as the characteristic integral equation with Cauchy kernel is related to the Riemann boundary value problem, so the characteristic equation (43) with Hilbert kernel can be analytically reduced to a Hilbert problem in a straightforward manner. In turn, the Hilbert problem can be reduced to the Riemann problem (see Subsection 14.3-12), and hence the solution of Eq. (43) can be constructed in a closed form. Forν> 0, the homogeneous equation (43) ( f(x)≡0) has 2ν linearly independent solutions, and the nonhomogeneous problem is unconditionally solvable and linearly depends on 2 νreal constants. Forν< 0, the homogeneous equation is unsolvable, and the nonhomogeneous equation is solvable only under –2 νreal solvability conditions. Taking into account the fact that any complex parameter contains two real parameters, and a complex solvability condition is equivalent to two real conditions, we see that, for ν≠0, the qualitative results of investigating the characteristic equation with Hilbert kernel completely agree with the corresponding results for the characteristic equation with Cauchy kernel. 15.2-6. Tricomi Equation. The singular integral Tricomi equation has the form ϕ(x)–λ⎝integraldisplay1 0⎝parenleftbigg1 ξ–x–1 x+ξ–2xξ⎝parenrightbigg ϕ(ξ)dξ=f(x), 0 ≤x≤1. (44) The kernel of this equation consists of two terms. The ﬁrst term is the Cauchy kernel. The second term is continuous if at least one of the variables xandξvaries strictly inside the interval [0, 1]; however, for x=ξ=0a n df o r x=ξ= 1, this kernel becomes inﬁnite and is nonintegrable in the square {0≤x≤1, 0≤ξ≤1}. By using the function Φ(z)=1 2πi⎝integraldisplay1 0⎝parenleftbigg1 ξ–z–1 z+ξ–2zξ⎝parenrightbigg ϕ(ξ)dξ, which is piecewise analytic in the upper and the lower half-plane, we can reduce Eq. (44) to the Riemann problem with boundary condition on the real axis. The solution of the Tricomi equation has the form y(x)=1 1+λ2π2⎝bracketleftbigg f(x)+⎝integraldisplay1 0ξα(1 –x)α xα(1 –ξ)α⎝parenleftbigg1 ξ–x–1 x+ξ–2xξ⎝parenrightbigg f(ξ)dξ⎝bracketrightbigg +C(1 –x)β x1+β, α=2 πarctan( λπ)( – 1 < α<1 ) , t a nβπ 2=λπ(–2 <β<0 ) , where Cis an arbitrary constant. References for Section 15.2: P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. D. Gakhov (1977, 1990), F. G. Tricomi (1985), N. I. Muskhelishvili (1992). 770 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS 15.3. Complete Singular Integral Equations Solvable in a Closed Form In contrast with characteristic equations and their transposed equations, complete singular integral equations cannot be solved in the closed form in general. However, there are some cases in which complete equations can be solved in a closed form. 15.3-1. Closed-Form Solutions in the Case of Constant Coefﬁcients. Consider the complete singular integral equation with Cauchy kernel in the form (see Subsec-tion 15.1-1) a(t)ϕ(t)+b(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ+⎝integraldisplay LK(t,τ)ϕ(τ)dτ=f(t), (1) where Lis an arbitrary closed contour. Let us show that Eq. (1) can be solved in a closed form if a(t)=aandb(t)=bare constants and K(t,τ) is an arbitrary function that has an analytic continuation to the domain Ω+with respect to each variable. Under the above assumptions, Eq. (1) has the form aϕ(t)+1 πi⎝integraldisplay LM(t,τ) τ–tϕ(τ)dτ=f(t), (2) where M(t,τ)=b+πi(t–τ)K(t,τ), so that M(t,t)=b= const. Let b≠0. We write ψ(t)=1 bπi⎝integraldisplay LM(t,τ) τ–tϕ(τ)dτ.( 3) According to Subsection 14.4-4, the function ϕ(t) can be expressed via ψ(t)a n d ψ(t) can be expressed via ϕ(t). Then we rewrite Eq. (2) as follows: aϕ(t)+bψ(t)=f(t). (4) On applying the operation (3) to this equation, we obtain aψ(t)+bϕ(t)=w(t), (5) where w(t)=1 bπi⎝integraldisplay LM(t,τ) τ–tf(τ)dτ. By solving system (4), (5) we ﬁnd ϕ(t): ϕ(t)=1 a2–b2⎝bracketleftbigg af(t)–1 πi⎝integraldisplay LM(t,τ) τ–tf(τ)dτ⎝bracketrightbigg (6) under the assumption that a≠±b. Thus, for a≠±band for a kernel K(t,τ) that can be analytically continued, Eq. (1) or (2) is solvable and has the unique solution given by formula (6). Equation (1) was studied above for b≠0. This assumption is natural because, for b≡0, Eq. (1) is no longer singular. However, the Fredholm equation obtained for b=0 ,t h a ti s , aϕ(t)+⎝integraldisplay LK(t,τ)ϕ(τ)dτ=f(t), a= const, (7) is solvable in a closed form for a kernel K(t,τ) that has analytic continuation. Let a function K(t,τ) have an analytic continuation to the domain Ω+with respect to each of the variables and continuous for t,τ∈L. In this case, the following assertions hold. 15.3. C OMPLETE SINGULAR INTEGRAL EQUATIONS SOLV ABLE IN A CLOSED FORM 771 1◦. The function Φ+(t)=⎝integraldisplay LK(t,τ)ϕ(τ)dτ has an analytic continuation to the domain Ω+for any function ϕ(t) satisfying the H ¨older condition. 2◦. If a function ϕ+(t) satisfying the H ¨older condition has an analytic continuation to the domain Ω+, then ⎝integraldisplay LK(t,τ)ϕ+(τ)dτ=0 . ( 8 ) This implies the relation ⎝integraldisplay LK(t,τ)⎝integraldisplay LK(τ,τ1)ϕ(τ1)dτ1dτ=0 ( 9 ) for each function ϕ(t) (satisfying the H ¨older condition). Therefore, it follows from (7) that a⎝integraldisplay LK(t,τ)ϕ(τ)dτ=⎝integraldisplay LK(t,τ)f(τ)dτ, and hence ϕ(t)=1 a2⎝bracketleftbigg af(t)–⎝integraldisplay LK(t,τ)f(τ)dτ⎝bracketrightbigg . (10) Therefore, if a kernel K(t,τ) is analytic in the domain Ω+with respect to each of the variables and continuous for t,τ∈L, then Eq. (7) is solvable for each right-hand side, and the solution is given by formula (10). 15.3-2. Closed-Form Solutions in the General Case. Let us pass to the general case of the solvability of Eq. (1) in a closed form under the condition that a function K(t,τ)[a(t)+b(t)]–1is analytic with respect to τand meromorphic with respect to tin the domain Ω+. For brevity, we write Kr[ϕ(t)] =⎝integraldisplay LK(t,τ)ϕ(τ)dτ and note that Kr[ϕ+(t)] = 0 (11) for each function ϕ+(t) that has an analytic continuation to the domain Ω+. By setting ϕ(t)= ϕ+(t)–ϕ–(t) and with regard to (11), we reduce Eq. (1) to a relation similar to that of the Riemann problem: ϕ+(t)–1 a(t)+b(t)Kr[ϕ–(t)] =D(t)ϕ–(t)+H(t), (12) where D(t)=a(t)–b(t) a(t)+b(t),H(t)=f(t) a(t)+b(t). By assumption, we have K(t,τ) a(t)+b(t)=A+(t,τ) Π+(t),Π+(t)=n⎝productdisplay k=1(t–zk)mk, (13) where zk∈Ω+andmkare positive integers and the function A+(t,τ) is analytic with respect to t and with respect to τonΩ+. 772 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS Relation (12) becomes Π+(t)ϕ+(t)+ A+[ϕ–(t)] =Π+(t)[D (t)ϕ–(t)+H(t)], (14) where A+is the integral operator with kernel A+(t,τ). Since the function A+[ϕ–(t)] is analytic onΩ+, it follows that the last relation is an ordinary Riemann problem for which the functions Π+(t)ϕ+(t)+ A+[ϕ–(t)] and ϕ–(t) can be deﬁned in a closed form, and hence the same holds for ϕ(t). Namely, let us rewrite the function D(t) in the form D(t)=X+(t)/X–(t), where X±(z)i st h e canonical function of the Riemann problem, and reduce relation (14) to the form in which the generalized Liouville theorem can be applied (see Subsection 14.3-1). We arrive at a polynomial of degree at most ν–1+n⎝summationtext k=1mkwith arbitrary coefﬁcients (for the case in which ν+n⎝summationtext k=1mk>0 ) . However, the presence of the factor Π+(t)( o nϕ+(t)), which vanishes in Ω+with total order of zeros n⎝summationtext k=1mk, clearly reduces the number of arbitrary constants in the general solution. Remark 1. Following the lines of the discussion in Subsection 15.3-2 we can treat the case in which the kernel K(t,τ) is meromorphic with respect to τas well. In this case, Eq. (1) can be reduced to a Riemann problem of the type (12) and a linear algebraic system. Remark 2. The solutions of a complete singular integral equation that are constructed in Sec- tion 15.3 can be applied for the case in which the contour Lis a collection of ﬁnitely many disjoint smooth closed contours. Example 1. Consider the equation λϕ(t)+1 πi⎝integraldisplay Lcos(τ–t) τ–tϕ(τ)dτ=f(t), (15) where Lis an arbitrary closed contour. Note that the function M(t,τ)=c o s ( τ–t) has the property M(t,t)≡1. Therefore, it remains to apply formula (6), and thus for (15) we have ϕ(t)=1 λ2–1⎝bracketleftbigg λf(t)–1 πi⎝integraldisplay Lcos(τ–t) τ–tf(τ)dτ⎝bracketrightbigg ,λ≠±1. Example 2. Consider the equation λϕ(t)+1 πi⎝integraldisplay Lsin(τ–t) (τ–t)2ϕ(τ)dτ=f(t), (16) where Lis an arbitrary closed contour. The function M(t,τ)=s i n ( τ–t)/(τ–t) has the property M(t,t)≡1. Therefore, applying formula (6), for (16) we obtain ϕ(t)=1 λ2–1⎝bracketleftbigg λf(t)–1 πi⎝integraldisplay Lsin(τ–t) (τ–t)2f(τ)dτ⎝bracketrightbigg ,λ≠±1. Reference for Section 15.3: F. D. Gakhov (1977, 1990). 15.4. Regularization Method for Complete Singular Integral Equations 15.4-1. Certain Properties of Singular Operators. Let K1and K2be singular operators, K1[ϕ(t)]≡a1(t)ϕ(t)+1 πi⎝integraldisplay LM1(t,τ) τ–tϕ(τ)dτ,( 1) K2[ω(t)]≡a2(t)ω(t)+1 πi⎝integraldisplay LM2(t,τ) τ–tω(τ)dτ.( 2) The operator K=K2K1deﬁned by the formula K[ϕ(t)] = K2⎝bracketleftbig K1[ϕ(t)]⎝bracketrightbig is called the composition or the product of the operators K1and K2. 15.4. R EGULARIZATION METHOD FOR COMPLETE SINGULAR INTEGRAL EQUATIONS 773 Let us form the expression for the operator K, K[ϕ(t)] = K2K1[ϕ(t)]≡a2(t)⎝bracketleftbigg a1(t)ϕ(t)+1 πi⎝integraldisplay LM1(t,τ) τ–tϕ(τ)dτ⎝bracketrightbigg +1 πi⎝integraldisplay LM2(t,τ) τ–t⎝bracketleftbigg a1(τ)ϕ(τ)+1 πi⎝integraldisplay LM1(τ,τ1) τ1–τϕ(τ1)dτ1⎝bracketrightbigg dτ,( 3 ) and select its characteristic part. To this end, we perform the following manipulations: ⎝integraldisplay LM1(t,τ) τ–tϕ(τ)dτ=M1(t,t)⎝integraldisplay Lϕ(τ) τ–tdτ+⎝integraldisplay LM1(t,τ)–M1(t,t) τ–tϕ(τ)dτ, ⎝integraldisplay La1(τ)M2(t,τ) τ–tϕ(τ)dτ=a1(t)M2(t,t)⎝integraldisplay Lϕ(τ) τ–tdτ+⎝integraldisplay La1(τ)M2(t,τ)–a1(t)M2(t,t) τ–tϕ(τ)dτ, ⎝integraldisplay LM2(t,τ) τ–tdτ⎝integraldisplay LM1(τ,τ1) τ1–τϕ(τ1)dτ1=–π2M2(t,t)M1(t,t)ϕ(t)+⎝integraldisplay Lϕ(τ1)dτ1⎝integraldisplay LM2(t,τ)M1(τ,τ1) (τ1–τ)(τ–t)dτ.(4) Here we applied the Poincar ´e–Bertrand formula (see Subsection 14.2-6). We can see that all kernels of the integrals of the last summands on the right-hand sides in (4) are Fredholm kernels. We write M1(t,t)=b1(t),M2(t,t)=b2(t)( 5) and see that the characteristic operator K◦of the composition (product) Kof two singular operators K1and K2can be expressed by the formula K◦[ϕ(t)] = (K 2K1)◦[ϕ(t)] = [a 2(t)a1(t)+b2(t)b1(t)]ϕ(t)+a2(t)b1(t)+b2(t)a1(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ.( 6 ) Let us write out the operator K1and K2in the form (3) with explicitly expressed characteristic parts: K1[ϕ(t)]≡a1(t)ϕ(t)+b1(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ+⎝integraldisplay LK1(t,τ)ϕ(τ)dτ,( 7 ) K2[ω(t)]≡a2(t)ω(t)+b2(t) πi⎝integraldisplay Lω(τ) τ–tdτ+⎝integraldisplay LK2(t,τ)ω(τ)dτ.( 8 ) Thus, the coefﬁcients a(t)a n d b(t) of the characteristic part of the product of the operators K1 and K2can be expressed by the formulas a(t)=a2(t)a1(t)+b2(t)b1(t),b(t)=a2(t)b1(t)+b2(t)a1(t). (9) These formulas do not contain regular kernels k1andk2and are symmetric with respect to the indices 1 and 2. This means that the characteristic part of the product of singular operators dependsneither on their regular parts nor on the order of these operators in the product. Thus, any change of order of the factors, as well as a change of the regular parts of the factors, inﬂuences the regular part of the product of the operators only and preserves the characteristic part of the product. Let us calculate the coefﬁcient of the Ri emann problem that corresponds to the characteristic operator ( K 2K1)◦: D(t)=a(t)–b(t) a(t)+b(t)=[a2(t)–b2(t)] [a 1(t)–b1(t)] [a2(t)+b2(t)] [a 1(t)+b1(t)]=D2(t)D1(t), (10) where we denote by D1(t)=a1(t)–b1(t) a1(t)+b1(t),D2(t)=a2(t)–b2(t) a2(t)+b2(t)(11) 774 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS the coefﬁcients of the Riemann problems that correspond to the operators K◦ 1and K◦ 2. This means that the coefﬁcient of the Riemann problem for the operator ( K2K1)◦is equal to the product of the coefﬁcients of the Riemann problems for the operators K◦ 1and K◦ 2, and hence the index of the product of singular operators is equal to the sum of indices of the factors: ν=ν1+ν2. (12) In its complete form, the operator K2K1is deﬁned by the expression K2K1[ϕ(t)]≡a(t)ϕ(t)+b(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ+⎝integraldisplay LK(t,τ)ϕ(τ)dτ, where a(t)a n d b(t) are deﬁned by formulas (9). For a regular kernel K(t,τ), on the basis of formulas (4) we can write out the explicit expression. For a singular operator Kand its transposed operator K∗(see Subsection 15.1-1), the following relations hold: ⎝integraldisplay Lψ(t)K[ϕ(t)]dt=⎝integraldisplay LϕK∗[ψ(t)]dt for any functions ϕ(t)a n d ψ(t) that satisfy the H ¨older condition, and (K2K1)∗=K∗ 1K∗2. 15.4-2. Regularizer. The regularization method is a reduction of a singular integral equation to a Fredholm equation. The reduction process itself is known as regularization . If a singular operator K2is such that the operator K2K1is regular (Fredholm), i.e., contains no singular integral ( b(t)≡0), then K2is called a regularizing operator with respect to the singular operator K1or, brieﬂy, a regularizer . Note that if K2is a regularizer, then the operator K1K2is regular as well. Let us ﬁnd the general form of a regularizer. By deﬁ nition, the following relation must hold: b(t)=a2(t)b1(t)+b2(t)a1(t) = 0, (13) which implies that a2(t)=g(t)a1(t),b2(t)=–g(t)b1(t), (14) where g(t) is an arbitrary function that vanishes nowhere and satisﬁes the H ¨older condition. Hence, if Kis a singular operator, K[ϕ(t)]≡a(t)ϕ(t)+b(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ+⎝integraldisplay LK(t,τ)ϕ(τ)dτ, (15) then, in general, the regularizer ˜Kcan be expressed as follows: ˜K[ω(t)]≡g(t)a(t)ω(t)–g(t)b(t) πi⎝integraldisplay Lω(τ) τ–tdτ+⎝integraldisplay L˜K(t,τ)ω(τ)dτ, (16) where ˜K(t,τ) is an arbitrary Fredholm kernel and g(t) is an arbitrary function satisfying the H ¨older condition. Since the index of a regular operator (b (t)≡0) is clearly equal to zero, it follows from the property of the product of operators that the index of the regularizer has the same modulus as the 15.4. R EGULARIZATION METHOD FOR COMPLETE SINGULAR INTEGRAL EQUATIONS 775 index of the original operator and the opposite sign. The same fact can be established directly by the form of a regularizer (16) from the formula ˜D(t)=˜a(t)–˜b(t) ˜a(t)+˜b(t)=a(t)+b(t) a(t)–b(t)=1 D(t). Thus, for any singular operator with Cauchy kernel (15) of the normal type ( a(t)±b(t)≠0), there exist inﬁnitely many regularizers (16) whose characteristic part depends on an arbitrary function g(t) that contains an arbitrary regular kernel ˜K(t,τ). Since the elements g(t)a n d ˜K(t,τ) are arbitrary, we can choose them so that the regularizer will satisfy some additional conditions. For instance, we can make the coefﬁcient of ϕ(t)i nt h e regularized equation be normalized, i.e., equal to one. To this end we must set g(t)=[a2(t)–b2(t)]–1. If no conditions are imposed, then it is natural to apply the simplest regularizers. These can be obtained by setting g(t)≡1a n d ˜K(t,τ)≡0 in formula (16), which gives the regularizer ˜K[ω(t)] = K∗◦[ω(t)]≡a(t)ω(t)–b(t) πi⎝integraldisplay Lω(τ) τ–tdτ, (17) or we can set g(t)≡1a n d ˜K(t,τ)=–1 πib(τ)–b(t) τ–tand obtain ˜K[ω(t)] = K◦∗[ω(t)]≡a(t)ω(t)–1 πi⎝integraldisplay Lb(τ)ω(τ) τ–tdτ. (18) The simplest operators K∗◦and K◦∗are most frequently used as regularizers. Since the multiplication of operators is not generally commutative, one should distinguish two forms of regularization: left regularization, which gives the operator ˜KK, and right regularization which leads to the operator K˜K. On the basis of the above remark we can claim that a right regularizer is simultaneously a left regularizer, and vice versa. Thus, the operation of regularization is commutative. If an operator ˜Kis a regularizer for an operator K, then, in turn, the operator Kis a regularizer for the operator ˜K. The operators K1K2and K2K1can differ by a regular part only. 15.4-3. Methods of Left and Right Regularization. Let a complete singular integral equation be given: K[ϕ(t)]≡a(t)ϕ(t)+b(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ+⎝integraldisplay LK(t,τ)ϕ(τ)dτ=f(t). (19) Three methods of regularization are used. The ﬁrst two methods are based on the composition of a given singular operator and its regularizer (left and right regularization). The third method differs essentially from the ﬁrst two, namely, the elimination of the singular integral is performed by solving the corresponding characteristic equation. 1◦.Left regularization. Let us take the regularizer (16): ˜K[ω(t)]≡g(t)a(t)ω(t)–g(t)b(t) πi⎝integraldisplay Lω(τ) τ–tdτ+⎝integraldisplay L˜K(t,τ)ω(τ)dτ. (20) On replacing the function ω(t)i n ˜K[ω(t)] with the expression K[ϕ(t)] –f(t) we arrive at the integral equation ˜KK[ϕ(t)] = ˜K[f(t)]. (21) By deﬁnition, ˜KKis a Fredholm operator, because ˜Kis a regularizer. Hence, Eq. (21) is a Fredholm equation. Thus, we have transformed the singular integral equation (19) into the Fredholm integralequation (21) for the same unknown function ϕ(t). This is the ﬁrst regularization method, which is called left regularization . 776 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS 2◦.Right Regularization. On replacing in Eq. (19) the desired function by the expression (20), ϕ(t)= ˜K[ω(t)], (22) where ω(t) is a new unknown function, we arrive at the integral equation K˜K[ω(t)] =f(t), (23) which is a Fredholm equation as well. Thus, from the singular integral equation (19) for the unknown function ϕ(t) we passed to the Fredholm integral equation for the new unknown function ω(t). On solving the Fredholm equation (23), we ﬁnd a solution of the original equation (19) by formula (22). The application of formula (22) requires integration only (a proper integral and a singular integral must be found). This is the second method of the regularization, which is called right regularization . 15.4-4. Problem of Equivalent Regularization. In the reduction of a singular integral equation to a regular one we perform a functional transformation over the corresponding equation. In general, this transf ormation can either introduce new irrelevant solutions that do not satisf y the original equation or imply a loss of some solutions. Therefore, in general, the resultant equation is not equivalent to the original equation. Consider the relationship between the solutions of these equations and ﬁnd out in what cases these equations are equivalent. 1◦.Left Regularization. Consider a singular equation K[ϕ(t)] =f(t) (24) and the corresponding regular equation ˜KK[ϕ(t)] = ˜K[f(t)]. (25) Let us write out Eq. (25) in the form ˜K⎝bracketleftbig K[ϕ(t)] –f(t)⎝bracketrightbig = 0. (26) Since the operator ˜Kis homogeneous, it follows that each solution of the original equation (24) (a function that vanishes the expression K[ϕ(t)] –f(t)) satisﬁes Eq. (26) as well. Hence, the left regularization implies no loss of solutions. However, a solution of the regularized equation need notbe a solution of the original equation. Consider the singular integral equation corresponding to the regularizer ˜K[ω(t)] = 0. (27) Letω 1(t),...,ωp(t) be a complete system of its solutions, i.e., a maximal collection of linearly independent eigenfunctions of the regularizer ˜K. We regard Eq. (26) as a singular equation of the form (27) with the unknown function ω(t)= K[ϕ(t)] –f(t). We obtain K[ϕ(t)] –f(t)=p⎝summationdisplay j=1αjωj(t), (28) where the αjare some constants. We see that the regularized equation is equivalent to Eq. (28) rather than the original equation (24). Thus, Eq. (25) is equivalent to Eq. (28) in which αjare arbitrary or deﬁnite constants. It may occur that Eq. (28) is solvable only under the assumption that all αjsatisfy the condition αj=0 . In this case, Eq. (25) is equivalent to the original equation (24), and the regularizer deﬁnes an equivalent transformation. In particular, if the regularizer has no eigenfunctions, then the right-hand side of Eq. (28) is identically zero, and it must be equivalent. This operator certainly exists forν≥0. For instance, we can take the regularizer K ∗◦, which has no eigenfunctions for the case under consideration because the index of the regularizer K∗◦is equal to – ν≤0. 15.4. R EGULARIZATION METHOD FOR COMPLETE SINGULAR INTEGRAL EQUATIONS 777 2◦.Right Regularization. Consider Eq. (24) and the corresponding regularized equation K˜K[ω(t)] =f(t), (29) which is obtained by substitution ˜K[ω(t)] =ϕ(t). (30) Ifωj(t) is a solution of Eq. (29), then formula (30) gives the corresponding solution of the original equation ϕj(t)= ˜K[ωj(t)]. Hence, the right regularization cannot lead to irrelevant solutions. Conversely, assume that ϕk(t) is a solution of the original equation. In this case a solution of the regularized equation (29) can be obtained as a solution of the nonhomogeneous singular equation ˜K[ω(t)] =ϕk(t); however, this solution may be unsolvable. Thus, the right regularization can lead to loss of solutions. We have no loss of solutions if Eq. (30) is solvable for each right-hand side. In this case the operator ˜K will be an equivalent right regularizer. 3◦.The Equivalent Regularization. The operator ˜K=K∗◦is an equivalent regularizer for any index; forν≥0, we must apply left regularization, while for ν≤0 we must use right regularization. In the latter case we obtain an equation for a new function ω(t), and if it is determined, then we can construct all solutions to the original equation in antiderivatives, and it follows from the properties of the right regularization that no irrelevant solutions can occur. For the other methods of equivalent regularization, see the references at the end of this section. 15.4-5. Fredholm Theorems. Let a complete singular integral equation be given: K[ϕ(t)] =f(t). (31) THEOREM 1.The number of solutions of the singular integral equation (31) is ﬁnite. THEOREM 2.A necessary and sufﬁcient solvability condition for the singular equation (31) is ⎝integraldisplay Lf(t)ψj(t)dt=0 , j=1 ,...,m, (32) where ψ1(t),...,ψm(t)is a maximal ﬁnite set of linearly independent solutions of the transposed homogeneous equation K∗[ψ(t)] = 0 . (Since the functions under consideration are complex, it follows that condition (32) is not the orthogonality condition for the functions f(t)andψj(t).) THEOREM 3.The difference between the number nof linearly independent solutions of the singular equation K[ϕ(t)] = 0 and the number mof linearly independent solutions of the transposed equation K∗[ψ(t)] = 0 depends on the characteristic part of the operator Konly and is equal to its index, i.e., n–m=ν. (33) Corollary. The number of linearly independent solutions of characteristic equations is minimal among all singular equations with given index ν. 778 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS 15.4-6. Carleman–Vekua Approach to the Regularization. Let us transfer the regular part of a singular equation to the right-hand side and rewrite the equation as follows: a(t)ϕ(t)+b(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ=f(t)–⎝integraldisplay LK(t,τ)ϕ(τ)dτ, (34) or, in the operator form, K◦[ϕ(t)] =f(t)– Kr[ϕ(t)]. (35) We regard the last equation as a characteristic one and solve it by temporarily assuming that the right-hand side is a known function. In this case (see Subsection 15.2-1) ϕ(t)=⎝bracketleftbigg a(t)f(t)–b(t)Z(t) πi⎝integraldisplay Lf(τ) Z(τ)dτ τ–t+b(t)Z(t)Pν–1(t)⎝bracketrightbigg –⎝bracketleftbigg a(t)⎝integraldisplay LK(t,τ)ϕ(τ)dτ–b(t)Z(t) πi⎝integraldisplay Ldτ1 Z(τ1)(τ1–t)⎝integraldisplay LK(τ1,τ)ϕ(τ)dτ⎝bracketrightbigg , (36) where for ν≤0w em u s ts e t Pν–1(t)≡0. Let us reverse the order of integration in the iterated integral and rewrite the expression in the last parentheses as follows: ⎝integraldisplay L⎝bracketleftbigg a(t)K(t,τ)–b(t)Z(t) πi⎝integraldisplay LK(τ1,τ) Z(τ1)(τ1–t)dτ1⎝bracketrightbigg ϕ(τ)dτ. Since Z(t) satisﬁes the H ¨older condition (and hence is bounded) and does not vanish and since K(τ1,τ) satisﬁes the estimate |K(τ1,τ)|<A|τ1–τ|–λ(with 0 ≤λ< 1) near the point τ1=τ, we can see that the entire integral⎝integraldisplay LK(τ1,τ) Z(τ1)(τ1–t)dτ1 satisﬁes an estimate similar to that for K(τ1,τ). Hence, the kernel N(t,τ)=a(t)K(t,τ)–b(t)Z(t) πi⎝integraldisplay LK(τ1,τ) Z(τ1)(τ1–t)dτ1 (37) is a Fredholm kernel. On transferring the terms with ϕ(t) to the right-hand side, we obtain ϕ(t)+⎝integraldisplay LN(t,τ)ϕ(τ)dτ=f1(t), (38) where N(t,τ) is the Fredholm kernel deﬁned by formula (37) and f1(t)h a st h ef o r m f1(t)=a(t)f(t)–b(t)Z(t) πi⎝integraldisplay Lf(τ) Z(τ)dτ τ–t+b(t)Z(t)Pν–1(t). (39) If the index of Eq. (34) νis negative, then the function must satisfy not only the Fredholm equation (38) but also the relations ⎝integraldisplay L⎝bracketleftbigg⎝integraldisplay LK(t,τ) Z(t)tk–1dt⎝bracketrightbigg ϕ(τ)dτ=⎝integraldisplay Lf(t) Z(t)tk–1dt,k=1 ,2 , ...,–ν. (40) Thus, if ν≥0, then the solution of a complete singular integral equation (34) is reduced to the solution of the Fredholm integral equation (38). If ν< 0, then Eq. (34) can be reduced to Eq. (38) (where we must set Pν–1(t)≡0) together with conditions (40), which can be rewritten in the form ⎝integraldisplay Lρk(τ)ϕ(τ)dτ=fk,k=1 ,2 , ...,–ν, ρk(τ)=⎝integraldisplay LK(t,τ) Z(t)tk–1dt,fk=⎝integraldisplay Lf(t) Z(t)tk–1dt,(41) where the ρk(τ) are known functions and the fkare known constants. 15.4. R EGULARIZATION METHOD FOR COMPLETE SINGULAR INTEGRAL EQUATIONS 779 Relations (41) are the solvability conditions for the regularized equation (38). However, they need not be the solvability conditions for the original singular integral equation (34). Some of themcan be the equivalence conditions for these two equations. Let us select the conditions of these two types. Assume that among the functions ρ k(t) there are precisely hlinearly independent functions. We can choose the numbering so that these are the functions ρ1(t),...,ρh(t). In this case we have ⎝integraldisplay Lρk(t)ϕ(t)dt=fk,k=1 ,2 , ...,h. (42) Moreover, the following η=|ν|–hlinearly independent relations must hold: αj1ρ1(t)+···+αj|ν|ρ|ν|(t)=0 , j=1 ,2 , ...,η. Let us multiply the relations in (40) successively by αj1,...,αj|ν|and sum the products. Taking into account the last relations, we have ⎝integraldisplay Lf(t)ψj(t)dt=0 , ψj(t)=1 Z(t)|ν|⎝summationdisplay k=1αjktk–1;j=1 ,2 , ...,η. (43) These relations, which do not involve the desired function ϕ(t), are the necessary solvability conditions on the right-hand side f(t) for the original singular equation and the regularized equation to be solvable. Relations (42) are the equivalence conditions for the origi nal singular equation and the regularized equation. The solution of the Fredholm equation (38) satisﬁes the original singularequation (34) if and only if it satisﬁes conditions (42). Thus, for ν≥0, the regularized equation (38) is equivalent to the original singular equation. Forν< 0, the original equation is equivalent to the regularized equation (with common solvability conditions (43)) together with conditions (42). Remark 1. If the kernel of the regular part of a complete singular integral equation with Cauchy kernel is degenerate, then by the Carleman–Vekua regularization this equation can be reduced to the investigation of a system of linear algebraic equations (see, e.g., S. G. Mikhlin and K. L. Smolitskiy (1967)). Remark 2. The Carleman–Vekua regularization is sometimes called the regularization by solv- ing the characteristic equation. 15.4-7. Regularization in Exceptional Cases. Consider the complete singular equation with Cauchy kernel K[ϕ(t)]≡a(t)ϕ(t)+b(t) πi⎝integraldisplay Lϕ(τ) τ–tdτ+⎝integraldisplay LK(t,τ)ϕ(τ)dτ=f(t) (44) under the same conditions on the functions a(t)±b(t) as above in Subsection 15.2-4. We represent this equation in the form K◦[ϕ(t)] =f(t)–⎝integraldisplay LK(t,τ)ϕ(τ)dτ, and apply the Carleman–Vekua regularization. In this case by formula (42) of Subsection 15.2-4 we obtain the equation ϕ(t)+ R1⎝bracketleftbigg⎝integraldisplay LK(t,τ)ϕ(τ)dτ⎝bracketrightbigg =R1[f(t)] +b(t)Z(t)Pν–p–1(t), (45) where the operator R1is deﬁned by formula (41) of Subsection 15.2-4. 780 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS In the expression for the second summand on the left-hand side in (45), the operation R1with respect to the variable tcommutes with the operation of integration with respect to τ. Therefore, Eq. (45) can be rewritten in the form ϕ(t)+⎝integraldisplay LRt 1⎝bracketleftbig K(t,τ)⎝bracketrightbig ϕ(τ)dτ=R1[f(t)] +b(t)Z(t)Pν–p–1(t), (46) where the superscript tat the symbol of the operator Rt 1means that the operation is performed with respect to the variable t. Since the operator R1is bounded, it follows that the resulting integral equation (46) is a Fredholm equation, and hence the regularization problem for the singular equation (44) is solved. It follows from the general theory of the regularization that Eq. (44) is equivalent to Eq. (46) for ν–p≥0 and to Eq. (46) and a system of functional equations for ν–p<0 . In conclusion we note that for the above cases of singular integral equations, the Fredholm theorems fail in general. Remark 3. Exceptional cases of singular integral equations with Cauchy kernel can be reduced to equations of the normal type. 15.4-8. Complete Equation with Hilbert Kernel. Consider the complete singular integral equation with Hilbert kernel (see Subsection 15.1-2) a(x)ϕ(x)–b(x) 2π⎝integraldisplay2π 0cot⎝parenleftbiggξ–x 2⎝parenrightbigg ϕ(ξ)dξ+⎝integraldisplay2π 0K(x,ξ)ϕ(ξ)dξ=f(x). (47) Let us show that Eq. (47) can be reduced to a complete singular integral equation with a kernel of the Cauchy type, and in this connection, the theory of the latter equation can be directly extended to Eq. (47). Since the regular parts of these two types of equations have the same character, it follows that it sufﬁces to apply the relationship between the Hilbert kernel and the Cauchy kernel (see Subsection 14.4-5): dτ τ–t=1 2cot⎝parenleftbiggξ–x 2⎝parenrightbigg dξ+i 2dξ. (48) Hence, 1 2cot⎝parenleftbiggξ–x 2⎝parenrightbigg dξ=dτ τ–t–1 2dτ τ, (49) where t=eixandτ=eiξare the complex coordinates of points of the contour L, that is, the unit circle. On replacing the Hilbert kernel in Eq. (47) with the expression (49) and on substituting x=–ilnt, ξ=–ilnτ,a n ddξ=–iτ–1dτ, after obvious manipulations we reduce Eq. (47) to a complete singular integral equation with Cauchy kernel of the form a1(t)ϕ1(t)–ib1(t) πi⎝integraldisplay Lϕ1(τ) τ–tdτ+⎝integraldisplay LK1(t,τ)dτ=f1(t). (50) The coefﬁcient of the Riemann problem corresponding to Eq. (50) is D(t)=a1(t)+ib1(t) a1(t)–ib1(t)=a(x)+ib(x) a(x)–ib(x), (51) and the index is expressed by the formula IndD(t)=2I n d [ a(x)+ib(x)]. (52) 15.4. R EGULARIZATION METHOD FOR COMPLETE SINGULAR INTEGRAL EQUATIONS 781 Example. Let us perform the regularization of the following singular integral equations in different ways: K[ϕ(t)]≡(t+t–1)ϕ(t)+t–t–1 πi⎝integraldisplay Lϕ(τ) τ–tdτ–1 2πi⎝integraldisplay L(t+t–1)(τ+τ–1)ϕ(τ)dτ=2t2, (53) where Lis the unit circle. The regular part of the kernel is degenerate. Therefore, in the same way as was applied in the solution of Fredholm equations with degenerate kernel (see Section 13.2), the equation can be reduced to the investigation of the characteristic equation and a linear algebraic equation, and hence it can be solved in a closed form. Thus, we need no regularization. However, the equation under consideration is useful in the illustration of general methods because all calculations can beperformed to the very end. For convenience of the subsequent discussion, we ﬁrst solve this equation. We write 1 2πi⎝integraldisplay L(τ+τ–1)ϕ(τ)dτ=A, (54) and write out the equation in the characteristic form: (t+t–1)ϕ(t)+t–t–1 πi⎝integraldisplay Lϕ(τ) τ–tdτ=2t2+A(t+t–1). For the corresponding Riemann boundary value problem Φ+(t)=t–2Φ–(t)+t+1 2A(1 +t–2), (55) we have the index ν= –2, and the solvability conditions (see Subsection 15.2-1) hold for A= 0 only. In this case, Φ+(z)=z andΦ–(z) = 0. This gives a solution to Eq. (53) in the form ϕ(t)=Φ+(t)–Φ–(t)=t. On substituting the last expression into Eq. (54) we see that this relation holds for A= 0. Hence, the given equation is solvable and has a unique solution of the form ϕ(t)=t. 1◦.Left Regularization. Since the equation index ν= –2 is negative, any regularizer of the equation has eigenfunctions (at least two linearly independent), and hence the left regularization leads, in general, to an equation that is not equivalent to the original one. We ﬁrst consider the left regularization by means of the simplest regularizer K∗◦. Let us ﬁnd the linearly independent eigenfunctions of the equation K∗◦[ω(t)]≡(t+t–1)ω(t)–t–t–1 πi⎝integraldisplay Lω(τ) τ–tdτ=0 . The corresponding Riemann boundary value problem Φ+(t)=t2Φ–(t) now has the index ν= 2. We can ﬁnd the eigenfunctions of the operator K∗◦by the formulas of Subsection 15.2-1 and obtain ω1(t)=1– t–2,ω2(t)=t–t–1. On the basis of the general theory (see Subsection 15.4-4), the regular equation K∗◦K[ϕ(t)] = K∗◦[f(t)] is equivalent to the singular equation: K[ϕ(t)] =f(t)+α1ω1(t)+α2ω2(t), (56) where α1andα2are constants that can be either arbitrary or deﬁnite. Taking into account Eq. (54), we write out Eq. (56) in the form of a characteristic equation: (t+t–1)ϕ(t)+t–t–1 πi⎝integraldisplay Lϕ(τ) τ–tdτ=2t2+A(t+t–1)+α1(1 –t–2)+α2(t–t–1). The corresponding Riemann boundary value problem has the form Φ+(t)=t–2Φ–(t)+t+1 2A(1 +t–2)+1 2α1(t–1+t–3)+1 2α2(1 –t–2). Its solution can be represented as follows: Φ+(z)=z+1 2A+1 2α2,Φ–(z)=1 2z2[α1z–3+(α2–A)z–2–α1z–1]. The solvability conditions give α1=0a n d α2=A. In this case, the solution of Eq. (56) is deﬁned by the formula ϕ(t)=Φ+(t)–Φ–(t)=t+A. On substituting the above expression for ϕ(t) into Eq. (54) we obtain the identity A=A. Hence, the constant α2=Aremains arbitrary, and the regularized equation is equivalent not to the original equation but to the equation K[ϕ(t)] =f(t)+α2ω2(t), which has the solution ϕ(t)=t+A,w h e r e Ais an arbitrary constant. The last function ϕsatisﬁes the original equation only forA=0 . 782 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS 2◦.Right Regularization. For a right regularizer we take the simplest operator K∗◦. By setting ϕ(t)=K∗◦[ω(t)]≡(t+t–1)ω(t)–t–t–1 πi⎝integraldisplay Lω(τ) τ–tdτ, (57) we obtain the following Fredholm equation with respect to the function ω(t): KK∗◦[ω(t)]≡ω(t)–1 4πi⎝integraldisplay L[t(τ2–1+τ–2)+2τ–1+t–1(τ2+3+τ–2)–2τ–2τ–1]ω(τ)dτ=1 2t2. (58) The last equation is degenerate. On solving it we obtain ω(t)=1 2t2+α(t–t–1)+β(1 –t–2), where αandβare arbitrary constants. Thus, the regularized equation for ω(t) has two linearly independent solutions, while the original equation (53) has a unique solution. On substituting the above expression for ω(t) into formula (57) we obtain ϕ(t)=K∗◦⎝bracketleftbig1 2t2+α(t–t–1)+β(1 –t–2)⎝bracketrightbig=t, where ϕ(t) is the (unique) solution of the original singular equation. The result agrees with the general theory because, for a negative index, the right regularization by means of the operator K∗◦is an equivalent regularization. 3◦.The Carleman–Vekua Regularization. This method of regularization is performed by formulas (36)–(39). However, we must recall that these formulas can be applied only for an equation such that a2(t)–b2(t) = 1. Therefore, we must ﬁrst divide Eq. (53) by two. In this case, we have a=1 2(t+t–1),b=1 2(t–t–1),f(t)=t2, K(t,τ)=–1 4πi(t+t–1)(τ+τ–1),X+(z)=1 , Z(t)=(a+b)X+=t, f1(t)=1 2(t+t–1)t2–(t–t–1)t 2πi⎝integraldisplay Lτ2 τdτ τ–t=t, N(t,τ)=–1 2(t+t–1)1 4πi(t+t–1)(τ+τ–1)+(t–t–1)t(τ+τ–1) 2πi⋅4πi⎝integraldisplay Lτ1+τ–1 1 τ1dτ1 τ1–t=–1 2πi(τ+τ–1). The regularized equation has the form ϕ(t)–1 2πi⎝integraldisplay L(τ+τ–1)ϕ(τ)dτ=t. (59) To this equation we must add conditions (41) for k= 1, 2. This equation is degenerate, and on solving it we ﬁnd the general solution ϕ(t)=t+A,w h e r e Ais an arbitrary constant. Let us write out conditions (42) and (43). Here we have ρk(τ)=⎝integraldisplay LK(t,τ) Z(t)tk–1dt=–τ+τ–1 4πi⎝integraldisplay L(1 +t–2)tk–1dt,k=1 ,2 , ρ1(τ)=0 , ρ2(τ)=–1 2(τ+τ–1),fk=⎝integraldisplay Lf(t) Z(t)tk–1dt=⎝integraldisplay Ltkdt,f1=f2=0 . The functions ρ1(t)a n dρ2(t) are linearly dependent. The dependence αj1ρ1(t)+···+αj|ν|ρ|ν|(t) = 0 (see Subsec- tion 15.4-6) has the form α1ρ1(t)+0 ⋅ρ2(t)=0 . Hence, the solvability condition (43) holds identically. The equivalence condition (42) ⎝integraldisplay Lρ2(τ)ϕ(τ)dτ=–1 2⎝integraldisplay L(τ+τ–1)(τ+A)dτ=0 holds for A= 0 only. Hence, among the solutions to the regularized equation, ϕ(t)=t+A, only the function ϕ(t)=tsatisﬁes the original equation. References for Section 15.4: F. D. Gakhov (1977, 1990), S. G. Mikhlin and S. Pr ¨ossdorf (1986), N. I. Muskhelishvili (1992). 15.5. A NALYSIS OF SOLUTIONS SINGULARITIES FOR COMPLETE INTEGRAL EQUATIONS 783 15.5. Analysis of Solutions Singularities for Complete Integral Equations with Generalized Cauchy Kernels∗ 15.5-1. Statement of the Problem and Preliminary Remarks. Consider a complete integral equation of the second kind in the form a(t)ϕ(t)+b(t) πi⎝integraldisplay1 –1ϕ(τ)dτ τ–t+⎝integraldisplay1 –1Kg(t,τ)ϕ(τ)dτ+⎝integraldisplay1 –1Lg(t,τ) ϕ(τ)dτ +⎝integraldisplay1 –1K(t,τ)ϕ(τ)dτ+⎝integraldisplay1 –1L(t,τ) ϕ(τ)dτ=f(t), –1 < t<1 , ( 1 ) where ϕ(τ) is an unknown function and ϕ(τ) is its complex conjugate; f(t) is a given continuous function on the closed interval [–1, 1]; the functions a(t),b(t) and the kernels K(t,τ),L(t,τ)a r e bounded and continuous (or satisfy the H ¨older condition in all their arguments), and the generalized kernels Kg(t,τ),Lg(t,τ) have ﬁxed singularities that are ﬁrst-order poles at the endpoints of the integration interval, as the (real) parameters τandtsimultaneously tend to either endpoint of the interval [–1, 1] ( τ=t→± 1). The kernels of equation (1), as well as the functions a(t),b(t),f(t), may be either real- or complex-valued. Note that th e method described below is suitable for the asymptotic analysis of equations of the second as well as ﬁrst kind for a(t)≡0. Assume that the solution of equation (1) belongs to the class of functions that have, at the endpoints of the integration interval, integrable singularities (generally complex) of power type due to both a “movable” singularity of the integral in the sense of the principal value (the ﬁrst integralin (1)) and ﬁxed singularities of the kernels K g(t,τ)a n dLg(t,τ). This type of asymptotic behavior of the unknown function can be taken into account by the introduction of a special weight function w(τ)=( 1– τ)α(1 +τ)β,– 1 ≤τ≤1, –1 < Re α,R eβ<0 , ( 2 ) which is present as a coefﬁcient in the unknown function, i.e., ϕ(τ)=u(τ)w(τ). (3) Here, u(τ) is a new unknown function satisfying the H ¨older condition and different from zero at the endpoints of the interval. The last requirement is connected with the fact that the sought weight function (2) should reﬂect the leading singular asymptotics of the unknown function (3). Note that the presence of ﬁxed singularities in the kernels Kg(t,τ)a n dLg(t,τ) signiﬁcantly effects the asymptotic behavior of the solution near the endpoints of the integration interval, which in this situation usually has the form (1 ∓τ)λ,– 1<R e λ<0 ,τ→± 1(λ=α,β), and Re λ≠–1/2. (Sometimes the weight function may be bounded on one end of the integration interval, whichcorresponds to Re λ≥0.) Assume that the generalized kernels can be represented in the form K g(t,τ)=⎝summationdisplay p,j,k,rAk(t)(1 +τ)p(1 +t)j (τ–z∗r)p+j+1+⎝summationdisplay l,m,n,sBl(t)(1 –τ)m(1 –t)n (τ–z∗∗s)m+n+1,( 4 ) Lg(t,τ)=⎝summationdisplay p,j,k,rCk(t)(1 +τ)p(1 +t)j (τ–z∗r)p+j+1+⎝summationdisplay l,m,n,sDl(t)(1 –τ)m(1 –t)n (τ–z∗∗s)m+n+1.( 5 ) * Section 15.5 was written by A. V . Andreev. 784 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS Here, the indices and the power exponents p,j,k,l,m,n,r,stake the values 0, 1, 2, ...and vary independently in each sum; the quantities z∗ randz∗∗ sdepend on the variable tas follows ( i2= –1): z∗ r(t)=– 1+( 1+ t)eiθr,0 < θr<2π; z∗∗ s(t)=1+( 1– t)eiθs,– π<θs<π.(6) It is also assumed that the functions Ak(t),Ck(t)a n d Bl(t),Dl(t) in (4)–(5) are ﬁnite and nonzero att=– 1a n d t=+ 1 . The geometrical meaning of expressions (6) is that for t→–1 (resp., t→+1) the points z∗ r (resp., z∗∗ s), on the complex plane, tend to –1 (resp., +1) along the ray obtained from the integration line by its rotation by the angle θr(resp., θs) about the point –1 (resp., +1). Remark. Note that in mathematical statements of some applied problems, the representations of z∗ r(t)a n d z∗∗ s(t) in the form (6) may involve θr<0o r θr>2π. In view of the inequality 0 < θr<2π indicating that the values of the angle θrin (6) are counted counterclockwise from the positive direction of the axis Ox, one should replace θrin each such case by a suitable equivalent value obtained by that rule. Similarly, if θs>πorθs<–π, this value of θsshould be replaced by a suitable acute angle. Such replacements ensure ﬁxed signs in the formulas for z∗ r(t)a n d z∗∗ s(t), which appear in the generalized kernels (4) and (5). 15.5-2. Auxiliary Results. Using (2) and (3), let us rewrite equation (1) in the equivalent form u(t)⎝bracketleftbigg a(t)w(t)+b(t) πi⎝integraldisplay1 –1w(τ)dτ τ–t+⎝integraldisplay1 –1Kg(t,τ)w(τ)dτ⎝bracketrightbigg + u(t)⎝integraldisplay1 –1Lg(t,τ) w(τ)dτ +⎝integraldisplay1 –1⎝bracketleftbiggb(t) πi1 τ–t+Kg(t,τ)⎝bracketrightbigg⎝bracketleftbigg u(τ)–u(t)⎝bracketrightbigg w(τ)dτ+⎝integraldisplay1 –1Lg(t,τ)[ u(τ)– u(t)] w(τ)dτ +⎝integraldisplay1 –1K(t,τ)u(τ)w(τ)dτ+⎝integraldisplay1 –1L(t,τ) u(τ) w(τ)dτ=f(t), –1 < t<1 . ( 7 ) It is easy to see from (7) that for H ¨older continuous u(τ), the characteristic part of equation (1), which goes to inﬁnity as t→± 1, has the form Is(t)=u(t)⎝bracketleftbigg a(t)w(t)+b(t) πi⎝integraldisplay1 –1w(τ)dτ τ–t+⎝integraldisplay1 –1Kg(t,τ)w(τ)dτ⎝bracketrightbigg + u(t)⎝integraldisplay1 –1Lg(t,τ) w(τ)dτ.( 8 ) The most general approach to solving an integral equation with conjugate unknown functions consists in regarding this equation as a system of equations for two unknown functions ϕ(τ)a n d ϕ(τ), where the second equation of the system is obtained by passing from (1) to conjugate values. Let us write out the characteristic part of the equation conjugate to (1) (to be used in the sequel): Is(t)=u(t)⎝integraldisplay1 –1 Lg(t,τ)w(τ)dτ+ u(t)⎝bracketleftbigg a(t) w(t)– b(t) πi⎝integraldisplay1 –1 w(τ)dτ τ–t+⎝integraldisplay1 –1 Kg(t,τ) w(τ)dτ⎝bracketrightbigg .( 9 ) Let us examine the asymptotic behavior of the characteristic part (8) and its conjugate (9) as t→± 1 (z∗ r→–1,z∗∗ s→+1). To that end, we obtain expressions for the leading terms of the integrals in the sense of the principal value and the int egrals containing genera lized kernels (4) , (5) as t→± 1. In order to calculate the integrals in (8), we use the integral representation of the zero-order Jacobi function of the second kind Q(α,β) 0(z): Q(α,β) 0(z)=1 w(z)⎝integraldisplay1 –1w(τ) τ–zdτ,z/∈[–1, 1]. (10) 15.5. A NALYSIS OF SOLUTIONS SINGULARITIES FOR COMPLETE INTEGRAL EQUATIONS 785 Using the representations of the Jacobi functions of the second kind in terms of the hypergeometric function F(a,b,c,ϑ), functional relations for this function, and the formula for the gamma function Γ(ζ), we obtain from (10) the following formal expressions of the Cauchy integral: ⎝integraldisplay1 –1w(τ) τ–zdτ=–πw(z) (–1)βsin(πβ)+2α+βΓ(α+1 )Γ(β) Γ(α+β+1 )F⎝parenleftbigg 1, –α –β,1–β,1+z 2⎝parenrightbigg =πw(z) (–1)αsin(πα)–2α+βΓ(α)Γ(β+1 ) Γ(α+β+1 )F⎝parenleftbigg 1, –α –β,1–α,1–z 2⎝parenrightbigg , (11) where z/∈[–1, 1]. Since F(a,b,c, 0) = 1, the singularities of the Cauchy integral for z→± 1a r e completely determined by the ﬁrst terms in the expressions (11). We ﬁx the multivaluedness of (–1)λ(λ=α,β) in (11) so that the resulting expressions correctly reﬂect the behavior of the Cauchy integral as z→± 1 from the complex plane cut along the segment [–1, 1]. The leading terms of the asymptotic expansion of the Cauchy integral near the endpoints of the integration interval are obtained from (11): ⎝braceleftBigg⎝integraldisplay1 –1w(τ) τ–zdτ⎝bracerightBigg z→–1=–2απe–iπβ sin(πβ)⎝braceleftbig (1 +z)β⎝bracerightbig z→–1,z/∈[–1, 1]; (12) ⎝braceleftBigg⎝integraldisplay1 –1w(τ) τ–zdτ⎝bracerightBigg z→+1=2βπeiπα sin(πα){(1 –z)α}z→+1,z/∈[–1, 1]. (13) Here and in subsequent asymptotic formulas, we use the notation {F(x)}x→a=F(x)|x→a, and only the leading term of the expansion is kept in the right-hand side. From (12) and (13), using the Sokhotski–Plemelj formula 2 Φ(x)=Φ+(x)+Φ–(x), we obtain the following asymptotic formulas for the leading part of the integral in the sense of t he principal value: ⎝braceleftBigg⎝integraldisplay1 –1w(τ) τ–tdτ⎝bracerightBigg t→–1=– 2απcot(πβ)⎝braceleftbig (1 +t)β⎝bracerightbig t→–1, (14) ⎝braceleftBigg⎝integraldisplay1 –1w(τ) τ–tdτ⎝bracerightBigg t→+1=2βπcot(πα){(1 –t)α}t→+1. (15) Here, it has been taken into account that the power functions (1 + t)βand (1 – t)αacquire the coefﬁcients e2iπβande–2iπα, respectively, as one goes around the points –1 and +1. Taking into account the explicit formulas (6) and using (12), (13), we get ⎝braceleftBigg⎝integraldisplay1 –1w(τ) τ–z∗r(t)dτ⎝bracerightBigg t→–1=–2απe–iπβeiθrβ sin(πβ)⎝braceleftbig (1 +t)β⎝bracerightbig t→–1, (16) ⎝braceleftBigg⎝integraldisplay1 –1w(τ) τ–z∗∗s(t)dτ⎝bracerightBigg t→+1=2βπeiθsα sin(πα){(1 –t)α}t→+1. (17) Note that when deriving the last expression, we have chosen the value e–iπαof the multi-valued quantity (–1)αlike for (13) (see also (11)). The representation of the integrals in (8) of the terms of the kernels (4) and (5) with denominators of degree > 1 are obtained by differentiating the Cauchy integral in the parameter z: ⎝integraldisplay1 –1w(τ)dτ (τ–z)s=1 (s–1 ) !ds–1 dzs–1⎝integraldisplay1 –1w(τ)dτ τ–z,s=2 ,3 , ... (18) 786 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS For the weight function (2) we introduce the notation w(τ)=( 1– τ)α(1 +τ)β≡w(α,β)(τ), w(τ)=w(¯α,¯β)(τ). (19) Consecutively differentiating relations (12), (13) and using (18), we obtain the following expressions for the leading terms of the expansions of the corresponding integrals ( s=2 ,3 , ...): ⎝braceleftBigg⎝integraldisplay1 –1w(α,β)(τ)dτ (τ–z)s⎝bracerightBigg z→–1=–2απe–iπβ sin(πβ)β(β– 1)...( β–s+2 ) (s–1 ) !⎝braceleftbig (1 +z)β–s+1⎝bracerightbig z→–1, (20) ⎝braceleftBigg⎝integraldisplay1 –1w(α,β)(τ)dτ (τ–z)s⎝bracerightBigg z→+1=2βπeiπα sin(πα)α(α–1 ) . . . ( α–s+2 ) (s–1 ) !⎝braceleftbig (1 –z)α–s+1⎝bracerightbig z→+1. (21) In view of (20) and (6), for the generic term of the ﬁrst series in (4) (the second series is bounded fort→–1), we have ⎝braceleftBigg⎝integraldisplay1 –1Ak(t)(1 +τ)p(1 +t)j (τ–z∗r(t))p+j+1w(α,β)(τ)dτ⎝bracerightBigg t→–1=⎝braceleftBigg Ak(t)(1 + t)j⎝integraldisplay1 –1w(α,β+p)(τ) (τ–z∗r(t))p+j+1dτ⎝bracerightBigg t→–1 =–Ak(–1)2απe–iπ(β+p) sin[π(β+p)](β+p)(β+p– 1)...( β–j+1 ) (p+j)!⎝braceleftbig (1 +t)j(1 +z∗ r(t))β–j⎝bracerightbig t→–1 =– 2αAk(–1)πe–iπβeiθr(β–j) sin(πβ)(β+p)(β+p– 1)...( β–j+1 ) (p+j)!⎝braceleftbig (1 +t)β⎝bracerightbig t→–1. (22) In a similar way, using (21), we obtain an expression for the leading term of the integral of the generic term of the second series in (4) that goes to inﬁnity as t→+1: ⎝braceleftBigg⎝integraldisplay1 –1Bl(t)(1 –τ)m(1 –t)n (τ–z∗∗s(t))m+n+1w(α,β)(τ)dτ⎝bracerightBigg t→+1 =2βBl(+1)(–1)nπeiθs(α–n) sin(πα)(α+m)(α+m– 1)...( α–n+1 ) (m+n)!{(1 –t)α}t→+1. (23) The expressions for the leading parts of the integrals of the generic terms of the series in (5) with the weight w(τ) (see (8)) can be obtained by replacing α(β)b y ¯α(¯β)a n dAk(Bl)b yCk(Dl)i n (22) and (23): ⎝braceleftBigg⎝integraldisplay1 –1Ck(t)(1 +τ)p(1 +t)j (τ–z∗r(t))p+j+1w(¯α,¯β)(τ)dτ⎝bracerightBigg t→–1 =– 2¯αCk(–1)πe–iπ¯βeiθr(¯β–j) sin(π¯β)(¯β+p)(¯β+p–1 ) . . . ( ¯β–j+1 ) (p+j)!⎝braceleftBig (1 +t)¯β⎝bracerightBig t→–1, (24) ⎝braceleftBigg⎝integraldisplay1 –1Dl(t)(1 –τ)m(1 –t)n (τ–z∗∗s(t))m+n+1w(¯α,¯β)(τ)dτ⎝bracerightBigg t→+1 =2¯βDl(+1)(–1)nπeiθs(¯α–n) sin(π¯α)(¯α+m)( ¯α+m– 1)...( ¯ α–n+1 ) (m+n)!⎝braceleftbig (1 –t)¯α⎝bracerightbig t→+1. (25) It can be seen that representations (22)–(25) of the integrals of the generic terms of the series in (4) and (5) also cover the cases p=j=0a n d m=n= 0 (see (16) and (17)). Summation of these representations with respect to the parameters of the corresponding series in (4) and (5), together 15.5. A NALYSIS OF SOLUTIONS SINGULARITIES FOR COMPLETE INTEGRAL EQUATIONS 787 with (14) and (15), allows us to obtain analytic formulas for the leading parts of all integrals in (8) and separate the factors that go to inﬁnity as t→± 1( – 1<R e α,R eβ<0 ) . Let us examine the representation of the conjugate characteristic part (9) of equation (1). The expression for the integral in the sense of principal value in (9) is easily obtained by the replacement ofαwith ¯αandβwith ¯βin (14) and (15). In order to obtain representations for the integrals of the generalized kernels (4) and (5) in (9), one should perform similar transformations α↔¯αand β↔¯βin (22)–(25). Moreover, it should be taken into account that passing to conjugates in these kernels is accompanied by the replacement of the coefﬁcients Ak,Bl,Ck,Dland the functions z∗ r(t),z∗∗ s(t) by their conjugates (see (4) and (5)), the last operation, in view of (6), being equivalent to the replacement of θrby 2π –θrandθsby –θ s. The ﬁnal expressions will not be written out here, since the comparison of the representations obtained in the above way with (22)–(25) shows that the former can be obtained from the latter by formal passage to the conjugate quantities. 15.5-3. Equations for the Exponents of Singularity of a Solution. In order to obtain an equation for the exponent β, we write the expressions of the characteristic part (8) of equation (1) and its complex conjugate (9) for t→–1: Is(t)=2α∆– 11(β)⎝braceleftbig (1 +t)β⎝bracerightbig t→–1u(–1) + 2¯α∆– 12(¯β)⎝braceleftBig (1 +t)¯β⎝bracerightBig t→–1 u(–1), Is(t)=2α∆– 21(β)⎝braceleftbig (1 +t)β⎝bracerightbig t→–1u(–1) + 2¯α∆– 22(¯β)⎝braceleftBig (1 +t)¯β⎝bracerightBig t→–1 u(–1).(26) Here, ∆– 11(β)=a(–1) + icot(πβ)b(–1) + 2–α⎝braceleftbig Kg,w⎝bracerightbig t→–1, ∆– 12(¯β)=2–¯α⎝braceleftbig Lg,¯w⎝bracerightbig t→–1, ∆– 21(β)= ∆– 12(¯β)=2–α⎝braceleftbig Lg,w⎝bracerightbig t→–1, ∆– 22(¯β)= ∆– 11(β)= a(–1) – icot(π¯β) b(–1) + 2–¯α⎝braceleftbig Kg,¯w⎝bracerightbig t→–1,(27) where⎝braceleftbig Kg,w⎝bracerightbig t→–1stands for the coefﬁcient of the leading term of the asymptotic expansion of the integral of kernel Kg(t,τ) with weight w(τ)a st→–1. In view of (4), this coefﬁcient is a sum of bounded factors of expressions calculated on the basis of (22), i.e., ⎝braceleftbig Kg,w⎝bracerightbig t→–1=–2απe–iπβ sin(πβ)⎝summationdisplay p,j,k,r⎝bracketleftbigg Ak(–1)eiθr(β–j)(β+p)(β+p– 1)...( β–j+1 ) (p+j)!⎝bracketrightbigg . (28) Using (24), we obtain a similar expression for the integral of the function Lg(t,τ) w(τ)i n( 8 ) : ⎝braceleftbig Lg,¯w⎝bracerightbig t→–1=–2¯απe–iπ¯β sin(π¯β)⎝summationdisplay p,j,k,r⎝bracketleftbigg Ck(–1)eiθr(¯β–j)(¯β+p)(¯β+p– 1)...( ¯β–j+1 ) (p+j)!⎝bracketrightbigg . (29) There is no need to write out the coefﬁcients of the leading asymptotic terms of the integrals of generalized kernels in (9), because of the above-mentioned fact that these coefﬁcients are thecomplex conjugates of the coefﬁcients (28) and (29). This fact is reﬂected in the relation between the functions ∆ – hq(β)(h,q= 1, 2) in (27). Let us rewrite the expression (26) in the form Is(t)=⎝braceleftBig (1 +t)Reβ⎝bracerightBig t→–1⎝bracketleftBig 2α∆– 11(β)⎝braceleftBig (1 +t)iImβ⎝bracerightBig t→–1u(–1) +2¯α∆– 12(¯β)⎝braceleftBig (1 +t)–iImβ⎝bracerightBig t→–1 u(–1)⎝bracketrightBig , Is(t)=⎝braceleftBig (1 +t)Reβ⎝bracerightBig t→–1⎝bracketleftBig 2α∆– 21(β)⎝braceleftBig (1 +t)iImβ⎝bracerightBig t→–1u(–1) +2¯α∆– 22(¯β)⎝braceleftBig (1 +t)–iImβ⎝bracerightBig t→–1 u(–1)⎝bracketrightBig . 788 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS It is easy to see that the ﬁrst factors in the right-hand sides go to inﬁnity as t→–1. Since the other terms of equations (7), not involved in (8), and the right-hand side of (7) are bounded for t→–1, it has to be required that the second factors (those in square brackets) be equal to zero. This brings us to a system of two homogeneous algebraic equations for the values u(–1) and u(–1): 2α∆– 11(β)⎝braceleftbig (1 +t)iImβ⎝bracerightbig t→–1u(–1) + 2¯α∆– 12(¯β)⎝braceleftbig (1 +t)–iImβ⎝bracerightbig t→–1 u(–1) = 0, 2α∆– 21(β)⎝braceleftbig (1 +t)iImβ⎝bracerightbig t→–1u(–1) + 2¯α∆– 22(¯β)⎝braceleftbig (1 +t)–iImβ⎝bracerightbig t→–1 u(–1) = 0.(30) According to the statement of the problem, the unknown function u(τ) does not vanish at the ends of the interval [–1, 1] (see (3)), and therefore, in order to satisfy this system it is necessary to require that its determinant be equal to zero. Interpreting an ambiguity of the form xix(x→0), dividing by equal factors, and taking into account the rela tion between functions (27), we ﬁnally obtain the following transcendental equation for the singularity exponent of the solution of equation (1) at the left endpoint of the integration interval: ∆– 11(β) ∆– 11(β)–∆– 12(¯β) ∆– 12(¯β) = 0, (31) where ∆– 11(β)=a(–1) + icot(πβ)b(–1) –πe–iπβ sin(πβ)⎝summationdisplay p,j,k,r⎝bracketleftbigg Ak(–1)eiθr(β–j)(β+p)(β+p– 1)...( β–j+1 ) (p+j)!⎝bracketrightbigg , (32) ∆– 12(¯β)=–πe–iπ¯β sin(π¯β)⎝summationdisplay p,j,k,r⎝bracketleftbigg Ck(–1)eiθr(¯β–j)(¯β+p)(¯β+p– 1)...( ¯β–j+1 ) (p+j)!⎝bracketrightbigg . (33) Note that the relation between u(–1) and its conjugate u(–1) represented by either equation (30) is actually only seeming, since for complex β(Imβ≠0) the limit⎝braceleftbig (1 +η)±iImβ⎝bracerightbig η→–1does not exist. To obtain an equation for the singularity exponent α, one should write the expressions for the singular part (8) of equation (1) and its conjugate (9) for t→+1, and then argue as above. Omitting intermediate calculations, we obtain the following transcendental equation for α: ∆+ 11(α) ∆+ 11(α)–∆+ 12(¯α) ∆+ 12(¯α) = 0, (34) where ∆+ 11(α)=a(+1) – icot(πα)b(+1) +π sin(πα)⎝summationdisplay l,m,n,s⎝bracketleftbigg Bl(+1)(–1)neiθs(α–n)(α+m)(α+m– 1)...( α–n+1 ) (m+n)!⎝bracketrightbigg , (35) ∆+ 12(¯α)=π sin(π¯α)⎝summationdisplay l,m,n,s⎝bracketleftbigg Dl(+1)(–1)neiθs(¯α–n)(¯α+m)( ¯α+m– 1)...( ¯ α–n+1 ) (m+n)!⎝bracketrightbigg . (36) Thus, the problem of ﬁnding the exponents of the asymptotic solution of equation (1) at the endpoints of the integration interval has been reduced to two independent transcendental equations (31) and (34) for these exponents. The roots of these equations lying in the strip –1 < Re α,R eβ<0 ,a r et h e desired singularity exponents in the weight function (2), and the corresponding root with the minimalreal part is the leading exponent of the singularity in the solution of equation (1) at a given endpoint of the integration interval. 15.5. A NALYSIS OF SOLUTIONS SINGULARITIES FOR COMPLETE INTEGRAL EQUATIONS 789 15.5-4. Analysis of Equations for Singularity Exponents. Let us give a theoretical analysis of possible solutions of equations (31) and (34). For deﬁniteness, consider equation (31). Using the relations between the terms involved in that equation, we can transform it to⎝parenleftbig |∆– 11(β)|–|∆– 12(¯β)|⎝parenrightbig⎝parenleftbig |∆– 11(β)|+|∆– 12(¯β)|⎝parenrightbig = 0. (37) It is easy to see that the left-hand side of the equation obtained is a real-valued function of the complex singularity exponent β, and the zeroes of this function can be found by equating to zero its ﬁrst and its second factors, which are also real-valued functions. Equating to zero the ﬁrst factor in (37), we obtain the equation |∆– 11(β)|–|∆– 12(¯β)|= 0, (38) which can be considered as an (implicit) equation g(x,y) = 0 of some curve on the plane xy,w h e r e xandyare, respectively, the real and the imaginary parts of the exponent β(x=R eβ,y=I mβ). Therefore, if at least some part of this curve lies in the strip –1 < x< 0, then equation (1) allows for the existence of inﬁnitely many singularity exponents of its solution. In this situation, the quantity βcan be ﬁxed only if the unique solvability of equation takes place only under an additional condition, and this condition, in its turn, imposes certain constraints on the singularity exponents. The solvability conditions occurring in applications impose no const raints of that kind (see the references at the end of this section), and the theory of equation (1) with generalized kernels, which might give a deﬁnite answer in regard to such a condition, has not been developed to a sufﬁcient extent,* in spite of the fact that equations of type (1) quite often occur in problems of mechanics and mathematical physics. It is apparent from (33) (see also (5) and (7)), that it is the integral of the function Lg(t,τ) ϕ(τ)i n( 1 ) that is responsible for the appearance of the term |∆– 12(¯β)|in (38). If Lg(t,τ)≡0, this term is absent and (38) reduces to the equation ∆– 11(β) = 0, (39) whose left-hand side is a complex-valued function. This means that in this case there is a system of two real equations Re⎝bracketleftbig ∆– 11(x+iy)⎝bracketrightbig ≡h(x,y)=0 , Im⎝bracketleftbig ∆– 11(x+iy)⎝bracketrightbig ≡p(x,y)=0(40) for the real and the imaginary parts of the singularity exponent. Of course, in some special cases the curves h(x,y)=0a n d p(x,y)=0m a yh a v ei n ﬁ n i t e l ym a n yc o m m o np o i n t s( i . e . ,c o i n c i d eo naﬁ n i t e arcL). However in actual applied problems as a rule, there are ﬁnitely many points of intersection of these curves, and therefore, ﬁnitely many solutions of system (40), which are admissible singularity exponents β=x+iyfor solutions of equation (1). Thus, analysis of equation (38) shows that the integral (with generalized kernel) of the conjugate of the unknown function in equation (1) leads to a qualitatively new behavior of the singularity exponent: equation (31) deﬁnes inﬁnitely many singularity exponents admissible for solutions of equation (1) (provided that a ﬁnite part of the curve g(x,y) = 0 associated with equation (38) belongs to the strip –1 < x<0 ) . Equating to zero the second factor in (37) brings us to the system of equations ∆– 11(β)=0 , ∆– 12(¯β) = 0, (41) whose left-hand sides are complex-valued functions. This system is overdetermined, since it imposes four real conditions on two real unknown variables xandy(β=x+iy). Although (41) is * One of the rare publications in this area is the monograph by Duduchava (1979) that dealt with singular equations with generalized kernels not containing integrals of the conjugate of the unknown function. 790 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS an overdetermined system, it cannot be excluded that there exist βin the strip –1 < x< 0 satisfying (41), and this shows that the second factor on the left-hand side of (37) should be taken into accountwhen solving speciﬁc problems. Going back to equation (31), we note that a more narrow class of its solutions can be obtained a priori by taking a real β(Imβ≡0). In this case, equation (31) becomes ∆ – 11(β) ∆– 11(β)–∆– 12(β) ∆– 12(β) = 0, (42) where the conjugation applies only to the functions (32) and (33), not to the parameter β(¯βin (33) must be replaced by β). This real equation serves to determine real singularity exponents admissible for equation (1). As a rule, there are ﬁnitely many such exponents. There is an important point that should be mentioned in connection with the a priori assumption ofβbeing real. By similarity with an algebraic equation with real coefﬁcients, equation (42) may admit (mutually conjugate) complex roots, in particular. Of course, such roots should be excluded from consideration, since neither they nor their real parts satisfy the original equation (31), and, therefore, are inadmissible for equation (1). Selecting a root with the minimal real part (–1 < Re β< 0) in the solutions of equations (37) and (42) allows us to determine the leading singularity exponent of the solution of equation (1) at the left endpoint of the integration interval. Prior to solving equation (31), it is convenient to perform a regularization by extracting the factors 1 /sin(πβ)a n d1 /sin(π¯β) in (32) and (33), respectively. After the division of equation (31) by the factor 1/ |sin(πβ)|, which does not vanish in the strip –1 < Re β< 0, the left-hand side of the equation becomes an analytic function in a ﬁnite region of the complex plane x+iy=β.T h i s allows us to use the methods of the theory of analytic functions for solving the equations constructedabove. Similar arguments and remarks are valid for equation (34) for the singularity exponent α. Table 10 summarizes the above analysis and other results known about exponents of singularity of solutions of singular integral equations of the form (1). TABLE 10 Singularity exponents for solutions of different cases of integral equation (1) Functions involved in equation (1) Singularity exponents Qualitative character of singularities a(t)=0, Kg(t,τ)=0, Lg(t,τ)=0 α=β=–1/2 or α=–β=±1/2 Real singularity, unique in the interval –1<α,β<0 Kg(t,τ)=0, Lg(t,τ)=0 α=–1/2+iω,β=–1/2–iω or α=–β=±1/2+iω, where ω=–1 2πlnb(±1) –a(±1) b(±1) –a(±1) (upper sign corresponds to the exponent α, lower corresponds to β) Complex singularity, unique in the region –1<Re α,R eβ<0 Lg(t,τ)=0 are determined by the equations: ∆+ 11(α)=0,∆– 11(β)=0 Singularities are complex (in general) and form a discrete set, Reα,R eβ≠–1/2 Complete equation (1) are determined by the equations: ∆+ 11(α) ∆+ 11(α)–∆+ 12(¯α) ∆+ 12(¯α)=0, ∆– 11(β) ∆– 11(β)–∆– 12(¯β) ∆– 12(¯β)=0 Complex singularities have continuous distribution.Under the a priori assumption Imα=Imβ=0, real singularities form a discrete set Remark . Singularity exponents αandβare independent of the Fredholm kernels K(t,τ)a n d L(t,τ) in the integral equation (1). 15.5. A NALYSIS OF SOLUTIONS SINGULARITIES FOR COMPLETE INTEGRAL EQUATIONS 791 15.5-5. Application to an Equation Arising in Fracture Mechanics. As an application, we use the above approach to determine singularity exponents for an equation that arises in a two-dimensional elasticity problem for a rectilinear crack of unit half-length with a vertex on the interface between two materials with different elastic properties (Linkov, 1999). This problem can be reduced to the integral equation ⎝integraldisplay1 –1ϕ(τ) τ–tdτ+⎝integraldisplay1 –1Kg(t,τ)ϕ(τ)dτ+⎝integraldisplay1 –1Lg(t,τ) ϕ(τ)dτ=f(t), –1 < t< 1, (43) where Kg(t,τ)=A0 τ–z∗ 0+A1 τ–z∗ 1+A21+τ (τ–z∗ 0)2+A3(1 +τ)(1 +t) (τ–z∗ 0)3, A0=–χ1 2e2iγ,A1=χ2 2e–2iγ,A2=χ2 2(1 –e–2iγ)2e4iγ,A3=2a2e2iγ; Lg(t,τ)=C01+τ (τ–z∗ 0)2+C11+τ (τ–z∗ 1)2,C0=–χ2 2(1 –e–2iγ)e4iγ,C1=–χ2 2(1 –e2iγ)e–2iγ; z∗ 0=– 1+( 1+ t)eiθ0,0 < θ0=2γ<2π;z∗ 1=– 1+( 1+ t)eiθ1,0 < θ1=2 (π–γ)<2π; χ1≡κ2µ1–κ1µ2 µ2+κ2µ1,χ2≡µ2–µ1 µ1+κ1µ2, p(t)=f(t)/πis a self-balanced load on the crack surface, κr=3–4 νrfor the plane-strain state, andκr=( 3– νr)/(1 +νr) for the plane-stress state; νris the Poisson ratio, µris the shear modulus (r= 1, 2). The index 2 in the last expressions refers to the upper half-plane (i.e., µ2andν2are its elastic constants), and the index 1 refers to the lower half-plane with the crack whose line forms angleγwith the positive direction of the axis Oxassociated with the interface (0 < γ<π). Equation (43) has a unique solution in the class of functions that may go to inﬁnity at the endpoints of the integration interval, provided that an additional condition is satisﬁed. The condition is that the displacement jump at the endpoints of the crack is zero: ⎝integraldisplay1 –1ϕ(τ)dτ=0 . Since the kernels in (43) are bounded for τ=t→+1, the expression (36) and the last term in (35) are equal to zero, and from (34) we obtain the equation cot( πα)=0( a(t)≡0). The solution of this equation with the minimal real part (Re α> –1) is the root α1=– 1/2 corresponding to the common root singularity of the unknown function. Calculating the expressions (32) and ( 33) for the kernels in (43), we have ∆– 11(β)=F(β)⎝braceleftBig cos(πβ ) +e–iπβ⎝bracketleftBig A0eiβθ 0+A1eiβθ 1+A2eiβθ 0(β+1 )+1 2A3ei(β–1)θ0(β+1 )β⎝bracketrightBig⎝bracerightBig , ∆– 12(¯β)=F(¯β)e–iπ¯β⎝bracketleftBig C0ei¯βθ0(¯β+1 )+ C1ei¯βθ1(¯β+1 )⎝bracketrightBig ,(44) where F(β)=–π/sin(πβ). The complex solution of equation (31) with the minimal real part was obtained with the help of graphical analysis and numerical methods (the M ¨uller method, the chord method, and the golden section method). Figure 7 shows the dependence of the leading complex singularity exponent βon the angle γ within the range 0 < γ<π/2(β(π–γ)= ¯β(γ)) for ν1=ν2= 0.3 (plane strain) for two cases 792 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS /c45/c48/c46/c51/c48 /c45/c48/c46/c53/c53 /c45/c48/c46/c56/c48 /c48 /c112/c47/c52 /c103 /c112/c47/c50Im/c98 /c45/c48/c46/c48/c56/c48/c46/c48/c56 Re/c98Im/c98/c49/c48 Im/c98/c48/c46/c49Re/c98/c49/c48Re/c98/c48/c46/c49 0 Figure 7. Dependence of the leading complex singular- ity exponent βon the angle γresulting from Eq. (38)./c45/c48/c46/c56/c45/c48/c46/c53/c48/c46/c50 Re/c98/c48/c46/c48/c57 /c48/c46/c48/c52/c53 /c48 /c112/c47/c52 /c103 /c112/c47/c500Im/c98/c49/c48Im/c98/c48/c46/c49 Re/c98/c48/c46/c49 Re/c98/c49/c48Im/c98 Figure 8. Dependence of the leading singularity expo- nentβon the angle γresulting from Eq. (42). µ1/µ2=0 . 1a n d µ1/µ2= 10 (the respective exponents are labelled by β0.1andβ10). Note that the function g(x,y) = 0 (see the interpretation of equation (38) in Section 15.5-4) has a kink at the point corresponding to the root with the minimal real part. Note also that for the parameters of theproblem under consideration, numerical experiments have shown that the equation ∆ – 11(β)=0 ,t h e ﬁrst equation in system (41), has suitable roots (with –1 < Re β< 0) and the equation ∆– 12(β)=0 has no roots. Figure 8 gives calculation results for the singularity exponent which were obtained using sim- pliﬁed equation (42) with the parameters of the problem being the same. The set of roots of this equation consists of two real values and, for some parameters of the problem, two complex-conjugateones. The minimal real root is always greater than the corresponding real part of the complex root (Fig. 7), except for the points at which its imaginary part changes sign. References for Section 15.5: F. E. Erdogan (1975), F. D. Gakhov (1977, 1990), R. Duduchava (1979), A. F. Nikiforov and V . B. Uvarov (1988), N. I. Muskhelishvili (1992), W. H. Press et al. (1992), M. P. Savruk et al. (1999), A. M. Linkov (2002), A. V . Andreev (2007). 15.6. Direct Numerical Solution of Singular Integral Equations with Generalized Kernels∗ 15.6-1. Preliminary Remarks. Below, we describe some approaches to the direct numerical solution of integral equations with generalized kernels of the Cauchy type (see Section 15.5). These approaches are based on the method of collocation and are more or less traditional, but due to the class of equations examinedhere have some speciﬁc features which require some special considerations. The ﬁrst characteristic feature of the class of equations considered here is the presence of nontrivial (generally complex) singularities of the solution at the endpoints of the integration interval. In order to obtain integral (nonlocal) characteristics of solutions of equations with generalized kernels, one can adopt well-known numerical approaches that do not take into account the asymptotic behavior of a solution near its singular points at the ends of the integration interval. On the otherhand, numerical experiments show that in some situations such methods (for instance, the method of discrete vortices) applied to int egral equations with generalized kernels give inadequate results, even if used to ﬁnd integral characteristics of a solution (see the next paragraph). Moreover, it is of special interest to obtain a fairly accurate local distribution of solution within the framework of the process of its numerical construction, which requires utilization of methods explicitly taking * Section 15.6 was written by A. V . Andreev. 15.6. D IRECT NUMERICAL SOLUTION OF SINGULAR INTEGRAL EQUATIONS WITH GENERALIZED KERNELS 793 into account asymptotic behavior of solutions. In particular, it is very important for construction of correct solution asymptotics near the endpoints of the integration interval. Thus, direct numericalsolution of singular integral equations with generalized kernels presumes that one has to ﬁnd a bounded function u(τ)*, while the Jacobi weight function w(τ) with singularities is supposed to be known from preliminary analysis, and its asymptotic behavior at the ends of the integration interval is explicitly taken into account in numerical approximations of integrals and other calculations. The second characteristic feature of equations with generalized kernels is that, as a rule, the analytic continuation of integral kernels (in the integration variable) has singularities outside the integration line, and this fact necessitates t he application of high-preci sion quadrature methods for the numerical approximation of integrals with su ch kernels. In this connection, quadrature formulas of the highest algebraic accuracy (like the Gauss method) are used below, and quadrature formulas of interpolation type are used to ensure greater ﬂexibility of the collocation method. The material presented below can be divided into two parts: ﬁrst, in Sections 15.6-2 to 15.6-4 we describe auxiliary numerical-analytical results, and then, in Sections 15.6-5, 15.6-6 we apply them to the construction of solutions to singular integral equations; in particular, we give examples of their numerical realization and compare its results with exact analytical solutions. 15.6-2. Quadrature Formulas for Integrals with the Jacobi Weight Function. For the numerical approximation of a nonsingular integral with the weight function w(τ) in the form of a sum we use the Gauss–Jacobi quadrature formula (of the highest algebraic precision): ⎝integraldisplay1 –1u(τ)w(τ)dτ=n⎝summationdisplay k=1Wku(τk), Re α,R eβ> –1. (1) Here q(α,β) n(t)=⎝integraldisplay1 –1w(τ)P(α,β) n(τ) τ–tdτ,Wk=q(α,β) n(τk) [P(α,β) n(τk)]/prime,( 2) andP(α,β) n(τ) is the Jacobi polynomial deﬁned by P(α,β) n(τ)=(–1)n 2nn!(1 –τ)–α(1 +τ)–βdn dτn⎝bracketleftBig (1 –τ)α+n(1 +τ)β+n⎝bracketrightBig =2–nn⎝summationdisplay m=0Cm n+αCn–m n+β(τ–1 )n–m(τ+1 )m,(3) Ca bare binomial coefﬁcients, and the nodes τkof the quadrature formulas form the set of roots of this polynomial, P(α,β) n(τk)=0 , k=1 ,2 , ...,n.( 4 ) Formula (1) is exact if u(τ) is a polynomial of a degree ≤2n– 1 (or brieﬂy, u(τ)∈ 2n–1). In formula (2) and below we use the notation [ F(τk)]/prime=dF dτ⎝vextendsingle⎝vextendsingle τ=τk. For Re α> –1, Re β> –1, and Im α=I mβ= 0, the roots of the Jacobi polynomial are simple and belong to the interval τ∈(–1, 1). The quadrature formula (1) remains valid in the case of complex values of αandβ, but in this case the roots of the Jacobi polynomial also turn out to be complex (Imτk≠0) and lie near the interval τ∈(–1, 1). * Recall that a solution of a singular integral equation is sought in the form of the product ϕ(τ)=u(τ)w(τ), where w(τ)=( 1– τ)α(1 +τ)β,– 1 ≤τ≤1, Re α,R eβ>– 1 (see formulas (2)–(3) in Subsection 15.5-1). 794 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS Note that the function q(α,β) n(t) can be easily expressed through the Jacobi function of the second kindQ(α,β) n(t) and the weight function w(τ): q(α,β) n(t)=w(t)Q(α,β) n(t), (5) and the zeroes of these functions coincide on the interval t∈(–1, 1). For a singular integral with the Cauchy kernel, the following modiﬁcation of the Gauss–Jacobi quadrature formula holds: ⎝integraldisplay1 –1u(τ)w(τ) τ–tdτ=n⎝summationdisplay k=1Wku(τk) τk–t+u(t)q(α,β) n(t) P(α,β) n(t),– 1 < t≠τk<1 , R e α,R eβ> –1. (6) This formula is exact if u(τ)∈ 2n(i.e.,u(τ) is a polynomial of degree ≤2n). For a discrete set of points tmsuch that (see (5)) Q(α,β) n(tm)=0 , tm≠τk,( 7) the quadrature formula (6) becomes similar to (1): ⎝integraldisplay1 –1u(τ)w(τ) τ–tmdτ=n⎝summationdisplay k=1Wku(τk) τk–tm.( 8) For the restoration of the values of the unknown function u(τ) on the entire interval τ∈[–1, 1] from its values on the discrete set τk(k=1 ,2 , ...,n), one can use the Lagrange interpolation polynomial, which it is convenient to write in the following form (since the interpolation is with respect to the zeroes of the Jacobi polynomial): u(τ)=P(α,β) n(τ)n⎝summationdisplay k=1u(τk) (τ–τk)[P(α,β) n(τk)]/prime.( 9) This interpolation representation is exact if u(τ) is a polynomial of a degree ≤n– 1. Note that the representation (9) may be useful for the approximation of the term outside the integral in a singular equation of the second kind. Moreover, on the basis of the approximation (9), one can construct quadrature formulas of interpolation type for a singular integral. Substituting (9) into (6), we obtain the following quadratureformula for the singular integral: ⎝integraldisplay 1 –1u(τ)w(τ) τ–tdτ=n⎝summationdisplay k=1W(s) k(t)u(τk), –1 < t<1 , R e α,R eβ> –1, (10) which is precise for u(τ)∈ n–1. Here, W(s) k(t)=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩q (α,β) n(τk)–q(α,β) n(t) [P(α,β) n(τk)]/prime(τk–t)ift≠τk, [q(α,β) n(τk)]/prime [P(α,β) n(τk)]/primeift=τk.(11) Note that the lower expression for the weight in (11) is obtained from the upper one by passing to the limit as t→τk. It can be seen that the quadrature formula (10) for the singular integral (unlike the similar formula (6)) holds also at t=τk. Moreover, formula (10) yields an expression which, in contrast to (6), is an approximation based only on the density values at the nodes of the quadrature formula (see also (8)). 15.6. D IRECT NUMERICAL SOLUTION OF SINGULAR INTEGRAL EQUATIONS WITH GENERALIZED KERNELS 795 15.6-3. Approximation of Solutions in Terms of a System of Orthogonal Polynomials. As mentioned in the previous subsection, for Im α≠0o rI m β≠0, the roots of the Jacobi polynomial are complex (Im τk≠0). This is an obstacle to the utilization of the above quadrature formulas for the approximation of integrals, since it becomes necessary to ﬁnd the unknown function of the integral equation outside its domain τ∈[–1, 1]. To construct a solution of an equation with complex asymptotics at the endpoints of the integra- tion interval, let us represent the unknown function u(τ) in the form of expansion in terms of a ﬁnite system of Jacobi polynomials which are orthogonal on the segment [–1, 1] with weight functionw(τ): u(τ)= n⎝summationdisplay k=0ckP(α,β) k(τ). (12) Here,ckare complex constants to be determined, and P(α,β) k(τ) is a Jacobi polynomial of real argu- mentτwith complex αandβ. Such a representation allows us to perfo rm analytical integration of the singular integral in terms of special functions on the basis of the following integral representation of the Jacobi function of the second kind Q(α,β) k(t)( s e e( 2 ) ,( 5 ) ) : ⎝integraldisplay1 –1w(τ)P(α,β) k(τ) τ–tdτ=w(t)Q(α,β) k(t), –1 < t< 1. (13) When using (12) for the approximation of a solution of an integral equation, one has to deal with integrals of the form hk(t)≡1 w(t)⎝integraldisplay1 –1k(t,τ)w(τ)P(α,β) k(τ)dτ,– 1 < t< 1, (14) where k(t,τ) is a generalized kernel. In general, such a kernel (see, for instance, (4) and (5) in Section 15.5) is nonanalytic for τin a neighborhood of the segment τ∈[–1, 1] on the complex plane. At the same time, if for a speciﬁc kernel one can separate its poles from the region of the complex roots of the Jacobi polynomial , then a direct and fairly precise approach to the calculation of integrals (14) can be realized by the method of mechanical Gauss–Jacobi quadratures (1). If such an operation is impossible or entails very difﬁcult calculations, the following technique can be used. Let us approximate the kernel k(t,τ) by a degenerate kernel in the form of a polynomial of degree Nwith respect to τ: k(t,τ)=N⎝summationdisplay s=0cs(t)τs,t∈(–1, 1). (15) Obviously, the Jacobi polynomials can be represented in a similar form P(α,β) k(τ)=k⎝summationdisplay l=0g(k) lτl. (16) Here, the superscript in the coefﬁcients g(k) lrefers to the highest degree of the polynomial. Using (15) and (16), we obtain the following expression for the integral (14): hk(t)=1 w(t)N+k⎝summationdisplay j=0d(k) j(t)⎝integraldisplay1 –1τjw(τ)dτ. (17) 796 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS Here, the coefﬁcients d(k) j(t) can be written in the form d(k) j(t)=j⎝summationdisplay s=0cs(t)g(k) j–s, (18) which is obtained on the basis of multiplication of the series (15) and (16). Note that in the sum (18), one should take cs(t)=0f o r s>Nandg(k) j–s=0f o r j–s>k. To calculate the integrals involved in (17), we use the identity ⎝integraldisplay1 –1τjw(τ)dτ=⎝integraldisplay1 –1τj+1w(τ) τ–t∗dτ–t∗⎝integraldisplay1 –1τjw(τ) τ–t∗dτ. (19) Here and in what follows, –1 < t∗< 1 is a ﬁxed auxiliary parameter. The integrals in the right-hand side of the last expression can be calculated with the help of (13), which, in view of (16), can be represented in the form Q(α,β) m(t∗)=1 w(t∗)m⎝summationdisplay l=0g(m) l⎝integraldisplay1 –1τlw(τ) τ–t∗dτ =1 w(t∗)⎝bracketleftbigg g(m) 0⎝integraldisplay1 –1w(τ) τ–t∗dτ+g(m) 1⎝integraldisplay1 –1τw(τ) τ–t∗dτ+···+g(m) m⎝integraldisplay1 –1τmw(τ) τ–t∗dτ⎝bracketrightbigg . The last expression implies that the integrals of the form Im(t∗)≡⎝integraldisplay1 –1τmw(τ) τ–t∗dτ=1 g(m) m⎝bracketleftbigg w(t∗)Q(α,β) m(t∗)–m–1⎝summationdisplay p=0g(m) pIp(t∗)⎝bracketrightbigg , (20) I0(t∗)=w(t∗)Q(α,β) 0(t∗) can be calculated on the basis of th e above recurrent relation (note that g(0) 0=1 ) . Let us introduce the function Sm(t∗)≡Im(t∗) w(t∗)=1 g(m) m⎝bracketleftbigg Q(α,β) m(t∗)–m–1⎝summationdisplay p=0g(m) pSp(t∗)⎝bracketrightbigg , (21) S0(t∗)=Q(α,β) 0(t∗), calculated on the basis of a similar recurrent relation. Substituting (19) into (17) and using (20), (21), we ﬁnally obtain hk(t)=w(t∗) w(t)N+k⎝summationdisplay j=0d(k) j(t)⎝bracketleftbig Sj+1(t∗)–t∗Sj(t∗)⎝bracketrightbig ,– 1 < t< 1. (22) Note that the above approach to the calculation of integrals (14) based on the expansion of the generalized kernel in power series (15) might be especially convenient if the kernel k(t,τ) cannot be expressed explicitly and one has to use its representation as an integral. In such a situation, therepresentation (15) can be obtained with the help of the expansion of the integrand in power series with respect to τand subsequent analytical integration of that series. 15.6. D IRECT NUMERICAL SOLUTION OF SINGULAR INTEGRAL EQUATIONS WITH GENERALIZED KERNELS 797 15.6-4. Some Special Functions and Their Calculations. In order to implement the above methods for the approximation of integrals with the Jacobi weight function, it is necessary to calculate special functions and their roots with great precision and efﬁciency. Next, we sum up the results necessary for the implementation of the methods and approaches proposed above (see also the references at the end of this section). To calculate Jacobi polynomials and their set, as well as the corresponding functions of the second kind, it is convenient to use a single recurrent procedure based on the relation anΨ(α,β) n+1(τ)=(bn+cnτ)Ψ(α,β) n(τ)–dnΨ(α,β) n–1(τ), (23) an=2 (n+1 ) (n+α+β+ 1)(2n +α+β), bn=( 2n+α+β+1 ) (α2–β2), cn=( 2n+α+β)(2n+α+β+ 1)(2n +α+β+2 ) , dn=2 (n+α)(n+β)(2n+α+β+2 ) . Here and henceforth in this subsection, we use the symbol Ψto denote the polynomial Pand the function of the second kind Qif they satisfy identical relations. For the coefﬁcients b(k) lof a Jacobi polynomial of the form (16), one can construct recurrent relations that can be used for the determination of the coefﬁcients of a polynomial of degree n+1 in terms of the coefﬁcients of polynomials of smaller degrees nandn– 1. Thus, substituting (16) into (23) and equating the coefﬁcients of equal powers of τ, we obtain anb(n+1) l=bnb(n) l+cnb(n) l–1–dnb(n–1) l,anb(n+1) n+1=cnb(n) n,b(n–1) n =b(n) –1=0 , l=0 , 1 , ...,n. (24) Since the derivative of a Jacobi polynomial (function of the second kind) is expressed through two consecutive polynomials (functions) of the corresponding orders with the same parameters α, βand argument τ: (1 –τ2)[Ψ(α,β) n(τ)]/prime=( ˜an+˜bnτ)Ψ(α,β) n(τ)+˜cnΨ(α,β) n–1(τ); (25) ˜an=n(α–β) 2n+α+β,˜bn=–n,˜cn=2(n+α)(n+β) (2n+α+β), (26) its calculation reduces to the calculation of coefﬁcients (26) and their substitution into (25) on the ﬁnal stage of the recurrent procedure (23). As the initial values in (23), one can use the Jacobi polynomials of the zero and the ﬁrst orders, P(α,β) 0(τ)=1 , P(α,β) 1(τ)=1 2(α–β)+1 2(2 +α+β)τ, and for a function of the second kind, use the initial values obtained from the explicit expression Q(α,β) n(τ)=–πP(α,β) n(τ) tan(πβ)+(–1)n2α+β w(τ)B(n+α+1 ,β)F⎝parenleftBig n+1 ,–n–α–β,1–β;1+τ 2⎝parenrightBig =πP(α,β) n(τ) tan(πα)–2α+β w(τ)B(n+β+1 ,α)F⎝parenleftBig n+1 ,–n–α–β,1–α;1–τ 2⎝parenrightBig , (27) where B(x,y) is the beta function, F(a,b,c;z) is the hypergeometric function, and –1 < τ<1 . 798 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS For the calculation of the hypergeometric function for –1 < z=( 1±τ)/2 < 1, one can use its representation as the Gauss series (see Supplement 11.10) F(a,b,c;z)=1+∞⎝summationdisplay m=1(a)m(b)m (c)mzm m!,(a)m=a(a+1 )...(a+m– 1). (28) The beta function is related to the gamma function Γ(x)b yB(x,y)=Γ(x)Γ(y)/Γ(x+y), and the latter can be calculated quite accurately by the Lanczos approximation Γ(z)=√ 2π(z+C1–1/2)z–1/2 ez+C1–1/2⎝parenleftbigg s1+m⎝summationdisplay k=2sk z+k–2⎝parenrightbigg ,z≠0, –1, –2, ... (29) Form= 15, the coefﬁcients of the approximation (29) have the form C1= 607/ 128, s1= 0.999999999999997, s2= 57.15623566586292, s3= –59.59796035547549, s4= 14.13609797474174, s5= –0.491913816097620, s6= 0.339946499848118 ×10–4,s7= 0.465236289270485 ×10–4, s8= –0.983744753048795 ×10–4,s9= 0.158088703224912 ×10–3, s10= –0.210264441724104 ×10–3,s11= 0.217439618115212 ×10–3, s12= –0.164318106536763 ×10–3,s13= 0.844182239838527 ×10–4, s14= –0.261908384015814 ×10–4,s15= 0.368991826595316 ×10–5. Note that the above methods for the calculation o f special functions are app licable for both real and complex parameters and arguments of these functions. Moreover, for real values one can obtain explicit expressions for the nodes and the weights in quadrature formulas. These expressions were obtained for large values of the discretizationparameter, n/greatermuch1. In this case, the Jacobi polynomials and the integral (2) can be written in terms of elementary functions: P (α,β) n(cosθ)=cos⎝braceleftbig⎝bracketleftbig n+(α+β+1 )/2⎝bracketrightbig θ–( 2α+1 )π/4⎝bracerightbig √ πn(sin(θ/2))α+1/2(cos(θ/2))β+1/2+O(n–3/2), 0 < θ<π, (30) q(α,β) n(cosθ)=2α+β⎝radicalbigg π nsin⎝braceleftbig⎝bracketleftbig n+(α+β+1 )/2⎝bracketrightbig θ–( 2α+1 )π/4⎝bracerightbig (sin(θ/2))–α+1/2(cos(θ/2))–β+1/2+O(n–3/2). (31) On the basis of these results, one obtains the following approximate expressions for the nodes and the weights in the quadrature formulas: τk≈cosθk,θk=2α–1+4 k 2n+α+β+1π 2,k=1 ,2 , ...,n; (32) Wk≈2π 2n+α+β+1⎝radicalBig 1–τ2 k(1 –τk)α(1 +τk)β. (33) Note that these expressions for the nodes and weights are precise for α=±1/2a n dβ=±1/2f o r anyn. In the general case, the real roots τkof a Jacobi polynomial (or function of the second kind) can be calculated by means of the following algorithm. Choosing a suitable initial approximation τ(1) kfor the kth root, its value can be found, quickly enough and with given accuracy, in an iteration 15.6. D IRECT NUMERICAL SOLUTION OF SINGULAR INTEGRAL EQUATIONS WITH GENERALIZED KERNELS 799 process based on the Newton method (of tangential lines), by consecutively reﬁning the position of a root with the help of the expression ( iis the number of the iteration) τ(i+1) k=τ(i) k–Ψ(α,β) n(τ(i) k) [Ψ(α,β) n(τ(i) k)]/prime,i=1 ,2 , ... (34) It is possible to choose initial approximations for (real) roots of Jacobi polynomials in the form τ(1) 1=1–(1 +α)⎝bracketleftbig 2.78/ (4 +n2)+0 . 7 6 8 α/n2⎝bracketrightbig 1 + 1.48α/n +0 . 9 6 β/n + 0.452α2/n2+0 . 8 3 αβ/n2, τ(1) 2=˜τ1–(1 – ˜τ1)(4.1 + α) (1 +α)(1 + 0.156α )⎝bracketleftbigg 1+0.06(n– 8)(1 + 0.12 α) n⎝bracketrightbigg⎝bracketleftbigg 1+0.012β (1 + 0.25 |α|) n⎝bracketrightbigg , τ(1) 3=˜τ2–(˜τ1–˜τ2)1.67 + 0.28α 1+0 . 3 7 α⎝bracketleftbigg 1+0.22(n–8 ) n⎝bracketrightbigg⎝bracketleftbigg 1+8β (6.28 + β)n2⎝bracketrightbigg , τ(1) k=3 ˜τk–1–3˜τk–2+˜τk–3,3 < k<n–1 , τ(1) n–1=˜τn–2+(˜τn–2–˜τn–3)1+0 . 2 3 5 β 0.766 + 0.119β⎝bracketleftbigg 1+1 + 0.639( n–4 ) 1+0 . 7 1 ( n–4 )⎝bracketrightbigg–1⎝bracketleftbigg 1+20α (7.5 + α)n2⎝bracketrightbigg–1 , τ(1) n=˜τn–1+(˜τn–1–˜τn–2)1 + 0.37β 1.67 + 0.28β⎝bracketleftbigg 1+0.22(n–8 ) n⎝bracketrightbigg–1⎝bracketleftbigg 1+8α (6.28 + α)n2⎝bracketrightbigg–1 . Here, the quantities marked with tilde denote approximate values of the roots of the polynomial which were obtained as a result of previous iteration processes (as regards this process). Initial approximations for (real) roots of a function of the second kind may be chosen in the form ˜t(1) 1=1–n–5/2(α+1/2)2(1 + ˜τ1),α>– 1/2; ˜t(1) k=( ˜τk+˜τk–1)/2,k=2 ,3 , ...,n; ˜t(1) n+1=– 1– n–5/2(β+1/2)2(–1 + ˜τn),β>– 1/2.(35) When calculating complex roots of Jacobi polynomials (Im α≠0o rI m β≠0), one can take as the initial approximation τ(1) kin (34) the roots of the real polynomial P(Reα,Reβ) n (τ), which can be found from the equation P(Reα,Reβ) n (τ(1) k)=0 , k=1 ,2 , ...,n. 15.6-5. Numerical Solution of Singular Integral Equations. Consider a complete singular integral equation of the ﬁrst kind ⎝integraldisplay1 –1ϕ(τ)dτ τ–t+⎝integraldisplay1 –1k(t,τ)ϕ(τ)dτ=f(t), –1 < t< 1. (36) Using approximations of the integrals (1) and (6), we write it in the form u(t)q(α,β) n(t) P(α,β) n(t)+n⎝summationdisplay k=1Wku(τk)⎝bracketleftbigg1 τk–t+k(t,τk)⎝bracketrightbigg =f(t), –1 < t<1 , t≠τk. (37) Next, one can realize several versions of the constr uction of a complete system of algebraic equations for the values u(τk)(k=1 ,2 , ...,n) on the basis of the collocation method. 800 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS If the function of the second kind Q(α,β) n(t)( s e e( 5 )a n d( 7 ) )h a s ≥nzeroes on the interval t∈(–1, 1), then, using these zeroes as collocation points (see also (8)), one can construct the complete system of algebraic equations n⎝summationdisplay k=1Wku(τk)⎝bracketleftbigg1 τk–tm+k(tm,τk)⎝bracketrightbigg =f(tm),m=1 ,2 , ...,n. (38) This situation takes place in the case of α>– 1/2o rβ>– 1/2 (see (35)). If neither of these conditions holds, then the function Q(α,β) n(t)h a sn– 1 zeroes on the interval t∈(–1, 1), and one should use the following approaches. First of all, we note that in actual applied problems,equation (36) whose solution has singularities in the region –1 < α,β≤–1/2 as a rule is accompanied by the condition ⎝integraldisplay1 –1ϕ(τ)dτ=A (39) (Abeing a known constant), whose quadrature analogue (see (1)) n⎝summationdisplay k=1Wku(τk)=A (40) allows us to complete the algebraic system of equations. At the same time, in cases not covered by this rule, one can use (9) for the interpolation of the unknown function in the ﬁrst term in (37) and the construction of a complete system of linear equations on an arbitrary set of collocation points –1 < tm≠τk<1(m=1 , 2 , ...,n). An equivalent approach is to use the quadrature formula (10) for the approximation of the singular integral in (36),and in the latter case, collocation points can be chosen coincident with the nodes of the quadrature formulas t m=τk(m,k=1 ,2 , ...,n): n⎝summationdisplay k=1u(τk)⎝bracketleftbigg W(s) k(tm)+Wkk(tm,τk)⎝bracketrightbigg =f(tm),k,m=1 ,2 , ...,n. (41) The last approach is especially convenient in that the special functions necessary for its realization are calculated only for a single system of points, namely, for the nodes of the quadrature formulas. Note that an important feature of all approaches described above and realized in the framework of the collocation method is the utilization of a quadrature formula of the highest algebraic precisionfor the approximation of an integral containing a generalized kernel k(t,τ). Consider the complete singular integral equation of the second kind with generalized kernel k(t,τ): a(t)ϕ(t)+b(t) πi⎝integraldisplay1 –1ϕ(τ)dτ τ–t+⎝integraldisplay1 –1k(t,τ)ϕ(τ)dτ=f(t), –1 < t< 1. (42) If the solution of this equation has real singularities at the ends of the integration interval, then a numerical approximate solution can be constructed by quadrature-collocation methods similar to those described above, with the interpolation polynomial (9) used for the term outside the integral. However, in many actual applied problems, singularities of a solution of equation (42) are complex,and this requires the approach described below. Using the approximation (12), from (13) and (42) we get: w(t)n⎝summationdisplay k=0ck⎝bracketleftbigg a(t)P(α,β) k(t)+b(t) πiQ(α,β) k(t)+hk(t)⎝bracketrightbigg =f(t), –1 < t< 1. (43) 15.6. D IRECT NUMERICAL SOLUTION OF SINGULAR INTEGRAL EQUATIONS WITH GENERALIZED KERNELS 801 To obtain a system of linear algebraic equations for the unknown coefﬁcients ck(k=0 ,1 , ...,n), the expression (43) can be written for the corresponding number of collocation points tr: w(tr)n⎝summationdisplay k=0ck⎝bracketleftbigg a(tr)P(α,β) k(tr)+b(tr) πiQ(α,β) k(tr)+hk(tr)⎝bracketrightbigg =f(tr), (44) r=0 ,1 , ...,n,– 1 < tr<1 . If equation (42) is accompanied by a condition of the form (39), it is necessary to use a slightly modiﬁed approach. Using (12), let us rewrite the add itional condition (39) in the form n⎝summationdisplay k=0ck⎝integraldisplay1 –1w(τ)P(α,β) k(τ)dτ=A. (45) From the orthogonality condition of the Jacobi polynomials on the interval τ∈[–1, 1] with the weight function w(τ), we have ⎝integraldisplay1 –1w(τ)P(α,β) k(τ)dτ=⎝braceleftbigg 2α+β+1B(α+1 ,β+1 ) i f k=0 , 0i fk>0 . Thus, condition (45) immediately allows us to ﬁnd one of the unknown constants: c0=2–1–α –β B(α+1 ,β+1 )A. (46) Therefore, in this situation it is necessary to take k= 1 in the lower limit of the sum (44), decrease the number of collocation points trby 1, and determine the unknown constant c0from (46). Note that in calculating expressions (14) according to (22), when constructing system (44), it is convenient to choose the auxiliary point t∗∈(–1, 1), introduced in (19), to be coincident with one of the points tr. This allows us to reduce calculations by using in (21) the values Q(α,β) k(tr) (r=0 , 1 , ...,n) obtained on the stage of calculations of the second term in the sum (44). Numerical experiments show that the accuracy of a solution is little affected by which point tris chosen as the auxiliary point. Thus, solving integral equations on the basis of the collocation method amounts to solving of systems of linear algebraic equations, which allows us to determine the coefﬁcients of the approximation of the unknown function or its values on a discrete set of points. Note that theapproaches described above can be directly extended to the case of an equation also containing an integral (with regular or generalized kernel) of the complex-conjugate of the unknown function (see (1) in Section 15.1). 15.6-6. Numerical Solutions of Singular Integral Equations of Bueckner Type. Example 1. Consider a singular integral equation of Bueckner type (1966): ⎝integraldisplay1 –1ϕ(τ)dτ τ–t+⎝integraldisplay1 –1ϕ(τ)dτ τ+t+2=πh(t), –1 < t<1 . (47) This equation with generalized kernel k(t,τ)=1/(τ+t+ 2) has a unique solution, which, for h(t)=q= const, can be expressed in terms of elementary functions, ϕ(τ)=q1+τ √ 1–τ√ 3+τ. (48) 802 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS It can be seen that in this case the weight function, which reﬂects the asymptotic behavior of the solution at the ends of the integration integral, has the form w(τ)=( 1– τ)–1/2(1 +τ). (49) Table 11 gives the results of numerical solution of equation (47) for h(t)=q= 1. This numerical solution was obtained in the framework of the quadrature-collocation approach (38) on the set of ten points τklisted in the second column. The third column lists the values of the exact analytical solution (48) calculated at these points for q=1 . TABLE 11 Comparison of exact and numerical solutions of the Bueckner equation (47) k Points τk Exact solution (48) Numerical solution 1 0.9893260 9.64037 9.64044 2 0.9052947 3.13291 3.13293 3 0.7443578 1.78291 1.78293 4 0.5201627 1.16966 1.16968 5 0.2517209 0.80245 0.80246 6 –0.0382037 0.54848 0.54850 7 –0.3250257 0.35852 0.35854 8 –0.5844217 0.21242 0.21246 9 –0.7943919 0.10335 0.10342 10 –0.9371120 0.03145 0.03172 As one can see from the table, the numerical solution is accurate to four or ﬁve signiﬁcant digits for the ﬁrst nine τk even for only n= 10. Note that the relative error increases towards the left endpoint of the interval (–1, 1), where it reaches εmax= 0.83%. This increase is due to the fact that as tdecreases, the pole of the function g(z)=1/(z+t+ 2) approaches the left endpoint of the interval, which worsens the quadrature approximation of the integrals. Example 2. Now consider a more general Bueckner equation ⎝integraldisplay1 –1ϕ(τ)dτ τ–t+D⎝integraldisplay1 –1ϕ(τ)dτ τ+t+2=πh(t), –1 < t<1 , (50) where Dis a complex constant. Numerical experiments show that, as a rule, the roots of the Jacobi polynomial (4) for complex αandβlie in the strip |Reτk|< 1 in a small neighborhood of the segment τ∈[–1, 1] ( |Imτk|/lessmuch1), and these roots approach this segment with the growth of n. Only if the imaginary parts of the singularity exponents are sufﬁciently large, the roots can lie outside the strip, but in this case the following estimate holds: |Reτk|<1+ε,0 <ε/lessmuch1. This justiﬁes the utilization of the ﬁrst method proposed in Subsection 15.6-2 for the calculation of integrals (14), since the (real) pole of the generalized kernel satisﬁes theinequality z ∗< –1. Note that in order to obtain a unique solution of equation (50) for D≠1, an additional condition should be introduced: ⎝integraldisplay1 –1ϕ(τ)dτ=0 . (51) Forh(t)≡q= const, the solution of equation (50) can be written in closed form, ϕ(τ)=q π√ 2(1 +D)⎝bracketleftbigg⎝parenleftbiggθ √ 1–x2+1⎝parenrightbigg⎝parenleftbiggx 1+√ 1–x2⎝parenrightbiggθ +⎝parenleftbiggθ √ 1–x2–1⎝parenrightbigg⎝parenleftbiggx 1+√ 1–x2⎝parenrightbigg–θ⎝bracketrightbigg , (52) where x=( 1+ τ)/2a n dθ= arccos(– D)/πis complex. We see that the asymptotic behavior of the solution near the left endpoint of the integration interval has the form ρ–θas ρ→0 and at the right endpoint the solution has a root singularity. This corresponds to the weight function w(τ)=( 1– τ)–1/2(1 +τ)–θ. (53) 15.6. D IRECT NUMERICAL SOLUTION OF SINGULAR INTEGRAL EQUATIONS WITH GENERALIZED KERNELS 803 For the construction of a numerical solution of equation (50) we use two methods: (i) integrals (14) are calculated according to the Gauss–Jacobi quadrature formulas (1): hk(t)=1 w(t)s⎝summationdisplay k=1Wkk(t,τk)P(α,β) k(τk),k(t,τ)=1 τ+t+2; (54) (ii) the method based on the expansion of this kernel into series (15). In case (ii), we use two types of polynomial approximation: Maclaurin series k(t,τ)=N⎝summationdisplay s=0as(t)τs,as(t)=(–1)s (t+2 )s+1, (55) and the expansion with respect to Chebyshev polynomials of the ﬁrst kind Tn(τ): k(t,τ)=–g0(t) 2+N⎝summationdisplay m=0gm(t)Tm(τ),gj(t)=2 N+1N+1⎝summationdisplay k=1k(t,τk)Tj(τk), (56) τk=c o s⎝parenleftbigg2k–1 N+1π 2⎝parenrightbigg ,Tj(τk)=c o s⎝parenleftbigg2k–1 N+1πj 2⎝parenrightbigg . Note that the coefﬁcients as(t) in the expansion (15) can be obtained from the coefﬁcients gm(t) (56) by the method used above for the derivation of (18). Astr(r=1 ,...,n) we use the uniform grid tr=( 1– δ)[2(r –1 )/(n–1 )–1 ] , where 0 < δ< 1 is a small parameter that ﬁxes the position of the minimal and the maximal collocation points (min tr=– 1+ δ, maxtr=1–δ). Table 12 gives calculation results for equation (50) (with the additional condition (51)) obtained for the following parameter values: D= 0.5 + 0.5 iin (72), q= 1 in (52), δ=0 . 2 , n= 10 in (12), s= 10 in (54), N= 25 in (55) and (56). For the given D,w eh a v e θ= 0.644 + 0.169 i, i.e., the solution has a sufﬁciently strong singularity near the endpoint τ=– 1 (Reβ<– 1/2; see (53)). TABLE 12 Comparison of exact and numerical solutions of the Bueckner equation (50). N.s. is shorthand notation for “numerical solution” τ Exact solution N.s. based on (54) N.s. based on (55) N.s. based on (56) –0.875 –0.62953 – 0.00548 i –0.62870 – 0.00542 i –0.62785 – 0.00758 i –0.62870 – 0.00541 i –0.750 –0.36772 + 0.03491 i –0.36747 + 0.03498 i –0.36711 + 0.03354 i –0.36747 + 0.03498 i –0.625 –0.24804 + 0.04166 i –0.24791 + 0.04172 i –0.24765 + 0.04101 i –0.24791 + 0.04172 i –0.500 –0.17148 + 0.04234 i –0.17139 + 0.04239 i –0.17118 + 0.04198 i –0.17139 + 0.04239 i –0.375 –0.11389 + 0.04151 i –0.11383 + 0.04155 i –0.11365 + 0.04125 i –0.11383 + 0.04155 i –0.250 –0.06598 + 0.04038 i –0.06592 + 0.04042 i –0.06577 + 0.04018 i –0.06592 + 0.04042 i –0.125 –0.02311 + 0.03937 i –0.02306 + 0.03940 i –0.02292 + 0.03921 i –0.02306 + 0.03940 i 0.000 0.01755 + 0.03865 i 0.01759 + 0.03868 i 0.01771 + 0.03851 i 0.01759 + 0.03868 i 0.125 0.05818 + 0.03832 i 0.05821 + 0.03834 i 0.05833 + 0.03819 i 0.05821 + 0.03834 i 0.250 0.10088 + 0.03848 i 0.10091 + 0.03851 i 0.10102 + 0.03837 i 0.10091 + 0.03851 i 0.375 0.14831 + 0.03930 i 0.14834 + 0.03933 i 0.14845 + 0.03920 i 0.14834 + 0.03933 i 0.500 0.20461 + 0.04108 i 0.20464 + 0.04111 i 0.20475 + 0.04098 i 0.20464 + 0.04111 i 0.625 0.27778 + 0.04447 i 0.27781 + 0.04450 i 0.27792 + 0.04436 i 0.27781 + 0.04450 i 0.750 0.38768 + 0.05118 i 0.38771 + 0.05122 i 0.38783 + 0.05104 i 0.38771 + 0.05122 i 0.875 0.61138 + 0.06818 i 0.61140 + 0.06824 i 0.61151 + 0.06794 i 0.61140 + 0.06824 i 804 METHODS FOR SOLVING COMPLETE SINGULAR INTEGRAL EQUATIONS As one can see, all three calculation techniques provide quite good agreement between the numerical and exact analytical solution. A slightly lower accuracy (to 3 or 4 signiﬁcant digits) is attained using the approximation (55), while the approximation (56), as well as the solution based on (54), provides a considerably higher accuracy (to 4 or 5 signiﬁcant digits) for the same N. This due to a higher accuracy of the approximation (56)—it is closer to the polynomial of best uniform approximation. Inparticular, the maximum relative error of the approximation (56) in the rectangle {mint r≤t≤maxtr,– 1 ≤τ≤1}for the above calculation parameters is max εchebyshev = 0.09%, while that of the approximation (55) is max εmaclaurin = 0.87%. The accuracy max εmaclaurin ≈0.1% can be attained using the calculations based on (55) by increasing NtoN= 37, while the approximation (56) provides the same accuracy for N= 25. This means that it is not the technique but the accuracy of approximation of the kernel that makes the main effect on the error of calculation of (14) using (22). It is noteworthy also that the calculation error slightly increases towards the left endpoint of the integration interval. This is due to the reason mentioned in Example 1 and, possibly, to the presence of the second asymptotic term in the expansionnear that endpoint. As one could expect, the quantity δhas a considerable effect on the calculation error. This is because it is δthat controls the position of the minimum and maximum points of collocation, and their positions determine the maximum error of the approximations (55) and (56) for t r∈[mintr,m a x tr]. However, a large increase in δmay result in ill-conditioning in the generated algebraic system. References for Section 15.6: H. F. Bueckner (1966), F. E. Erdogan, G. D. Gupta and T. S. Cook (1973), P. S. Theocaris and N. I. Ioakimidis (1979), M. P. Savruk et al. (1989, 1999), W. H. Press, S. A. Teukolsky et al. (1992), N. G. Moiseyev andG. Ya. Popov (1994), S. M. Belotserkovskii and I. K. Lifanov (1993), A. M. Linkov (2002), A.V . Andreev (2005, 2006). Chapter 16 Methods for Solving Nonlinear Integral Equations 16.1. Some Deﬁnitions and Remarks 16.1-1. Nonlinear Equations with Variable Limit of Integration (V olterra Equations). Nonlinear V olterra integral equations can be represented in the form ⎝integraldisplayx aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=F⎝parenleftbig x,y(x)⎝parenrightbig ,( 1 ) where K⎝parenleftbig x,t,y(t)⎝parenrightbig is the kernel of the integral equation and y(x) is the unknown function ( a≤x≤b). All functions in (1) are usually assumed to be continuous. The form (1) does not cover all possible forms of nonlinear V olterra integral equations; however, it includes the types of nonlinear equations which are most frequently used and studied. A nonlinear integral equation (1) is called a V olterra integral equation in the Urysohn form. In some cases, Eq. (1) can be rewritten in the form ⎝integraldisplayx aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=f(x). (2) Equation (2) is called a V olterra equation of the ﬁrst kind in the Urysohn form . Similarly, the equation y(x)–⎝integraldisplayx aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=f(x)( 3) is called a V olterra equation of the second kind in the Urysohn form . By the substitution u(x)=y(x)–f(x), Eq. (3) can be reduced to the canonical form u(x)=⎝integraldisplayx aK⎝parenleftbig x,t,u(t)⎝parenrightbig dt,( 4 ) whereK⎝parenleftbig x,t,u(t)⎝parenrightbig is the kernel* of the canonical integral equation. The kernel K⎝parenleftbig x,t,y(t)⎝parenrightbig is said to be degenerate if K⎝parenleftbig x,t,y(t)⎝parenrightbig =n⎝summationdisplay k=1gk(x)hk⎝parenleftbig t,y(t)⎝parenrightbig . * There are other ways of reducing Eq. (3) to the form (4) for which the form of the function Kmay be different. 805 806 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS If in Eq. (1) the kernel is K⎝parenleftbig x,t,y(t)⎝parenrightbig =Q(x,t)Φ⎝parenleftbig t,y(t)⎝parenrightbig ,w h e r e Q(x,t)a n dΦ(t,y) are known functions, then we obtain the V olterra integral equation in the Hammerstein form: ⎝integraldisplayx aQ(x,t)Φ⎝parenleftbig t,y(t)⎝parenrightbig dt=F⎝parenleftbig x,y(x)⎝parenrightbig .( 5) In some cases Eq. (5) can be rewritten in the form ⎝integraldisplayx aQ(x,t)Φ⎝parenleftbig t,y(t)⎝parenrightbig dt=f(x). (6) Equation (6) is called a V olterra equation of the ﬁrst kind in the Hammerstein form . Similarly, an equation of the form y(x)–⎝integraldisplayx aQ(x,t)Φ⎝parenleftbig t,y(t)⎝parenrightbig dt=f(x), (7) is called a V olterra equation of the second kind in the Hammerstein form . It is possible to reduce Eq. (7) to the canonical form u(x)=⎝integraldisplayx aQ(x,t)Φ∗⎝parenleftbig t,u(t)⎝parenrightbig dt,( 8) where u(x)=y(x)–f(x). Remark 1. Since a V olterra equation in the Hammerstein form is a special case of a V olterra equation in the Urysohn form, the methods discussed below for the latter are certainly applicable to the former. Remark 2. Some other types of nonlinear integral equations with variable limits of integration are considered in Chapters 5–6. 16.1-2. Nonlinear Equations with Constant Integration Limits (Urysohn Equations). Nonlinear integral equations with constant integration limits can be represented in the form ⎝integraldisplayb aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=F(x,y(x)), α≤x≤β,( 9 ) where K⎝parenleftbig x,t,y(t)⎝parenrightbig is the kernel of the integral equation and y(x) is the unknown function. Usually, all functions in (9) are assumed to be continuous and the case of α=aandβ=bis considered. The form (9) does not cover all possible forms of nonlinear integral equations with constant integration limits; however, ju st as the form (1) for the V olterra equations, it includes the most frequently used and most studied types of these equations. A nonlinear integral equation (9) with constant limits of integration is called an integral equation of the Urysohn type . If Eq. (9) can be rewritten in the form ⎝integraldisplayb aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=f(x), (10) then (10) is called an Urysohn equation of the ﬁrst kind. Similarly, the equation y(x)–⎝integraldisplayb aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=f(x) (11) is called an Urysohn equation of the second kind. 16.1. S OME DEFINITIONS AND REMARKS 807 An Urysohn equation of the second kind can be rewritten in the canonical form u(x)=⎝integraldisplayb aK⎝parenleftbig x,t,u(t)⎝parenrightbig dt. (12) Remark 3. Conditions for existence and uniqueness of the solution of an Urysohn equation are discussed below in Section 16.6. If in Eq. (9) the kernel is K⎝parenleftbig x,t,y(t)⎝parenrightbig =Q(x,t)Φ⎝parenleftbig t,y(t)⎝parenrightbig ,a n dQ(x,t)a n d Φ(t,y)a r eg i v e n functions, then we obtain an integral equation of the Hammerstein type : ⎝integraldisplayb aQ(x,t)Φ⎝parenleftbig t,y(t)⎝parenrightbig dt=F⎝parenleftbig x,y(x)⎝parenrightbig , (13) where, as usual, all functions in the equation are assumed to be continuous. If Eq. (13) can be rewritten in the form ⎝integraldisplayb aQ(x,t)Φ⎝parenleftbig t,y(t)⎝parenrightbig dt=f(x), (14) then (14) is called a Hammerstein equation of the ﬁrst kind . Similarly, an equation of the form y(x)–⎝integraldisplayb aQ(x,t)Φ⎝parenleftbig t,y(t)⎝parenrightbig dt=f(x) (15) is called a Hammerstein equation of the second kind . A Hammerstein equation of the second kind can be rewritten in the canonical form u(x)=⎝integraldisplayb aQ(x,t)Φ∗⎝parenleftbig t,u(t)⎝parenrightbig dt. (16) The existence of the canonical forms (4), (8), (12), and (16) means that the distinction between the inhomogeneous and homogeneous nonlinear integral equations is unessential, unlike the case of linear equations. Another speciﬁc feature of a nonlinear equation is that it frequently has several solutions. Remark 4. Since a Hammerstein equation is a special case of an Urysohn equation, the methods discussed below for the latter are certainly applicable to the former. Remark 5. Some other types of nonlinear integral equati ons with constant limits of integration are considered in Chapters 7–8. 16.1-3. Some Special Features of Nonlinear Integral Equations. Even simplest nonlinear equations, such as those of V olterra or Hammerstein, exhibit some new phenomena characteristic only of nonlinear equations and having no analogues in the theory of linearintegral equations. Example 1. Consider the V olterra integral equation with power nonlinearity y(x)=a⎝integraldisplayx 0yn(t)dt+b,a>0 ,b≥0,n>0 . (17) By the differentiation in x, this equation is reduced to the Cauchy problem for the ﬁrst-order ODE: y/prime x=ayn,y(0) =b. (18) 808 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS The solution of problem (18) depends on the parameter nand, for b> 0, is deﬁned by the formulas y(x)=⎧ ⎪⎨ ⎪⎩[b1–n+a(1 –n)x]1 1–n if 0 < n<1 , beaxifn=1 , [b1–n–a(n–1 )x]1 1–n ifn>1 .(19) It is easy to see that for 0 < n≤1 the solution exists for all x≥0, and for 0 < n< 1 and large x, the function yhas power growth, while for n= 1 it has exponential growth. For n> 1, a continuous solution exists only on the ﬁnite interval 0≤x<x∗=b1–n a(n–1 ). Such a situation is not observed for linear V olterra equations. Now consider the limit case b= 0. Then for any 0 < n<∞,0<a<∞, equation (17) has the trivial solution y(x)≡0. Moreover, for 0 < n<1 ,0< a<∞, equation (17) admits another real solution, y(x)=[a(1 –n)x]1 1–n. Example 2. Consider the Hammerstein integral equation with a quadratic nonlinearity y(x)=λ⎝integraldisplay1 0x2ty2(t)dt, (20) where λis a free parameter. Setting A=⎝integraldisplay1 0ty2(t)dt, (21) let us represent equation (20) in the form y(x)=Aλx2. Substituting this expression into (21), we obtain a quadratic equation for the determination of the constant A: A=1 6A2λ2. (22) Its solutions are A1=0a n d A2=6λ–2. Therefore, the original integral equation (20) has two solutions for any λ≠0: y1(x)≡0,y2(x)=6 λx2. Note that the linear homogeneous integral equation y(x)=λ⎝integraldisplay1 0x2ty(t)dt (23) with the same kernel K(x,t)=x2thas a nontrivial solution only for a single value of λ, namely, λ= 4, which is a characteristic value of the kernel K(x,t). Therefore, if we follow the terminology of linear equations and say that λis a characteristic value of a nonlinear equation if this equation has a nontrivial solution for that λ, it turns out that equation (23) has inﬁnite intervals of characteristic values (– ∞,0 )a n d( 0 ,∞ ). Example 3. Consider another integral equation of Hammerstein’s type with a quadratic nonlinearity y(x)=λ⎝integraldisplay1 0y2(t)dt+1 . (24) This equation can be written as y(x)=Aλ+1 , (25) where A=⎝integraldisplay1 0y2(t)dt. (26) Substituting (25) into (26), we obtain the quadratic equation λ2A2+( 2λ–1 )A+1=0 with the roots A=1–2λ±√ 1–4λ 2λ2. Thus, equation (24) has real solutions only for λ≤1/4. It has two solutions for λ<1/4 and one solution for λ=1/4 (forλ= 0 there is one bounded solution y(x)=1 ) . 16.2. E XACT METHODS FOR NONLINEAR EQUATIONS WITH VARIABLE LIMIT OF INTEGRATION 809 The corresponding equation with no free term y(x)=λ⎝integraldisplay1 0ty2(t)dt, for any λ≠0, admits the nontrivial solution y(x)=1/λ. Obviously, this does not mean that equation (24) with a free term had inﬁnitely many solutions. Example 4. Now consider an integral equation of Hammerstein’s type with a transcendental nonlinearity y(x)=λ⎝integraldisplay1 0f(x)g(t)s i n⎝parenleftbiggy(t) f(t)⎝parenrightbigg y(t)dt. (27) Its solutions are sought in the form y(x)=Af(x), where the constant Ais determined from the transcendental equation* 1=λσsinA,σ=⎝integraldisplay1 0f(t)g(t)dt. (28) For|λ|<1/|σ|, equation (28), and therefore equation (27), has no real solutions (the case σ= 0 is included). For any λsatisfying the inequality |λ|>1/|σ|, equation (28), and therefore equation (27), has inﬁnitely many real solutions. References for Section 16.1: N. S. Smirnov (1951), M. A. Krasnosel’skii (1964), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), M. L. Krasnov (1975), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. G. Tricomi (1985), A. F. Verlan’ and V . S. Sizikov (1986). 16.2. Exact Methods for Nonlinear Equations with Variable Limit of Integration 16.2-1. Method of Integral Transforms. Consider a V olterra integral equation with quadratic nonlinearity µy(x)–λ⎝integraldisplayx 0y(x–t)y(t)dt=f(x). (1) This equation can be solved using the Laplace transform. In doing so, one applies the convolution theorem (see Section 9.2) to obtain a quadratic equation for the transform ˜ y(p)=L{y(x)}: µ˜y(p)–λ˜y2(p)=˜f(p). This implies ˜y(p)=µ±⎝radicalbig µ2–4λ˜f(p) 2λ.( 2) The inverse Laplace transform y(x)=L–1{˜y(p)}, if it exists, is a solution to Eq. (1). Note that for the two different signs in formula (2), there are two corresponding solutions of the original equation. Example. Consider the integral equation ⎝integraldisplayx 0y(x–t)y(t)dt=Axm,m> –1. Applying the Laplace transform to this equation and taking into account the relation L{xm}=Γ(m+1 )p–m–1, we obtain ˜y2(p)=AΓ(m+1 )p–m–1, whereΓ(m) is the Gamma function. Taking the square root of both sides of the equation, we obtain ˜y(p)=±⎝radicalbig AΓ(m+1 )p–m+1 2. Applying the Laplace inversion formula, we obtain two solutions to the original integral equation y1(x)=–√ AΓ(m+1 ) Γ⎝parenleftBigm+1 2⎝parenrightBigxm–1 2,y2(x)=√ AΓ(m+1 ) Γ⎝parenleftBigm+1 2⎝parenrightBigxm–1 2. * The trivial solution corresponding to A= 0 is not taken into account. 810 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS 16.2-2. Method of Differentiation for Nonlinear Equations with Degenerate Kernel. Sometimes, differentiation (possibly multiple) of a nonlinear integral equation with subsequent elimination of the integral term using the original equation makes it possible to reduce this equationto a nonlinear ordinary differential equation. Listed below are some equations of this type. 1 ◦. The equation y(x)+⎝integraldisplayx af⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x)( 3 ) can be reduced by differentiation to the nonlinear ﬁrst-order equation y/prime x+f(x,y)–g/prime x(x)=0 ( 4 ) with the initial condition y(a)=g(a). 2◦. The equation y(x)+⎝integraldisplayx a(x–t)f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x)( 5 ) can be reduced by double differentiation (with the subsequent elimination of the integral term using the original equation) to the nonlinear second-order equation: y/prime/prime xx+f(x,y)–g/prime/prime xx(x)=0 . ( 6 ) The initial conditions for the function y=y(x) have the form y(a)=g(a),y/prime x(a)=g/prime x(a). 3◦. The equation y(x)+⎝integraldisplayx aeλ(x–t)f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x)( 7) can be reduced by differentiation to the nonlinear ﬁrst-order equation y/prime x+f(x,y)–λy+λg(x)–g/prime x(x)=0 . ( 8 ) The desired function y=y(x) must satisfy the initial condition y(a)=g(a). Remark 1. A considerable number of exact solutions to the ordinary differential equations (4), (6), and (8) for various functions f(x,y)a n dg(x) can be found in the book by Polyanin and Zaitsev (2003). 4◦. Equations of the form y(x)+⎝integraldisplayx acosh⎝bracketleftbig λ(x–t)⎝bracketrightbig f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x), y(x)+⎝integraldisplayx asinh⎝bracketleftbig λ(x–t)⎝bracketrightbig f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x), y(x)+⎝integraldisplayx acos⎝bracketleftbig λ(x–t)⎝bracketrightbig f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x), y(x)+⎝integraldisplayx asin⎝bracketleftbig λ(x–t)⎝bracketrightbig f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x) can also be reduced to second-order ordinary differential equations by double differentiation. For these equations, see Section 6.8 in the ﬁrst part of the book (Eqs. 20, 21, 22, and 23, respectively). 16.3. A PPROXIMA TE METHODS FOR NONLINEAR EQUATIONS WITH VARIABLE LIMIT OF INTEGRATION 811 5◦. Consider the nonlinear V olterra equation of the second kind with the general degenerate kernel y(x)–n⎝summationdisplay m=1ϕm(x)⎝integraldisplayx afm(t,y(t))dt=g(x). (9) Let us introduce the notation wj(x)=⎝integraldisplayx afj(t,y(t))dt,j=1 ,...,n, (10) and rewrite Eq. (9) as follows: y(x)=g(x)+n⎝summationdisplay m=1ϕm(x)wm(x). (11) On differentiating the expressions (10) with regard to formula (11), we arrive at the following system of nonlinear differential equations for the functions wj=wj(x): w/prime j=fj⎝parenleftBig x,g(x)+n⎝summationdisplay m=1ϕm(x)wm⎝parenrightBig ,j=1 ,...,n, with the initial conditions wj(a)=0 , j=1 ,...,n. Once a solution of this system is found,the corresponding solution of the original integral equation (9) is deﬁned by formula (11). Remark 2. Equations (3), (5), and (7) are special cases of equation (9). The equations of Item 4◦ can be reduced to (9) using hyperbolic and trigonometric formulas (see the addition formulas in Supplements 1.4-7 and 1.2-7, respectively). References for Section 16.2: M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), A. F. Verlan’ and V . S. Sizikov (1986), A. D. Polyanin and A. V . Manzhirov (1998). 16.3. Approximate and Numerical Methods for Nonlinear Equations with Variable Limit of Integration 16.3-1. Successive Approximation Method. 1◦. In many cases, the successive approximation method can be successfully applied to solve various types of integral equations. The principles of constructing the iteration process are the same as in the case of linear equations. For V olterra equations of the second kind in the Urysohn form y(x)–⎝integraldisplayx aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=f(x), a≤x≤b,( 1) the corresponding recursive expression has the form yn+1(x)=f(x)+⎝integraldisplayx aK⎝parenleftbig x,t,yn(t)⎝parenrightbig dt,n=0 ,1 ,2 ,... (2) It is customary to take the initial approximation either in the form y0(x)≡0o ri nt h ef o r my 0(x)=f(x). 812 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS In contrast to the case of linear equations, the successive approximation method has a smaller domain of convergence. Let us present the convergence conditions for the iteration process (2),which are simultaneous ly the existence conditions for a so lution of Eq. (1). To be speciﬁc, we assume that y 0(x)=f(x). If for any z1andz2the relations |K(x,t,z1)–K(x,t,z2)|≤ϕ(x,t)|z1–z2| and ⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝integraldisplay x aK⎝parenleftbig x,t,f(t)⎝parenrightbig dt⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle≤ψ(x) hold, with⎝integraldisplay x aψ2(t)dt≤N2,⎝integraldisplayb a⎝integraldisplayx aϕ2(x,t)dt dx ≤M2, where NandMare some constants, then the successive approximations converge to a unique solution of Eq. (1) almost everywhere absolutely and uniformly. Example 1. Let us apply the successive approximation method to solve the equation y(x)=⎝integraldisplayx 01+y2(t) 1+t2dt. Ify0(x)≡0, then y1(x)=⎝integraldisplayx 0dt 1+t2=a r c t a n x, y2(x)=⎝integraldisplayx 01 + arctan2t 1+t2dt=a r c t a n x+1 3arctan3x, y3(x)=⎝integraldisplayx 01 + arctan t+1 3arctan3t 1+t2dt=a r c t a n x+1 3arctan3x+2 3⋅5arctan5x+1 7⋅9arctan7x. On continuing this process, we can observe that yn(x)→tan(arctan x)=xasn→∞ , i.e.,y(x)=x. This result is validated by substituting it into the original equation. Example 2. For the nonlinear equation y(x)=⎝integraldisplayx 0[ty2(t)–1 ]dt, we wish to obtain the ﬁrst three approximations. If we set y0(x)=0 ,t h e n y1(x)=⎝integraldisplayx 0(–1)dt=–x, y2(x)=⎝integraldisplayx 0(t3–1 )dt=–x+1 4x4, y3(x)=⎝integraldisplayx 0⎝bracketleftbigt⎝parenleftbig1 16t8–1 2t5+t2⎝parenrightbig–1⎝bracketrightbigdt=–x+1 4x4–1 14x7+1 160x10. 2◦. Suppose that in the nonlinear V olterra equation y(x)=⎝integraldisplayx 0K(x,t,y(t))dt, the function K(x,t,y) and its partial derivative K/prime y(x,t,y) are continuous in the domain x,t≥0, –∞<y<∞, and the following inequality holds: |K(x,t,y)|≤ϕ(y), 16.3. A PPROXIMA TE METHODS FOR NONLINEAR EQUATIONS WITH VARIABLE LIMIT OF INTEGRATION 813 where ϕ(y) is a nondecreasing function on the half-line [0, ∞). If the Cauchy problem for the differential equation u/prime x=ϕ(|u|), u(0) = 0 has a solution on the interval [0, ω], then the above V olterra equation has a solution on [0, ω]. For y0(x)≡0 as the initial function, the successive approximations yn(x)=⎝integraldisplayx 0K(x,t,yn–1(t))dt (n=1 ,2 , ...) are uniformly convergent on [0, ω] to a solution of the V olterra equation. Note that all approximations do not abandon the domain – u(x)≤yn(x)≤u(x) and satisfy the inequality |yn(x)–yn–1(x)|≤M L(Lt)n n!(n=1 ,2 , ...), where LandMare constants such that |K(x,t,0 )|≤M for 0 ≤x,t≤ω, |K(x,t,y1)–K(x,t,y2)|≤L|y1–y2|for 0 ≤x,t≤ω,–u(x)≤y1,y2≤u(x). 3◦. The successive approximation method can be applied to solve other forms of nonlinear equations, for instance, equations of the form y(x)=F⎝parenleftbigg x,⎝integraldisplayx aK(x,t)y(t)dt⎝parenrightbigg solved for y(x) in which the integral has xas the upper integration limit. This makes it possible to obtain a numerical solution by applyi ng small steps with respect to xand by linearization at each step, which usually provides the uniqueness of the result of the iterations for an arbitrary initial approximation. 4◦. The initial approximation has a substantial effect on the number of iterations required to obtain the result with a prescribed accuracy. Therefore, when choosing this approximation, some additionalarguments are usually applied. Namely, for the equation Ay(x)–⎝integraldisplay x 0Q(x–t)Φ⎝parenleftbig y(t)⎝parenrightbig dt=f(x), where Ais a constant, a good initial approximation y0(x) can sometimes be found from the solution of the following (in general, transcendental) equation for ˜ y0(p): A˜y0(p)–˜Q(p)Φ⎝parenleftbig ˜y0(p)⎝parenrightbig =˜f(p), where ˜ y0(p),˜Q(p), and ˜f(p) are the Laplace transforms of the respective functions. If ˜ y0(p)i s deﬁned, then the initial approximation can be found by applying the Laplace inversion formula: y0(x)=L–1{˜y0(p)}. 16.3-2. Newton–Kantorovich Method. A merit of the iteration methods when applied to V olterra linear equations of the second kind is their unconditional convergence under weak restr ictions on the kernel and the right-hand side. When solving nonlinear equations, the applicability domain of the method of simple iterations is smaller, and if the process is still convergent, then, in many cases, the rate of convergence can be 814 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS very low. An effective method that makes it possible to overcome the indicated complications is the Newton–Kantorovichmethod. The main objective of this method is the solution of nonlinear integralequations of the second kind with constant limits of integration. Nevertheless, this method is useful in the solution of many problems for the V olterra equations and makes it possible to signiﬁcantly increase the rate of convergence compared with the successive approximation method. Let us apply the Newton–Kantorovich method to solve a V olterra equation of the second kind in the Urysohn form y(x)=f(x)+⎝integraldisplay x aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt.( 3) We obtain the following iteration process: yn(x)=yn–1(x)+ϕn–1(x), n=1 ,2 , ...,( 4 ) ϕn–1(x)=εn–1(x)+⎝integraldisplayx aK/prime y⎝parenleftbig x,t,yn–1(t)⎝parenrightbig ϕn–1(t)dt,( 5) εn–1(x)=f(x)+⎝integraldisplayx aK⎝parenleftbig x,t,yn–1(t)⎝parenrightbig dt–yn–1(x). (6) The algorithm is based on the solution of the linear integral equation (5) for the correction ϕn–1(x) with the kernel and right-hand side that vary from step to step. This process has a high rate of convergence, but it is rather complicated because we must solve a new equation at each step ofiteration. To simplify the problem, we can replace Eq. (5) by the equation ϕ n–1(x)=εn–1(x)+⎝integraldisplayx aK/prime y⎝parenleftbig x,t,y0(t)⎝parenrightbig ϕn–1(t)dt (7) or by the equation ϕn–1(x)=εn–1(x)+⎝integraldisplayx aK/prime y⎝parenleftbig x,t,ym(t)⎝parenrightbig ϕn–1(t)dt,( 8) whose kernels do not vary. In Eq. (8), mis ﬁxed and satisﬁes the condition m<n–1 . It is reasonable to apply Eq. (7) with an appropriately chosen initial approximation. Otherwise we can stop at some mth approximation and, beginning with this approximation, apply the simpliﬁed equation (8). The iteration process thus obtained is the modiﬁed Newton–Kantorovich method. In principle, it converges somewhat slower than the original process (4)–(6); however, it is not so cumbersome in the calculations. Example 3. Let us apply the Newton–Kantorovich method to solve the equation y(x)=⎝integraldisplayx 0[ty2(t)–1 ]dt. The derivative of the integrand with respect to yhas the form K/prime y⎝parenleftbigt,y(t)⎝parenrightbig=2ty(t). For the zero approximation we take y0(x)≡0. According to (5) and (6) we obtain ϕ0(x)=–xandy1(x)=–x.F u r t h e r m o r e , y2(x)=y1(x)+ϕ1(x). By (6) we have ε1(x)=⎝integraldisplayx 0[t(–t)2–1 ]dt+x=1 4x4. The equation for the correction has the form ϕ1(x)=– 2⎝integraldisplayx 0t2ϕ1(t)dt+1 4x4 16.3. A PPROXIMA TE METHODS FOR NONLINEAR EQUATIONS WITH VARIABLE LIMIT OF INTEGRATION 815 and can be solved by any of the known methods for V olterra linear equations of the second kind. In the case under consideration, we apply the successive approximation method, which leads to the following results (the number of the stepis indicated in the superscript): ϕ(0) 1=1 4x4, ϕ(1) 1=1 4x4–2⎝integraldisplayx 01 4t6dt=1 4x4–1 14x7, ϕ(2) 1=1 4x4–2⎝integraldisplayx 0t2⎝parenleftbig1 4t4–1 14x7⎝parenrightbigdt=1 4x4–1 14x7+1 70x10. We restrict ourselves to the second approximation and obtain y2(x)=–x+1 4x4–1 14x7+1 70x10 and then pass to the third iteration step of the Newton–Kantorovich method: y3(x)=y2(x)+ϕ2(x), ε2(x)=1 160x10–1 1820x13–1 7840x16+1 9340x19+1 107800x22, ϕ2(x)=ε2(x)+2⎝integraldisplayx 0t⎝parenleftbig–t+1 4t4–1 14t7+1 70t10⎝parenrightbigϕ2(t)dt. When solving the last equation, we restrict ourselves to the zero approximation and obtain y3(x)=–x+1 4x4–1 14x7+23 112x10–1 1820x13–1 7840x16+1 9340x19+1 107800x22. The application of the successive approximation method to the original equation leads to the same result at the fourth step. As usual, in the numerical solution the integral is replaced by a quadrature formula. The main difﬁculty of the implementation of the method in this case is in evaluating the derivative of the kernel. The problem can be simpliﬁed if the kernel is given as an analytic expression that can be differentiated in the analytic form. However, if the kernel is given by a table, then the evaluation must be performed numerically. 16.3-3. Collocation Method. When applied to the solution of a V olterra equation of the ﬁrst kind in the Urysohn form ⎝integraldisplayx aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=f(x), a≤x≤b,( 9) thecollocation method is as follows. The interval [a ,b] is divided into Nparts on each of which the desired solution can be presented by a function of a certain form ˜y(x)=Φ(x,A1,...,Am), (10) involving free parameters Ai,i=1 ,...,m. On the ( k+1 ) s tp a r t xk≤x≤xk+1,w h e r e k=0 ,1 , ...,N– 1, the solution can be written in the form ⎝integraldisplayx xkK⎝parenleftbig x,t,˜y(t)⎝parenrightbig dt=f(x)–Ψk(x), (11) where the integral Ψk(x)=⎝integraldisplayxk aK⎝parenleftbig x,t,˜y(t)⎝parenrightbig dt (12) can always be calculated for the approximate solution ˜ y(x), which is known on the interval a≤x≤xk and was previously obtained for k– 1 parts. The initial value y(a) of the desired solution can be found by an auxiliary method or is assumed to be given. 816 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS To solve Eq. (11), representation (10) is applied, and the free parameters Ai(i=1 ,...,m) can be deﬁned from the condition that the residuals vanish: ε(Ai,xk,j)=⎝integraldisplayxk,j xkK⎝parenleftbig xk,j,t,Φ(t,A1,...,Am)⎝parenrightbig dt–f(xk,j)–Ψk(xk,j), (13) where the xk,j(j=1 ,...,m) are the nodes that correspond to the partition of the interval [ xk,xk+1] intomparts (subintervals). System (13) is a system of mequations for A1,...,Am. For convenience of the calculations, it is reasonable to pr esent the desired so lution on any part as a polynomial ˜y(x)=m⎝summationdisplay i=1Aiϕi(x), (14) where the ϕi(x) are linearly independent coordinate functions. For the functions ϕi(x), power and trigonometric polynomials are frequently used; for instance, ϕi(x)=xi–1. In applications, the concrete form of the functions ϕi(x) in formula (14), as well as the form of the functions Φin (10), can sometimes be given on the basis of physical reasoning or deﬁned by the structure of the solution of a simpler model equation. 16.3-4. Quadrature Method. To solve a nonlinear V olterra equation, we can apply the method based on the use of quadrature formulas. The procedure of constructing the approximate system of equations is the same as in the linear case (see Subsection 11.10-1). 1◦. We consider the nonlinear V olterra equation of the second kind in the Urysohn form y(x)–⎝integraldisplayx aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=f(x) (15) on an interval a≤x≤b. Assume that K⎝parenleftbig x,t,y(t)⎝parenrightbig andf(x) are continuous functions. From Eq. (15) we ﬁnd that y(a)=f(a). Let us choose a constant integration step hand consider the discrete set of points xi=a+h(i–1 ) ,w h e r e i=1 ,...,n.F o rx=xi, Eq. (15) becomes y(xi)–⎝integraldisplayxi aK⎝parenleftbig xi,t,y(t)⎝parenrightbig dt=f(xi). (16) Applying the quadrature formula (see Subsection 10.7-1) to the integral in (16), choosing xj (j=1 ,...,i) to be the nodes in t, and neglecting the truncation error, we arrive at the following system of nonlinear algebraic (or transcendental) equations: y1=f1,yi–i⎝summationdisplay j=1AijKij(yj)=fi,i=2 ,...,n, (17) where the Aijare the coefﬁcients of the quadrature formula on the interval [a ,xi], theyiare the approximate values of the solution y(x) at the nodes xi,fi=f(xi), and Kij(yj)=K(xi,tj,yj). Relations (17) can be rewritten as a sequence of recursive nonlinear equations, y1=f1,yi–AiiKii(yi)=fi+i–1⎝summationdisplay j=1AijKij(yj), i=2 ,...,n, (18) for the approximate values of the desired solution at the nodes. 16.4. E XACT METHODS FOR NONLINEAR EQUATIONS WITH CONSTANT INTEGRATION LIMITS 817 2◦. When applied to the V olterra equation of the second kind in the Hammerstein form y(x)–⎝integraldisplayx aQ(x,t)Φ⎝parenleftbig t,y(t)⎝parenrightbig dt=f(x), (19) the main relations of the quadrature method have the form ( x1=a) y1=f1,yi–i⎝summationdisplay j=1AijQijΦj(yj)=fi,i=2 ,...,n, (20) where Qij=Q(xi,tj)a n dΦj(yj)=Φ(tj,yj). These relations lead to the sequence of nonlinear recursive equations y1=f1,yi–AiiQiiΦi(yi)=fi+i–1⎝summationdisplay j=1AijQijΦj(yj), i=2 ,...,n, (21) whose solutions give approximate values of the desired function. Example 4. In the solution of the equation y(x)–⎝integraldisplayx 0e–(x–t)y2(t)dt=e–x,0 ≤x≤0.1, where Q(x,t)=e–(x–t),Φ⎝parenleftbig t,y(t)⎝parenrightbig =y2(t), and f(x)=e–x, the approximate expression has the form y(xi)–⎝integraldisplayxi 0e–(xi–t)y2(t)dt=e–xi. On applying the trapezoidal rule to evaluate the integral (with step h= 0.02) and ﬁnding the solution at the nodes xi=0 , 0.02, 0.04, 0.06, 0.08, 0.1, we obtain, according to (21), the following system of computational relations: y1=f1,yi–0 . 0 1 Qiiy2 i=fi+i–1⎝summationdisplay j=10.02Qijy2 j,i=2 ,...,6 . Thus, to ﬁnd an approximate solution, we must solve a quadratic equation for each value yi, which makes it possible to write out the answer yi=5 0±50⎝bracketleftbigg 1–0 . 0 4⎝parenleftBig fi+i–1⎝summationdisplay j=10.02Qijy2 j⎝parenrightBig⎝bracketrightbigg1/2 ,i=2 ,...,6 . References for Section 16.3: M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), P. P. Zabreyko, A. I. Koshelev, et al. (1975), A. F. Verlan’ and V . S. Sizikov (1986). 16.4. Exact Methods for Nonlinear Equations with Constant Integration Limits 16.4-1. Nonlinear Equations with Degenerate Kernels. 1◦. Consider a Hammerstein equation of the second kind in the canonical form y(x)=⎝integraldisplayb aQ(x,t)Φ⎝parenleftbig t,y(t)⎝parenrightbig dt,( 1) where Q(x,t)a n dΦ(t,y) are given functions and y(x) is the unknown function. 818 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS Let the kernel Q(x,t) be degenerate, i.e., Q(x,t)=m⎝summationdisplay k=1gk(x)hk(t). (2) In this case Eq. (1) becomes y(x)=m⎝summationdisplay k=1gk(x)⎝integraldisplayb ahk(t)Φ⎝parenleftbig t,y(t)⎝parenrightbig dt.( 3) We write Ak=⎝integraldisplayb ahk(t)Φ⎝parenleftbig t,y(t)⎝parenrightbig dt,k=1 ,...,m,( 4) where the constants Akare yet unknown. Then it follows from (3) that y(x)=m⎝summationdisplay k=1Akgk(x). (5) On substituting the expression (5) for y(x) into relations (4), we obtain (in the general case) mtranscendental equations of the form Ak=Ψk(A1,...,Am), k=1 ,...,m,( 6) which contain munknown numbers A1,...,Am. For the case in which Φ(t,y) is a polynomial in y,i . e . , Φ(t,y)=p0(t)+p1(t)y+···+pn(t)yn,( 7) where p0(t),...,pn(t) are, for instance, continuous functions of ton the interval [ a,b], system (6) becomes a system of nonlinear algebraic equations for A1,...,Am. The number of solutions of the integral equation (3) is equal to the number of solutions of system (6). Each solution of system (6) generates a solution (5) of the integral equation. 2◦. Consider the Urysohn equation of the second kind with the simpliﬁed degenerate kernel of the following form: y(x)+⎝integraldisplayb a⎝braceleftbiggn⎝summationdisplay k=1gk(x)fk⎝parenleftbig t,y(t)⎝parenrightbig⎝bracerightbigg dt=h(x). (8) Its solution has the form y(x)=h(x)+n⎝summationdisplay k=1λkgk(x), (9) where the constants λkcan be deﬁned by solving the al gebraic (or transcendental) system of equations λm+⎝integraldisplayb afm⎝parenleftBig t,h(t)+n⎝summationdisplay k=1λkgk(t)⎝parenrightBig dt=0 , m=1 ,...,n. (10) To different roots of this system, there are different corresponding solutions of the nonlinear integral equation. It may happen that (real) solutions are absent. 16.4. E XACT METHODS FOR NONLINEAR EQUATIONS WITH CONSTANT INTEGRATION LIMITS 819 A solution of an Urysohn equation of the second kind with degenerate kernel in the general form f⎝parenleftbig x,y(x)⎝parenrightbig +⎝integraldisplayb a⎝braceleftbiggn⎝summationdisplay k=1gk⎝parenleftbig x,y(x)⎝parenrightbig hk⎝parenleftbig t,y(t)⎝parenrightbig⎝bracerightbigg dt= 0 (11) can be represented in the implicit form f⎝parenleftbig x,y(x)⎝parenrightbig +n⎝summationdisplay k=1λkgk⎝parenleftbig x,y(x)⎝parenrightbig = 0, (12) where the parameters λkare determined from the system of al gebraic (or transcendental) equations: λk–Hk(/vectorλ)=0 , k=1 ,...,n, Hk(/vectorλ)=⎝integraldisplayb ahk⎝parenleftbig t,y(t)⎝parenrightbig dt,/vectorλ={λ1,...,λn}.(13) Into system (13), we must substitute the function y(x)=y(x,/vectorλ), which can be obtained by solving Eq. (12). The number of solutions of the integral equation is deﬁned by the number of solutions obtained from (12) and (13). It can occur that there is no solution. Example 1. Let us solve the integral equation y(x)=λ⎝integraldisplay1 0xty3(t)dt (14) with parameter λ. We write A=⎝integraldisplay1 0ty3(t)dt. (15) In this case, it follows from (14) that y(x)=λAx . (16) On substituting y(x) in the form (16) into relation (15), we obtain A=⎝integraldisplay1 0tλ3A3t3dt. Hence, A=1 5λ3A3. (17) Forλ> 0, Eq. (17) has three solutions: A1=0 , A2=⎝parenleftBig5 λ3⎝parenrightBig1/2 ,A3=–⎝parenleftBig5 λ3⎝parenrightBig1/2 . Hence, the integral equation (14) also has three solutions for any λ>0 : y1(x)≡0,y2(x)=⎝parenleftBig5 λ3⎝parenrightBig1/2 x,y3(x)=–⎝parenleftBig5 λ3⎝parenrightBig1/2 x. Forλ≤0, Eq. (17) has only the trivial solution y(x)≡0. 16.4-2. Method of Integral Transforms. 1◦. Consider the following nonlinear integral equation with quadratic nonlinearity on a semi-axis: µy(x)–λ⎝integraldisplay∞ 01 ty⎝parenleftBigx t⎝parenrightBig y(t)dt=f(x). (18) 820 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS To solve this equation, the Mellin transform can be applied, which, with regard to the convolution theorem (see Section 9.3), leads to a quadratic equation for the transform ˆ y(s)=M{y(x)}: µˆy(s)–λˆy2(s)=ˆf(s). This implies ˆy(s)=µ±⎝radicalBig µ2–4λˆf(s) 2λ. (19) The inverse transform y(x)=M–1{ˆy(s)}obtained by means of the Mellin inversion formula (if it exists) is a solution of Eq. (18). To different signs in the formula for the images (19), there are twocorresponding solutions of the original equation. 2 ◦. By applying the Mellin transform, one can solve nonlinear integral equations of the form y(x)–λ⎝integraldisplay∞ 0tβy(xt)y(t)dt=f(x). (20) The Mellin transform (see Table 3 in Section 9.3) reduces (20) to the following functional equation for the transform ˆ y(s)=M{y(x)}: ˆy(s)–λˆy(s)ˆy(1 –s+β)=ˆf(s). (21) On replacing sby 1 – s+βin (21), we obtain the relationship ˆy(1 –s+β)–λˆy(s)ˆy(1 –s+β)=ˆf(1 –s+β). (22) On eliminating the quadratic term from (21) and (22), we obtain ˆy(s)–ˆf(s)= ˆy(1 –s+β)–ˆf(1–s+β). We express ˆ y(1 –s+β) from this relation and substitute it into (21). We arrive at the quadratic equation λˆy2(s)–⎝bracketleftbig 1+ˆf(s)–ˆf(1 –s+β)⎝bracketrightbig ˆy(s)+ˆf(s)=0 . On solving this equation for ˆ y(s), by means of the Mellin inversion formula we can ﬁnd a solution of the original integral equation (20). 16.4-3. Method of Differentiating for Integral Equations. 1◦. The nonlinear integral equation y(x)+⎝integraldisplayb a|x–t|f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x), a≤x≤b (23) can be reduced to a nonlinear second-order equation by double differentiation. Let us remove the modulus in the integrand: y(x)+⎝integraldisplayx a(x–t)f⎝parenleftbig t,y(t)⎝parenrightbig dt+⎝integraldisplayb x(t–x)f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x). (24) Differentiating (24) with respect to xyields y/prime x(x)+⎝integraldisplayx af⎝parenleftbig t,y(t)⎝parenrightbig dt–⎝integraldisplayb xf⎝parenleftbig t,y(t)⎝parenrightbig dt=g/prime x(x). (25) Differentiating (25), we arrive at a second-order ordinary differential equation for y=y(x): y/prime/prime xx+2f(x,y)=g/prime/prime xx(x). (26) For the boundary conditions for this equation, see Section 8.8 in the ﬁrst part of the book (Eq. 8.8.15). 16.4. E XACT METHODS FOR NONLINEAR EQUATIONS WITH CONSTANT INTEGRATION LIMITS 821 2◦. The equation y(x)+⎝integraldisplayb aeλ|x–t|f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x) (27) can also be reduced to a nonlinear second-order equation by double differentiation (with subsequent elimination of the integral term by using the original equation): y/prime/prime xx+2λf(x,y)–λ2y=g/prime/prime xx(x)–λ2g(x). (28) For the boundary conditions for this equation, see Section 8.8 of the ﬁrst part of the book (Eq. 8.8.16). Remark. A considerable number of exact solutions to ordinary differential equations (26) and (28) for various functions f(x,y)a n dg(x) can be found in the book by Polyanin and Zait- sev (2003). 3◦. The equations y(x)+⎝integraldisplayb asinh⎝parenleftbig λ|x–t|⎝parenrightbig f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x), y(x)+⎝integraldisplayb asin⎝parenleftbig λ|x–t|⎝parenrightbig f⎝parenleftbig t,y(t)⎝parenrightbig dt=g(x), can also be reduced to second-order ordinary differential equations by means of the differentiation. For these equations, see Section 8.8 of the ﬁrst part of the book (Eqs. 8.8.17 and 8.8.18). 16.4-4. Method for Special Urysohn Equations of the First Kind. 1◦. Consider the linear integral equation of the ﬁrst kind ⎝integraldisplayb aK(x,t)Y(t)dt=f(x). (29) Suppose equation (29) can be solved for any f(x) from some class of functions LF.L e tYf(t) denote the corresponding solution. Now consider the more complex nonlinear Urysohn equation of the ﬁrst kind ⎝integraldisplayb a[K(x,t)y(t)+ϕ(x)Ψ(t,y(t))]dt=f(x) (30) with its kernel containing an additional nonlinear term ϕ(x)Ψ(t,y(t)). A solution to equation (30) will be sought in the form y(t)=Yf(t)+AYϕ(t), (31) where Yϕ(t) is the solution to equation (29) in which f(x) must be replaced with ϕ(x). Substituting (31) into (30) we have the following algebraic (transcendental) equation for the coefﬁcient A: A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt= 0. (32) Formulas (31)–(32) can deﬁne one, several, or inﬁnitely many solutions (or even none) to equation (30). In addition, the condition ϕ(x)∈LFmust be satisﬁed. 822 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS Example 2. The solution of the linear integral equation of the ﬁrst kind ⎝integraldisplay∞ 0sin(xt)Y(t)dt=f(x) (33) is expressed as (see equation 3.5.8 in Section 3.5) Yf(t)=2 π⎝integraldisplay∞ 0sin(xt)f(x)dx. (34) Up to constant factors, the function f(x) and the solution Yf(t) in (33)–(34) are the Fourier sine transform pair. Now consider the more complex integral equation with quadratic nonlinearity ⎝integraldisplay∞ 0[sin(xt)y(t)+ϕ(x)ψ(t)y2(t)]dt=f(x). (35) In terms of equation (30), we have Ψ(t,y(t)) =ψ(t)y2(t) in (35). The corresponding solution (34) to equation (33) with ϕ(x) is written as Yϕ(t)=2 π⎝integraldisplay∞ 0sin(xt)ϕ(x)dx. (36) Hence, equation (35) has the two solutions y(t)=Yf(t)+A1,2Yϕ(t), where A1,2are roots of the quadratic equation pA2+qA+r=0 , p=⎝integraldisplay∞ 0ψ(t)Y2 ϕ(t)dt,q=1+2⎝integraldisplay∞ 0ψ(t)Yf(t)Yϕ(t)dt,r=⎝integraldisplay∞ 0ψ(t)Y2 f(t)dt. Here all integrals are supposed to converge. 2◦. The integral equation ⎝integraldisplayb a⎝bracketleftbigg K(x,t)y(t)+n⎝summationdisplay m=1ϕm(x)Ψm(t,y(t))⎝bracketrightbigg dt=f(x) (37) whose kernel is the sum of the kernel of equation (29) and an arbitrary degenerate nonlinear kernel can be solved in a similar manner. The solution is sought in the additive form y(t)=Yf(t)+n⎝summationdisplay m=1AmYϕm(t), (38) where Yϕm(x) is the solution to equation (29) in which f(x) must be replaced with ϕm(x). Substi- tuting (38) into (37) results in the following algebraic (transcendental) system of equations for the coefﬁcients Am: Am+⎝integraldisplayb aΨm⎝parenleftbigg t,Yf(t)+n⎝summationdisplay j=1AjYϕj(t)⎝parenrightbigg dt=0 , m=1 ,...,n. (39) 16.4-5. Method for Special Urysohn Equations of the Second Kind. 1◦. Consider the linear equation of the second kind Y(x)+⎝integraldisplayb aK(x,t)Y(t)dt=f(x). (40) Suppose equation (40) can be solved for any f(x) from some class of functions LF.L e tYf(x) denote the corresponding solution. 16.4. E XACT METHODS FOR NONLINEAR EQUATIONS WITH CONSTANT INTEGRATION LIMITS 823 Now consider the more complex nonlinear Urysohn equation of the second kind y(x)+⎝integraldisplayb a[K(x,t)y(t)+ϕ(x)Ψ(t,y(t))]dt=f(x), (41) with its kernel containing an additional term ϕ(x)Ψ(t,y(t)). A solution to equation (41) will be sought in the form y(x)=Yf(x)+AYϕ(x), (42) where Yϕ(x) is the solution to equation (40) in which f(x) must be replaced with ϕ(x). Substituting (42) into (41) we have the following algebraic (transcendental) equation for the coefﬁcient A: A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt= 0. (43) Formulas (42)–(43) can deﬁne one, several, or inﬁnitely many solutions (or even none) to equation (41). In addition, the condition ϕ(x)∈LFmust be satisﬁed. Example 3. The solution of the linear integral equation of the second kind Y(x)+λ⎝integraldisplay∞ –∞e–|x–t|Y(t)dt=f(x), λ>–1 2, (44) is expressed as (see equation 4.2.14 in Section 4.2) Yf(x)=f(x)–λ √ 1+2λ⎝integraldisplay∞ –∞exp⎝parenleftbig –√ 1+2λ|x–t|⎝parenrightbig f(t)dt. (45) Now consider the more complex integral equation with quadratic nonlinearity y(x)+⎝integraldisplay∞ –∞⎝bracketleftbigλe–|x–t|y(t)+ϕ(x)ψ(t)y2(t)]dt=f(x). (46) In terms of equation (41), we have Ψ(t,y(t)) =ψ(t)y2(t) in (46). The corresponding solution (45) to equation (44) with ϕ(x) is written as Yϕ(x)=ϕ(x)–λ √ 1+2λ⎝integraldisplay∞ –∞exp⎝parenleftbig –√ 1+2λ|x–t|⎝parenrightbig ϕ(t)dt. Hence, equation (46) has the two solutions y(t)=Yf(t)+A1,2Yϕ(t), where A1,2are roots of the quadratic equation pA2+qA+r=0 , p=⎝integraldisplay∞ 0ψ(t)Y2 ϕ(t)dt,q=1+2⎝integraldisplay∞ 0ψ(t)Yf(t)Yϕ(t)dt,r=⎝integraldisplay∞ 0ψ(t)Y2 f(t)dt. 2◦. The integral equation y(x)+⎝integraldisplayb a⎝bracketleftbigg K(x,t)y(t)+n⎝summationdisplay m=1ϕm(x)Ψm(t,y(t))⎝bracketrightbigg dt=f(x), (47) with its kernel being the sum of the kernel of equation (40) and an arbitrary degenerate nonlinear kernel, can be solved in a similar manner. The solution is sought in the additive form y(x)=Yf(x)+n⎝summationdisplay m=1AmYϕm(x), (48) where Yϕm(x) is the solution to equation (40) in which f(x) must be replaced with ϕm(x). Substi- tuting (48) into (47) results in the following algebraic (transcendental) system of equations for thecoefﬁcients A m: Am+⎝integraldisplayb aΨm⎝parenleftbigg t,Yf(t)+n⎝summationdisplay j=1AjYϕj(t)⎝parenrightbigg dt=0 , m=1 ,...,n. (49) Remark 1. Formulas (38)–(39) and (48)–(49), which deﬁne solutions to the special Urysohn equations of the ﬁrst and second kind (37) and (47), respectively, are coincident (but the functions Yf(x)a n dYϕm(x) are different). 824 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS Remark 2. The method outlined may be used for approximate solution of nonlinear integral equations of the form y(x)+⎝integraldisplayb a⎝bracketleftbig K(x,t)y(t)+Ψ(x,t,y(t))⎝bracketrightbig dt=f(x) by appropriately selecting an approximation of the nonlinear part of the kernel, Ψ(x,t,y(t))≈ n⎝summationdisplay m=1ϕm(x)Ψm(t,y(t)). 16.4-6. Some Generalizations. The method presented in Subsections 16.4-4 and 16.4-5 for the special Urysohn equations of the ﬁrst and second kind admits generalizations. Consider an abstract nonlinear equation for the function y=y(x): L[y]+n⎝summationdisplay m=1ϕm(x)Im[y]=f(x), (50) where L[y] is a linear operator (it can be integral, functional, differential,* or other) and Im[y]a r e some nonlinear functionals (i.e., numbers for any given y(x)). Examples of nonlinear functionals: I1[y]=ay2(0) +by(1), I2[y]= m a x 0≤x≤1|y(x)|,I3[y]=⎝integraldisplayb aK⎝parenleftbig t,y(t),y/prime t(t),y/prime/prime tt(t)⎝parenrightbig dt. Suppose the truncated linear equation L[Y]=f(x), (51) obtained from (50) by setting ϕm(x)=0(m=1 ,...,n), can be solved for any f(x) from some class of functions LF.L e tYf(x) denote the corresponding solution. Let the conditions ϕm(x)∈LF(m=1 ,...,n) be satisﬁed. Solutions to the nonlinear equation (50) are sought in the form y(x)=Yf(x)+n⎝summationdisplay m=1AmYϕm(x), (52) where Yϕm(x) is the solution to equation (51) in which f(x) must be replaced with ϕm(x). Substi- tuting (52) into (50) results in the following algebraic (transcendental) system of equations for thecoefﬁcients A m: Am+Im⎝bracketleftbigg Yf(x)+n⎝summationdisplay j=1AjYϕj(x)⎝bracketrightbigg ,m=1 ,...,n. (53) Formulas (52)–(53) can deﬁne one, several, or inﬁnitely many solutions (or even none) to equa- tion (50). * In this case, the equation must be supplemented with appropriate homogeneous boundary conditions. 16.4. E XACT METHODS FOR NONLINEAR EQUATIONS WITH CONSTANT INTEGRATION LIMITS 825 Example 4. Consider the nonlinear functional integral equation y(x)+λy(a–x)+⎝integraldisplaya 0ϕ(x)Ψ(t,y(t),y(a–t))dt=f(x),λ≠±1, (54) where 0 ≤x≤a,0≤t≤a. The truncated linear functional equations (54), with ϕ(x) = 0, has the solution Yf(x)=f(x)–λf(a–x) 1–λ2. Therefore, solutions to the nonlinear equation (54) are sought in the form y(x)=Yf(x)+AYϕ(x),Yϕ(x)=ϕ(x)–λϕ(a–x) 1–λ2, where the constant Ais determined from the algebraic (transcendental) equation A+⎝integraldisplaya 0Ψ(t,Yf(t)+AYϕ(t),Yf(a–t)+AYϕ(a–t))dt=0 . Example 5. Consider the nonlinear integro-functional-differential equation ⎝integraldisplayπ/2 0[y(xsint)+ϕ(x)Ψ(t,y(t),y/prime t(t))]dt=f(x). (55) Forϕ(x) = 0, it is the Schl ¨omilch equation. Its solution is given in Subsection 3.5 (see Eq. 3.5.40). It should be noted that equation (50) contains the unknown function with different arguments, y(xsint)a n dy(t). Following the method described above, we look for solutions to equation (55) in the form ym(z)=Yf(z)+AmYϕ(z), where Yf(z)=2 π⎝bracketleftbigg f(0) +z⎝integraldisplayπ/2 0f/prime ξ(ξ)dτ⎝bracketrightbigg ,Yϕ(z)=2 π⎝bracketleftbigg ϕ(0) +z⎝integraldisplayπ/2 0ϕ/prime ξ(ξ)dτ⎝bracketrightbigg ,ξ=zsinτ, andAmare roots of the algebraic (transcendental) equation A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t),Y/prime f(t)+AY/prime ϕ(t))dt=0 . Example 6. Consider a boundary value problem for the nonlinear integro-differential equation y/prime/prime xx+ϕ(x)⎝integraldisplay1 0Ψ(t,y(t))dt=f(x) (56) with homogeneous boundary conditions y(0) =y(1) = 0. (57) The solution of an auxiliary linear boundary value problem Y/prime/prime xx=f(x); Y(0) =Y(1) = 0 has the form Yf(x)=⎝integraldisplay1 0G(x,ξ)f(ξ)dξ, (58) G(x,ξ)=⎝braceleftbigg(ξ–1 )xfor 0 ≤x≤ξ≤1, (x–1 )ξfor 0 ≤ξ≤x≤1. Therefore solutions of the original boundary value problem for nonlinear integro-differential equation (56) with boundary conditions (57) can be constructed in the form y(x)=Yf(x)+AYϕ(x), (59) where Yϕ(x) is determined by the right-hand side of formula (58), in which function f(x) is changed by function ϕ(x). Substitution of (59) to (56) leads to the following algebraic (transcendental) equation for determining of A: A+⎝integraldisplayb aΨ(t,Yf(t)+AYϕ(t))dt=0 . (60) In particular case of f(x)=0a n d ϕ(x) = 1 in formulas (59)–(60) it is necessary to set Yf(x)=0 ,Yϕ(x)=1 2x(x–1 ) . 826 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS Remark. In the case of inhomogeneous boundary conditions for integro-differential equation it is necessary to change variables using relation y(x)= ¯y(x)+g(x), where g(x)i sa na r b i t r a r y sufﬁciently smooth function that satisﬁes boundary conditions. Finally we obtain the problem with homogeneous boundary conditions. For example, for integro-differential equation (56) with inhomogeneous boundary conditions y(0) =a,y(1) =bone can take g(x)=a+(b–a)x. References for Section 16.4: M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), A. D. Polyanin and A. V . Manzhi- rov (1998), A. D. Polyanin and A. I. Zhurov (2007). 16.5. Approximate and Numerical Methods for Nonlinear Equations with Constant Integration Limits 16.5-1. Successive Approximation Method. Consider the nonlinear Urysohn integral equation in the canonical form: y(x)=⎝integraldisplayb aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt,a≤x≤b.( 1) The iteration process for this equation is constructed by the formula yk(x)=⎝integraldisplayb aK⎝parenleftbig x,t,yk–1(t)⎝parenrightbig dt,k=1 ,2 , ... (2) If the function K(x,t,y) is jointly continuous together with the derivative K/prime y(x,t,y) (with respect to the variables x,t,a n d ρ,a≤x≤b,a≤t≤b,a n d |y|≤ρ)a n di f ⎝integraldisplayb asup y|K(x,t,y)|dt≤ρ,⎝integraldisplayb asup y|K/prime y(x,t,y)|dt≤β<1 , ( 3 ) then for any continuous function y0(x) of the initial approximation from the domain {|y|≤ρ,a≤x≤b}, the successive approximations (2) converge to a continuous solution y∗(x), which lies in the same domain and is unique in this domain. The rate of convergence is deﬁned by the inequality |y∗(x)–yk(x)|≤βk 1–βsup x|y1(x)–y0(x)|,a≤x≤b,( 4 ) which gives an ap r i o r i estimate for the error of the kth approximation. The a posteriori estimate (which is, in general, more precise) has the form |y∗(x)–yk(x)|≤β 1–βsup x|yk(x)–yk–1(x)|,a≤x≤b.( 5 ) A solution of an equation of the form (1) with an additional term f(x) on the right-hand side can be constructed in a similar manner. Example 1. Let us apply the successive approximation method to solve the equation y(x)=⎝integraldisplay1 0xty2(t)dt–5 12x+1 . The recursive formula has the form yk(x)=⎝integraldisplay1 0xty2 k–1(t)dt–5 12x+1 , k=1 ,2 , ... For the initial approximation we take y0(x) = 1. The calculation yields y1(x) = 1 + 0.083 x, y8(x) = 1 + 0.27 x, y16(x) = 1 + 0.318 x,y2(x) = 1 + 0.14 x, y9(x) = 1 + 0.26 x, y17(x) = 1 + 0.321 x,y3(x) = 1 + 0.18 x, y10(x) = 1 + 0.29 x, y18(x) = 1 + 0.323 x,..., ..., ... Thus, the approximations tend to the exact solution y(x)=1+1 3x. We see that the rate of convergence of the iteration process is fairly small. Note that in Subsection 16.5-2, the equation in question is solved by a more efﬁcient method. 16.5. A PPROXIMA TE METHODS FOR NONLINEAR EQUATIONS WITH CONSTANT INTEGRATION LIMITS 827 16.5-2. Newton–Kantorovich Method. The solution of nonlinear integral equations is a complicated problem of computational mathematics, which is related to difﬁculties of both a principal and computational character. In this connection,methods are developed that are especially designed for solving nonlinear equations, including the Newton–Kantorovich method, which makes it possible to provide and accelerate the convergence of iteration processes in many cases. We consider this method in connection with the Urysohn equation in the canonical form (1). The iteration process is constructed as follows: y k(x)=yk–1(x)+ϕk–1(x), k=1 ,2 , ...,( 6 ) ϕk–1(x)=εk–1(x)+⎝integraldisplayb aK/prime y⎝parenleftbig x,t,yk–1(t)⎝parenrightbig ϕk–1(t)dt,( 7) εk–1(x)=⎝integraldisplayb aK⎝parenleftbig x,t,yk–1(t)⎝parenrightbig dt–yk–1(x). (8) At each step of the algorithm, a linear integral equation for the correction ϕk–1(x) is solved. Under some conditions, the process (6) has high rate of co nvergence; however, it is rather complicated, because at each iteration we must obtain the new kernel K/prime y⎝parenleftbig x,t,yk–1(t)⎝parenrightbig for Eqs. (7). The algorithm can be simpliﬁed by using the equation ϕk–1(x)=εk–1(x)+⎝integraldisplayb aK/prime y⎝parenleftbig x,t,y0(t)⎝parenrightbig ϕk–1(t)dt (9) instead of (7). If the initial approximation is chosen successfully, then the difference between the integral operators in (7) and (9) is small, and the kernel in (9) remains the same in the course of thesolution. The successive approximation method that consists in the application of formulas (6), (8), and (9) is called the modiﬁed Newton–Kantorovich method. In principle, its rate of convergence is less than that of the original (unmodiﬁed) method; however, this version of the method is less complicated in calculations, and therefore it is frequently preferable. Let the function K(x,t,y) be jointly continuous together with the derivatives K /prime y(x,t,y)a n d K/prime/prime yy(x,t,y) with respect to the variables x,t,y,w h e r e a≤x≤banda≤t≤b, and let the following conditions hold: 1◦. For the initial approximation y0(x), the resolvent R(x,t) of the linear integral equation (7) with the kernel K/prime y⎝parenleftbig x,t,y0(t)⎝parenrightbig satisﬁes the condition ⎝integraldisplayb a|R(x,t)|dt≤A<∞, a≤x≤b. 2◦. The residual ε0(x) of Eq. (8) for the approximation y0(x) satisﬁes the inequality |ε0(x)|=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝integraldisplay b aK⎝parenleftbig x,t,y0(t)⎝parenrightbig dt–y0(x)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle≤B<∞. 3◦. In the domain |y(x)–y0(x)|≤2(1 +A)B, the following relation holds: ⎝integraldisplayb asup y⎝vextendsingle⎝vextendsingleK/prime/prime yy(x,t,y)⎝vextendsingle⎝vextendsingledt≤D<∞. 4◦. The constants A,B,a n dDsatisfy the condition H=( 1+ A)2BD ≤1 2. 828 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS In this case, under assumptions 1◦–4◦, the process (6) converges to a solution y∗(x)o fE q .( 1 )i nt h e domain |y(x)–y0(x)|≤(1 –√ 1–2H)H–1(1 –A)B,a≤x≤b. This solution is unique in the domain |y(x)–y0(x)|≤2(1 +A)B,a≤x≤b. The rate of convergence is determined by the estimate |y∗(x)–yk(x)|≤21–k(2H)2k–1(1 –A)B,a≤x≤b. Thus, the above conditions establish the convergence of the algorithm and the existence, the position, and the uniqueness domain of a solution of the nonlinear equation (1). These conditions impose certain restrictions on the initial approximation y0(x) whose choice is an important independent problem that has no uniﬁed approach. As usual, the initial approximation is determined either by more detailed ap r i o r i analysis of the equation under consideration or by physical reasoning implied by the essence of the problem described by this equation. Under a successful choice of the initialapproximation, the Newton–Kantorovich method provides a high rate of convergence of the iteration process to obtain an approximate solution with given accuracy. Remark. Let the right-hand side of Eq. (1) contain an additional term f(x). Then such an equation can be represented in the form (1), where the integrand is K⎝parenleftbig x,t,y(t)⎝parenrightbig +(b–a)–1f(x). Example 2. Let us apply the Newton–Kantorovich method to solve the equation y(x)=⎝integraldisplay1 0xty2(t)dt–5 12x+1 . (10) For the initial approximation we take y0(x) = 1. According to (8), we ﬁnd the residual ε0(x)=⎝integraldisplay1 0xty2 0(t)dt–5 12x+1–y0(x)=x⎝integraldisplay1 0td t–5 12x+1–1=1 12x. They-derivative of the kernel K(x,t,y)=xty2(t), which is needed in the calculations, has the form K/prime y(x,t,y)=2xty(t). According to (7), we form the following equation for ϕ0(x): ϕ0(x)=1 12x+2x⎝integraldisplay1 0ty0(t)ϕ0(t)dt, where the kernel turns out to be degenerate, which makes it possible to obtain the solution ϕ0(x)=1 4xdirectly. Now we deﬁne the ﬁrst approximation to the desired function: y1(x)=y0(x)+ϕ0(x)=1+1 4x. We continue the iteration process and obtain ε1(x)=⎝integraldisplay1 0xt⎝parenleftbig 1+1 4t⎝parenrightbig dt+⎝parenleftbig 1–5 12x⎝parenrightbig –⎝parenleftbig 1+1 4x⎝parenrightbig =1 64x. The equation for ϕ1(x)h a st h ef o r m ϕ1(x)=1 64x+2x⎝integraldisplay1 0t⎝parenleftbig 1+1 4t⎝parenrightbig dt+⎝parenleftbig 1–5 12x⎝parenrightbig –⎝parenleftbig 1+1 4x⎝parenrightbig , and the solution is ϕ1(x)=3 40x. Hence, y2(x)=1+1 4x+3 40x= 1 + 0.325 x. The maximal difference between the exact solution y(x)=1+1 3xand the approximate solution y2(x) is observed at x= 1 and is less than 0.5%. This solution is not unique. The other solution can be obtained by taking the function y0(x)=1+0 . 8 xfor the initial approximation. In this case we can repeat the above sequence of approximations and obtain the following results (thenumerical coefﬁcient of xis rounded): y 1(x) = 1 + 0.82 x,y2(x) = 1 + 1.13 x,y3(x) = 1 + 0.98 x,..., and the subsequent approximations tend to the exact solution y(x)=1+ x. 16.5. A PPROXIMA TE METHODS FOR NONLINEAR EQUATIONS WITH CONSTANT INTEGRATION LIMITS 829 We see that the rate of convergence of the iteration process performed by the Newton–Kantorovich method is signiﬁcantly higher than that performed by the method of successive approximations (see Example 1 in Subsection 16.5-1). To estimate the rate of convergence of the performed iteration process, we can compare the above results with the realization of the modiﬁed Newton–Kantorovich method. In connection with the latter, for the above versions of theapproximations we can obtain y n(x)=1+ knx; k0 k1 k2 k3 k4 k5 k6 k7 k8 ... 0 0.25 0.69 0.60 0.51 0.44 0.38 0.36 0.345 ... . The iteration process converges to the exact solution y(x)=1+1 3x. We see that the modiﬁed Newton–Kantorovich method is less efﬁcient than the Newton–Kantorovich method, but more efﬁcient than the method of successive approximations (see Example 1 in Subsection 16.5-1). 16.5-3. Quadrature Method. To solve an arbitrary nonlinear equation, we can apply the method based on the application of quadrature formulas. The procedure of composing the approximating system of equations is the same as in the linear case (see Subsection 13.19-1). We consider this procedure for an example of the Urysohn equation of the second kind: y(x)–⎝integraldisplayb aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=f(x), a≤x≤b. (11) We set x=xi(i=1 ,...,n). Then we obtain y(xi)–⎝integraldisplayb aK⎝parenleftbig xi,t,y(t)⎝parenrightbig dt=f(xi), i=1 ,...,n, (12) On applying the quadrature formula from Subsection 13.19-1 and neglecting the approximation error, we transform relations (12) into the system of nonlinear equations yi–n⎝summationdisplay j=1AjKij(yj)=fi,i=1 ,...,n, (13) for the approximate values yiof the solution y(x) at the nodes x1,...,xn,w h e r e fi=f(xi)a n d Kij(yj)=K(xi,tj,yj), and Ajare the coefﬁcients of the quadrature formula. The solution of the nonlinear system (13) gives values y1,...,ynfor which by interpolation we ﬁnd an approximate solution of the integral equation (11) on the entire interval [ a,b]. For the analytic expression of an approximate solution, we can take the function ˜y(x)=f(x)+n⎝summationdisplay j=1AjK(x,xj,yj). (14) 16.5-4. Tikhonov Regularization Method. In connection with the nonlinear Urysohn integral equation of the ﬁrst kind ⎝integraldisplayb aK⎝parenleftbig x,t,y(t)⎝parenrightbig dt=f(x), c≤x≤d, (15) 830 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS where f(x)∈L2(c,d)a n dy(t)∈L2(a,b), the Tikhonov regularization method leads to a regularized nonlinear integral equation in the form αyα(x)+⎝integraldisplayb aM⎝parenleftbig t,x,yα(t),yα(x)⎝parenrightbig dt=F⎝parenleftbig x,yα(x)⎝parenrightbig ,a≤x≤b, (16) M⎝parenleftbig t,x,y(t),y(x)⎝parenrightbig =⎝integraldisplayd cK⎝parenleftbig s,t,y(t)⎝parenrightbig K/prime y⎝parenleftbig s,x,y(x)⎝parenrightbig ds, (17) F⎝parenleftbig x,y(x)⎝parenrightbig =⎝integraldisplayd cK/prime y⎝parenleftbig t,x,y(x)⎝parenrightbig f(t)dt, (18) where αis a regularization parameter. For instance, by applying the quadrature method on the basis of the trapezoidal rule, we can reduce Eq. (16) to a system of nonlinear algebraic equations. An approximate solution of (15) is constructed by the principle described above for linear equations (see Section 12.11). References for Section 16.5: N. S. Smirnov (1951), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. G. Tricomi (1985), A. F. Verlan’ and V . S. Sizikov (1986). 16.6 Existence and Uniqueness Theorems for Nonlinear Equations 16.6-1. Hammerstein Equations. 1◦. Consider the Hammerstein equation y(x)=⎝integraldisplayb aK(x,t)Φ(t,y(t))dt,a≤x≤b.( 1) Assume that the function Φ(t,y) is continuous, and the kernel K(x,t) is positive deﬁnite, continuous, and symmetric, K(x,t)=K(t,x). THEOREM 1.Suppose that the inequality |Φ(t,y)|≤C1|y|+C2 holds with some positive constants C1andC2such that C1<λ1,a n dλ1is the smallest characteristic value of the kernel K(x,t). Then the nonlinear integral equation (1) has at least one continuous solution. THEOREM 2.If for any ﬁxed t∈[a,b], the function Φ(t,y)is nondecreasing with respect to y, then the nonlinear integral equation (1) has at most one solution. THEOREM 3.The nonlinear integral equation (1) has at most one solution if the function Φ(t,y) satisﬁes the uniform Lip schitz condition |Φ(t,y2)–Φ(t,y1)|≤σ|y2–y1|, where 0<σ<λ1,λ1is the smallest characteristic value of the kernel K(x,t). THEOREM 4(ON NONEXISTENCE OF SOLUTIONS ).Suppose that K(x,t)≥0,K(x,t)/ ≡0,a n dt h e eigenfunction y1(x)of the kernel K(x,t)corresponding to the smallest characteristic value λ1does not change sign in the domain a≤x,t≤b. Then the condition Φ(t,y(t)) >λ1y(t)( for all t∈[a,b]) ensures that equation (1) has no solutions. 2◦. Assume now that the kernel K(x,t) of equation (1) is continuous and positive deﬁnite (possibly, nonsymmetric, K(x,t)≠K(t,x)), and the function Φ(t,y) is continuous. 16.6 E XISTENCE AND UNIQUENESS THEOREMS FOR NONLINEAR EQUATIONS 831 THEOREM 5.Suppose that the inequality ⎝integraldisplayy 0Φ(t,y)dy≤1 2Ay2+B (t∈Ω,|y|<∞)( 2) holds with a constant A<λ1,w h e r e λ1is the smallest characteristic value of the kernel K(x,t). Then equation (1) has at least one continuous solution. Now consider the case of an unbounded positive-deﬁnite kernel K(x,t). Then the following result holds. THEOREM 6.Suppose that the kernel K(x,t)satisﬁes the condition ⎝integraldisplayb a⎝integraldisplayb a|K(x,t)|pdx dt <∞, p≥2, and the function Φ(t,y)satisﬁes the inequality (2) and the condition |Φ(t,y)|≤a+b|u|p–1(a≤x,t≤b,|y|<∞). Then equation (1) has at least one solution. THEOREM 7.LetK(x,t)be positive and continuous in the domain a≤x,t≤b. Suppose that the function Φ(t,y)is continuous in the domain a≤t≤b,y>0, nonnegative for y≥0and strictly positive for y>0and almost all t. Suppose also that one of the following conditions holds: 1)Φ(t,y)does not decrease in y,a n d y–βΦ(t,y)does not increase in y,w h e r e βis a point from the interval (0, 1); 2)Φ(t,y)does not increase in y,a n d yβΦ(t,y)increases in y,w h e r e βis a point from the interval. Then equation (1) has one and only one positive solution. This solution is the uniform limit of the successive approximations yn(x)=⎝integraldisplayb aK(x,t)Φ(t,yn–1(t))dt (n=1 ,2 , ...), where y0(x)is an arbitrary nonzero nonnegative initial function. 3◦. Consider a system of integral equations of Hammerstein’s type yi(x)=⎝integraldisplayb aKi(x,t)Φi(t,y1(t),...,yn(t))dt (i=1 ,...,n)( 3 ) with continuous symmetric positive-deﬁnite kernels Ki(x,t), where Φi(t,y1,...,yn) are continuous functions in all their arguments. THEOREM 8.Suppose that the functions Φi(t,y1,...,yn)satisfy the inequality n⎝summationdisplay i=1yiΦi(t,y1,...,yn)≤An⎝summationdisplay i=1y2 i+B, where A<λ0andλ0is the smallest characteristic value of the kernels Ki(x,t). Then system (3) has at least one continuous solution. 832 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS 16.6-2. Urysohn Equations. Consider the nonlinear Urysohn equation y(x)=λ⎝integraldisplayb aK(x,t,y(t))dt+f(x). (4) THEOREM 1.LetK(x,t,y)andf(x)be continuous functions of their arguments a≤x,t≤b, –∞<y<∞, and suppose that K(x,t,y)satisﬁes the Lipsch itz condition in y, |K(x,t,y2)–K(x,t,y1)|≤L|y2–y1|, where Lis a constant independent of y1andy2. Then the condition λ<b–a L ensures that equation (4) has one and only one continuous solution. This solution can be found by the method of successive approximations yn+1(x)=λ⎝integraldisplayb aK(x,t,yn(t))dt+f(x), n=0 ,1 , ..., with an arbitrary continuous function y0(x). Remark. If the function K(x,t,y) has a bounded partial derivative in y: ⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle∂K ∂y⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle≤Λ (a≤x,t≤b,–∞<y<∞), thenK(x,y,t) satisﬁes the Lipsc hitz condition in ywith a constant L≤Λ. THEOREM 2.Letf(x)∈L2(a,b)and suppose that |K(x,t,y1)–K(x,t,y2)|≤M(x,t)|y1–y2| (a≤x,t≤b,–∞<y<∞), where⎝integraldisplayb a⎝integraldisplayb a|M(x,t)|2dx dt =B2<∞. Then, for λ<1 B, equation (4) has one and only one solution in L2(a,b). THEOREM 3.Suppose that the function K(x,t,y)is continuous in yand satisﬁes the inequality |K(x,t,y)|≤M(x,t)(A +B|u|p)( a≤x,t≤b,–∞<y<∞), where A,B,p>0,a n d⎝integraldisplayb a⎝integraldisplayb a|M(x,t)|p+1dx dt <∞. Then, for any sufﬁciently small |λ|and any f(x)∈Lp+1(a,b), equation (4) has a solution y(x)∈ Lp+1(a,b).I fp<1, then a solution exists for any λ. THEOREM 4.Suppose that the function K(x,t,y)is continuous in the domain Ω={a≤x≤b, a≤t≤b,|y|≤ρ}and its partial derivative in yis bounded, ⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle∂K ∂y⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle≤C (x,t,y∈Ω). 16.6 E XISTENCE AND UNIQUENESS THEOREMS FOR NONLINEAR EQUATIONS 833 Then the equation y(x)=λ⎝integraldisplayb aK(x,t,y(t))dt,( 5) where |λ|C|b–a|<1 , |λ|max a≤x≤b⎝integraldisplayb amax |y|≤ρ|K(x,t,y)|dt≤ρ, has one and only one continuous solution y(x)(a≤x≤b) satisfying the inequality |y(x)|≤ρ. Ify0(x)is an arbitrary continuous function satisfying the inequality |y0(x)|≤ρ(a≤x≤b), then the successive approximations yn+1(x)=λ⎝integraldisplayb aK(x,t,yn(t))dt (n=0 ,1 , ...) are uniformly convergent on [a,b]to this solution. THEOREM 5.Suppose that the function K(x,t,y)is continuous in the domain Ω={a≤x≤b, a≤t≤b,|y|≤ρ}. Then the condition |λ|≤ρ (b–a)m a x x,t,y∈Ω|K(x,t,y)| ensures that the integral equation (5) has at least one continuous solution satisfying the inequality |y(x)|≤ρ. THEOREM 6.LetK(x,t,y)be continuous in the domain Ω={a≤x≤b,a≤t≤b,–∞<y<∞} and let ϕ(r)= m a x a≤x,t≤b,y≤r|K(x,t,y)|, Λ=s u p 0<r<∞r (b–a)ϕ(r). Then, for |λ|<Λ, the integral equation (5) has at least one continuous solution. In particular, if lim r→∞ϕ(r) r=0 , then the integral equation (5) has a solution for any λ. In this case, equation (4), too, has a solution for any λand any continuous f(x). THEOREM 7.Suppose a function K(x,t,y)is continuous in x,t,yand satisﬁes the inequalities 0≤K(x,t,y)≤a+L(x,t)y (a≤x,t≤b,y≥0), where L(x,t)is a nonnegative kernel the smallest characteristic value of which satisﬁes the condition λ>1. Then the equation y(x)=⎝integraldisplayb aK(x,t,y(t))dt has at least one continuous nonnegative solution. References for Section 16.6: M. A. Krasnosel’skii (1964), M. L. Krasnov (1975), P. P. Zabreyko, A. I. Koshelev, et al. (1975), R. Precup (2006). 834 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS 16.7. Nonlinear Equations with a Parameter: Eigenfunctions, Eigenvalues, Bifurcation Points 16.7-1. Eigenfunctions and Eigenvalues of Nonlinear Integral Equations. Consider a nonlinear integral equation y=λA[y]( 1) with a parameter λ. An example of such an equation is provided by the Urysohn equation (5) from Subsection 16.6-2, which has a solution for sufﬁciently small λ(this solution tends to zero as λ→0). Of most importance for applications are nontrivial solutions corresponding to ﬁnite values of λ.B y analogy with the theory of linear integral equations, nontrivial solutions of the nonlinear equation (1) are called eigenfunctions of the corresponding nonlinear operator A(or eigenfunctions of the nonlinear equation (1)), and λfor which such solutions exist are called characteristic values (and the numbers µ=1/λare called eigenvalues ). For nonlinear integral equations (1) with a paramet er (1), a typical situation is the existence of nontrivial solutions for all λfrom some interval (α ,β) (thus, there is a continuous spectrum of characteristic values corresponding to a continuum of eigenfunctions; see Section 16.1-3 for relevant examples), which is in harsh contrast with the situation typical for linear integral equations with a discrete spectrum of characteristic values. THEOREM 1.Suppose that Φ(t,y)is continuousand the kernel K(x,t)is continuous, symmetric, and positive deﬁnite. Then the Hammerstein equation y(x)=λ⎝integraldisplayb aK(x,t)Φ(t,y(t))dt,a≤x≤b,( 2) has a continuum of eigenfunctions. THEOREM 2.LetK(x,t)be symmetric, positive deﬁnite, and satisfying the inequality ⎝integraldisplayb a⎝integraldisplayb a|K(x,t)|pdx dt <∞, p≥2, and suppose that Φ(t,y)is continuous and satisﬁes the inequality |Φ(t,y)|≤C1+C2|y|p–1. Then the Hammerstein equation (2) has a continuum of eigenfunctions. THEOREM 3.Suppose that the kernel K(x,t)is continuous, symmetric, and has ﬁnitely many negative characteristic values. Suppose also that Φ(t,y)is continuous and satisﬁes the inequality yΦ(t,y)≥Ay2–B (a≤x,t≤b,–∞<y<∞), forA>0. Then the Hammerstein equation (2) has a continuum of eigenfunctions. THEOREM 4.Suppose that K(x,t,y)is continuous and satisﬁes the inequality K(x,t,y)≥L(x,t)y (a≤x,t≤b,y>0 ) , where L(x,t)is a positive continuous kernel. Then the Urysohn equation y(x)=λ⎝integraldisplayb aK(x,t,y(t))dt has a continuum of eigenfunctions. 16.7. N ONLINEAR EQUATIONS WITH A PARAMETER :EIGENFUNCTIONS ,EIGENV ALUES ,BIFURCATION POINTS 835 16.7-2. Local Solutions of a Nonlinear Integral Equation with a Parameter. Consider the Urysohn equation y(x)=⎝integraldisplayb aK(x,t,y(t);λ)dt (a≤x≤b)( 3 ) with the integrand depending on the parameter λin an arbitrary manner. Assume that a function y0(x) is a solution of equation (3) for λ=λ0. It is important to know the conditions under which equation (3) has solutions y(x) close to y0(x)f o rλ close to λ0. THEOREM .Suppose that the function K(x,t,y;λ)and its partial derivative K/prime y(x,t,y;λ)are continuous in all their arguments and a continuous function y0(x)is a solution of equation (3) for λ=λ0, y0(x)=⎝integraldisplayb aK(x,t,y0(t);λ0)dt (a≤x≤b). (4) IfΛ=1is not a characteristic value of the kernel K/prime y(x,t,y0(t);λ0), then equation (3) has one and only one continuous solution y=y(x,λ)close to y0(x)forλclose to λ0. Remark. The solution of equation (3) can be sought in the form of expansion in powers of (λ–λ0): y(x,λ)=y0(x)+(λ–λ0)y1(x)+(λ–λ0)2y2(x)+···. Foryk(x) one obtains the triangular system of linear integral equations yk(x)=⎝integraldisplayb aK/prime y(x,t,y0(t);λ0)yk(t)dt+⎝integraldisplayb aFk(x,t,y0(t),...,yk–1(t);λ0)dt,k=1 ,2 , ..., which can be solved in consecutive manner with y0(x) being the solution of equation (4). 16.7-3. Bifurcation Points of Nonlinear Integral Equations. Here it is assumed that for all values of the parameter λ, the integral equation (3) admits the trivial solution y(x)≡0, i.e., K(x,t,0 ;λ)=0 . Av a l u e λ∗is called a bifurcation point for equation (3) if for any ε>0t h e r ei s λ∈(λ∗–ε,λ∗+ε) for which the equation has a nontrivial solution y(x)=y(x,λ) that satisﬁes the inequality /bardbly(x)/bardbl<ε. In simple words, the meaning of a bifurcation point λ∗is that the number of solutions changes asλcrosses that point. Example 1. Consider the linear integral equation with a continuous kernel y(x)=λ⎝integraldisplayb aK(x,t)y(t)dt. (5) For any λ, this equation admits the trivial solution y(x)≡0. Letλ=λ∗be a characteristic value of equation (1) corresponding to a nontrivial solution y∗(x). Since this solution is deﬁned to within a constant coefﬁcient, it can be made arbitrarily small in the norm of C(a,b), i.e., for any ε>0 ,t h e r ei sa solution y∗(x)s u c ht h a t /bardbly∗(x)/bardbl=m a x a≤x≤b|y∗(x)|<ε. Thus, for any ε>0t h e r ei s λ∈(λ∗–ε,λ∗+ε) (in this case λ=λ∗) for which equation (5) has a nontrivial solution y∗(x) satisfying the condition /bardbly∗(x)/bardbl<ε. By deﬁnition, the characteristic value λ∗of the kernel K(x,t) is a bifurcation point of equation (5). 836 METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS Example 2. Consider the Hammerstein equation with a degenerate kernel containing a quadratic nonlinearity: y(x)=λ⎝integraldisplay1 0xt[y(t)+y2(t)]dt. (6) For any value of the parameter λ, equation (6) the trivial solution y(x)≡0. Denote A1=⎝integraldisplay1 0ty(t)dt,A2=⎝integraldisplay1 0ty2(t)dt. (7) With this notation, equation (6) can be rewritten as y(t)=λ(A1+A2)x. (8) Substituting (8) into (7), one obtains a second-order algebraic system for the determination of the coefﬁcients A1andA2: A1=1 3λ(A1+A2), A1=1 4λ2(A1+A2)2. The solution of this system leads us to two solutions of the integral equation (6), one of which is trivial, y(x)≡0, and the other has the form y(x)=4(3 –λ) 3λx. (9) Let us show that λ= 3 is a bifurcation point for equation (6). Indeed, for any ε>0t h e r ei s λ∈(3 –ε,3+ε) (for instance, any λ≠3 from this interval) for which equation (6) has the nontrivial solution (9) satisfying the condition /bardbly∗(x)/bardbl=m a x 0≤x≤1⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle4(3 –λ) 3λ⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle<ε. THEOREM 1.Letλ∗be a bifurcation point for the nonlinear integral equation (3). Then 1 is a characteristic value of the kernel L(x,t)=K/prime y(x,t,0 ;λ∗).(It is assumed that the integrand K(x,t,y;λ)and its partial derivative K/prime y(x,t,y;λ)are continuous in their arguments. ) In other words, a n ecessary condition for the existen ce of a bifurcation point λ=λ∗for equation (3) is the existence of a nontrivial solution of the homogeneous linear integral equation u(x)=⎝integraldisplayb aK/prime y(x,t,0 ;λ∗)u(t)dt (a≤x≤b). (10) Theorem 1 suggests which values λ=λ∗can be expected to be bifurcation points. THEOREM 2.Let1be a characteristic value of the kernel L(x,t)=K/prime y(x,t,0 ;λ∗)and let the multiplicity of this value be equal to 1(is odd). Then λ∗is a bifurcation point of the nonlinear integral equation (3). Example 3. Consider the Nekrasov equation y(x)=λ⎝integraldisplay2π 0K(x,t)s i ny(t) 1+⎝integraltextt 0siny(s)dsdt,K(x,t)=1 3∞⎝summationdisplay n=1sin(nx)s i n (nt) n, that describes waves on the surface of ideal incompressible ﬂuid. For all values of the parameter λ, this equation admits the trivial solution y(x)≡0. Its bifurcation points correspond to the values λ=λ∗for which waves are produced. The linearized equation (10) in this case can be written as u(x)=λ∗⎝integraldisplay2π 0K(x,t)u(t)dt. Its characteristic values and the corresponding eigenfunctions have the form λn=3n,un(x)=s i n ( nx). All characteristic values are simple, and therefore, these and only these are bifurcation points of the Nekrasov equation. 16.7. N ONLINEAR EQUATIONS WITH A PARAMETER :EIGENFUNCTIONS ,EIGENV ALUES ,BIFURCATION POINTS 837 THEOREM 3.Let the kernel K(x,t)be continuous and positive deﬁnite and let the function Φ(t,y)and its derivative Φ/prime y(t,y)be continuous, with Φ(t,0 ) ≡0. Then the bifurcation points of the Hammerstein equation (2) with parameter λcoincide with the characteristic values of the kernel L(x,t)=K(x,t)Φ/prime y(t,0 ). Example 4. For the Hammerstein equation with degenerate kernel (6), we have K(x,t)=xt,Φ(t,y)=y+y2,Φ/prime y(t,y)=1+2 y,Φ(t,0 )≡0. Therefore, the assumptions of Theorem 3 hold. Since Φ/prime y(t, 0) = 1, we see that the bifurcation points of equation (6) coincide with the characteristic values of the kernel K(x,t)=xt, u(x)=λ∗⎝integraldisplay1 0xtu(t)dt. Substituting u(x)=Aλ∗xinto this equation, we obtain the unique characteristic value λ∗= 3, which is a bifurcation point of the Hammerstein equation. Remark. The bifurcation point λ∗= 3 in Example 2 was obtained in a more difﬁcult way, by direct examination of equation (6). References for Section 16.7: M. A. Krasnosel’skii (1964), M. G. Krein (1972), M. L. Krasnov (1975), P. P. Zabreyko, A. I. Koshelev, et al. (1975), R. Precup (2006). Chapter 17 Methods for Solving Multidimensional Mixed Integral Equations 17.1. Some Deﬁnition and Remarks 17.1-1. Basic Classes of Functions. Integral equations containing both the V olterra kernels (see Subsection 10.1-1) and the Fredholm kernels (see Subsections 12.1-1 and 12.1-2) are called mixed integral equations . Such integral equations arise in applications and are a fairly new object of mathematical studies. So far, nodeﬁnite classiﬁcation of such equations has been given, and such a classiﬁcation is likely to be vast and ramiﬁed. Here, we consider some integral equations and related problems that have been studied in more detail. Mixed integral equations are multidimensional (at least two-dimensional). In the integral terms of such equations with V olterra kernels, the unknown function of several variables is integrated inthe variable that has the meaning of the time; and in the integral terms with Fredholm kernels, the integration of the same unknown function is over some (one- or multi-dimensional) domain. Let us describe the main classes of multidimensional real-valued functions that appear in mixed integral equations. For a bounded closed domain ΩinR n,t h es e t L2(Ω) consists of all real-valued functions f(/vector x) deﬁned in Ωand having their squared absolute value |f(/vector x)|2integrable in Ω.T h e s e t L2(Ω)i sa Hilbert space with the following scalar product and the norm (see Supplement 12.5-2): (f,g)=⎝integraldisplay Ωf(/vector x)g(/vector x)dΩx,/bardblf/bardbl=(f,f)1/2=⎝radicalBigg ⎝integraldisplay Ω|f(/vector x)|2dΩx, where /vector x=(x1,...,xn)∈Rn. Example. 1◦.I fn=2a n d Ωis the ring ω={a≤r≤b,0≤ϕ≤2π},t h e n L2(Ω)≡L2(ω),f(/vector x)≡f(r,ϕ),⎝integraldisplay Ω≡⎝integraldisplay ω≡⎝integraldisplayb a⎝integraldisplay2π 0,dΩ≡rd rd ϕ . 2◦. The subspace of L2(ω) consisting of functions that depend only on the radial coordinate ris denoted by⎝hatwideL2(ω). In this case, f(/vector x)≡f(r),⎝integraldisplay Ωf(/vector x)dΩ=⎝integraldisplay ωf(r)rdr dϕ =2π⎝integraldisplayb arf(r)dr, The space⎝hatwideL2(ω) is a Hilbert space with the scalar product (f,g)=2π⎝integraldisplayb af(r)g(r)rd r. 839 840 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Frequently the space ⎝hatwideL2(ω) is considered as the space of functions of the form ⎝hatwidef(r)=√ 2πf(r) for which the scalar product has the like form but without factor 2 π, (⎝hatwidef,⎝hatwideg)=⎝integraldisplayb a⎝hatwidef(r)⎝hatwideg(r)rd r. In what follows we will use the latter deﬁnition of the scalar product. To simplify notation we will omit the hat symbol over functions. LetF(t) be a function of the real argument t∈[τ0,T] with values in a Banach space B, i.e., F:[τ0,T]→B. The function F(t)i sc a l l e d continuous on [τ0,T]i ff o ra n y t∈[τ0,T], we have /bardblF(t)–F(t1)/bardblB→0a s t1→t, where/bardbl⋅/bardblBis the norm in B. The space of such continuous functions is denoted by C([τ0,T],B). For example, if B=L2(Ω) with the above norm /bardbl⋅/bardbl, we can consider a function y(/vector x,t)s u c h that for each t∈[τ0,T] its value belongs to L2(Ω). Regarded as a function of twith values in L2(Ω), such a function is called continuous inton the interval [ τ0,T]i ff o ra n y t∈[τ0,T], we have /bardbly(/vector x,t)–y(/vector x,t1)/bardbl→ 0a s t1→t. Accordingly, the space of such functions is denoted by C([τ0,T],L2(Ω)). For a function y(/vector x,t)∈C([τ0,T],L2(Ω)), the following properties hold: 1) the norm /bardbly(/vector x,t)/bardblis continuous in t∈[τ0,T]; 2) for any f(/vector x)∈L2(Ω), the scalar product ( y(/vector x,t),f(/vector x)) is continuous in t∈[τ0,T]; 3)y(/vector x,t)∈L2(Ω×(τ0,T)) if the interval ( τ0,T) is ﬁnite. In what follows, we consider the cases of Ωbeing a ﬁnite interval, a circle, or an arbitrary closed bounded set. 17.1-2. Mixed Equations on a Finite Interval. 1◦. For continuous functions of t∈[τ0,T] with values in L2[a,b], the mixed two-dimensional integral equation with symmetric Fredholm kernel has the form σ(t)⎝bracketleftbigg y(x,t)–⎝integraldisplayt τ0V1(t,τ)y(x,τ)dτ⎝bracketrightbigg +⎝integraldisplayb aF(x,ξ)y(ξ,t)dξ –⎝integraldisplayt τ0V2(t,τ)⎝integraldisplayb aF(x,ξ)y(ξ,τ)dξ dτ =f(x,t),a≤x≤b,τ0≤t≤T,( 1 ) where σ(t) is a known continuous positive function; y(x,t) is the unknown function of class C([τ0,T],L2[a,b]);f(x,t)∈C([τ0,T],L2[a,b]) is a given function; V1(t,τ)a n dV2(t,τ) are V olterra kernels (see Subsection 10.1-1); and F(x,ξ) is a Fredholm kernel, so that ⎝integraldisplayb a⎝integraldisplayb aF2(x,ξ)dx dξ =B2<∞. Assume, in addition, that the kernel F(x,ξ) is symmetric and positive deﬁnite. Such a kernel is also called a Hilbert–Schmidt kernel (see also Subsection 13.6-2). 2◦. For functions of class C([τ0,T],L2[a,b]), a mixed two-dimensional integral equation with a Schmidt kernel has the form σ(t)⎝bracketleftbigg y(x,t)–⎝integraldisplayt τ0V1(t,τ)y(x,τ)dτ⎝bracketrightbigg +⎝integraldisplayb aS(x,ξ)y(ξ,t)dξ –⎝integraldisplayt τ0V2(t,τ)⎝integraldisplayb aS(x,ξ)y(ξ,τ)dξ dτ =f(x,t) h(x), S(x,ξ)=F(x,ξ) h(x),a≤x≤b,τ0≤t≤T,(2) 17.1. S OME DEFINITION AND REMARKS 841 where h(x) > 0 is a given function of class L2[a,b], and the other quantities are similar to those introduced for equation (1). The kernel S(x,ξ)i sc a l l e da Schmidt kernel . This kernel is nonsym- metric, but has all the properties of symmetric kernels. Equation (2) is often written in the following equivalent form: σ(t)h(x)⎝bracketleftbigg y(x,t)–⎝integraldisplayt τ0V1(t,τ)y(x,τ)dτ⎝bracketrightbigg +⎝integraldisplayb aF(x,ξ)y(ξ,t)dξ –⎝integraldisplayt τ0V2(t,τ)⎝integraldisplayb aF(x,ξ)y(ξ,τ)dξ dτ =f(x,t),a≤x≤b,τ0≤t≤T.( 3 ) Together with equations (1), (2), we consider some problems that contain, in addition to (1) or (2), some auxiliary integral conditions on the unknown function. Such conditions are introduced in some cases if the right hand-side of the equation is not determined completely. Often, such conditions have the form ⎝integraldisplayb ay(ξ,t)dξ=M1(t),⎝integraldisplayb a⎝parenleftBig ξ–a+b 2⎝parenrightBig y(ξ,t)dξ=M2(t). (4) 17.1-3. Mixed Equation on a Ring-Shaped (Circular) Domain. 1◦. A mixed two-dimensional integral equation with Fredholm kernel for functions of class C⎝parenleftbig [τ0,T],⎝hatwideL2(ω)⎝parenrightbig has the form σ(t)⎝bracketleftbigg y(r,t)–⎝integraldisplayt τ0V1(t,τ)y(r,τ)dτ⎝bracketrightbigg +⎝integraldisplayb aFω(r,ρ)y(ρ,t)ρd ρ –⎝integraldisplayt τ0V2(t,τ)⎝integraldisplayb aFω(r,ρ)y(ρ,τ)ρd ρd τ =f(r,t),a≤r≤b,τ0≤t≤T,( 5 ) where ωis the ring of internal radius aand external radius b(fora= 0, the domain ωis the circle of radius b);σ(t) is a given positive continuous function; y(r,t) is the unknown function of class C⎝parenleftbig [τ0,T],⎝hatwideL2(ω)⎝parenrightbig ;f(r,t) is a given right-hand side of class C⎝parenleftbig [τ0,T],⎝hatwideL2(ω)⎝parenrightbig ;V1(t,τ)a n dV2(t,τ) are V olterra kernels (see Subsection 10.1-1); and Fω(r,ρ) is a Fredholm kernel, so that ⎝integraldisplayb a⎝integraldisplayb aF2 ω(r,ρ)rρ dr dρ =B2 ω<∞. Assume, in addition, that the kernel Fω(r,ρ) is symmetric and positive deﬁnite (see also Subsec- tion 13.6-2), i.e., Fω(r,ρ)=Fω(ρ,r),⎝integraldisplayb a⎝integraldisplayb aFω(r,ρ)ϕ(r)ϕ(ρ)dr dρ ≥0, and the second relation holds as equality only for ϕ(r) = 0. As above, a symmetric positive Fredholm kernel will be called a Hilbert–Schmidt kernel. 2◦. A mixed two-dimensional integral equation with a Schmidt kernel for functions of class C⎝parenleftbig [τ0,T],⎝hatwideL2(ω)⎝parenrightbig has the form σ(t)⎝bracketleftbigg y(r,t)–⎝integraldisplayt τ0V1(t,τ)y(r,τ)dτ⎝bracketrightbigg +⎝integraldisplayb aSω(r,ρ)y(ρ,t)ρd ρ –⎝integraldisplayt τ0V2(t,τ)⎝integraldisplayb aSω(r,ρ)y(ρ,τ)ρd ρd τ =f(r,t) h(r), Sω(r,ρ)=Fω(r,ρ) h(r),a≤r≤b,τ0≤t≤T,(6) 842 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS where h(r) > 0 is a given function in ⎝hatwideL2(ω), and the other notations are similar to those introduced for equation (5). The kernel Sω(r,ρ) is a Schmidt kernel, which possesses all the properties of Hilbert–Schmidt kernels. Equation (6) is often written in the following equivalent form: σ(t)h(r)⎝bracketleftbigg y(r,t)–⎝integraldisplayt τ0V1(t,τ)y(r,τ)dτ⎝bracketrightbigg +⎝integraldisplayb aFω(r,ρ)y(ρ,t)ρd ρ –⎝integraldisplayt τ0V2(t,τ)⎝integraldisplayb aFω(r,ρ)y(ρ,τ)ρd ρd τ =f(r,t),a≤r≤b,τ0≤t≤T.( 7 ) Together with equations (5)–(7), we consider some problems with an auxiliary integral condition on the unknown function. Such conditions are introduced if there is not enough information about the right-hand side of the equation. For equations (5)–(7), this condition often has the form ⎝integraldisplayb ay(ρ,t)ρd ρ =M(t). (8) 17.1-4. Mixed Equations on a Closed Bounded Set. 1◦. A mixed multi-dimensional integral equation with a symmetric Fredholm kernel for functions of class C([τ0,T],L2(Ω)) has the form σ(t)(I–V1)y(/vector x,t)+( I–V2)Fy(/vector x,t)=f(/vector x,t),/vector x∈Ω,τ0≤t≤T, Fy(/vector x,t)=⎝integraldisplay ΩFΩ(/vector x,/vectorξ)y(/vectorξ,t)dΩξ,Vpy(/vector x,t)=⎝integraldisplayt τ0Vp(t,τ)y(/vector x,τ)dτ,(9) where /vector x=(x1,...,xn)∈Rn;Ωis a closed bounded set in Rn;σ(t) is a continuous function of t on [τ0,T];y(/vector x,t)∈C([τ0,T],L2(Ω)) is the unknown function; f(/vector x,t)∈C([τ0,T],L2(Ω)) is a given right-hand side of the equation; Iis the identity operator; Vp(p= 1, 2) are V olterra integral operators with continuous or polar kernels Vp(t,τ); and Fis a Fredholm integral operator, which is a compact operator from L2(Ω)t oL2(Ω) (see Supplement 12.5-3). Its properties are determined by the kernel FΩ(/vector x,/vectorξ), which is assumed to satisfy the condition ⎝integraldisplay Ω⎝integraldisplay ΩF2 Ω(/vector x,/vectorξ)dΩxdΩξ=B2 Ω<∞. (10) Relation (10) is a sufﬁcient conditio n for the compactness of the integral operator F. If the kernel of an integral operator satisﬁes the relation FΩ(/vector x,/vectorξ)=FΩ(/vectorξ,/vector x), (11) then this operator is self-adjoint. If, moreover,⎝integraldisplay Ω⎝integraldisplay ΩFΩ(/vector x,/vectorξ)ϕ(/vector x)ϕ(/vectorξ)dΩxdΩξ≥0, (12) and (12) holds as equality only for ϕ(/vector x) = 0, then the integral operator is called positive deﬁnite. Compact self-adjoint positive deﬁnite operators are called Hilbert–Schmidt operators . 17.2. M ETHODS OF SOLUTION OF MIXED INTEGRAL EQUATIONS ON A FINITE INTERV AL 843 2◦. A mixed multi-dimensional integral equation with a Schmidt operator for functions of class C([τ0,T],L2(Ω)) has the form σ(t)(I–V1)y(/vector x,t)+( I–V2)Sy(/vector x,t)=f(/vector x,t) h(/vector x), Sy(/vector x,t)=⎝integraldisplay ΩSΩ(/vector x,/vectorξ)y(/vectorξ,t)dΩξ,SΩ(/vector x,/vectorξ)=FΩ(/vector x,/vectorξ) h(/vector x), Vpy(/vector x,t)=⎝integraldisplayt τ0Vp(t,τ)y(/vector x,τ)dτ,/vector x∈Ω,τ0≤t≤T,(13) where h(/vector x) > 0 is a given function in L2(Ω);Sis a Schmidt integral operator; and the other notations are similar to those introduced for equation (12). Equation (13) is often written in the followingequivalent form: σ(t)h(/vector x)(I–V 1)y(/vector x,t)+( I–V2)Fy(/vector x,t)=f(/vector x,t),/vector x∈Ω,τ0≤t≤T, Fy(/vector x,t)=⎝integraldisplay ΩFΩ(/vector x,/vectorξ)y(/vectorξ,t)dΩξ,Vpy(/vector x,t)=⎝integraldisplayt τ0Vp(t,τ)y(/vector x,τ)dτ.(14) Together with equations (9), (13), and (14), we consider some problems with auxiliary conditions on the unknown function. Such conditions are introduced if there is not enough information about the right-hand side of the equation. Such conditions usually have the form ⎝integraldisplay Ωy(/vector x,t)fi(/vector x)dΩx=Mi(t), i=1 ,...,N, (15) where fi(/vector x)i sas y s t e mo f Nlinearly independent functions of class L2(Ω). Remark 1. Any equation with a Schmidt kernel (integral operator) can always be reduced (by changing the variables) to an equation with a symmetric Hilbert–Schmidt kernel (self-adjoint integraloperator). Remark 2. A compact operator is a generalization of a Fredholm integral operator. Equations with compact operators are studied in the framework of the Riesz–Schauder theory. Remark 3. A compact self-adjoint operator is a generalization of a Fredholm integral operator with a symmetric kernel. If its kernel is positive deﬁnite, then the corresponding operator is also positive deﬁnite (see Supplement 12.5-3). Equations with compact self-adjoint and positive deﬁnite operators are studied in the framework of the Hilbert–Schmidt theory. References for Section 17.1: E. Goursat (1923), F. Riesz and B. Sz.-Nagy (1955), V . S. Vladimirov (1981), V . M. Alek- sandrov and S. M. Mkhitaryan (1983), N. Kh. Arutynyan, A. V . Manzhirov, and V .E. Naumov (1991), A. N. Kolmogorov and S. V . Fomin (1999), A. V . Manzhirov (2001, 2005). 17.2. Methods of Solution of Mixed Integral Equations on a Finite Interval 17.2-1. Equation with a Hilbert–Schmidt Kernel and a Given Right-Hand Side. Consider the mixed integral equation (1) of Subsection 17.1-2 with a Hilbert–Schmidt kernel. By changing the variables, this equation can always be reduced to a similar equation with the parameters a= –1,b=1 ,τ0=1 : σ(t)⎝bracketleftbigg y(x,t)–⎝integraldisplayt 1V1(t,τ)y(x,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 –1F(x,ξ)y(ξ,t)dξ –⎝integraldisplayt τ0V2(t,τ)⎝integraldisplay1 –1F(x,ξ)y(ξ,τ)dξ dτ =f(x,t), –1 ≤x≤1, 1 ≤t≤T.( 1 ) 844 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Suppose that the right-hand side f(x,t) of (1) is known and we have to ﬁnd the function y(x,t). Heref(x,t)a n d y(x,t) are functions of class C([1,T],L2(–1, 1)). Let us seek a solution of equation (1) in the form of a series y(x,t)=∞⎝summationdisplay k=1yk(t)ϕk(x), (2) where ϕk(x) are eigenfunctions of the kernel F(x,ξ) corresponding to the eigenvalues µk> 0, i.e., ⎝integraldisplay1 –1F(x,ξ)ϕk(ξ)dξ=µkϕk(x), k=1 ,2 , ... (3) The representation of a solution in the form (2) is possible, since eigenfunctions of the kernel F(x,ξ) form a complete orthonormal system of functions in L2[–1, 1] (a basis in L2[–1, 1]; see Subsec- tion 13.6-1 and Supplement 12.5-3). For the same reason, the right-hand side of the equation can be represented in the form f(x,t)=∞⎝summationdisplay k=1fk(t)ϕk(x),fk(t)=⎝integraldisplay1 –1f(x,t)ϕk(x)dx.( 4 ) Substituting (2) into (1) and taking into account (3) and (4), we obtain the following sequence of V olterra equations for the unknown functions yk(t): yk(t)–⎝integraldisplayt 1Vk(t,τ)yk(τ)dτ=δk(t),δk(t)=fk(t) σ(t)+µk,( 5) Vk(t,τ)=σ(t)V1(t,τ)+µkV2(t,τ) σ(t)+µk,k=1 ,2 , ...,( 6 ) where Vk(t,τ) are V olterra kernels which belong to the same class of functions as the kernels V1(t,τ), V2(t,τ), since µk→0a sk→∞ . A solution of the inﬁnite sequence of V olterra equations (5) can be constructed by analytical and numerical methods of Chapter 11. This solution can be written in the form yk(t)=δk(t)+⎝integraldisplayt 1Rk(t,τ)δk(τ)dτ,( 7) where Rk(t,τ) is the resolvent of the kernel Vk(t,τ). Thus, the desired solution has been constructed. The series (2) converges in L2[–1, 1] uniformly int∈[1,T], and its sum is a continuous function of t∈[1,T] with values in L2[–1, 1]. In order to justify the above method of constructing a solution, it remains to construct the eigenfunctions and calculate the eigenvalues of the Hilbert–Schmidt integral operator. Let us represent the kth eigenfunction as a series in terms of any basis pi(x)o fL2[–1, 1]. For deﬁniteness, we take the orthonormal Legendre polynomials P∗ i–1(x) as the basis. Then ϕk(x)=∞⎝summationdisplay i=1ϕi(k)pi(x),pi(x)=P∗ i–1(x). (8) Let us expand the Hilbert–Schmidt kernel in double series with respect to the chosen basis: F(x,ξ)=∞⎝summationdisplay m=1∞⎝summationdisplay i=1Fmnpm(x)pn(ξ), Fmn=⎝integraldisplay1 –1⎝integraldisplay1 –1F(x,ξ)pm(x)pn(ξ)dx dξ ,Fmn=Fnm.(9) 17.2. M ETHODS OF SOLUTION OF MIXED INTEGRAL EQUATIONS ON A FINITE INTERV AL 845 Substituting (8) and (9) into (3),we obtain a linear system of algebraic equations for the determination of eigenvalues and eigenfunction expansion coefﬁci ents. This algebraic system has a symmetric matrix and can be written as follows: ∞⎝summationdisplay n=1Fmnϕn(k)=µkϕm(k),m=1 ,2 , ... (10) Of course, for practical calculations, one has to limit the number of expansion terms. For instance, taking Northonormal Legendre polynomials, we obtain the Nth approximation of the solution. In this case, to construct eigenfunctions and eigenvalues of the Hilbert–Schmidt kernel, it is necessary to ﬁnd eigenvalues and orthonormal eigenvectors of the matrix [FNN]=⎛ ⎜⎜⎜⎜⎝F11F12F13···F1N F12F22F23···F2N F13F23F33···F3N ............... F1NF2NF3N···FNN⎞ ⎟⎟⎟⎟⎠. (11) Eigenvalues of the matrix (11) give approximate values of the ﬁrst Neigenvalues of the Hilbert– Schmidt kernel, and the components of orthonormal eigenvectors of the matrix give expansion coefﬁcients of the ﬁrst Neigenfunctions of the Hilbert–Schmidt kernel with respect to Northonormal Legendre polynomials. 17.2-2. Equation with Hilbert–Schmidt Kernel and Auxiliary Conditions. Consider equation (1) with the right-hand side of the form f(x,t)=α1(t)+α2(t)x–g(x,t)a n dt w o auxiliary integral conditions of the form (4) of Subsection 17.1-2. The problem is to ﬁnd a solution of the mixed integral equation σ(t)⎝bracketleftbigg y(x,t)–⎝integraldisplayt 1V1(t,τ)y(x,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 –1F(x,ξ)y(ξ,t)dξ –⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 –1F(x,ξ)y(ξ,τ)dξ dτ =α1(t)+α2(t)x–g(x,t), –1≤x≤1, 1 ≤t≤T(12) with the auxiliary conditions ⎝integraldisplay1 –1y(ξ,t)dξ=M1(t),⎝integraldisplay1 –1ξy(ξ,t)dξ=M2(t), (13) regarding y(x,t),α1(t), and α2(t) as unknown. All other functions in (12) are assumed given andg(x,t)i so fc l a s s C([1,T],L2[–1, 1]). Note that the Hilbert space L2[–1, 1] can be represented as the direct sum of its orthogonal subspaces, L2[–1, 1] = L◦ 2[–1, 1] ⊕L∗ 2[–1, 1] (see Supplement 12.5-3.), where L◦ 2[–1, 1] is the Euclidean space with the basis p1(x)=P∗ 0(x)=1/√ 2,p2(x)=P∗ 1(x)=⎝radicalbig 3/2x,a n dL∗ 2[–1, 1] is the Hilbert space with the basis pk(x)=P∗ k–1(x)(k=3 ,4 , ...). Note also that the integrand and the right -hand side can be represented as a sum of functions continuous in t∈[1,T] with values in L◦ 2[–1, 1] and L∗ 2[–1, 1], respectively, i.e., y(x,t)=y◦(x,t)+y∗(x,t),f(x,t)=f◦(x,t)+f∗(x,t), (14) 846 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS where y◦(x,t)=y◦ 1(t)p1(x)+y◦ 2(t)p2(x),y◦ 1(t)=1 √ 2M1(t),y◦ 2(t)=⎝radicalBig 3 2M2(t) f◦(x,t)=⎝bracketleftbig√ 2α1(t)–g◦ 1(t)⎝bracketrightbig p1(x)+⎝bracketleftBig⎝radicalBig 2 3α2(t)–g◦ 2(t)⎝bracketrightBig p2(x), (15) f∗(x,t)=–g∗(x,t),g(x,t)=g◦(x,t)+g∗(x,t),g◦(x,t)=g◦ 1(t)p1(x)+g◦ 2(t)p2(x), g◦ 1(t)=1 √ 2⎝integraldisplay1 –1g(x,t)dx,g◦ 2(t)=⎝radicalBig 3 2⎝integraldisplay1 –1g(x,t)dx. Note that in the representation (14) of y(x,t), the term y◦(x,t) is known and is determined by the auxiliary conditions; and the term y∗(x,t) is to be found. Conversely, for the right-hand side, f◦(x,t) is the unknown and f∗(x,t) is determined by the function g(x,t). These features allow us to classify the resulting problem as a special case of th e general projection problem formulated and solved in Subsection 17.4-3. Applying the general method to the present case, we introduce an operator of orthogonal projection that maps L2[–1, 1] onto L◦ 2[–1, 1]: P◦f(x,t)=⎝integraldisplay1 –1f(ξ,t)[p 1(x)p1(ξ)+p2(x)p2(ξ)]dξ. (16) Obviously, P∗=I–P◦is the orthogonal projector of L2[–1, 1] onto L∗ 2[–1, 1]. Moreover, the following relations hold: P◦y(x,t)=y◦(x,t), P∗y(x,t)=y∗(x,t), P◦f(x,t)=f◦(x,t), P∗f(x,t)=f∗(x,t).(17) Following the method of Section 17.4, let us apply the projection operator P∗to equation (12). As a result, for y∗(x,t) we obtain an integral equation on the space L∗ 2[–1, 1] with a known right-hand side: σ(t)⎝bracketleftbigg y∗(x,t)–⎝integraldisplayt 1V1(t,τ)y∗(x,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 –1F∗(x,ξ)y∗(ξ,t)dξ–⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 –1F∗(x,ξ)y∗(ξ,τ)dξ dτ =–g∗(x,t)–⎝integraldisplay1 –1F∗(x,ξ)y◦(ξ,t)dξ+⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 –1F∗(x,ξ)y◦(ξ,τ)dξ dτ , –1≤x≤1, 1 ≤t≤T,(18) where the kernel of the integral equation F∗(x,ξ)=F(x,ξ)–⎝integraldisplay1 –1F(s,ξ)[p1(x)p1(s)+p2(x)p2(s)]ds (19) is of Fredholm type and, moreover, is symmetric and positive deﬁnite. Let us construct a solution of equation (18) in the form of a series with eigenfunctions of the kernel (19). These eigenfunctionform a basis in the Hilbert space L ∗ 2[–1, 1]. We start, however, with the construction of the said eigenfunctions. Letϕ∗ k(x) be eigenfunctions and µ∗ kthe corresponding eigenvalues of the kernel F∗(x,ξ). Then ⎝integraldisplay1 –1F∗(x,ξ)ϕ∗ k(ξ)dξ=µ∗ kϕ∗k(x), k=3 ,4 , ... (20) 17.2. M ETHODS OF SOLUTION OF MIXED INTEGRAL EQUATIONS ON A FINITE INTERV AL 847 Let us represent the eigenfunction ϕ∗ k(x) in the form of a series with respect to the basis pi(x) (i≥3): ϕ∗ k(x)=∞⎝summationdisplay i=3ϕ∗ i(k)pi(x),pi(x)=P∗ i–1(x), k=3 ,4 , ... (21) Using (9) and (19), we obtain a double series expansion for the kernel F∗(x,ξ): F∗(x,ξ)=∞⎝summationdisplay m=3∞⎝summationdisplay n=3Fmnpm(x)pn(ξ)+∞⎝summationdisplay n=3F1npn(x)p1(ξ)+∞⎝summationdisplay n=3F2npn(x)p2(ξ). (22) Note that the coefﬁcients of the expansion of the kernel F∗(x,ξ) in (22) coincide with the coefﬁcients of the expansion of the kernel F(x,ξ), which allows us to avoid recalculation of the coefﬁcients of the new problem and use the already available data. Substituting (21) and (22) into (20), we obtain the following inﬁnite system of algebraic equa- tions (with a symmetric matrix) for the determination of eigenvalues and eigenfunction expansion coefﬁcients: ∞⎝summationdisplay n=3Fmnϕ∗ n(k)=µ∗ kϕ∗m(k),m=3 ,4 , ... (23) Now, let us construct a solution of equation (18). For this purpose, we represent the func- tionsy∗(x,t)a n d g∗(x,t) in the form of series with respect to eigenfunctions of the kernel F∗(x,ξ): y∗(x,t)=∞⎝summationdisplay k=3y∗ k(t)ϕ∗ k(x),g∗(x,t)=∞⎝summationdisplay k=3g∗ k(t)ϕ∗ k(x),g∗ k(t)=⎝integraldisplay1 –1g∗(x,t)ϕ∗ k(x)dx, (24) and substitute these into (18). Then, taking into account (15), (19)–(22), we obtain the following sequence of independent V olterra equations: y∗ k(t)–⎝integraldisplayt 1V∗ k(t,τ)y∗ k(τ)dτ=f∗ k(t),V∗ k(t,τ)=σ(t)V1(t,τ)+µ∗ kV2(t,τ) σ(t)+µ∗ k, f∗ k(t)=–1 σ(t)+µ∗ k⎝bracketleftbigg g∗ k(t)+2⎝summationdisplay i=1Fk(i)y◦ i(t)–⎝integraldisplayt 1V2(t,τ)2⎝summationdisplay i=1Fk(i)y◦ i(t)dτ⎝bracketrightbigg , Fk(i)=∞⎝summationdisplay n=3Finϕ∗ n(k),i=1 ,2 , k=3 ,4 , ...(25) Resolving (25) with respect to y∗ k(t) by the methods of Chapter 11, we obtain y∗ k(t)=f∗ k(t)+⎝integraldisplayt 1R∗ k(t,τ)f∗ k(τ)dτ, (26) where R∗ k(t,τ) is the resolvent of the kernel V∗ k(t,τ). Now, in view of (24)–(26), the function y∗(x,t) has been determined, and therefore, the func- tiony(x,t) has also been found, since y◦(x,t) is known by assumption (see (14) and (15)). Before passing to the determination of the other unknown quantities of the problem, we make some remarks that may be useful in practice. For practical calculations one should restrict the number of expansion terms. For instance, taking the orthonormal Legendre polynomials from the third to the Nth, we obtain the Nth approximation of the desired solution. In this case, for the construction of eigenvalues and eigenfunctions of the 848 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Hilbert–Schmidt kernel F∗(x,ξ) one should ﬁnd the eigenvalues and orthonormal eigenvectors of the matrix [F∗ NN]=⎛ ⎜⎜⎜⎜⎝F 33F34F35···F3N F34F44F45···F4N F35F45F55···F5N ............... F 3NF4NF5N···FNN⎞ ⎟⎟⎟⎟⎠. (27) The eigenvalues of the matrix (27) give approximate values of the ﬁrst N– 2 eigenvalues of the Fredholm operator,and the components of the orthonormal eigenvectors of this matrix give expansioncoefﬁcients for the ﬁrst N– 2 eigenfunctions of the Hilgert–Schmidt operator with respect to the chosen orthonormal Legendre polynomials. Recall that the ﬁrst two terms (i.e., two terms of y ◦(x,t)) of the function y(x,t) are known by assumption. Therefore, constructing the next N–2t e r m so f the expansion, we obtain the Nth approximation. It is important to emphasize the relation b etween the matrices (11) and (27), which, in general, correspond to two different problems. The matrix (27) can be obtained from the matrix (11) by deleting its ﬁrst two rows and columns. This allows us to construct the expansion of the original kernel only once, and then use that data for the examination of the new kernel arising in the problem with auxiliary conditions. Now, let us ﬁnd the functions α1(t)a n d α2(t). To that end, we apply the orthogonal projection operators P∗to equation (12). As a result, we obtain the following formulas: α1(t)=1 √ 2⎝braceleftbigg g◦ 1(t)+σ(t)⎝bracketleftbigg y◦ 1(t)–⎝integraldisplayt 1V1(t,τ)y◦ 1(τ)dτ⎝bracketrightbigg +F11y◦ 1(t)+F12y◦ 2(t) +∞⎝summationdisplay k=3Fk(1)y∗ k(t)–⎝integraldisplayt 1V1(t,τ)⎝bracketleftbigg F11y◦ 1(τ)+F12y◦ 2(τ)+∞⎝summationdisplay k=3Fk(1)y∗ k(τ)⎝bracketrightbigg dτ⎝bracerightbigg , (28) α2(t)=⎝radicalbigg 3 2⎝braceleftbigg g◦ 2(t)+σ(t)⎝bracketleftbigg y◦ 2(t)–⎝integraldisplayt 1V1(t,τ)y◦ 2(τ)dτ⎝bracketrightbigg +F12y◦ 1(t)+F22y◦ 2(t) +∞⎝summationdisplay k=3Fk(2)y∗ k(t)–⎝integraldisplayt 1V1(t,τ)⎝bracketleftbigg F12y◦ 1(τ)+F22y◦ 2(τ)+∞⎝summationdisplay k=3Fk(2)y∗ k(τ)⎝bracketrightbigg dτ⎝bracerightbigg . (29) Thus, we have obtained a complete solution of the integral equation (12) with the auxiliary conditions (13). 17.2-3. Equation with a Schmidt Kernel and a Given Right-Hand Side on an Interval. Consider a mixed integral equation of the form (17.1.3) with a Schmidt kernel. Changing the variables, we can always reduce this equation to the following equation with the parameters a= –1, b=1 ,τ0=1 : σ(t)⎝bracketleftbigg y(x,t)–⎝integraldisplayt 1V1(t,τ)y(x,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 –1S(x,ξ)y(ξ,t)dξ –⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 –1S(x,ξ)y(ξ,τ)dξ dτ =f(x,t) h(x), S(x,ξ)=F(x,ξ) h(x),– 1 ≤x≤1, 1 ≤t≤T.(30) Suppose that the right-hand side f(x,t) in (30) is known, and it is required to ﬁnd the func- tiony(x,t). Here, f(x,t)a n d y(x,t) are functions of class C([1,T],L2[–1, 1]); σ(t)i sag i v e n positive continuous function; h(x)>0i sag i v e nf u n c t i o ni nL 2[a,b];V1(t,τ)a n d V2(t,τ)a r e 17.2. M ETHODS OF SOLUTION OF MIXED INTEGRAL EQUATIONS ON A FINITE INTERV AL 849 V olterra kernels; S(x,ξ)i saS c h m i d tk e r n e l ; F(x,ξ) is a symmetric positive deﬁnite Fredholm kernel. Let us transform the equation with the Schmidt kernel to a Hilbert–Schmidt equation. To that end, we multiply (30) by√ h(x) and change the variables as follows: q(x,t)=⎝radicalbig h(x)y(x,t),Fh(x,ξ)=S(x,ξ)√ h(x) √ h(ξ)=F(x,ξ) √ h(x)√ h(ξ). (31) Then we have σ(t)⎝bracketleftbigg q(x,t)–⎝integraldisplayt 1V1(t,τ)q(x,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 –1Fh(x,ξ)q(ξ,t)dξ –⎝integraldisplayt τ0V2(t,τ)⎝integraldisplay1 –1Fh(x,ξ)q(ξ,τ)dξ dτ =f(x,t) √ h(x),– 1 ≤x≤1, 1 ≤t≤T, (32) where q(x,t)a n d f(x,t)/√ h(x) are functions of class C([1,T],L2[–1, 1]); Fh(x,ξ) is a symmetric positive deﬁnite kernel of Hilbert–Schmidt type (due to the properties of Schmidt kernels); and the other functions have been speciﬁed above. Suppose that the right-hand side of equation (32) is known, and it is required to ﬁnd the function q(x,t). Let us seek a solution of the mixed equation (32) in the form or a series q(x,t)=∞⎝summationdisplay k=1qk(t)ϕh k(x), (33) where ϕh k(x) are eigenfunctions of the kernel Fh(x,ξ) corresponding to eigenvalues µh k>0 ,i . e . , ⎝integraldisplay1 –1Fh(x,ξ)ϕh k(ξ)dξ=µh kϕhk(x), k=1 ,2 , ... (34) The representation of a solution in the form (33) is possible, since the system of eigenfunctions of the kernel Fh(x,ξ) forms a basis in L2[–1, 1]. Here, in contrast to the above case, we construct the basis functions in the form ϕh k(x)=Φh k(x) √ h(x)k=1 ,2 , ... (35) with an explicit function h(x), where ⎝integraldisplay1 –1ϕh i(ξ)ϕh j(ξ)dξ=⎝integraldisplay1 –1Φh i(ξ)Φh j(ξ) h(ξ)dξ=δij=⎝braceleftbigg 1f o r i=j, 0f o r i≠j.(36) In order to construct such eigenfunctions, we ﬁrst construct a certain basis pn(x)i nL2[–1, 1] for which⎝integraldisplay1 –1ph i(ξ)ph j(ξ)dξ=δij,ph n(x)=Ph n–1(x) √ h(x),n=1 ,2 , ... (37) Such a basis can be constructed with the help of the formulas Ph 0(x)=1 √ J0,Ph n(x)=1 √ ∆n–1∆n⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleJ 0J1··· Jn J1J2···Jn+1 ............ Jn–1Jn···J2n–1 1x··· xn⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle, ∆ –1=1 , ∆0=J0,∆n=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleJ 0J1··· Jn J1J2···Jn+1 ............ J nJn+1···J2n⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,J n=⎝integraldisplay1 –1ξn h(ξ)dξ.(38) 850 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Note that for h(x) = 1, the basis functions of the Hilbert space L2[–1, 1] obtained from (37) and (38) coincide with orthonormal Legendre polynomials. Let us represent the kth eigenfunction as a series in terms of the basis ph i(x)o fL2[–1, 1]. Then ϕh k(x)=∞⎝summationdisplay i=1ϕh i(k)ph i(x),ph i(x)=Ph i–1(x) √ h(x),Φh k(x)=∞⎝summationdisplay i=1ϕh i(k)Ph i–1(x). (39) Expanding the Hilbert–Schmidt kernel Fh(x,ξ) as a double series in terms of the chosen basis, we can write Fh(x,ξ)=∞⎝summationdisplay m=1∞⎝summationdisplay i=1Fh mnphm(x)ph n(ξ), Fh mn=⎝integraldisplay1 –1⎝integraldisplay1 –1Fh(x,ξ)ph m(x)ph n(ξ)dx dξ ,Fh mn=Fh nm.(40) Substituting (39) and (40) into (34), we obtain the following inﬁnite system of linear algebraic equations (with a symmetric matrix) for the determination of the eigenvalues and the eigenfunctionexpansion coefﬁcients: ∞⎝summationdisplay n=1Fh mnϕhn(k)=µh kϕhm(k),m=1 ,2 , ... (41) To calculate approximations of Neigenvalues and eigenfunctions of the Hilbert–Schmidt kernel, one should ﬁnd the eigenvalues and orthonormal eigenvectors of the matrix [Fh NN]=⎛ ⎜⎜⎜⎜⎝F h 11Fh 12Fh 13···Fh 1N Fh 12Fh 22Fh 23···Fh 2N Fh 13Fh 23Fh 33···Fh 3N............... Fh 1NFh 2NFh 3N···Fh NN⎞ ⎟⎟⎟⎟⎠. (42) The eigenvalues of the matrix (42) give approximate values of the ﬁrst Neigenvalues of the Hilbert– Schmidt kernel, and the components of its orthonormal eigenvectors give the coefﬁcients in the expansion of the ﬁrst Neigenfunctions of that kernel in terms of Northonormal functions of the basis. Now, let us expand the right-hand side of equation (32) into the following series: f(x,t) √ h(x)=∞⎝summationdisplay k=1fh k(t)ϕh k(x)=∞⎝summationdisplay k=1fh k(t)Φh k(x) √ h(x), fh k(t)=⎝integraldisplay1 –1f(x,t) √ h(x)ϕh k(x)dx=⎝integraldisplay1 –1f(x,t) h(x)Φh k(x)dx.(43) Substituting (33) and (43) into (32) and taking into account (34), we obtain the following sequence of V olterra equations for the unknown functions qk(t): qk(t)–⎝integraldisplayt 1Vh k(t,τ)qk(τ)dτ=δh k(t),δh k(t)=fk(t) σ(t)+µh k, (44) Vh k(t,τ)=σ(t)V1(t,τ)+µh kV2(t,τ) σ(t)+µh k,k=1 ,2 , ..., (45) where Vh k(t,τ) are V olterra kernels belonging to the same class as the kernels V1(t,τ)a n dV2(t,τ), sinceµh k→0a sk→∞ . 17.2. M ETHODS OF SOLUTION OF MIXED INTEGRAL EQUATIONS ON A FINITE INTERV AL 851 A solution of the inﬁnite system of V olterra equations (44) can be constructed by analytical and numerical methods of Chapter 11. This solution can be written in the form qk(t)=δh k(t)+⎝integraldisplayt 1Rh k(t,τ)δh k(τ)dτ, (46) where Rh k(t,τ)i st h er e s o l v e n to ft h ek e r n e l Vh k(t,τ). The series (33) converges in L2[–1, 1] uniformly in t∈[1,T], and its sum is a continuous function of t∈[1,T] with values in L2[–1, 1]. Inserting (45) into (33) and taking into account (43), one can also represent the solution in the form q(x,t)=f(x) √ h(x)+∞⎝summationdisplay k=1⎝integraldisplayt 1Rh k(t,τ)fh k(τ)dτ ϕh k(x). Finally, in view of the transformation of the variable (31) and the formula for eigenfunctions (35), we have y(x,t)=1 h(x)∞⎝summationdisplay k=1qk(t)Φh k(x). (47) Note that the solution (47) involves the function h(x) in an explicit manner, which allows us to solve equation (30) with great accuracy by preserving a small number of terms. In the case of a strongly oscillating function h(x), the other known methods can hardly be used for the construction of solutions. 17.2-4. Equation with a Schmidt Kernel and Auxiliary Conditions. Consider equation (30) with the right-hand side of the form f(x,t)=α1(t)+α2(t)x–g(x,t)a n dt w o auxiliary integral conditions of the form (17.1.4) on the unknown function y(x,t). The problem is to ﬁnd a solution of the mixed integral equation σ(t)⎝bracketleftbigg y(x,t)–⎝integraldisplayt 1V1(t,τ)y(x,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 –1S(x,ξ)y(ξ,t)dξ –⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 –1S(x,ξ)y(ξ,τ)dξ dτ =α1(t) h(x)+α2(t)x h(x)–g(x,t) h(x), S(x,ξ)=F(x,ξ) h(x),– 1 ≤x≤1, 1 ≤t≤T.(48) with the auxiliary conditions ⎝integraldisplay1 –1y(ξ,t)dξ=M1(t),⎝integraldisplay1 –1ξy(ξ,t)dξ=M2(t), (49) the unknown functions being y(x,t),α1(t), and α2(t). The other functions in (48) are assumed given, and g(x,t) is a continuous function of t∈[1,T] with values in L2[–1, 1]. Let us transform the equation with the Schmidt kernel to an equation with a Hilbert–Schmidt kernel by changing the variables as in (31). As a result, equation (48) and the auxiliary conditions (49)become σ(t)⎝bracketleftbigg q(x,t)–⎝integraldisplay t 1V1(t,τ)q(x,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 –1Fh(x,ξ)q(ξ,t)dξ –⎝integraldisplayt τ0V2(t,τ)⎝integraldisplay1 –1Fh(x,ξ)q(ξ,τ)dξ dτ =α1(t) √ h(x)+α2(t)x √ h(x)–g(x,t) √ h(x), (50) ⎝integraldisplay1 –1q(ξ,t) √ h(ξ)dξ=M1(t),⎝integraldisplay1 –1q(ξ,t) √ h(ξ)ξd ξ =M2(t), –1 ≤x≤1, 1 ≤t≤T. (51) 852 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS In order to construct a solution of the mixed integral equation (48) with the auxiliary condi- tions (49), we use the basis ph n(x)i nL2[–1, 1] (see (37) and (38)) and note that the space L2[–1, 1] can be represented as a direct sum of its orthogonal subspaces, L2[–1, 1] = Lh◦ 2[–1, 1] ⊕Lh∗ 2[–1, 1] (see Supplement 12.5-3), where Lh◦ 2[–1, 1] is the Euclidean space with the basis ph 1(x)a n dph 2(x), andLh∗ 2[–1, 1] is the Hilbert space with the basis ph k(x)(k=3 ,4 , ...). It can be seen that the integrand and the right-hand side can be represented as a sum of continuous functions of t∈[1,T] with values in Lh◦ 2[–1, 1] and Lh∗ 2[–1, 1], respectively, i.e, q(x,t)=q◦(x,t)+q∗(x,t),f(x,t) √ h(x)=f◦ h(x,t)+f∗ h(x,t), (52) and the following representations hold: q◦(x,t)=q◦ 1(t)ph 1(x)+q◦ 2(t)ph 2(x),q◦ 1(t)=M1(t) √ J0,q◦ 2(t)=J0M2(t)–J1M1(t) ⎝radicalbig J0(J0J2–J2 1), f(x,t) √ h(x)=α1(t) √ h(x)+α2(t)x √ h(x)–g(x,t) √ h(x),g(x,t) √ h(x)=g◦ h(x,t)+g∗ h(x,t), (53) f◦ h(x,t)=⎝bracketleftbigg⎝radicalbig J0α1(t)+J1 √ J0α2(t)–g1(t)⎝bracketrightbigg ph 1(x)+⎝bracketleftBigg⎝radicalbig J0J2–J2 1 √ J0α(t)–g1⎝bracketrightBigg ph 2(x), f∗ h(x,t)=–g∗ h(x),g◦ h(x,t)=gh◦ 1(t)ph 1(x)+gh◦ 2(t)ph 2(x), g◦ 1(t)=⎝integraldisplay1 –1g(x,t) √ h(x)ph 1(x)dx,g◦ 2(t)=⎝integraldisplay1 –1g(x,t) √ h(x)ph 2(x)dx. Note that in the representation (52) for q(x,t), the function q◦(x,t) is known as determined by the auxiliary conditions, and the term q∗(x,t) should be found. Conversely, for the right-hand side,f◦ h(x,t) should be found and f∗ h(x,t) is determined by the function g(x,t)/√ h(x). The facts mentioned above allow us to classify the resulting problem as a special case of the general projection problem whose solution is given in Subsection 17.4-3. According to the general method, in the present case one can introduce an operator of orthogonal projection that maps the space L2[–1, 1] onto Lh◦ 2[–1, 1]: P◦ hφ(x,t)=⎝integraldisplay1 –1φ(ξ,t)[ph 1(x)ph 1(ξ)+ph 2(x)ph 2(ξ)]dξ. (54) Obviously, the projector P∗ h=I–P◦ hmapsL2[–1, 1] onto Lh∗ 2[–1, 1]. Moreover, the following relations hold: P◦ hq(x,t)=q◦(x,t), P∗ hq(x,t)=q∗(x,t), P◦ hf(x,t) √ h(x)=f◦ h(x,t), P∗ hf(x,t) √ h(x)=f∗ h(x,t).(55) According to Subsections 17.4-2 and 17.4-3, let us apply the projection operator P∗ hto equation (50). Then, for the determination of q∗(x,t), we obtain the following integral equation in Lh∗ 2[–1, 1] with a known right-hand side: σ(t)⎝bracketleftbigg q∗(x,t)–⎝integraldisplayt 1V1(t,τ)q∗(x,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 –1F∗ h(x,ξ)q∗(ξ,t)dξ–⎝integraldisplayt τ0V2(t,τ)⎝integraldisplay1 –1F∗ h(x,ξ)q∗(ξ,τ)dξ dτ =–g∗ h(x,t)–⎝integraldisplay1 –1F∗ h(x,ξ)q◦(ξ,t)dξ+⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 –1F∗ h(x,ξ)q◦(ξ,τ)dξ dτ , –1≤x≤1, 1 ≤t≤T,(56) 17.2. M ETHODS OF SOLUTION OF MIXED INTEGRAL EQUATIONS ON A FINITE INTERV AL 853 where the kernel of the integral equation F∗ h(x,ξ)=Fh(x,ξ)–⎝integraldisplay1 –1Fh(s,ξ)[ph 1(x)ph 1(s)+ph 2(x)ph 2(s)]ds (57) is a Hilbert–Schmidt kernel. A solution of equation (56) can be constructed in the form of a series in terms of eigenfunctions of the kernel (57). These form a basis in the Hilbert space Lh∗ 2[–1, 1]. Let us construct a system of these eigenfunctions. For an eigenfunction ϕh∗ k(x), letµh∗ kbe the corresponding eigenvalue of the kernel F∗ p(x,ξ). Then⎝integraldisplay1 –1F∗ h(x,ξ)ϕh∗ k(ξ)dξ=µh∗ kϕh∗ k(x), k=3 ,4 , ... (58) Let us represent the eigenfunction ϕh∗ i(x) as a series in terms of the basis functions ph i(x)(i≥3): ϕh∗ k(x)=∞⎝summationdisplay i=3ϕh∗ i(k)ph i(x), k=3 ,4 , ... (59) The double series expansion of the kernel F∗ h(x,ξ) is obtained with the help of (40) and (57): F∗ h(x,ξ)=∞⎝summationdisplay m=3∞⎝summationdisplay n=3Fh mnphm(x)ph n(ξ)+∞⎝summationdisplay n=3Fh 1nph n(x)ph 1(ξ)+∞⎝summationdisplay n=3Fh 2nph n(x)ph 2(ξ). (60) Note that the coefﬁcients of the expansion of the kernel F∗ h(x,ξ) in (60) coincide with coefﬁcients of the expansion of the kernel Fh(x,ξ), which allows us to avoid recalculation of the coefﬁcients of the new problem and use the existing data. Substituting (59) and (60) into (58), we obtain the following inﬁnite system of linear algebraic equations (with a symmetric matrix) for the eigenvalues and the eigenfunction expansion coefﬁcients: ∞⎝summationdisplay n=3Fh mnϕh∗ n(k)=µh∗ kϕh∗ m(k),m=3 ,4 , ... (61) Now, let us construct a solution of equation (56). To that end, we represent the functions q∗(x,t) andg∗ h(x,t) as series in terms of eigenfunctions of the kernel F∗ h(x,ξ): q∗(x,t)=∞⎝summationdisplay k=3q∗ k(t)ϕh∗ k(x),g∗ h(x,t)=∞⎝summationdisplay k=3gh∗ k(t)ϕh∗ k(x),gh∗ k(t)=⎝integraldisplay1 –1g∗ h(x,t)ϕh∗ k(x)dx. (62) Substituting these into (56) and taking into account (53), (57)–(60), we obtain the following sequence of independent V olterra equations of the second kind: q∗ k(t)–⎝integraldisplayt 1Vh∗ k(t,τ)q∗ k(τ)dτ=δh∗ k(t),Vh∗ k(t,τ)=σ(t)V1(t,τ)+µh∗ kV2(t,τ) σ(t)+µh∗ k, δh∗ k(t)=–1 σ(t)+µh∗ k⎝bracketleftbigg gh∗ k(t)+2⎝summationdisplay i=1Fh k(i)q◦ i(t)–⎝integraldisplayt 1V2(t,τ)2⎝summationdisplay i=1Fh k(i)q◦ i(τ)dτ⎝bracketrightbigg , Fh k(i)=∞⎝summationdisplay n=3Fh inϕh∗ n(k),i=1 ,2 , k=3 ,4 , ...(63) 854 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Resolving (63) with respect to q∗ k(t) by the methods of Chapter 11, we obtain q∗ k(t)=fh∗ k(t)+⎝integraldisplayt 1Rh∗ k(t,τ)fh∗ k(τ)dτ, (64) where Rh∗ k(t,τ)i st h er e s o l v e n to ft h ek e r n e l Vh∗ k(t,τ). Now, in view of (62)–(64), the function q∗(x,t) has been determined, and therefore, we easily ﬁndq(x,t), since q◦(x,t) is known by assumption (see (52) and (53)). In practical calculations, the number of expansion terms is naturally limited. For instance, taking the basis functions ph k(x)f o rk =3 ,...,N, we obtain the Nth approximation of the desired solution. In this case, for the construction of eigenvalues and eigenfunctions of the Hilbert–Schmidt kernel F∗ h(x,ξ), one should ﬁnd the eigenvalues and orthonormal eigenvectors of the matrix [Fh NN]=⎛ ⎜⎜⎜⎜⎝F h 33Fh 34Fh 35···Fh 3N Fh 34Fh 44Fh 45···Fh 4N Fh 35Fh 45Fh 55···Fh 5N............... Fh 3NFh 4NFh 5N···Fh NN⎞ ⎟⎟⎟⎟⎠. (65) The eigenvalues of the matrix (65) give approximations for the ﬁrst N– 2 eigenvalues if the Hilbert– Schmidt operator and the components of the orthonormal eigenvectors of the matrix give expansion coefﬁcients of the ﬁrst N– 2 eigenfunctions of this operator in terms of the chosen orthonormal Legendre polynomials. Recall that the ﬁrst two terms of the expansion (i.e., two terms of q ◦(x,t)) of the function q(x,t) are known by assumption. Therefore, constructing the next N–2t e r m so f the expansion, we obtain the Nth approximation of the solution. It is important to keep in mind the relation between the matrices (42) and (65). The matrix (65) can be obtained from (42) by deleting its ﬁrst two rows and columns. This allows us to construct the expansion of the original series only once, and then use this information for studying the new kernel in the problem with auxiliary conditions. Now, let us ﬁnd the functions α1(t), and α2(t). To this end, we apply the projection operator P◦ h to equation (56). As a result, we obtain the following formulas: α1(t)=1 √ J0⎝braceleftBig gh◦ 1(t)–J1 √ J0α2(t)+σ(t)⎝bracketleftBig q◦ 1(t)–⎝integraldisplayt 1V1(t,τ)q◦ 1(τ)dτ⎝bracketrightBig +Fh 11q◦ 1(t) +Fh 12q◦ 2(t)+∞⎝summationdisplay k=3Fh k(1)q∗ k(t)–⎝integraldisplayt 1V1(t,τ)⎝bracketleftBig Fh 11q◦ 1(τ)+Fh 12q◦ 2(τ)+∞⎝summationdisplay k=3Fh k(1)q∗ k(τ)⎝bracketrightBig dτ⎝bracerightBig , (66) α2(t)=⎝radicalBigg J0 J0J2–J2 1⎝braceleftBig gh◦ 2(t)+σ(t)⎝bracketleftBig q◦ 2(t)–⎝integraldisplayt 1V1(t,τ)q◦ 2(τ)dτ⎝bracketrightBig +Fh 12q◦ 1(t) +Fh 22q◦ 2(t)+∞⎝summationdisplay k=3Fh k(2)q∗ k(t)–⎝integraldisplayt 1V1(t,τ)⎝bracketleftBig Fh 12q◦ 1(τ)+Fh 22q◦ 2(τ)+∞⎝summationdisplay k=3Fh k(2)q∗ k(τ)⎝bracketrightBig dτ⎝bracerightBig . (67) Note that relations (66) and (67) form a system of two linear algebraic equations (with a triangular matrix) for the determination of the unknown quantities α1(t)a n d α2(t). Thus, we have constructed a solution of the integral equation (48) with the auxiliary condi- tions (49). References for Section 17.2: E. Goursat (1923), G. Szeg ¨o (1975), V . M. Aleksandrov and S. M. Mkhitaryan (1983), V . M. Aleksandrov and A. V . Manzhirov (1987), A. V . Manzhirov (2001, 2005), A. V . Manzhirov and K. E. Kazakov (2006). 17.3. M ETHODS OF SOLVING MIXED INTEGRAL EQUATIONS ON A RING-SHAPED DOMAIN 855 17.3. Methods of Solving Mixed Integral Equations on a Ring-Shaped Domain 17.3-1. Equation with a Hilbert–Schmidt Kernel and a Given Right-Hand Side. Consider a mixed integral equation with a Hilbert–Schmidt kernel on a ring-shape domain (see Subsection 17.1-3). By a suitable transformation of the variables, this equation can be reduced to asimilar equation with the parameters a=0 ,b=1 ,τ 0= 1:approximation of the solution σ(t)⎝bracketleftBig y(r,t)–⎝integraldisplayt 1V1(t,τ)y(r,τ)dτ⎝bracketrightBig +⎝integraldisplay1 0G(r,ρ)y(ρ,t)ρd ρ –⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 0G(r,ρ)y(ρ,τ)ρd ρd τ =f(r,t), 0 ≤r≤1, 1 ≤t≤T.( 1 ) Suppose that the right-hand side f(r,t) of equation (1) is known and it is required to ﬁnd the function y(r,t). Here, f(r,t),y(r,t) are supposed to be continuous functions of t∈[1,T] with values in⎝hatwideL2(ω). Let us seek a solution of equation (1) in the form of a series y(x,t)=∞⎝summationdisplay k=1yk(t)ψk(r), (2) where ψk(r) are eigenfunctions of the kernel F(r,ρ) corresponding to eigenvalues νk>0 ,i . e . , ⎝integraldisplay1 0G(r,ρ)ψk(ρ)ρd ρ =νkψk(r), k=1 ,2 , ... (3) The representation (2) is justiﬁed by the fact that the system of eigenfunctions of the kernel F(r,ρ) forms a complete orthonormal system in ⎝hatwideL2(ω), in other words, an orthonormal basis in ⎝hatwideL2(ω)( s e e Subsection 13.6-1 and Supplement 12.5-3). This fact also allows us to represent the right-hand sideof the equation in the form f(r,t)= ∞⎝summationdisplay k=1fk(t)ψk(r),fk(t)=⎝integraldisplay1 0f(ρ,t)ψk(ρ)ρd ρ.( 4) Substituting (2) into (1) and taking into account (3) and (4), we obtain the following sequence of V olterra equations of the second kind for the unknown functions yk(t): yk(t)–⎝integraldisplayt 1Vk(ν)(t,τ)yk(τ)dτ=γk(t),γk(t)=fk(t) σ(t)+νk,( 5) Vk(ν)(t,τ)=σ(t)V1(t,τ)+νkV2(t,τ) σ(t)+νk,k=1 ,2 , ...,( 6 ) where Vν k(t,τ) are V olterra kernels of the same class as V1(t,τ)a n dV2(t,τ), since νk→0a sk→∞ . A solution of the sequence of V olterra equations (5) can be constructed by analytical and numerical methods of Chapter 11. This solution can be written in the form yk(t)=γk(t)+⎝integraldisplayt 1Rk(ν)(t,τ)γk(τ)dτ,( 6) where Rν k(t,τ) is the resolvent of the kernel Vk(t,τ). 856 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Thus, we have found a solution of equation (1). The series (2) converges in ⎝hatwideL2(ω) uniformly in t∈[1,T], and its sum is a continuous function of twith values in⎝hatwideL2(ω). Now, let us construct eigenfunctions and eigenvalues of the Hilbert–Schmidt kernel. We rep- resent the kth eigenfunction as a series in terms of any orthonormal basis zi(r)i n⎝hatwideL2(ω). For deﬁniteness, we take zk(r)=√ 4k+2Pk–1( 1–2r2), k=1 ,2 , ...,( 7 ) where Pn(x) are the Legendre polynomials. Now we can represent the eigenfunctions of the kernel G(r,ρ) in the form of a series ψk(r)=∞⎝summationdisplay i=1ψi(k)zi(r), k=1 ,2 , ... (8) We write a double series expansion (in terns of the chosen basis) for the kernel of the equation: G(r,ρ)=∞⎝summationdisplay m=1∞⎝summationdisplay i=1Gmnzm(r)zn(ρ), Gmn=⎝integraldisplay1 0⎝integraldisplay1 0G(r,ρ)zm(r)zn(ρ)rρ dr dρ ,Gmn=Gnm.(9) Substituting (8) and (9) into (3), we obtain an inﬁnite system of linear algebraic equations (with a symmetric matrix) for the d etermination of the eigenfunctions. This system has the form ∞⎝summationdisplay n=1Gmnψn(k)=νkψm(k),m=1 ,2 , ... (10) Naturally, in practical calculations one has to limit the number of expansion terms. For instance, taking Northonormal Legendre polynomials, we obtain the Nth approximation of the solution. And in order to construct the eigenvalues and eigenfunctions of the Hilbert–Schmidt kernel, in this case, one should ﬁnd the eigenvalues and orthonormal eigenvect ors of the matrix [GNN]=⎛ ⎜⎜⎜⎜⎝G11G12G13···G1N G12G22G23···G2N G13G23G33···G3N ............... G1NG2NG3N···GNN⎞ ⎟⎟⎟⎟⎠. (11) Eigenvalues of the matrix (11) give approximations for the ﬁrst Neigenvalues of the Hilbert–Schmidt kernel, and the components of its orthonormal eigenvectors give the coefﬁcients in the expansion of the ﬁrst Neigenfunctions of the Hilbert–Schmidt kernel in terms of Northonormal Legendre polynomials. 17.3-2. Equation with a Hilbert–Schmidt Kernel and Auxiliary Conditions. Consider equation (1) with the right-hand side of the form f(r,t)=β(t)–w(r,t) and an auxiliary condition of the form (8) of Subsection 17.1-3 for the unknown function y(r,t). The problem is to ﬁnd a solution of the mixed integral equation σ(t)⎝bracketleftbigg y(r,t)–⎝integraldisplayt 1V1(t,τ)y(r,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 0G(r,ρ)y(ρ,t)ρd ρ –⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 0G(r,ρ)y(ρ,τ)ρd ρd τ =β(t)–w(r,t), 0≤r≤1, 1 ≤t≤T(12) 17.3. M ETHODS OF SOLVING MIXED INTEGRAL EQUATIONS ON A RING-SHAPED DOMAIN 857 with the auxiliary condition⎝integraldisplay1 0y(ρ,t)ρd ρ =M(t), (13) the unknown functions being y(r,t)a n d β(t). All other functions in (12) are assumed known, and w(r,t) is a continuous function of t∈[1,T] with values in⎝hatwideL2(ω). Note that the Hilbert space ⎝hatwideL2(ω) can be represented as the direct sum of its orthogonal subspaces: ⎝hatwideL2(ω)=L◦ 2(ω)⊕L∗ 2(ω) (see Supplement 12.5-3), where ⎝hatwideL◦ 2(ω) is the Euclidean space with the basis z1(r)=√ 2P0(r)=√ 2a n d⎝hatwideL∗ 2(ω) is the Hilbert space with the basis zk(r)(k=2 ,3 , ...). Note that the integrand and the right-hand side can be represented as a sum of functions that are continuous in t∈[1,T] and take values in ⎝hatwideL◦ 2(ω)a n d⎝hatwideL∗ 2(ω), respectively. Thus, we can write y(r,t)=y◦(r,t)+y∗(r,t),f(r,t)=f◦(r,t)+f∗(r,t), (14) and the following expansions hold: y◦(r,t)=y◦ 1(t)z1(r),y◦ 1(t)=√ 2M(t), f◦(r,t)=⎝bracketleftbiggβ(t) √ 2–w◦ 1(t)⎝bracketrightbigg z1(r),f∗(r,t)=–w∗(r,t), (15) w(x,t)=w◦(x,t)+w∗(x,t),w◦(x,t)=w◦ 1(t)z1(r),w◦ 1(t)=√ 2⎝integraldisplay1 0w(ρ,t)ρd ρ. Note that in the representation (14) for y(r,t), the term y◦(r,t) is known (as deﬁned by the auxiliary conditions), and the term y∗(r,t) should be found. Conversely, for the right-hand side, one should ﬁnd f◦(r,t)a n dt h et e r m f∗(r,t) is determined by the function w(r,t). These considerations allow us to classify the problem as a special case of the general projection problem examined inSubsection 17.4-3. According to the general method, in the present case we can introduce an operator of orthogonal projection that maps the space ⎝hatwideL 2(ω) onto⎝hatwideL◦ 2(ω): Q◦f(r,t)=⎝integraldisplay1 0f(ρ,t)z1(r)z1(ρ)ρd ρ =2⎝integraldisplay1 0f(ρ,t)ρd ρ. (16) Obviously, the orthogonal projector Q∗=I–Q◦maps the space⎝hatwideL2(ω) onto⎝hatwideL∗ 2(ω). Moreover, the following relations hold: Q◦y(r,t)=y◦(r,t), Q∗y(r,t)=y∗(r,t), Q◦f(r,t)=f◦(r,t), Q∗f(r,t)=f∗(r,t).(17) Following Section 17.4, we apply the projection operator Q∗to equation (12). As a result, we obtain an integral equation for y∗(r,t)i nt h es p a c e⎝hatwideL∗ 2(ω) with a known right-hand side: σ(t)⎝bracketleftbigg y∗(r,t)–⎝integraldisplayt 1V1(t,τ)y∗(r,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 0G∗(r,ρ)y∗(ρ,t)ρd ρ –⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 0G∗(r,ρ)y∗(ρ,τ)ρd ρd τ =–w∗(r,t)–⎝integraldisplay1 0G∗(r,ρ)y◦(ρ,t)ρd ρ +⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 0G∗(r,ρ)y◦(ρ,τ)ρd ρd τ , 0≤r≤1, 1 ≤t≤T,(18) 858 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS where the kernel of the integral equation G∗(r,ρ)=G(r,ρ)–2⎝integraldisplay1 0G(r,ρ)rd r (19) is of Hilbert–Schmidt type. Let us construct a solution of equation (18) in the form of a series in terms of eigenfunctions of the kernel (19). These eigenfunctions form a basis in the Hilbert space ⎝hatwideL∗ 2(ω). Let us construct the system of these eigenfunctions. For an eigenfunction ψ∗ k(r)o ft h ek e r n e l G∗(r,ρ), letν∗ kbe the corresponding eigenvalue. Then, ⎝integraldisplay1 0G∗(r,ρ)ψ∗ k(ρ)ρd ρ =ν∗ kψ∗ k(r), k=2 ,3 , ... (20) Let us represent the eigenfunction ψ∗ k(r) as a series with respect to the basis zi(r)(i≥2): ψ∗ k(r)=∞⎝summationdisplay i=2ψ∗ i(k)zi(r), k=2 ,3 , ... (21) For the kernel G∗(r,ρ), we construct a double series expansion with the help of (9) and (19): G∗(r,ρ)=∞⎝summationdisplay m=2∞⎝summationdisplay n=2Gmnzm(r)zn(ρ)+∞⎝summationdisplay n=2√ 2G1nzn(r). (22) Note that the coefﬁcients in the expansion of the kernel G∗(r,ρ) in (22) coincide with those in the expansion of G(r,ρ), and this allows us to avoid recalculating the coefﬁcients of the new problem and use the available information. Substituting (21) and (22) into (20), we obtain an inﬁnite system of linear algebraic equations for the eigenvalues and the eigenfunction expansion coefﬁcients. This system has a symmetric matrix and can be written as follows: ∞⎝summationdisplay n=2Gmnψ∗ n(k)=ν∗ kψ∗ m(k),m=2 ,3 , ... (23) Now let us construct a solution of equation (18). For this purpose, we represent the func- tionsy∗(r,t)a n d w∗(r,t) in the form of series in terns of eigenfunctions of the kernel G∗(r,ρ): y∗(x,t)=∞⎝summationdisplay k=2y∗ k(t)ψ∗ k(r),w∗(r,t)=∞⎝summationdisplay k=2w∗ k(t)ψ∗ k(r),w∗ k(t)=⎝integraldisplay1 0w∗(r,t)ψ∗ k(ρ)ρdρ. (24) Substituting these into (18) and taking into account (15), (19)–(22), we obtain the following sequence of independent V olterra equations: y∗ k(t)–⎝integraldisplayt 1V∗ k(ν)(t,τ)y∗ k(τ)dτ=γ∗ k(t),V∗ k(ν)(t,τ)=σ(t)V1(t,τ)+ν∗ kV2(t,τ) σ(t)+ν∗ k, γ∗ k(t)=–1 σ(t)+ν∗ k⎝bracketleftbigg w∗ k(t)+Gky◦ 1(t)–⎝integraldisplayt 1V2(t,τ)Gky◦ 1(τ)dτ⎝bracketrightbigg , Gk=∞⎝summationdisplay n=1G1nψ∗ n(k),k=2 ,3 , ...(25) 17.3. M ETHODS OF SOLVING MIXED INTEGRAL EQUATIONS ON A RING-SHAPED DOMAIN 859 Resolving (25) with respect to y∗ k(t) by the methods of Chapter 11, we get y∗ k(t)=γ∗ k(t)+⎝integraldisplayt 1R∗ k(ν)(t,τ)γ∗ k(τ)dτ, (26) where R∗ k(ν)(t,τ) is the resolvent of the kernel V∗ k(ν)(t,τ). Thus, in view of (24)–(26), the function y∗(r,t) has been determined, and we easily ﬁnd y(r,t), sincey◦(r,t) is known by assumption (see (14) and (15)). Before we go on to ﬁnd the other unknown quantities of the problem, we make some practical recommendations. Naturally, in practical calculations the number of expansion terms should belimited. For instance, taking the Legendre polynomials from the second to the Nth, we obtain theNth approximation of the desired solution. In this case, for the construction of eigenvalues and eigenfunctions of the Hilbert–Schmidt kernel G ∗(r,ρ), one should ﬁnd the eigenvalues and the orthonormal eigenvectors of the matrix [G∗ NN]=⎛ ⎜⎜⎜⎜⎝G 22G23G24···G2N G23G33G34···G3N G24G34G44···G4N............... G 2NG3NG4N···GNN⎞ ⎟⎟⎟⎟⎠. (27) The eigenvalues of the matrix (27) give approximations of the ﬁrst Neigenvalues of the Hilbert– Schmidt kernel, and the components of its eigenvectors give the coefﬁcients in the expansion of the ﬁrstN– 1 eigenfunctions of that kernel in terms of the chosen Legendre polynomials. Recall that the ﬁrst term y◦(x,t) of the expansion of y(x,t) is known by assumption. Therefore, constructing the next N– 1 terms of the expansion, we obtain the Nth approximation. Note that the matrix (27) can be obtained from the matrix (11) by deleting its ﬁrst two rows and columns. This allows us to construct the expansion of the original kernel only once and then usethat data for the examination of the new kernel arising in the problem with auxiliary conditions. Now, let us ﬁnd the function β(t). To that end, we apply the orthogonal projection operator Q ∗ to equation (12). As a result we obtain the following formula: β(t)=√ 2⎝braceleftbigg w◦ 1(t)+σ(t)⎝bracketleftbigg y◦ 1(t)–⎝integraldisplayt 1V1(t,τ)y◦ 1(τ)dτ⎝bracketrightbigg +G11y◦ 1(t) +∞⎝summationdisplay k=2Gk(1)y∗ k(t)–⎝integraldisplayt 1V1(t,τ)⎝bracketleftbigg G11y◦ 1(τ)+∞⎝summationdisplay k=2Gky∗ k(τ)⎝bracketrightbigg dτ⎝bracerightbigg . (28) Thus, we have obtained a complete solution of the integral equation (12) with the auxiliary conditions (13). 17.3-3. Equation with a Schmidt Kernel and a Given Right-Hand Side. Consider a mixed integral equation of the form (6) from Subsection 17.1-3 with a Schmidt ker-nel. Changing the variables, we can easily transform this equation to a similar equation with theparameters a=0 ,b=1 ,τ 0=1 : σ(t)⎝bracketleftBig y(r,t)–⎝integraldisplayt 1V1(t,τ)y(ρ,τ)dτ⎝bracketrightBig +⎝integraldisplay1 0Sω(r,ρ)y(ρ,t)ρd ρ –⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 0Sω(r,ρ)y(ρ,τ)ρd ρd τ =f(r,t) h(r), Sω(r,ρ)=G(r,ρ) h(r),0 ≤r≤1, 1 ≤t≤T.(29) 860 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Suppose that the right-hand side f(r,t)/h(r) of (29) is known, and it is required to ﬁnd the function y(r,t). Here, f(r,t)a n d y(r,t) are continuous functions of t∈[1,T] with values in⎝hatwideL2(ω); σ(t) is a given positive continuous function, h(r) > 0 is a given function in ⎝hatwideL2(ω);V1(t,τ)a n dV2(t,τ) are V olterra kernels; Sω(r,ρ) is a Schmidt kernel; G(r,ρ) is a symmetric positive deﬁnite Fredholm kernel. Let us transform the equation with the Schmidt kernel to a an equation with Hilbert–Schmidt kernel. To this end, we multiply equation (28) by√ h(r) and change the variables as follows: q(r,t)=⎝radicalbig h(r)y(r,t),Gh(r,ρ)=S(r,ρ)√ h(r) √ h(ρ)=G(r,ρ) √ h(r)h(ρ). (30) Then, we have σ(t)⎝bracketleftBig q(r,t)–⎝integraldisplayt 1V1(t,τ)q(r,τ)dτ⎝bracketrightBig +⎝integraldisplay1 0Gh(r,ρ)q(ρ,t)ρd ρ –⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 0Gh(r,ρ)q(ρ,τ)ρd ρd τ =f(r,t) √ h(r),0 ≤r≤1, 1 ≤t≤T, (31) where q(r,t)a n d f(r,t)/√ h(r) are continuous functions of t∈[1,T] with values in⎝hatwideL2(ω);Gh(r,ρ) is a symmetric positive deﬁnite Hilb ert–Schmidt kernel; and the other functions are the same as above. Suppose that the right hand side of equation (31) is known and it is required to ﬁnd the function q(r,t). Let us seek a solution of the mixed equation (31) in the form of a series q(r,t)=∞⎝summationdisplay k=1qk(t)ϕh k(r), (32) where ψh k(r) are eigenfunctions of the kernel Gh(r,ρ) corresponding to eigenvalues νh k>0 ,i . e . , ⎝integraldisplay1 0Gh(r,ρ)ψh k(ρ)ρd ρ =νh kψh k(r), k=1 ,2 , ... (33) The representation (33) is possible, since the system of eigenfunctions of the kernel Gh(r,ρ)f o r m s a basis in⎝hatwideL2(ω). Here, in contrast to the above cases, we construct the basis in the form ψh k(r)=Ψh k(r) √ h(r)k=1 ,2 , ... (34) with explicit dependence on the function h(r), where ⎝integraldisplay1 0ψh i(ρ)ψh j(ρ)ρd ρ =⎝integraldisplay1 0Ψh i(ρ)Ψh j(ρ) h(ρ)ρd ρ =δij=⎝braceleftbigg 1i f i=j, 0i f i≠j.(35) In order to construct such eigenfunctions, we ﬁrst construct a basis zh n(r)i n⎝hatwideL2(ω)f o rw h i c h ⎝integraldisplay1 0zh i(ρ)zh j(ρ)ρd ρ =δij,zh n(r)=Qh n–1(r) √ h(r),n=1 ,2 , ... (36) 17.3. M ETHODS OF SOLVING MIXED INTEGRAL EQUATIONS ON A RING-SHAPED DOMAIN 861 Such a basis can be constructed by the formulas Qh 0(r)=1 √ I0,Qh n(r)=1 √ Dn–1Dn⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleI 0I1··· In I1I2···In+1............ I n–1In···I2n–1 1r2···r2n⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle, D –1=1 , D0=I0,Dn=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleI 0I1··· In I1I2···In+1............ InIn+1···I2n⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,I n=⎝integraldisplay1 0ρ2n+1 h(ρ)dρ.(37) Let us represent the kth eigenfunction as a series in terms of the basis zh i(r)i n⎝hatwideL2(ω). Then ψh k(r)=∞⎝summationdisplay i=1ψh i(k)zh i(r),zh i(r)=Qh i–1(r) √ h(r),Ψh k(r)=∞⎝summationdisplay i=1ψh i(k)Qh i–1(r). (38) For the Hilbert–Schmidt kernel Qh(r,ρ) we use the double series expansion with respect to the chosen basis: Qh(r,ρ)=∞⎝summationdisplay m=1∞⎝summationdisplay i=1Qh mnzh m(r)zh n(ρ), Qh mn=⎝integraldisplay1 0⎝integraldisplay1 0Qh(r,ρ)zh m(r)zh n(ρ)rρ dr dρ ,Qh mn=Qh nm.(39) Substituting (38) and (39) into (33), we obtain an inﬁnite system of linear algebraic equations for the determination of the eigenvalues and the eigenfunction expansion coefﬁcients. This system hasa symmetric matrix and can be written in the form ∞⎝summationdisplay n=1Qh mnψh n(k)=νh kψh m(k),m=1 ,2 , ... (40) In order to calculate approximations for Neigenvalues and eigenfunctions of the Hilbert– Schmidt kernel, it is necessary to ﬁnd the eigenvalues and orthonormal eigenv ectors of the matrix [Gh NN]=⎛ ⎜⎜⎜⎜⎝Gh 11Gh12Gh13···Gh 1N Gh 12Gh22Gh23···Gh 2N Gh 13Gh23Gh33···Gh 3N............... Gh 1NGh 2NGh 3N···Gh NN⎞ ⎟⎟⎟⎟⎠. (41) The eigenvalues of the matrix (41) give approximations of the ﬁrst Neigenvalues of the Hilbert– Schmidt kernel, and the components of its orthonormal eigenvectors give the coefﬁcients in the expansion of the ﬁrst Neigenfunctions of this kernel in terms of Northonormal basis functions of the space⎝hatwideL2(ω). Now, consider the expansion of the right-hand side of equation (31) into the following series: f(r,t) √ h(r)=∞⎝summationdisplay k=1fh k(t)ψh k(r)=∞⎝summationdisplay k=1fh k(t)Ψh k(r) √ h(r), fh k(t)=⎝integraldisplay1 0f(ρ,t) √ h(ρ)ψh k(ρ)ρd ρ =⎝integraldisplay1 0f(ρ,t) h(ρ)Ψh k(ρ)ρd ρ.(42) 862 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Substituting (32), (42) into (31) and taking into account (33), we obtain the following sequence of V olterra equations for the unknown functions qk(t): qk(t)–⎝integraldisplayt 1Vh k(ν)(t,τ)qk(τ)dτ=γh k(t),γh k(t)=fh k(t) σ(t)+νh k(43) Vh k(ν)(t,τ)=σ(t)V1(t,τ)+νh kV2(t,τ) σ(t)+νh k,k=1 ,2 , ..., (44) where Vh k(ν)(t,τ) are V olterra kernels of the same class as V1(t,τ)a n d V2(t,τ), since νh k→0 ask→∞ . A solution of the sequence of V olterra equations (44) can be constructed by analytical and numerical methods of Chapter 11. This solution can be written in the form qk(t)=γh k(t)+⎝integraldisplayt 1Rh k(ν)(t,τ)γh k(τ)dτ, (45) where Rh k(ν)(t,τ) is the resolvent of the kernel Vh k(ν)(t,τ). The series (32) converges in ⎝hatwideL2(ω) uniformly with respect to t∈[1,T], and its sum is a continuous function of t∈[1,T] with values in⎝hatwideL2(ω). Finally, in view of the transformation of the variables (30) and formula (34) for the eigenfunc- tions, we have y(r,t)=1 h(r)∞⎝summationdisplay k=1qk(t)Ψh k(r). (46) Note that the solution (46) explicitly depends on the function h(r), which allows us to solve equation (29) with great accuracy by keeping a small number of terms of the series. In the case of a strongly oscillating function h(r), it is hardly possible to construct a solution by other known methods. 17.3-4. Equation with a Schmidt Kernel and Auxiliary Conditions on Ring-Shaped Domain. Consider equation (29) with the right-hand side of the form f(r,t)=β(t)–w(r,t) and an integral condition of the form (8) from Subsection 17.1-3 on the unknown function y(r,t). The problem is to ﬁnd a solution of the mixed integral equation σ(t)⎝bracketleftbigg y(r,t)–⎝integraldisplayt 1V1(t,τ)y(r,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 0Sω(r,ρ)y(ρ,t)ρd ρ –⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 0Sω(r,ρ)y(ρ,τ)ρd ρd τ =β(t) h(r)–w(r,t) h(r), Sω(r,ρ)=G(r,ρ) h(r),0 ≤r≤1, 1 ≤t≤T(47) with the auxiliary condition⎝integraldisplay1 0y(ρ,t)ρd ρ =M(t), (48) where the unknown functions are the following: y(r,t),β(t). All the other functions in (47) and (48) are assumed known, and w(r,t) is a continuous function of t∈[1,T] with values in⎝hatwideL2(ω). 17.3. M ETHODS OF SOLVING MIXED INTEGRAL EQUATIONS ON A RING-SHAPED DOMAIN 863 Let us transform the equation with the Schmidt kernel to an equation with a Hilbert–Schmidt ker- nel by changing the variables according to (30). Then, equation (47) and the auxiliary conditions (48)become σ(t)⎝bracketleftbigg q(r,t)–⎝integraldisplay t 1V1(t,τ)q(r,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 0Gh(r,ρ)q(ρ,t)ρd ρ –⎝integraldisplayt τ0V2(t,τ)⎝integraldisplay1 0Gh(r,ρ)q(ρ,τ)ρd ρd τ =β(t) √ h(r)–w(r,t) √ h(r), (49) ⎝integraldisplay1 0q(ρ,t) √ h(ρ)ρd ρ =M(t), 0 ≤r≤1, 1 ≤t≤T. (50) To construct a solution of the mixed integral equation (47) with the auxiliary conditions (48), we use the basis zh n(r)o f⎝hatwideL2(ω) (see (36) and (37)) and note that the space ⎝hatwideL2(ω) can be represented as a direct sum of its orthogonal subspaces: ⎝hatwideL2(ω)=Lh◦ 2(ω)⊕Lh∗ 2(ω) (see Supplement 12.5-3), where⎝hatwideLh◦ 2(ω) is the Euclidean space with the basis zh 1(r), and⎝hatwideLh∗ 2(ω) is the Hilbert space with the basisph k(r)(k=2 ,3 , ...). Note also that the integrand and the right-hand side can be represented as a sum of continuous functions of t∈[1,T] with values in⎝hatwideLh◦ 2(ω)a n d⎝hatwideLh∗ 2(ω), respectively, i.e., q(r,t)=q◦(r,t)+q∗(r,t),f(r,t) √ h(r)=f◦ h(r,t)+f∗ h(r,t), (51) where the following representations hold: q◦(r,t)=q◦ 1(t)zh 1(r),q◦ 1(t)=M(t) √ I0, f(r,t) √ h(r)=β(t) √ h(r)–w(r,t) √ h(r),w(r,t) √ h(r)=w◦ h(r,t)+w∗ h(r,t), (52) f◦ h(r,t)=⎝bracketleftbig⎝radicalbig I0β(t)–w1(t)⎝bracketrightbig zh 1(r),f∗ h(r,t)=–w∗ h(r,t), w◦ h(r,t)=wh◦ 1(t)zh 1(r),wh◦ 1(t)=⎝integraldisplay1 0w(ρ,t) √ h(ρ)zh 1(ρ)ρd ρ. Note that in the representation (52) for q(r,t), the term q◦(r,t) is known (as determined by the auxiliary conditions), and the term q∗(r,t) is to be found. For the right-hand side, the term f◦ h(r,t) should be found and f∗ h(r,t)i sg i v e nb yg (r,t)/√ h(r). Thus, we have come to a special case of the general projection problem whose solution is constructed in Subsection 17.4-3. According to the general method, in this case, one can introduce an operator of orthogonal projection that maps ⎝hatwideL2(ω) onto⎝hatwideLh◦ 2(ω): Q◦ hφ(r,t)=⎝integraldisplay1 0φ(ρ,t)zh 1(r)zh 1(ρ)ρd ρ. (53) Obviously, the orthogonal projector Q∗ h=I–Q◦ hmaps⎝hatwideL2(ω) onto⎝hatwideLh∗ 2(ω). Moreover, the following relations hold: Q◦ hq(r,t)=q◦(r,t), Q∗ hq(r,t)=q∗(r,t), Q◦ hf(r,t) √ h(r)=f◦ h(r,t), Q∗ hf(r,t) √ h(r)=f∗ h(r,t).(54) 864 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Following Section 17.4, we apply the projection operator Q∗ hto equation (49) and obtain an integral equation in the space ⎝hatwideLh∗ 2(ω) with a known right-hand side. This is the equation for the determination of q∗(x,t): σ(t)⎝bracketleftbigg q∗(r,t)–⎝integraldisplayt 1V1(t,τ)q∗(r,τ)dτ⎝bracketrightbigg +⎝integraldisplay1 0G∗ h(r,ρ)q∗(ρ,t)ρd ρ –⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 0G∗ h(r,ρ)q∗(ρ,τ)ρd ρd τ =–g∗ h(r,t)–⎝integraldisplay1 0G∗ h(r,ρ)q◦(ρ,t)ρd ρ +⎝integraldisplayt 1V2(t,τ)⎝integraldisplay1 0G∗ h(r,ρ)q◦(ρ,τ)ρd ρd τ , 0≤r≤1, 1 ≤t≤T,(55) where the kernel of the integral equation G∗ h(r,ρ)=Gh(r,ρ)–⎝integraldisplay1 0Gh(s,ρ)zh 1(r)zh 1(s)sds (56) is of Hilbert–Schmidt type. Let us construct a solution of equation (55) in the form of a series with respect to eigenfunctions of the kernel (56). These eigenfunctions form a basis in the Hilbert space ⎝hatwideLh∗ 2(ω). Let us construct a system of these functions. Letψh∗ k(r) be eigenfunctions and νh∗ kthe corresponding eigenvalues of the kernel G∗ h(r,ρ), i.e., ⎝integraldisplay1 0G∗ h(r,ρ)ψh∗ k(ρ)ρd ρ =νh∗ kψh∗ k(r), k=2 ,3 , ... (57) Let us represent the eigenfunction ψh∗ i(r) in the form of a series with respect to the basis zh i(r) (i≥2): ψh∗ k(r)=∞⎝summationdisplay i=2ψh∗ i(k)zh i(r),zh i(r)=Qh i–1(r) √ h(r),Ψh∗ k(r)=∞⎝summationdisplay i=2ψh∗ i(k)Qh i–1(r). (58) Using (39) and (56), we obtain a double series expansion for the kernel G∗ h(r,ρ): G∗ h(x,ξ)=∞⎝summationdisplay m=2∞⎝summationdisplay n=2Gh mnzh m(r)zh n(ρ)+∞⎝summationdisplay n=2Gh 1nzh n(r)zh 1(ρ). (59) Note that the coefﬁcients of the expansion of G∗ h(r,ρ) in (59) coincide with those of the expansion ofGh(r,ρ), which allows us to avoid recalculation of the coefﬁcients of the new problem and use the available data. Substituting (58) and (59) into (57), we obtain an inﬁnite system of linear algebraic equations for the determination of the eigenvalues and eigenfunction expansion coefﬁcients. This system has a symmetric matrix and can be written as follows: ∞⎝summationdisplay n=2Gh mnψh∗ n(k)=νh∗ kψh∗ m(k),m=2 ,3 , ... (60) 17.3. M ETHODS OF SOLVING MIXED INTEGRAL EQUATIONS ON A RING-SHAPED DOMAIN 865 Now, let us construct a solution of equation (55). For this purpose, we represent the functions q∗(r,t)a n d w∗ h(r,t) in the form of series with eigenfunctions of the kernel G∗ h(r,ρ): q∗(r,t)=∞⎝summationdisplay k=2q∗ k(t)ψh∗ k(r),w∗ h(r,t)=∞⎝summationdisplay k=2wh∗ k(t)ψh∗ k(r),wh∗ k(t)=⎝integraldisplay1 0w∗ h(ρ,t)ψh∗ k(ρ)ρd ρ, (61) and substitute these into (55). Then, taking into account (52), (56)–(59), we obtain a sequence of independent V olterra equations of the second kind: q∗ k(t)–⎝integraldisplayt 1Vh∗ k(ν)(t,τ)q∗ k(τ)dτ=γh∗ k(t),Vh∗ k(ν)(t,τ)=σ(t)V1(t,τ)+νh∗ kV2(t,τ) σ(t)+νh∗ k, γh∗ k(t)=–1 σ(t)+µh∗ k⎝bracketleftbigg gh∗ k(t)+Gh kq◦ 1(t)–⎝integraldisplayt 1V2(t,τ)Gh kq◦ 1(τ)dτ⎝bracketrightbigg , Gh k=∞⎝summationdisplay n=2Gh nψh∗ n(k),k=2 ,3 , ...(62) Resolving (62) with respect to q∗ k(t) by the methods of Chapter 11, we get q∗ k(t)=γh∗ k(t)+⎝integraldisplayt 1Rh∗ k(ν)(t,τ)γh∗ k(τ)dτ, (63) where Rh∗ k(ν)(t,τ) is the resolvent of the kernel Vh∗ k(ν)(t,τ). Now, in view of (61)–(63), the function q∗(r,t) has been found, as well as the function q(r,t), sinceq◦(r,t) is known by assumption (see (51) and (52)). Hence, using (30), we ﬁnally obtain y(r,t)=1 h(r)⎝bracketleftbiggM(t) √ I0Qh 0(r)+∞⎝summationdisplay k=2q∗ k(t)Ψh∗ k(r)⎝bracketrightbigg . (64) Note that the function h(r) enters solution (64) explicitly, which allows us to solve equation (47) with high accuracy by keeping a relatively small number of terms of the series even in the case of arapidly oscillating h(r). In practical calculations, the number of terms in the series is taken ﬁnite. For instance, taking the functions z h k(r) of the basis with k=2 ,...,N, we obtain the Nth approximation of the desired solution. In this case, for the construction of eigenvalues and eigenfunctions of the Hilbert–Schmidt kernel G∗ h(r,ρ) one should ﬁnd the eigenvalues and the orthonormal eigenfunctions of the matrix [Gh NN]=⎛ ⎜⎜⎜⎜⎝G h 33Gh34Gh 35···Gh 3N Gh 34Gh44Gh 45···Gh 4N Gh 35Gh45Gh55···Gh 5N............... Gh 3NGh 4NGh 5N···Gh NN⎞ ⎟⎟⎟⎟⎠. (65) The eigenvalues of the matrix (65) give approximations for the ﬁrst N– 1 eigenvalues of the Hilbert–Schmidt operator, and the components of its orthonormal eigenvectors give the expansioncoefﬁcients for the ﬁrst N– 1 eigenfunctions of that operator. Recall that the ﬁrst term q ◦(r,t)i n the expansion of q(r,t) is known by assumption. Therefore, constructing the next N–1t e r m so f the expansion, we obtain the Nth approximation of the solution. It is important to keep in mind the relation between the matrices (41) and (65). The matrix (64) is obtained from the matrix (41) by deleting its ﬁrst row and column. This allows us to construct 866 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS an expansion of the original kernel only once and then use this data for the examination of the new kernel arising in the problem with auxiliary conditions. Now, let us ﬁnd the function β(t). To this end, we apply the projection operator Q◦ hto equation (55). As a result, we obtain the following formulas: β(t)=1 √ I0⎝braceleftbigg wh◦ 1(t)+σ(t)⎝bracketleftbigg q◦ 1(t)–⎝integraldisplayt 1V1(t,τ)q◦ 1(τ)dτ⎝bracketrightbigg +Gh 11q◦ 1(t) +∞⎝summationdisplay k=3Gh kq∗ k(t)–⎝integraldisplayt 1V2(t,τ)⎝bracketleftbigg Gh 11q◦ 1(τ)+∞⎝summationdisplay k=2Gh kq∗ k(τ)⎝bracketrightbigg dτ⎝bracerightbigg . (66) Thus, we have constructed a complete solution of the integral equation (47) with the auxiliary conditions (48). References for Section 17.3: E. Goursat (1923), G. Szeg ¨o (1975), A. V . Manzhirov (1985, 2005), N. Kh. Arutynyan, A. V . Manzhirov, and V .E. Naumov (1991), N. Kh. Arutynyan, A. V . Manzhirov (1999), A. V . Manzhirov and K. E. Kaza-kov (2006). 17.4. Projection Method for Solving Mixed Equations on a Bounded Set 17.4-1. Mixed Operator Equation with a Given Right-Hand Side. Consider a mixed multi-dimensional equation of the form (10) of Subsection 17.1-4 with integral operators of V olterra and Schmidt types: σ(t)(I–V1)y(/vector x,t)+( I–V2)Sy(/vector x,t)=f(/vector x,t) h(/vector x), Sy(/vector x,t)=⎝integraldisplay ΩS(/vector x,/vectorξ)y(/vectorξ,t)dΩξ,S(/vector x,/vectorξ)=F(/vector x,/vectorξ) h(/vector x), Vpy(/vector x,t)=⎝integraldisplayt τ0Vp(t,τ)y(/vector x,τ)dτ,/vector x∈Ω,τ0≤t≤T.(1) In this section, we consider some general questions of the theory of mixed equations. For this reason we do not single out equations with the Hilbert–Schmidt integral operator of the form (9) from Subsection 17.1-4 and only mention that this equation is a special case of equation (1) with the Schmidt integral operator with h(/vector x)=1 . Let the right-hand side f(/vector x,t)/h(/vector x) of equation (1) be known. It is required to ﬁnd the function y(/vector x,t). Here, f(/vector x,t)a n d y(/vector x,t) are continuous functions of t∈[1,T] with values in L2(Ω); σ(t) is a given positive continuous function; h(/vector x) > 0 is a given function of class L2(Ω);F(/vector x,/vectorξ)i s a symmetric positive deﬁnite Fredholm kernel; V1and V2are V olterra operators; and Sis a Schmidt operator. Let us transform the equation with the Schmidt operator to an equation with a Hilbert–Schmidt operator. To this end, we multiply (1) by√ h(/vector x) and change the variables as follows: q(/vector x,t)=⎝radicalbig h(/vector x)y(/vector x,t),Fh(/vector x,/vectorξ)=S(/vector x,/vectorξ)√ h(/vector x) ⎝radicalBig h(/vectorξ)=F(/vector x,/vectorξ) ⎝radicalBig h(/vector x)h(/vectorξ).( 2) Then σ(t)(I–V1)q(/vector x,t)+( I–V2)Fhq(/vector x,t)=f(/vector x,t) √ h(/vector x), Fhq(/vector x,t)=⎝integraldisplay ΩFh(/vector x,/vectorξ)q(/vectorξ,t)dΩξ,/vector x∈Ω,τ0≤t≤T.(3) 17.4. P ROJECTION METHOD FOR SOLVING MIXED EQUATIONS ON A BOUNDED SET 867 where q(/vector x,t)a n d f(/vector x,t)/√ h(/vector x) are continuous functions of t∈[τ0,T] with values in the Hilbert spaceL2(Ω);Fhis a Hilbert–Schmidt operator; and the other functions have been speciﬁed above. Suppose that the right-hand side of equation (3) is known and we have to ﬁnd the function q(/vector x,t). Let us seek a solution of the mixed equation (3) in the form of a series q(/vector x,t)=∞⎝summationdisplay k=1qk(t)ϕh k(/vector x), (4) where ϕh k(/vector x) are eigenfunctions of the operator Fhcorresponding to eigenvalues µh k> 0, i.e., Fhϕh k(/vector x)dξ=µh kϕhk(/vector x), k=1 ,2 , ... (5) The representation (4) is possible, since the system of eigenfunctions of the operator Fhforms a basis in L2(Ω). Let us construct the functions of the basis in the form ϕh k(/vector x)=Φh k(/vector x) √ h(/vector x),k=1 ,2 , ... (6) with explicit dependence on the function h(/vector x), where ⎝integraldisplay Ωϕh i(/vectorξ)ϕh j(/vectorξ)dΩξ=⎝integraldisplay ΩΦh i(/vectorξ)Φh j(/vectorξ) h(/vectorξ)dΩξ=δij=⎝braceleftbigg1f o r i=j, 0f o r i≠j.(7) In order to construct such eigenfunctions, we ﬁrst construct a basis ph n(/vector x)i nL2(Ω)f o rw h i c h ⎝integraldisplay Ωph i(/vectorξ)ph j(/vectorξ)dΩξ=δij,ph n(/vector x)=Ph n(/vector x) √ h(/vector x),n=1 ,2 , ... (8) Such a basis can be constructed by the formulas Ph 1(/vector x)=f1(/vector x) √ H11,Ph n(/vector x)=1 √ ∆n–1∆n=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleH 11H12... H 1n H21H22... H 2n............ f 1(/vector x),f2(/vector x)... f n(/vector x)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle, ∆ 0=1 , ∆1=H11,∆n=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleH 11H12... H 1n H21H22... H 2n............ Hn1Hn2... H nn⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,H ij=⎝integraldisplay Ωfi(/vectorξ)fj(/vectorξ) h(/vectorξ)dΩξ,(9) where fi(/vector x) is an arbitrary complete system of linearly independent function in L2(Ω). Let us represent the kth eigenfunction in the form of a series with respect to the basis ph i(/vector x) ofL2(Ω). We have ϕh k(/vector x)=∞⎝summationdisplay i=1ϕh i(k)ph i(/vector x),ph i(/vector x)=Ph i(/vector x) √ h(/vector x),Φh k(/vector x)=∞⎝summationdisplay i=1ϕh i(k)Ph i(/vector x). (10) The Hilbert–Schmidt kernel Fh(/vector x,/vectorξ) can be expanded into double series with respect to the chosen basis: Fh(/vector x,/vectorξ)=∞⎝summationdisplay m=1∞⎝summationdisplay n=1Fh mnphm(/vector x)ph n(/vectorξ), Fh mn=⎝integraldisplay Ω⎝integraldisplay ΩFh(/vector x,/vectorξ)ph m(/vector x)ph n(/vectorξ)dΩxdΩξ,Fh mn=Fh nm.(11) 868 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Substituting (10) and (11) into (5), we obtain an inﬁnite system of linear algebraic equations for the determination of the eigenval ues and the eigenfunction expansion coefﬁcients. This system has a symmetric matrix and can be written as follows: ∞⎝summationdisplay n=1Fh mnϕhn(k)=µh kϕhm(k),m=1 ,2 , ... (12) In order to calculate approximations for Neigenvalues and eigenfunctions of the Hilbert– Schmidt operator Fh, it is necessary to ﬁnd the eigenvalues and orthonormal eigenvectors of the matrix [Fh NN]=⎛ ⎜⎜⎜⎜⎝Fh 11Fh 12Fh 13···Fh 1N Fh 12Fh 22Fh 23···Fh 2N Fh 13Fh 23Fh 33···Fh 3N............... Fh 1NFh 2NFh 3N···Fh NN⎞ ⎟⎟⎟⎟⎠. (13) The eigenvalues of the matrix (13) give approximations for the ﬁrst Neigenvalues of the Hilbert– Schmidt operator, and the components of its orthonormal eigenvectors give approximate values of the expansion coefﬁcients for the ﬁrst Neigenfunctions of that series with Northonormal functions of the basis. Let us write the right-hand side of equation (3) in the form f(/vector x,t) √ h(/vector x)=∞⎝summationdisplay k=1fh k(t)ϕh k(/vector x)=∞⎝summationdisplay k=1fh k(t)Φh k(/vector x) √ h(/vector x), fh k(t)=⎝integraldisplay Ωf(/vectorξ,t) ⎝radicalBig h(/vectorξ)ϕh k(/vectorξ)dΩξ=⎝integraldisplay1 –1f(/vector x,t) h(/vector x)Φh k(/vector x)dx.(14) Substituting (4), (14) into (3) and taking into account (5), we obtain the following sequence of V olterra equations for the unknown functions qk(t): (I–Vh k)qk(t)=δh k(t),δh k(t)=fh k(t) σ(t)+µh k, Vh k=σ(t)V1+µh kV2 σ(t)+µh k,Vh kf(t)=⎝integraldisplayt τ0Vh k(t,τ)f(τ)dτ, Vh k=σ(t)V1(t,τ)+µh kV2(t,τ) σ(t)+µh k,k=1 ,2 , ...,(15) where all operators Vh kare of V olterra type, just as the operators V1and V2,s i n c e µh k→0a sk→∞ . A solution of the sequence of V olterra equations (15) can be constructed by analytical and numerical methods of Chapter 11. This solution can be written in the form qk(t)=( I+Rh k)δh k(t), (I –Vh k)–1=(I+Rh k), Rh kf(t)=⎝integraldisplayt τ0Rh k(t,τ)f(τ)dτ, (16) where Rh kis the resolvent operator for Vh k,a n dRk(t,τ) is the resolvent of the kernel Vh k(t,τ). The series (13) converges in L2(Ω) uniformly in t∈[τ0,t], and its sum is a continuous function of twith values in L2(Ω). Finally, taking into account (2), (4), and (6), (16), we ﬁnd that y(/vector x,t)=1 h(/vector x)∞⎝summationdisplay k=1(I+Rh k)δh k(t)Φh k(/vector x). (17) Note that the function h(/vector x) enters the solution (17) in explicit form, which allows us to solve equation (1) with high accuracy, even for a rapidly oscillating function h(/vector x). 17.4. P ROJECTION METHOD FOR SOLVING MIXED EQUATIONS ON A BOUNDED SET 869 17.4-2. Mixed Operator Equations with Auxiliary Conditions. Consider equation (1) with the right-hand side f(/vector x,t)=N⎝summationtext i=1αi(t)fi(/vector x)–g(/vector x,t)a n d Nauxiliary integral conditions (of the form (12) from Subsection 17.1-4) on the unknown function y(x,t). The problem is to ﬁnd a solution of the operator equation σ(t)(I–V1)y(/vector x,t)+( I–V2)Sy(/vector x,t)=N⎝summationdisplay i=1αi(t)fi(/vector x) h(/vector x)–g(/vector x,t) h(/vector x),/vector x∈Ω,τ0≤t≤T(18) with the auxiliary conditions ⎝integraldisplay Ωy(/vectorξ,t)fi(/vectorξ)dΩξ=Mi(t), i=1 ,...,N, (19) regarding y(/vector x,t)a n dα1(t),...,αN(t) as unknown functions. All other functions in (18) are assumed given, and g(/vector x,t) is a continuous function of twith values in L2(Ω);fi(/vector x) is a system of Nlinearly independent functions in L2(Ω). Let us transform the equation with the Schmidt operator to an equation with a Hilbert–Schmidt operator by changing the variables as in (2). Then, equation (18) and the auxiliary conditions (19) become σ(t)(I–V1)q(/vector x,t)+( I–V2)Fhq(/vector x,t)=N⎝summationdisplay i=1αi(t)fi(/vector x) √ h(/vector x)–g(/vector x,t) √ h(/vector x), (20) /vector x∈Ω,τ0≤t≤T, ⎝integraldisplay Ωq(/vectorξ,t)fi(/vectorξ) ⎝radicalBig h(/vectorξ)dΩξ=Mi(t), i=1 ,...,N. (21) In order to construct a solution of the mixed integral equation (18) with the auxiliary condi- tions (19), we construct a special basis in L2(Ω) with explicit dependence on the function 1 /√ h(/vector x). To this end, we complement the system of Nlinearly independent functions fi(/vector x), so as to obtain a complete system in L2(Ω), and then use formulas (8) and (9). As a result, we obtain a basis ph n(/vector x) inL2(Ω) for which (in view of (8)and (9)) the following expansion holds: ph i(/vector x)=Ph i(/vector x) √ h(/vector x)=i⎝summationdisplay k=1aikfk(/vector x) √ h(/vector x),i=1 ,...,N. (22) Resolving the system of algebraic equations (22), we obtain fi(/vector x) √ h(/vector x)=i⎝summationdisplay k=1bikph k(/vector x), i=1 ,...,N, (23) the matrix of system (23) being the inverse of the matrix corresponding to system (22). Let us represent the Hilbert space L2(Ω) as the direct sum of its orthogonal subspaces: L2(Ω)=L◦ 2(Ω)⊕L∗ 2(Ω), (24) where L◦ 2(Ω) is the Euclidean space with the basis p1(/vector x),...,pN(/vector x), and L∗ 2(Ω) is the Hilbert space with the basis {pk(/vector x)}(k=N+1 ,N+2 ,...). 870 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS Note that any continuous function of twith values in L2(Ω) can be represented as a sum of continuous functions of twith values in L◦ 2(Ω)a n dL∗ 2(Ω). Let us write such a representation for the integrand: q(/vector x,t)=q◦(/vector x,t)+q∗(/vector x,t),q◦(/vector x,t)=N⎝summationdisplay n=1q◦ n(t)ph n(/vector x). (25) Using the auxiliary conditions (21), together with (23) and (25), we obtain the following system of equations: i⎝summationdisplay k=1bikq◦ n(t)=Mi(t), i=1 ,...,N. (26) The solution of this system determines the coefﬁcients of the ﬁrst term in the expansion (25) ofq◦(/vector x,t): q◦ i(t)=i⎝summationdisplay k=1aikMk(t), i=1 ,...,N. (27) In view of (23), the right-hand side of the equation can be written in the form f(/vector x,t) √ h(x)=N⎝summationdisplay i=1αi(t)fi(/vector x) √ h(/vector x)–g(/vector x,t) √ h(/vector x)=f◦ h(/vector x,t)+f∗ h(/vector x,t), f◦ h(/vector x,t)=N⎝summationdisplay k=1N⎝summationdisplay i=k[αi(t)bik–gh◦ k(t)]ph k(/vector x),f∗ h(/vector x,t)=–g∗ h(/vector x,t), g(/vector x,t) √ h(/vector x)=g◦ h(/vector x,t)+g∗ h(/vector x,t),g◦ h(/vector x,t)=N⎝summationdisplay k=1gh◦ k(t)ph k(/vector x), gh◦ k(t)=⎝integraldisplay Ωg(/vectorξ,t) ⎝radicalBig h(/vectorξ)ph k(/vectorξ)dΩξ,k=1 ,...,N.(28) Note that in the representations (25)–(28) for q(/vector x,t), the function q◦(/vector x,t) is known (as determined by the auxiliary conditions), and the term q∗(/vector x,t) is to be found. Conversely, for the right-hand side, we should ﬁnd f◦ h(/vector x,t), and f∗ h(/vector x,t)i sg i v e nb yg (/vector x,t)/√ h(/vector x). The facts mentioned above allow us to classify the resulting problem as a special case of the general projection problem consideredin Subsection 17.4-3. According to the general method, in the present case, one can introduce an operator of orthogonal projection that maps the space L 2(Ω) onto Lh◦ 2(Ω): P◦ hf(/vector x)=⎝integraldisplay Ωf(/vectorξ)N⎝summationdisplay i=1ph i(/vector x)ph i(/vectorξ)dΩξ. (29) Obviously, the orthogonal projector P∗ h=I–P◦ hmaps L2(Ω) onto Lh∗ 2(Ω). Moreover, the following relations hold: P◦ hq(/vector x,t)=q◦(/vector x,t), P∗ hq(/vector x,t)=q∗(/vector x,t), P◦ hf(/vector x,t) √ h(/vector x)=f◦ h(/vector x,t), P∗ hf(/vector x,t) √ h(/vector x)=f∗ h(/vector x,t).(30) 17.4. P ROJECTION METHOD FOR SOLVING MIXED EQUATIONS ON A BOUNDED SET 871 Following Section 17.4, we apply the projection operator P∗ hto equation (20) and obtain an integral equation in Lh∗ 2(Ω) (with a known right-hand side) for the determination of q∗(/vector x,t): σ(t)(I–V1)q∗(/vector x,t)+( I–V2)P∗ hFhq∗(/vector x,t)=–g∗(/vector x,t)–( I–V2)P∗ hFhq◦(/vector x,t), P∗ hFhφ(/vector x,t)=⎝integraldisplay ΩF∗ h(/vector x,/vectorξ)φ(/vectorξ,t)dΩξ,/vector x∈Ω,τ0≤t≤T, F∗ h(/vector x,/vectorξ)=Fh(/vector x,/vectorξ)–⎝integraldisplay ΩFh(/vectors,/vectorξ)N⎝summationdisplay i=1ph i(/vector x)ph i(/vectors)dΩs.(31) The operator P∗ hFhis a Hilbert–Schmidt operator from Lh∗ 2(Ω)t oLh∗ 2(Ω). Let us construct a solution of equation (31) in the form of a series with respect to its eigenfunctions that form a basis inLh∗ 2(Ω). Let us construct the system of these functions. Letϕh∗ k(/vector x) be eigenfunctions of the operator P∗ hFhandµh∗ kthe corresponding eigenvalues. We have P∗ hFhϕh∗ k(/vector x)=µh∗ kϕh∗ k(/vector x), k=N+1 ,N+2 ,... (32) Let us represent the eigenfunction ϕh∗ i(/vector x) as a series with respect to the basis ph i(/vector x)(i≥N+1 ) : ϕh∗ k(/vector x)=∞⎝summationdisplay i=N+1ϕh∗ i(k)ph i(/vector x),ph i(/vector x)=Ph i(/vector x) √ h(/vector x),Φh∗ k(/vector x)=∞⎝summationdisplay i=N+1ϕh∗ i(k)Ph i(/vector x). (33) Using (11) and (31), we obtain the following double series expansion for the kernel F∗ h(/vector x,/vectorξ): F∗ h(/vector x,/vectorξ)=∞⎝summationdisplay m=N+1∞⎝summationdisplay n=N+1Fh mnphm(/vector x)ph n(/vectorξ)+N⎝summationdisplay i=1∞⎝summationdisplay n=N+1Fh inphn(/vector x)ph i(/vectorξ). (34) Note that the coefﬁcients in the expansion of the kernel F∗ h(/vector x,/vectorξ) in (34) coincide with those in the expansion of the kernel Fh(/vector x,/vectorξ), and this allows us to use the available data instead of recalculating the coefﬁcients of the new problem. Substituting (33) and (34) into (32), we obtain an inﬁnite system of linear algebraic equations for the determination of the eigenvalues and the eigenfunction expansion coefﬁcients. This systemhas a symmetric matrix and can be written in the form ∞⎝summationdisplay n=N+1Fh mnϕh∗ n(k)=µh∗ kϕh∗ m(k),m=N+1 ,N+2 ,... (35) Now, let us construct a solution of equation (31). To this end, we represent the functions q∗(/vector x,t) andg∗ h(/vector x,t) in the form of series with eigenfunctions of the operator P∗ hFh: q∗(/vector x,t)=∞⎝summationdisplay k=N+1q∗ k(t)ϕh∗ k(/vector x), g∗ h(/vector x,t)=∞⎝summationdisplay k=N+1gh∗ k(t)ϕh∗ k(/vector x),gh∗ k(t)=⎝integraldisplay1 –1g∗ h(/vectorξ,t)ϕh∗ k(/vectorξ)dΩξ,(36) 872 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS and substitute these into (31). Then, taking into account (25)–(28) and (30)–(34), we obtain the following sequence of independent V olterra equations of the second kind: (I–Vh∗ k)q∗ k(t)=δh∗ k(t), Vh∗ k=σ(t)V1+µh∗ kV2 σ(t)+µh∗ k, Vh∗ kf(t)=⎝integraldisplayt τ0Vh∗ k(t,τ)f(τ)dτ,Vh∗ k(t,τ)=σ(t)V1(t,τ)+µh∗ kV2(t,τ) σ(t)+µh∗ k, δh∗ k(t)=–1 σ(t)+µh∗ k⎝bracketleftbigg gh∗ k(t)+( I–V2)N⎝summationdisplay i=1Fh k(i)q◦ i(t)⎝bracketrightbigg , Fh k(i)=∞⎝summationdisplay n=N+1Fh inϕh∗ n(k),i=1 ,...,N,k=N+1 ,N+2 ,...(37) Resolving (37) with respect to q∗ k(t) by the methods of Chapter 11, we get q∗ k(t)=( I+Rh∗ k)δh∗ k(t), (I –Vh∗ k)–1=(I+Rh∗ k), Rh∗ kf(t)=⎝integraldisplayt τ0Rh∗ k(t,τ)f(τ)dτ, (38) where Rh∗ k(t,τ)i st h er e s o l v e n to ft h ek e r n e l Vh∗ k(t,τ). We see that in view of (35)–(38) the function q∗(/vector x,t) has been determined, and it is easy to ﬁndq(/vector x,t), since q◦(/vector x,t) is known by assumption (see (25)–(27)). Hence, taking into account the transformation of the variables (2), we ﬁnally obtain y(/vector x,t)=1 h(/vector x)⎝bracketleftBigN⎝summationdisplay n=1n⎝summationdisplay k=1ankMk(t)Ph n(/vector x)+∞⎝summationdisplay k=N+1q∗ k(t)Φh∗ k(/vector x)⎝bracketrightBig . (39) The solution (39) depends on the function h(x) in explicit manner, and this allows us to solve equation (48) with high accuracy by keeping a relatively small number of terms in the series even for a rapidly oscillating h(/vector x). In practical calculations, the number of terms in the expansions has to be limited. For instance, taking the basis functions ph k(/vector x) with k=N+1 ,...,M, we obtain the Mth approximation of the desired solution. In this case, for the construction of eigenvalues and eigenfunctions of the Hilbert–Schmidt operator P∗ hFhone should ﬁnd the eigenvalues and orthonormal eigenvectors of the matrix [Fh MM]=⎛ ⎜⎜⎜⎜⎝F h N+1N+1Fh N+1N+2 Fh 35 ···Fh N+1M Fh N+1N+2Fh N+2N+2Fh N+2N+3···Fh N+2M Fh N+1N+3Fh N+2N+3Fh N+3N+3···Fh N+3M............... F h N+1M Fh N+2M Fh N+3M···Fh MM⎞ ⎟⎟⎟⎟⎠. (40) The eigenvalues of the matrix (65) give approximations of the ﬁrst M–Neigenvalues of the Hilbert– Schmidt operator, and the components of its orthonormal eigenvectors approximate the coefﬁcients in the expansion of the ﬁrst M–Neigenfunctions of this operator. Recall that the ﬁrst Nterms of the expansion (25) of q ◦(/vector x,t)o ft h ef u n c t i o n q(/vector x,t) are known by assumption. Therefore, constructing the next M–Nterms of the expansion (36) of q∗(/vector x,t), we obtain the Mth approximation of the solution q(/vector x,t) (see (25)). It is important to observe that the matrix (40) can be obtained from the matrix (13) by deleting its ﬁrstNrows and columns. This allows us to construct an expansion of the original kernel only once and then use these data for the examination of the new kernel arising in the problem with auxiliary conditions. 17.4. P ROJECTION METHOD FOR SOLVING MIXED EQUATIONS ON A BOUNDED SET 873 Now, in order to ﬁnd the functions αi(t)(i=1 ,...,N), we apply the projection operator P◦ hto equation (31). As a result, we get αk(t)=N⎝summationdisplay i=kaik⎝braceleftbigg gh◦ i(t)+σ(t)(I–V1)i⎝summationdisplay m=1aimMm(t) +(I–V2)⎝bracketleftbiggN⎝summationdisplay j=1Fh jij⎝summationdisplay m=1ajmMm(t)+∞⎝summationdisplay j=N+1Fh j(i)q∗ j(t)⎝bracketrightbigg⎝bracerightbigg . (41) Note that relations (41) form a system of Nlinear algebraic equations (with a triangular matrix) for the determination of the unknown quantities α1(t),...,αN(t). Thus, we have constructed a complete solution of the integral equation (18) with the auxiliary conditions (19). 17.4-3. General Projection Problem for Operator Equation. Consider the equation c(t)(I–V1)y(t)+( I–V2)Fy(t)=f(t), (42) where y(t)a n d f(t) are continuous functions of twith values in an abstract Hilbert space H;c(t)>0 is a continuous scalar function of t;Iis the identity operator; Fis a compact self-adjoint positive operator from HtoH;V1and V2are V olterra operators (with respect to t) such that the operators (I–V1), (I –V2), and (I –(ω1(t)V1+ω2(t)V2)) and their inverse operators preserve the class of continuous functions, provided that ω1(t)a n d ω2(t) are continuous in t. Let us represent the Hilbert space Has a sum of its orthogonal subspaces H=H◦⊕H∗.F o r continuous functions of twith values in Hthe following representations hold: f(t)=f◦(t)+f∗(t),y(t)=y◦(t)+y∗(t), (43) where f(t)◦,y(t)◦are continuous functions of twith values in H◦,a n df(t)∗,y(t)∗are continuous functions of twith values in H∗. Consider the operator P◦of orthogonal projection from HontoH◦. The operator P∗=I–P◦ projects HontoH∗. Obviously, P◦f(t)=f(t)◦,P∗f(t)=f(t)∗,P◦y(t)=y(t)◦,P∗y(t)=y(t)∗. (44) General projection problem. Lety(t)andf(t)satisfy equation (42). For given y◦(t)and f∗(t), it is required to ﬁnd the unknown y∗(t)andf◦(t). Let us apply the operator P∗to equation (42). As a result, we obtain a new equation which, after simple transformations, can be written in the form c(t)(I–V1)y∗(t)+( I–V2)P∗Fy∗(t)=f∗(t)–( I–V2)P∗Fy◦(t). (45) THEOREM 1.The operator P∗Fis compact, self-adjoint, and positive deﬁnite as an operator fromH∗toH∗. Letϕibe eigenfunctions of the operator P∗Fcorresponding to its eigenvalues µi,i . e . , P∗Fϕi=µiϕi. (46) 874 METHODS FOR SOLVING MULTIDIMENSIONAL MIXED INTEGRAL EQUATIONS All these eigenfunctions form a basis in H∗. Then, for continuous functions of twith values in H∗ the following representations hold: y∗(t)=⎝summationdisplay iai(t)ϕi,f∗(t)=⎝summationdisplay ifi(t)ϕi, g∗(t)=( I–V2)P∗Fy∗(t)=⎝summationdisplay igi(t)ϕi,(47) where ai(t),fi(t),gi(t) are continuous in t. Substituting (47) into (45) and taking into account (46), we get (I–Vi)ai(t)=Φi(t), Vi=c(t)V1+αiV2 c(t)+αi, Φi(t)=fi(t)–gi(t) c(t)+αi,ai(t)=( I+Ri)Φi(t),(48) where Riis the resolvent V olterra operator for Vi. Note that ViandΦi(t) are always deﬁned since c(t)>0 ,α i>0 ,a n d αi→0,Vi→V1asi→∞ (see Supplement 12.5-3). Due to the conditions imposed on the functions and the operators, the series (47) for y∗(t)c o n v e r g e si n Huniformly with respect to t, and its sum is a continuous function oftwith values in H∗. Equation (45) is linear and for f∗(t)=0 , y◦(t) = 0 has the trivial solutions. THEOREM 2.In the above classes of continuous functions, equation (45) has one and only one solution. Thus, we have found y∗(t). In order to ﬁnd f◦(t), let us apply the operator P◦to equation (42). Then f◦(t)=c(t)(I–V1)y◦(t)+( I–V2)P◦F⎝parenleftbig y◦(t)+y∗(t)⎝parenrightbig , (49) which immediately yields an expression for f◦(t), since y◦(t) is given and y∗(t) has been found. The question about the existence and the uniqueness of the solution f◦(t) is solved simultaneously with that of the existence and the uniqueness of the solution y∗(t). For the justiﬁcation of this method the following theorem is needed. THEOREM 3.Functions y(t)andf(t)satisfy equation (42) for given projections P◦y(t)and P∗f(t)if and only if relations (45) and(49) hold. THEOREM 4.A solution of equation (42) for given y◦(t)andf∗(t)exists and is unique if and only if equation (45) has one and only one solution. Remark 1. For P∗=I, the above projection problem reduces to the classical problem for an equation with a given right-hand side. Thus, the problem considered here is a generalization of the classical approach to more complex cases of equations with auxiliary conditions. Remark 2. The projection method considered here can be regarded as an extension of the Hilbert–Schmidt method to multidimensional equations with auxiliary conditions. Remark 3. For given auxiliary conditions, the basic operator of the problem is P∗Fand not F, which makes the problem considered here essentially different from the problem with a givenright-hand side. Remark 4. Eigenfunctions and eigenvalues of the kernels and operators can be found by various methods described in literature, and not only those represented in Chapter 17. References for Section 17.4: E. Goursat (1923), F. Riesz and B. Sz.-Nagy (1955), G. Szeg ¨o (1975), V . S. Vladimirov (1981), A. N. Kolmogorov and S. V . Fomin (1999), A. V . Manzhirov (2005). Chapter 18 Application of Integral Equations for the Investigation of Differential Equations /trianglerightsldPreliminary remarks. Integral equations play an important role in the theory of ordinary and partial differential equations and boundary value problems. The reduction of boundary valueproblems to integral equations allows for the application of iteration and ﬁnite-difference methods of solving integral equations. These methods are, as a rule, substantially simpler than those used for solving differential equations. Moreover, many delicate proofs and qualitative results of thetheory of differential equations have been obtained by the investigation of the corresponding integral equations. 18.1. Reduction of the Cauchy Problem for ODEs to Integral Equations 18.1-1. Cauchy Problem for First-Order ODEs. Uniqueness and Existence Theorems. The Cauchy problem : ﬁnd a solution of the equation y/prime x=f(x,y)( 1 ) that satisﬁes the initial condition y(x0)=y0 (2) for given y0andx0. Geometrical meaning of the Cauchy problem: ﬁnd an integral curve of equation (1) passing through the point (x 0,y0). THEOREM (EXISTENCE ,PEANO ).Let the function f(x,y)be continuous in an open domain D of the xy-plane. Then there is at least one integral curve of equation (1) that passes through each point (x0,y0)∈D;each of these curves can be extended at both ends up to the boundary of any closed domain D0⊂Dsuch that (x0,y0)belongs to the interior of D0. THEOREM (UNIQUENESS ).Let the function f(x,y)be continuous in an open domain Dand have a bounded partial derivative in Dwith respect to y(or satisfy the Lipschitz condition: |f(x,y)– f(x,z)|≤M|y–z|,where M>0is a constant). Then there is a unique solution of equation (1) satisfying condition (2). 875 876 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS 18.1-2. Cauchy Problem for First-Order ODEs. Method of Successive Approximations. The method of successive approximations (the Picard method) consists of two stages. On the ﬁrst stage, the Cauchy problem (1)–(2) is reduced to the equivalent integral equation: y(x)=y0+⎝integraldisplayx x0f(t,y(t))dt.( 3 ) Then a solution of equation (3) is sought using the formula of successive approximations: yn+1(x)=y0+⎝integraldisplayx x0f(t,yn(t))dt;n=0 ,1 ,2 , ... The initial approximation y0(x) can be chosen arbitrarily; the simplest way is to take y0a con- stant. The iterative process converges as n→∞ , provided the assumptions of the theorems in Subsection 18.1-1 are satisﬁed. 18.1-3. Cauchy Problem for Second-Order ODEs. Method of Successive Approximations. The method of successive approximations is implemented in two steps. First, the Cauchy problem y/prime/prime xx=f(x,y,y/prime x) (equation), y(x0)=y0,y/prime x(x0)=y/prime 0(initial conditions) is reduced to an equivalent system of integral equations by the introduction of the new variable u(x)=y/prime x. These integral equations have the form u(x)=y/prime 0+⎝integraldisplayx x0f⎝parenleftbig t,y(t),u(t)⎝parenrightbig dt,y(x)=y0+⎝integraldisplayx x0u(t)dt.( 4) Then the solution of system (4) is sought by means of successive approximations deﬁned by the following recurrence formulas: un+1(x)=y/prime 0+⎝integraldisplayx x0f⎝parenleftbig t,yn(t),un(t)⎝parenrightbig dt,yn+1(x)=y0+⎝integraldisplayx x0un(t)dt;n=0 ,1 ,2 , ... As the initial approximation, one can take y0(x)=y0andu0(x)=y/prime 0. The iterative process converges asn→∞ , under assumptions similar to those formulated in the theorems of Subsection 18.1-1. Remark. In a similar way, the Cauchy problem for an nth order ODE can be reduced to a system of integral equations. 18.1-4. Cauchy Problem for a Special n-Order Linear ODE. Consider the Cauchy problem for the following linear nth order ODE: y(n) x+fn–1(x)y(n–1) x +···+f1(x)y/prime x+f0(x)y=g(x)( 5 ) with the homogeneous initial conditions at the point x=a: y(a)=y/prime x(a)=···=y(n–1) x(a)=0 . ( 6 ) Introducing a new unknown function by y(x)=1 (n–1 ) !⎝integraldisplayx a(x–t)n–1u(t)dt (7) 18.2. R EDUCTION OF BOUNDARY VALUE PROBLEMS FOR ODE ST O VOLTERRA INTEGRAL EQUATIONS 877 and differentiating (7) ntimes, we get y(k) x(x)=1 (n–k–1 ) !⎝integraldisplayx a(x–t)n–k–1u(t)dt,k=1 ,...,n–1 ; y(n) x(x)=u(x).(8) Obviously, the function (7) satisﬁes the initial conditions (6). Substituting (8) into the left-hand side of equation (5), we obtain u(x)+⎝integraldisplayx aK(x,t)u(t)dt=g(x), (9) where K(x,t)=fn–1(x)+fn–2(x)x–t 1!+···+f0(x)(x–t)n–1 (n–1 ) !. (10) Thus, the Cauchy problem (5)–(6) has been reduced to the integral equation (9)–(10), which is a V olterra equation of the second kind. Finding the function u(x) from (9) and using formula (7) we obtain the desired solution y(x). Remark. The Cauchy problem for equation (5) with nonhomogeneous boundary conditions y(a)=b0,y/prime x(a)=b1,...,y(n–1) x(a)=bn–1 can be reduced to a Cauchy problem with homogeneous boundary conditions for another function w(x) with the help of the substitution y(x)=w(x)+n–1⎝summationdisplay k=1bk(x–a)k k!. References for Section 18.1: W. V . Lovitt (1950), E. Kamke (1977), R. P. Kanwal (1996), A. D. Polyanin and A. V . Manzhi- rov (2007). 18.2. Reduction of Boundary Value Problems for ODEs to Volterra Integral Equations. Calculation of Eigenvalues 18.2-1. Reduction of Differential Equations to V olterra Integral Equations. 1◦. Consider a linear nonhomogeneous ODE for the function y=y(x): Ln[y]=h(x)( a<x<b), (1) where Ln[y]=n⎝summationdisplay k=0fk(x)y(k) x,fn(x)≠0. (2) Letϕ1(x),...,ϕn(x) be a fundamental system of solutions of the truncated homogeneous equation Ln[ϕ]=0 . ( 3 ) 878 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS Denote by W(x) its Wronskian determinant W(x)=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleϕ 1(x)··· ϕn(x) ϕ/prime 1(x)··· ϕ/prime n(x)......... ϕ(n–2) 1(x)···ϕ(n–2) n(x) ϕ(n–1) 1(x)···ϕ(n–1) n(x)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle, and by W ν(x) the determinant obtained from W(x) by replacing the νth column by 0, ...,0 ,h(x). The general solution of equation (1) can be written in the form y(x)=n⎝summationdisplay ν=1ϕν(x)⎝integraldisplayx aWν(ξ) fn(ξ)W(ξ)dξ+n⎝summationdisplay ν=1Cνϕν(x), (4) where the ﬁrst sum is a particu lar solution of equation (1), the second sum is the general solution of the homogeneous equation (3), and Cνare arbitrary constants. For boundary value problems, the constants Cνare found from the corresponding boundary conditions, and for the Cauchy problem, Cνare obtained from the initial conditions. 2◦. Consider the linear ODE for the function y=y(x) with a parameter λ: Ln[y]=h(x)–λg(x)y (a<x<b), (5) where Lnis the differential operator (2). Equation ( 5) differs from (1) only by an additional term in the right-hand side. Therefore, replacing the function h(x)b yh(x)–λg(x)y(x) in the solution (4) and performing simple transformations, we come to the V olterra integral equation y(x)+λ⎝integraldisplayx aK(x,ξ)y(ξ)dξ=F(x), (6) where K(x,ξ)=g(ξ) fn(ξ)W(ξ)D(x,ξ),F(x)=⎝integraldisplayx ah(ξ) fn(ξ)W(ξ)D(x,ξ)dξ+n⎝summationdisplay ν=1Cνϕν(x), (7) and D(x,ξ)=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleϕ 1(ξ)··· ϕn(ξ) ϕ/prime 1(ξ)··· ϕ/prime n(ξ)......... ϕ(n–2) 1(ξ)···ϕ(n–2) n(ξ) ϕ1(x)··· ϕn(x)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle.( 8 ) The V olterra integral equation (6)–(8) is equivalent to the differential equation (5). 3 ◦. In a similar way, one can approach nonlinear ODEs of the form Ln[y]=h(x,y)( a<x<b)( 9 ) with the same differential operator (2). This equation can be reduced to the nonlinear V olterra integral equation y(x)=⎝integraldisplayx ah(ξ,y(ξ)) fn(ξ)W(ξ)D(x,ξ)dξ+n⎝summationdisplay ν=1Cνϕν(x), (10) where the function D(x,ξ)i sd e ﬁ n e db y( 8 ) . Note that both sides of equation (9) may depend on the spectral parameter λ. The linear equation corresponds to the right-hand side h(x,y)=h1(x)y+h0(x). 18.2. R EDUCTION OF BOUNDARY VALUE PROBLEMS FOR ODE ST O VOLTERRA INTEGRAL EQUATIONS 879 18.2-2. Application of V olterra Equations to the Calculation of Eigenvalues. 1◦. The V olterra integral equation (6) can be used for the calculation of the smallest eigenvalue and the corresponding eigenfunction of various boundary value problems for the ODE (5). For this purpose, one utilizes the method of successive approximations: the ﬁrst term y(x) of the integral equation is replaced by yn(x), and y(ξ) in the integrand is replaced by yn–1(ξ). On each step, the parameter λis chosen such that the function yn(x) would satisfy the boundary conditions. This procedure can be illustrated by the following example. Example 1. Consider the equation y/prime/prime xx+λg(x)y=0 ( 0<x <1 ) (11) with the homogeneous boundary conditions of the ﬁrst kind y(0) =y(1) = 0. (12) Equation (11) is a special case of (5) for n=2 ,L2[y]=y/prime/prime xx,h(x)≡0,a=0 ,b=0 . The fundamental system of solutions of the truncated equation L2[ϕ]=0h a st h ef o r m ϕ1(x)=1 , ϕ2(x)=x. (13) Simple transformations with the help of (8) yield W(x)=ϕ1(x)[ϕ2(x)]/prime x–ϕ2(x)[ϕ1(x)]/prime x=1 , D(x,ξ)=ϕ1(ξ)ϕ2(x)–ϕ1(x)ϕ2(ξ)=x–ξ.(14) Substituting (13)–(14) into (6)–(7), we come to the V olterra equation y(x)=C1+C2x–λ⎝integraldisplayx 0(x–ξ)g(ξ)y(ξ)dξ. (15) From the ﬁrst boundary condition in (12), we get C1= 0. Since eigenfunctions are deﬁned to within a constant coefﬁcient, we can take C2= 1 in (15). As a result we get y(x)=x–λ⎝integraldisplayx 0(x–ξ)g(ξ)y(ξ)dξ. (16) This equation can be solved by the method of successive approximations based on the formula yn(x)=x–λ⎝integraldisplayx 0(x–ξ)g(ξ)yn–1(ξ)dξ,n=1 ,2 , ... (17) Next, consider more closely the simplest case g(x) = 1. As the zero approximation, we take y0= 1 and ﬁnd that y1(x)=x–λ⎝integraldisplayx 0(x–ξ)dξ=x–1 2λx2. From the second boundary condition in (12), we get y1(1) = 0, and therefore, λ=λ1= 2. It follows that y1(x)=x–x2. Let us insert this function into the right-hand side of (17), where g(x) = 1. We have y2(x)=x–λ⎝integraldisplayx 0(x–ξ)(ξ–ξ2)dξ=x–λ⎝parenleftbigg1 6x3–1 12x4⎝parenrightbigg . Satisfying the second boundary condition in (12), i.e., y2(1) = 0, we obtain λ2= 12, y2(x)=x–2x3+x4. In a similar way, we ﬁnd that λ3= 10, y2(x)=x–5 3x3+3x5–1 3x6. (18) The exact smallest eigenvalue for g(x) = 1 is equal to λ=π2≈9.87, and the corresponding eigenfunction has the form y(x)=1 πsin(πx)≈x–1 . 6x3+0 . 8x5. 880 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS In the case under consideration, the choice of the initial approximation, y0(x) = 1, was not quite good, since both boundary conditions in (12) do not hold for this function. The convergence ratemay be increased by taking the initial approximation of the form y 0(x)=x–x2, in which case both boundary conditions in (12) are satisﬁed. For an arbitrary continuous function g(x), in the absence of information about eigenfunctions, it is convenient to take the initial approximation in (17) of the form y0(x)=x–x2ory0(x)=1 πsin(πx), since these functions satisfy the boundary conditions (12). In the special case of g(x)=1 ,t h eﬁ r s t initial function ensures fast convergence of the expression (17) to the exact result, and the second yields the exact result immediately. 2◦. The V olterra integral equation (6) can be used for o btaining asymptotic expansions of eigenvalues and eigenfunctions of the corresponding boundary value problems for the ODE (5) for large λ.N o t e that in this situation, various cases are possible; for instance, the operator Ln[y] may involve the spectral parameter λ, while the right-hand side of the differential equation is independent on λ. Example 2. Consider the equation y/prime/prime xx+[f(x)+λ2]y=0 ( a<x<b) (19) with the homogeneous boundary condition of the ﬁrst kind y(a)=y(b)=0 . (20) The function f(x) is assumed continuous on the ﬁnite segment [ a,b]. Let us write equation (19), using the notation from (9). As the d ifferential operator and the right-hand side of the equation we take L2[y]=y/prime/prime xx+λ2y, h(x,y)=–f(x)y(x).(21) The fundamental system of solutions of the truncated equation L2[ϕ]=0h a st h ef o r m ϕ1=c o s ( λx),ϕ2=s i n (λx). (22) After elementary calculations with the help of (8), we get W(x)=ϕ1(x)[ϕ2(x)]/prime x–ϕ2(x)[ϕ1(x)]/prime x=λ, D(x,ξ)=ϕ1(ξ)ϕ2(x)–ϕ1(x)ϕ2(ξ)=s i n [ λ(x–ξ)].(23) Substituting the second expression from (21), as well as (22) and (23), into (10), we come to the V olterra integral equation y(x)=–1 λ⎝integraldisplayx asin[λ(x–ξ)]f(ξ)y(ξ)dξ+C1cos(λx)+C2sin(λx). The ﬁrst boundary condition in (20) yields C1cos(λa)+C2sin(λa) = 0. Therefore, C1cos(λx)+C2sin(λx)=Csin[λ(x–a)]. Since eigenfunctions are deﬁned to within an arbitrary constant coefﬁcient, we come to the integral equation y(x)=s i n [ λ(x–a)] –1 λ⎝integraldisplayx asin[λ(x–ξ)]f(ξ)y(ξ)dξ. (24) It is easy to see that the functions y(x) are uniformly bounded for sufﬁciently small λ> 0. From the second boundary condition in (20), using (24), we ﬁnd that sin[λ(b–a)] =1 λ⎝integraldisplayb asin[λ(x–ξ)]f(ξ)y(ξ)dξ=O⎝parenleftbigg1 λ⎝parenrightbigg . (25) Hence, we obtain the following asymptotic formula for the eigenvalues λ=λn: λn=πn b–a+O⎝parenleftbigg1 λ⎝parenrightbigg =πn b–a+O⎝parenleftbigg1 n⎝parenrightbigg , (26) 18.3. R EDUCTION OF BOUNDARY VALUE PROBLEMS FOR ODE ST O FREDHOLM INTEGRAL EQUATIONS 881 where nis a large positive integer. The corresponding eigenfunctions are obtained by substituting the values (26) into (24): yn(x)=s i n [ λn(x–a)] +O⎝parenleftbigg1 λ⎝parenrightbigg =s i n [λn(x–a)] +O⎝parenleftbigg1 n⎝parenrightbigg . Inserting this function into (25), one can reﬁne the asymptotic formula (26), etc. A similar approach can be taken with regard to other boundary conditions for equation (19). References for Section 18.2: E. Kamke (1977), A. D. Polyanin and V . F. Zaitsev (2003). 18.3. Reduction of Boundary Value Problems for ODEs to Fredholm Integral Equations with the Help of the Green’s Function 18.3-1. Linear Ordinary Differential Equations. Fundamental Solutions. Consider a homogeneous linear ordinary differential equation L[y]≡n⎝summationdisplay k=0fk(x)y(k) x=0 ( a<x<b), (1) where fk(x) are continuous functions on the segment a≤x≤bandfn(x)≠0. Afundamental solution of the differential equation (1) is a function of two variables g(x,ξ) deﬁned on the square a≤x,ξ≤band having the following properties: (a) in each of the triangles a≤x≤ξ≤banda≤ξ≤x≤b, the function g(x,ξ) has partial derivatives in xof the orders ≤n, and these derivatives are continuous in xandξin each triangle; (b)g(x,ξ), as a function of x, satisﬁes equation (1) in each of the triangles; (c) on the entire square a≤x,ξ≤b, the function g(x,ξ) is continuous and has partial derivative inxup to the order ( n– 2), and these derivatives are continuous in xandξon that square; (d) for a<ξ<b, the following relation holds: ∂n–1g ∂xn–1⎝vextendsingle⎝vextendsingle⎝vextendsingle x=ξ+0–∂n–1g ∂xn–1⎝vextendsingle⎝vextendsingle⎝vextendsingle x=ξ–0=1 fn(ξ).( 2) Fundamental solutions exist always. For instance, one can take g(x,ξ)=sign(x–ξ) 2fn(ξ)W(ξ)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley 1(ξ)··· yn(ξ) y/prime 1(ξ)··· y/prime n(ξ)......... y(n–2) 1(ξ)···y(n–2) n(ξ) y1(x)··· yn(x)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,W(x)=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingley 1(x)··· yn(x) y/prime 1(x)··· y/prime n(x)......... y(n–2) 1(x)···y(n–2) n(x) y(n–1) 1(x)···y(n–1) n(x)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,( 3 ) where y 1(x),...,yn(x) is a fundamental system of solutions of equation (1), and W(x) is its Wronskian determinant. This special fundamental solution has the following property: g(ξ,ξ)=g/prime ξ(ξ,ξ)=···=g(n–2) x(ξ,ξ)=0 . The set of all fundamental solutions can be described by the sum g(x,ξ)+C1(ξ)y1(x)+···+Cn(ξ)yn(x), where Ck(ξ) are arbitrary continuous functions. Fundamental solutions play an important role in the theory of linear differential equations, since the function y(x)=⎝integraldisplayb ag(x,ξ)ϕ(ξ)dξ is a particular solution of the nonhomogeneous linear ODE L[y]=ϕ(x). 882 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS 18.3-2. Boundary Value Problems for nth Order Differential Equations. Green’s Function. Consider a homogeneous linear boundary value problem for equation (1) with the boundary condition Γm[y]=0 , m=1 ,...,n.( 4 ) Assume that the functions fk(x) are continuous on the segment [ a,b] and the left-hand sides of the boundary conditions have the form Γm[y]=Γm,a[y]+Γm,b[y], (5) whereΓm,a[y]a n dΓm,b[y] are linear differential forms of an order ≤n–1 calculated at the endpoints x=aandx=b. A function G(x,ξ) deﬁned on the square a≤x,ξ≤bis called the Green’s function or the inﬂuence function for problem (1), (4) if it is a fundamental solution of equation (1) and for any ﬁxedξ(a<ξ<b) satisﬁes boundary conditions (4) as a function of x. If the boundary value problem (1), (4) admits only the trivial solution y≡0, then there is only one Green’s function for this problem. Knowing a fundamental system of solutions y1(x),...,yn(x) of equation (1), one can construct the Green’s function as follows. For each ξ(a≤ξ≤b), we ﬁnd a solution c1=c1(ξ),...,cn=cn(ξ) of the system of linear algebraic equations n⎝summationdisplay ν=1cνy(σ) ν(ξ)=0 , σ=0 ,...,n–2 , n⎝summationdisplay ν=1cνy(n–1) ν(ξ)=1 fn(ξ), and then a solution b1=b1(ξ),...,bn=bn(ξ)o fa n o t h e rs y s t e m n⎝summationdisplay ν=1bνΓm[yν]=n⎝summationdisplay ν=1cνΓm,a[yν], m=1 ,...,n. The Green’s function can be deﬁned by the formula G(x,ξ)=⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩n⎝summationdisplay ν=1aν(ξ)yν(x)f o r a≤x≤ξ≤b, n⎝summationdisplay ν=1bν(ξ)yν(x)f o r a≤ξ≤x≤b,(6) where aν(ξ)=bν(ξ)–cν(ξ). The Green’s function can be expressed in terms of a fundamental system of solutions y1(x),..., yn(x) of equation (1), the fundamental solution g(x,ξ), and the differential forms (4): G(x,ξ)=Z(x,ξ) ∆,Z(x,ξ)=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleg(x,ξ)y 1(x)···yn(x) Γ1[g]Γ1[y1]···Γ1[yn] ............ Γ n[g]Γn[y1]···Γn[yn]⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,( 7) where∆stands for the determinant ∆=d e t |Γ i[yj]|,i,j=1 ,...,n. 18.3. R EDUCTION OF BOUNDARY VALUE PROBLEMS FOR ODE ST O FREDHOLM INTEGRAL EQUATIONS 883 Remark. Forξ=aandξ=b, the representation (7), is not valid, in general. However, in this case the following limit relations hold: G(x,a) = lim ξ→aG(x,ξ),G(x,b) = lim ξ→bG(x,ξ). The Green’s function plays an important role in the theory of linear boundary value problems, since the function y(x)=⎝integraldisplayb aG(x,ξ)ϕ(ξ)dξ is a solution of the linear nonhomogeneous differential equation L[y]=ϕ(x) with the homogeneous boundary conditions (4). A nonlinear boundary value problem for a nonlinear ODE of the form n⎝summationdisplay k=0fk(x)y(k) x=Φ(x,y)( a<x<b), (8) with the boundary conditions (4) can be reduced, with the help of the Green’s function, to the nonlinear integral equation y(x)=⎝integraldisplayb aG(x,ξ)Φ(ξ,y(ξ))dξ,( 9) whose investigation is, as a rule, much simpler than that of the original boundary value problem (8), (4). Remark. In applications, one often has to deal with linear eigenvalue problems in which equation (8) is considered with Φ(x,y)=λp(x)y, where λis a spectral parameter. 18.3-3. Boundary Value Problems for Second-Order Differential Equations. Green’s Function. The Green’s function for the boundary value problem for the linear second-order equation f2(x)y/prime/prime xx+f1(x)y/prime xx+f0(x)y= 0 (10) with the homogeneous boundary conditions k1y/prime x+s1y=0 a t x=a, k2y/prime x+s2y=0 a t x=b,(11) can be written as G(x,ξ)=⎧ ⎪⎪⎨ ⎪⎪⎩y 1(x)y2(ξ) f2(ξ)W(ξ)ifa≤x≤ξ≤b, y1(ξ)y2(x) f2(ξ)W(ξ)ifa≤ξ≤x≤b,(12) where y1(x) is any nontrivial solution of equation (10) satisfying the ﬁrst boundary condition in (11), and y2(x) is any nontrivial solution of equation (10) satisfying the second boundary condition in (11); W(x)=y1(x)y/prime 2(x)–y/prime 1(x)y2(x) is the Wronskian determinant. The Green’s function (12) can be used for constructing solutions of nonhomogeneous linear or nonlinear boundary value problems for second-order ODEs. 884 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS Example. Consider the boundary value problem for the nonlinear second-order equation y/prime/prime xx=Φ(x,y(x)) (13) with the homogeneous boundary conditions of the ﬁrst kind y(0) = 0, y(1) = 0. (14) Let us construct the Green’s function for the linear equation y/prime/prime xx=0 (15) with boundary conditions (14). Taking into account that the general solution of equation (15) has the form y=C1+C2x, we take in (12) the solutions y1(x)=x,y2(x)=1– x[each of these satisﬁes one of the boundary conditions from (14)] and f2(x)=1 ,W(x) = –1, a=0 ,b= 1. As a result, we get G(x,ξ)=⎝braceleftbigg(ξ–1 )xif 0 ≤x≤ξ≤1, (x–1 )ξif 0 ≤ξ≤x≤1. Regarding the right-hand side of equation (13) as known, we obtain y(x)=⎝integraldisplay1 0G(x,ξ)Φ(ξ,y(ξ))dξ. Thus, solving the boundary value problem (13)–(14) amounts to solving a nonlinear integral equation of Hammerstein type with the kernel being the Green’s function for problem (15), (14). Table 13 contains simplest examples of Green’s functions G(x,ξ) for some linear boundary value problems for ODEs. In all these examples, G(x,ξ)=G(ξ,x), and therefore the Green’s function is speciﬁed only in the domain x≤ξ. For equations with the operator L[y]=– [f(x)y/prime x]/primex,i ti sa s s u m e d thatf(x)>0a n d q(x)=⎝integraldisplayx 0dt f(t). 18.3-4. Nonlinear Problem of Nonisothermal Flow in Plane Channel. It is known that the dynamic viscosity of a ﬂuid µessentially depends on temperature T(µdecreases asTincreases) and the other physical parameters of the ﬂuid have small variation. For high-viscosity ﬂuids (like glycerol, liquid oil, or petroleum) it is common to assume the exponential dependence µ=µ0exp[–β(T–T0)], (16) where µ0,β,a n dT0are empirical constants. Stationary nonisothermal ﬂows of viscous incompressible ﬂuid are described by the following system of equations: 3⎝summationdisplay j=1⎝parenleftbigg uj∂ui ∂Xj–1 ρ∂pij ∂Xj⎝parenrightbigg =0 , pij=–pδij+µ⎝parenleftbigg∂ui ∂Xj+∂uj ∂Xi⎝parenrightbigg ,i=1 ,2 ,3 ; ( 1 7 ) ∂u1 ∂X 1+∂u2 ∂X 2+∂u3 ∂X 3= 0, (18) u1∂T ∂X 1+u2∂T ∂X 2+u3∂T ∂X 3=σ⎝parenleftbigg∂2T ∂X2 1+∂2T ∂X2 2+∂2T ∂X2 3⎝parenrightbigg . (19) Hereujare ﬂuid velocity components, Xjare Cartesian coordinates, ρis density, pis pressure, σis the heat transfer coefﬁcient, and δij=⎝braceleftbigg1f o r i=j, 0f o r i≠j. Stationary rectilinear ﬂows in a plane channel correspond to solutions of the form u1=u2=0 , u3=u(X),p=p(X,Z),T=T(X,Z), (20) where Z=X3is the longitudinal coordinate in the channel and X=X1is the transverse coordinate. 18.3. R EDUCTION OF BOUNDARY VALUE PROBLEMS FOR ODE ST O FREDHOLM INTEGRAL EQUATIONS 885 TABLE 13 Green’s function for some boundary value problems for linear ODEs L[y]=0 Differential operator, L[y] Boundary conditions Green’s function, G(x,ξ) –y/prime/prime xx y(0) =y(a)=0 x⎝parenleftBig 1–ξ a⎝parenrightBig –y/prime/prime xx y(0) =y/prime x(a)=0 ξ –y/prime/prime xx y/prime x(0) =y(a)=0 a–ξ –y/prime/prime xx–k2y y(0) =y(1) = 0 sin(kx)s i n [k(1 –ξ)] ksink –y/prime/prime xx+k2y y(0) =y(1) = 0 sinh(kx) sinh[ k(1 –ξ)] ksinhk –xy/prime/prime xx–y/prime x y(0)≠∞,y(a)=0 –l nξ a –(xy/prime x)/primex+n2 xy y(0)≠∞,y(a)=0 1 2n⎝parenleftBigx a⎝parenrightBign –(xξ)n 2na2n(n=1 ,2 , ...) –[f(x)y/prime x]/primex y(0) =y(a)=0 q(x)–q(x)q(ξ) q(a) –[f(x)y/prime x]/primex y(0) =y/prime x(a)=0 q(x) –[f(x)y/prime x]/primex y(0) = 0, ky/prime x(a)+y(a)=0 q(x)–f(a)q(x)q(ξ) f(a)q(a)+k(k>0 ) y/prime/prime/prime/prime xxxx y(0) =y/prime x(0) = 0, y(1) =y/prime x(1) = 0 ⎝parenleftbig1 2ξ–ξ2+1 2ξ3⎝parenrightbig x2–⎝parenleftbig1 6ξ–1 2ξ2+1 3ξ3⎝parenrightbig x3 y/prime/prime/prime/prime xxxx y(0) =y/prime x(0) = 0, y/prime/prime xx(1) =y/prime/prime/prime xxx(1) = 0 1 6x2(3ξ–x) Substituting expressions (20) into equations (17)–(19) and letting u/prime X=du/dX , we obtain the following three equations: ∂p ∂X=u/prime X∂µ ∂Z,∂p ∂Z=∂ ∂X(µu/prime X), (21) u(X)∂T ∂Z=σ⎝parenleftbigg∂2T ∂X2+∂2T ∂Z2⎝parenrightbigg . (22) Using differentiation, we eliminate the pressure pfrom (21) and obtain ⎝parenleftbigg∂2 ∂Z2–∂2 ∂X2⎝parenrightbigg (µu/prime X) = 0. (23) The general solution of equation (23) can be written in the form µu/prime X=Φ(Z+X)+Ψ(Z–X), (24) whereΦandΨare arbitrary functions. 886 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS LetX= 0 correspond to the middle line of the plane channel of width 2 h, i.e., – h≤X≤h.O n the walls we have the conditions of adhesion u=0 f o r X=±h. (25) Moreover, assume that the temperature has linear variation on the channel walls, T=T0–EZ forX=±h. (26) Instead of the domain – h≤X≤h, we can consider its half 0 ≤X≤hwith the symmetry condition on the middle line: u/prime X=TX=0 f o r X= 0. (27) Equations (22) and (24) (with the viscosity deﬁned by (16)) and the boundary conditions (25), (26) (for x=h), and (27) can be satisﬁed if one seeks a solution in the form Φ(ζ)=–AeβEζ,Ψ(ζ)=AeβEζ,u=u(X),T=T0–EZ+E σθ(X). (28) As a result, we obtain a system of ODEs, u/prime X=–A µ0exp⎝parenleftbiggβE σθ⎝parenrightbigg⎝bracketleftbig eβEX–e–βEX⎝bracketrightbig , u=–θ/prime/prime XX.(29) Eliminating ufrom these equations and taking into account (25)–(26), (28), we come to the following nonlinear boundary value problem for the excessive temperature: w/prime/prime/prime xxx=γewsinh(εx), (30) w/prime x=0 f o r x=0 , w=0 f o r x=1 , w/prime/prime xx=0 f o r x= 1, (31) where we have set x=X h,w=βEθ σ,γ=2Ah3βE σµ0,ε=βEh . The last boundary condition in (31) has been derived by passing to the limit for X→hin the second equation in (29) with the adhesion condition (25) on the walls taken into account. Note that the volume rate of ﬂow Qis calculated in terms of the heat ﬂow on the walls by the formula Q=– 2θ/prime X(h)=– 2σ(βEh )–1w/prime x(1). Let us prove that for sufﬁciently large γ> 0, the boundary value problem (30)–(31) has no solutions. It is not difﬁcult to show that the Green’s function for the linear boundary value problem (30)–(31) with γ= 0 (see Subsection 18.3-2) has the form G(x,ξ)=⎝braceleftbiggξ–1 2(x2+ξ2)f o r 0 ≤x≤ξ≤1, ξ–xξ for 0 ≤ξ≤x≤1.(32) Therefore, the nonlinear boundary value problem (30)–(31) is equivalent to the nonlinear integral equation w(x)=γ⎝integraldisplay1 0G(x,ξ)ew(ξ)sinh(εξ)dξ. (33) 18.4. R EDUCTION OF PDE SW I T H BOUNDARY CONDITIONS OF THE THIRD KIND TO INTEGRAL EQUATIONS 887 SinceG(x,ξ)≥0, it follows that for γ>0w eh a v e w(x) > 0. This inequality has a clear physical meaning: if the walls are cooled by the environment, th e temperature in the channel is larger than that of the walls. Consider an auxiliary linear boundary value problem for eigenvalues: y/prime/prime/prime xxx=–λsinh(εx)y, (34) y=0 f o r x=0 , y/prime/prime xx=0 f o r x=0 , y/prime x=0 f o r x= 1. (35) This problem is equivalent to the linear Fredholm integral equation y(x)=λ⎝integraldisplay1 0G(ξ,x)y(ξ)s i n h ( εξ)dξ. (36) Here the Green’s function G(ξ,x) corresponds to the transposition of the variables xandξin (32). Since the kernel of the integral operator (33) is positive, the generalized Jentzch theorem implies that the smallest eigenvalue is positive, λ0> 0, and the corresponding eigenfunction y0(x) does not change sign on the interval [0, 1]. Let us multiply both sides of equation (30) by y0(x) and integrate the resulting expression in xfrom 0 to 1. Taking into account the relations y0w/prime/prime/prime xxx=(y0w/prime/prime xx)/primex–(y/prime 0xw/prime x)/primex+(y/prime/prime 0xxw)/prime x–y/prime/prime/prime 0xxxw, y/prime/prime/prime 0xxx=–λ0sinh(εx)y0, and the boundary conditions (31) and (35) for the functions wandy0, we come to the relation λ0 γ=⎝integraldisplay1 0y0(ξ)ew(ξ)sinh(εξ)dξ ⎝integraldisplay1 0y0(ξ)w(ξ)s i n h ( εξ)dξ. (37) Sincew≥0, we have ew≥ew. This inequality, together with (37), implies the estimate λ0/γ≥e. Therefore, for γ>λ0/e, the boundary value problem (30)–(31) has no solutions, and for the critical value γ∗we have γ∗<λ0/e. Remark. It can be shown that for 0 < γ<γ∗, the boundary value problem (30)–(31) has two solutions (one stable and another unstable). For γ=γ∗, there is only one solution. References for Section 18.3: P. P. Zabreyko, A. I. Koshelev, et al. (1975), E. Kamke (1977), V . I. Naidenov and A. D. Polyanin (1990), R. P. Agarwal, D. O’Regan, and P. J. Y . Wong (1998), A. D. Polyanin and V . F. Zaitsev (2003). 18.4. Reduction of PDEs with Boundary Conditions of the Third Kind to Integral Equations 18.4-1. Usage of Particular Solutions of PDEs for the Construction of Other Solutions. Let L[w]=0 ( 1 ) be an arbitrary homogeneous linear partial differential equation of any order in the variables x,t with sufﬁciently smooth coefﬁcients ( tmay stand for the time or a spatial variable). 888 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS There is an effective way to construct solutions of this equation. Suppose that equation (1) has a particular solution ˜w(x,t;µ)( 2) depending on a parameter µ, and the coefﬁcients of the linear differential operator Lare independent ofµ. Multiplying the particular solution (2) by an arbitrary function ϕ(µ) and integrating the result with respect to µover some interval [ α,β], we obtain the new function w=⎝integraldisplayβ α˜w(x,t;µ)ϕ(µ)dµ,( 3) which is also a solution of the original homogeneous linear equation (1). Let us mention some useful facts applied for the construction of solutions of boundary value problems with the help of integral representations like (3). 1. The domain of integration in (3) usually coincides with the domain of one of the independent variables of the PDEs under consideration (in particular, if tis the time, one often takes α=0a n d β=∞). 2. Let ˜ w(x,t) be a particular solution of equation (1) with the coefﬁcients independent of t. Then, for any constant µ, the function ˜ w(x,t–µ) is also a solution of equation (1). 3. Suppose that the particular solution (2) satisﬁes one or several initial or boundary conditions of the form w=0 f o r t=0 , wt=0 f o r t=0 , w=0 f o r x=a,wx=0 f o r x=b.(4) Then the function (3) also satisﬁes equation (1) with the same initial or boundary conditions. Consider a boundary value problem for equation (1). Suppose that one of the boundary conditions has the form (generalized boundary conditions of the third kind) wx=F(t,w)f o rx =0 , ( 5 ) and the other boundary and initial conditions are homogeneous and have the form (4). Suppose that the special solution (2) of equation (1) satisﬁes all the homogeneous initial and boundary conditions except (5). Let us seek a solution of the corresponding boundary value problem in the form of the integral (3). Substituting this integral into the boundary condition (5), we obtain an integral equation for the function ϕ(µ). It is important to make a proper choice of the particular solution (2). As an illustration, consider the following example. 18.4-2. Mass Transfer to a Particle in Fluid Flow Complicated by a Surface Reaction. Consider steady-state diffusion to a particle in laminar viscous incompressible ﬂuid ﬂow. Assumethat on the surface of the particle a chemical reaction occurs with rate F ∗(C), where Cis the mass concentration of a reactant. In particular, for a reaction of order nwe have F∗(C)=KCn,( 6) where Kis the reaction rate coefﬁcient. It is assumed that the velocity ﬁeld in the ﬂuid is known from the solution of the corresponding hydrodynamic problem and can be speciﬁed in terms of a ﬂow function ψ(a ﬂow function can be introduced, for instance, for plane and axisymmetric ﬂows). In the diffusion boundary layer approximation, the dimensionless equation of stationary convective diffusion and the boundary 18.4. R EDUCTION OF PDE SW I T H BOUNDARY CONDITIONS OF THE THIRD KIND TO INTEGRAL EQUATIONS 889 conditions in curvilinear orthogonal coordinates ξ,η,ζassociated with the body surface ξ=ξsand the lines of ﬂow have the form 1 √ gs⎝parenleftbigg∂ψ ∂ξ∂w ∂η–∂ψ ∂η∂w ∂ξ⎝parenrightbigg =1 Pe∂2w ∂ξ2,( 7) –∂w ∂ξ=F(w)f o r ξ=ξs,w→0f o r ξ→∞ ,( 8) where ψ=(ξ–ξs)mf(η),gs=gs(η)=(gξξgηηgζζ)|ξ=ξs, w=C∞–C C∞,P e =aU D,F(w)=aF∗(C) DC∞, ais the characteristic size of the particle (radius), C∞is concentration far away from the particle, U is the characteristic ﬂow velocity (far away from the particle), Dis the diffusion coefﬁcient, Pe is the Peclet number, and gξξ,gηη,gζζare the metric tensor components; the value m= 1 corresponds to drops or bubbles and m= 2 corresponds to solid particles. In the problem stated in terms of (7)–(8), the boundary condition for η= 0 has been dropped [for f(0) = 0, one imposes the condition that the solution is bounded for η= 0]. When writing equation (7) and the ﬁrst boundary condition in (8), it has been assumed that the coordinate ξnear the surface ξ=ξsis chosen such that the difference ξ–ξsdetermines the distance between the point ( ξs,η) on the surface of the body and the point ( ξ,η) in the ﬂow (i.e., it is assumed that gξξ|ξ=ξs=1 ) . For a reaction of order n(6), the dimensionless rate of surface chemical reaction is described by the expression F(w)=k(1 –w)n,k=aKCn–1 ∞/D.( 9 ) Further, it is assumed that the domain under consideration is speciﬁed by the inequalities ξs≤ξ<∞,0 ≤η≤η0, and also that the inequality f(η) > 0 holds for 0 < η<η0,a n df(0)≥0. Introducing the new variables t=t(η)=1 n⎝integraldisplayη 0f1/n(η)[gs(η)]1/2dη,x=2 n+1Pe1/2ψ(n+1)/(2n), (10) we reduce (7)–(8) to the following boundary value problem for the unknown function w(x,t): ∂w ∂t=∂2w ∂x2+1–2ν x∂w ∂x, (11) w=0 f o r t=0 , w→0a s x→∞ , (12) x1–2ν∂w ∂x+P e–ν(2ν)1–2νhν(t)F(w)=0 f o r x= 0, (13) where ν=(n+1 )–1,hν(t)=f–1/n(η(t)). The function hν(t) in the boundary condition on the particle surface (13) is found from the parametric relations hν=f–1/n(η),t=t(η) (see the ﬁrst formula in (10)). Simple veriﬁcation shows that equation (11) admits the particular solution ˜w(x,t;µ)=⎧ ⎨ ⎩A(t–µ)ν–1exp⎝bracketleftbigg –x2 4(t–µ)⎝bracketrightbigg fort>µ, 0f ort≤µ.(14) Note that for ν=1 2equation (11) turns into the classical heat transfer equation. In this case, the function (14) for µ=0a n d A=1 2π–1/2coincides with the fundamental solution of the heat equation. 890 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS Let us seek a solution of problem (11)–(13) in the form of the integral (3) (for α=0a n d β=∞), into which the function (14) should be inserted. As a result, we obtain w(x,t)=22ν–1 Γ(1 –ν)⎝integraldisplayt 0(t–µ)ν–1exp⎝bracketleftbigg –x2 4(t–µ)⎝bracketrightbigg ϕ(µ)dµ, (15) whereΓ(ν) is the gamma function (here, for the sake of deﬁniteness, we have taken a speciﬁc value of the constant A). Obviously, the function (15) satisﬁes the initial and the boundary conditions (12). As shown by Sutton (1943), the function (15) has the following limit properties: lim x→0w=22ν–1 Γ(1 –ν)⎝integraldisplayt 0(t–µ)ν–1ϕ(µ)dµ, lim x→0⎝parenleftBig x1–2ν∂w ∂x⎝parenrightBig =–ϕ(t). (16) Substituting (15) into the boundary condition (13) and taking into account (16), we come to the integral equation for the function ϕ(t): ϕ(t)=P e–ν(2ν)1–2νhν(t)F⎝parenleftbigg22ν–1 Γ(1 –ν)⎝integraldisplayt 0(t–µ)ν–1ϕ(µ)dµ⎝parenrightbigg . (17) The replacement ϕ(t)=P e–ν(2ν)1–2νhν(t)F⎝parenleftbig ψ(t)⎝parenrightbig reduces equation (17) to a more common form (it is assumed that the function Fis invertible) ψ(t)=ν2ν–1Pe–ν Γ(1 –ν)⎝integraldisplayt 0(t–µ)ν–1hν(µ)F⎝parenleftbig ψ(µ)⎝parenrightbig dµ. (18) After solving equation (17) or (18), formulas (10), ( 15) can be used to obtain the distribution of concentration in the diffusive boundary layer of the particle. 18.4-3. Integral Equations for Surface Concentration and Diffusion Flux. Instead of equation (17) for the function ϕ, it is convenient to consider directly the equations for surface concentration or local diffusion ﬂux—the quantities with a clear physical meaning (in most practical problems these two are the desired quantities). In view of (10) and (15), surface concentration is determined by the expression ws=ws(t)≡w(0,t)=22ν–1 Γ(1 –ν)⎝integraldisplayt 0(t–µ)ν–1ϕ(µ)dµ=Λν(0,t)∗ϕ(t). (19) Note that the operator Λν(0,t) coincides, to within a constant coefﬁcient, with an integral of fractional orderν(see Subsection 10.5-1). Applying the operator Λν(0,t) to both terms of equation (17) and using (17), we obtain the equation for surface concentration: ws=P e–ν(2ν)1–2νΛν(0,t)∗[hn(t)F(ws)],ws=ws(t). (20) Applying the inverse operator Λ–1 ν(0,t) to both sides of this equation, we come to the following equivalent equation: j∞(t)tνΞν(t)∗ws(t)=F(ws(t)),Ξν(t)≡Γ(ν)22ν–1Λ–1 ν(0,t); (21) Ξν(t)∗z(t)≡d dt⎝integraldisplayt 0z(λ)(t–λ)–νdλ=z(0) tν+⎝integraldisplayt 0dz(λ) dλ(t–λ)–νdλ, 18.4. R EDUCTION OF PDE SW I T H BOUNDARY CONDITIONS OF THE THIRD KIND TO INTEGRAL EQUATIONS 891 where j∞=j∞(t) is the local diffusion ﬂux corresponding to the diffusion mode of the reaction on the surface (i.e., the boundary condition w=1f o r ξ=ξs): j∞(t)=ν2ν–1[Γ(ν)]–1Peνh–1 n(t)t–ν=ν2ν–1[Γ(ν)]–1Peνf1/n(η)t–ν(η). (22) From equation (20), combined with the formula j=F(ws) (23) and the identity Γ(ν)Γ(1 –ν)=π/sin(πν), we obtain a relation between surface concentration and local diffusion ﬂux: ws(t)=sin(πν) π⎝integraldisplayt 0j(λ) j∞(λ)λ–ν(t–λ)ν–1dλ. (24) Substituting (24) into the right-hand side of (23), we obtain an integral equation for local diffusion ﬂux on the particle surface: j=F⎝parenleftbiggsin(πν) π⎝integraldisplayt 0j(λ) j∞(λ)λ–ν(t–λ)ν–1dλ⎝parenrightbigg . (25) It is not difﬁcult to show that if the limit local diffusion ﬂux on a part of the body is constant, j∞(t)=j∞= const (0 ≤t≤t0), (26) then the solution of the nonlinear integral equation (25) reduces to the solution of the algebraic (transcendental) equation j=F(j/j∞)( 0 ≤t≤t0). (27) In view of (23), (27), surface concentration, under the condition (26), is also determined by solving an algebraic equation, ws=j–1 ∞F(ws). (28) In the general case, for j∞=j∞(t)≠0, it is impossible to obtain an exact analytical solution of integral equations for surface concentration and local diffusion ﬂux (21)–(22) and (25). Therefore, one has to resort to the methods of numerical or approximate integration of these equations. In engineering,approximations of surface concentration and local ﬂux are sometimes constructed by the method of equidistant surface . The essence of this method can be described as follows. First, formula (22) is used to determine the limit local diffusion ﬂux j=j∞(t), and then this expression is inserted into equations (27) and (28), i.e., instead of the original integral equations (21) and (25), one solves algebraic (transcendental) equations. Comparison of the approximate results obtained by this method with those of numerical analysis for many typical cases shows that the method of equidistant surface is fairly accurate (for relatively simple reactions, the error does not exceed 20%; see the references at the end of this section). Therefore, when using iteration methods for solving integralequations, it is reasonable to take a solution obtained by the said method as the initial approximation. 18.4-4. Method of Numerical Integration of the Equation for Surface Concentration. Consider more closely a method of numerical inte gration of the equation for surface concentration (21)–(22); local diffusion ﬂux in this case is found with the help of (23). Let us represent equation (21) in the form j∞(t)⎝bracketleftbigg ws(0) +tν⎝integraldisplayt 0dws(λ) dλ(t–λ)–νdλ⎝bracketrightbigg =F(ws(t)). (29) 892 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS Numerical integration of equation (29) is carried out as follows. First, the segment [0, t0], t0=t(η0), is split into Mequal parts [(i–1 )∆t,i∆t]( i=1 ,...,M) of length ∆t=t0/Mand equation (29) is transformed to ws(0) + ( m∆t)νm⎝summationdisplay i=1⎝integraldisplayi∆t (i–1)∆tdws(λ) dλ(m∆t–λ)–νdλ=F(ws) j∞(m∆t). (30) Then, the derivative dws/dλshould be approximated on every segment (i –1 )∆t≤λ≤i∆tby the expression dws(λ) dλ=1 ∆t⎝bracketleftbig ws(i∆t)–ws((i–1 )∆t)⎝bracketrightbig , and then one integrates equation (30). After suitable transformations, one obtains 1–ν mνws(0) –ws(0)⎝braceleftbig m1–ν–(m–1 )1–ν⎝bracerightbig +m–1⎝summationdisplay i=1ws(i∆t)Am–i =–ws(m∆t)+1–ν mνF(ws(m∆t)) j∞(m∆t),m=1 ,...,M– 1, (31) ν=1 n+1,Am=– 2m1–ν+(m+1 )1–ν+(m–1 )1–ν. The algebraic equation (31) is solved for m=1 ,2 ,3 , ...with respect to surface concentration ws(m∆t), which corresponds to t=m∆t. The left-hand side of (31) contains the values of ws(i∆t) for 1 ≤i≤m– 1, which have already been calculated, while the right-hand side of (31) contains the unknown quantity ws(m∆t). Numerical integration of equatio n (31) starts with preliminary determination of surface concentration ws(0) at the point of diffusion boundary layer initiation (for t= 0) by solving the auxiliary equation ws(0) =j–1 ∞(0)F(ws(0)), (32) which coincides with (28) for t=0 .I f j∞(0) =∞,t h e n ws(0) = 0 (this situation occurs at the front critical point of a plate streamlined by ﬂuid). Then the solution procedure goes on in successive order for m=1 ,2 ,3 ,... , and this process is direct in the sense that no repeated calculations are needed. Naturally, the precision of these calculations depends on the value of ∆t. Remark. The algebraic (transcendental) equation (32) may have several roots, depending on the structure of the function F(w) (thus, there may exist several stationary regimes of reaction on the particle surface). In this case, one has to examine the stability of the solutions. References for Section 18.4: W. G. L. Sutton (1943), A. Acrivos and P. L. Shambre (1957), A. D. Polyanin and Yu. A. Sergeev (1980), D. A. Frank-Kamenetskii (1987), Yu. P. Gupalo, A. D. Polyanin, and Yu. S. Ryazantsev (1985). 18.5. Representation of Linear Boundary Value Problems in Terms of Potentials 18.5-1. Basic Types of Potentials for the Laplace Equation and Their Properties. 1◦.L e t Sbe a smooth closed surface in the n-dimensional Euclidean space Rn(n≥2) that coincides with the boundary of a ﬁnite domain G=G+,a n dl e t G–be the exterior inﬁnite domain (G+∪S∪G–=Rn). 18.5. R EPRESENTATION OF LINEAR BOUNDARY VALUE PROBLEMS IN TERMS OF POTENTIALS 893 Consider the n-dimensional Laplace equation ∆w≡n⎝summationdisplay k=1∂2w ∂x2 k=0 . ( 1 ) The fundamental solution of equation (1) has the form (x,y)= (|x–y|)=⎧ ⎪⎪⎨ ⎪⎪⎩1 Ωn(n–2 )1 |x–y|n–1ifn≥3, 1 2πln1 |x–y|ifn=2 ,(2) where |x–y|=⎝bracketleftbiggn⎝summationdisplay k=1(xk–yk)2⎝bracketrightbigg1/2 ,Ωn=2πn/2 Γ(n/2), |x–y|is the distance between points x=(x1,...,xn)a n dy=(y1,...,yn),Ωnis the area of the unit sphere in Rn,a n dΓ(z) is the gamma function. Three integrals depending on xas a parameter deﬁne different potentials: V(x)=⎝integraldisplay Sµ(y)(x,y)dSy (single layer potential ), W(x)=⎝integraldisplay Sν(y)∂ ∂ny(x,y)dSy(double layer potential ), Z(x)=⎝integraldisplay Gρ(y)(x,y)dy (volume potential ).(3) Herenyis the direction of the outward (with respect to G+) normal to the surface Sat the point y∈S. The functions µ(y),ν(y), and ρ(y) are called densities of the respective potentials. In what follows, these densities are always assumed absolutely integrable on SorG. 2◦.L e tµ(y)∈C1(S). The single layer potential V(x)i sa harmonic function [i.e., a function satisfying the Laplace equation (1)] for x∉S,a n d lim |x|→∞V(x) (x,0 )=M1,M1=⎝integraldisplay Sµ(y)dSy; in particular, lim |x|→∞V(x)=0f o r n≥3, but lim |x|→∞V(x)=0f o r n= 2, if and only if⎝integraltext Sµ(y)dSy=0 . The single layer potential is continuous everywhere in Rn. Moreover, V(x) and its tangential derivatives are continuous across the surface S. The normal derivative of the single layer potential has a jump across the surface S: ⎝parenleftbigg∂V ∂nx⎝parenrightbigg+ =1 2µ(x)+∂V ∂nx,⎝parenleftbigg∂V ∂nx⎝parenrightbigg– =–1 2µ(x)+∂V ∂nx.( 4) Here the superscripts + and – in the left-hand sides mark the limit values of the normal derivatives from the direction of G+andG–, respectively, i.e., ⎝parenleftbigg∂V ∂nx⎝parenrightbigg+ = lim x/prime→x,x/prime∈G+∂V ∂nx,⎝parenleftbigg∂V ∂nx⎝parenrightbigg– = lim x/prime→x,x/prime∈G–∂V ∂nx. 894 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS And in the right-hand sides of (4), the normal derivative is calculated directly on the surface S,i . e . , ∂V ∂nx=⎝integraldisplay Sµ(y)∂ ∂nx(x,y)dSy,x∈S, which is a continuous function of x∈S, and the kernel has a weak singularity on S:⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle∂ ∂nx(x,y)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle≤const |x–y|n–2,x,y∈S. 3◦.L e tν(y)∈C1(S). The double layer potential W(x) is a harmonic function of x∉Sand lim |x|→∞Ωn|x|n–1W(x)=M2,M2=⎝integraldisplay Sν(y)dSy. Across the surface S, the double layer potential has a jump: W+(x)=–1 2ν(x)+W(x),W–(x)=1 2ν(x)+W(x),x∈S,( 5) where W+(x)a n dW–(x) are the limit values of the double layer potential in the directions from G+ andG–,i . e . , W+(x) = lim x/prime→x,x/prime∈G+W(x/prime),W–(x) = lim x/prime→x,x/prime∈G–W(x/prime). The right-hand sides of (5) involve the direct value of the double layer potential on the surface S, W(x)=⎝integraldisplay Sν(y)∂ ∂ny(x,y)dSy,x∈S, which is a continuous function of x∈S, and the kernel has a weak singularity on S:⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle∂ ∂ny(x,y)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle≤const |x–y|n–2,x,y∈S. The tangential derivatives of the double layer potential also have a jump across the surface S, but its normal derivative preserves its value across S:⎝parenleftbigg∂W ∂nx⎝parenrightbigg+ =⎝parenleftbigg∂W ∂nx⎝parenrightbigg– ,x∈S. In the case of constant density ν=1 ,t h e Gauss formula –⎝integraldisplay S∂ ∂ny(x,y)dSy=q(x)≡⎝braceleftBigg1i f x∈G+, 1/2i f x∈S, 0i f x∈G–(6) holds for the double layer potential. The integral on the left is interpreted as the solid angle, divided byΩn(n– 2), under which the surface Sis seen from the point x. 4◦.F o r ρ(y)∈C1(G∪S), the volume potential and its ﬁrst-order derivatives are continuous everywhere in Rnand can be calculated by the differentiation under the sign of the integral. Thus, Z∈C1(Rn). Moreover, lim |x|→∞Z(x) (x,0 )=M3,M3=⎝integraldisplay Gρ(y)dy. Its second-order derivatives are continuous outside S, but have a jump on S. In the interior domain G+the Poisson equation holds ∆Z=ρ(x), x∈G+, and in the exterior domain G–the volume potential sa tisﬁes the Laplace equation ∆Z=0 , x∈G–. For a ﬁnite domain G1inRnwith the boundary S1=∂G 1of class C1,t h e Gauss formula for the volume potential holds:⎝integraldisplay S1∂Z ∂nxdS1x=–⎝integraldisplay G∩G1ρ(y)dy. The integration in the ﬁrst integral is over the variable x. 18.5. R EPRESENTATION OF LINEAR BOUNDARY VALUE PROBLEMS IN TERMS OF POTENTIALS 895 18.5-2. Integral Identities. Green’s Formula. LetΦ(x) be a function of class C2(G∪S), where Sis a surface of class C2. Then the following integral identity, called the Green’s formula , holds: –⎝integraldisplay G∆Φ(y)(x,y)dy+⎝integraldisplay S⎝bracketleftbigg∂Φ(y) ∂ny(x,y)–Φ(y)∂ ∂ny(x,y)⎝bracketrightbigg dSy=q(x)Φ(x). (7) Hereq(x) is the function deﬁned by (6). Formula (7) implies that in the domain Gthe function Φ(x) can be represented as the sum of a single layer potential, a double layer potential, and a volume potential with the respective densities µ(y)=∂Φ(y) ∂ny,ν(y)=–Φ(y),ρ(y)=–∆Φ(y). For a function u(x) which is harmonic in the domain Gand belongs to the class C1(G∪S), the following identity holds: ⎝integraldisplay S⎝bracketleftbigg∂w(y) ∂ny(x,y)–w(y)∂ ∂ny(x,y)⎝bracketrightbigg dSy=q(x)w(x), (8) and thus, w(x) can be represented in Gas the sum of a single layer potential and a double layer potential with the respective densities µ(y)=∂w(y) ∂ny,ν(y)=–w(y). However, the densities in (8) cannot be chosen arbitrary on S, because they are related by the integral identity obtained from (8) for x∈G+. 18.5-3. Reduction of Interior Dirichlet and Neumann Problems to Integral Equations. 1◦.Interior Dirichlet problem (ﬁrst boundary value problem ): ﬁnd a function w(x) that satisﬁes equation (1) in G+and the boundary condition w(x)=ϕ+(x)f o rx ∈S,( 9) where ϕ+(x) is a given continuous function on S. Problem (1), (9) has a solution.* This solution is unique and can be represented in the form of the double layer potential w(x)=⎝integraldisplay Sν(y)∂ ∂ny(x,y)dSy with density ν(y) which is found as the unique solution of the following Fredholm integral equation of the second kind: –1 2ν(x)+⎝integraldisplay Sν(y)∂ ∂ny(x,y)dSy=ϕ+(x), x∈S. 2◦.Interior Neumann problem (second boundary value problem ): ﬁnd a function w(x) that satisﬁes equation (1) in G+and the boundary condition ∂w(x) ∂nx=ψ+(x)f o r x∈S, (10) * In Subsections 18.5-3 and 18.5-4 it is assumed that the surface Sis sufﬁciently smooth. 896 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS where ψ+(x) is a given continuous function on S. This problem has a solution if and only if the function ψ+(x) satisﬁes the compatibility condition ⎝integraldisplay Sψ+(x)dSx= 0. (11) A solution of problem (1), (10) with the conditio n (11) is deﬁned to within an additive constant, w(x)=V(x)+C, where V(x)=⎝integraldisplay Sµ(y)(x,y)dSy is the single layer potential with density µfound by solving the Fredholm integral equation of the second kind 1 2µ(x)+⎝integraldisplay Sµ(y)∂ ∂nx(x,y)dSy=ψ+(x), x∈S. (12) The corresponding homogeneous equation (with ψ+(x) = 0) has a nontrivial solution µ0(x), and the nonhomogeneous equation (12) has a solution if the condition (11) is satisﬁed. The general solutionof equation (12) has the form µ(x)+Aµ 0(x), where Ais an arbitrary constant. 18.5-4. Reduction of Exterior Dirichlet and Neumann Problems to Integral Equations. 1◦.Exterior Dirichlet problem (ﬁrst boundary value problem ): ﬁnd a function w(x) that satisﬁes equation (1) in G–(0∉G–) and the boundary condition w(x)=ϕ–(x)f o rx ∈S, (13) where ϕ–(x) is a given continuous function on S, and it is also required that the following regularity condition holds at inﬁnity: lim |x|→∞|x|n–2w(x) = const . (14) This problem has a solution; this solution is unique and can be represented in the form w(x)=W(x)+A |x|n–2,W(x)=⎝integraldisplay Sν(y)∂ ∂ny(x,y)dSy, where Ais a constant, W(x) is a double layer potential, ν(y) its density, which is found by solving the Fredholm integral equation of the second kind: 1 2ν(x)+⎝integraldisplay Sν(y)∂ ∂ny(x,y)dSy=ϕ–(x)–A |x|n–2,x∈S. (15) The corresponding homogeneous equation has the nontrivial solution ⎝tildewideν0= 1. For suitable A,t h e solution of the nonhomogeneous equation (15) has the form ν(y)=ν–(y)+C, where Cis an arbitrary constant, ν–(y) is a particular solution of equation (15). The constant Ais chosen of the form A=–⎝integraldisplay Sϕ–(x)ν0(x)dSx, 18.5. R EPRESENTATION OF LINEAR BOUNDARY VALUE PROBLEMS IN TERMS OF POTENTIALS 897 where the auxiliary density ν0(x) should satisfy the normalization condition⎝integraldisplay Sν0(y) |y|n–2dSy= 1. (16) The density ν0(x) is a nontrivial solution of the integral equation (12) for the interior Neumann problem with the Neumann boundary values ψ+(x)=0 , x∈S, and this density satisﬁes the following normalization conditio n equivalent to (16) for n≥3: V0(x)≡⎝integraldisplay Sν0(y)(x,y)dSy=1 , x∈G+∪S. The single layer potential V0(x) with density ν0(x)i sc a l l e dt h e equilibrium potential or the Roben potential . The density ν0(x) yields the solution of the Roben electrostatic problem for charge distribution in a conductor Sthat produces an equilibrium potential which is constant in the domain G+. A certain complexity of the solution of the external Dirichlet problem is due to the fact that a harmonic function w(x) satisfying the regularity condition at inﬁnity generally has a slower decay rate (as |x|→∞ ) than the double layer potential. Therefore, in the general case, w(x) cannot be represented merely in terms of the double layer potential. 2◦.External Neumann problem (second boundary value problem ): ﬁnd a function w(x) that satisﬁes equation (1) in G–(0∉G–) and the boundary condition ∂w(x) ∂nx=ψ–(x)f o r x∈S, where ψ–(x) is a given continuous function on S, and it is also required that the regularity condition (14) hold at inﬁnity. Forn≥3, a solution of this problem exists and is unique. For n= 2, a solution exists if and only if the function ψ–(x) satisﬁes the compatibility condition⎝integraldisplay Sψ–(x)dSx= 0; (17) and the solution is deﬁned to within an arbitrary additive constant. The solution of the external Neumann problem can be represented as the single layer potential w(x)=⎝integraldisplay Sµ(y)(x,y)dSy whose density µ(y) is determined by solving the Fredholm integral equation of the second kind: –1 2µ(x)+⎝integraldisplay Sµ(y)∂ ∂nx(x,y)dSy=ψ–(x), x∈S. (18) Forn≥3, this equation has one and only one solution. For n= 2, the corresponding homoge- neous integral equation (with ψ–(x) = 0) admits the nontrivial solution µ0(x), and therefore, the nonhomogeneous equation (18), with the solvability condition (17), has a unique solution ⎝tildewideµ(x)s u c h that ⎝integraldisplay S⎝tildewideµ(x)dSx=0 , and its general solution has the form µ(x)=⎝tildewideµ(x)+cµ0(x), where cis an arbitrary constant. Remark. In a similar way, potentials can be introduced for the heat equation and other equations of mathematical physics. These potentials can also be used for the reduction of the corresponding stationary and nonstationary linear problems to integral equations. References for Section 18.5: S. G. Mikhlin (1967), P. P. Zabreyko, A. I. Koshelev et al. (1975), R. Courant and D. Hilbert (1989), A. N. Tikhonov and A. A. Samarskii (1990), I. G. Petrovsky (1991), R. B. Guenther and J. W. Lee (1996), W. McLean (2000). 898 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS 18.6. Representation of Solutions of Nonlinear PDEs in Terms of Solutions of Linear Integral Equations (Inverse Scattering) 18.6-1. Description of the Zakharov–Shabat Method. Solutions of some nonlinear PDEs can be expressed through solutions of linear integral equations. Below we outline an approach based on the application of linear integral equations of the form* K(x,y)=F(x,y)+⎝integraldisplay∞ xK(x,z)N(x;z,y)dz,y≥x,( 1) where the functions F,N,a n d Kmay depend on some additional parameters other than the speciﬁed arguments. In each speciﬁc case, the function Nis explicitly expressed through F,a n d both functions FandNare solutions of some linear PDEs. Deﬁne an operator A xby Axf(y)=⎝braceleftbigg⎝integraltext∞ xf(z)N(x;z,y)dz ify≥x, 0i fy<x and assume that for each chosen N, it is possible to prove that the operator I – A xis invertible and its inverse, (I – A x)–1, is continuous, where I is the identity operator. The following three steps represent an algorithm for ﬁnding a nonlinear equation that can then be solved by the inversescattering method. 1 ◦. A speciﬁc structure is chosen for the integral equation (1). To that end, one prescribes a relation between the functions NandF(Nis expressed through F). 2◦. Two suitable linear differential (ordinary or partial) equations are introduced for the function F: LmF=0 , m=1 , 2 . ( 2 ) 3◦. The function Kis related to Fby equation (1), which can be rewritten as (I – A x)K=F.( 3) Applying the operators L minvolved in (2) to equation (3), we obtain Lm(I – A x)K=0 , m=1 ,2 . This equation can be rewritten in the form (I – A x)(LmK)=R m,m=1 ,2 , where R mcontains all nonzero terms of the commutator [L m,( I–A x)]. Moreover, (1) and (2) should be chosen so that R mcould be represented in the form Rm=( I–A x)Mm(K),m=1 ,2 , where M m(K) is a nonlinear functional of K. But the operator I – A xis invertible, and therefore, the function Ksatisﬁes the nonlin ear differential equations LmK–Mm(K)=0 , m=1 ,2 . ( 4 ) It follows that each solution of the linear integral equation (1) is a solution of nonlinear differential equations (4). Of most interest, as a rule, are special cases of one of the equations in (4) or equationsderived from (4). Remark. The ﬁrst two steps of the algorithm are fundamental and most difﬁcult. Linear differ- ential equations (2) usually correspond to a linear eigenvalue problem (for m= 1) and a problem of time-evolution of eigenfunctions (for m=2 ) . * Such equations are called integral equations of the Gel’fand–Levitan–Marchenko type. 18.6. R EPRESENTATION OF SOLUTIONS OF NONLINEAR PDE SV I A SOLUTIONS OF LINEAR INTEGRAL EQUATIONS 899 18.6-2. Korteweg–de Vries Equation and Other Nonlinear Equations. To clarify basic features of the above algorithm, consider some examples. Example 1. Let us consider the integral equation K(x,y)=F(x,y)+⎝integraldisplay∞ xK(x,z)F(z,y)dz (5) and write out some identities to be used in the sequel, ∂n x⎝integraldisplay∞ xK(x,z)F(z,y)dz=⎝integraldisplay∞ xF(z,y)∂n xK(x,z)dz+An,( 6) ⎝integraldisplay∞ xK(x,z)∂n xF(z,y)dz= (–1)n⎝integraldisplay∞ xF(z,y)∂n zK(x,z)dz+Bn,( 7) where Anare deﬁned by the recurrence relations A1=–K(x,x)F(x,y),An=(An–1)x–F(x,y)[∂n–1 xK(x,z)]z=x, and B1=–K(x,x)F(x,y),B2=–K(x,x)∂xF(x,y)+[∂zK(x,z)]z=xF(x,y),... Let us introduce an operator L1and require that Fsatisfy the linear equation L1F≡(∂2 x–∂2 y)F(x,y)=0 . (8) Applying the operator L1to (5) and taking into account (6), (7), we obtain (∂2 x–∂2 y)K(x,y)=⎝integraldisplay∞ xF(x,z)(∂2 x–∂2 y)K(x,z)dz–2F(x,y)d dxK(x,x). Using the equation F=( I–A x)Kand taking into account that the operator I – A xis invertible, we ﬁnally get (∂2 x–∂2 y)K(x,y)+u(x)K(x,y)=0 , (9) where the function u(x)i sd e ﬁ n e db y u(x)=2d dxK(x,x). (10) Let us require that Fsatisfy the linear equation L2F=(∂t+(∂x+∂y)3)F=0 (11) and apply the operator L2to (5). We thus obtain ⎝parenleftbig ∂t+(∂x+∂y)3⎝parenrightbig K(x,y)=⎝parenleftbig ∂t+(∂x+∂y)3⎝parenrightbig⎝integraldisplay∞ xK(x,z)F(z,y)dz. A procedure similar to the above calculations for the operator L1yields Kt+(∂x+∂y)3K+3u(∂x+∂y)K=0 . (12) For the characteristic y=x, equation (12) can be rewritten in terms of u=2 (d/dx )K(x,x). Differentiating (12) with respect toxand rearranging terms, we arrive at the Korteweg–de Vries equation ut+6uux+uxxx =0 . Any function Fsatisfying the linear equations (8), (11) and rapidly decaying as x→+∞generates a solution of the Korteweg–de Vries equation. To this end, one should solve the linear integral equation (5) for the function Kand express u through Kby (10). Example 2. Consider the integral equation K(x,y)=F(x,y)+σ 4⎝integraldisplay∞ x⎝integraldisplay∞ xK(x,z)F(z,u)F(u,y)dz du , (13) where σ=±1. Here and in what follows, the coefﬁcients are chosen with a view to simplifying the calculations. Let the operator L1have the form L1F=(∂x–∂y)F=0 , (14) 900 APPLICATION OF INTEGRAL EQUATIONS FOR THE INVESTIGATION OF DIFFERENTIAL EQUATIONS which implies that F(x,y)=F⎝parenleftBigx+y 2⎝parenrightBig . Shifting the lower limit of integration to zero, we rewrite equation (13) in the form K(x,y)=F⎝parenleftBigx+y 2⎝parenrightBig +σ 4⎝integraldisplay∞ 0⎝integraldisplay∞ 0K(x,x+ζ)F⎝parenleftBig2x+ζ+η 2⎝parenrightBig F⎝parenleftBigx+η+y 2⎝parenrightBig dζ dη , (15) or, equivalently, [(I –σAx)K](x,y)=F⎝parenleftBigx+y 2⎝parenrightBig , where the operator A xis deﬁned by Axf(y)=1 4⎝integraldisplay∞ 0⎝integraldisplay∞ 0f(ζ)F⎝parenleftBig2x+ζ+η 2⎝parenrightBig F⎝parenleftBigx+η+y 2⎝parenrightBig dζ dη . Introducing the function K2(x,z)=⎝integraldisplay∞ 0K(x,x+ζ)F⎝parenleftBigx+ζ+z 2⎝parenrightBig dζ, (16) we can rewrite equation (13) as K(x,y)=F⎝parenleftBigx+y 2⎝parenrightBig +σ 4⎝integraldisplay∞ 0K2(x,x+η)F⎝parenleftBigx+η+y 2⎝parenrightBig dη. (17) Applying the operator L1of (14) to equation (17), and the operator ∂x+∂zto (16), and taking into account the invertibility of I –σAx, we ﬁnd, after appropriate calculations, that (∂x+∂y)K2(x,y)=– 2K(x,x)K(x,y), (18) (∂x–∂y)K(x,y)=–σ 2K(x,x)K2(x,y). (19) Applying the operator ∂x+∂yto (15), we get F/prime⎝parenleftBigx+y 2⎝parenrightBig =( I– σAx)⎝bracketleftBig (∂x+∂y)K(x,y)+σ 2K2(x,x)K(x,y)⎝bracketrightBig . (20) Let us require that the function Fsatisfy the second linear equation L2F=(∂t+(∂x+∂y)3)F=0 . (21) Applying the operator L2to equation (15) and taking into account the above auxiliary relations (18)–(20), we ultimately ﬁnd that [∂t+(∂x+∂y)3]K(x,y)=3σK(x,x)K(x,y)∂xK(x,x)+3σK2(x,x)(∂x+∂y)K(x,y) (22) fory≥x. Now, by setting q(x,t)=K(x,x;t), we rewrite equation (22), for y=x, in terms of the dependent variable qto obtain the modiﬁed Korteweg–de Vries equation qt+qxxx =6σq2qx. (23) Thus, each solution of the equations L iF=0 ,i= 1, 2, with a sufﬁciently fast decay rate as x→∞ determines a solution of equation (23). Note that we have to solve the linear integral equation (13) at an intermediate step. Example 3. Consider the Boussinesq equation wtt+(wwx)x+wxxxx =0 . This equation arises in several physical applications: propagation of long waves in shallow water, one-dimensional nonlinear lattice-waves, vibrations in a nonlinear string, and ion sound waves in plasma. It can be shown that any rapidly decaying (as x→+∞) function F=F(x,y;t), which simultaneously satisﬁes the following two linear partial differential equations: Ft+√ 3(Fxx–Fyy)=0 , Fxxx+Fyyy=0 , generates a solution w=1 2d dxK(x,x;t) of the Boussinesq equation, where K(x,y;t) is a solution of the linear Gel’fand–Levitan–Marchenko integral equation K(x,y;t)+F(x,y;t)+⎝integraldisplay∞ xK(x,s;t)F(s,y;t)ds=0 . Timetappears here as a parameter. 18.6. R EPRESENTATION OF SOLUTIONS OF NONLINEAR PDE SV I A SOLUTIONS OF LINEAR INTEGRAL EQUATIONS 901 Example 4. Consider the Kadomtsev–Petviashvili equation (wt+wxxx–6wwx)x+3awyy=0 . It can be shown that any rapidly decaying (as x→+∞) function F=F(x,z;y,t), which simultaneously satisﬁes the following two linear partial differential equations: √ aFy+Fxx–Fzz=0 , Ft+4Fxxx+4Fzzz=0 , generates a solution w=– 2d dxK(x,x;y,t) of the Kadomtsev–Petviashvili equation, where K=K(x,z;y,t) is a solution of the linear Gel’fand–Levitan–Marchenko integral equation K(x,z;y,t)+F(x,z;y,t)+⎝integraldisplay∞ xK(x,s;y,t)F(s,z;y,t)ds=0 . Here the variables yandtare regarded as parameters. References for Section 18.6: V . E. Zakharov and A. B. Shabat (1974), S. P. Novikov, S. V . Manakov, L. B. Pitaevskii, and V . E. Zakharov (1984), M. J. Ablowitz and P. A. Clarkson (1991), A. D. Polyanin and V . F. Zaitsev (2004). Supplements Supplement 1 Elementary Functions and Their Properties /trianglerightsldThroughout Supplement 1 it is assumed that nis a positive integer , unless otherwise speciﬁed. 1.1. Power, Exponential, and Logarithmic Functions 1.1-1. Properties of the Power Function. Basic properties of the power function: xαxβ=xα+β,(x1x2)α=xα 1xα2,(xα)β=xαβ, for any αandβ,w h e r e x>0 ,x1>0 ,x2>0 . Differentiation and integration formulas: (xα)/prime=αxα–1,⎝integraldisplay xαdx=⎧ ⎨ ⎩xα+1 α+1+Cifα≠–1, ln|x|+C ifα= –1. The Taylor series expansion in a neighborhood of an arbitrary point: xα=∞⎝summationdisplay n=0Cn αxα–n 0(x–x0)nfor |x–x0|<|x0|, where Cn α=α(α–1 )...(α–n+1 ) n!are binomial coefﬁcients. 1.1-2. Properties of the Exponential Function. Basic properties of the exponential function: ax1ax2=ax1+x2,axbx=(ab)x,(ax1)x2=ax1x2, where a>0a n d b>0 . Number e,base of natural (Napierian) logarithms , and the function ex: e= lim n→∞⎝parenleftBig 1+1 n⎝parenrightBign = 2.718281 ...,ex= lim n→∞⎝parenleftBig 1+x n⎝parenrightBign . 905 906 ELEMENTARY FUNCTIONS AND THEIR PROPERTIES The formula for passing from an arbitrary base ato the base eof natural logarithms: ax=exlna. The inequality ax1>ax2⇐⇒⎝braceleftbigg x1>x2ifa>1 , x1<x2if 0 < a<1 . The limit relations for any a>1a n d b>0 : lim x→+∞ax |x|b=∞, lim x→–∞ax|x|b=0 . Differentiation and integration formulas: (ex)/prime=ex,⎝integraldisplay exdx=ex+C; (ax)/prime=axlna,⎝integraldisplay axdx=ax lna+C. The expansion in power series: ex=1+x 1!+x2 2!+x3 3!+···+xn n!+···=∞⎝summationdisplay k=0xk k!. 1.1-3. Properties of the Logarithmic Function. By deﬁnition, the logarithmic function is the inverse of the exponential function. The following equivalence relation holds: y=l o gax⇐⇒ x=ay, where a>0 ,a≠1. Basic properties of the logarithmic function: alogax=x,l o ga(x1x2)=l o gax1+l o gax2, loga(xk)=klogax,l o gax=logbx logba, where x>0 ,x1>0 ,x2>0 ,a>0 ,a≠1,b>0 ,b≠1. The simplest inequality: logax1>l o gax2⇐⇒⎝braceleftbigg x1>x2ifa>1 , x1<x2if 0 < a<1 . For any b> 0, the following limit relations hold: lim x→+∞logax xb= 0, lim x→+0xblogax=0 . The logarithmic function with the base e(base of natural logarithms orNapierian base )i s denoted by logex=l nx, where e= lim n→∞⎝parenleftBig 1+1 n⎝parenrightBign = 2.718281 ... 1.2. T RIGONOMETRIC FUNCTIONS 907 Formulas for passing from an arbitrary base ato the Napierian base e: logax=lnx lna. Differentiation and integration formulas: (lnx)/prime=1 x,⎝integraldisplay lnxd x =xlnx–x+C. Expansion in power series: ln(1 + x)=x–x2 2+x3 3–x4 4+···=∞⎝summationdisplay k=1(–1)k–1xk k, |x|<1 ; ln⎝parenleftbiggx+1 x–1⎝parenrightbigg =2 x+2 3x3+2 5x5+···=2∞⎝summationdisplay k=11 (2k–1 )x2k–1, |x|>1 ; lnx=2⎝parenleftbiggx–1 x+1⎝parenrightbigg +2 3⎝parenleftbiggx–1 x+1⎝parenrightbigg3 +2 5⎝parenleftbiggx–1 x+1⎝parenrightbigg5 +···=2∞⎝summationdisplay k=11 2k–1⎝parenleftbiggx–1 x+1⎝parenrightbigg2k–1 ,x>0 . 1.2. Trigonometric Functions 1.2-1. Simplest Relations. sin2x+c o s2x=1 , t a n xcotx=1 , sin(–x)=–s i n x,c o s ( – x)=c o s x, tanx=sinx cosx,c o t x=cosx sinx, tan(–x)=–t a n x,c o t ( – x)=–c o t x, 1+t a n2x=1 cos2x,1 + c o t2x=1 sin2x. 1.2-2. Reduction Formulas. sin(x±2nπ)=s i n x, sin(x±nπ) = (–1)nsinx, sin⎝parenleftBig x±2n+1 2π⎝parenrightBig =±(–1)ncosx, sin⎝parenleftBig x±π 4⎝parenrightBig =√ 2 2(sinx±cosx), tan(x±nπ)=t a n x, tan⎝parenleftBig x±2n+1 2π⎝parenrightBig =–c o t x, tan⎝parenleftBig x±π 4⎝parenrightBig =tanx±1 1∓tanx,cos(x±2nπ)=c o s x, cos(x±nπ) = (–1)ncosx, cos⎝parenleftBig x±2n+1 2π⎝parenrightBig =∓(–1)nsinx, cos⎝parenleftBig x±π 4⎝parenrightBig =√ 2 2(cosx∓sinx), cot(x±nπ)=c o t x, cot⎝parenleftBig x±2n+1 2π⎝parenrightBig =–t a n x, cot⎝parenleftBig x±π 4⎝parenrightBig =cotx∓1 1±cotx, where n=1 ,2 , ... 908 ELEMENTARY FUNCTIONS AND THEIR PROPERTIES 1.2-3. Relations Between Trigonometric Functions of Single Argument. sinx=±√ 1–c o s2x=±tanx √ 1+t a n2x=±1 √ 1+c o t2x, cosx=±√ 1–s i n2x=±1 √ 1+t a n2x=±cotx √ 1+c o t2x, tanx=±sinx √ 1–s i n2x=±√ 1–c o s2x cosx=1 cotx, cotx=±√ 1–s i n2x sinx=±cosx √ 1–c o s2x=1 tanx. The sign before the radical is determined by the quarter in which the argument takes its values. 1.2-4. Addition and Subtraction of Trigonometric Functions. sinx+s i ny=2s i n⎝parenleftBigx+y 2⎝parenrightBig cos⎝parenleftBigx–y 2⎝parenrightBig , sinx–s i ny=2s i n⎝parenleftBigx–y 2⎝parenrightBig cos⎝parenleftBigx+y 2⎝parenrightBig , cosx+c o sy=2c o s⎝parenleftBigx+y 2⎝parenrightBig cos⎝parenleftBigx–y 2⎝parenrightBig , cosx–c o sy=– 2s i n⎝parenleftBigx+y 2⎝parenrightBig sin⎝parenleftBigx–y 2⎝parenrightBig , sin2x–s i n2y=c o s2y–c o s2x=s i n (x+y)s i n (x–y), sin2x–c o s2y=–c o s ( x+y)c o s (x–y), tanx±tany=sin(x±y) cosxcosy,c o t x±coty=sin(y±x) sinxsiny, acosx+bsinx=rsin(x+ϕ)=rcos(x –ψ). Herer=√ a2+b2,s i nϕ=a/r,c o sϕ=b/r,s i nψ=b/r,a n dc o s ψ=a/r. 1.2-5. Products of Trigonometric Functions. sinxsiny=1 2[cos(x–y)–c o s ( x+y)], cosxcosy=1 2[cos(x–y)+c o s ( x+y)], sinxcosy=1 2[sin(x –y)+s i n ( x+y)]. 1.2-6. Powers of Trigonometric Functions. cos2x=1 2cos 2x+1 2, cos3x=1 4cos 3x+3 4cosx, cos4x=1 8cos 4x+1 2cos 2x+3 8, cos5x=1 16cos 5x+5 16cos 3x+5 8cosx,sin2x=–1 2cos 2x+1 2, sin3x=–1 4sin 3x+3 4sinx, sin4x=1 8cos 4x–1 2cos 2x+3 8, sin5x=1 16sin 5x–5 16sin 3x+5 8sinx, 1.2. T RIGONOMETRIC FUNCTIONS 909 cos2nx=1 22n–1n–1⎝summationdisplay k=0Ck 2ncos[2( n–k)x]+1 22nCn 2n, cos2n+1x=1 22nn⎝summationdisplay k=0Ck 2n+1cos[(2 n–2k+1 )x], sin2nx=1 22n–1n–1⎝summationdisplay k=0(–1)n–kCk 2ncos[2( n–k)x]+1 22nCn 2n, sin2n+1x=1 22nn⎝summationdisplay k=0(–1)n–kCk 2n+1sin[(2n–2k+1 )x]. Heren=1 ,2 , ...andCk m=m! k!(m–k)!are binomial coefﬁcients (0! = 1). 1.2-7. Addition Formulas. sin(x±y)=s i n xcosy±cosxsiny, tan(x±y)=tanx±tany 1∓tanxtany,cos(x±y)=c o s xcosy∓sinxsiny, cot(x±y)=1∓tanxtany tanx±tany. 1.2-8. Trigonometric Functions of Multiple Arguments. cos 2x=2c o s2x–1=1–2s i n2x, cos 3x=– 3c o s x+4c o s3x, cos 4x=1–8c o s2x+8c o s4x, cos 5x=5c o s x–2 0c o s3x+1 6c o s5x,sin 2x=2s i n xcosx, sin 3x=3s i n x–4s i n3x, sin 4x=4c o s x(sinx–2s i n3x), sin 5x=5s i n x–2 0s i n3x+1 6s i n5x, cos(2nx)=1+n⎝summationdisplay k=1(–1)kn2(n2–1 )...[n2–(k–1 )2] (2k)!4ksin2kx, cos[(2 n+1)x]=c o s x⎝braceleftbigg 1+n⎝summationdisplay k=1(–1)k[(2n+1)2–1][(2 n+1)2–32]...[(2n+1)2–(2k–1)2] (2k)!sin2kx⎝bracerightbigg , sin(2nx)=2ncosx⎝bracketleftbigg sinx+n⎝summationdisplay k=1(–4)k(n2–1 ) (n2–22)...(n2–k2) (2k–1 ) !sin2k–1x⎝bracketrightbigg , sin[(2n+1)x ]=(2n+1)⎝braceleftbigg sinx+n⎝summationdisplay k=1(–1)k[(2n+1)2–1][(2 n+1)2–32]...[(2n+1)2–(2k–1)2] (2k+1)!sin2k+1x⎝bracerightbigg , tan 2x=2t a nx 1–t a n2x,t a n 3 x=3t a nx–t a n3x 1–3t a n2x,t a n 4 x=4t a nx–4t a n3x 1–6t a n2x+t a n4x, where n=1 ,2 , ... 1.2-9. Trigonometric Functions of Half Argument. sin2x 2=1–c o s x 2,c o s2x 2=1+c o s x 2, 910 ELEMENTARY FUNCTIONS AND THEIR PROPERTIES tanx 2=sinx 1+c o s x=1–c o s x sinx,c o tx 2=sinx 1–c o s x=1+c o s x sinx, sinx=2t a nx 2 1+t a n2x 2,c o s x=1–t a n2x 2 1+t a n2x 2,t a n x=2t a nx 2 1–t a n2x 2. 1.2-10. Differentiation Formulas. dsinx dx=c o sx,dcosx dx=–s i n x,dtanx dx=1 cos2x,dcotx dx=–1 sin2x. 1.2-11. Integration Formulas. ⎝integraldisplay sinxd x =–c o s x+C,⎝integraldisplay cosxd x =s i nx+C, ⎝integraldisplay tanxd x =–l n |cosx|+C,⎝integraldisplay cotxd x =l n|sinx|+C, where Cis an arbitrary constant. 1.2-12. Expansion in Power Series. cosx=1–x2 2!+x4 4!–x6 6!+···+ (–1)nx2n (2n)!+··· (|x|<∞), sinx=x–x3 3!+x5 5!–x7 7!+···+ (–1)nx2n+1 (2n+1 ) !+··· (|x|<∞), tanx=x+x3 3+2x5 15+17x7 315+···+22n(22n–1 )|B2n| (2n)!x2n–1+··· (|x|<π/2), cotx=1 x–⎝parenleftbiggx 3+x3 45+2x5 945+···+22n|B2n| (2n)!x2n–1+···⎝parenrightbigg (0 < |x|<π), 1 cosx=1+x2 2+5x4 24+61x6 720+···+(–1)nE2n (2n)!x2n+··· (|x|<π/2), 1 sinx=1 x+x 6+7x3 360+···+(–1)n–12(22n–1–1 )B2n (2n)!x2n–1+··· (0 < |x|<π), where BnandEnare Bernoulli and Euler numbers (see Supplements 11.1-3 and 11.1-4). 1.2-13. Representation in the Form of Inﬁnite Products. sinx=x⎝parenleftbigg 1–x2 π2⎝parenrightbigg⎝parenleftbigg 1–x2 4π2⎝parenrightbigg⎝parenleftbigg 1–x2 9π2⎝parenrightbigg ...⎝parenleftbigg 1–x2 n2π2⎝parenrightbigg ... cosx=⎝parenleftbigg 1–4x2 π2⎝parenrightbigg⎝parenleftbigg 1–4x2 9π2⎝parenrightbigg⎝parenleftbigg 1–4x2 25π2⎝parenrightbigg ...⎝parenleftbigg 1–4x2 (2n+1 )2π2⎝parenrightbigg ... 1.3. I NVERSE TRIGONOMETRIC FUNCTIONS 911 1.2-14. Euler and de Moivre Formulas. Relationship with Hyperbolic Functions. ey+ix=ey(cosx+isinx), (cos x+isinx)n=c o s ( nx)+isin(nx),i2= –1, sin(ix)=isinhx,c o s ( ix)=c o s h x,t a n ( ix)=itanhx,c o t ( ix)=–icothx. 1.3. Inverse Trigonometric Functions 1.3-1. Deﬁnitions of Invers e Trigonometric Functions. Inverse trigonometric functions (arc functions ) are the functions that are inverse to the trigonometric functions. Since the trigonometric functions sin x,c o sx,t a nx,c o txare periodic, the correspond- ing inverse functions, denoted by Arcsin x, Arccos x,A r c t a n x, Arccot x, are multi-valued. The following relations deﬁne the multi-valued inverse trigonometric functions: sin⎝parenleftbig Arcsin x⎝parenrightbig =x,c o s⎝parenleftbig Arccos x⎝parenrightbig =x, tan⎝parenleftbig Arctan x⎝parenrightbig =x,c o t⎝parenleftbig Arccot x⎝parenrightbig =x. These functions admit the followi ng verbal deﬁnitions: Arcsin xis the angle whose sine is equal tox; Arccos xis the angle whose cosine is equal to x;A r c t a n xis the angle whose tangent is equal tox; Arccot xis the angle whose cotangent is equal to x. The principal (single-valued) branches of the inverse trigonometric functions are denoted by arcsin x≡sin–1x (arcsine is the inverse of sine), arccos x≡cos–1x(arccosine is the inverse of cosine), arctan x≡tan–1x(arctangent is the inverse of tangent), arccot x≡cot–1x(arccotangent is the inverse of cotangent) and are determined by the inequalities –π 2≤arcsin x≤π 2,0 ≤arccos x≤π (–1≤x≤1); –π 2<a r c t a n x<π 2, 0 < arccot x<π (–∞<x<∞). The following equivalent relations can be taken a s deﬁnitions of single-v alued inverse trigono- metric functions: y=a r c s i n x,– 1 ≤x≤1 ⇐⇒ x=s i ny,–π 2≤y≤π 2; y= arccos x,– 1 ≤x≤1 ⇐⇒ x=c o sy,0 ≤y≤π; y=a r c t a n x,–∞<x<+∞⇐ ⇒ x=t a ny,–π 2<y<π 2; y= arccot x,–∞<x<+∞⇐ ⇒ x=c o ty,0 < y<π. The multi-valued and the single-valued inverse trigonometric functions are related by the for- mulas Arcsin x= (–1)narcsin x+πn, Arccos x=±arccos x+2πn, Arctan x=a r c t a n x+πn, Arccot x= arccot x+πn, where n=0 ,±1,±2,... 912 ELEMENTARY FUNCTIONS AND THEIR PROPERTIES 1.3-2. Simplest Formulas. sin(arcsin x)=x, cos(arccos x)=x, tan(arctan x)=x, cot(arccot x)=x. 1.3-3. Some Properties. arcsin(– x)=–a r c s i n x, arctan(– x)=–a r c t a n x,arccos(– x)=π– arccos x, arccot(– x)=π– arccot x, arcsin(sin x)=⎝braceleftbiggx–2nπ if 2nπ–π 2≤x≤2nπ+π 2, –x+2 (n+1 )πif (2n+1 )π–π 2≤x≤2(n+1 )π+π 2, arccos(cos x)=⎝braceleftbigg x–2nπ if 2nπ≤x≤(2n+1 )π, –x+2 (n+1 )πif (2n+1 )π≤x≤2(n+1 )π, arctan(tan x)=x–nπ ifnπ–π 2<x<nπ+π 2, arccot(cot x)=x–nπ ifnπ<x<(n+1 )π. 1.3-4. Relations Between Inverse Trigonometric Functions. arcsin x+arccos x=π 2,a r c t a n x+arccot x=π 2; arcsin x=⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩arccos√ 1–x2 if 0 ≤x≤1, – arccos√ 1–x2 if –1 ≤x≤0, arctanx √ 1–x2if –1 < x<1 , arccot√ 1–x2 x–πif –1 ≤x<0 ;arccos x=⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩arcsin√ 1–x2 if 0 ≤x≤1, π–arcsin√ 1–x2if –1 ≤x≤0, arctan√ 1–x2 xif 0 < x≤1, arccotx √ 1–x2if –1 < x<1 ; arctan x=⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩arcsinx √ 1+x2for any x, arccos1 √ 1+x2ifx≥0, – arccos1 √ 1+x2ifx≤0, arccot1 xifx>0 ;arccot x=⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩arcsin1 √ 1+x2ifx>0 , π–arcsin1 √ 1+x2ifx<0 , arctan1 xifx>0 , π+arctan1 xifx<0 . 1.3-5. Addition and Subtraction of Inverse Trigonometric Functions. arcsin x+a r c s i n y=a r c s i n⎝parenleftbig x⎝radicalbig 1–y2+y√ 1–x2⎝parenrightbig forx2+y2≤1, arccos x±arccos y=±arccos⎝bracketleftbig xy∓⎝radicalbig (1 –x2)(1 –y2)⎝bracketrightbig forx±y≥0, arctan x+a r c t a n y=a r c t a nx+y 1–xyforxy<1 , arctan x–a r c t a n y=a r c t a nx–y 1+xyforxy> –1. 1.4. H YPERBOLIC FUNCTIONS 913 1.3-6. Differentiation Formulas. d dxarcsin x=1 √ 1–x2,d dxarccos x=–1 √ 1–x2, d dxarctan x=1 1+x2,d dxarccot x=–1 1+x2. 1.3-7. Integration Formulas. ⎝integraldisplay arcsin xd x =xarcsin x+√ 1–x2+C,⎝integraldisplay arccos xd x =xarccos x–√ 1–x2+C, ⎝integraldisplay arctan xd x =xarctan x–1 2ln(1 + x2)+C,⎝integraldisplay arccot xd x =xarccot x+1 2ln(1 + x2)+C, where Cis an arbitrary constant. 1.3-8. Expansion in Power Series. arcsin x=x+1 2x3 3+1×3 2×4x5 5+1×3×5 2×4×6x7 7+···+1×3×···×(2n–1 ) 2×4×···×(2n)x2n+1 2n+1+··· (|x|<1 ) , arctan x=x–x3 3+x5 5–x7 7+···+ (–1)n–1x2n–1 2n–1+··· (|x|≤1), arctan x=π 2–1 x+1 3x3–1 5x5+···+ (–1)n 1 (2n–1 )x2n–1+··· (|x|>1 ) . The expansions for arccos xand arccot xcan be obtained from the relations arccos x=π 2–arcsin x and arccot x=π 2–a r c t a n x. 1.4. Hyperbolic Functions 1.4-1. Deﬁnitions of Hyperbolic Functions. Hyperbolic functions are deﬁned in terms of the exponential functions as follows: sinhx=ex–e–x 2,c o s h x=ex+e–x 2, tanhx=ex–e–x ex+e–x,c o t h x=ex+e–x ex–e–x. 1.4-2. Simplest Relations. cosh2x–s i n h2x=1 , sinh(– x)=–s i n h x, tanhx=sinhx coshx, tanh(– x)=–t a n h x, 1–t a n h2x=1 cosh2x,tanhxcothx=1 , cosh(– x)=c o s h x, cothx=coshx sinhx, coth(– x)=–c o t h x, coth2x–1=1 sinh2x. 914 ELEMENTARY FUNCTIONS AND THEIR PROPERTIES 1.4-3. Relations Between Hyperbolic Functions of Single Argument ( x≥0). sinhx=⎝radicalbig cosh2x–1=tanhx √ 1–t a n h2x=1 √ coth2x–1, coshx=⎝radicalbig sinh2x+1=1 √ 1–t a n h2x=cothx √ coth2x–1, tanhx=sinhx √ sinh2x+1=√ cosh2x–1 coshx=1 cothx, cothx=√ sinh2x+1 sinhx=coshx √ cosh2x–1=1 tanhx. 1.4-4. Addition and Subtraction of Hyperbolic Functions. sinhx+s i n h y=2s i n h⎝parenleftBigx+y 2⎝parenrightBig cosh⎝parenleftBigx–y 2⎝parenrightBig , sinhx–s i n h y=2s i n h⎝parenleftBigx–y 2⎝parenrightBig cosh⎝parenleftBigx+y 2⎝parenrightBig , coshx+c o s h y=2c o s h⎝parenleftBigx+y 2⎝parenrightBig cosh⎝parenleftBigx–y 2⎝parenrightBig , coshx–c o s h y=2s i n h⎝parenleftBigx+y 2⎝parenrightBig sinh⎝parenleftBigx–y 2⎝parenrightBig , sinh2x–s i n h2y=c o s h2x–c o s h2y=s i n h ( x+y)s i n h ( x–y), sinh2x+c o s h2y=c o s h ( x+y)c o s h ( x–y), (coshx±sinhx)n=c o s h ( nx)±sinh(nx), tanhx±tanhy=sinh(x±y) coshxcoshy,c o t h x±cothy=±sinh(x±y) sinhxsinhy, where n=0 ,±1,±2,... 1.4-5. Products of Hyperbolic Functions. sinhxsinhy=1 2[cosh( x+y)–c o s h ( x–y)], coshxcoshy=1 2[cosh( x+y)+c o s h ( x–y)], sinhxcoshy=1 2[sinh(x+y)+s i n h ( x–y)]. 1.4-6. Powers of Hyperbolic Functions. cosh2x=1 2cosh 2 x+1 2, cosh3x=1 4cosh 3 x+3 4coshx, cosh4x=1 8cosh 4 x+1 2cosh 2 x+3 8, cosh5x=1 16cosh 5 x+5 16cosh 3 x+5 8coshx,sinh2x=1 2cosh 2 x–1 2, sinh3x=1 4sinh 3x–3 4sinhx, sinh4x=1 8cosh 4 x–1 2cosh 2 x+3 8, sinh5x=1 16sinh 5x–5 16sinh 3x+5 8sinhx, 1.4. H YPERBOLIC FUNCTIONS 915 cosh2nx=1 22n–1n–1⎝summationdisplay k=0Ck 2ncosh[2(n –k)x]+1 22nCn 2n, cosh2n+1x=1 22nn⎝summationdisplay k=0Ck 2n+1cosh[(2 n–2k+1)x], sinh2nx=1 22n–1n–1⎝summationdisplay k=0(–1)kCk 2ncosh[2(n –k)x]+(–1)n 22nCn 2n, sinh2n+1x=1 22nn⎝summationdisplay k=0(–1)kCk 2n+1sinh[(2 n–2k+1)x]. Heren=1 ,2 , ...andCk mare binomial coefﬁcients. 1.4-7. Addition Formulas. sinh(x±y)=s i n h xcoshy±sinhycoshx,c o s h ( x±y)=c o s h xcoshy±sinhxsinhy, tanh(x±y)=tanhx±tanhy 1±tanhxtanhy,c oth(x±y)=cothxcothy±1 cothy±cothx. 1.4-8. Hyperbolic Functions of Multiple Argument. cosh 2 x=2c o s h2x–1 , cosh 3 x=– 3c o s h x+4c o s h3x, cosh 4 x=1–8c o s h2x+8c o s h4x, cosh 5 x=5c o s h x–2 0c o s h3x+1 6c o s h5x,sinh 2x=2s i n h xcoshx, sinh 3x=3s i n h x+4s i n h3x, sinh 4x=4c o s h x(sinhx+2s i n h3x), sinh 5x=5s i n h x+2 0s i n h3x+1 6s i n h5x. cosh(nx )=2n–1coshnx+n 2[n/2]⎝summationdisplay k=0(–1)k+1 k+1Ck–2 n–k–22n–2k–2(coshx)n–2k–2, sinh(nx)=s i n h x[(n–1)/2]⎝summationdisplay k=02n–k–1Ck n–k–1(coshx)n–2k–1. HereCk mare binomial coefﬁcients and [ A] stands for the integer part of the number A. 1.4-9. Hyperbolic Functions of Half Argument. sinhx 2=s i g n x⎝radicalbigg coshx–1 2,c o s hx 2=⎝radicalbigg coshx+1 2, tanhx 2=sinhx coshx+1=coshx–1 sinhx,c o t hx 2=sinhx coshx–1=coshx+1 sinhx. 916 ELEMENTARY FUNCTIONS AND THEIR PROPERTIES 1.4-10. Differentiation Formulas. dsinhx dx=c o s h x,dcoshx dx=s i n h x, dtanhx dx=1 cosh2x,dcothx dx=–1 sinh2x. 1.4-11. Integration Formulas. ⎝integraldisplay sinhxd x =c o s h x+C,⎝integraldisplay coshxd x =s i n h x+C, ⎝integraldisplay tanhxd x =l nc o s h x+C,⎝integraldisplay cothxd x =l n|sinhx|+C, where Cis an arbitrary constant. 1.4-12. Expansion in Power Series. coshx=1+x2 2!+x4 4!+x6 6!+···+x2n (2n)!+··· (|x|<∞), sinhx=x+x3 3!+x5 5!+x7 7!+···+x2n+1 (2n+1 ) !+··· (|x|<∞), tanhx=x–x3 3+2x5 15–17x7 315+···+ (–1)n–122n(22n–1 )|B2n|x2n–1 (2n)!+··· (|x|<π/2), cothx=1 x+x 3–x3 45+2x5 945–···+ (–1)n–122n|B2n|x2n–1 (2n)!+··· (|x|<π), 1 coshx=1–x2 2+5x4 24–61x6 720+···+E2n (2n)!x2n+··· (|x|<π/2), 1 sinhx=1 x–x 6+7x3 360–31x5 15120+···+2(22n–1–1 )B2n (2n)!x2n–1+··· (0 < |x|<π), where BnandEnare Bernoulli and Euler numbers (see Supplements 11.1-3 and 11.1-4). 1.4-13. Representation in the Form of Inﬁnite Products. sinhx=x⎝parenleftbigg 1+x2 π2⎝parenrightbigg⎝parenleftbigg 1+x2 4π2⎝parenrightbigg⎝parenleftbigg 1+x2 9π2⎝parenrightbigg ...⎝parenleftbigg 1+x2 n2π2⎝parenrightbigg ... coshx=⎝parenleftbigg 1+4x2 π2⎝parenrightbigg⎝parenleftbigg 1+4x2 9π2⎝parenrightbigg⎝parenleftbigg 1+4x2 25π2⎝parenrightbigg ...⎝parenleftbigg 1+4x2 (2n+1 )2π2⎝parenrightbigg ... 1.4-14. Relationship with Trigonometric Functions. sinh(ix)=isinx,c o s h ( ix)=c o s x,t a n h ( ix)=itanx,c o t h ( ix)=–icotx,i2= –1. 1.5. I NVERSE HYPERBOLIC FUNCTIONS 917 1.5. Inverse Hyperbolic Functions 1.5-1. Deﬁnitions of Inverse Hyperbolic Functions. Inverse hyperbolic functions are the functions that are inverse to hyperbolic functions. The following notation is used for inverse hyperbolic functions: arcsinh x≡sinh–1x (inverse of hyperbolic sine), arccosh x≡cosh–1x(inverse of hyperbolic cosine), arctanh x≡tanh–1x(inverse of hyperbolic tangent), arccoth x≡coth–1x(inverse of hyperbolic cotangent). Inverse hyperbolic functions can be expressed in terms of logarithmic functions: arcsinh x=l n⎝parenleftbig x+√ x2+1⎝parenrightbig (xis any); arccosh x=l n⎝parenleftbig x+√ x2–1⎝parenrightbig (x≥1); arctanh x=1 2ln1+x 1–x(|x|< 1); arccoth x=1 2lnx+1 x–1(|x|>1 ) . Here only one (principal) branch of the function arccosh xis listed, the function itself being double- valued. In order to write out both branches of arccosh x, the symbol ±should be placed before the logarithm on the right-hand side of the formula. 1.5-2. Simplest Relations. arcsinh(– x)=–a r c s i n h x, arctanh(–x )=–a r c t a n h x, arccoth(– x) = – arccoth x. 1.5-3. Relations Between Inverse Hyperbolic Functions. arcsinh x= arccosh√ x2+1=a r c t a n hx √ x2+1, arccosh x=a r c s i n h√ x2–1=a r c t a n h√ x2–1 x, arctanh x=a r c s i n hx √ 1–x2= arccosh1 √ 1–x2= arccoth1 x. 1.5-4. Addition and Subtraction of Inverse Hyperbolic Functions. arcsinh x±arcsinh y=a r c s i n h⎝parenleftbig x⎝radicalbig 1+y2±y√ 1+x2⎝parenrightbig , arccosh x±arccosh y= arccosh⎝bracketleftbig xy±⎝radicalbig (x2–1 ) (y2–1 )⎝bracketrightbig , arcsinh x±arccosh y=a r c s i n h⎝bracketleftbig xy±⎝radicalbig (x2+1 ) (y2–1 )⎝bracketrightbig , arctanh x±arctanh y=a r c t a n hx±y 1±xy,a r c t a n h x±arccoth y=a r c t a n hxy±1 y±x. 1.5-5. Differentiation Formulas. d dxarcsinh x=1 √ x2+1, d dxarctanh x=1 1–x2(x2<1 ) ,d dxarccosh x=1 √ x2–1, d dxarccoth x=1 1–x2(x2>1 ) . 918 ELEMENTARY FUNCTIONS AND THEIR PROPERTIES 1.5-6. Integration Formulas. ⎝integraldisplay arcsinh xd x =xarcsinh x–√ 1+x2+C, ⎝integraldisplay arccosh xd x =xarccosh x–√ x2–1+C, ⎝integraldisplay arctanh xd x =xarctanh x+1 2ln(1 – x2)+C, ⎝integraldisplay arccoth xd x =xarccoth x+1 2ln(x2–1 )+ C, where Cis an arbitrary constant. 1.5-7. Expansion in Power Series. arcsinh x=x–1 2x3 3+1×3 2×4x5 5–···+ (–1)n1×3×···×(2n–1 ) 2×4×···×(2n)x2n+1 2n+1+··· (|x|<1 ) , arcsinh x=l n ( 2x)+1 21 2x2+1×3 2×41 4x4+···+1×3×···×(2n–1 ) 2×4×···×(2n)1 2nx2n+··· (|x|>1 ) , arccosh x=l n ( 2x)–1 21 2x2–1×3 2×41 4x4–···–1×3×···×(2n–1 ) 2×4×···×(2n)1 2nx2n–··· (|x|>1 ) , arctanh x=x+x3 3+x5 5+x7 7+···+x2n+1 2n+1+··· (|x|<1 ) , arccoth x=1 x+1 3x3+1 5x5+1 7x7+···+1 (2n+1 )x2n+1+··· (|x|>1 ) . References for Supplement 1: M. Abramowitz and I. A. Stegun (1964), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986), D. G. Zill and J. M. Dewar (1990), M. Kline (1998), R. Courant and F. John (1999), I. S. Gradshteynand I. M. Ryzhik (2000), G. A. Korn and T. M. Korn (2000), C. H. Edwards and D. Penney (2002), D. Zwillinger (2002), E. W. Weisstein (2003), I. N. Bronshtein and K. A. Semendyayev (2004), M. Sullivan (2004), H. Anton, I. Bivens, and S. Davis (2005), R. Adams (2006). Supplement 2 Finite Sums and Inﬁnite Series 2.1. Finite Numerical Sums 2.1-1. Progressions. Arithmetic progression: 1.n–1⎝summationdisplay k=0(a+bk)=an+bn(n–1 ) 2. Geometric progression: 2.n⎝summationdisplay k=1aqk–1=aqn–1 q–1. Arithmetic-geometric progression: 3.n–1⎝summationdisplay k=0(a+bk)qk=a(1 –qn)–b(n–1 )qn 1–q+bq(1 –qn–1) (1 –q)2. 2.1-2. Sums of Powers of Natural Numbers Having the Form⎝summationtextkm. 1.n⎝summationdisplay k=1k=n(n+1 ) 2. 2.n⎝summationdisplay k=1k2=1 6n(n+ 1)(2n +1 ) . 3.n⎝summationdisplay k=1k3=1 4n2(n+1 )2. 4.n⎝summationdisplay k=1k4=1 30n(n+ 1)(2n + 1)(3n2+3n–1 ) . 5.n⎝summationdisplay k=1k5=1 12n2(n+1 )2(2n2+2n–1 ) . 6.n⎝summationdisplay k=1km=nm+1 m+1+nm 2+1 2C1 mB2nm–1+1 4C3 mB4nm–3+1 6C5 mB6nm–5+···. Here the Ck mare binomial coefﬁcients and the B2kare Bernoulli numbers (see Supplement 11.1-3); the last term in the sum contains norn2. 919 920 FINITE SUMS AND INFINITE SERIES 2.1-3. Alternating Sums of Powers of Natural Numbers,⎝summationtext(–1)kkm. 1.n⎝summationdisplay k=1(–1)kk= (–1)n⎝bracketleftBign–1 2⎝bracketrightBig ;[m] stands for the integer part of m. 2.n⎝summationdisplay k=1(–1)kk2= (–1)nn(n+1 ) 2. 3.n⎝summationdisplay k=1(–1)kk3=1 8⎝bracketleftbig 1 + (–1)n(4n3+6n2–1 )⎝bracketrightbig . 4.n⎝summationdisplay k=1(–1)kk4= (–1)n1 2(n4+2n3–n). 5.n⎝summationdisplay k=1(–1)kk5=1 4⎝bracketleftbig –1 + (–1)n(2n5+5n4–5n2+1 )⎝bracketrightbig . 2.1-4. Other Sums Containing Integers. 1.n⎝summationdisplay k=0(2k+1 )=( n+1 )2. 2.n⎝summationdisplay k=0(2k+1 )2=1 3(n+ 1)(2n + 1)(2n +3 ) . 3.n⎝summationdisplay k=1k(k+1 )=1 3n(n+1 ) (n+2 ) . 4.n⎝summationdisplay k=1(k+a)(k+b)=1 6n(n+ 1)(2n +1+3 a+3b)+nab. 5.n⎝summationdisplay k=1kk!=(n+1 ) !–1 . 6.n⎝summationdisplay k=0(–1)k(2k+ 1) = (–1)n(n+1 ) . 7.n⎝summationdisplay k=0(–1)k(2k+1 )2= 2(–1)n(n+1 )2–1 2⎝bracketleftbig 1 + (–1)n⎝bracketrightbig . 2.1-5. Sums Containing Binomial Coefﬁcients. /trianglerightsldThroughout Supplement 2.1-5, it is assumed that m=1 ,2 ,3 , ... 1.n⎝summationdisplay k=0Ck n=2n. 2.n⎝summationdisplay k=0Cm m+k=Cm+1 n+m+1. 3.n⎝summationdisplay k=0(–1)kCk m= (–1)nCn m–1. 2.1. F INITE NUMERICAL SUMS 921 4.n⎝summationdisplay k=0(k+1 )Ck n=2n–1(n+2 ) . 5.n⎝summationdisplay k=1(–1)k+1kCk n=0 . 6.n⎝summationdisplay k=1(–1)k+1 kCk n=n⎝summationdisplay m=11 m. 7.n⎝summationdisplay k=1(–1)k+1 k+1Ck n=n n+1. 8.n⎝summationdisplay k=01 k+1Ck n=2n+1–1 n+1. 9.n⎝summationdisplay k=0ak+1 k+1Ck n=(a+1 )n+1–1 n+1. 10.p⎝summationdisplay k=0Ck nCp–k m=Cp n+m;mandpare natural numbers. 11.n–p⎝summationdisplay k=0Ck nCp+k n=(2n)! (n–p)! (n+p)!. 12.n⎝summationdisplay k=0(Ck n)2=Cn 2n. 13.2n⎝summationdisplay k=0(–1)k(Ck 2n)2= (–1)nCn 2n. 14.2n+1⎝summationdisplay k=0(–1)k(Ck 2n+1)2=0 . 15.n⎝summationdisplay k=1k(Ck n)2=(2n–1 ) ! [(n– 1)!]2. 2.1-6. Other Numerical Sums. 1.n–1⎝summationdisplay k=1sinπk n=c o tπ 2n. 2.n⎝summationdisplay k=1sin2mπk 2n=n 22mCm 2m+1 2,m<2n. 3.n–1⎝summationdisplay k=0(–1)kcosmπk n=1 2⎝bracketleftbig 1–( – 1 )m+n⎝bracketrightbig ,m=0 ,1 , ...,n–1 . 4.n–1⎝summationdisplay k=0(–1)kcosnπk n=n 2n–1. 922 FINITE SUMS AND INFINITE SERIES 2.2. Finite Functional Sums 2.2-1. Sums Involving Hyperbolic Functions. 1.n–1⎝summationdisplay k=0sinh(kx+a)=s i n h⎝parenleftbiggn–1 2x+a⎝parenrightbiggsinh(nx/2) sinh(x/2). 2.n–1⎝summationdisplay k=0cosh(kx +a)=c o s h⎝parenleftbiggn–1 2x+a⎝parenrightbiggsinh(nx/2) sinh(x/2). 3.n–1⎝summationdisplay k=0(–1)ksinh(kx+a)=1 2c o s h ( x/2)⎝bracketleftbigg sinh⎝parenleftbigg a–x 2⎝parenrightbigg + (–1)nsinh⎝parenleftbigg2n–1 2x+a⎝parenrightbigg⎝bracketrightbigg . 4.n–1⎝summationdisplay k=0(–1)kcosh(kx +a)=1 2c o s h ( x/2)⎝bracketleftbigg cosh⎝parenleftbigg a–x 2⎝parenrightbigg + (–1)ncosh⎝parenleftbigg2n–1 2x+a⎝parenrightbigg⎝bracketrightbigg . 5.n–1⎝summationdisplay k=1ksinh(kx+a)=–1 sinh2(x/2)⎝braceleftBig nsinh[(n–1 )x+a]–(n–1 )s i n h ( nx+a)–s i n h a⎝bracerightBig . 6.n–1⎝summationdisplay k=1kcosh(kx +a)=–1 sinh2(x/2)⎝braceleftBig ncosh[(n –1 )x+a]–(n–1 )c o s h ( nx+a)–c o s h a⎝bracerightBig . 7.n–1⎝summationdisplay k=1(–1)kksinh(kx+a)=1 cosh2(x/2)⎝braceleftBig (–1)n–1nsinh[(n–1 )x+a] + (–1)n–1(n–1 )s i n h ( nx+a)–s i n h a⎝bracerightBig . 8.n–1⎝summationdisplay k=1(–1)kkcosh(kx +a)=1 cosh2(x/2)⎝braceleftBig (–1)n–1ncosh[(n –1 )x+a] + (–1)n–1(n–1 )c o s h ( nx+a)–c o s h a⎝bracerightBig . 9.n⎝summationdisplay k=0Ck nsinh(kx+a)=2ncoshnx 2sinh⎝parenleftbiggnx 2+a⎝parenrightbigg . 10.n⎝summationdisplay k=0Ck ncosh(kx+a)=2ncoshnx 2cosh⎝parenleftbiggnx 2+a⎝parenrightbigg . 11.n–1⎝summationdisplay k=1aksinh(kx)=asinhx–ansinh(nx)+an+1sinh[(n–1 )x] 1–2acoshx+a2. 12.n–1⎝summationdisplay k=0akcosh(kx )=1–a coshx–ancosh(nx )+an+1cosh[(n –1 )x] 1–2acoshx+a2. 13.n⎝summationdisplay k=11 2ktanhx 2k=c o t h x–1 2ncothx 2n. 14.n–1⎝summationdisplay k=02ktanh(2kx)=2ncoth(2nx)–c o t h x. 2.2-2. Sums Involving Trigonometric Functions. 1.n⎝summationdisplay k=1sin(2kx)=s i n [ ( n+1 )x]s i n (nx)c o s e c x. 2.2. F INITE FUNCTIONAL SUMS 923 2.n⎝summationdisplay k=0cos(2kx)=s i n [ ( n+1 )x]c o s (nx)c o s e c x. 3.n⎝summationdisplay k=1sin[(2k–1 )x]=s i n2(nx)c o s e c x. 4.n⎝summationdisplay k=1cos[(2 k–1 )x]=s i n ( nx)c o s (nx)c o s e c x. 5.n–1⎝summationdisplay k=0sin(kx+a)=s i n⎝parenleftbiggn–1 2x+a⎝parenrightbigg sinnx 2cosecx 2. 6.n–1⎝summationdisplay k=0cos(kx +a)=c o s⎝parenleftbiggn–1 2x+a⎝parenrightbigg sinnx 2cosecx 2. 7.2n–1⎝summationdisplay k=0(–1)kcos(kx +a)=s i n⎝parenleftbigg2n–1 2x+a⎝parenrightbigg sin(nx)s e cx 2. 8.n⎝summationdisplay k=1(–1)k+1sin[(2k–1 )x] = (–1)n+1sin(2nx) 2c o sx. 9.n⎝summationdisplay k=1(–1)kcos(2kx)=–1 2+ (–1)ncos[(2 n+1 )x] 2c o sx. 10.n⎝summationdisplay k=1sin2(kx)=n 2–cos[(n +1 )x]s i n (nx) 2s i nx. 11.n⎝summationdisplay k=1cos2(kx)=n 2+cos[(n +1 )x]s i n (nx) 2s i nx. 12.n–1⎝summationdisplay k=1ksin(2kx)=sin(2nx) 4s i n2x–ncos[(2 n–1 )x] 2s i nx. 13.n–1⎝summationdisplay k=1kcos(2kx)=nsin[(2n–1 )x] 2s i nx–1–c o s ( 2 nx) 4s i n2x. 14.n–1⎝summationdisplay k=1aksin(kx)=asinx–ansin(nx)+an+1sin[(n–1 )x] 1–2acosx+a2. 15.n–1⎝summationdisplay k=0akcos(kx )=1–a cosx–ancos(nx )+an+1cos[(n –1 )x] 1–2acosx+a2. 16.n⎝summationdisplay k=0Ck nsin(kx+a)=2ncosnx 2sin⎝parenleftbiggnx 2+a⎝parenrightbigg . 17.n⎝summationdisplay k=0Ck ncos(kx +a)=2ncosnx 2cos⎝parenleftbiggnx 2+a⎝parenrightbigg . 18.n⎝summationdisplay k=0(–1)kCk nsin(kx+a)=( – 2 )nsinnx 2sin⎝parenleftbiggnx 2+πn 2+a⎝parenrightbigg . 19.n⎝summationdisplay k=0(–1)kCk ncos(kx +a)=( – 2 )nsinnx 2cos⎝parenleftbiggnx 2+πn 2+a⎝parenrightbigg . 924 FINITE SUMS AND INFINITE SERIES 20.n⎝summationdisplay k=1⎝parenleftbigg 2ksin2x 2k⎝parenrightbigg2 =⎝parenleftbigg 2nsin2x 2n⎝parenrightbigg2 –s i n2x. 21.n⎝summationdisplay k=01 2ktanx 2k=1 2ncotx 2n–2c o t ( 2 x). 2.3. Inﬁnite Numerical Series 2.3-1. Progressions. 1.∞⎝summationdisplay k=0aqk=a 1–q,|q|<1 . 2.∞⎝summationdisplay k=0(a+bk)qk=a 1–q+bq (1 –q)2,|q|<1 . 2.3-2. Other Numerical Series. 1.∞⎝summationdisplay n=0(–1)n n+1=l n2 . 2.∞⎝summationdisplay n=0(–1)n 2n+1=π 4. 3.∞⎝summationdisplay n=11 n(n+1 )=1 . 4.∞⎝summationdisplay n=1(–1)n n(n+1 )=1–2l n2 . 5.∞⎝summationdisplay n=11 n(n+2 )=3 4. 6.∞⎝summationdisplay n=1(–1)n n(n+2 )=–1 4. 7.∞⎝summationdisplay n=11 (2n– 1)(2n+1 )=1 2. 8.∞⎝summationdisplay n=11 n2=π2 6. 9.∞⎝summationdisplay n=1(–1)n+1 n2=π2 12. 10.∞⎝summationdisplay n=11 (2n–1 )2=π2 8. 11.∞⎝summationdisplay n=11 n2+a2=π 2acoth(πa)–1 2a2. 2.4. I NFINITE FUNCTIONAL SERIES 925 12.∞⎝summationdisplay n=11 n2–a2=–π 2acot(πa)+1 2a2. 13.∞⎝summationdisplay k=11 k2n=22n–1π2n (2n)!|B2n|;t h e B2nare Bernoulli numbers (see Supplement 11.1-3). 14.∞⎝summationdisplay k=1(–1)k+1 k2n=(22n–1–1 )π2n (2n)!|B2n|;t h e B2nare Bernoulli numbers. 15.∞⎝summationdisplay k=11 (2k–1 )2n=(22n–1–1 )π2n 2(2n)!|B2n|;t h e B2nare Bernoulli numbers. 16.∞⎝summationdisplay k=11 k2k=l n2 . 17.∞⎝summationdisplay k=0(–1)k n2k=n2 n2+1. 18.∞⎝summationdisplay k=01 k!=e= 2.71828 ... 19.∞⎝summationdisplay k=0(–1)k k!=1 e= 0.36787 ... 20.∞⎝summationdisplay k=1k (k+1 ) !=1 . 2.4. Inﬁnite Functional Series 2.4-1. Power Series. 1.∞⎝summationdisplay k=0xk=1 1–x,|x|<1 . 2.∞⎝summationdisplay k=1kxk=x (1 –x)2,|x|<1 . 3.∞⎝summationdisplay k=1k2xk=x(x+1 ) (1 –x)3,|x|<1 . 4.∞⎝summationdisplay k=1k3xk=x(1 + 4x+x2) (1 –x)4,|x|<1 . 5.∞⎝summationdisplay k=0(±1)kknxk=⎝parenleftbigg xd dx⎝parenrightbiggn1 1∓x,|x|<1 . 6.∞⎝summationdisplay k=1xk k=–l n ( 1– x), –1 ≤x<1 . 7.∞⎝summationdisplay k=1(–1)k–1xk k=l n ( 1+ x),|x|<1 . 8.∞⎝summationdisplay k=1x2k–1 2k–1=1 2ln1+x 1–x,|x|<1 . 926 FINITE SUMS AND INFINITE SERIES 9.∞⎝summationdisplay k=1(–1)k–1x2k–1 2k–1=a r c t a n x,|x|≤1. 10.∞⎝summationdisplay k=1xk k2=–⎝integraldisplayx 0ln(1 – t) tdt,|x|≤1. 11.∞⎝summationdisplay k=1xk+1 k(k+1 )=x+( 1– x)l n ( 1– x),|x|≤1. 12.∞⎝summationdisplay k=1xk+2 k(k+2 )=x 2+x2 4+1 2(1 –x2)l n ( 1– x),|x|≤1. 13.∞⎝summationdisplay k=0xk k!=ex,xis any number. 14.∞⎝summationdisplay k=0x2k (2k)!=c o s h x,xis any number. 15.∞⎝summationdisplay k=0(–1)kx2k (2k)!=c o sx,xis any number. 16.∞⎝summationdisplay k=0x2k+1 (2k+1 ) !=s i n h x,xis any number. 17.∞⎝summationdisplay k=0(–1)kx2k+1 (2k+1 ) !=s i nx,xis any number. 18.∞⎝summationdisplay k=0xk+1 k!(k+1 )=ex–1 , xis any number. 19.∞⎝summationdisplay k=0xk+2 k!(k+2 )=(x–1 )ex+1 , xis any number. 20.∞⎝summationdisplay k=0(–1)kx2k+1 k!( 2k+1 )=√ π 2erfx,xis any number. 21.∞⎝summationdisplay k=0(k+a)n k!xk=⎝bracketleftbiggdn dtnexp(at+xet)⎝bracketrightbigg t=0,xis any number. 22.∞⎝summationdisplay k=122k(22k–1 )|B2k| (2k)!x2k–1=t a nx;t h e B2kare Bernoulli numbers, |x|<π/2. 23.∞⎝summationdisplay k=1(–1)k–122k(22k–1 )|B2k| (2k)!x2k–1=t a n h x;t h e B2kare Bernoulli numbers, |x|<π/2. 24.∞⎝summationdisplay k=122k|B2k| (2k)!x2k–1=1 x–c o tx;t h e B2kare Bernoulli numbers, 0 < |x|<π. 25.∞⎝summationdisplay k=1(–1)k–122k|B2k| (2k)!x2k–1=c o t h x–1 x;t h e B2kare Bernoulli numbers, |x|<π. 2.4. I NFINITE FUNCTIONAL SERIES 927 2.4-2. Trigonometric Series in One Variable Involving Sine. 1.∞⎝summationdisplay k=11 ksin(kx)=1 2(π–x), 0 < x<2π. 2.∞⎝summationdisplay k=1(–1)k–1 ksin(kx)=1 2x,–π<x<π. 3.∞⎝summationdisplay k=1ak ksin(kx) = arctanasinx 1–acosx,0 < x<2π,|a|≤1. 4.∞⎝summationdisplay k=01 2k+1sin(kx)=π 4cosx 2–s i nx 2ln⎝parenleftbigg cot2x 4⎝parenrightbigg ,0 < x<2π. 5.∞⎝summationdisplay k=0(–1)k 2k+1sin(kx)=–1 4cosx 2ln⎝parenleftbigg cot2x+π 4⎝parenrightbigg –π 4sinx 2,–π<x<π. 6.∞⎝summationdisplay k=11 k2sin(kx)=–⎝integraldisplayx 0ln⎝parenleftbigg 2s i nt 2⎝parenrightbigg dt,0 ≤x<π. 7.∞⎝summationdisplay k=1(–1)k k2sin(kx)=–⎝integraldisplayx 0ln⎝parenleftbigg 2c o st 2⎝parenrightbigg dt,–π<x<π. 8.∞⎝summationdisplay k=11 k(k+1 )sin(kx)=(π–x)s i n2x 2+s i nxln⎝parenleftbigg 2s i nx 2⎝parenrightbigg ,0 ≤x≤2π. 9.∞⎝summationdisplay k=1(–1)k k(k+1 )sin(kx)=–xcos2x 2+s i nxln⎝parenleftbigg 2c o sx 2⎝parenrightbigg ,–π≤x≤π. 10.∞⎝summationdisplay k=1k k2+a2sin(kx)=π 2s i n h ( πa)sinh[a(π–x)], 0 < x<2π. 11.∞⎝summationdisplay k=1(–1)k+1k k2+a2sin(kx)=π 2s i n h ( πa)sinh(ax), –π<x<π. 12.∞⎝summationdisplay k=1k k2–a2sin(kx)=π 2s i n (πa)sin[a(π–x)], 0 < x<2π. 13.∞⎝summationdisplay k=1(–1)k+1k k2–a2sin(kx)=π 2s i n (πa)sin(ax), –π<x<π. 14.∞⎝summationdisplay k=2(–1)kk k2–1sin(kx)=1 4sinx+1 2xcosx,–π<x<π. 15.∞⎝summationdisplay k=11 k2n+1sin(kx)=(–1)n–1(2π)2n+1 2(2n+1 ) !B2n+1⎝parenleftbiggx 2π⎝parenrightbigg ,w h e r e 0 ≤x≤2πforn=1 ,2 , ...; 0<x<2πforn=0 ;a n dt h e Bn(x) are Bernoulli polynomials (see Supplement 11.18-1). 16.∞⎝summationdisplay k=1(–1)k k2n+1sin(kx)=(–1)n–1(2π)2n+1 2(2n+1 ) !B2n+1⎝parenleftbiggx+π 2π⎝parenrightbigg ,w h e r e – π<x≤πforn=0 ,1 , ...; theBn(x) are Bernoulli polynomials. 17.∞⎝summationdisplay k=11 k!sin(kx)=e x p ( c o s x) sin(sin x),xis any number. 928 FINITE SUMS AND INFINITE SERIES 18.∞⎝summationdisplay k=1(–1)k k!sin(kx)=–e x p ( –c o s x) sin(sin x),xis any number. 19.∞⎝summationdisplay k=01 (2k)!sin(kx)=s i n⎝parenleftbigg sinx 2⎝parenrightbigg sinh⎝parenleftbigg cosx 2⎝parenrightbigg ,xis any number. 20.∞⎝summationdisplay k=0(–1)k (2k)!sin(kx)=–s i n⎝parenleftbigg cosx 2⎝parenrightbigg sinh⎝parenleftbigg sinx 2⎝parenrightbigg ,xis any number. 21.∞⎝summationdisplay k=0ak k!sin(kx)=e x p ( kcosx)s i n (ksinx),|a|≤1, xis any number. 22.∞⎝summationdisplay k=0aksin(kx)=asinx 1–2acosx+a2,|a|<1 ,xis any number. 23.∞⎝summationdisplay k=1kaksin(kx)=a(1 –a2)s i nx (1 – 2acosx+a2)2,|a|<1 ,xis any number. 24.∞⎝summationdisplay k=11 ksin(kx+a)=1 2(π–x)c o sa–l n⎝parenleftbigg 2s i nx 2⎝parenrightbigg sina,0 < x<2π. 25.∞⎝summationdisplay k=1(–1)k–1 ksin(kx+a)=1 2xcosa+l n⎝parenleftbigg 2c o sx 2⎝parenrightbigg sina,–π<x<π. 26.∞⎝summationdisplay k=1sin[(2k–1 )x] 2k–1=π 4,0 < x<π. 27.∞⎝summationdisplay k=1(–1)k–1sin[(2k–1 )x] 2k–1=1 2ln tan⎝parenleftbiggx 2+π 4⎝parenrightbigg ,–π 2<x<π 2. 28.∞⎝summationdisplay k=1a2k–1sin[(2k–1 )x] 2k–1=1 2arctan2asinx 1–a2,0 < x<2π,|a|≤1. 29.∞⎝summationdisplay k=1(–1)k–1a2k–1sin[(2k–1 )x] 2k–1=1 4ln1+2asinx+a2 1–2asinx+a2,0 < x<π,|a|≤1. 30.∞⎝summationdisplay k=1(–1)ksin[(k+1 )x] k(k+1 )=s i nx–1 2x(1 + cos x)–s i n xln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle2c o sx 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle. 31.∞⎝summationdisplay k=0a2k+1sin[(2k+1 )x]=a(1 +a2)s i nx (1 +a2)2–4a2cos2x,|a|<1 ,xis any number. 32.∞⎝summationdisplay k=0(–1)ka2k+1sin[(2k+1 )x]=a(1 –a2)s i nx (1 +a2)2–4a2sin2x,|a|<1 ,xis any number. 33.∞⎝summationdisplay k=1sin[2(k+1 )x] k(k+1 )=s i n ( 2 x)–(π–2x)s i n2x–s i nxcosxln(4 sin2x), 0 ≤x≤π. 34.∞⎝summationdisplay k=1(–1)ksin[(2k+1 )x] (2k+1 )2=⎝braceleftbigg1 4πx if –1 2π≤x≤1 2π, 1 4π(π–x)i f1 2π≤x≤3 2π. 2.4-3. Trigonometric Series in One Variable Involving Cosine. 1.∞⎝summationdisplay k=11 kcos(kx )=–l n⎝parenleftbigg 2s i nx 2⎝parenrightbigg ,0 < x<2π. 2.4. I NFINITE FUNCTIONAL SERIES 929 2.∞⎝summationdisplay k=1(–1)k–1 kcos(kx )=l n⎝parenleftbigg 2c o sx 2⎝parenrightbigg ,–π<x<π. 3.∞⎝summationdisplay k=1ak kcos(kx )=l n1 √ 1–2acosx+a2,0 < x<2π,|a|≤1. 4.∞⎝summationdisplay k=01 2k+1cos(kx )=π 4sinx 2+c o sx 2ln⎝parenleftbigg cot2x 4⎝parenrightbigg ,0 < x<2π. 5.∞⎝summationdisplay k=0(–1)k 2k+1cos(kx )=–1 4sinx 2ln⎝parenleftbigg cot2x+π 4⎝parenrightbigg +π 4cosx 2,–π<x<π. 6.∞⎝summationdisplay k=11 k2cos(kx )=1 12(3x2–6πx+2π2), 0 ≤x≤2π. 7.∞⎝summationdisplay k=1(–1)k k2cos(kx )=1 12(3x2–π2), –π≤x≤π. 8.∞⎝summationdisplay k=11 k(k+1 )cos(kx )=1 2(x–π)s i nx–2s i n2x 2ln⎝parenleftbigg 2s i nx 2⎝parenrightbigg +1 , 0 ≤x≤2π. 9.∞⎝summationdisplay k=1(–1)k k(k+1 )cos(kx )=–1 2xsinx–2c o s2x 2ln⎝parenleftbigg 2c o sx 2⎝parenrightbigg +1 , – π≤x≤π. 10.∞⎝summationdisplay k=11 k2+a2cos(kx )=π 2asinh(πa)cosh[a (π–x)] –1 2a2,0 ≤x≤2π. 11.∞⎝summationdisplay k=11 k2–a2cos(kx )=–π 2asin(πa)cos[a (π–x)] +1 2a2,0 ≤x≤2π. 12.∞⎝summationdisplay k=2(–1)k k2–1cos(kx )=1 2–1 4cosx–1 2xsinx,–π≤x≤π. 13.∞⎝summationdisplay k=2k k2–1cos(kx )=–1 2–1 4cosx–c o sxln⎝parenleftbigg 2s i nx 2⎝parenrightbigg ,0 < x<2π. 14.∞⎝summationdisplay k=11 k2ncos(kx )=(–1)n–1(2π)2n 2(2n)!B2n⎝parenleftbiggx 2π⎝parenrightbigg ,w h e r e 0 ≤x≤2πforn=1 ,2 , ...; theBn(x) are Bernoulli polynomials (see Supplement 11.18-1). 15.∞⎝summationdisplay k=1(–1)k k2ncos(kx )=(–1)n–1(2π)2n 2(2n)!B2n⎝parenleftbiggx+π 2π⎝parenrightbigg ,w h e r e – π≤x≤πforn=1 ,2 , ...; theBn(x) are Bernoulli polynomials. 16.∞⎝summationdisplay k=01 k!cos(kx )=e x p ( c o s x)c o s ( s i n x),xis any number. 17.∞⎝summationdisplay k=0(–1)k k!cos(kx )=e x p ( –c o s x)c o s ( s i n x),xis any number. 18.∞⎝summationdisplay k=01 (2k)!cos(kx )=c o s⎝parenleftbigg sinx 2⎝parenrightbigg cosh⎝parenleftbigg cosx 2⎝parenrightbigg ,xis any number. 19.∞⎝summationdisplay k=0(–1)k (2k)!cos(kx )=c o s⎝parenleftbigg cosx 2⎝parenrightbigg cosh⎝parenleftbigg sinx 2⎝parenrightbigg ,xis any number. 930 FINITE SUMS AND INFINITE SERIES 20.∞⎝summationdisplay k=0ak k!cos(kx )=e x p ( acosx)c o s (asinx),|a|≤1,xis any number. 21.∞⎝summationdisplay k=0akcos(kx )=1–a cosx 1–2acosx+a2,|a|<1 ,xis any number. 22.∞⎝summationdisplay k=1kakcos(kx )=a(1 +a2)c o sx–2a2 (1 – 2acosx+a2)2,|a|<1 ,xis any number. 23.∞⎝summationdisplay k=11 kcos(kx +a)=1 2(x–π)s i na–l n⎝parenleftbigg 2s i nx 2⎝parenrightbigg cosa,0 < x<2π. 24.∞⎝summationdisplay k=1(–1)k–1 kcos(kx +a)=–1 2xsina+l n⎝parenleftbigg 2c o sx 2⎝parenrightbigg cosa,–π<x<π. 25.∞⎝summationdisplay k=1cos[(2 k–1 )x] 2k–1=1 2ln cotx 2,0 < x<π. 26.∞⎝summationdisplay k=1(–1)k–1cos[(2 k–1 )x] 2k–1=π 4,0 < x<π. 27.∞⎝summationdisplay k=1a2k–1cos[(2 k–1 )x] 2k–1=1 4ln1+2acosx+a2 1–2acosx+a2,0 < x<2π,|a|≤1. 28.∞⎝summationdisplay k=1(–1)k–1a2k–1cos[(2 k–1 )x] 2k–1=1 2arctan2acosx 1–a2,0 < x<π,|a|≤1. 29.∞⎝summationdisplay k=1cos[(2 k–1 )x] (2k–1 )2=π 4⎝parenleftbiggπ 2–|x|⎝parenrightbigg ,–π≤x≤π. 30.∞⎝summationdisplay k=1(–1)kcos[(k +1 )x] k(k+1 )=c o sx–1 2xsinx–( 1+c o s x)l n⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle2c o sx 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle. 31. ∞⎝summationdisplay k=0a2k+1cos[(2 k+1 )x]=a(1 –a2)c o sx (1 +a2)2–4a2cos2x,|a|<1 ,xis any number. 32.∞⎝summationdisplay k=0(–1)ka2k+1cos[(2 k+1 )x]=a(1 +a2)c o sx (1 +a2)2–4a2sin2x,|a|<1 ,xis any number. 33.∞⎝summationdisplay k=1cos[2( k+1 )x] k(k+1 )=c o s ( 2 x)–⎝parenleftbiggπ 2–x⎝parenrightbigg sin(2x)+s i n2xln(4 sin2x), 0 ≤x≤π. 2.4-4. Trigonometric Series in Two Variables. 1.∞⎝summationdisplay k=11 ksin(kx)s i n (ky)=1 2ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglesinx+y 2cosecx–y 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,x±y≠0, 2π ,4π,... 2.∞⎝summationdisplay k=1(–1)k ksin(kx)s i n (ky)=1 2ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsinglecosx+y 2secx–y 2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,x±y≠π,3π,5π,... 3. ∞⎝summationdisplay k=11 k2sin(kx)s i n (ky)=⎝braceleftBigg1 2x(π–y)i f – y≤x≤y, 1 2y(π–x)i f y≤x≤2π–y.Here 0 < y<π. 2.4. I NFINITE FUNCTIONAL SERIES 931 4.∞⎝summationdisplay k=1(–1)k+1 k2sin(kx)s i n (ky)=1 2xy,|x±y|≤π. 5.∞⎝summationdisplay k=1ak ksin(kx)s i n (ky)=1 4ln4asin2[(x+y)/2] + (a–1 )2 4asin2[(x–y)/2] + (a–1 )2,0 < a<1 . 6.∞⎝summationdisplay k=11 k2sin2(kx)s i n2(ky)=1 2πx,0 ≤x≤y≤π 2. 7.∞⎝summationdisplay k=11 kcos(kx )c o s (ky)=–1 2ln⎝vextendsingle⎝vextendsingle2(cosx–c o sy)⎝vextendsingle⎝vextendsingle,x±y≠0, 2π ,4π,... 8.∞⎝summationdisplay k=1(–1)k kcos(kx )c o s (ky)=–1 2ln⎝vextendsingle⎝vextendsingle2(cosx+c o sy)⎝vextendsingle⎝vextendsingle,x±y≠π,3π,5π,... 9.∞⎝summationdisplay k=11 ksin(kx)c o s (ky)=⎧ ⎪⎨ ⎪⎩–1 2if 0 < x<y, 1 4(π–2y)i f x=y, 1 2(π–x)i f y<x<π.Here 0 < y<π. 10.∞⎝summationdisplay k=11 k2cos(kx )c o s (ky)=⎝braceleftBigg1 12⎝bracketleftbig 3x2+3 (y–π)2–π2⎝bracketrightbig if 0 ≤x≤y, 1 12⎝bracketleftbig 3y2+3 (x–π)2–π2⎝bracketrightbig ify≤x≤π. Here 0 < y<π. 11.∞⎝summationdisplay k=1(–1)k k2cos(kx )c o s (ky)=⎝braceleftBigg1 12⎝bracketleftbig 3(x2+y2)–π2⎝bracketrightbig if –(π–y)≤x≤π–y, 1 12⎝bracketleftbig 3(x–π)2+3(y–π)2–π2⎝bracketrightbig ifπ–y≤x≤π+y. Here 0 < y<π. References for Supplement 2: H. B. Dwight (1961), V . Mangulis (1965), E. R. Hansen (1975), I. S. Gradshteyn and I. M. Ryzhik (2000), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986), D. Zwillinger (2002). Supplement 3 Tables of Indeﬁnite Integrals /trianglerightsldThroughout Supplement 3, the integration constant Cis omitted for brevity . 3.1. Integrals Involving Rational Functions 3.1-1. Integrals Involving a+bx. 1.⎝integraldisplaydx a+bx=1 bln|a+bx|. 2.⎝integraldisplay (a+bx)ndx=(a+bx)n+1 b(n+1 ),n≠–1. 3.⎝integraldisplayxd x a+bx=1 b2⎝parenleftbig a+bx–aln|a+bx|⎝parenrightbig . 4.⎝integraldisplayx2dx a+bx=1 b3⎝bracketleftBig1 2(a+bx)2–2a(a+bx)+a2ln|a+bx|⎝bracketrightBig . 5.⎝integraldisplaydx x(a+bx)=–1 aln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+bx x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 6.⎝integraldisplaydx x2(a+bx)=–1 ax+b a2ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+bx x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 7.⎝integraldisplayxd x (a+bx)2=1 b2⎝parenleftBig ln|a+bx|+a a+bx⎝parenrightBig . 8.⎝integraldisplayx2dx (a+bx)2=1 b3⎝parenleftBig a+bx–2aln|a+bx|–a2 a+bx⎝parenrightBig . 9.⎝integraldisplaydx x(a+bx)2=1 a(a+bx)–1 a2ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+bx x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 10.⎝integraldisplayxd x (a+bx)3=1 b2⎝bracketleftBig –1 a+bx+a 2(a+bx)2⎝bracketrightBig . 3.1-2. Integrals Involving a+xandb+x. 1.⎝integraldisplaya+x b+xdx=x+(a–b)l n|b+x|. 2.⎝integraldisplaydx (a+x)(b+x)=1 a–bln⎝vextendsingle⎝vextendsingle⎝vextendsingleb+x a+x⎝vextendsingle⎝vextendsingle⎝vextendsingle,a≠b.F o ra=b, see Integral 2 with n=– 2i n Supplement 3.1-1. 3.⎝integraldisplayxd x (a+x)(b+x)=1 a–b⎝parenleftbig aln|a+x|–bln|b+x|⎝parenrightbig . 933 934 TABLES OF INDEFINITE INTEGRALS 4.⎝integraldisplaydx (a+x)(b+x)2=1 (b–a)(b+x)+1 (a–b)2ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x b+x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 5.⎝integraldisplayxd x (a+x)(b+x)2=b (a–b)(b+x)–a (a–b)2ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x b+x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 6.⎝integraldisplayx2dx (a+x)(b+x)2=b2 (b–a)(b+x)+a2 (a–b)2ln|a+x|+b2–2ab (b–a)2ln|b+x|. 7.⎝integraldisplaydx (a+x)2(b+x)2=–1 (a–b)2⎝parenleftBig1 a+x+1 b+x⎝parenrightBig +2 (a–b)3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x b+x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 8.⎝integraldisplayxd x (a+x)2(b+x)2=1 (a–b)2⎝parenleftBiga a+x+b b+x⎝parenrightBig +a+b (a–b)3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x b+x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 9.⎝integraldisplayx2dx (a+x)2(b+x)2=–1 (a–b)2⎝parenleftBiga2 a+x+b2 b+x⎝parenrightBig +2ab (a–b)3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x b+x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 3.1-3. Integrals Involving a2+x2. 1.⎝integraldisplaydx a2+x2=1 aarctanx a. 2.⎝integraldisplaydx (a2+x2)2=x 2a2(a2+x2)+1 2a3arctanx a. 3.⎝integraldisplaydx (a2+x2)3=x 4a2(a2+x2)2+3x 8a4(a2+x2)+3 8a5arctanx a. 4.⎝integraldisplaydx (a2+x2)n+1=x 2na2(a2+x2)n+2n–1 2na2⎝integraldisplaydx (a2+x2)n;n=1 ,2 , ... 5.⎝integraldisplayxd x a2+x2=1 2ln(a2+x2). 6.⎝integraldisplayxd x (a2+x2)2=–1 2(a2+x2). 7.⎝integraldisplayxd x (a2+x2)3=–1 4(a2+x2)2. 8.⎝integraldisplayxd x (a2+x2)n+1=–1 2n(a2+x2)n;n=1 ,2 , ... 9.⎝integraldisplayx2dx a2+x2=x–aarctanx a. 10.⎝integraldisplayx2dx (a2+x2)2=–x 2(a2+x2)+1 2aarctanx a. 11.⎝integraldisplayx2dx (a2+x2)3=–x 4(a2+x2)2+x 8a2(a2+x2)+1 8a3arctanx a. 12.⎝integraldisplayx2dx (a2+x2)n+1=–x 2n(a2+x2)n+1 2n⎝integraldisplaydx (a2+x2)n;n=1 ,2 , ... 13.⎝integraldisplayx3dx a2+x2=x2 2–a2 2ln(a2+x2). 14.⎝integraldisplayx3dx (a2+x2)2=a2 2(a2+x2)+1 2ln(a2+x2). 15.⎝integraldisplayx3dx (a2+x2)n+1=–1 2(n–1 ) (a2+x2)n–1+a2 2n(a2+x2)n;n=2 ,3 , ... 16.⎝integraldisplaydx x(a2+x2)=1 2a2lnx2 a2+x2. 3.1. I NTEGRALS INVOLVING RATI ONAL FUNCTIONS 935 17.⎝integraldisplaydx x(a2+x2)2=1 2a2(a2+x2)+1 2a4lnx2 a2+x2. 18.⎝integraldisplaydx x(a2+x2)3=1 4a2(a2+x2)2+1 2a4(a2+x2)+1 2a6lnx2 a2+x2. 19.⎝integraldisplaydx x2(a2+x2)=–1 a2x–1 a3arctanx a. 20.⎝integraldisplaydx x2(a2+x2)2=–1 a4x–x 2a4(a2+x2)–3 2a5arctanx a. 21.⎝integraldisplaydx x3(a2+x2)2=–1 2a4x2–1 2a4(a2+x2)–1 a6lnx2 a2+x2. 22.⎝integraldisplaydx x2(a2+x2)3=–1 a6x–x 4a4(a2+x2)2–7x 8a6(a2+x2)–15 8a7arctanx a. 23.⎝integraldisplaydx x3(a2+x2)3=–1 2a6x2–1 a6(a2+x2)–1 4a4(a2+x2)2–3 2a8lnx2 a2+x2. 3.1-4. Integrals Involving a2–x2. 1.⎝integraldisplaydx a2–x2=1 2aln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x a–x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 2.⎝integraldisplaydx (a2–x2)2=x 2a2(a2–x2)+1 4a3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x a–x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 3.⎝integraldisplaydx (a2–x2)3=x 4a2(a2–x2)2+3x 8a4(a2–x2)+3 16a5ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x a–x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 4.⎝integraldisplaydx (a2–x2)n+1=x 2na2(a2–x2)n+2n–1 2na2⎝integraldisplaydx (a2–x2)n;n=1 ,2 , ... 5.⎝integraldisplayxd x a2–x2=–1 2ln|a2–x2|. 6.⎝integraldisplayxd x (a2–x2)2=1 2(a2–x2). 7.⎝integraldisplayxd x (a2–x2)3=1 4(a2–x2)2. 8.⎝integraldisplayxd x (a2–x2)n+1=1 2n(a2–x2)n;n=1 ,2 , ... 9.⎝integraldisplayx2dx a2–x2=–x+a 2ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x a–x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 10.⎝integraldisplayx2dx (a2–x2)2=x 2(a2–x2)–1 4aln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x a–x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 11.⎝integraldisplayx2dx (a2–x2)3=x 4(a2–x2)2–x 8a2(a2–x2)–1 16a3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x a–x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 12.⎝integraldisplayx2dx (a2–x2)n+1=x 2n(a2–x2)n–1 2n⎝integraldisplaydx (a2–x2)n;n=1 ,2 , ... 13.⎝integraldisplayx3dx a2–x2=–x2 2–a2 2ln|a2–x2|. 14.⎝integraldisplayx3dx (a2–x2)2=a2 2(a2–x2)+1 2ln|a2–x2|. 936 TABLES OF INDEFINITE INTEGRALS 15.⎝integraldisplayx3dx (a2–x2)n+1=–1 2(n–1 ) (a2–x2)n–1+a2 2n(a2–x2)n;n=2 ,3 , ... 16.⎝integraldisplaydx x(a2–x2)=1 2a2ln⎝vextendsingle⎝vextendsingle⎝vextendsinglex2 a2–x2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 17.⎝integraldisplaydx x(a2–x2)2=1 2a2(a2–x2)+1 2a4ln⎝vextendsingle⎝vextendsingle⎝vextendsinglex2 a2–x2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 18.⎝integraldisplaydx x(a2–x2)3=1 4a2(a2–x2)2+1 2a4(a2–x2)+1 2a6ln⎝vextendsingle⎝vextendsingle⎝vextendsinglex2 a2–x2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 3.1-5. Integrals Involving a3+x3. 1.⎝integraldisplaydx a3+x3=1 6a2ln(a+x)2 a2–ax+x2+1 a2√ 3arctan2x–a a√ 3. 2.⎝integraldisplaydx (a3+x3)2=x 3a3(a3+x3)+2 3a3⎝integraldisplaydx a3+x3. 3.⎝integraldisplayxd x a3+x3=1 6alna2–ax+x2 (a+x)2+1 a√ 3arctan2x–a a√ 3. 4.⎝integraldisplayxd x (a3+x3)2=x2 3a3(a3+x3)+1 3a3⎝integraldisplayxd x a3+x3. 5.⎝integraldisplayx2dx a3+x3=1 3ln|a3+x3|. 6.⎝integraldisplaydx x(a3+x3)=1 3a3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglex3 a3+x3⎝vextendsingle⎝vextendsingle⎝vextendsingle. 7.⎝integraldisplaydx x(a3+x3)2=1 3a3(a3+x3)+1 3a6ln⎝vextendsingle⎝vextendsingle⎝vextendsinglex3 a3+x3⎝vextendsingle⎝vextendsingle⎝vextendsingle. 8.⎝integraldisplaydx x2(a3+x3)=–1 a3x–1 a3⎝integraldisplayxd x a3+x3. 9.⎝integraldisplaydx x2(a3+x3)2=–1 a6x–x2 3a6(a3+x3)–4 3a6⎝integraldisplayxd x a3+x3. 3.1-6. Integrals Involving a3–x3. 1.⎝integraldisplaydx a3–x3=1 6a2lna2+ax+x2 (a–x)2+1 a2√ 3arctan2x+a a√ 3. 2.⎝integraldisplaydx (a3–x3)2=x 3a3(a3–x3)+2 3a3⎝integraldisplaydx a3–x3. 3.⎝integraldisplayxd x a3–x3=1 6alna2+ax+x2 (a–x)2–1 a√ 3arctan2x+a a√ 3. 4.⎝integraldisplayxd x (a3–x3)2=x2 3a3(a3–x3)+1 3a3⎝integraldisplayxd x a3–x3. 5.⎝integraldisplayx2dx a3–x3=–1 3ln|a3–x3|. 6.⎝integraldisplaydx x(a3–x3)=1 3a3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglex3 a3–x3⎝vextendsingle⎝vextendsingle⎝vextendsingle. 7.⎝integraldisplaydx x(a3–x3)2=1 3a3(a3–x3)+1 3a6ln⎝vextendsingle⎝vextendsingle⎝vextendsinglex3 a3–x3⎝vextendsingle⎝vextendsingle⎝vextendsingle. 3.2. I NTEGRALS INVOLVING IRRATIONAL FUNCTIONS 937 8.⎝integraldisplaydx x2(a3–x3)=–1 a3x+1 a3⎝integraldisplayxd x a3–x3. 9.⎝integraldisplaydx x2(a3–x3)2=–1 a6x–x2 3a6(a3–x3)+4 3a6⎝integraldisplayxd x a3–x3. 3.1-7. Integrals Involving a4±x4. 1.⎝integraldisplaydx a4+x4=1 4a3√ 2lna2+ax√ 2+x2 a2–ax√ 2+x2+1 2a3√ 2arctanax√ 2 a2–x2. 2.⎝integraldisplayxd x a4+x4=1 2a2arctanx2 a2. 3.⎝integraldisplayx2dx a4+x4=–1 4a√ 2lna2+ax√ 2+x2 a2–ax√ 2+x2+1 2a√ 2arctanax√ 2 a2–x2. 4.⎝integraldisplaydx a4–x4=1 4a3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x a–x⎝vextendsingle⎝vextendsingle⎝vextendsingle+1 2a3arctanx a. 5.⎝integraldisplayxd x a4–x4=1 4a2ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea2+x2 a2–x2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 6.⎝integraldisplayx2dx a4–x4=1 4aln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x a–x⎝vextendsingle⎝vextendsingle⎝vextendsingle–1 2aarctanx a. 3.2. Integrals Involving Irrational Functions 3.2-1. Integrals Involving x1/2. 1.⎝integraldisplayx1/2dx a2+b2x=2 b2x1/2–2a b3arctanbx1/2 a. 2.⎝integraldisplayx3/2dx a2+b2x=2x3/2 3b2–2a2x1/2 b4+2a3 b5arctanbx1/2 a. 3.⎝integraldisplayx1/2dx (a2+b2x)2=–x1/2 b2(a2+b2x)+1 ab3arctanbx1/2 a. 4.⎝integraldisplayx3/2dx (a2+b2x)2=2x3/2 b2(a2+b2x)+3a2x1/2 b4(a2+b2x)–3a b5arctanbx1/2 a. 5.⎝integraldisplaydx (a2+b2x)x1/2=2 abarctanbx1/2 a. 6.⎝integraldisplaydx (a2+b2x)x3/2=–2 a2x1/2–2b a3arctanbx1/2 a. 7.⎝integraldisplaydx (a2+b2x)2x1/2=x1/2 a2(a2+b2x)+1 a3barctanbx1/2 a. 8.⎝integraldisplayx1/2dx a2–b2x=–2 b2x1/2+2a b3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+bx1/2 a–bx1/2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 9.⎝integraldisplayx3/2dx a2–b2x=–2x3/2 3b2–2a2x1/2 b4+a3 b5ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+bx1/2 a–bx1/2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 10.⎝integraldisplayx1/2dx (a2–b2x)2=x1/2 b2(a2–b2x)–1 2ab3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+bx1/2 a–bx1/2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 938 TABLES OF INDEFINITE INTEGRALS 11.⎝integraldisplayx3/2dx (a2–b2x)2=3a2x1/2–2b2x3/2 b4(a2–b2x)–3a 2b5ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+bx1/2 a–bx1/2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 12.⎝integraldisplaydx (a2–b2x)x1/2=1 abln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+bx1/2 a–bx1/2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 13.⎝integraldisplaydx (a2–b2x)x3/2=–2 a2x1/2+b a3ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+bx1/2 a–bx1/2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 14.⎝integraldisplaydx (a2–b2x)2x1/2=x1/2 a2(a2–b2x)+1 2a3bln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+bx1/2 a–bx1/2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 3.2-2. Integrals Involving ( a+bx)p/2. 1.⎝integraldisplay (a+bx)p/2dx=2 b(p+2 )(a+bx)(p+2)/2. 2.⎝integraldisplay x(a+bx)p/2dx=2 b2⎝bracketleftbigg(a+bx)(p+4)/2 p+4–a(a+bx)(p+2)/2 p+2⎝bracketrightbigg . 3.⎝integraldisplay x2(a+bx)p/2dx=2 b3⎝bracketleftbigg(a+bx)(p+6)/2 p+6–2a(a+bx)(p+4)/2 p+4+a2(a+bx)(p+2)/2 p+2⎝bracketrightbigg . 3.2-3. Integrals Involving ( x2+a2)1/2. 1.⎝integraldisplay (x2+a2)1/2dx=1 2x(a2+x2)1/2+a2 2ln⎝bracketleftbig x+(x2+a2)1/2⎝bracketrightbig . 2.⎝integraldisplay x(x2+a2)1/2dx=1 3(a2+x2)3/2. 3.⎝integraldisplay (x2+a2)3/2dx=1 4x(a2+x2)3/2+3 8a2x(a2+x2)1/2+3 8a4ln⎝vextendsingle⎝vextendsinglex+(x2+a2)1/2⎝vextendsingle⎝vextendsingle. 4.⎝integraldisplay1 x(x2+a2)1/2dx=(a2+x2)1/2–aln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+(x2+a2)1/2 x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 5.⎝integraldisplaydx √ x2+a2=l n⎝bracketleftbig x+(x2+a2)1/2⎝bracketrightbig . 6.⎝integraldisplayxd x √ x2+a2=(x2+a2)1/2. 7.⎝integraldisplay (x2+a2)–3/2dx=a–2x(x2+a2)–1/2. 3.2-4. Integrals Involving ( x2–a2)1/2. 1.⎝integraldisplay (x2–a2)1/2dx=1 2x(x2–a2)1/2–a2 2ln⎝vextendsingle⎝vextendsinglex+(x2–a2)1/2⎝vextendsingle⎝vextendsingle. 2.⎝integraldisplay x(x2–a2)1/2dx=1 3(x2–a2)3/2. 3.⎝integraldisplay (x2–a2)3/2dx=1 4x(x2–a2)3/2–3 8a2x(x2–a2)1/2+3 8a4ln⎝vextendsingle⎝vextendsinglex+(x2–a2)1/2⎝vextendsingle⎝vextendsingle. 4.⎝integraldisplay1 x(x2–a2)1/2dx=(x2–a2)1/2–aarccos⎝vextendsingle⎝vextendsingle⎝vextendsinglea x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 3.2. I NTEGRALS INVOLVING IRRATIONAL FUNCTIONS 939 5.⎝integraldisplaydx √ x2–a2=l n⎝vextendsingle⎝vextendsinglex+(x2–a2)1/2⎝vextendsingle⎝vextendsingle. 6.⎝integraldisplayxd x √ x2–a2=(x2–a2)1/2. 7.⎝integraldisplay (x2–a2)–3/2dx=–a–2x(x2–a2)–1/2. 3.2-5. Integrals Involving ( a2–x2)1/2. 1.⎝integraldisplay (a2–x2)1/2dx=1 2x(a2–x2)1/2+a2 2arcsinx a. 2.⎝integraldisplay x(a2–x2)1/2dx=–1 3(a2–x2)3/2. 3.⎝integraldisplay (a2–x2)3/2dx=1 4x(a2–x2)3/2+3 8a2x(a2–x2)1/2+3 8a4arcsinx a. 4.⎝integraldisplay1 x(a2–x2)1/2dx=(a2–x2)1/2–aln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+(a2–x2)1/2 x⎝vextendsingle⎝vextendsingle⎝vextendsingle. 5.⎝integraldisplaydx √ a2–x2=a r c s i nx a. 6.⎝integraldisplayxd x √ a2–x2=– (a2–x2)1/2. 7.⎝integraldisplay (a2–x2)–3/2dx=a–2x(a2–x2)–1/2. 3.2-6. Integrals Involving Arbitrary Powers. Reduction Formulas. 1.⎝integraldisplaydx x(axn+b)=1 bnln⎝vextendsingle⎝vextendsingle⎝vextendsinglexn axn+b⎝vextendsingle⎝vextendsingle⎝vextendsingle. 2.⎝integraldisplaydx x√ xn+a2=2 anln⎝vextendsingle⎝vextendsingle⎝vextendsinglexn/2 √ xn+a2+a⎝vextendsingle⎝vextendsingle⎝vextendsingle. 3.⎝integraldisplaydx x√ xn–a2=2 anarccos⎝vextendsingle⎝vextendsingle⎝vextendsinglea xn/2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 4.⎝integraldisplaydx x√ ax2n+bxn=–2√ ax2n+bxn bnxn. /trianglerightsldThe parameters a,b,p,m,andnbelow in Integrals 5–8 can assume arbitrary values, except for those at which denominators vanish in successive applications of a formula. Notation: w=axn+b. 5.⎝integraldisplay xm(axn+b)pdx=1 m+np+1⎝parenleftBig xm+1wp+npb⎝integraldisplay xmwp–1dx⎝parenrightBig . 6.⎝integraldisplay xm(axn+b)pdx=1 bn(p+1 )⎝bracketleftBig –xm+1wp+1+(m+n+np+1 )⎝integraldisplay xmwp+1dx⎝bracketrightBig . 7.⎝integraldisplay xm(axn+b)pdx=1 b(m+1 )⎝bracketleftBig xm+1wp+1–a(m+n+np+1 )⎝integraldisplay xm+nwpdx⎝bracketrightBig . 8.⎝integraldisplay xm(axn+b)pdx=1 a(m+np+1 )⎝bracketleftBig xm–n+1wp+1–b(m–n+1 )⎝integraldisplay xm–nwpdx⎝bracketrightBig . 940 TABLES OF INDEFINITE INTEGRALS 3.3. Integrals Involving Exponential Functions 1.⎝integraldisplay eaxdx=1 aeax. 2.⎝integraldisplay axdx=ax lna. 3.⎝integraldisplay xeaxdx=eax⎝parenleftBigx a–1 a2⎝parenrightBig . 4.⎝integraldisplay x2eaxdx=eax⎝parenleftBigx2 a–2x a2+2 a3⎝parenrightBig . 5.⎝integraldisplay xneaxdx=eax⎝bracketleftBig1 axn–n a2xn–1+n(n–1 ) a3xn–2–···+(–1)n–1n! anx+(–1)nn! an+1⎝bracketrightBig ,n=1, 2, ... 6.⎝integraldisplay Pn(x)eaxdx=eaxn⎝summationdisplay k=0(–1)k ak+1dk dxkPn(x), where Pn(x) is an arbitrary polynomial of degree n. 7.⎝integraldisplaydx a+bepx=x a–1 apln|a+bepx|. 8.⎝integraldisplaydx aepx+be–px=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩1 p√ abarctan⎝parenleftbigg epx⎝radicalbigg a b⎝parenrightbigg ifab>0 , 1 2p√ –abln⎝parenleftbiggb+epx√ –ab b–epx√ –ab⎝parenrightbigg ifab<0 . 9.⎝integraldisplaydx √ a+bepx=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩1 p√ aln√ a+bepx–√ a √ a+bepx+√ aifa>0 , 2 p√ –aarctan√ a+bepx √ –aifa<0 . 3.4. Integrals Involving Hyperbolic Functions 3.4-1. Integrals Involving cosh x. 1.⎝integraldisplay cosh(a +bx)dx=1 bsinh(a+bx). 2.⎝integraldisplay xcoshxd x =xsinhx–c o s h x. 3.⎝integraldisplay x2coshxd x =(x2+2 )s i n h x–2xcoshx. 4.⎝integraldisplay x2ncoshxd x =( 2n)!n⎝summationdisplay k=1⎝bracketleftbiggx2k (2k)!sinhx–x2k–1 (2k–1 ) !coshx⎝bracketrightbigg . 5.⎝integraldisplay x2n+1coshxd x =( 2n+1 ) !n⎝summationdisplay k=0⎝bracketleftbiggx2k+1 (2k+1 ) !sinhx–x2k (2k)!coshx⎝bracketrightbigg . 6.⎝integraldisplay xpcoshxd x =xpsinhx–pxp–1coshx+p(p–1 )⎝integraldisplay xp–2coshxd x. 7.⎝integraldisplay cosh2xd x =1 2x+1 4sinh 2x. 8.⎝integraldisplay cosh3xd x =s i n h x+1 3sinh3x. 3.4. I NTEGRALS INVOLVING HYPERBOLIC FUNCTIONS 941 9.⎝integraldisplay cosh2nxd x =Cn 2nx 22n+1 22n–1n–1⎝summationdisplay k=0Ck 2nsinh[2( n–k)x] 2(n–k),n=1 ,2 , ... 10.⎝integraldisplay cosh2n+1xd x =1 22nn⎝summationdisplay k=0Ck 2n+1sinh[(2 n–2k+1 )x] 2n–2k+1=n⎝summationdisplay k=0Ck nsinh2k+1x 2k+1,n=1 ,2 , ... 11.⎝integraldisplay coshpxd x =1 psinhxcoshp–1x+p–1 p⎝integraldisplay coshp–2xd x. 12.⎝integraldisplay coshaxcoshbxdx =1 a2–b2(acoshbxsinhax–bcoshaxsinhbx). 13.⎝integraldisplaydx coshax=2 aarctan⎝parenleftbig eax⎝parenrightbig . 14.⎝integraldisplaydx cosh2nx=sinhx 2n–1⎝bracketleftbigg1 cosh2n–1x +n–1⎝summationdisplay k=12k(n–1 ) (n–2 )...(n–k) (2n– 3)(2n–5 )...(2n–2k–1 )1 cosh2n–2k–1x⎝bracketrightbigg ,n=1 ,2 , ... 15.⎝integraldisplaydx cosh2n+1x=sinhx 2n⎝bracketleftbigg1 cosh2nx +n–1⎝summationdisplay k=1(2n– 1)(2n–3 )...(2n–2k+1 ) 2k(n–1 ) (n–2 )...(n–k)1 cosh2n–2kx⎝bracketrightbigg +(2n– 1)!! (2n)!!arctan sinh x,n=1 ,2 , ... 16.⎝integraldisplaydx a+bcoshx=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩–signx √ b2–a2arcsinb+acoshx a+bcoshxifa2<b2, 1 √ a2–b2lna+b+√ a2–b2tanh(x/2) a+b–√ a2–b2tanh(x/2)ifa2>b2. 3.4-2. Integrals Involving sinh x. 1.⎝integraldisplay sinh(a+bx)dx=1 bcosh(a+bx). 2.⎝integraldisplay xsinhxd x =xcoshx–s i n h x. 3.⎝integraldisplay x2sinhxd x =(x2+2 )c o s h x–2xsinhx. 4.⎝integraldisplay x2nsinhxd x =( 2n)!⎝bracketleftbiggn⎝summationdisplay k=0x2k (2k)!coshx–n⎝summationdisplay k=1x2k–1 (2k–1 ) !sinhx⎝bracketrightbigg . 5.⎝integraldisplay x2n+1sinhxd x =( 2n+1 ) !n⎝summationdisplay k=0⎝bracketleftbiggx2k+1 (2k+1 ) !coshx–x2k (2k)!sinhx⎝bracketrightbigg . 6.⎝integraldisplay xpsinhxd x =xpcoshx–pxp–1sinhx+p(p–1 )⎝integraldisplay xp–2sinhxd x. 7.⎝integraldisplay sinh2xd x =–1 2x+1 4sinh 2x. 8.⎝integraldisplay sinh3xd x =–c o s h x+1 3cosh3x. 9.⎝integraldisplay sinh2nxd x = (–1)nCn 2nx 22n+1 22n–1n–1⎝summationdisplay k=0(–1)kCk 2nsinh[2( n–k)x] 2(n–k),n=1 ,2 , ... 942 TABLES OF INDEFINITE INTEGRALS 10.⎝integraldisplay sinh2n+1xd x =1 22nn⎝summationdisplay k=0(–1)kCk 2n+1cosh[(2 n–2k+1 )x] 2n–2k+1 =n⎝summationdisplay k=0(–1)n+kCk ncosh2k+1x 2k+1,n=1 ,2 , ... 11.⎝integraldisplay sinhpxd x =1 psinhp–1xcoshx–p–1 p⎝integraldisplay sinhp–2xd x. 12.⎝integraldisplay sinhaxsinhbxdx =1 a2–b2⎝parenleftbig acoshaxsinhbx–bcoshbxsinhax⎝parenrightbig . 13.⎝integraldisplaydx sinhax=1 aln⎝vextendsingle⎝vextendsingle⎝vextendsingletanhax 2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 14.⎝integraldisplaydx sinh2nx=coshx 2n–1⎝bracketleftbigg –1 sinh2n–1x +n–1⎝summationdisplay k=1(–1)k–1 2k(n–1 ) (n–2 )...(n–k) (2n– 3)(2n–5 )...(2n–2k–1 )1 sinh2n–2k–1x⎝bracketrightbigg ,n=1 ,2 , ... 15.⎝integraldisplaydx sinh2n+1x=coshx 2n⎝bracketleftbigg –1 sinh2nx +n–1⎝summationdisplay k=1(–1)k–1(2n–1)(2n–3)...(2n–2k+1) 2k(n–1)(n–2)...(n–k)1 sinh2n–2kx⎝bracketrightbigg +(–1)n(2n–1)!! (2n)!!ln tanhx 2, n=1 ,2 , ... 16.⎝integraldisplaydx a+bsinhx=1 √ a2+b2lnatanh(x/2) –b+√ a2+b2 atanh(x/2) –b–√ a2+b2. 17.⎝integraldisplayAx+Bsinhx a+bsinhxdx=B bx+Ab–Ba b√ a2+b2lnatanh(x/2) –b+√ a2+b2 atanh(x/2) –b–√ a2+b2. 3.4-3. Integrals Involving tanh xor coth x. 1.⎝integraldisplay tanhxd x =l nc o s h x. 2.⎝integraldisplay tanh2xd x =x–t a n h x. 3.⎝integraldisplay tanh3xd x =–1 2tanh2x+l nc o s h x. 4.⎝integraldisplay tanh2nxd x =x–n⎝summationdisplay k=1tanh2n–2k+1x 2n–2k+1,n=1 ,2 , ... 5.⎝integraldisplay tanh2n+1xd x =l nc o s h x–n⎝summationdisplay k=1(–1)kCk n 2kcosh2kx=l nc o s h x–n⎝summationdisplay k=1tanh2n–2k+2x 2n–2k+2,n=1 ,2 , ... 6.⎝integraldisplay tanhpxd x =–1 p–1tanhp–1x+⎝integraldisplay tanhp–2xd x. 7.⎝integraldisplay cothxd x =l n|sinhx|. 8.⎝integraldisplay coth2xd x =x–c o t h x. 9.⎝integraldisplay coth3xd x =–1 2coth2x+l n|sinhx|. 3.5. I NTEGRALS INVOLVING LOGARITHMIC FUNCTIONS 943 10.⎝integraldisplay coth2nxd x =x–n⎝summationdisplay k=1coth2n–2k+1x 2n–2k+1,n=1 ,2 , ... 11.⎝integraldisplay coth2n+1xd x =l n|sinhx|–n⎝summationdisplay k=1Ck n 2ksinh2kx=l n|sinhx|–n⎝summationdisplay k=1coth2n–2k+2x 2n–2k+2,n=1 , 2 , ... 12.⎝integraldisplay cothpxd x =–1 p–1cothp–1x+⎝integraldisplay cothp–2xd x. 3.5. Integrals Involving Logarithmic Functions 1.⎝integraldisplay lnaxdx =xlnax–x. 2.⎝integraldisplay xlnxd x =1 2x2lnx–1 4x2. 3.⎝integraldisplay xplnaxdx =⎧ ⎨ ⎩1 p+1xp+1lnax–1 (p+1 )2xp+1ifp≠–1, 1 2ln2ax ifp= –1. 4.⎝integraldisplay (lnx)2dx=x(lnx)2–2xlnx+2x. 5.⎝integraldisplay x(lnx)2dx=1 2x2(lnx)2–1 2x2lnx+1 4x2. 6.⎝integraldisplay xp(lnx)2dx=⎧ ⎪⎨ ⎪⎩xp+1 p+1(lnx)2–2xp+1 (p+1 )2lnx+2xp+1 (p+1 )3ifp≠–1, 1 3ln3x ifp= –1. 7.⎝integraldisplay (lnx)ndx=x n+1n⎝summationdisplay k=0(–1)k(n+1 )n... (n–k+ 1)(ln x)n–k,n=1 ,2 , ... 8.⎝integraldisplay (lnx)qdx=x(lnx)q–q⎝integraldisplay (lnx)q–1dx,q≠–1. 9.⎝integraldisplay xn(lnx)mdx=xn+1 m+1m⎝summationdisplay k=0(–1)k (n+1 )k+1(m+1 )m... (m–k+ 1)(ln x)m–k,n,m=1 ,2 , ... 10.⎝integraldisplay xp(lnx)qdx=1 p+1xp+1(lnx)q–q p+1⎝integraldisplay xp(lnx)q–1dx,p,q≠–1. 11.⎝integraldisplay ln(a+bx)dx=1 b(ax+b)l n (ax+b)–x. 12.⎝integraldisplay xln(a+bx)dx=1 2⎝parenleftbigg x2–a2 b2⎝parenrightbigg ln(a+bx)–1 2⎝parenleftbiggx2 2–a bx⎝parenrightbigg . 13.⎝integraldisplay x2ln(a+bx)dx=1 3⎝parenleftbigg x3–a3 b3⎝parenrightbigg ln(a+bx)–1 3⎝parenleftbiggx3 3–ax2 2b+a2x b2⎝parenrightbigg . 14.⎝integraldisplaylnxd x (a+bx)2=–lnx b(a+bx)+1 ablnx a+bx. 15.⎝integraldisplaylnxd x (a+bx)3=–lnx 2b(a+bx)2+1 2ab(a+bx)+1 2a2blnx a+bx. 16.⎝integraldisplaylnxd x √ a+bx=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩2 b⎝bracketleftbigg (lnx–2 )√ a+bx+√ aln√ a+bx+√ a √ a+bx–√ a⎝bracketrightbigg ifa>0 , 2 b⎝bracketleftbigg (lnx–2 )√ a+bx+2√ –aarctan√ a+bx √ –a⎝bracketrightbigg ifa<0 . 944 TABLES OF INDEFINITE INTEGRALS 17.⎝integraldisplay ln(x2+a2)dx=xln(x2+a2)–2x+2aarctan( x/a). 18.⎝integraldisplay xln(x2+a2)dx=1 2⎝bracketleftbig (x2+a2)l n (x2+a2)–x2⎝bracketrightbig . 19.⎝integraldisplay x2ln(x2+a2)dx=1 3⎝bracketleftbig x3ln(x2+a2)–2 3x3+2a2x–2a3arctan( x/a)⎝bracketrightbig . 3.6. Integrals Involving Trigonometric Functions 3.6-1. Integrals Involving cos x(n=1 ,2 , ...). 1.⎝integraldisplay cos(a +bx)dx=1 bsin(a+bx). 2.⎝integraldisplay xcosxd x =c o sx+xsinx. 3.⎝integraldisplay x2cosxd x =2xcosx+(x2–2 )s i n x. 4.⎝integraldisplay x2ncosxd x =( 2n)!⎝bracketleftbiggn⎝summationdisplay k=0(–1)kx2n–2k (2n–2k)!sinx+n–1⎝summationdisplay k=0(–1)kx2n–2k–1 (2n–2k–1 ) !cosx⎝bracketrightbigg . 5.⎝integraldisplay x2n+1cosxd x =( 2n+1 ) !n⎝summationdisplay k=0⎝bracketleftbigg (–1)kx2n–2k+1 (2n–2k+1 ) !sinx+x2n–2k (2n–2k)!cosx⎝bracketrightbigg . 6.⎝integraldisplay xpcosxd x =xpsinx+pxp–1cosx–p(p–1 )⎝integraldisplay xp–2cosxd x. 7.⎝integraldisplay cos2xd x =1 2x+1 4sin 2x. 8.⎝integraldisplay cos3xd x =s i nx–1 3sin3x. 9.⎝integraldisplay cos2nxd x =1 22nCn 2nx+1 22n–1n–1⎝summationdisplay k=0Ck 2nsin[(2n–2k)x] 2n–2k. 10.⎝integraldisplay cos2n+1xd x =1 22nn⎝summationdisplay k=0Ck 2n+1sin[(2n–2k+1 )x] 2n–2k+1. 11.⎝integraldisplaydx cosx=l n⎝vextendsingle⎝vextendsingle⎝vextendsingletan⎝parenleftBigx 2+π 4⎝parenrightBig⎝vextendsingle⎝vextendsingle⎝vextendsingle. 12.⎝integraldisplaydx cos2x=t a nx. 13.⎝integraldisplaydx cos3x=sinx 2c o s2x+1 2ln⎝vextendsingle⎝vextendsingle⎝vextendsingletan⎝parenleftBigx 2+π 4⎝parenrightBig⎝vextendsingle⎝vextendsingle⎝vextendsingle. 14.⎝integraldisplaydx cosnx=sinx (n–1 )c o sn–1x+n–2 n–1⎝integraldisplaydx cosn–2x,n>1 . 15.⎝integraldisplayxd x cos2nx=n–1⎝summationdisplay k=0(2n–2)(2n–4)...(2n–2k+2) (2n–1)(2n–3)...(2n–2k+3)(2n–2k)xsinx–cosx (2n–2k+1)(2n–2k)c o s2n–2k+1x +2n–1(n–1)! (2n–1)!!⎝parenleftbig xtanx+ln|cosx|⎝parenrightbig . 16.⎝integraldisplay cosaxcosbxdx =sin⎝bracketleftbig (b–a)x⎝bracketrightbig 2(b–a)+sin⎝bracketleftbig (b+a)x⎝bracketrightbig 2(b+a),a≠±b. 3.6. I NTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS 945 17.⎝integraldisplaydx a+bcosx=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩2 √ a2–b2arctan(a–b)t a n (x/2) √ a2–b2ifa2>b2, 1 √ b2–a2ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle√ b2–a2+(b–a)t a n (x/2) √ b2–a2–(b–a)t a n (x/2)⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleifb2>a2. 18.⎝integraldisplaydx (a+bcosx)2=bsinx (b2–a2)(a+bcosx)–a b2–a2⎝integraldisplaydx a+bcosx. 19.⎝integraldisplaydx a2+b2cos2x=1 a√ a2+b2arctanatanx √ a2+b2. 20.⎝integraldisplaydx a2–b2cos2x=⎧ ⎪⎪⎨ ⎪⎪⎩1 a√ a2–b2arctanatanx √ a2–b2ifa2>b2, 1 2a√ b2–a2ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle√ b2–a2–atanx √ b2–a2+atanx⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleifb 2>a2. 21.⎝integraldisplay eaxcosbxdx =eax⎝parenleftbiggb a2+b2sinbx+a a2+b2cosbx⎝parenrightbigg . 22.⎝integraldisplay eaxcos2xd x =eax a2+4⎝parenleftbigg acos2x+2s i n xcosx+2 a⎝parenrightbigg . 23.⎝integraldisplay eaxcosnxd x =eaxcosn–1x a2+n2(acosx+nsinx)+n(n–1 ) a2+n2⎝integraldisplay eaxcosn–2xd x. 3.6-2. Integrals Involving sin x(n=1 ,2 , ...). 1.⎝integraldisplay sin(a+bx)dx=–1 bcos(a +bx). 2.⎝integraldisplay xsinxd x =s i nx–xcosx. 3.⎝integraldisplay x2sinxd x =2xsinx–(x2–2 )c o s x. 4.⎝integraldisplay x3sinxd x =( 3x2–6 )s i n x–(x3–6x)c o sx. 5.⎝integraldisplay x2nsinxd x =( 2n)!⎝bracketleftbiggn⎝summationdisplay k=0(–1)k+1x2n–2k (2n–2k)!cosx+n–1⎝summationdisplay k=0(–1)kx2n–2k–1 (2n–2k–1 ) !sinx⎝bracketrightbigg . 6.⎝integraldisplay x2n+1sinxd x =( 2n+1 ) !n⎝summationdisplay k=0⎝bracketleftbigg (–1)k+1x2n–2k+1 (2n–2k+1 ) !cosx+ (–1)kx2n–2k (2n–2k)!sinx⎝bracketrightbigg . 7.⎝integraldisplay xpsinxd x =–xpcosx+pxp–1sinx–p(p–1 )⎝integraldisplay xp–2sinxd x. 8.⎝integraldisplay sin2xd x =1 2x–1 4sin 2x. 9.⎝integraldisplay xsin2xd x =1 4x2–1 4xsin 2x–1 8cos 2x. 10.⎝integraldisplay sin3xd x =–c o s x+1 3cos3x. 11.⎝integraldisplay sin2nxd x =1 22nCn 2nx+(–1)n 22n–1n–1⎝summationdisplay k=0(–1)kCk 2nsin[(2n–2k)x] 2n–2k, where Ck m=m! k!(m–k)!are binomial coefﬁcients (0! = 1). 946 TABLES OF INDEFINITE INTEGRALS 12.⎝integraldisplay sin2n+1xd x =1 22nn⎝summationdisplay k=0(–1)n+k+1Ck 2n+1cos[(2 n–2k+1 )x] 2n–2k+1. 13.⎝integraldisplaydx sinx=l n⎝vextendsingle⎝vextendsingle⎝vextendsingletanx 2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 14.⎝integraldisplaydx sin2x=–c o t x. 15.⎝integraldisplaydx sin3x=–cosx 2s i n2x+1 2ln⎝vextendsingle⎝vextendsingle⎝vextendsingletanx 2⎝vextendsingle⎝vextendsingle⎝vextendsingle. 16.⎝integraldisplaydx sinnx=–cosx (n–1 )s i nn–1x+n–2 n–1⎝integraldisplaydx sinn–2x,n>1 . 17.⎝integraldisplayxd x sin2nx=–n–1⎝summationdisplay k=0(2n–2)(2n–4)...(2n–2k+2) (2n–1)(2n–3)...(2n–2k+3)sinx+(2n–2k)xcosx (2n–2k+1)(2n–2k)s i n2n–2k+1x +2n–1(n–1)! (2n–1)!!⎝parenleftbig ln|sinx|–xcotx⎝parenrightbig . 18.⎝integraldisplay sinaxsinbxdx =sin[(b–a)x] 2(b–a)–sin[(b+a)x] 2(b+a),a≠±b. 19.⎝integraldisplaydx a+bsinx=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩2 √ a2–b2arctanb+atanx/2 √ a2–b2ifa2>b2, 1 √ b2–a2ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleb–√ b2–a2+atanx/2 b+√ b2–a2+atanx/2⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleifb 2>a2. 20.⎝integraldisplaydx (a+bsinx)2=bcosx (a2–b2)(a+bsinx)+a a2–b2⎝integraldisplaydx a+bsinx. 21.⎝integraldisplaydx a2+b2sin2x=1 a√ a2+b2arctan√ a2+b2tanx a. 22.⎝integraldisplaydx a2–b2sin2x=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩1 a√ a2–b2arctan√ a2–b2tanx aifa2>b2, 1 2a√ b2–a2ln⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle√ b2–a2tanx+a √ b2–a2tanx–a⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingleifb 2>a2. 23.⎝integraldisplaysinxd x √ 1+k2sin2x=–1 karcsinkcosx √ 1+k2. 24.⎝integraldisplaysinxd x √ 1–k2sin2x=–1 kln⎝vextendsingle⎝vextendsinglekcosx+√ 1–k2sin2x⎝vextendsingle⎝vextendsingle. 25.⎝integraldisplay sinx√ 1+k2sin2xd x =–cosx 2√ 1+k2sin2x–1+k2 2karcsinkcosx √ 1+k2. 26.⎝integraldisplay sinx√ 1–k2sin2xd x =–cosx 2√ 1–k2sin2x–1–k2 2kln⎝vextendsingle⎝vextendsinglekcosx+√ 1–k2sin2x⎝vextendsingle⎝vextendsingle. 27.⎝integraldisplay eaxsinbxdx =eax⎝parenleftBiga a2+b2sinbx–b a2+b2cosbx⎝parenrightBig . 28.⎝integraldisplay eaxsin2xd x =eax a2+4⎝parenleftBig asin2x–2s i n xcosx+2 a⎝parenrightBig . 29.⎝integraldisplay eaxsinnxd x =eaxsinn–1x a2+n2(asinx–ncosx)+n(n–1 ) a2+n2⎝integraldisplay eaxsinn–2xd x. 3.6. I NTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS 947 3.6-3. Integrals Involving sin xand cos x. 1.⎝integraldisplay sinaxcosbxdx =–cos[(a +b)x] 2(a+b)–cos⎝bracketleftbig (a–b)x⎝bracketrightbig 2(a–b),a≠±b. 2.⎝integraldisplaydx b2cos2ax+c2sin2ax=1 abcarctan⎝parenleftBigc btanax⎝parenrightBig . 3.⎝integraldisplaydx b2cos2ax–c2sin2ax=1 2abcln⎝vextendsingle⎝vextendsingle⎝vextendsinglectanax+b ctanax–b⎝vextendsingle⎝vextendsingle⎝vextendsingle. 4.⎝integraldisplaydx cos2nxsin2mx=n+m–1⎝summationdisplay k=0Ck n+m–1tan2k–2m +1x 2k–2m+1,n,m=1 ,2 , ... 5.⎝integraldisplaydx cos2n+1xsin2m+1x=Cm n+mln|tanx|+n+m⎝summationdisplay k=0Ck n+mtan2k–2mx 2k–2m,n,m=1 ,2 , ... 3.6-4. Reduction Formulas. /trianglerightsldThe parameters pandqbelow can assume any values,except for those at which the denominators on the right-hand side vanish . 1.⎝integraldisplay sinpxcosqxd x =–sinp–1xcosq+1x p+q+p–1 p+q⎝integraldisplay sinp–2xcosqxd x. 2.⎝integraldisplay sinpxcosqxd x =sinp+1xcosq–1x p+q+q–1 p+q⎝integraldisplay sinpxcosq–2xd x. 3.⎝integraldisplay sinpxcosqxd x =sinp–1xcosq–1x p+q⎝parenleftBig sin2x–q–1 p+q–2⎝parenrightBig +(p–1 ) (q–1 ) (p+q)(p+q–2 )⎝integraldisplay sinp–2xcosq–2xd x. 4.⎝integraldisplay sinpxcosqxd x =sinp+1xcosq+1x p+1+p+q+2 p+1⎝integraldisplay sinp+2xcosqxd x. 5.⎝integraldisplay sinpxcosqxd x =–sinp+1xcosq+1x q+1+p+q+2 q+1⎝integraldisplay sinpxcosq+2xd x. 6.⎝integraldisplay sinpxcosqxd x =–sinp–1xcosq+1x q+1+p–1 q+1⎝integraldisplay sinp–2xcosq+2xd x. 7.⎝integraldisplay sinpxcosqxd x =sinp+1xcosq–1x p+1+q–1 p+1⎝integraldisplay sinp+2xcosq–2xd x. 3.6-5. Integrals Involving tan xand cot x. 1.⎝integraldisplay tanxd x =–l n |cosx|. 2.⎝integraldisplay tan2xd x =t a nx–x. 3.⎝integraldisplay tan3xd x =1 2tan2x+l n|cosx|. 4.⎝integraldisplay tan2nxd x = (–1)nx–n⎝summationdisplay k=1(–1)k(tanx)2n–2k+1 2n–2k+1,n=1 ,2 , ... 948 TABLES OF INDEFINITE INTEGRALS 5.⎝integraldisplay tan2n+1xd x = (–1)n+1ln|cosx|–n⎝summationdisplay k=1(–1)k(tanx)2n–2k+2 2n–2k+2,n=1 ,2 , ... 6.⎝integraldisplaydx a+btanx=1 a2+b2⎝parenleftbig ax+bln|acosx+bsinx|⎝parenrightbig . 7.⎝integraldisplaytanxd x √ a+btan2x=1 √ b–aarccos⎝parenleftbigg⎝radicalbigg 1–a bcosx⎝parenrightbigg ,b>a,b>0 . 8.⎝integraldisplay cotxd x =l n|sinx|. 9.⎝integraldisplay cot2xd x =–c o t x–x. 10.⎝integraldisplay cot3xd x =–1 2cot2x–l n|sinx|. 11.⎝integraldisplay cot2nxd x = (–1)nx+n⎝summationdisplay k=1(–1)k(cotx)2n–2k+1 2n–2k+1,n=1 ,2 , ... 12.⎝integraldisplay cot2n+1xd x = (–1)nln|sinx|+n⎝summationdisplay k=1(–1)k(cotx)2n–2k+2 2n–2k+2,n=1 ,2 , ... 13.⎝integraldisplaydx a+bcotx=1 a2+b2⎝parenleftbig ax–bln|asinx+bcosx|⎝parenrightbig . 3.7. Integrals Involving Inverse Trigonometric Functions 1.⎝integraldisplay arcsinx adx=xarcsinx a+√ a2–x2. 2.⎝integraldisplay⎝parenleftBig arcsinx a⎝parenrightBig2 dx=x⎝parenleftBig arcsinx a⎝parenrightBig2 –2x+2√ a2–x2arcsinx a. 3.⎝integraldisplay xarcsinx adx=1 4(2x2–a2)a r c s i nx a+x 4√ a2–x2. 4.⎝integraldisplay x2arcsinx adx=x3 3arcsinx a+1 9(x2+2a2)√ a2–x2. 5.⎝integraldisplay arccosx adx=xarccosx a–√ a2–x2. 6.⎝integraldisplay⎝parenleftBig arccosx a⎝parenrightBig2 dx=x⎝parenleftBig arccosx a⎝parenrightBig2 –2x–2√ a2–x2arccosx a. 7.⎝integraldisplay xarccosx adx=1 4(2x2–a2) arccosx a–x 4√ a2–x2. 8.⎝integraldisplay x2arccosx adx=x3 3arccosx a–1 9(x2+2a2)√ a2–x2. 9.⎝integraldisplay arctanx adx=xarctanx a–a 2ln(a2+x2). 10.⎝integraldisplay xarctanx adx=1 2(x2+a2)a r c t a nx a–ax 2. 11.⎝integraldisplay x2arctanx adx=x3 3arctanx a–ax2 6+a3 6ln(a2+x2). 3.7. I NTEGRALS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS 949 12.⎝integraldisplay arccotx adx=xarccotx a+a 2ln(a2+x2). 13.⎝integraldisplay xarccotx adx=1 2(x2+a2) arccotx a+ax 2. 14.⎝integraldisplay x2arccotx adx=x3 3arccotx a+ax2 6–a3 6ln(a2+x2). References for Supplement 3: H. B. Dwight (1961), I. S. Gradshteyn and I. M. Ryzhik (2000), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986, 1988), D. Zwillinger (2002), I. N. Bronshtein and K. A. Semendyayev (2004). Supplement 4 Tables of Deﬁnite Integrals /trianglerightsldThroughout Supplement 4 it is assumed that nis a positive integer , unless otherwise speciﬁed . 4.1. Integrals Involving Power-Law Functions 4.1-1. Integrals Over a Finite Interval. 1.⎝integraldisplay1 0xndx x+1= (–1)n⎝bracketleftbigg ln 2 +n⎝summationdisplay k=1(–1)k k⎝bracketrightbigg . 2.⎝integraldisplay1 0dx x2+2xcosβ+1=β 2s i nβ. 3.⎝integraldisplay1 0⎝parenleftbig xa+x–a⎝parenrightbig dx x2+2xcosβ+1=πsin(aβ) sin(πa)s i nβ,|a|<1 ,β≠(2n+1 )π. 4.⎝integraldisplay1 0xa(1 –x)1–adx=πa(1 –a) 2s i n (πa),– 1 < a<1 . 5.⎝integraldisplay1 0dx xa(1 –x)1–a=π sin(πa),0 < a<1 . 6.⎝integraldisplay1 0xadx (1 –x)a=πa sin(πa),– 1 < a<1 . 7.⎝integraldisplay1 0xp–1(1 –x)q–1dx≡B(p,q)=Γ(p)Γ(q) Γ(p+q),p,q>0 . 8.⎝integraldisplay1 0xp–1(1 –xq)–p/qdx=π qsin(πp/q ),q>p>0 . 9.⎝integraldisplay1 0xp+q–1(1 –xq)–p/qdx=πp q2sin(πp/q ),q>p. 10.⎝integraldisplay1 0xq/p–1(1 –xq)–1/pdx=π qsin(π/p),p>1 ,q>0 . 11.⎝integraldisplay1 0xp–1–x–p 1–xdx=πcot(πp),|p|<1 . 12.⎝integraldisplay1 0xp–1–x–p 1+xdx=π sin(πp),|p|<1 . 951 952 TABLES OF DEFINITE INTEGRALS 13.⎝integraldisplay1 0xp–x–p x–1dx=1 p–πcot(πp),|p|<1 . 14.⎝integraldisplay1 0xp–x–p 1+xdx=1 p–π sin(πp),|p|<1 . 15.⎝integraldisplay1 0x1+p–x1–p 1–x2dx=π 2cot⎝parenleftBigπp 2⎝parenrightBig –1 p,|p|<1 . 16.⎝integraldisplay1 0x1+p–x1–p 1+x2dx=1 p–π 2s i n (πp/2),|p|<1 . 17.⎝integraldisplay1 0dx ⎝radicalbig (1 +a2x)(1 –x)=2 aarctan a. 18.⎝integraldisplay1 0dx ⎝radicalbig (1 –a2x)(1 –x)=1 aln1+a 1–a. 19.⎝integraldisplay1 –1dx (a–x)√ 1–x2=π √ a2–1,1 < a. 20.⎝integraldisplay1 0xndx √ 1–x=2( 2n)!! (2n+ 1)!!,n=1 ,2 , ... 21.⎝integraldisplay1 0xn–1/2dx √ 1–x=π(2n– 1)!! (2n)!!,n=1 ,2 , ... 22.⎝integraldisplay1 0x2ndx √ 1–x2=π 21×3×...×(2n–1 ) 2×4×...×(2n),n=1 ,2 , ... 23.⎝integraldisplay1 0x2n+1dx √ 1–x2=2×4×...×(2n) 1×3×...×(2n+1 ),n=1 ,2 , ... 24.⎝integraldisplay1 0xλ–1dx (1 +ax)(1 –x)λ=π (1 +a)λsin(πλ),0 < λ<1 , a> –1. 25.⎝integraldisplay1 0xλ–1/2dx (1 +ax)λ(1 –x)λ=2π–1/2Γ⎝parenleftbig λ+1 2⎝parenrightbig Γ⎝parenleftbig 1–λ⎝parenrightbig cos2λksin[(2λ–1 )k] (2λ–1 )s i n k,k=a r c t a n√ a, –1 2<λ<1 , a>0 . 4.1-2. Integrals Over an Inﬁnite Interval. 1.⎝integraldisplay∞ 0dx ax2+b=π 2√ ab. 2.⎝integraldisplay∞ 0dx x4+1=π√ 2 4. 3.⎝integraldisplay∞ 0xa–1dx x+1=π sin(πa),0 < a<1 . 4.⎝integraldisplay∞ 0xλ–1dx (1 +ax)2=π(1 –λ) aλsin(πλ),0 < λ<2 . 5.⎝integraldisplay∞ 0xλ–1dx (x+a)(x+b)=π(aλ–1–bλ–1) (b–a)s i n (πλ),0 < λ<2 . 4.1. I NTEGRALS INVOLVING POWER -LAWFUNCTIONS 953 6.⎝integraldisplay∞ 0xλ–1(x+c)dx (x+a)(x+b)=π sin(πλ)⎝parenleftbigga–c a–baλ–1+b–c b–abλ–1⎝parenrightbigg ,0 < λ<1 . 7.⎝integraldisplay∞ 0xλdx (x+1 )3=πλ(1 –λ) 2s i n (πλ),– 1 < λ<2 . 8.⎝integraldisplay∞ 0xλ–1dx (x2+a2)(x2+b2)=π⎝parenleftbig bλ–2–aλ–2⎝parenrightbig 2⎝parenleftbig a2–b2⎝parenrightbig sin(πλ/2),0 < λ<4 . 9.⎝integraldisplay∞ 0xp–1–xq–1 1–xdx=π[cot(πp)–c o t ( πq)],p,q>0 . 10.⎝integraldisplay∞ 0xλ–1dx (1 +ax)n+1= (–1)nπCn λ–1 aλsin(πλ),0 < λ<n+1 ,Cn λ–1=(λ–1 ) (λ–2 )...(λ–n) n!. 11.⎝integraldisplay∞ 0xmdx (a+bx)n+1/2=2m+1m!(2n–2m– 3)!! (2n– 1)!!am–n+1/2 bm+1,a,b>0 , n,m=1 ,2 , ..., m<b–1 2. 12.⎝integraldisplay∞ 0dx (x2+a2)n=π 2(2n– 3)!! (2n– 2)!!1 a2n–1,n=1 ,2 , ... 13.⎝integraldisplay∞ 0(x+1 )λ–1 (x+a)λ+1dx=1–a–λ λ(a–1 ),a>0 . 14.⎝integraldisplay∞ 0xa–1dx xb+1=π bsin(πa/b ),0 < a≤b. 15.⎝integraldisplay∞ 0xa–1dx (xb+1 )2=π(a–b) b2sin[π(a–b)/b],a<2b. 16.⎝integraldisplay∞ 0xλ–1/2dx (x+a)λ(x+b)λ=√ π⎝parenleftbig√ a+√ b⎝parenrightbig1–2λΓ(λ–1/2) Γ(λ),λ>0 . 17.⎝integraldisplay∞ 01–xa 1–xbxc–1dx=πsinA bsinCsin(A+C),A=πa b,C=πc b;a+c<b,c>0 . 18.⎝integraldisplay∞ 0xa–1dx (1 +x2)1–b=1 2B⎝parenleftbig1 2a,1–b–1 2a⎝parenrightbig ,1 2a+b<1 , a>0 . 19.⎝integraldisplay∞ 0x2mdx (ax2+b)n=π(2m– 1)!! (2 n–2m– 3)!! 2( 2n– 2)!!ambn–m–1√ ab,a,b>0 , n>m+1 . 20.⎝integraldisplay∞ 0x2m+1dx (ax2+b)n=m!(n–m–2 ) ! 2(n–1 ) !am+1bn–m–1,ab>0 , n>m+1≥1. 21.⎝integraldisplay∞ 0xµ–1dx (1 +axp)ν=1 paµ/pB⎝parenleftBigµ p,ν–µ p⎝parenrightBig ,p>0 , 0< µ<pν. 22.⎝integraldisplay∞ 0⎝parenleftbig√ x2+a2–x⎝parenrightbigndx=nan+1 n2–1,n=2 ,3 , ... 23.⎝integraldisplay∞ 0dx ⎝parenleftbig x+√ x2+a2⎝parenrightbign=n an–1(n2–1 ),n=2 ,3 , ... 24.⎝integraldisplay∞ 0xm⎝parenleftbig√ x2+a2–x⎝parenrightbigndx=m!nan+m+1 (n–m–1 ) (n–m+1 )...(n+m+1 ),n,m=1 ,2 , ..., 0≤m≤n–2 . 25.⎝integraldisplay∞ 0xmdx ⎝parenleftbig x+√ x2+a2⎝parenrightbign=m!n (n–m–1 ) (n–m+1 )...(n+m+1 )an–m–1,n=2 ,3 , ... 954 TABLES OF DEFINITE INTEGRALS 4.2. Integrals Involving Exponential Functions 1.⎝integraldisplay∞ 0e–axdx=1 a,a>0 . 2.⎝integraldisplay1 0xne–axdx=n! an+1–e–an⎝summationdisplay k=0n! k!1 an–k+1,a>0 ,n=1 ,2 , ... 3.⎝integraldisplay∞ 0xne–axdx=n! an+1,a>0 ,n=1 ,2 , ... 4.⎝integraldisplay∞ 0e–ax √ xdx=⎝radicalbigg π a,a>0 . 5.⎝integraldisplay∞ 0xν–1e–µxdx=Γ(ν) µν,µ,ν>0 . 6.⎝integraldisplay∞ 0dx 1+eax=ln 2 a. 7.⎝integraldisplay∞ 0x2n–1dx epx–1= (–1)n–1⎝parenleftBig2π p⎝parenrightBig2nB2n 4n,n=1 ,2 , ...;t h e Bmare Bernoulli numbers (see Supplement 11.1-3). 8.⎝integraldisplay∞ 0x2n–1dx epx+1=( 1–21–2n)⎝parenleftBig2π p⎝parenrightBig2n|B2n| 4n,n=1 ,2 , ...;t h eBmare Bernoulli numbers. 9.⎝integraldisplay∞ –∞e–pxdx 1+e–qx=π qsin(πp/q ),q>p>0 o r 0>p >q. 10.⎝integraldisplay∞ 0eax+e–ax ebx+e–bxdx=π 2bcos⎝parenleftBigπa 2b⎝parenrightBig,b>a. 11.⎝integraldisplay∞ 0e–px–e–qx 1–e–(p+q)xdx=π p+qcotπp p+q,p,q>0 . 12.⎝integraldisplay∞ 0⎝parenleftbig 1–e–βx⎝parenrightbigνe–µxdx=1 βB⎝parenleftBigµ β,ν+1⎝parenrightBig . 13.⎝integraldisplay∞ 0exp⎝parenleftbig –ax2⎝parenrightbig dx=1 2⎝radicalbigg π a,a>0 . 14.⎝integraldisplay∞ 0x2n+1exp⎝parenleftbig –ax2⎝parenrightbig dx=n! 2an+1,a>0 , n=1 ,2 , ... 15.⎝integraldisplay∞ 0x2nexp⎝parenleftbig –ax2⎝parenrightbig dx=1×3×...×(2n–1 )√ π 2n+1an+1/2,a>0 , n=1 ,2 , ... 16.⎝integraldisplay∞ –∞exp⎝parenleftbig –a2x2±bx⎝parenrightbig dx=√ π |a|exp⎝parenleftBigb2 4a2⎝parenrightBig . 17.⎝integraldisplay∞ 0exp⎝parenleftBig –ax2–b x2⎝parenrightBig dx=1 2⎝radicalbigg π aexp⎝parenleftbig –2√ ab⎝parenrightbig ,a,b>0 . 18.⎝integraldisplay∞ 0exp⎝parenleftbig –xa⎝parenrightbig dx=1 aΓ⎝parenleftBig1 a⎝parenrightBig ,a>0 . 4.4. I NTEGRALS INVOLVING LOGARITHMIC FUNCTIONS 955 4.3. Integrals Involving Hyperbolic Functions 1.⎝integraldisplay∞ 0dx coshax=π 2|a|. 2.⎝integraldisplay∞ 0dx a+bcoshx=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩2 √ b2–a2arctan√ b2–a2 a+bif|b|>|a|, 1 √ a2–b2lna+b+√ a2–b2 a+b–√ a2+b2if|b|<|a|. 3.⎝integraldisplay∞ 0x2ndx coshax=⎝parenleftBigπ 2a⎝parenrightBig2n+1 |E2n|,a>0 ; t h e Emare Euler numbers (see Supplement 11.1-4). 4.⎝integraldisplay∞ 0x2ndx cosh2ax=π2n(22n–2 ) |a|(2a)2n|B2n|,t h eBmare Bernoulli numbers (see Supplement 11.1-3). 5.⎝integraldisplay∞ 0coshax coshbxdx=π 2bcos⎝parenleftBigπa 2b⎝parenrightBig,b>|a|. 6.⎝integraldisplay∞ 0x2ncoshax coshbxdx=π 2bd2n da2n1 cos⎝parenleftbig1 2πa/b⎝parenrightbig,b>|a|,n=1 ,2 , ... 7.⎝integraldisplay∞ 0coshaxcoshbx cosh(cx )dx=π ccos⎝parenleftBigπa 2c⎝parenrightBig cos⎝parenleftBigπb 2c⎝parenrightBig cos⎝parenleftBigπa c⎝parenrightBig +c o s⎝parenleftBigπb c⎝parenrightBig,c>|a|+|b|. 8.⎝integraldisplay∞ 0xd x sinhax=π2 2a2,a>0 . 9.⎝integraldisplay∞ 0dx a+bsinhx=1 √ a2+b2lna+b+√ a2+b2 a+b–√ a2+b2,ab≠0. 10.⎝integraldisplay∞ 0sinhax sinhbxdx=π 2btan⎝parenleftBigπa 2b⎝parenrightBig ,b>|a|. 11.⎝integraldisplay∞ 0x2nsinhax sinhbxdx=π 2bd2n dx2ntan⎝parenleftBigπa 2b⎝parenrightBig ,b>|a|,n=1 ,2 , ... 12.⎝integraldisplay∞ 0x2n sinh2axdx=π2n a2n+1|B2n|,a>0 ; t h e Bmare Bernoulli numbers. 4.4. Integrals Involving Logarithmic Functions 1.⎝integraldisplay1 0xa–1lnnxd x = (–1)nn!a–n–1,a>0 , n=1 ,2 , ... 2.⎝integraldisplay1 0lnx x+1dx=–π2 12. 3.⎝integraldisplay1 0xnlnx x+1dx= (–1)n+1⎝bracketleftbiggπ2 12+n⎝summationdisplay k=1(–1)k k2⎝bracketrightbigg ,n=1 ,2 , ... 4.⎝integraldisplay1 0xµ–1lnx x+adx=πaµ–1 sin(πµ)⎝bracketleftbig lna–πcot(πµ)⎝bracketrightbig ,0 < µ<1 . 956 TABLES OF DEFINITE INTEGRALS 5.⎝integraldisplay1 0|lnx|µdx=Γ(µ+1 ) , µ> –1. 6.⎝integraldisplay∞ 0xµ–1ln(1 + ax)dx=π µaµsin(πµ),– 1 < µ<0 . 7.⎝integraldisplay1 0x2n–1ln(1 + x)dx=1 2n2n⎝summationdisplay k=1(–1)k–1 k,n=1 ,2 , ... 8.⎝integraldisplay1 0x2nln(1 + x)dx=1 2n+1⎝bracketleftbigg ln 4 +2n+1⎝summationdisplay k=1(–1)k k⎝bracketrightbigg ,n=0 ,1 , ... 9.⎝integraldisplay1 0xn–1/2ln(1 + x)dx=2l n2 2n+1+4(–1)n 2n+1⎝bracketleftbigg π–n⎝summationdisplay k=0(–1)k 2k+1⎝bracketrightbigg ,n=1 ,2 , ... 10.⎝integraldisplay∞ 0lna2+x2 b2+x2dx=π(a–b),a,b>0 . 11.⎝integraldisplay∞ 0xp–1lnx 1+xqdx=–π2cos(πp/q ) q2sin2(πp/q ),0 < p<q. 12.⎝integraldisplay∞ 0e–µxlnxd x =–1 µ(C+l nµ),µ>0 ,C= 0.5772 ... 4.5. Integrals Involving Trigonometric Functions 4.5-1. Integrals Over a Finite Interval. 1.⎝integraldisplayπ/2 0cos2nxd x =π 21×3×···×(2n–1 ) 2×4×···×(2n),n=1 ,2 , ... 2.⎝integraldisplayπ/2 0cos2n+1xd x =2×4×···×(2n) 1×3×···×(2n+1 ),n=1 ,2 , ... 3.⎝integraldisplayπ/2 0xcosnxd x =–m–1⎝summationdisplay k=0(n–2k+1 ) (n–2k+3 )...(n–1 ) (n–2k)(n–2k+2 )...n1 n–2k +⎧ ⎪⎪⎨ ⎪⎪⎩π 2(2m– 2)!! (2m– 1)!!ifn=2m–1 , π2 8(2m– 1)!! (2m)!!ifn=2m,m=1 ,2 , ... 4.⎝integraldisplayπ 0dx (a+bcosx)n+1=π 2n(a+b)n√ a2–b2n⎝summationdisplay k=0(2n–2k–1)!! (2 k–1)!! (n–k)!k!⎝parenleftBiga+b a–b⎝parenrightBigk ,a>|b|. 5.⎝integraldisplayπ/2 0sin2nxd x =π 21×3×···×(2n–1 ) 2×4×···×(2n),n=1 ,2 , ... 6.⎝integraldisplayπ/2 0sin2n+1xd x =2×4×···×(2n) 1×3×···×(2n+1 ),n=1 ,2 , ... 7.⎝integraldisplayπ 0xsinµxd x =π2 2µ+1Γ(µ+1 ) ⎝bracketleftbig Γ⎝parenleftbig µ+1 2⎝parenrightbig⎝bracketrightbig2,µ> –1. 4.5. I NTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS 957 8.⎝integraldisplayπ/2 0sinxd x √ 1–k2sin2x=1 2kln1+k 1–k. 9.⎝integraldisplayπ/2 0sin2n+1xcos2m+1xd x =n!m! 2(n+m+1 ) !,n,m=1 ,2 , ... 10.⎝integraldisplayπ/2 0sinp–1xcosq–1xd x =1 2B⎝parenleftbig1 2p,1 2q⎝parenrightbig . 11.⎝integraldisplay2π 0(asinx+bcosx)2ndx=2π(2n– 1)!! (2n)!!⎝parenleftbig a2+b2⎝parenrightbign,n=1 ,2 , ... 12.⎝integraldisplayπ 0sinxd x √ a2+1–2 acosx=⎝braceleftbigg2i f 0 ≤a≤1, 2/a if 1 < a. 13.⎝integraldisplayπ/2 0(tanx)±λdx=π 2c o s⎝parenleftbig1 2πλ⎝parenrightbig,|λ|<1 . 14.⎝integraldisplaya 0cos(xt )dt √ a2–t2=π 2J0(ax),J0(z) is the Bessel function (see Supplement 11.6). 15.⎝integraldisplaya 0tsin(xt)dt √ a2–t2=π 2aJ1(ax),J1(z) is the Bessel function. 4.5-2. Integrals Over an Inﬁnite Interval. 1.⎝integraldisplay∞ 0cosax √ xdx=⎝radicalbigg π 2a,a>0 . 2.⎝integraldisplay∞ 0cosax–c o sbx xdx=l n⎝vextendsingle⎝vextendsingle⎝vextendsingleb a⎝vextendsingle⎝vextendsingle⎝vextendsingle,ab≠0. 3.⎝integraldisplay∞ 0cosax–c o sbx x2dx=1 2π(b–a),a,b≥0. 4.⎝integraldisplay∞ 0xµ–1cosaxdx =a–µΓ(µ)c o s⎝parenleftbig1 2πµ⎝parenrightbig ,a>0 , 0< µ<1 . 5.⎝integraldisplay∞ 0cosax b2+x2dx=π 2be–ab,a,b>0 . 6.⎝integraldisplay∞ 0cosax b4+x4dx=π√ 2 4b3exp⎝parenleftbigg –ab √ 2⎝parenrightbigg⎝bracketleftbigg cos⎝parenleftbiggab √ 2⎝parenrightbigg +s i n⎝parenleftBigab √ 2⎝parenrightbigg⎝bracketrightbigg ,a,b>0 . 7.⎝integraldisplay∞ 0cosax (b2+x2)2dx=π 4b3(1 +ab)e–ab,a,b>0 . 8.⎝integraldisplay∞ 0cosaxdx (b2+x2)(c2+x2)=π⎝parenleftbig be–ac–ce–ab⎝parenrightbig 2bc⎝parenleftbig b2–c2⎝parenrightbig,a,b,c>0 . 9.⎝integraldisplay∞ 0cos⎝parenleftbig ax2⎝parenrightbig dx=1 2⎝radicalbigg π 2a,a>0 . 10.⎝integraldisplay∞ 0cos⎝parenleftbig axp⎝parenrightbig dx=Γ(1/p) pa1/pcosπ 2p,a>0 , p>1 . 11.⎝integraldisplay∞ 0sinax xdx=π 2signa. 958 TABLES OF DEFINITE INTEGRALS 12.⎝integraldisplay∞ 0sin2ax x2dx=π 2|a|. 13.⎝integraldisplay∞ 0sinax √ xdx=⎝radicalbigg π 2a,a>0 . 14.⎝integraldisplay∞ 0xµ–1sinaxdx =a–µΓ(µ)s i n⎝parenleftbig1 2πµ⎝parenrightbig ,a>0 , 0< µ<1 . 15.⎝integraldisplay∞ 0sin⎝parenleftbig ax2⎝parenrightbig dx=1 2⎝radicalbigg π 2a,a>0 . 16.⎝integraldisplay∞ 0sin⎝parenleftbig axp⎝parenrightbig dx=Γ(1/p) pa1/psinπ 2p,a>0 , p>1 . 17.⎝integraldisplay∞ 0sinxcosax xdx=⎧ ⎨ ⎩π 2if|a|<1 , π 4if|a|=1 , 0i f 1 < |a|. 18.⎝integraldisplay∞ 0tanax xdx=π 2signa. 19.⎝integraldisplay∞ 0e–axsinbxdx =b a2+b2,a>0 . 20.⎝integraldisplay∞ 0e–axcosbxdx =a a2+b2,a>0 . 21.⎝integraldisplay∞ 0exp⎝parenleftbig –ax2⎝parenrightbig cosbxdx =1 2⎝radicalbigg π aexp⎝parenleftBig –b2 4a⎝parenrightBig . 22.⎝integraldisplay∞ 0cos(ax2)c o sbxdx =⎝radicalbigg π 8a⎝bracketleftbigg cos⎝parenleftbiggb2 4a⎝parenrightbigg +s i n⎝parenleftbiggb2 4a⎝parenrightbigg⎝bracketrightbigg ,a,b>0 . 23.⎝integraldisplay∞ 0(cosax+s i nax)c o s (b2x2)dx=1 b⎝radicalbigg π 8exp⎝parenleftbigg –a2 2b⎝parenrightbigg ,a,b>0 . 24.⎝integraldisplay∞ 0⎝bracketleftbig cosax+s i nax⎝bracketrightbig sin(b2x2)dx=1 b⎝radicalbigg π 8exp⎝parenleftBig –a2 2b⎝parenrightBig ,a,b>0 . 4.6. Integrals Involving Bessel Functions 4.6-1. Integrals Over an Inﬁnite Interval. 1.⎝integraldisplay∞ 0Jν(ax)dx=1 a,a>0 , R e ν> –1. 2.⎝integraldisplay∞ 0cos(xu )J0(tu)du=⎝braceleftBigg1 √ t2–x2ifx<t, 0i f x>t. 3.⎝integraldisplay∞ 0sin(xu)J0(tu)du=⎝braceleftBigg0i f x<t, 1 √ x2–t2ifx>t. 4.⎝integraldisplay∞ 0cos(xu )J1(tu)du=⎧ ⎪⎨ ⎪⎩1 tifx<t, –t √ x2–t2(x+√ x2–t2)ifx>t. 5.⎝integraldisplay∞ 0sin(tu)J0(au) u2+b2du=sinh(bt) bK0(ab),b>0 , 0< t<a,K0(z) is the modiﬁed Bessel function (see Supplement 11.7). 4.6. I NTEGRALS INVOLVING BESSEL FUNCTIONS 959 6.⎝integraldisplay∞ 0usin(tu)J0(au) u2+b2du=π 2e–btI0(ab),b>0 ,a<t<∞,I0(z) is the modiﬁed Bessel function. 7.⎝integraldisplay∞ 0sin(tu)J1(au) u2+b2du=π 2be–btI1(ab),b>0 ,a<t<∞,I1(z) is the modiﬁed Bessel function. 8.⎝integraldisplay∞ 0usin(tu)J1(au) u2+b2du=s i n h ( bt)K1(ab),b>0 , 0< t<a,K1(z) is the modiﬁed Bessel function. 9.⎝integraldisplay∞ 0J1(au) √ u2+b2du=1–e–ab ab,a>0 , R e b>0 . 4.6-2. Other Integrals. 1.⎝integraldisplay1 0uJ0(xu)du=J1(x) x. 2.⎝integraldisplaya 0J1(bx)dx √ a2–x2=1–c o s ( ab) ab,a>0 . 3.⎝integraldisplayt 0uJ0(xu)du √ t2–u2=sin(xt) x. 4.⎝integraldisplay∞ tJ1(xu)du √ u2–t2=sin(xt) x,x>0 ,t>0 . References for Supplement 4: H. B. Dwight (1961), I. S. Gradshteyn and I. M. Ryzhik (2000), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986, 1988), D. Zwillinger (2002), I. N. Bronshtein and K. A. Semendyayev (2004). Supplement 5 Tables of Laplace Transforms 5.1. General Formulas No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 1 af1(x)+bf2(x) a˜f1(p)+b˜f2(p) 2 f(x/a),a>0 a˜f(ap) 3 ⎝braceleftbigg0i f 0 < x<a, f(x–a)i fa<x, e–ap˜f(p) 4 xnf(x);n=1 ,2 , ... (–1)ndn dpn˜f(p) 5 1 xf(x) ⎝integraldisplay∞ p˜f(q)dq 6 eaxf(x) ˜f(p–a) 7 sinh(ax)f(x) 1 2⎝bracketleftbig˜f(p–a)–˜f(p+a)⎝bracketrightbig 8 cosh(ax )f(x) 1 2⎝bracketleftbig˜f(p–a)+˜f(p+a)⎝bracketrightbig 9 sin(ωx)f(x) –i 2⎝bracketleftbig˜f(p–iω)–˜f(p+iω)⎝bracketrightbig,i2=– 1 10 cos(ωx )f(x) 1 2⎝bracketleftbig˜f(p–iω)+˜f(p+iω)⎝bracketrightbig ,i2=– 1 11 f(x2) 1 √ π⎝integraldisplay∞ 0exp⎝parenleftBig –p2 4t2⎝parenrightBig ˜f(t2)dt 12 xa–1f⎝parenleftBig1 x⎝parenrightBig ,a>– 1 ⎝integraldisplay∞ 0(t/p)a/2Ja⎝parenleftbig 2√ pt⎝parenrightbig˜f(t)dt 13 f(asinhx),a>0 ⎝integraldisplay∞ 0Jp(at)˜f(t)dt 14 f(x+a)=f(x) (periodic function) 1 1–eap⎝integraldisplaya 0f(x)e–pxdx 15 f(x+a)=–f(x) (antiperiodic function) 1 1+e–ap⎝integraldisplaya 0f(x)e–pxdx 16 f/prime x(x) p˜f(p)–f(+0) 17 f(n) x(x) pn˜f(p)–n⎝summationdisplay k=1pn–kf(k–1) x(+0) 961 962 TABLES OF LAPLACE TRANSFORMS No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 18 xmf(n) x(x),m≥n ⎝parenleftBig –d dp⎝parenrightBigm⎝bracketleftbig pn˜f(p)⎝bracketrightbig 19 dn dxn⎝bracketleftbig xmf(x)⎝bracketrightbig ,m≥n (–1)mpndm dpm˜f(p) 20 ⎝integraldisplayx 0f(t)dt ˜f(p) p 21 ⎝integraldisplayx 0(x–t)f(t)dt 1 p2˜f(p) 22 ⎝integraldisplayx 0(x–t)νf(t)dt,ν>– 1 Γ(ν+1 )p–ν–1˜f(p) 23 ⎝integraldisplayx 0e–a(x–t)f(t)dt 1 p+a˜f(p) 24 ⎝integraldisplayx 0sinh⎝bracketleftbig a(x–t)⎝bracketrightbig f(t)dt a˜f(p) p2–a2 25 ⎝integraldisplayx 0sin⎝bracketleftbig a(x–t)⎝bracketrightbig f(t)dt a˜f(p) p2+a2 26 ⎝integraldisplayx 0f1(t)f2(x–t)dt ˜f1(p)˜f2(p) 27 ⎝integraldisplayx 01 tf(t)dt 1 p⎝integraldisplay∞ p˜f(q)dq 28 ⎝integraldisplay∞ x1 tf(t)dt 1 p⎝integraldisplayp 0˜f(q)dq 29 ⎝integraldisplay∞ 01 √ tsin⎝parenleftbig 2√ xt⎝parenrightbig f(t)dt √ π p√ p˜f⎝parenleftBig1 p⎝parenrightBig 30 1 √ x⎝integraldisplay∞ 0cos⎝parenleftbig 2√ xt⎝parenrightbig f(t)dt √ π √ p˜f⎝parenleftBig1 p⎝parenrightBig 31 ⎝integraldisplay∞ 01 √ πxexp⎝parenleftBig –t2 4x⎝parenrightBig f(t)dt 1 √ p˜f⎝parenleftbig√ p⎝parenrightbig 32 ⎝integraldisplay∞ 0t 2√ πx3exp⎝parenleftBig –t2 4x⎝parenrightBig f(t)dt ˜f⎝parenleftbig√ p⎝parenrightbig 33 f(x)–a⎝integraldisplayx 0f⎝parenleftbig√ x2–t2⎝parenrightbig J1(at)dt ˜f⎝parenleftbig⎝radicalbig p2+a2⎝parenrightbig 34 f(x)+a⎝integraldisplayx 0f⎝parenleftbig√ x2–t2⎝parenrightbig I1(at)dt ˜f⎝parenleftbig⎝radicalbig p2–a2⎝parenrightbig 5.3. E XPRESSIONS WITH EXPONENTIAL FUNCTIONS 963 5.2. Expressions with Power-Law Functions No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 1 1 1 p 2 ⎝braceleftBigg0i f 0 < x<a, 1i f a<x<b, 0i f b<x. 1 p⎝parenleftbig e–ap–e–bp⎝parenrightbig 3 x 1 p2 4 1 x+a –eapEi(–ap) 5 xn,n=1 ,2 , ... n! pn+1 6 xn–1/2,n=1 ,2 , ... 1⋅3...(2n–1 )√ π 2npn+1/2 7 1 √ x+a ⎝radicalbigg π peaperfc⎝parenleftbig√ ap⎝parenrightbig 8 √ x x+a ⎝radicalbigg π p–π√ aeaperfc⎝parenleftbig√ ap⎝parenrightbig 9 (x+a)–3/2 2a–1/2–2 (πp)1/2eaperfc⎝parenleftbig√ ap⎝parenrightbig 10 x1/2(x+a)–1 (π/p)1/2–πa1/2eaperfc⎝parenleftbig√ ap⎝parenrightbig 11 x–1/2(x+a)–1 πa–1/2eaperfc⎝parenleftbig√ ap⎝parenrightbig 12 xν,ν>– 1 Γ(ν+1 )p–ν–1 13 (x+a)ν,ν>– 1 p–ν–1e–apΓ(ν+1 ,ap) 14 xν(x+a)–1,ν>– 1 keapΓ(–ν,ap), k=aνΓ(ν+1 ) 15 (x2+2ax)–1/2(x+a) aeapK1(ap) 5.3. Expressions with Exponential Functions No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 1 e–ax (p+a)–1 2 xe–ax (p+a)–2 3 xν–1e–ax,ν>0 Γ(ν)(p+a)–ν 4 1 x⎝parenleftbig e–ax–e–bx⎝parenrightbig ln(p+b)–l n (p+a) 964 TABLES OF LAPLACE TRANSFORMS No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 5 1 x2⎝parenleftbig 1–e–ax⎝parenrightbig2 (p+2a)l n (p+2a)+plnp–2 (p+a)l n (p+a) 6 exp⎝parenleftbig –ax2⎝parenrightbig ,a>0 (πb)1/2exp⎝parenleftbig bp2⎝parenrightbig erfc(p√ b), a=1 4b 7 xexp⎝parenleftbig –ax2⎝parenrightbig 2b–2π1/2b3/2perfc(p√ b), a=1 4b 8 exp(–a/x), a≥0 2⎝radicalbig a/pK1⎝parenleftbig 2√ ap⎝parenrightbig 9 √ xexp(–a/x), a≥0 1 2⎝radicalbig π/p3⎝parenleftbig 1+2√ ap⎝parenrightbig exp⎝parenleftbig –2√ ap⎝parenrightbig 10 1 √ xexp(–a/x), a≥0 ⎝radicalbig π/pexp⎝parenleftbig –2√ ap⎝parenrightbig 11 1 x√ xexp(–a/x), a>0 ⎝radicalbig π/aexp⎝parenleftbig –2√ ap⎝parenrightbig 12 xν–1exp(–a/x), a>0 2(a/p)ν/2Kν⎝parenleftbig 2√ ap⎝parenrightbig 13 exp⎝parenleftbig –2√ ax⎝parenrightbig p–1–(πa)1/2p–3/2ea/perfc⎝parenleftbig⎝radicalbig a/p⎝parenrightbig 14 1 √ xexp⎝parenleftbig –2√ ax⎝parenrightbig (π/p)1/2ea/perfc⎝parenleftbig⎝radicalbig a/p⎝parenrightbig 5.4. Expressions with Hyperbolic Functions No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 1 sinh(ax) a p2–a2 2 sinh2(ax) 2a2 p3–4a2p 3 1 xsinh(ax) 1 2lnp+a p–a 4 xν–1sinh(ax), ν>– 1 1 2Γ(ν)⎝bracketleftbig (p–a)–ν–(p+a)–ν⎝bracketrightbig 5 sinh⎝parenleftbig 2√ ax⎝parenrightbig √ πa p√ pea/p 6 √ xsinh⎝parenleftbig 2√ ax⎝parenrightbig π1/2p–5/2⎝parenleftbig1 2p+a⎝parenrightbig ea/perf⎝parenleftbig⎝radicalbig a/p⎝parenrightbig –a1/2p–2 7 1 √ xsinh⎝parenleftbig 2√ ax⎝parenrightbig π1/2p–1/2ea/perf⎝parenleftbig⎝radicalbig a/p⎝parenrightbig 8 1 √ xsinh2⎝parenleftbig√ ax⎝parenrightbig 1 2π1/2p–1/2⎝parenleftbig ea/p–1⎝parenrightbig 9 cosh(ax ) p p2–a2 5.5. E XPRESSIONS WITH LOGARITHMIC FUNCTIONS 965 No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 10 cosh2(ax) p2–2a2 p3–4a2p 11 xν–1cosh(ax ), ν>0 1 2Γ(ν)⎝bracketleftbig (p–a)–ν+(p+a)–ν⎝bracketrightbig 12 cosh⎝parenleftbig 2√ ax⎝parenrightbig 1 p+√ πa p√ pea/perf⎝parenleftbig⎝radicalbig a/p⎝parenrightbig 13 √ xcosh⎝parenleftbig 2√ ax⎝parenrightbig π1/2p–5/2⎝parenleftbig1 2p+a⎝parenrightbig ea/p 14 1 √ xcosh⎝parenleftbig 2√ ax⎝parenrightbig π1/2p–1/2ea/p 15 1 √ xcosh2⎝parenleftbig√ ax⎝parenrightbig 1 2π1/2p–1/2⎝parenleftbig ea/p+1⎝parenrightbig 5.5. Expressions with Logarithmic Functions No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 1 lnx –1 p(lnp+C), C= 0.5772 ...is the Euler constant 2 ln(1 + ax) –1 pep/aEi(–p/a) 3 ln(x+a) 1 p⎝bracketleftbig lna–eapEi(–ap)⎝bracketrightbig 4 xnlnx,n=1 ,2 , ... n! pn+1⎝parenleftbig 1+1 2+1 3+···+1 n–l np–C⎝parenrightbig , C= 0.5772 ...is the Euler constant 5 1 √ xlnx –⎝radicalbig π/p⎝bracketleftbig ln(4p)+C⎝bracketrightbig 6 xn–1/2lnx,n=1 ,2 , ... kn pn+1/2⎝bracketleftbig 2+2 3+2 5+···+2 2n–1–l n ( 4p)–C⎝bracketrightbig , kn=1⋅3⋅5...(2n–1 )√ π 2n,C= 0.5772 ... 7 xν–1lnx,ν>0 Γ(ν)p–ν⎝bracketleftbig ψ(ν)–l np⎝bracketrightbig ,ψ(ν) is the logarithmic derivative of the gamma function 8 (lnx)2 1 p⎝bracketleftbig (lnx+C)2+1 6π2⎝bracketrightbig ,C= 0.5772 ... 9 e–axlnx –ln(p+a)+C p+a,C= 0.5772 ... 966 TABLES OF LAPLACE TRANSFORMS 5.6. Expressions with Trigonometric Functions No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 1 sin(ax) a p2+a2 2 |sin(ax)|,a>0 a p2+a2coth⎝parenleftBigπp 2a⎝parenrightBig 3 sin2n(ax), n=1 ,2 , ... a2n(2n)! p⎝bracketleftbig p2+( 2a)2⎝bracketrightbig⎝bracketleftbig p2+( 4a)2⎝bracketrightbig ...⎝bracketleftbig p2+( 2na)2⎝bracketrightbig 4 sin2n+1(ax), n=1 ,2 , ... a2n+1(2n+1 ) ! ⎝bracketleftbig p2+a2⎝bracketrightbig⎝bracketleftbig p2+32a2⎝bracketrightbig ...⎝bracketleftbig p2+( 2n+1 )2a2⎝bracketrightbig 5 xnsin(ax), n=1 ,2 , ... n!pn+1 ⎝parenleftbig p2+a2⎝parenrightbign+1⎝summationdisplay 0≤2k≤n(–1)kC2k+1 n+1⎝parenleftBiga p⎝parenrightBig2k+1 6 1 xsin(ax) arctan⎝parenleftBiga p⎝parenrightBig 7 1 xsin2(ax) 1 4ln⎝parenleftbig 1+4a2p–2⎝parenrightbig 8 1 x2sin2(ax) aarctan(2 a/p)–1 4pln⎝parenleftbig 1+4a2p–2⎝parenrightbig 9 sin⎝parenleftbig 2√ ax⎝parenrightbig √ πa p√ pe–a/p 10 1 xsin⎝parenleftbig 2√ ax⎝parenrightbig πerf⎝parenleftbig⎝radicalbig a/p⎝parenrightbig 11 cos(ax ) p p2+a2 12 cos2(ax) p2+2a2 p⎝parenleftbig p2+4a2⎝parenrightbig 13 xncos(ax ), n=1 ,2 , ... n!pn+1 ⎝parenleftbig p2+a2⎝parenrightbign+1⎝summationdisplay 0≤2k≤n+1(–1)kC2k n+1⎝parenleftBiga p⎝parenrightBig2k 14 1 x⎝bracketleftbig 1–c o s ( ax)⎝bracketrightbig 1 2ln⎝parenleftbig1+a2p–2⎝parenrightbig 15 1 x⎝bracketleftbig cos(ax )–c o s ( bx)⎝bracketrightbig 1 2lnp2+b2 p2+a2 16 √ xcos⎝parenleftbig 2√ ax⎝parenrightbig 1 2π1/2p–5/2(p–2a)e–a/p 17 1 √ xcos⎝parenleftbig 2√ ax⎝parenrightbig ⎝radicalbig π/pe–a/p 18 sin(ax)s i n (bx) 2abp ⎝bracketleftbig p2+(a+b)2⎝bracketrightbig⎝bracketleftbig p2+(a–b)2⎝bracketrightbig 19 cos(ax )s i n (bx) b⎝parenleftbig p2–a2+b2⎝parenrightbig ⎝bracketleftbig p2+(a+b)2⎝bracketrightbig⎝bracketleftbig p2+(a–b)2⎝bracketrightbig 5.7. E XPRESSIONS WITH SPECIAL FUNCTIONS 967 No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 20 cos(ax )c o s (bx) p⎝parenleftbig p2+a2+b2⎝parenrightbig ⎝bracketleftbig p2+(a+b)2⎝bracketrightbig⎝bracketleftbig p2+(a–b)2⎝bracketrightbig 21 axcos(ax )–s i n ( ax) x2 parctana x–a 22 ebxsin(ax) a (p–b)2+a2 23 ebxcos(ax ) p–b (p–b)2+a2 24 sin(ax)s i n h ( ax) 2a2p p4+4a4 25 sin(ax)c o s h ( ax) a⎝parenleftbig p2+2a2⎝parenrightbig p4+4a4 26 cos(ax )s i n h ( ax) a⎝parenleftbig p2–2a2⎝parenrightbig p4+4a4 27 cos(ax )c o s h ( ax) p3 p4+4a4 5.7. Expressions with Special Functions No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 1 erf(ax) 1 pexp⎝parenleftbig b2p2⎝parenrightbig erfc(bp ), b=1 2a 2 erf⎝parenleftbig√ ax⎝parenrightbig √ a p√ p+a 3 eaxerf⎝parenleftbig√ ax⎝parenrightbig √ a √ p(p–a) 4 erf⎝parenleftbig1 2⎝radicalbig a/x⎝parenrightbig 1 p⎝bracketleftbig 1–e x p⎝parenleftbig –√ ap⎝parenrightbig⎝bracketrightbig 5 erfc⎝parenleftbig√ ax⎝parenrightbig √ p+a–√ a p√ p+a 6 eaxerfc⎝parenleftbig√ ax⎝parenrightbig 1 p+√ ap 7 erfc⎝parenleftbig1 2⎝radicalbig a/x⎝parenrightbig 1 pexp⎝parenleftbig –√ ap⎝parenrightbig 8 Ci(x) 1 2pln(p2+1 ) 968 TABLES OF LAPLACE TRANSFORMS No Original function, f(x) Laplace transform, ˜f(p)=⎝integraldisplay∞ 0e–pxf(x)dx 9 Si(x) 1 parccot p 10 Ei(–x) –1 pln(p+1 ) 11 J0(ax) 1 ⎝radicalbig p2+a2 12 Jν(ax), ν>– 1 aν ⎝radicalbig p2+a2⎝parenleftbig p+⎝radicalbig p2+a2⎝parenrightbigν 13 xnJn(ax), n=1 ,2 , ... 1⋅3⋅5...(2n–1 )an⎝parenleftbig p2+a2⎝parenrightbig–n–1/2 14 xνJν(ax), ν>–1 2 2νπ–1/2Γ⎝parenleftbig ν+1 2⎝parenrightbig aν⎝parenleftbig p2+a2⎝parenrightbig–ν–1/2 15 xν+1Jν(ax), ν>– 1 2ν+1π–1/2Γ⎝parenleftbig ν+3 2⎝parenrightbig aνp⎝parenleftbig p2+a2⎝parenrightbig–ν–3/2 16 J0⎝parenleftbig 2√ ax⎝parenrightbig 1 pe–a/p 17 √ xJ1⎝parenleftbig 2√ ax⎝parenrightbig √ a p2e–a/p 18 xν/2Jν⎝parenleftbig 2√ ax⎝parenrightbig ,ν>– 1 aν/2p–ν–1e–a/p 19 I0(ax) 1 ⎝radicalbig p2–a2 20 Iν(ax), ν>– 1 aν ⎝radicalbig p2–a2⎝parenleftbig p+⎝radicalbig p2–a2⎝parenrightbigν 21 xνIν(ax), ν>–1 2 2νπ–1/2Γ⎝parenleftbig ν+1 2⎝parenrightbig aν⎝parenleftbig p2–a2⎝parenrightbig–ν–1/2 22 xν+1Iν(ax), ν>– 1 2ν+1π–1/2Γ⎝parenleftbig ν+3 2⎝parenrightbig aνp⎝parenleftbig p2–a2⎝parenrightbig–ν–3/2 23 I0⎝parenleftbig 2√ ax⎝parenrightbig 1 pea/p 24 1 √ xI1⎝parenleftbig 2√ ax⎝parenrightbig 1 √ a⎝parenleftbig ea/p–1⎝parenrightbig 25 xν/2Iν⎝parenleftbig 2√ ax⎝parenrightbig ,ν>– 1 aν/2p–ν–1ea/p 26 Y0(ax) –2 πArsinh( p/a) ⎝radicalbig p2+a2 27 K0(ax) ln⎝parenleftbig p+⎝radicalbig p2–a2⎝parenrightbig –l na ⎝radicalbig p2–a2 References for Supplement 5: G. Doetsch (1950, 1956, 1958), H. Bateman and A. Erd ´elyi (1954), V . A. Ditkin and A. P. Prudnikov (1965), F. Oberhettinger and L. Badii (1973), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, Vo l . 4 ) . Supplement 6 Tables of Inverse Laplace Transforms 6.1. General Formulas No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 1 ˜f(p+a) e–axf(x) 2 ˜f(ap), a>0 1 af⎝parenleftBigx a⎝parenrightBig 3 ˜f(ap+b),a>0 1 aexp⎝parenleftBig –b ax⎝parenrightBig f⎝parenleftBigx a⎝parenrightBig 4 ˜f(p–a)+˜f(p+a) 2f(x)c o s h ( ax) 5 ˜f(p–a)–˜f(p+a) 2f(x)s i n h ( ax) 6 e–ap˜f(p),a≥0 ⎝braceleftbigg 0i f 0 ≤x<a, f(x–a)i fa<x. 7 p˜f(p) df(x) dx,i ff(+0) = 0 8 1 p˜f(p) ⎝integraldisplayx 0f(t)dt 9 1 p+a˜f(p) e–ax⎝integraldisplayx 0eatf(t)dt 10 1 p2˜f(p) ⎝integraldisplayx 0(x–t)f(t)dt 11 ˜f(p) p(p+a) 1 a⎝integraldisplayx 0⎝bracketleftbig 1–ea(x–t)⎝bracketrightbig f(t)dt 12 ˜f(p) (p+a)2 ⎝integraldisplayx 0(x–t)e–a(x–t)f(t)dt 13 ˜f(p) (p+a)(p+b) 1 b–a⎝integraldisplayx 0⎝bracketleftbig e–a(x–t)–e–b(x–t)⎝bracketrightbig f(t)dt 14 ˜f(p) (p+a)2+b2 1 b⎝integraldisplayx 0e–a(x–t)sin⎝bracketleftbig b(x–t)⎝bracketrightbig f(t)dt 15 1 pn˜f(p),n=1 ,2 , ... 1 (n–1 ) !⎝integraldisplayx 0(x–t)n–1f(t)dt 16 ˜f1(p)˜f2(p) ⎝integraldisplayx 0f1(t)f2(x–t)dt 969 970 TABLES OF INVERSE LAPLACE TRANSFORMS No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 17 1 √ p˜f⎝parenleftBig1 p⎝parenrightBig ⎝integraldisplay∞ 0cos⎝parenleftbig 2√ xt⎝parenrightbig √ πxf(t)dt 18 1 p√ p˜f⎝parenleftBig1 p⎝parenrightBig ⎝integraldisplay∞ 0sin⎝parenleftbig 2√ xt⎝parenrightbig √ πtf(t)dt 19 1 p2ν+1˜f⎝parenleftBig1 p⎝parenrightBig ⎝integraldisplay∞ 0(x/t)νJ2ν⎝parenleftbig 2√ xt⎝parenrightbig f(t)dt 20 1 p˜f⎝parenleftBig1 p⎝parenrightBig ⎝integraldisplay∞ 0J0⎝parenleftbig 2√ xt⎝parenrightbig f(t)dt 21 1 p˜f⎝parenleftBig p+1 p⎝parenrightBig ⎝integraldisplayx 0J0⎝parenleftbig 2√ xt–t2⎝parenrightbig f(t)dt 22 1 p2ν+1˜f⎝parenleftBig p+a p⎝parenrightBig ,–1 2<ν≤0 ⎝integraldisplayx 0⎝parenleftBigx–t at⎝parenrightBigν J2ν⎝parenleftbig 2√ axt–at2⎝parenrightbig f(t)dt 23 ˜f⎝parenleftbig√ p⎝parenrightbig ⎝integraldisplay∞ 0t 2√ πx3exp⎝parenleftBig –t2 4x⎝parenrightBig f(t)dt 24 1 √ p˜f⎝parenleftbig√ p⎝parenrightbig 1 √ πx⎝integraldisplay∞ 0exp⎝parenleftBig –t2 4x⎝parenrightBig f(t)dt 25 ˜f⎝parenleftbig p+√ p⎝parenrightbig 1 2√ π⎝integraldisplayx 0t (x–t)3/2exp⎝bracketleftBig –t2 4(x–t)⎝bracketrightBig f(t)dt 26 ˜f⎝parenleftbig⎝radicalbig p2+a2⎝parenrightbig f(x)–a⎝integraldisplayx 0f⎝parenleftbig√ x2–t2⎝parenrightbig J1(at)dt 27 ˜f⎝parenleftbig⎝radicalbig p2–a2⎝parenrightbig f(x)+a⎝integraldisplayx 0f⎝parenleftbig√ x2–t2⎝parenrightbig I1(at)dt 28 ˜f⎝parenleftbig⎝radicalbig p2+a2⎝parenrightbig ⎝radicalbig p2+a2 ⎝integraldisplayx 0J0⎝parenleftbig a√ x2–t2⎝parenrightbig f(t)dt 29 ˜f⎝parenleftbig⎝radicalbig p2–a2⎝parenrightbig ⎝radicalbig p2–a2 ⎝integraldisplayx 0I0⎝parenleftbig a√ x2–t2⎝parenrightbig f(t)dt 30 ˜f⎝parenleftbig⎝radicalbig (p+a)2–b2⎝parenrightbig e–axf(x)+be–ax⎝integraldisplayx 0f⎝parenleftbig√ x2–t2⎝parenrightbig I1(bt)dt 31 ˜f(lnp) ⎝integraldisplay∞ 0xt–1 Γ(t)f(t)dt 32 1 p˜f(lnp) ⎝integraldisplay∞ 0xt Γ(t+1 )f(t)dt 33 ˜f(p–ia)+˜f(p+ia),i2=– 1 2f(x)c o s (ax) 34 i⎝bracketleftbig˜f(p–ia)–˜f(p+ia)⎝bracketrightbig ,i2=– 1 2f(x)s i n (ax) 35 d˜f(p) dp –xf(x) 36 dn˜f(p) dpn (–x)nf(x) 6.2. E XPRESSIONS WITH RATI ONAL FUNCTIONS 971 No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 37 pndm˜f(p) dpm,m≥n (–1)mdn dxn⎝bracketleftbig xmf(x)⎝bracketrightbig 38 ⎝integraldisplay∞ p˜f(q)dq 1 xf(x) 39 1 p⎝integraldisplayp 0˜f(q)dq ⎝integraldisplay∞ xf(t) tdt 40 1 p⎝integraldisplay∞ p˜f(q)dq ⎝integraldisplayx 0f(t) tdt 6.2. Expressions with Rational Functions No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 1 1 p 1 2 1 p+a e–ax 3 1 p2 x 4 1 p(p+a) 1 a⎝parenleftbig 1–e–ax⎝parenrightbig 5 1 (p+a)2 xe–ax 6 p (p+a)2 (1 –ax)e–ax 7 1 p2–a2 1 asinh(ax) 8 p p2–a2 cosh(ax) 9 1 (p+a)(p+b) 1 a–b⎝parenleftbig e–bx–e–ax⎝parenrightbig 10 p (p+a)(p+b) 1 a–b⎝parenleftbig ae–ax–be–bx⎝parenrightbig 11 1 p2+a2 1 asin(ax) 12 p p2+a2 cos(ax ) 13 1 (p+b)2+a2 1 ae–bxsin(ax) 14 p (p+b)2+a2 e–bx⎝bracketleftBig cos(ax )–b asin(ax)⎝bracketrightBig 972 TABLES OF INVERSE LAPLACE TRANSFORMS No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 15 1 p3 1 2x2 16 1 p2(p+a) 1 a2⎝parenleftbig e–ax+ax–1⎝parenrightbig 17 1 p(p+a)(p+b) 1 ab(a–b)⎝parenleftbig a–b+be–ax–ae–bx⎝parenrightbig 18 1 p(p+a)2 1 a2⎝parenleftbig 1–e–ax–axe–ax⎝parenrightbig 19 1 (p+a)(p+b)(p+c) (c–b)e–ax+(a–c)e–bx+(b–a)e–cx (a–b)(b–c)(c–a) 20 p (p+a)(p+b)(p+c) a(b–c)e–ax+b(c–a)e–bx+c(a–b)e–cx (a–b)(b–c)(c–a) 21 p2 (p+a)(p+b)(p+c) a2(c–b)e–ax+b2(a–c)e–bx+c2(b–a)e–cx (a–b)(b–c)(c–a) 22 1 (p+a)(p+b)2 1 (a–b)2⎝bracketleftbig e–ax–e–bx+(a–b)xe–bx⎝bracketrightbig 23 p (p+a)(p+b)2 1 (a–b)2⎝braceleftbig –ae–ax+[a+b(b–a)x⎝bracketrightbig e–bx⎝bracerightbig 24 p2 (p+a)(p+b)2 1 (a–b)2⎝bracketleftbig a2e–ax+b(b–2a–b2x+abx)e–bx⎝bracketrightbig 25 1 (p+a)3 1 2x2e–ax 26 p (p+a)3 x⎝parenleftbig 1–1 2ax⎝parenrightbig e–ax 27 p2 (p+a)3 ⎝parenleftbig1–2ax+1 2a2x2⎝parenrightbige–ax 28 1 p(p2+a2) 1 a2⎝bracketleftbig 1–c o s ( ax)⎝bracketrightbig 29 1 p⎝bracketleftbig (p+b)2+a2⎝bracketrightbig 1 a2+b2⎝braceleftbigg 1–e–bx⎝bracketleftBig cos(ax )+b asin(ax)⎝bracketrightBig⎝bracerightbigg 30 1 (p+a)(p2+b2) 1 a2+b2⎝bracketleftBig e–ax+a bsin(bx)–c o s ( bx)⎝bracketrightBig 31 p (p+a)(p2+b2) 1 a2+b2⎝bracketleftbig –ae–ax+acos(bx )+bsin(bx)⎝bracketrightbig 32 p2 (p+a)(p2+b2) 1 a2+b2⎝bracketleftbig a2e–ax–absin(bx)+b2cos(bx )⎝bracketrightbig 33 1 p3+a3 1 3a2e–ax–1 3a2eax/2⎝bracketleftbig cos(kx )–√ 3s i n (kx)⎝bracketrightbig , k=1 2a√ 3 34 p p3+a3 –1 3ae–ax+1 3aeax/2⎝bracketleftbig cos(kx )+√ 3s i n (kx)⎝bracketrightbig , k=1 2a√ 3 6.2. E XPRESSIONS WITH RATI ONAL FUNCTIONS 973 No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 35 p2 p3+a3 1 3e–ax+2 3eax/2cos(kx ),k=1 2a√ 3 36 1 ⎝parenleftbig p+a)⎝bracketleftbig (p+b)2+c2] e–ax–e–bxcos(cx )+ke–bxsin(cx) (a–b)2+c2,k=a–b c 37 p ⎝parenleftbig p+a)⎝bracketleftbig (p+b)2+c2] –ae–ax+ae–bxcos(cx )+ke–bxsin(cx) (a–b)2+c2, k=b2+c2–ab c 38 p2 ⎝parenleftbig p+a)⎝bracketleftbig (p+b)2+c2] a2e–ax+(b2+c2–2ab)e–bxcos(cx )+ke–bxsin(cx) (a–b)2+c2, k=–ac–bc+ab2–b3 c 39 1 p4 1 6x3 40 1 p3(p+a) 1 a3–1 a2x+1 2ax2–1 a3e–ax 41 1 p2(p+a)2 1 a2x⎝parenleftbig 1+e–ax⎝parenrightbig +2 a3⎝parenleftbig e–ax–1⎝parenrightbig 42 1 p2(p+a)(p+b) –a+b a2b2+1 abx+1 a2(b–a)e–ax+1 b2(a–b)e–bx 43 1 (p+a)2(p+b)2 1 (a–b)2⎝bracketleftBig e–ax⎝parenleftBig x+2 a–b⎝parenrightBig +e–bx⎝parenleftBig x–2 a–b⎝parenrightBig⎝bracketrightBig 44 1 (p+a)4 1 6x3e–ax 45 p (p+a)4 1 2x2e–ax–1 6ax3e–ax 46 1 p2(p2+a2) 1 a3⎝bracketleftbig ax–s i n (ax)⎝bracketrightbig 47 1 p4–a4 1 2a3⎝bracketleftbig sinh(ax)–s i n ( ax)⎝bracketrightbig 48 p p4–a4 1 2a2⎝bracketleftbig cosh(ax )–c o s ( ax)⎝bracketrightbig 49 p2 p4–a4 1 2a⎝bracketleftbig sinh(ax)+s i n ( ax)⎝bracketrightbig 50 p3 p4–a4 1 2⎝bracketleftbig cosh(ax )+c o s ( ax)⎝bracketrightbig 51 1 p4+a4 1 a3√ 2⎝parenleftbig coshξsinξ–s i n h ξcosξ⎝parenrightbig ,ξ=ax √ 2 52 p p4+a4 1 a2sin⎝parenleftBigax √ 2⎝parenrightBig sinh⎝parenleftBigax √ 2⎝parenrightBig 53 p2 p4+a4 1 a√ 2⎝parenleftbig cosξsinhξ+s i nξcoshξ⎝parenrightbig ,ξ=ax √ 2 974 TABLES OF INVERSE LAPLACE TRANSFORMS No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 54 1 (p2+a2)2 1 2a3⎝bracketleftbig sin(ax)–axcos(ax )⎝bracketrightbig 55 p (p2+a2)2 1 2axsin(ax) 56 p2 (p2+a2)2 1 2a⎝bracketleftbig sin(ax)+axcos(ax )⎝bracketrightbig 57 p3 (p2+a2)2 cos(ax )–1 2axsin(ax) 58 1 ⎝bracketleftbig (p+b)2+a2⎝bracketrightbig2 1 2a3e–bx⎝bracketleftbig sin(ax)–axcos(ax )⎝bracketrightbig 59 1 (p2–a2)(p2–b2) 1 a2–b2⎝bracketleftBig1 asinh(ax)–1 bsinh(bx)⎝bracketrightBig 60 p (p2–a2)(p2–b2) cosh(ax )–c o s h ( bx) a2–b2 61 p2 (p2–a2)(p2–b2) asinh(ax)–bsinh(bx) a2–b2 62 p3 (p2–a2)(p2–b2) a2cosh(ax )–b2cosh(bx ) a2–b2 63 1 (p2+a2)(p2+b2) 1 b2–a2⎝bracketleftBig1 asin(ax)–1 bsin(bx)⎝bracketrightBig 64 p (p2+a2)(p2+b2) cos(ax )–c o s ( bx) b2–a2 65 p2 (p2+a2)(p2+b2) –asin(ax)+bsin(bx) b2–a2 66 p3 (p2+a2)(p2+b2) –a2cos(ax )+b2cos(bx ) b2–a2 67 1 pn,n=1 ,2 , ... 1 (n–1 ) !xn–1 68 1 (p+a)n,n=1 ,2 , ... 1 (n–1 ) !xn–1e–ax 69 1 p(p+a)n,n=1 ,2 , ... a–n⎝bracketleftbig 1–e–axen(ax)⎝bracketrightbig ,en(z)=1+z 1!+···+zn n! 70 1 p2n+a2n,n=1 ,2 , ... –1 na2nn⎝summationdisplay k=1exp(akx)⎝bracketleftbig akcos(b kx)–bksin(bkx)⎝bracketrightbig , ak=acosϕk,bk=asinϕk,ϕk=π(2k–1 ) 2n 71 1 p2n–a2n,n=1 ,2 , ... 1 na2n–1sinh(ax)+1 na2nn⎝summationdisplay k=2exp(akx) ×⎝bracketleftbig akcos(b kx)–bksin(bkx)⎝bracketrightbig , ak=acosϕk,bk=asinϕk,ϕk=π(k–1 ) n 6.3. E XPRESSIONS WITH SQUARE ROOTS 975 No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 72 1 p2n+1+a2n+1,n=0 ,1 , ... e–ax (2n+1 )a2n–2 (2n+1 )a2n+1n⎝summationdisplay k=1exp(akx) ×⎝bracketleftbig akcos(b kx)–bksin(bkx)⎝bracketrightbig , ak=acosϕk,bk=asinϕk,ϕk=π(2k–1 ) 2n+1 73 1 p2n+1–a2n+1,n=0 ,1 , ... eax (2n+1 )a2n+2 (2n+1 )a2n+1n⎝summationdisplay k=1exp(akx) ×⎝bracketleftbig akcos(b kx)–bksin(bkx)⎝bracketrightbig , ak=acosϕk,bk=asinϕk,ϕk=2πk 2n+1 74 Q(p) P(p), P(p)=(p–a1)...(p–an); Q(p) is a polynomial of degree ≤n–1 ;ai≠ajifi≠j n⎝summationdisplay k=1Q(ak) P/prime(ak)exp⎝parenleftbig akx⎝parenrightbig , (the prime stand for the differentiation) 75 Q(p) P(p), P(p)=(p–a1)m1...(p–an)mn; Q(p) is a polynomial of degree <m1+m2+···+mn–1 ; ai≠ajifi≠j n⎝summationdisplay k=1mk⎝summationdisplay l=1Φkl(ak) (mk–l)! (l–1 ) !xmk–lexp⎝parenleftbig akx⎝parenrightbig , Φkl(p)=dl–1 dpl–1⎝bracketleftbiggQ(p) Pk(p)⎝bracketrightbigg ,Pk(p)=P(p) (p–ak)mk 76 Q(p)+pR(p) P(p), P(p)=(p2+a2 1)...(p2+a2 n); Q(p)a n dR(p) are polynomials of degree ≤2n–2 ;al≠aj,l≠j n⎝summationdisplay k=1Q(iak)s i n (akx)+akR(iak)c o s (akx) akPk(iak), Pm(p)=P(p) p2+a2m,i2=– 1 6.3. Expressions with Square Roots No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 1 1 √ p 1 √ πx 2 √ p–a–⎝radicalbig p–b ebx–eax 2√ πx3 3 1 √ p+a 1 √ πxe–ax 4 ⎝radicalbigg p+a p–1 1 2ae–ax/2⎝bracketleftbig I1⎝parenleftbig1 2ax⎝parenrightbig +I0⎝parenleftbig1 2ax⎝parenrightbig⎝bracketrightbig 5 √ p+a p+b e–ax √ πx+(a–b)1/2e–bxerf⎝bracketleftbig (a–b)1/2x1/2⎝bracketrightbig 976 TABLES OF INVERSE LAPLACE TRANSFORMS No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 6 1 p√ p 2⎝radicalbigg x π 7 1 (p+a)√ p+b (b–a)–1/2e–axerf⎝bracketleftbig (b–a)1/2x1/2⎝bracketrightbig 8 1 √ p(p–a) 1 √ aeaxerf⎝parenleftbig√ ax⎝parenrightbig 9 1 p3/2(p–a) a–3/2eaxerf⎝parenleftbig√ ax⎝parenrightbig –2a–1π–1/2x1/2 10 1 √ p+a π–1/2x–1/2–aea2xerfc⎝parenleftbig a√ x⎝parenrightbig 11 a p⎝parenleftbig√ p+a⎝parenrightbig 1–ea2xerfc⎝parenleftbig a√ x⎝parenrightbig 12 1 p+a√ p ea2xerfc⎝parenleftbig a√ x⎝parenrightbig 13 1 ⎝parenleftbig√ p+√ a⎝parenrightbig2 1–2 √ π(ax)1/2+( 1–2 ax)eax⎝bracketleftbig erf⎝parenleftbig√ ax⎝parenrightbig –1⎝bracketrightbig 14 1 p⎝parenleftbig√ p+√ a⎝parenrightbig2 1 a+⎝parenleftBig 2x–1 a⎝parenrightBig eaxerfc⎝parenleftbig√ ax⎝parenrightbig –2 √ πa√ x 15 1 √ p⎝parenleftbig√ p+a⎝parenrightbig2 2π–1/2x1/2–2axea2xerfc⎝parenleftbig a√ x⎝parenrightbig 16 1 ⎝parenleftbig√ p+a⎝parenrightbig3 2 √ π(a2x+1 )√ x–ax(2a2x+3 )ea2xerfc⎝parenleftbig a√ x⎝parenrightbig 17 p–n–1/2,n=1 ,2 , ... 2n 1⋅3...(2n–1 )√ πxn–1/2 18 (p+a)–n–1/2 2n 1⋅3...(2n–1 )√ πxn–1/2e–ax 19 1 ⎝radicalbig p2+a2 J0(ax) 20 1 ⎝radicalbig p2–a2 I0(ax) 21 1 ⎝radicalbig p2+ap+b exp⎝parenleftbig –1 2ax⎝parenrightbig J0⎝bracketleftbig (b–1 4a2⎝parenrightbig1/2x⎝bracketrightbig 22 ⎝parenleftbig⎝radicalbig p2+a2–p⎝parenrightbig1/2 1 √ 2πx3sin(ax) 23 1 ⎝radicalbig p2+a2⎝parenleftbig⎝radicalbig p2+a2+p⎝parenrightbig1/2 √ 2 √ πxcos(ax ) 24 1 ⎝radicalbig p2–a2⎝parenleftbig⎝radicalbig p2–a2+p⎝parenrightbig1/2 √ 2 √ πxcosh(ax) 6.4. E XPRESSIONS WITH ARBITRARY POWERS 977 No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 25 ⎝parenleftbig⎝radicalbig p2+a2+p⎝parenrightbig–n na–nx–1Jn(ax) 26 ⎝parenleftbig⎝radicalbig p2–a2+p⎝parenrightbig–n na–nx–1In(ax) 27 ⎝parenleftbig p2+a2⎝parenrightbig–n–1/2 (x/a)nJn(ax) 1⋅3⋅5...(2n–1 ) 28 ⎝parenleftbig p2–a2⎝parenrightbig–n–1/2 (x/a)nIn(ax) 1⋅3⋅5...(2n–1 ) 6.4. Expressions with Arbitrary Powers No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 1 (p+a)–ν,ν>0 1 Γ(ν)xν–1e–ax 2 ⎝bracketleftbig (p+a)1/2+(p+b)1/2⎝bracketrightbig–2ν,ν>0 ν (a–b)νx–1exp⎝bracketleftbig –1 2(a+b)x⎝bracketrightbig Iν⎝bracketleftbig1 2(a–b)x⎝bracketrightbig 3 ⎝bracketleftbig (p+a)(p+b)⎝bracketrightbig–ν,ν>0 √ π Γ(ν)⎝parenleftBigx a–b⎝parenrightBigν–1/2 exp⎝parenleftBig –a+b 2x⎝parenrightBig Iν–1/2⎝parenleftBiga–b 2x⎝parenrightBig 4 ⎝parenleftbig p2+a2⎝parenrightbig–ν–1/2,ν>–1 2 √ π (2a)νΓ(ν+1 2)xνJν(ax) 5 ⎝parenleftbig p2–a2⎝parenrightbig–ν–1/2,ν>–1 2 √ π (2a)νΓ(ν+1 2)xνIν(ax) 6 p⎝parenleftbig p2+a2⎝parenrightbig–ν–1/2,ν>0 a√ π (2a)νΓ⎝parenleftbig ν+1 2⎝parenrightbigxνJν–1(ax) 7 p⎝parenleftbig p2–a2⎝parenrightbig–ν–1/2,ν>0 a√ π (2a)νΓ⎝parenleftbig ν+1 2⎝parenrightbigxνIν–1(ax) 8 ⎝bracketleftbig (p2+a2)1/2+p⎝bracketrightbig–ν= a–2ν⎝bracketleftbig (p2+a2)1/2–p⎝bracketrightbigν,ν>0 νa–νx–1Jν(ax) 9 ⎝bracketleftbig (p2–a2)1/2+p⎝bracketrightbig–ν= a–2ν⎝bracketleftbig p–(p2–a2)1/2⎝bracketrightbigν,ν>0 νa–νx–1Iν(ax) 10 p⎝bracketleftbig (p2+a2)1/2+p⎝bracketrightbig–ν,ν>1 νa1–νx–1Jν–1(ax)–ν(ν+1 )a–νx–2Jν(ax) 11 p⎝bracketleftbig (p2–a2)1/2+p⎝bracketrightbig–ν,ν>1 νa1–νx–1Iν–1(ax)–ν(ν+1 )a–νx–2Iν(ax) 12 ⎝parenleftbig⎝radicalbig p2+a2+p⎝parenrightbig–ν ⎝radicalbig p2+a2,ν>– 1 a–νJν(ax) 13 ⎝parenleftbig⎝radicalbig p2–a2+p⎝parenrightbig–ν ⎝radicalbig p2–a2,ν>– 1 a–νIν(ax) 978 TABLES OF INVERSE LAPLACE TRANSFORMS 6.5. Expressions with Exponential Functions No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 1 p–1e–ap,a>0 ⎝braceleftBig0i f 0 < x<a, 1i f a<x. 2 p–1⎝parenleftbig 1–e–ap⎝parenrightbig ,a>0 ⎝braceleftBig1i f 0 < x<a, 0i f a<x. 3 p–1⎝parenleftbig e–ap–e–bp⎝parenrightbig ,0 ≤a0 ⎝braceleftBig0i f 0 < x<a, e–b(x–a)ifa<x. 6 p–νe–ap,ν>0 ⎝braceleftBigg0i f 0 < x<a, (x–a)ν–1 Γ(ν)ifa<x. 7 p–1⎝parenleftbig eap–1⎝parenrightbig–1,a>0 f(x)=nifna<x<(n+1 )a;n=0 ,1 ,2 , ... 8 ea/p–1 ⎝radicalbigg a xI1⎝parenleftbig 2√ ax⎝parenrightbig 9 p–1/2ea/p 1 √ πxcosh⎝parenleftbig 2√ ax⎝parenrightbig 10 p–3/2ea/p 1 √ πasinh⎝parenleftbig 2√ ax⎝parenrightbig 11 p–5/2ea/p ⎝radicalbigg x πacosh⎝parenleftbig 2√ ax⎝parenrightbig –1 2√ πa3sinh⎝parenleftbig 2√ ax⎝parenrightbig 12 p–ν–1ea/p,ν>– 1 (x/a)ν/2Iν(2√ ax⎝parenrightbig 13 1–e–a/p ⎝radicalbigg a xJ1⎝parenleftbig 2√ ax⎝parenrightbig 14 p–1/2e–a/p 1 √ πxcos⎝parenleftbig 2√ ax⎝parenrightbig 15 p–3/2e–a/p 1 √ πasin⎝parenleftbig 2√ ax⎝parenrightbig 16 p–5/2e–a/p 1 2√ πa3sin⎝parenleftbig 2√ ax⎝parenrightbig –⎝radicalbigg x πacos⎝parenleftbig 2√ ax⎝parenrightbig 17 p–ν–1e–a/p,ν>– 1 (x/a)ν/2Jν(2√ ax⎝parenrightbig 18 exp⎝parenleftbig –√ ap⎝parenrightbig ,a>0 √ a 2√ πx–3/2exp⎝parenleftBig –a 4x⎝parenrightBig 19 pexp⎝parenleftbig –√ ap⎝parenrightbig ,a>0 √ a 8√ π(a–6x)x–7/2exp⎝parenleftBig –a 4x⎝parenrightBig 20 1 pexp⎝parenleftbig –√ ap⎝parenrightbig ,a≥0 erfc⎝parenleftBig√ a 2√ x⎝parenrightBig 6.6. E XPRESSIONS WITH HYPERBOLIC FUNCTIONS 979 No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 21 √ pexp⎝parenleftbig –√ ap⎝parenrightbig ,a>0 1 4√ π(a–2x)x–5/2exp⎝parenleftBig –a 4x⎝parenrightBig 22 1 √ pexp⎝parenleftbig –√ ap⎝parenrightbig ,a≥0 1 √ πxexp⎝parenleftBig –a 4x⎝parenrightBig 23 1 p√ pexp⎝parenleftbig –√ ap⎝parenrightbig ,a≥0 2√ x √ πexp⎝parenleftBig –a 4x⎝parenrightBig –√ aerfc⎝parenleftBig√ a 2√ x⎝parenrightBig 24 exp⎝parenleftbig –k⎝radicalbig p2+a2⎝parenrightbig ⎝radicalbig p2+a2,k>0 ⎝braceleftbigg0i f0<x<k, J0⎝parenleftbig a√ x2–k2⎝parenrightbig ifk<x. 25 exp⎝parenleftbig –k⎝radicalbig p2–a2⎝parenrightbig ⎝radicalbig p2–a2,k>0 ⎝braceleftbigg0i f0<x<k, I0⎝parenleftbig a√ x2–k2⎝parenrightbig ifk<x. 6.6. Expressions with Hyperbolic Functions No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 1 1 psinh(ap),a>0 f(x)=2nifa(2n–1 )< x<a(2n+1 ) ; n=0 ,1 ,2 , ...(x>0 ) 2 1 p2sinh(ap),a>0 f(x)=2n(x–an)i f a(2n–1 )< x<a(2n+1 ) ; n=0 ,1 ,2 , ...(x>0 ) 3 sinh(a/p) √ p 1 2√ πx⎝bracketleftbig cosh⎝parenleftbig 2√ ax⎝parenrightbig –c o s⎝parenleftbig 2√ ax⎝parenrightbig⎝bracketrightbig 4 sinh(a/p) p√ p 1 2√ πa⎝bracketleftbig sinh⎝parenleftbig 2√ ax⎝parenrightbig –s i n⎝parenleftbig 2√ ax⎝parenrightbig⎝bracketrightbig 5 p–ν–1sinh(a/p),ν>– 2 1 2(x/a)ν/2⎝bracketleftbig Iν⎝parenleftbig 2√ ax⎝parenrightbig –Jν⎝parenleftbig 2√ ax⎝parenrightbig⎝bracketrightbig 6 1 pcosh(ap),a>0 f(x)=⎝braceleftbigg 0i f a(4n–1 )< x<a(4n+1 ) , 2i f a(4n+1 )< x<a(4n+3 ) , n=0 ,1 ,2 , ...(x>0 ) 7 1 p2cosh(ap),a>0 x– (–1)n(x–2an)i f2 n–1<x/a <2n+1 ; n=0 ,1 ,2 , ...(x>0 ) 8 cosh(a/p ) √ p 1 2√ πx⎝bracketleftbig cosh⎝parenleftbig 2√ ax⎝parenrightbig +c o s⎝parenleftbig 2√ ax⎝parenrightbig⎝bracketrightbig 9 cosh(a/p ) p√ p 1 2√ πa⎝bracketleftbig sinh⎝parenleftbig 2√ ax⎝parenrightbig +s i n⎝parenleftbig 2√ ax⎝parenrightbig⎝bracketrightbig 10 p–ν–1cosh(a/p ),ν>– 1 1 2(x/a)ν/2⎝bracketleftbig Iν⎝parenleftbig 2√ ax⎝parenrightbig +Jν⎝parenleftbig 2√ ax⎝parenrightbig⎝bracketrightbig 11 1 ptanh(ap), a>0 f(x)=( – 1 )n–1if 2a(n–1 )< x<2an; n=1 ,2 , ... 980 TABLES OF INVERSE LAPLACE TRANSFORMS No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 12 1 pcoth(ap), a>0 f(x)=( 2n– 1 )i f2 a(n–1 )< x<2an; n=1 ,2 , ... 13 Arcoth( p/a) 1 xsinh(ax) 6.7. Expressions with Logarithmic Functions No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 1 1 plnp –l nx–C, C= 0.5772 ...is the Euler constant 2 p–n–1lnp ⎝parenleftbig 1+1 2+1 3+···+1 n–l nx–C⎝parenrightbigxn n!, C= 0.5772 ...is the Euler constant 3 p–n–1/2lnp kn⎝bracketleftbig 2+2 3+2 5+···+2 2n–1–l n ( 4x)–C⎝bracketrightbig xn–1/2, kn=2n 1⋅3⋅5...(2n–1 )√ π,C= 0.5772 ... 4 p–νlnp,ν>0 1 Γ(ν)xν–1⎝bracketleftbig ψ(ν)–l nx⎝bracketrightbig ,ψ(ν) is the logarithmic derivative of the gamma function 5 1 p(lnp)2 (lnx+C)2–1 6π2,C= 0.5772 ... 6 1 p2(lnp)2 x⎝bracketleftbig (lnx+C–1 )2+1–1 6π2⎝bracketrightbig 7 ln(p+b) p+a e–ax⎝braceleftbig ln(b–a)–E i⎝bracketleftbig (a–b)x⎝bracketrightbig } 8 lnp p2+a2 1 acos(ax )S i (ax)+1 asin(ax)⎝bracketleftbig lna–C i (ax)⎝bracketrightbig 9 plnp p2+a2 cos(ax )⎝bracketleftbig lna–C i (ax)⎝bracketrightbig –s i n (ax)S i (ax)⎝bracketrightbig 10 lnp+b p+a 1 x⎝parenleftbig e–ax–e–bx⎝parenrightbig 11 lnp2+b2 p2+a2 2 x⎝bracketleftbig cos(ax )–c o s ( bx)⎝bracketrightbig 12 plnp2+b2 p2+a2 2 x⎝bracketleftbig cos(bx )+bxsin(bx)–c o s ( ax)–axsin(ax)⎝bracketrightbig 13 ln(p+a)2+k2 (p+b)2+k2 2 xcos(kx )(e–bx–e–ax⎝parenrightbig 14 pln⎝parenleftBig1 p⎝radicalbig p2+a2⎝parenrightBig 1 x2⎝bracketleftbig cos(ax )–1⎝bracketrightbig +a xsin(ax) 15 pln⎝parenleftBig1 p⎝radicalbig p2–a2⎝parenrightBig 1 x2⎝bracketleftbig cosh(ax )–1⎝bracketrightbig –a xsinh(ax) 6.9. E XPRESSIONS WITH SPECIAL FUNCTIONS 981 6.8. Expressions with Trigonometric Functions No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 1 sin(a/p) √ p 1 √ πxsinh⎝parenleftbig√ 2ax⎝parenrightbig sin⎝parenleftbig√ 2ax⎝parenrightbig 2 sin(a/p) p√ p 1 √ πacosh⎝parenleftbig√ 2ax⎝parenrightbig sin⎝parenleftbig√ 2ax⎝parenrightbig 3 cos(a/p ) √ p 1 √ πxcosh⎝parenleftbig√ 2ax⎝parenrightbig cos⎝parenleftbig√ 2ax⎝parenrightbig 4 cos(a/p ) p√ p 1 √ πasinh⎝parenleftbig√ 2ax⎝parenrightbig cos⎝parenleftbig√ 2ax⎝parenrightbig 5 1 √ pexp⎝parenleftbig –√ ap⎝parenrightbig sin⎝parenleftbig√ ap⎝parenrightbig 1 √ πxsin⎝parenleftBiga 2x⎝parenrightBig 6 1 √ pexp⎝parenleftbig –√ ap⎝parenrightbig cos⎝parenleftbig√ ap⎝parenrightbig 1 √ πxcos⎝parenleftBiga 2x⎝parenrightBig 7 arctana p 1 xsin(ax) 8 1 parctana p Si(ax) 9 parctana p–a 1 x2⎝bracketleftbig axcos(ax )–s i n ( ax)⎝bracketrightbig 10 arctan2ap p2+b2 2 xsin(ax)c o s⎝parenleftbig x√ a2+b2⎝parenrightbig 6.9. Expressions with Special Functions No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 1 exp⎝parenleftbig ap2⎝parenrightbig erfc⎝parenleftbig p√ a⎝parenrightbig 1 √ πaexp⎝parenleftBig –x2 4a⎝parenrightBig 2 1 pexp⎝parenleftbig ap2⎝parenrightbig erfc⎝parenleftbig p√ a⎝parenrightbig erf⎝parenleftBigx 2√ a⎝parenrightBig 3 erfc⎝parenleftbig√ ap⎝parenrightbig ,a>0 ⎝braceleftBigg0i f 0 < x<a,√ a πx√ x–aifa<x. 4 eaperfc⎝parenleftbig√ ap⎝parenrightbig √ a π√ x(x+a) 5 1 √ peaperfc⎝parenleftbig√ ap⎝parenrightbig 1 √ π(x+a) 6 erf⎝parenleftbig⎝radicalbig a/p⎝parenrightbig 1 πxsin⎝parenleftbig 2√ ax⎝parenrightbig 982 TABLES OF INVERSE LAPLACE TRANSFORMS No Laplace transform, ˜f(p) Inverse transform, f(x)=1 2πi⎝integraldisplayc+i∞ c–i∞epx˜f(p)dp 7 1 √ pexp(a/p)e r f⎝parenleftbig⎝radicalbig a/p⎝parenrightbig 1 √ πxsinh⎝parenleftbig 2√ ax⎝parenrightbig 8 1 √ pexp(a/p) erfc⎝parenleftbig⎝radicalbig a/p⎝parenrightbig 1 √ πxexp⎝parenleftbig –2√ ax⎝parenrightbig 9 p–aγ(a,bp),a,b>0 ⎝braceleftbigg xa–1if 0 < x<b, 0i f b<x. 10 γ(a,b/p),a>0 ba/2xa/2–1Ja⎝parenleftbig 2√ bx⎝parenrightbig 11 a–pγ(p,a) exp⎝parenleftbig –ae–x⎝parenrightbig 12 K0(ap), a>0 ⎝braceleftbigg 0i f 0 < x<a, (x2–a2)–1/2ifa<x. 13 Kν(ap), a>0 ⎧ ⎨ ⎩0i f0<x<a, cosh⎝bracketleftbig νArcosh( x/a)⎝bracketrightbig √ x2–a2ifa<x. 14 K0⎝parenleftbig a√ p⎝parenrightbig 1 2xexp⎝parenleftBig –a2 4x⎝parenrightBig 15 1 √ pK1⎝parenleftbig a√ p⎝parenrightbig 1 aexp⎝parenleftBig –a2 4x⎝parenrightBig References for Supplement 6: G. Doetsch (1950, 1956, 1958), H. Bateman and A. Erd ´elyi (1954), I. I. Hirschman and D. V . Widder (1955), V . A. Ditkin and A. P. Prudnikov (1965), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992,Vo l . 5 ) . Supplement 7 Tables of Fourier Cosine Transforms 7.1. General Formulas No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 1 af1(x)+bf2(x) aˇf1c(u)+bˇf2c(u) 2 f(ax),a>0 1 aˇfc⎝parenleftBigu a⎝parenrightBig 3 x2nf(x),n=1 ,2 , ... (–1)nd2n du2nˇfc(u) 4 x2n+1f(ax),n=0 ,1 , ... (–1)nd2n+1 du2n+1ˇfs(u),ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (xu)dx 5 f(ax)c o s (bx),a,b>0 1 2a⎝bracketleftBig ˇfc⎝parenleftBigu+b a⎝parenrightBig +ˇfc⎝parenleftBigu–b a⎝parenrightBig⎝bracketrightBig 7.2. Expressions with Power-Law Functions No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 1 ⎝braceleftBig1i f 0 < x<a, 0i f a<x 1 usin(au) 2 ⎝braceleftBiggx if 0 < x<1 , 2–x if 1 < x<2 , 0i f 2 < x 4 u2cosusin2u 2 3 1 a+x,a>0 –s i n (au)s i (au)–c o s ( au)C i (au) 4 1 a2+x2,a>0 π 2ae–au(the integral is understood in the sense of Cauchy principal value) 5 1 a2–x2,a>0 πsin(au) 2u 6 a a2+(b+x)2+a a2+(b–x)2 πe–aucos(bu ) 7 b+x a2+(b+x)2+b–x a2+(b–x)2 πe–ausin(bu) 8 1 a4+x4,a>0 1 2πa–3exp⎝parenleftBig –au √ 2⎝parenrightBig sin⎝parenleftBigπ 4+au √ 2⎝parenrightBig 983 984 TABLES OF FOURIER COSINE TRANSFORMS No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 9 1 (a2+x2)(b2+x2),a,b>0 π 2ae–bu–be–au ab(a2–b2) 10 x2m (x2+a)n+1, n,m=1 ,2 , ...;n+1>m≥0 (–1)n+mπ 2n!∂n ∂an⎝parenleftbig a1/√ me–u√ a⎝parenrightbig 11 1 √ x ⎝radicalbigg π 2u 12 ⎝braceleftBigg1 √ xif 0 < x<a, 0i f a<x 2⎝radicalbigg π 2uC(au),C(u) is the Fresnel integral 13 ⎝braceleftBigg0i f 0 < x<a, 1 √ xifa<x ⎝radicalbigg π 2u⎝bracketleftbig 1–2C(au)⎝bracketrightbig ,C(u) is the Fresnel integral 14 ⎝braceleftBigg0i f 0 < x<a, 1 √ x–aifa<x ⎝radicalbigg π 2u⎝bracketleftbig cos(au )–s i n ( au)⎝bracketrightbig 15 1 √ a2+x2 K0(au) 16 ⎝braceleftBigg1 √ a2–x2if 0 < x<a, 0i f a<x π 2J0(au) 17 x–ν,0 < ν<1 sin⎝parenleftbig1 2πν⎝parenrightbigΓ(1 –ν)uν–1 7.3. Expressions with Exponential Functions No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 1 e–ax a a2+u2 2 1 x⎝parenleftbig e–ax–e–bx⎝parenrightbig 1 2lnb2+u2 a2+u2 3 √ xe–ax 1 2√ π(a2+u2)–3/4cos⎝parenleftBig 3 2arctanu a⎝parenrightBig 4 1 √ xe–ax ⎝radicalbigg π 2⎝bracketleftBiga+(a2+u2)1/2 a2+u2⎝bracketrightBig1/2 5 xne–ax,n=1 ,2 , ... an+1n! (a2+u2)n+1⎝summationdisplay 0≤2k≤n+1(–1)kC2k n+1⎝parenleftBigu a⎝parenrightBig2k 6 xn–1/2e–ax,n=1 ,2 , ... knu∂n ∂an1 r√ r–a, where r=√ a2+u2,kn= (–1)n⎝radicalbig π/2 7 xν–1e–ax Γ(ν)(a2+u2)–ν/2cos⎝parenleftBig νarctanu a⎝parenrightBig 7.5. E XPRESSIONS WITH LOGARITHMIC FUNCTIONS 985 No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 8 x eax–1 1 2u2–π2 2a2sinh2⎝parenleftbig πa–1u⎝parenrightbig 9 1 x⎝parenleftBig1 2–1 x+1 ex–1⎝parenrightBig –1 2ln⎝parenleftbig 1–e–2πu⎝parenrightbig 10 exp⎝parenleftbig –ax2⎝parenrightbig 1 2⎝radicalbigg π aexp⎝parenleftBig –u2 4a⎝parenrightBig 11 1 √ xexp⎝parenleftBig –a x⎝parenrightBig ⎝radicalbigg π 2ue–√ 2au⎝bracketleftbig cos⎝parenleftbig√ 2au⎝parenrightbig –s i n⎝parenleftbig√ 2au⎝parenrightbig⎝bracketrightbig 12 1 x√ xexp⎝parenleftBig –a x⎝parenrightBig ⎝radicalbigg π ae–√ 2aucos⎝parenleftbig√ 2au⎝parenrightbig 7.4. Expressions with Hyperbolic Functions No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 1 1 cosh(ax ),a>0 π 2acosh⎝parenleftbig1 2πa–1u⎝parenrightbig 2 1 cosh2(ax),a>0 πu 2a2sinh⎝parenleftbig1 2πa–1u⎝parenrightbig 3 cosh(ax ) cosh(bx ),|a|<b π b⎝bracketleftbiggcos⎝parenleftbig1 2πab–1⎝parenrightbig cosh⎝parenleftbig1 2πb–1u⎝parenrightbig cos⎝parenleftbig πab–1⎝parenrightbig +c o s h⎝parenleftbig πb–1u⎝parenrightbig⎝bracketrightbigg 4 1 cosh(ax )+c o s b πsinh⎝parenleftbig a–1bu⎝parenrightbig asinbsinh⎝parenleftbig πa–1u⎝parenrightbig 5 exp⎝parenleftbig–ax2⎝parenrightbigcosh(bx ),a>0 1 2⎝radicalbigg π aexp⎝parenleftBigb2–u2 4a⎝parenrightBig cos⎝parenleftBigabu 2⎝parenrightBig 6 x sinh(ax) π2 4a2cosh2⎝parenleftbig1 2πa–1u⎝parenrightbig 7 sinh(ax) sinh(bx),|a|0 ln⎝bracketleftbig coth⎝parenleftbig1 4πa–1u⎝parenrightbig⎝bracketrightbig 7.5. Expressions with Logarithmic Functions No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 1 ⎝braceleftBiglnxif 0 < x<1 , 0i f 1 < x –1 uSi(u) 986 TABLES OF FOURIER COSINE TRANSFORMS No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 2 lnx √ x –⎝radicalbigg π 2u⎝bracketleftbig ln(4u)+C+π 2⎝bracketrightbig , C= 0.5772 ...is the Euler constant 3 xν–1lnx,0 < ν<1 Γ(ν)c o s⎝parenleftbigπν 2⎝parenrightbig u–ν⎝bracketleftBig ψ(ν)–π 2tan⎝parenleftBigπν 2⎝parenrightBig –l nu⎝bracketrightBig 4 ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x a–x⎝vextendsingle⎝vextendsingle⎝vextendsingle,a>0 2 u⎝bracketleftbig cos(au )S i (au)–s i n ( au)C i (au)⎝bracketrightbig 5 ln⎝parenleftbig 1+a2/x2⎝parenrightbig ,a>0 π u⎝parenleftbig 1–e–au⎝parenrightbig 6 lna2+x2 b2+x2,a,b>0 π u⎝parenleftbig e–bu–e–au⎝parenrightbig 7 e–axlnx,a>0 –aC+1 2aln(u2+a2)+uarctan( u/a) u2+a2 8 ln⎝parenleftbig 1+e–ax⎝parenrightbig ,a>0 a 2u2–π 2usinh⎝parenleftbig πa–1u⎝parenrightbig 9 ln⎝parenleftbig 1–e–ax⎝parenrightbig ,a>0 a 2u2–π 2ucoth⎝parenleftbig πa–1u⎝parenrightbig 7.6. Expressions with Trigonometric Functions No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 1 sin(ax) x,a>0 ⎧ ⎨ ⎩1 2πifu<a, 1 4πifu=a, 0i f u>a 2 xν–1sin(ax),a>0 , |ν|<1 π(u+a)–ν–|u+a|–νsign(u–a) 4Γ(1 –ν)c o s⎝parenleftbig1 2πν⎝parenrightbig 3 xsin(ax) x2+b2,a,b>0 ⎝braceleftbigg1 2πe–abcosh(bu )i f u<a, –1 2πe–businh(ab)i fu>a 4 sin(ax) x(x2+b2),a,b>0 ⎝braceleftbigg1 2πb–2⎝bracketleftbig 1–e–abcosh(bu )⎝bracketrightbig ifu<a, 1 2πb–2e–businh(ab)i f u>a 5 e–bxsin(ax),a,b>0 1 2⎝bracketleftBiga+u (a+u)2+b2+a–u (a–u)2+b2⎝bracketrightBig 6 1 xsin2(ax),a>0 1 4ln⎝vextendsingle⎝vextendsingle⎝vextendsingle1–4a2 u2⎝vextendsingle⎝vextendsingle⎝vextendsingle 7 1 x2sin2(ax),a>0 ⎝braceleftbigg 1 4π(2a–u)i fu<2a, 0i fu>2a 8 1 xsin⎝parenleftBiga x⎝parenrightBig ,a>0 π 2J0⎝parenleftbig 2√ au⎝parenrightbig 7.7. E XPRESSIONS WITH SPECIAL FUNCTIONS 987 No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 9 1 √ xsin⎝parenleftbig a√ x⎝parenrightbig sin⎝parenleftbig b√ x⎝parenrightbig ,a,b>0 ⎝radicalbigg π usin⎝parenleftBigab 2u⎝parenrightBig sin⎝parenleftBiga2+b2 4u–π 4⎝parenrightBig 10 sin⎝parenleftbigax2⎝parenrightbig,a>0 ⎝radicalbigg π 8a⎝bracketleftBig cos⎝parenleftBigu2 4a⎝parenrightBig –s i n⎝parenleftBigu2 4a⎝parenrightBig⎝bracketrightBig 11 exp⎝parenleftbig –ax2⎝parenrightbig sin⎝parenleftbig bx2⎝parenrightbig ,a>0 √ π (A2+B2)1/4exp⎝parenleftBig –Au2 A2+B2⎝parenrightBig sin⎝parenleftBig ϕ–Bu2 A2+B2⎝parenrightBig , A=4a,B=4b,ϕ=1 2arctan( b/a) 12 1–c o s ( ax) x,a>0 1 2ln⎝vextendsingle⎝vextendsingle⎝vextendsingle1–a2 u2⎝vextendsingle⎝vextendsingle⎝vextendsingle 13 1–c o s ( ax) x2,a>0 ⎝braceleftbigg 1 2π(a–u)i fu<a, 0i f u>a 14 xν–1cos(ax ),a>0 ,0<ν <1 1 2Γ(ν)c o s⎝parenleftbig1 2πν⎝parenrightbig⎝bracketleftbig |u–a|–ν+(u+a)–ν⎝bracketrightbig 15 cos(ax ) x2+b2,a,b>0 ⎝braceleftbigg1 2πb–1e–abcosh(bu )i fu<a, 1 2πb–1e–bucosh(ab)i fu>a 16 e–bxcos(ax ),a,b>0 b 2⎝bracketleftBig1 (a+u)2+b2+1 (a–u)2+b2⎝bracketrightBig 17 1 √ xcos⎝parenleftbig a√ x⎝parenrightbig ⎝radicalbigg π usin⎝parenleftBiga2 4u+π 4⎝parenrightBig 18 1 √ xcos⎝parenleftbig a√ x⎝parenrightbig cos⎝parenleftbig b√ x⎝parenrightbig ⎝radicalbigg π ucos⎝parenleftBigab 2u⎝parenrightBig sin⎝parenleftBiga2+b2 4u+π 4⎝parenrightBig 19 exp⎝parenleftbig –bx2⎝parenrightbig cos(ax ),b>0 1 2⎝radicalbigg π bexp⎝parenleftBig –a2+u2 4b⎝parenrightBig cosh⎝parenleftBigau 2b⎝parenrightBig 20 cos⎝parenleftbig ax2⎝parenrightbig ,a>0 ⎝radicalbigg π 8a⎝bracketleftbig cos⎝parenleftbig1 4a–1u2⎝parenrightbig +s i n⎝parenleftbig1 4a–1u2⎝parenrightbig⎝bracketrightbig 21 exp⎝parenleftbig–ax2⎝parenrightbigcos⎝parenleftbigbx2⎝parenrightbig,a>0 √ π (A2+B2)1/4exp⎝parenleftBig –Au2 A2+B2⎝parenrightBig cos⎝parenleftBig ϕ–Bu2 A2+B2⎝parenrightBig , A=4a,B=4b,ϕ=1 2arctan( b/a) 7.7. Expressions with Special Functions No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 1 Ei(–ax) –1 uarctan⎝parenleftBigu a⎝parenrightBig 2 Ci(ax) ⎝braceleftbigg0i f 0 < u<a, –π 2uifa<u 3 si(ax) –1 2uln⎝vextendsingle⎝vextendsingle⎝vextendsingleu+a u–a⎝vextendsingle⎝vextendsingle⎝vextendsingle,u≠a 988 TABLES OF FOURIER COSINE TRANSFORMS No Original function, f(x) Cosine transform, ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (ux)dx 4 J0(ax),a>0 ⎝braceleftBigg1 √ a2–u2if 0 < u<a, 0i f a0 ,ν>– 1 ⎧ ⎪⎪⎨ ⎪⎪⎩cos⎝bracketleftbig νarcsin( u/a)⎝bracketrightbig √ a2–u2if 0 < u<a, –aνsin(πν/2) ξ(u+ξ)νifa<u, where ξ=√ u2–a2 6 1 xJν(ax),a>0 ,ν>0 ⎧ ⎨ ⎩ν–1cos⎝bracketleftbig νarcsin( u/a)⎝bracketrightbig if 0 < u<a, aνcos(πν/ 2) ν⎝parenleftbig u+√ u2–a2⎝parenrightbigν ifa0 ,ν>–1 2 ⎧ ⎨ ⎩√ π⎝parenleftbig a2–u2⎝parenrightbigν–1/2 (2a)νΓ⎝parenleftbig ν+1 2⎝parenrightbig if 0 < u<a, 0i fa0 , – 1< ν<–1 2 ⎧ ⎨ ⎩0i f 0 < u<a, 2ν+1√ πaνu Γ⎝parenleftbig –ν–1 2⎝parenrightbig (u2–a2⎝parenrightbigν+3/2ifa0 1 usin⎝parenleftBiga2 4u⎝parenrightBig 10 1 √ xJ1⎝parenleftbig a√ x⎝parenrightbig ,a>0 4 asin2⎝parenleftBiga2 8u⎝parenrightBig 11 xν/2Jν⎝parenleftbig a√ x⎝parenrightbig ,a>0 , – 1< ν<1 2 ⎝parenleftBiga 2⎝parenrightBigν u–ν–1sin⎝parenleftBiga2 4u–πν 2⎝parenrightBig 12 J0⎝parenleftbig a√ x2+b2⎝parenrightbig ⎧ ⎨ ⎩cos⎝parenleftbig b√ a2–u2⎝parenrightbig √ a2–u2if 0 < u<a, 0i fa0 ⎝braceleftBigg0i f 0 < u<a, –1 √ u2–a2ifa0 , |ν|<1 2 ⎧ ⎨ ⎩0i f0<u<a, –(2a)ν√ π Γ⎝parenleftbig1 2–ν⎝parenrightbig (u2–a2⎝parenrightbigν+1/2ifa0 π 2√ u2+a2exp⎝parenleftbig –b√ u2+a2⎝parenrightbig References for Supplement 7: G. Doetsch (1950, 1956, 1958), H. Bateman and A. Erd ´elyi (1954), V . A. Ditkin and A. P. Prudnikov (1965), F. Oberhettinger (1980). Supplement 8 Tables of Fourier Sine Transforms 8.1. General Formulas No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 1 af1(x)+bf2(x) aˇf1s(u)+bˇf2s(u) 2 f(ax),a>0 1 aˇfs⎝parenleftBigu a⎝parenrightBig 3 x2nf(x),n=1 ,2 , ... (–1)nd2n du2nˇfs(u) 4 x2n+1f(ax),n=0 ,1 , ... (–1)n+1d2n+1 du2n+1ˇfc(u),ˇfc(u)=⎝integraldisplay∞ 0f(x)c o s (xu)dx 5 f(ax)c o s (bx),a,b>0 1 2a⎝bracketleftBig ˇfs⎝parenleftBigu+b a⎝parenrightBig +Fs⎝parenleftBigu–b a⎝parenrightBig⎝bracketrightBig 8.2. Expressions with Power-Law Functions No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 1 ⎝braceleftBig1i f 0 < x<a, 0i f a<x 1 u⎝bracketleftbig 1–c o s ( au)⎝bracketrightbig 2 ⎝braceleftBiggx if 0 < x<1 , 2–x if 1 < x<2 , 0i f 2 < x 4 u2sinusin2u 2 3 1 x π 2 4 1 a+x,a>0 sin(au)C i (au)–c o s ( au)s i (au) 5 x a2+x2,a>0 π 2e–au 6 1 x(a2+x2),a>0 π 2a2⎝parenleftbig 1–e–au⎝parenrightbig 7 a a2+(x–b)2–a a2+(x+b)2 πe–ausin(bu) 8 x+b a2+(x+b)2–x–b a2+(x–b)2 πe–aucos(bu ) 989 990 TABLES OF FOURIER SINETRANSFORMS No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 9 x (x2+a2)n,a>0 ,n=1 ,2 , ... πue–au 22n–2(n–1 ) !a2n–3n–2⎝summationdisplay k=0(2n–k–4 ) ! k!(n–k–2 ) !(2au)k 10 x2m+1 (x2+a)n+1, n,m=0 ,1 , ...;0≤m≤n (–1)n+mπ 2n!∂n ∂an⎝parenleftbig ame–u√ a⎝parenrightbig 11 1 √ x ⎝radicalbigg π 2u 12 1 x√ x √ 2πu 13 x(a2+x2)–3/2 uK 0(au) 14 ⎝parenleftbig√ a2+x2–a⎝parenrightbig1/2 √ a2+x2 ⎝radicalbigg π 2ue–au 15 x–ν,0 < ν<2 cos⎝parenleftbig1 2πν⎝parenrightbig Γ(1 –ν)uν–1 8.3. Expressions with Exponential Functions No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 1 e–ax,a>0 u a2+u2 2 xne–ax,a>0 ,n=1 ,2 , ... n!⎝parenleftBiga a2+u2⎝parenrightBign+1[n/2]⎝summationdisplay k=0(–1)kC2k+1 n+1⎝parenleftBigu a⎝parenrightBig2k+1 3 1 xe–ax,a>0 arctanu a 4 √ xe–ax,a>0 √ π 2(a2+u2)–3/4sin⎝parenleftBig3 2arctanu a⎝parenrightBig 5 1 √ xe–ax,a>0 ⎝radicalbigg π 2⎝parenleftbig√ a2+u2–a)1/2 √ a2+u2 6 1 x√ xe–ax,a>0 √ 2π⎝parenleftbig√ a2+u2–a)1/2 7 xn–1/2e–ax,a>0 ,n=1 ,2 , ... (–1)n⎝radicalbigg π 2∂n ∂an⎝bracketleftBigg⎝parenleftbig√ a2+u2–a⎝parenrightbig1/2 √ a2+u2⎝bracketrightBigg 8 xν–1e–ax,a>0 ,ν>– 1 Γ(ν)(a2+u2)–ν/2sin⎝parenleftBig νarctanu a⎝parenrightBig 9 x–2⎝parenleftbig e–ax–e–bx⎝parenrightbig ,a,b>0 u 2ln⎝parenleftBigu2+b2 u2+a2⎝parenrightBig +barctan⎝parenleftBigu b⎝parenrightBig –aarctan⎝parenleftBigu a⎝parenrightBig 8.4. E XPRESSIONS WITH HYPERBOLIC FUNCTIONS 991 No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 10 1 eax+1,a>0 1 2u–π 2asinh(πu/a ) 11 1 eax–1,a>0 π 2acoth⎝parenleftBigπu a⎝parenrightBig –1 2u 12 ex/2 ex–1 –1 2tanh(πu) 13 xexp⎝parenleftbig –ax2⎝parenrightbig √ π 4a3/2uexp⎝parenleftBig –u2 4a⎝parenrightBig 14 1 xexp⎝parenleftbig –ax2⎝parenrightbig π 2erf⎝parenleftBigu 2√ a⎝parenrightBig 15 1 √ xexp⎝parenleftBig –a x⎝parenrightBig ⎝radicalbigg π 2ue–√ 2au⎝bracketleftbig cos⎝parenleftbig√ 2au⎝parenrightbig +s i n⎝parenleftbig√ 2au⎝parenrightbig⎝bracketrightbig 16 1 x√ xexp⎝parenleftBig –a x⎝parenrightBig ⎝radicalbigg π ae–√ 2ausin⎝parenleftbig√ 2au⎝parenrightbig 8.4. Expressions with Hyperbolic Functions No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 1 1 sinh(ax),a>0 π 2atanh⎝parenleftbig1 2πa–1u⎝parenrightbig 2 x sinh(ax),a>0 π2sinh⎝parenleftbig1 2πa–1u⎝parenrightbig 4a2cosh2⎝parenleftbig1 2πa–1u⎝parenrightbig 3 1 xe–bxsinh(ax),b>|a| 1 2arctan⎝parenleftBig2au u2+b2–a2⎝parenrightBig 4 1 xcosh(ax ),a>0 arctan⎝bracketleftbig sinh⎝parenleftbig1 2πa–1u⎝parenrightbig⎝bracketrightbig 5 1–t a n h⎝parenleftbig1 2ax⎝parenrightbig ,a>0 1 u–π asinh⎝parenleftbig πa–1u⎝parenrightbig 6 coth⎝parenleftbig1 2ax⎝parenrightbig –1 , a>0 π acoth⎝parenleftbig πa–1u⎝parenrightbig –1 u 7 cosh(ax ) sinh(bx),|a|<b π 2bsinh⎝parenleftbig πb–1u⎝parenrightbig cos⎝parenleftbig πab–1⎝parenrightbig +c o s h⎝parenleftbig πb–1u⎝parenrightbig 8 sinh(ax) cosh(bx ),|a|<b π bsin⎝parenleftbig1 2πab–1⎝parenrightbig sinh⎝parenleftbig1 2πb–1u⎝parenrightbig cos⎝parenleftbig πab–1⎝parenrightbig +c o s h⎝parenleftbig πb–1u⎝parenrightbig 992 TABLES OF FOURIER SINETRANSFORMS 8.5. Expressions with Logarithmic Functions No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 1 ⎝braceleftBiglnxif 0 < x<1 , 0i f 1 < x 1 u⎝bracketleftbig Ci(u)–l nu–C⎝bracketrightbig , C= 0.5772 ...is the Euler constant 2 lnx x –1 2π(lnu+C) 3 lnx √ x –⎝radicalbigg π 2u⎝bracketleftbig ln(4u)+C–π 2⎝bracketrightbig 4 xν–1lnx,|ν|<1 πu–ν⎝bracketleftbig ψ(ν)+π 2cot⎝parenleftbigπν 2⎝parenrightbig –l nu⎝bracketrightbig 2Γ(1 –ν)c o s⎝parenleftbigπν 2⎝parenrightbig 5 ln⎝vextendsingle⎝vextendsingle⎝vextendsinglea+x a–x⎝vextendsingle⎝vextendsingle⎝vextendsingle,a>0 π usin(au) 6 ln(x+b)2+a2 (x–b)2+a2,a,b>0 2π ue–ausin(bu) 7 e–axlnx,a>0 aarctan( u/a)–1 2uln(u2+a2)–eCu u2+a2 8 1 xln⎝parenleftbig 1+a2x2⎝parenrightbig ,a>0 –πEi⎝parenleftBig –u a⎝parenrightBig 8.6. Expressions with Trigonometric Functions No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 1 sin(ax) x,a>0 1 2ln⎝vextendsingle⎝vextendsingle⎝vextendsingleu+a u–a⎝vextendsingle⎝vextendsingle⎝vextendsingle 2 sin(ax) x2,a>0 ⎝braceleftbigg1 2πu if 0 < u<a, 1 2πa ifu>a 3 xν–1sin(ax),a>0 , – 2< ν<1 π|u–a|–ν–|u+a|–ν 4Γ(1 –ν)s i n⎝parenleftbig1 2πν⎝parenrightbig,ν≠0 4 sin(ax) x2+b2,a,b>0 ⎝braceleftbigg1 2πb–1e–absinh(bu)i f 0 < u<a, 1 2πb–1e–businh(ab)i fu>a 5 sin(πx) 1–x2 ⎝braceleftBigsinuif 0 < u<π, 0i f u>π 6 e–axsin(bx),a>0 a 2⎝bracketleftbigg1 a2+(b–u)2–1 a2+(b+u)2⎝bracketrightbigg 7 x–1e–axsin(bx),a>0 1 4ln(u+b)2+a2 (u–b)2+a2 8 1 xsin2(ax),a>0 ⎧ ⎨ ⎩1 4πif 0 < u<2a, 1 8πifu=2a, 0i f u>2a 8.7. E XPRESSIONS WITH SPECIAL FUNCTIONS 993 No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 9 1 x2sin2(ax),a>0 1 4(u+2a)l n|u+2a|+1 4(u–2a)l n|u–2a| –1 2ulnu 10 exp⎝parenleftbig –ax2⎝parenrightbig sin(bx),a>0 1 2⎝radicalbigg π aexp⎝parenleftBig –u2+b2 4a⎝parenrightBig sinh⎝parenleftBigbu 2a⎝parenrightBig 11 1 xsin(ax)s i n (bx),a≥b>0 ⎝braceleftBigg0i f 0 < u<a–b, π 4ifa–b<u<a+b, 0i f a+b0 π√ a 2√ uJ1⎝parenleftbig 2√ au⎝parenrightbig 13 1 √ xsin⎝parenleftBiga x⎝parenrightBig ,a>0 ⎝radicalbigg π 8u⎝bracketleftbig sin⎝parenleftbig 2√ au⎝parenrightbig –c o s⎝parenleftbig 2√ au⎝parenrightbig +e x p⎝parenleftbig –2√ au⎝parenrightbig⎝bracketrightbig 14 exp⎝parenleftbig –a√ x⎝parenrightbig sin⎝parenleftbig a√ x⎝parenrightbig ,a>0 a⎝radicalbigg π 8u–3/2exp⎝parenleftBig –a2 2u⎝parenrightBig 15 cos(ax ) x,a>0 ⎧ ⎨ ⎩0i f 0 < u<a, 1 4πifu=a, 1 2πifa0 , |ν|<1 π(u+a)–ν– sign( u–a)|u–a|–ν 4Γ(1 –ν)c o s⎝parenleftbig1 2πν⎝parenrightbig 17 xcos(ax ) x2+b2,a,b>0 ⎝braceleftbigg–1 2πe–absinh(bu)i fu<a, 1 2πe–bucosh(ab)i f u>a 18 1–c o s ( ax) x2,a>0 u 2ln⎝vextendsingle⎝vextendsingle⎝vextendsingleu2–a2 u2⎝vextendsingle⎝vextendsingle⎝vextendsingle+a 2ln⎝vextendsingle⎝vextendsingle⎝vextendsingleu+a u–a⎝vextendsingle⎝vextendsingle⎝vextendsingle 19 1 √ xcos⎝parenleftbig a√ x⎝parenrightbig ⎝radicalbigg π ucos⎝parenleftBiga2 4u+π 4⎝parenrightBig 20 1 √ xcos⎝parenleftbig a√ x⎝parenrightbig cos⎝parenleftbig b√ x⎝parenrightbig ,a,b>0 ⎝radicalbigg π ucos⎝parenleftBigab 2u⎝parenrightBig cos⎝parenleftBiga2+b2 4u+π 4⎝parenrightBig 8.7. Expressions with Special Functions No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 1 erfc(ax ),a>0 1 u⎝bracketleftBig 1–e x p⎝parenleftBig –u2 4a2⎝parenrightBig⎝bracketrightBig 2 ci(ax),a>0 –1 2uln⎝vextendsingle⎝vextendsingle⎝vextendsingle1–u2 a2⎝vextendsingle⎝vextendsingle⎝vextendsingle 3 si(ax),a>0 ⎝braceleftbigg0i f 0 < u<a, –1 2πu–1ifa0 ⎝braceleftBigg0i f 0 < u<a, 1 √ u2–a2ifa0 ,ν>– 2 ⎧ ⎪⎪⎨ ⎪⎪⎩sin⎝bracketleftbig νarcsin( u/a)⎝bracketrightbig √ a2–u2if 0 < u<a, aνcos(πν/ 2) ξ(u+ξ)νifa<u, where ξ=√ u2–a2 6 1 xJ0(ax),a>0 ,ν>0 ⎝braceleftbigg arcsin( u/a)i f 0 < u<a, π/2i f a0 ,ν>– 1 ⎧ ⎪⎨ ⎪⎩ν–1sin⎝bracketleftbig νarcsin( u/a)⎝bracketrightbig if 0 < u<a, aνsin(πν/2) ν⎝parenleftbig u+√ u2–a2⎝parenrightbigν ifa0 , – 1< ν<1 2 ⎧ ⎨ ⎩0i f0<u<a,√ π(2a)ν Γ⎝parenleftbig1 2–ν⎝parenrightbig⎝parenleftbig u2–a2⎝parenrightbigν+1/2ifa0 arcsin⎝parenleftBigg 2u ⎝radicalbig (u+b)2+a2+⎝radicalbig (u–b)2+a2⎝parenrightBigg 10 J0(ax) x2+b2,a,b>0 ⎝braceleftbigg b–1sinh(bu)K0(ab)i f 0 < u<a, 0i f a0 ⎝braceleftbigg0i f0<u<a, 1 2πe–buI0(ab)i fa0 , n=0 ,1 ,2 , ... ⎝braceleftBig(–1)nsinh(bu)K2n+1/2(ab)i f 0 < u<a, 0i fa0 , – 1< ν<5 2 ⎝braceleftbigg bν–1sinh(bu)Kν(ab)i f 0 < u<a, 0i fa0 , ν>–3 2 ⎝braceleftbigg0i f0<u<a, 1 2πb–νe–buIν(ab)i fa0 1 ucos⎝parenleftBiga2 4u⎝parenrightBig 16 1 √ xJ1⎝parenleftbig a√ x⎝parenrightbig ,a>0 2 asin⎝parenleftBiga2 4u⎝parenrightBig 17 xν/2Jν⎝parenleftbig a√ x⎝parenrightbig , a>0 , – 2< ν<1 2 aν 2νuν+1cos⎝parenleftBiga2 4u–πν 2⎝parenrightBig 8.7. E XPRESSIONS WITH SPECIAL FUNCTIONS 995 No Original function, f(x) Sine transform, ˇfs(u)=⎝integraldisplay∞ 0f(x)s i n (ux)dx 18 Y0(ax),a>0 ⎧ ⎪⎪⎨ ⎪⎪⎩2a r c s i n ( u/a) π√ a2–u2if 0 < u<a, 2⎝bracketleftbig ln⎝parenleftbig u–√ u2–a2⎝parenrightbig –l na⎝bracketrightbig π√ u2–a2ifa0 ⎝braceleftbigg0i f 0 < u<a, –u a√ u2–a2ifa0 ln⎝parenleftbig u+√ u2+a2⎝parenrightbig –l na √ u2+a2 21 xK 0(ax),a>0 πu 2(u2+a2)3/2 22 xν+1Kν(ax),a>0 ,ν>–3 2 √ π(2a)νΓ⎝parenleftbig ν+3 2⎝parenrightbig u(u2+a2)–ν–3/2 References for Supplement 8: G. Doetsch (1950, 1956, 1958), H. Bateman and A. Erd ´elyi (1954), I. I. Hirschman and D. V . Widder (1955), V . A. Ditkin and A. P. Prudnikov (1965), F. Oberhettinger (1980). Supplement 9 Tables of Mellin Transforms 9.1. General Formulas No Original function, f(x) Mellin transform, ˆf(s)=⎝integraldisplay∞ 0f(x)xs–1dx 1 af1(x)+bf2(x) aˆf1(s)+bˆf2(s) 2 f(ax),a>0 a–sˆf(s) 3 xaf(x) ˆf(s+a) 4 f(1/x) ˆf(–s) 5 f⎝parenleftbig xβ⎝parenrightbig ,β>0 1 βˆf⎝parenleftBigs β⎝parenrightBig 6 f⎝parenleftbig x–β⎝parenrightbig ,β>0 1 βˆf⎝parenleftBig –s β⎝parenrightBig 7 xλf⎝parenleftbig axβ⎝parenrightbig ,a,β>0 1 βa–s+λ βˆf⎝parenleftBigs+λ β⎝parenrightBig 8 xλf⎝parenleftbig ax–β⎝parenrightbig ,a,β>0 1 βas+λ βˆf⎝parenleftBig –s+λ β⎝parenrightBig 9 f/prime x(x) –(s–1 )ˆf(s–1 ) 10 xf/prime x(x) –sˆf(s) 11 f(n) x(x) (–1)nΓ(s) Γ(s–n)ˆf(s–n) 12 ⎝parenleftBig xd dx⎝parenrightBign f(x) (–1)nsnˆf(s) 13 ⎝parenleftBigd dxx⎝parenrightBign f(x) (–1)n(s–1 )nˆf(s) 14 xα⎝integraldisplay∞ 0tβf1(xt)f2(t)dt ˆf1(s+α)ˆf2(1 –s–α+β) 15 xα⎝integraldisplay∞ 0tβf1⎝parenleftBigx t⎝parenrightBig f2(t)dt ˆf1(s+α)ˆf2(s+α+β+1 ) 997 998 TABLES OF MELLIN TRANSFORMS 9.2. Expressions with Power-Law Functions No Original function, f(x) Mellin transform, ˆf(s)=⎝integraldisplay∞ 0f(x)xs–1dx 1 ⎝braceleftBiggx if 0 < x<1 , 2–x if 1 < x<2 , 0i f 2 < x ⎝braceleftBigg2(2s–1 ) s(s+1 )ifs≠0, 2l n2 i f s=0 ,Res>– 1 2 1 x+a,a>0 πas–1 sin(πs),0 < R e s<1 3 1 (x+a)(x+b),a,b>0 π⎝parenleftbig as–1–bs–1⎝parenrightbig (b–a)s i n (πs),0 < R e s<2 4 x+a (x+b)(x+c),b,c>0 π sin(πs)⎝bracketleftBig⎝parenleftBigb–a b–c⎝parenrightBig bs–1+⎝parenleftBigc–a c–b⎝parenrightBig cs–1⎝bracketrightBig , 0<R e s<1 5 1 x2+a2,a>0 πas–2 2s i n⎝parenleftbig1 2πs⎝parenrightbig,0 < R e s<2 6 1 x2+2axcosβ+a2,a>0 , |β|<π –πas–2sin⎝bracketleftbig β(s–1 )⎝bracketrightbig sinβsin(πs),0 < R e s<2 7 1 (x2+a2)(x2+b2),a,b>0 π⎝parenleftbig as–2–bs–2⎝parenrightbig 2(b2–a2)s i n⎝parenleftbig1 2πs⎝parenrightbig,0 < R e s<4 8 1 (1 +ax)n+1,a>0 ,n=1 ,2 , ... (–1)nπ assin(πs)Cn s–1,0 < R e s<n+1 9 1 xn+an,a>0 ,n=1 ,2 , ... πas–n nsin(πs/n),0 < R e s<n 10 1–x 1–xn,n=2 ,3 , ... πsin(π/n) nsin(πs/n)s i n⎝bracketleftbig π(s+1 )/n⎝bracketrightbig,0 < R e s<n–1 11 ⎝braceleftBigxνif 0 < x<1 , 0i f 1 < x 1 s+ν,R e s>–ν 12 1–xν 1–xnν,n=2 ,3 , ... πsin(π/n) nνsin⎝parenleftbigπs nν⎝parenrightbig sin⎝bracketleftbigπ(s+ν) nν⎝bracketrightbig,0 < R e s<(n–1 )ν 9.3. Expressions with Exponential Functions No Original function, f(x) Mellin transform, ˆf(s)=⎝integraldisplay∞ 0f(x)xs–1dx 1 e–ax,a>0 a–sΓ(s), Re s>0 2 ⎝braceleftbigg e–bxif 0 < x<a, 0i f a<x,b>0 b–sγ(s,ab), Re s>0 3 ⎝braceleftbigg 0i f 0 < x<a, e–bxifa<x,b>0 b–sΓ(s,ab) 4 e–ax x+b,a,b>0 eabbs–1Γ(s)Γ(1 –s,ab), Re s>0 5 exp⎝parenleftbig –axβ⎝parenrightbig ,a,β>0 β–1a–s/βΓ(s/β), Re s>0 9.5. E XPRESSIONS WITH TRIGONOMETRIC FUNCTIONS 999 No Original function, f(x) Mellin transform, ˆf(s)=⎝integraldisplay∞ 0f(x)xs–1dx 6 exp⎝parenleftbig –ax–β⎝parenrightbig ,a,β>0 β–1as/βΓ(–s/β), Re s<0 7 1–e x p⎝parenleftbig –axβ⎝parenrightbig ,a,β>0 –β–1a–s/βΓ(s/β), –β<R es<0 8 1–e x p⎝parenleftbig –ax–β⎝parenrightbig ,a,β>0 –β–1as/βΓ(–s/β), 0 < Re s<β 9.4. Expressions with Logarithmic Functions No Original function, f(x) Mellin transform, ˆf(s)=⎝integraldisplay∞ 0f(x)xs–1dx 1 ⎝braceleftBiglnxif 0 < x<a, 0i f a<x slna–1 s2as,R e s>0 2 ln(1 + ax),a>0 π sassin(πs),– 1 < R e s<0 3 ln|1–x| π scot(πs), –1 < Re s<0 4 lnx x+a,a>0 πas–1⎝bracketleftbig lna–πcot(πs)⎝bracketrightbig sin(πs),0 < R e s<1 5 lnx (x+a)(x+b),a,b>0 π⎝bracketleftbig as–1lna–bs–1lnb–πcot(πs)(as–1–bs–1)⎝bracketrightbig (b–a)s i n (πs), 0<R e s<1 6 ⎝braceleftBigxνlnxif 0 < x<1 , 0i f 1 < x –1 (s+ν)2,R e s>–ν 7 ln2x x+1 π3⎝bracketleftbig 2–s i n2(πs)⎝bracketrightbig sin3(πs),0 < R e s<1 8 ⎝braceleftbigg lnν–1xif 0 < x<1 , 0i f 1 < x Γ(ν)(–s)–ν,R e s<0 ,ν>0 9 ln⎝parenleftbig x2+2xcosβ+1⎝parenrightbig ,|β|<π 2πcos(βs) ssin(πs),– 1 < R e s<0 10 ln⎝vextendsingle⎝vextendsingle⎝vextendsingle1+x 1–x⎝vextendsingle⎝vextendsingle⎝vextendsingle π stan⎝parenleftbig1 2πs⎝parenrightbig ,– 1 < R e s<1 11 e–xlnnx,n=1 ,2 , ... dn dsnΓ(s), Re s>0 9.5. Expressions with Trigonometric Functions No Original function, f(x) Mellin transform, ˆf(s)=⎝integraldisplay∞ 0f(x)xs–1dx 1 sin(ax),a>0 a–sΓ(s)s i n⎝parenleftbig1 2πs⎝parenrightbig ,– 1 < R e s<1 2 sin2(ax),a>0 –2–s–1a–sΓ(s)c o s⎝parenleftbig1 2πs⎝parenrightbig ,– 2 < R e s<0 3 sin(ax)s i n (bx),a,b>0 ,a≠b 1 2Γ(s)c o s⎝parenleftbig1 2πs⎝parenrightbig⎝bracketleftbig |b–a|–s–(b+a)–s⎝bracketrightbig , –2 < Re s<1 1000 TABLES OF MELLIN TRANSFORMS No Original function, f(x) Mellin transform, ˆf(s)=⎝integraldisplay∞ 0f(x)xs–1dx 4 cos(ax ),a>0 a–sΓ(s)c o s⎝parenleftbig1 2πs⎝parenrightbig ,0 < R e s<1 5 sin(ax)c o s (bx),a,b>0 Γ(s) 2sin⎝parenleftBigπs 2⎝parenrightBig⎝bracketleftbig (a+b)–s+|a–b|–ssign(a–b)⎝bracketrightbig , –1 < Re s<1 6 e–axsin(bx),a>0 Γ(s)s i n⎝bracketleftbig sarctan( b/a)⎝bracketrightbig (a2+b2)s/2,– 1 < R e s 7 e–axcos(bx ),a>0 Γ(s)c o s⎝bracketleftbig sarctan( b/a)⎝bracketrightbig (a2+b2)s/2,0 < R e s 8 ⎝braceleftBigsin(alnx)i f 0 < x<1 , 0i f 1 < x –a s2+a2,R e s>0 9 ⎝braceleftBigcos(a lnx)i f 0 < x<1 , 0i f 1 < x s s2+a2,R e s>0 10 arctan x –π 2scos⎝parenleftbig1 2πs⎝parenrightbig,– 1 < R e s<0 11 arccot x π 2scos⎝parenleftbig1 2πs⎝parenrightbig,0 < R e s<1 9.6. Expressions with Special Functions No Original function, f(x) Mellin transform, ˆf(s)=⎝integraldisplay∞ 0f(x)xs–1dx 1 erfcx Γ⎝parenleftbig1 2s+1 2⎝parenrightbig √ πs,R e s>0 2 Ei(–x) –s–1Γ(s), Re s>0 3 Si(x) –s–1sin⎝parenleftbig1 2πs⎝parenrightbig Γ(s), –1 < Re s<0 4 si(x) –4s–1sin⎝parenleftbig1 2πs⎝parenrightbig Γ(s), –1 < Re s<0 5 Ci(x) –s–1cos⎝parenleftbig1 2πs⎝parenrightbig Γ(s), 0 < Re s<1 6 Jν(ax),a>0 2s–1Γ⎝parenleftbig1 2ν+1 2s⎝parenrightbig asΓ⎝parenleftbig1 2ν–1 2s+1⎝parenrightbig,–ν<R es<3 2 7 Yν(ax),a>0 –2s–1 πasΓ⎝parenleftBigs 2+ν 2⎝parenrightBig Γ⎝parenleftBigs 2–ν 2⎝parenrightBig cos⎝bracketleftBigπ(s–ν) 2⎝bracketrightBig , |ν|<R es<3 2 8 e–axIν(ax),a>0 Γ(1/2–s)Γ(s+ν) √ π(2a)sΓ(1 +ν–s),–ν<R es<1 2 9 Kν(ax),a>0 2s–2 asΓ⎝parenleftBigs 2+ν 2⎝parenrightBig Γ⎝parenleftBigs 2–ν 2⎝parenrightBig ,|ν|<R es 10 e–axKν(ax),a>0 √ πΓ(s–ν)Γ(s+ν) (2a)sΓ(s+1/2),|ν|<R es References for Supplement 9: H. Bateman and A. Erd ´elyi (1954), V . A. Ditkin and A. P. Prudnikov (1965), F. Oberhet- tinger (1974). Supplement 10 Tables of Inverse Mellin Transforms See Section 9.1 of Supplement 9 for general formulas. 10.1. Expressions with Power-Law Functions No Direct transform, ˆf(s) Inverse transform, f(x)=1 2πi⎝integraldisplayσ+i∞ σ–i∞ˆf(s)x–sds 1 1 s,R e s>0 ⎝braceleftBig1i f 0 < x<1 , 0i f 1 < x 2 1 s,R e s<0 ⎝braceleftBig0i f 0 < x<1 , –1 if 1 < x 3 1 s+a,R e s>–a ⎝braceleftBigxaif 0 < x<1 , 0i f 1 < x 4 1 s+a,R e s<–a ⎝braceleftBig0i f 0 < x<1 , –xaif 1 < x 5 1 (s+a)2,R e s>–a ⎝braceleftBig–xalnxif 0 < x<1 , 0i f 1 < x 6 1 (s+a)2,R e s<–a ⎝braceleftBig0i f 0 < x<1 , xalnxif 1 < x 7 1 (s+a)(s+b),R e s>–a,–b ⎝braceleftBigg xa–xb b–aif 0 < x<1 , 0i f 1 < x 8 1 (s+a)(s+b),–a<R es<–b ⎧ ⎪⎨ ⎪⎩xa b–aif 0 < x<1 , xb b–aif 1 < x 9 1 (s+a)(s+b),R e s<–a,–b ⎝braceleftBigg0i f 0 < x<1 , xb–xa b–aif 1 < x 10 1 (s+a)2+b2,R e s>–a ⎝braceleftBigg1 bxasin⎝parenleftBig bln1 x⎝parenrightBig if 0 < x<1 , 0i f1<x 11 s+a (s+a)2+b2,R e s>–a ⎝braceleftBigxacos(b lnx)i f 0 < x<1 , 0i f1<x 1001 1002 TABLES OF INVERSE MELLIN TRANSFORMS No Direct transform, ˆf(s) Inverse transform, f(x)=1 2πi⎝integraldisplayσ+i∞ σ–i∞ˆf(s)x–sds 12 √ s2–a2–s,R e s>|a| ⎝braceleftbigg –a lnxI1(–alnx)i f 0 < x<1 , 0i f1<x 13 ⎝radicalbigg s+a s–a–1 , R e s>|a| ⎝braceleftBigaI0(–alnx)+aI1(–alnx)i f 0 < x<1 , 0i f1<x 14 (s+a)–ν,R e s>–a,ν>0 ⎝braceleftBigg1 Γ(ν)xa(– lnx)ν–1if 0 < x<1 , 0i f1<x 15 s–1(s+a)–ν, Res>0 , R e s>–a,ν>0 ⎝braceleftbigg a–ν⎝bracketleftbig Γ(ν)⎝bracketrightbig–1γ(ν,–alnx)i f 0 < x<1 , 0i f1<x 16 s–1(s+a)–ν, –a<R es<0 ,ν>0 ⎝braceleftbigg –a–ν⎝bracketleftbig Γ(ν)⎝bracketrightbig–1Γ(ν,–alnx)i f 0 < x<1 , –a–νif 1 < x 17 (s2–a2)–ν,R e s>|a|,ν>0 ⎝braceleftBigg√ π(– lnx)ν–1/2Iν–1/2(–alnx) Γ(ν)(2a)ν–1/2if 0 < x<1 , 0i f1<x 18 (a2–s2)–ν,R e s<|a|,ν>0 ⎧ ⎪⎪⎨ ⎪⎪⎩(– lnx)ν–1/2Kν–1/2(–alnx) √ πΓ(ν)(2a)ν–1/2if 0 < x<1 , (lnx)ν–1/2Kν–1/2(alnx) √ πΓ(ν)(2a)ν–1/2if 1 < x 10.2. Expressions with Exponential and Logarithmic Functions No Direct transform, ˆf(s) Inverse transform, f(x)=1 2πi⎝integraldisplayσ+i∞ σ–i∞ˆf(s)x–sds 1 exp(as2),a>0 1 2√ πaexp⎝parenleftBig –ln2x 4a⎝parenrightBig 2 s–νe–a/s,R e s>0 ;a,ν>0 ⎧ ⎨ ⎩⎝vextendsingle⎝vextendsingle⎝vextendsinglea lnx⎝vextendsingle⎝vextendsingle⎝vextendsingle1–ν 2Jν–1⎝parenleftbig 2⎝radicalbig a|lnx|⎝parenrightbig if 0 < x<1 , 0i f1<x 3 exp⎝parenleftbig –√ as⎝parenrightbig ,R e s>0 ,a>0 ⎧ ⎨ ⎩(a/π)1/2 2|lnx|3/2exp⎝parenleftBig –a 4|lnx|⎝parenrightBig if 0 < x<1 , 0i f1<x 4 1 sexp⎝parenleftbig –a√ s⎝parenrightbig ,R e s>0 ⎝braceleftBigg erfc⎝parenleftBiga 2√ |lnx|⎝parenrightBig if 0 < x<1 , 0i f1<x 5 1 s⎝bracketleftbig exp⎝parenleftbig –a√ s⎝parenrightbig –1⎝bracketrightbig ,R e s>0 ⎝braceleftBigg –e r f⎝parenleftBiga 2√ |lnx|⎝parenrightBig if 0 < x<1 , 0i f1<x 10.3. E XPRESSIONS WITH TRIGONOMETRIC FUNCTIONS 1003 No Direct transform, ˆf(s) Inverse transform, f(x)=1 2πi⎝integraldisplayσ+i∞ σ–i∞ˆf(s)x–sds 6 √ sexp⎝parenleftbig –√ as⎝parenrightbig ,R e s>0 ⎧ ⎨ ⎩a–2|lnx| 4⎝radicalbig π|lnx|5exp⎝parenleftBig –a 4|lnx|⎝parenrightBig if 0 < x<1 , 0i f1<x 7 1 √ sexp⎝parenleftbig –√ as⎝parenrightbig ,R e s>0 ⎧ ⎨ ⎩1 √ π|lnx|exp⎝parenleftBig –a 4|lnx|⎝parenrightBig if 0 < x<1 , 0i f1<x 8 lns+a s+b,R e s>–a,–b ⎝braceleftBiggxa–xb lnxif 0 < x<1 , 0i f 1 < x 9 s–νlns,R e s>0 ,ν>0 ⎝braceleftBigg |lnx|ν–1ψ(ν)–l n |lnx| Γ(ν)if 0 < x<1 , 0i f1<x 10.3. Expressions with Trigonometric Functions No Direct transform, ˆf(s) Inverse transform, f(x)=1 2πi⎝integraldisplayσ+i∞ σ–i∞ˆf(s)x–sds 1 π sin(πs),0 < R e s<1 1 x+1 2 π sin(πs),–n<R es<1–n, n=..., –1, 0, 1, 2, ... (–1)nxn x+1 3 π2 sin2(πs),0 < R e s<1 lnx x–1 4 π2 sin2(πs),n<R es<n+1 , n=..., –1, 0, 1, 2, ... lnx xn(x–1 ) 5 2π3 sin3(πs),0 < R e s<1 π2+l n2x x+1 6 2π3 sin3(πs),n<R es<n+1 , n=..., –1, 0, 1, 2, ... π2+l n2x (–x)n(x+1 ) 7 sin⎝parenleftbig s2/a⎝parenrightbig ,a>0 1 2⎝radicalbigg a πsin⎝parenleftbig1 4a|lnx|2–1 4π⎝parenrightbig 8 π cos(πs),–1 2<R es<1 2 √ x x+1 9 π cos(πs),n–1 2<R es<n+1 2 n=..., –1, 0, 1, 2, ... (–1)nx1/2–n x+1 10 cos(βs) scos(πs),– 1 < R e s<0 , |β|<π 1 2πln(x2+2xcosβ+1 ) 1004 TABLES OF INVERSE MELLIN TRANSFORMS No Direct transform, ˆf(s) Inverse transform, f(x)=1 2πi⎝integraldisplayσ+i∞ σ–i∞ˆf(s)x–sds 11 cos⎝parenleftbig s2/a⎝parenrightbig ,a>0 1 2⎝radicalbigg a πcos⎝parenleftbig1 4a|lnx|2–1 4π⎝parenrightbig 12 arctan⎝parenleftBiga s+b⎝parenrightBig ,R e s>–b ⎧ ⎨ ⎩xb |lnx|sin⎝parenleftbig a|lnx|⎝parenrightbig if 0 < x<1 , 0i f1<x 10.4. Expressions with Special Functions No Direct transform, ˆf(s) Inverse transform, f(x)=1 2πi⎝integraldisplayσ+i∞ σ–i∞ˆf(s)x–sds 1 Γ(s), Re s>0 e–x 2 Γ(s), –1 < Re s<0 e–x–1 3 sin⎝parenleftbig1 2πs⎝parenrightbig Γ(s), –1 < Re s<1 sinx 4 sin(as)Γ(s), Res> –1, |a|<π 2 exp(–xcosa)s i n (xsina) 5 cos⎝parenleftbig1 2πs⎝parenrightbig Γ(s), 0 < Re s<1 cosx 6 cos⎝parenleftbig1 2πs⎝parenrightbig Γ(s), –2 < Re s<0 –2 sin2(x/2) 7 cos(as)Γ(s), Re s>0 , |a|<π 2 exp(–xcosa)c o s (xsina) 8 Γ(s) cos(πs),0 < R e s<1 2 exerfc⎝parenleftbig√ x⎝parenrightbig 9 Γ(a+s)Γ(b–s), –a<R es<b,a+b>0 Γ(a+b)xa(x+1 )–a–b 10 Γ(a+s)Γ(b+s), Res>–a,–b 2x(a+b)/2Ka–b⎝parenleftbig 2√ x⎝parenrightbig 11 Γ(s) Γ(s+ν),R e s>0 ,ν>0 ⎝braceleftBigg(1 –x)ν–1 Γ(ν)if 0 < x<1 , 0i f 1 < x 12 Γ(1 –ν–s) Γ(1 –s), Res<1–ν,ν>0 ⎝braceleftBigg0i f 0 < x<1 , (x–1 )ν–1 Γ(ν)if 1 < x 13 Γ(s) Γ(ν–s+1 ), 0<R e s<ν 2+3 4 x–ν/2Jν⎝parenleftbig 2√ x⎝parenrightbig 14 Γ(s+ν)Γ(s–ν) Γ(s+1/2),R e s>|ν| π–1/2e–x/2Kν(x/2) 10.4. E XPRESSIONS WITH SPECIAL FUNCTIONS 1005 No Direct transform, ˆf(s) Inverse transform, f(x)=1 2πi⎝integraldisplayσ+i∞ σ–i∞ˆf(s)x–sds 15 Γ(s+ν)Γ(1/2–s) Γ(1 +ν–s), –ν<R es<1 2 π1/2e–x/2Iν(x/2) 16 ψ(s+a)–ψ(s+b), Res>–a,–b ⎝braceleftBigg xb–xa 1–xif 0 < x<1 , 0i f 1 < x 17 Γ(s)ψ(s), Re s>0 e–xlnx 18 Γ(s,a),a>0 ⎝braceleftBig0i f 0 < x<a, e–xifa<x 19 Γ(s)Γ(1 –s,a), Re s>0 ,a>0 (x+1 )–1e–a(x+1) 20 γ(s,a), Re s>0 ,a>0 ⎝braceleftBige–xif 0 < x<a, 0i f a<x 21 J0⎝parenleftbig a√ b2–s2⎝parenrightbig ,a>0 ⎧ ⎪⎨ ⎪⎩0i f 0 < x<e–a, cos⎝parenleftbig b√ a2–l n2x⎝parenrightbig π√ a2–l n2xife–a<x<ea, 0i fea<x 22 s–1I0(s), Re s>0 ⎝braceleftBigg1i f0<x<e–1, π–1arccos(ln x)i fe–1<x<e, 0i fe<x 23 Iν(s), Re s>0 ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩–2 νsin(πν) πF(x)√ ln2x–1if 0 < x<e–1, cos⎝bracketleftbig νarccos(ln x)⎝bracketrightbig π√ 1–l n2xife–1<x<e, 0i fe<x, F(x)=⎝parenleftbig√ – 1–l n x+√ 1–l nx⎝parenrightbig2ν 24 s–1Iν(s), Re s>0 ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩2 νsin(πν) πνF (x)if 0 < x<e–1, sin⎝bracketleftbig νarccos(ln x)⎝bracketrightbig πνife–1<x<e, 0i fe<x, F(x)=⎝parenleftbig√ – 1–l n x+√ 1–l nx⎝parenrightbig2ν 25 s–νIν(s), Re s>–1 2 ⎧ ⎪⎨ ⎪⎩0i f0<x<e–1, ( 1–l n2x)ν–1/2 √ π2νΓ(ν+1/2)ife–1<x<e, 0i fe<x 26 s–1K0(s), Re s>0 ⎝braceleftbigg Arcosh(– ln x)i f 0 < x<e–1, 0i f e–1<x 27 s–1K1(s), Re s>0 ⎝braceleftbigg√ ln2x–1 i f 0< x<e–1, 0i f e–1<x 1006 TABLES OF INVERSE MELLIN TRANSFORMS No Direct transform, ˆf(s) Inverse transform, f(x)=1 2πi⎝integraldisplayσ+i∞ σ–i∞ˆf(s)x–sds 28 Kν(s), Re s>0 ⎧ ⎨ ⎩cosh⎝bracketleftbig νArcosh(– ln x)⎝bracketrightbig √ ln2x–1if 0 < x<e–1, 0i fe–1<x 29 s–1Kν(s), Re s>0 ⎝braceleftBigg1 νsinh⎝bracketleftbig νArcosh(– ln x)⎝bracketrightbig if 0 < x<e–1, 0i f e–1<x 30 s–νKν(s), Re s>0 ,ν>–1 2 ⎧ ⎨ ⎩√ π(ln2x–1 )ν–1/2 2νΓ(ν+1/2)if 0 < x<e–1, 0i f e–1<x References for Supplement 10: H. Bateman and A. Erd ´elyi (1954), V . A. Ditkin and A. P. Prudnikov (1965). Supplement 11 Special Functions and Their Properties /trianglerightsldThroughout Supplement 11 it is assumed that nis a positive integer , unless otherwise speciﬁed. 11.1. Some Coefﬁcients, Symbols, and Numbers 11.1-1. Binomial Coefﬁcients. Deﬁnitions (special cases): Ck n=⎝parenleftBign k⎝parenrightBig =n! k!(n–k)!,w h e r e k=1 ,...,n; C0 a=1 , Ck a=⎝parenleftBiga k⎝parenrightBig = (–1)k(–a)k k!=a(a–1 )...(a–k+1 ) k!,w h e r e k=1 ,2 , ... Hereais an arbitrary real number. Deﬁnition (general case): Cb a=Γ(a+1 ) Γ(b+1 )Γ(a–b+1 ),w h e r e Γ(x) is the gamma function. Properties: C0 a=1 , Ck n=0 f o r k= –1, –2, ...ork>n, Cb+1 a=a b+1Cb a–1=a–b b+1Cb a,Cb a+Cb+1 a=Cb+1 a+1, Cn –1/2=(–1)n 22nCn 2n= (–1)n(2n– 1)!! (2n)!!, Cn 1/2=(–1)n–1 n22n–1Cn–1 2n–2=(–1)n–1 n(2n– 3)!! (2n– 2)!!, C2n+1 n+1/2= (–1)n2–4n–1Cn 2n,Cn 2n+1/2=2–2nC2n 4n+1, C1/2 n=22n+1 πCn 2n,Cn/2 n=22n πC(n–1)/2 n , 1+C1 n+C2 n+···+Cn n=2n, 1–C1 n+C2 n–···+ (–1)nCn n=0 . Here (2n )!! = 2 ⋅4⋅6...(2n), (2n– 1)!! = 1 ⋅3⋅5...(2n–1 ) ,w h e r e n= 1 ,2 ,3 , ...( 0!! = 1!! = 1). 11.1-2. Pochhammer Symbol. Deﬁnition: (a)n=a(a+1 )...(a+n–1 )=Γ(a+n) Γ(a)= (–1)nΓ(1 –a) Γ(1 –a–n). 1007 1008 SPECIAL FUNCTIONS AND THEIR PROPERTIES Some properties (k =1 ,2 , ...): (a)0=1 , ( a)n+k=(a)n(a+n)k,(n)k=(n+k–1 ) ! (n–1 ) !, (a)–n=Γ(a–n) Γ(a)=(–1)n (1 –a)n,w h e r e a≠1,...,n; (1)n=n!, (1 /2)n=2–2n(2n)! n!,( 3/2)n=2–2n(2n+1 ) ! n!, (a+mk)nk=(a)mk+nk (a)mk,(a+n)n=(a)2n (a)n,(a+n)k=(a)k(a+k)n (a)n. 11.1-3. Bernoulli Numbers. The Bernoulli numbers are deﬁned by the recurrence relation B0=1 ,n–1⎝summationdisplay k=0Ck nBk=0 , n=2 ,3 , ... Numerical values: B0=1 , B1=–1 2,B2=1 6,B4=–1 30,B6=1 42,B8=–1 30,B10=5 66,..., B2m+1=0 f o r m=1 ,2 , ... All odd-numbered Bernoulli numbers but B1are zero; all even-numbered Bernoulli numbers have alternating signs. The Bernoulli numbers are the values of Bernoulli polynomials at x=0 :Bn=Bn(0). Generating function: x ex–1=∞⎝summationdisplay n=0Bnxn n!,|x|<2π. This relation may be regarded as a deﬁnition of the Bernoulli numbers. The following expansions may be used to calculate the Bernoulli numbers: tanx=∞⎝summationdisplay n=1|B2n|22n(22n–1 ) (2n)!x2n,|x|<π 2; cotx=∞⎝summationdisplay n=0(–1)nB2n22n (2n)!x2n–1,|x|<π. 11.1-4. Euler Numbers. The Euler numbers Enare deﬁned by the recurrence relation n⎝summationdisplay k=0C2k 2nE2k= 0 (even numbered), E2n+1= 0 (odd numbered), where n=0 ,1 , ... 11.2. E RROR FUNCTIONS .EXPONENTIAL AND LOGARITHMIC INTEGRALS 1009 Numerical values: E0=1 , E2= –1, E4=5 , E6= –61, E8= 1385, E10= –50251, ..., E2n+1=0 f o r n=0 ,1 , ... All Euler numbers are integer, the odd-numbered Euler numbers are zero, and the even-numbered Euler numbers have alternating signs. The Euler numbers are expressed via the values of Euler polynomials at x=1/2:En= 2nEn(1/2), where n=0 ,1 , ... Generating function: ex e2x+1=∞⎝summationdisplay n=0Enxn n!,|x|<2π. This relation may be regarded as a deﬁnition of the Euler numbers. Representation via a deﬁnite integral: E2n= (–1)n22n+1⎝integraldisplay∞ 0t2ndt cosh(πt ). 11.2. Error Functions. Exponential and Logarithmic Integrals 11.2-1. Error Function and Complementary Error Function. Deﬁnitions: erfx=2 √ π⎝integraldisplayx 0exp(–t2)dt (error function, also called probability integral), erfcx=1–e r f x=2 √ π⎝integraldisplay∞ xexp(–t2)dt (complementary error function). Properties: erf(–x)=–e r f x; erf(0) = 0, erf( ∞) = 1; erfc(0) = 1, erfc( ∞)=0 . Expansion of erf xinto series in powers of xasx→0: erfx=2 √ π∞⎝summationdisplay k=0(–1)kx2k+1 k!( 2k+1 )=2 √ πexp⎝parenleftbig –x2⎝parenrightbig∞⎝summationdisplay k=02kx2k+1 (2k+ 1)!!. Asymptotic expansion of erfc xasx→∞ : erfcx=1 √ πexp⎝parenleftbig –x2⎝parenrightbig⎝bracketleftbiggM–1⎝summationdisplay m=0(–1)m⎝parenleftbig1 2⎝parenrightbig m x2m+1+O⎝parenleftbig |x|–2M –1⎝parenrightbig⎝bracketrightbigg ,M=1 ,2 , ... Integral:⎝integraldisplayx 0erftd t=xerfx–1 2+1 2exp(–x2). 1010 SPECIAL FUNCTIONS AND THEIR PROPERTIES 11.2-2. Exponential Integral. Deﬁnition: Ei(x)=⎝integraldisplayx –∞et tdt=–⎝integraldisplay∞ –xe–t tdt forx<0 , Ei(x) = lim ε→+0⎝parenleftbigg⎝integraldisplay–ε –∞et tdt+⎝integraldisplayx εet tdt⎝parenrightbigg forx>0 . Other integral representations: Ei(–x)=–e–x⎝integraldisplay∞ 0xsint+tcost x2+t2dt forx>0 , Ei(–x)=e–x⎝integraldisplay∞ 0xsint–tcost x2+t2dt forx<0 , Ei(–x)=–x⎝integraldisplay∞ 1e–xtlntd t forx>0 , Ei(x)=C+l nx+⎝integraldisplayx 0et–1 tdt forx>0 , whereC= 0.5772 ...is the Euler constant. Expansion into series in powers of xasx→0: Ei(x)=⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩C+l n ( –x)+∞⎝summationdisplay k=1xk k!kifx<0 , C+l nx+∞⎝summationdisplay k=1xk k!kifx>0 . Asymptotic expansion as x→∞ : Ei(–x)=e–xn⎝summationdisplay k=1(–1)k(k–1 ) ! xk+Rn,Rn<n! xn. 11.2-3. Logarithmic Integral. Deﬁnition: li(x)=⎧ ⎪⎪⎨ ⎪⎪⎩⎝integraldisplay x 0dt lntif 0 < x<1 , lim ε→+0⎝parenleftbigg⎝integraldisplay1–ε 0dt lnt+⎝integraldisplayx 1+εdt lnt⎝parenrightbigg ifx>1 . For small x, li(x)≈x ln(1/x). For large x, li(x)≈x lnx. Asymptotic expansion as x→1: li(x)=C+l n|lnx|+∞⎝summationdisplay k=1lnkx k!k. Relation to the exponential integral: lix= Ei(ln x),x<1 ; li(ex)=E i ( x),x<0 . 11.3. S INEINTEGRAL AND COSINE INTEGRAL .FRESNEL INTEGRALS 1011 11.3. Sine Integral and Cosine Integral. Fresnel Integrals 11.3-1. Sine Integral. Deﬁnition: Si(x)=⎝integraldisplayx 0sint tdt,s i ( x)=–⎝integraldisplay∞ xsint tdt=S i (x)–π 2. Speciﬁc values: Si(0) = 0, Si( ∞)=π 2,s i (∞)=0 . Properties: Si(–x)=–S i ( x), si(x )+s i ( – x)=–π, lim x→–∞si(x)=–π. Expansion into series in powers of xasx→0: Si(x)=∞⎝summationdisplay k=1(–1)k+1x2k–1 (2k–1 )( 2 k–1 ) !. Asymptotic expansion as x→∞ : si(x)=–c o s x⎝bracketleftbiggM–1⎝summationdisplay m=0(–1)m(2m)! x2m+1+O⎝parenleftbig |x|–2M –1⎝parenrightbig⎝bracketrightbigg +s i nx⎝bracketleftbiggN–1⎝summationdisplay m=1(–1)m(2m–1 ) ! x2m+O⎝parenleftbig |x|–2N⎝parenrightbig⎝bracketrightbigg , where M,N=1 ,2 , ... 11.3-2. Cosine Integral. Deﬁnition: ci(x)=–⎝integraldisplay∞ xcost tdt=C+l nx+⎝integraldisplayx 0cost–1 tdt, whereC= 0.5772 ...is the Euler constant. Expansion into series in powers of xasx→0: ci(x)=C+l nx+∞⎝summationdisplay k=1(–1)kx2k 2k(2k)!. Asymptotic expansion as x→∞ : ci(x)=c o s x⎝bracketleftbiggM–1⎝summationdisplay m=1(–1)m(2m–1 ) ! x2m+O⎝parenleftbig |x|–2M⎝parenrightbig⎝bracketrightbigg +s i nx⎝bracketleftbiggN–1⎝summationdisplay m=0(–1)m(2m)! x2m+1+O⎝parenleftbig |x|–2N –1⎝parenrightbig⎝bracketrightbigg , where M,N=1 ,2 , ... 1012 SPECIAL FUNCTIONS AND THEIR PROPERTIES 11.3-3. Fresnel Integrals and Generalized Fresnel Integrals. Fresnel sine and cosine integrals : S(x)=1 √ 2π⎝integraldisplayx 0sint √ tdt=⎝radicalbigg 2 π⎝integraldisplay√ x 0sint2dt, C(x)=1 √ 2π⎝integraldisplayx 0cost √ tdt=⎝radicalbigg 2 π⎝integraldisplay√ x 0cost2dt. Expansion into series in powers of xasx→0: S(x)=⎝radicalbigg 2 πx∞⎝summationdisplay k=0(–1)kx2k+1 (4k+3 )( 2 k+1 ) !, C(x)=⎝radicalbigg 2 πx∞⎝summationdisplay k=0(–1)kx2k (4k+1 )( 2 k)!. Asymptotic expansion as x→∞ : S(x)=1 2–cosx √ 2πxP(x)–sinx √ 2πxQ(x), C(x)=1 2+sinx √ 2πxP(x)–cosx √ 2πxQ(x), P(x)=1–1×3 (2x)2+1×3×5×7 (2x)4–···,Q(x)=1 2x–1×3×5 (2x)3+···. Generalized Fresnel sine and cosine integrals: S(x,ν)=⎝integraldisplay∞ xtν–1sintd t,R e ν<1 ; C(x,ν)=⎝integraldisplay∞ xtν–1costd t,R e ν<1 . 11.4. Gamma Function, Psi Function, and Beta Function 11.4-1. Gamma Function. The gamma function ,Γ(z), is an analytic function of the complex argument zeverywhere except for the points z= 0, –1, –2, ... For Re z>0 , Γ(z)=⎝integraldisplay∞ 0tz–1e–tdt. For –(n+1 )<R e z<–n,w h e r e n=0 ,1 ,2 , ..., Γ(z)=⎝integraldisplay∞ 0⎝bracketleftbigg e–t–n⎝summationdisplay m=0(–1)m m!⎝bracketrightbigg tz–1dt. Simplest properties: Γ(z+1 )= zΓ(z),Γ(n+1 )= n!,Γ(1) =Γ(2) = 1. 11.4. G AMMA FUNCTION ,PSIFUNCTION ,AND BETAFUNCTION 1013 Fractional values of the argument: Γ⎝parenleftBig1 2⎝parenrightBig =√ π, Γ⎝parenleftBig –1 2⎝parenrightBig =– 2√ π,Γ⎝parenleftBig n+1 2⎝parenrightBig =√ π 2n(2n– 1)!!, Γ⎝parenleftBig1 2–n⎝parenrightBig = (–1)n2n√ π (2n– 1)!!. Euler formula Γ(z) = lim n→∞n!nz z(z+1 )...(z+n)(z≠0, –1, –2, ...). Symmetry formulas: Γ(z)Γ(–z)=–π zsin(πz),Γ(z)Γ(1 –z)=π sin(πz), Γ⎝parenleftBig1 2+z⎝parenrightBig Γ⎝parenleftBig1 2–z⎝parenrightBig =π cos(πz ). Multiple argument formulas: Γ(2z)=22z–1 √ πΓ(z)Γ⎝parenleftBig z+1 2⎝parenrightBig , Γ(3z)=33z–1/2 2πΓ(z)Γ⎝parenleftBig z+1 3⎝parenrightBig Γ⎝parenleftBig z+2 3⎝parenrightBig , Γ(nz)=( 2π)(1–n)/2nnz–1/2n–1⎝productdisplay k=0Γ⎝parenleftBig z+k n⎝parenrightBig . Asymptotic expansion ( Stirling formula ): Γ(z)=√ 2πe–zzz–1/2⎝bracketleftbig 1+1 12z–1+1 288z–2+O(z–3)⎝bracketrightbig (|argz|<π). 11.4-2. Psi Function (Digamma Function). Deﬁnition: ψ(z)=dlnΓ(z) dz=Γ/prime z(z) Γ(z). The psi function is the logarithmic derivative of the gamma function and is also called the digamma function . Integral representations (Re z>0 ) : ψ(z)=⎝integraldisplay∞ 0⎝bracketleftbig e–t–( 1+ t)–z⎝bracketrightbig t–1dt, ψ(z)=l nz+⎝integraldisplay∞ 0⎝bracketleftbig t–1–( 1–e–t)–1⎝bracketrightbig e–tzdt, ψ(z)=–C+⎝integraldisplay1 01–tz–1 1–tdt, whereC=–ψ(1) = 0.5772 ...is the Euler constant. 1014 SPECIAL FUNCTIONS AND THEIR PROPERTIES Values for integer argument: ψ(1) = –C,ψ(n)=–C+n–1⎝summationdisplay k=1k–1(n=2 ,3 , ...). Functional relations: ψ(z)–ψ(1 +z)=–1 z, ψ(z)–ψ(1 –z)=–πcot(πz), ψ(z)–ψ(–z)=–πcot(πz)–1 z, ψ⎝parenleftbig1 2+z⎝parenrightbig –ψ⎝parenleftbig1 2–z⎝parenrightbig =πtan(πz), ψ(mz)=l nm+1 mm–1⎝summationdisplay k=0ψ⎝parenleftBig z+k m⎝parenrightBig . Asymptotic expansion as z→∞ (|argz|<π): ψ(z)=l nz–1 2z–1 12z2+1 120z4–1 252z6+···=l nz–1 2z–∞⎝summationdisplay n=1B2n 2nz2n, where the B2nare Bernoulli numbers. 11.4-3. Beta Function. Deﬁnition: B(x,y)=⎝integraldisplay1 0tx–1(1 –t)y–1dt, where Re x>0a n dR e y>0 . Relationship with the gamma function: B(x,y)=Γ(x)Γ(y) Γ(x+y). Some properties: B(x,y)=B(y,x); B(x,y+1 )=y xB(x+1 ,y)=y x+yB(x,y); B(x,1–x)=π sin(πx),0 < x<1 ; 1 B(n,m)=mCn–1 n+m–1=nCm–1 n+m–1, where nandmare positive integers. 11.5. Incomplete Gamma and Beta Functions 11.5-1. Incomplete Gamma Function. Deﬁnitions: γ(α,x)=⎝integraldisplayx 0e–ttα–1dt,R e α>0 , Γ(α,x)=⎝integraldisplay∞ xe–ttα–1dt=Γ(α)–γ(α,x). 11.5. I NCOMPLETE GAMMA AND BETAFUNCTIONS 1015 Recurrence formulas: γ(α+1 ,x)=αγ(α,x)–xαe–x, γ(α+1 ,x)=(x+α)γ(α,x)+( 1– α)xγ(α–1 ,x), Γ(α+1 ,x)=αΓ(α,x)+xαe–x. Special cases: γ(n+1 ,x)=n!⎝bracketleftbigg 1–e–x⎝parenleftbiggn⎝summationdisplay k=0xk k!⎝parenrightbigg⎝bracketrightbigg , n=0 ,1 , ...; Γ(n+1 ,x)=n!e–xn⎝summationdisplay k=0xk k!, n=0 ,1 , ...; Γ(–n,x)=(–1)n n!⎝bracketleftbigg Γ(0,x)–e–xn–1⎝summationdisplay k=0(–1)kk! xk+1⎝bracketrightbigg ,n=1 ,2 , ... Asymptotic expansions as x→0: γ(α,x)=∞⎝summationdisplay n=0(–1)nxα+n n!(α+n), Γ(α,x)=Γ(α)–∞⎝summationdisplay n=0(–1)nxα+n n!(α+n). Asymptotic expansions as x→∞ : γ(α,x)=Γ(α)–xα–1e–x⎝bracketleftbiggM–1⎝summationdisplay m=0(1 –α)m (–x)m+O⎝parenleftbig |x|–M⎝parenrightbig⎝bracketrightbigg , Γ(α,x)=xα–1e–x⎝bracketleftbiggM–1⎝summationdisplay m=0(1 –α)m (–x)m+O⎝parenleftbig |x|–M⎝parenrightbig⎝bracketrightbigg⎝parenleftbig –3 2π<a r gx<3 2π⎝parenrightbig . Asymptotic formulas as α→∞ : γ(x,α)=Γ(α)⎝bracketleftBig Φ⎝parenleftbig 2√ x–√ α–1⎝parenrightbig +O⎝parenleftBig1 √ α⎝parenrightBig⎝bracketrightBig ,Φ(x)=1 √ 2π⎝integraldisplayx –∞exp⎝parenleftBig –1 2t2⎝parenrightBig dt; γ(x,α)=Γ(α)⎝bracketleftBig Φ⎝parenleftbig 3√ αz⎝parenrightbig +O⎝parenleftBig1 α⎝parenrightBig⎝bracketrightBig ,z=⎝parenleftBigx α⎝parenrightBig1/3 –1+1 9α. Representation of the error function, complementary error function, and exponential integral in terms of the gamma functions: erfx=1 √ πγ⎝parenleftBig1 2,x2⎝parenrightBig , erfc x=1 √ πΓ⎝parenleftBig1 2,x2⎝parenrightBig ,E i ( – x)=–Γ(0,x). 11.5-2. Incomplete Beta Function. Deﬁnitions: Bx(a,b)=⎝integraldisplayx 0ta–1(1 –t)b–1dt,Ix(a,b)=Bx(a,b) B(a,b), where Re a>0a n dR e b>0 ,a n d B(a,b)=B1(a,b) is the beta function. 1016 SPECIAL FUNCTIONS AND THEIR PROPERTIES Symmetry property: Ix(a,b)+I1–x(b,a)=1 . Recurrence formulas: Ix(a,b)=xIx(a–1 ,b)+( 1– x)Ix(a,b–1 ) , (a+b)Ix(a,b)=aIx(a+1 ,b)+bIx(a,b+1 ) , (a+b–ax)Ix(a,b)=a(1 –x)Ix(a+1 ,b–1 )+ bIx(a,b+1 ) . 11.6. Bessel Functions (Cylindrical Functions) 11.6-1. Deﬁnitions and Basic Formulas. The Bessel function of the ﬁrst kind ,Jν(x), and the Bessel function of the second kind ,Yν(x)( a l s o called the Neumann function ), are solutions of the Bessel equation x2y/prime/prime xx+xy/prime x+(x2–ν2)y=0 and are deﬁned by the formulas Jν(x)=∞⎝summationdisplay k=0(–1)k(x/2)ν+2k k!Γ(ν+k+1 ),Yν(x)=Jν(x)c o sπν–J–ν(x) sinπν.( 1) The formula for Yν(x) is valid for ν≠0,±1,±2,...(the cases ν≠0,±1,±2,...are discussed in what follows). The general solution of the Bessel equation has the form Zν(x)=C1Jν(x)+C2Yν(x)a n di s called the cylinder function . Some formulas: 2νZν(x)=x[Zν–1(x)+Zν+1(x)], d dxZν(x)=1 2[Zν–1(x)–Zν+1(x)] =±⎝bracketleftBigν xZν(x)–Zν±1(x)⎝bracketrightBig , d dx[xνZν(x)] =xνZν–1(x),d dx[x–νZν(x)] = –x–νZν+1(x), ⎝parenleftbigg1 xd dx⎝parenrightbiggn [xνJν(x)] =xν–nJν–n(x),⎝parenleftbigg1 xd dx⎝parenrightbiggn [x–νJν(x)] = (–1)nx–ν–nJν+n(x), J–n(x)=( – 1 )nJn(x),Y–n(x) = (–1)nYn(x), n=0 ,1 ,2 ,... Bessel functions for ν=±n±1 2(n=0 ,1 ,2 , ...): J1/2(x)=⎝radicalbigg 2 πxsinx, J3/2(x)=⎝radicalbigg 2 πx⎝parenleftbigg1 xsinx–c o sx⎝parenrightbigg ,J–1/2(x)=⎝radicalbigg 2 πxcosx, J–3/2(x)=⎝radicalbigg 2 πx⎝parenleftbigg –1 xcosx–s i nx⎝parenrightbigg , 11.6. B ESSEL FUNCTIONS (CYLINDRICAL FUNCTIONS ) 1017 Jn+1/2(x)=⎝radicalbigg 2 πx⎝bracketleftbigg sin⎝parenleftBig x–nπ 2⎝parenrightBig[n/2]⎝summationdisplay k=0(–1)k(n+2k)! (2k)! (n–2k)! (2x)2k +c o s⎝parenleftBig x–nπ 2⎝parenrightBig[(n–1)/2]⎝summationdisplay k=0(–1)k(n+2k+1 ) ! (2k+1 ) !(n–2k– 1)! (2 x)2k+1⎝bracketrightbigg , J–n–1/2(x)=⎝radicalbigg 2 πx⎝bracketleftbigg cos⎝parenleftBig x+nπ 2⎝parenrightBig[n/2]⎝summationdisplay k=0(–1)k(n+2k)! (2k)! (n–2k)! (2x)2k –s i n⎝parenleftBig x+nπ 2⎝parenrightBig[(n–1)/2]⎝summationdisplay k=0(–1)k(n+2k+1 ) ! (2k+1 ) !(n–2k– 1)! (2 x)2k+1⎝bracketrightbigg , Y1/2(x)=–⎝radicalbigg 2 πxcosx, Yn+1/2(x) = (–1)n+1J–n–1/2(x),Y–1/2(x)=⎝radicalbigg 2 πxsinx, Y–n–1/2(x)=( – 1 )nJn+1/2(x), where [ A] is the integer part of the number A. Letν=nbe an arbitrary integer. The relations J–n(x)=( – 1 )nJn(x),Y–n(x)=( – 1 )nYn(x) are valid. The function Jn(x) is given by the ﬁrst formula in (1) with ν=n,a n dYn(x) can be obtained from the second formula in (1) by proceeding to the limit ν→n. For nonnegative n,Yn(x) can be represented in the form Yn(x)=2 πJn(x)l nx 2–1 πn–1⎝summationdisplay k=0(n–k–1 ) ! k!⎝parenleftBig2 x⎝parenrightBign–2k –1 π∞⎝summationdisplay k=0(–1)k⎝parenleftBigx 2⎝parenrightBign+2kψ(k+1 )+ ψ(n+k+1 ) k!(n+k)!, where ψ(1) = –C ,ψ(n)=–C+n–1⎝summationtext k=1k–1,C= 0.5772 ...is the Euler constant, and ψ(x)=[ l nΓ(x)]/prime xis the logarithmic derivative of the gamma function, also known as the digamma function. Wronskians and similar formulas: W(Jν,J–ν)=–2 πxsin(πν),W(Jν,Yν)=2 πx, Jν(x)J–ν+1(x)+J–ν(x)Jν–1(x)=2s i n (πν) πx,Jν(x)Yν+1(x)–Jν+1(x)Yν(x)=–2 πx. Here the notation W(f,g)=fg/prime x–f/prime xgis used. 11.6-2. Integral Representations and Asymptotic Expansions. The functions Jν(x)a n dYν(x) can be represented in the form of deﬁnite integrals (for x>0 ) : πJν(x)=⎝integraldisplayπ 0cos(x sinθ–νθ)dθ–s i nπν⎝integraldisplay∞ 0exp(–xsinht–νt)dt, πYν(x)=⎝integraldisplayπ 0sin(xsinθ–νθ)dθ–⎝integraldisplay∞ 0(eνt+e–νtcosπν)e–xsinhtdt. 1018 SPECIAL FUNCTIONS AND THEIR PROPERTIES For|ν|<1 2,x>0 , Jν(x)=21+νx–ν π1/2Γ(1 2–ν)⎝integraldisplay∞ 1sin(xt)dt (t2–1 )ν+1/2, Yν(x)=–21+νx–ν π1/2Γ(1 2–ν)⎝integraldisplay∞ 1cos(xt )dt (t2–1 )ν+1/2. Forν>–1 2, Jν(x)=2(x/2)ν π1/2Γ(1 2+ν)⎝integraldisplayπ/2 0cos(x cost)s i n2νtd t (Poisson’s formula ). Forν=0 ,x>0 , J0(x)=2 π⎝integraldisplay∞ 0sin(xcosht)dt,Y0(x)=–2 π⎝integraldisplay∞ 0cos(x cosht)dt. For integer ν=n=0 ,1 ,2 ,... , Jn(x)=1 π⎝integraldisplayπ 0cos(nt –xsint)dt (Bessel’s formula ), J2n(x)=2 π⎝integraldisplayπ/2 0cos(x sint)c o s ( 2 nt)dt, J2n+1(x)=2 π⎝integraldisplayπ/2 0sin(xsint) sin[(2 n+1 )t]dt. Asymptotic expansions as |x|→∞ : Jν(x)=⎝radicalbigg 2 πx⎝braceleftbigg cos⎝parenleftBig4x–2νπ–π 4⎝parenrightBig⎝bracketleftbiggM–1⎝summationdisplay m=0(–1)m(ν,2m)(2x)–2m+O(|x|–2M)⎝bracketrightbigg –s i n⎝parenleftBig4x–2νπ–π 4⎝parenrightBig⎝bracketleftbiggM–1⎝summationdisplay m=0(–1)m(ν,2m+ 1)(2x )–2m –1+O(|x|–2M –1)⎝bracketrightbigg⎝bracerightbigg , Yν(x)=⎝radicalbigg 2 πx⎝braceleftbigg sin⎝parenleftBig4x–2νπ–π 4⎝parenrightBig⎝bracketleftbiggM–1⎝summationdisplay m=0(–1)m(ν,2m)(2x)–2m+O(|x|–2M)⎝bracketrightbigg +c o s⎝parenleftBig4x–2νπ–π 4⎝parenrightBig⎝bracketleftbiggM–1⎝summationdisplay m=0(–1)m(ν,2m+ 1)(2x )–2m –1+O(|x|–2M –1)⎝bracketrightbigg⎝bracerightbigg , where ( ν,m)=1 22mm!(4ν2– 1)(4ν2–32)...[4ν2–( 2m–1 )2]=Γ(1 2+ν+m) m!Γ(1 2+ν–m). For nonnegative integer nand large x, √ πxJ 2n(x)=( – 1 )n(cosx+s i nx)+O(x–2), √ πxJ 2n+1(x)=( – 1 )n+1(cosx–s i nx)+O(x–2). Asymptotic for large ν(ν→∞ ): Jν(x)/similarequal1 √ 2πν⎝parenleftBigex 2ν⎝parenrightBigν ,Yν(x)/similarequal–⎝radicalbigg 2 πν⎝parenleftBigex 2ν⎝parenrightBig–ν , 11.6. B ESSEL FUNCTIONS (CYLINDRICAL FUNCTIONS ) 1019 where xis ﬁxed, and Jν(ν)/similarequal21/3 32/3Γ(2/3)1 ν1/3,Yν(ν)/similarequal–21/3 31/6Γ(2/3)1 ν1/3. Integrals with Bessel functions: ⎝integraldisplayx 0xλJν(x)dx=xλ+ν+1 2ν(λ+ν+1 )Γ(ν+1 )F⎝parenleftbiggλ+ν+1 2,λ+ν+3 2,ν+1; –x2 4⎝parenrightbigg ,R e ( λ+ν) > –1, where F(a,b,c;x) is the hypergeometric series (see Supplement 11.10.1), ⎝integraldisplayx 0xλYν(x)dx=–cos(νπ )Γ(–ν) 2νπ(λ+ν+1 )xλ+ν+1F⎝parenleftbiggλ+ν+1 2,ν+1 ,λ+ν+3 2;–x2 4⎝parenrightbigg –2νΓ(ν) λ–ν+1xλ–ν+1F⎝parenleftbiggλ–ν+1 2,1 –ν,λ–ν+3 2;–x2 4⎝parenrightbigg ,R e λ>|Reν|–1 . 11.6-3. Zeros of Bessel Functions. Each of the functions Jν(x)a n dYν(x) has inﬁnitely many real zeros (for real ν). All zeros are simple, except possibly for the point x=0 . The zeros γmofJ0(x), i.e., the roots of the equation J0(γm) = 0, are approximately given by γm=2 . 4+3 . 1 3( m–1 ) ( m=1 ,2 , ...), with a maximum error of 0.2%. 11.6-4. Orthogonality Properties of Bessel Functions. 1◦.L e tµ=µmbe positive roots of the Bessel function Jν(µ), where ν>– 1a n d m=1 ,2 ,3 , ... Then the set of functions Jν(µmr/a) is orthogonal on the interval 0 ≤r≤awith weight r: ⎝integraldisplaya 0Jν⎝parenleftBigµmr a⎝parenrightBig Jν⎝parenleftBigµkr a⎝parenrightBig rd r =⎝braceleftbigg0i f m≠k, 1 2a2⎝bracketleftbig J/prime ν(µm)⎝bracketrightbig2=1 2a2J2 ν+1(µm)i fm=k. 2◦.L e t µ=µmbe positive zeros of the Bessel function derivative J/prime ν(µ), where ν>– 1a n d m=1 ,2 ,3 , ...Then the set of functions Jν(µmr/a) is orthogonal on the interval 0 ≤r≤awith weight r: ⎝integraldisplaya 0Jν⎝parenleftBigµmr a⎝parenrightBig Jν⎝parenleftBigµkr a⎝parenrightBig rd r =⎧ ⎨ ⎩0i fm≠k, 1 2a2⎝parenleftbigg 1–ν2 µ2m⎝parenrightbigg J2 ν(µm)i fm=k. 3◦.L e tµ=µmbe positive roots of the transcendental equation µJ/prime ν(µ)+sJν(µ)=0 ,w h e r e ν>– 1 andm=1 ,2 ,3 ,... Then the set of functions Jν(µmr/a) is orthogonal on the interval 0 ≤r≤a with weight r: ⎝integraldisplaya 0Jν⎝parenleftBigµmr a⎝parenrightBig Jν⎝parenleftBigµkr a⎝parenrightBig rd r =⎧ ⎨ ⎩0i fm≠k, 1 2a2⎝parenleftbigg 1+s2–ν2 µ2m⎝parenrightbigg J2 ν(µm)i fm=k. 1020 SPECIAL FUNCTIONS AND THEIR PROPERTIES 4◦.L e tµ=µmbe positive roots of the transcendental equation Jν(λmb)Yν(λma)–Jν(λma)Yν(λmb)=0 ( ν> –1, m=1 ,2 ,3 , ...). Then the set of functions Zν(λmr)=Jν(λmr)Yν(λma)–Jν(λma)Yν(λmr), m=1 ,2 ,3 , ..., satisfying th e conditions Zν(λma)=Zν(λmb) = 0 is orthogonal on the interval a≤r≤bwith weight r: ⎝integraldisplayb aZν(λmr)Zν(λkr)rd r =⎧ ⎨ ⎩0i fm≠k, 2 π2λ2mJ2 ν(λma)–J2 ν(λmb) J2ν(λmb)ifm=k. 5◦.L e tµ=µmbe positive roots of the transcendental equation J/prime ν(λmb)Y/prime ν(λma)–J/prime ν(λma)Y/prime ν(λmb)=0 ( ν> –1, m=1 ,2 ,3 , ...). Then the set of functions Zν(λmr)=Jν(λmr)Y/prime ν(λma)–J/prime ν(λma)Yν(λmr), m=1 ,2 ,3 , ..., satisfying th e conditions Z/prime ν(λma)=Z/prime ν(λmb) = 0 is orthogonal on the interval a≤r≤bwith weight r: ⎝integraldisplayb aZν(λmr)Zν(λkr)rd r =⎧ ⎪⎨ ⎪⎩0i fm≠k, 2 π2λ2m⎝bracketleftbigg⎝parenleftbigg 1–ν2 b2λ2m⎝parenrightbigg⎝bracketleftbig J/prime ν(λma)⎝bracketrightbig2 ⎝bracketleftbig J/primeν(λmb)⎝bracketrightbig2–⎝parenleftbigg 1–ν2 a2λ2m⎝parenrightbigg⎝bracketrightbigg ifm=k. 11.6-5. Hankel Functions (Bessel Functions of the Third Kind). The Hankel functions of the ﬁrst kind and the second kind are related to Bessel functions by H(1) ν(z)=Jν(z)+iYν(z), H(2) ν(z)=Jν(z)–iYν(z), where i2= –1. Asymptotics for z→0: H(1) 0(z)/similarequal2i πlnz,H(1) ν(z)/similarequal–i πΓ(ν) (z/2)ν(Reν>0 ) , H(2) 0(z)/similarequal–2i πlnz,H(2) ν(z)/similarequali πΓ(ν) (z/2)ν(Reν>0 ) . Asymptotics for |z|→∞ : H(1) ν(z)/similarequal⎝radicalbigg 2 πzexp⎝bracketleftbig i⎝parenleftbig z–1 2πν–1 4π⎝parenrightbig⎝bracketrightbig (–π<a r gz<2π), H(2) ν(z)/similarequal⎝radicalbigg 2 πzexp⎝bracketleftbig –i⎝parenleftbig z–1 2πν–1 4π⎝parenrightbig⎝bracketrightbig (–2π<a r gz<π). 11.7. M ODIFIED BESSEL FUNCTIONS 1021 11.7. Modiﬁed Bessel Functions 11.7-1. Deﬁnitions. Basic Formulas. The modiﬁed Bessel functions of the ﬁrst kind ,Iν(x), and the modiﬁed Bessel functions of the second kind,Kν(x) (also called the MacDonald function ), of order νare solutions of the modiﬁed Bessel equation x2y/prime/prime xx+xy/prime x–(x2+ν2)y=0 and are deﬁned by the formulas Iν(x)=∞⎝summationdisplay k=0(x/2)2k+ν k!Γ(ν+k+1 ),Kν(x)=π 2I–ν(x)–Iν(x) sin(πν) (see below for Kν(x) with ν=0 ,1 ,2 ,... ). The modiﬁed Bessel functions possess the properties K–ν(x)=Kν(x); I–n(x) = (–1)nIn(x),n=0 ,1 ,2 ,... 2νIν(x)=x[Iν–1(x)–Iν+1(x)], 2 νKν(x)=–x[Kν–1(x)–Kν+1(x)], d dxIν(x)=1 2[Iν–1(x)+Iν+1(x)],d dxKν(x)=–1 2[Kν–1(x)+Kν+1(x)]. Modiﬁed Bessel functions for ν=±n±1 2(n=0 ,1 ,2 ,... ): I1/2(x)=⎝radicalbigg 2 πxsinhx,I–1/2(x)=⎝radicalbigg 2 πxcoshx, I3/2(x)=⎝radicalbigg 2 πx⎝parenleftbigg –1 xsinhx+c o s h x⎝parenrightbigg ,I–3/2(x)=⎝radicalbigg 2 πx⎝parenleftbigg –1 xcoshx+s i n h x⎝parenrightbigg , In+1/2(x)=1 √ 2πx⎝bracketleftbigg exn⎝summationdisplay k=0(–1)k(n+k)! k!(n–k)! (2x)k– (–1)ne–xn⎝summationdisplay k=0(n+k)! k!(n–k)! (2x)k⎝bracketrightbigg , I–n–1/2(x)=1 √ 2πx⎝bracketleftbigg exn⎝summationdisplay k=0(–1)k(n+k)! k!(n–k)! (2x)k+ (–1)ne–xn⎝summationdisplay k=0(n+k)! k!(n–k)! (2x)k⎝bracketrightbigg , K±1/2(x)=⎝radicalbigg π 2xe–x,K±3/2(x)=⎝radicalbigg π 2x⎝parenleftBig 1+1 x⎝parenrightBig e–x, Kn+1/2(x)=K–n–1/2(x)=⎝radicalbigg π 2xe–xn⎝summationdisplay k=0(n+k)! k!(n–k)! (2x)k. Ifν=nis a nonnegative integer, then Kn(x) = (–1)n+1In(x)l nx 2+1 2n–1⎝summationdisplay m=0(–1)m⎝parenleftBigx 2⎝parenrightBig2m–n(n–m–1 ) ! m! +1 2(–1)n∞⎝summationdisplay m=0⎝parenleftBigx 2⎝parenrightBign+2mψ(n+m+1 )+ ψ(m+1 ) m!(n+m)!;n=0 ,1 ,2 ,... , where ψ(z) is the logarithmic derivative of the gamma function; for n= 0, the ﬁrst sum is dropped. Wronskians and similar formulas: W(Iν,I–ν)=–2 πxsin(πν),W(Iν,Kν)=–1 x, Iν(x)I–ν+1(x)–I–ν(x)Iν–1(x)=–2s i n (πν) πx,Iν(x)Kν+1(x)+Iν+1(x)Kν(x)=1 x, where W(f,g)=fg/prime x–f/prime xg. 1022 SPECIAL FUNCTIONS AND THEIR PROPERTIES Modiﬁed Bessel functions can be expressed in terms of Bessel functions: Iν(z)=e–πνi/ 2Jν(zeπi/2)( – π<a r gz≤π/2); Iν(z)=e3πνi/ 2Jν(ze–3πi/ 2)( π/2<a r g z≤π); Kν(z)=1 2πieπνi/ 2H(1) ν(zeπi/2)( – π<a r gz≤π/2); Kν(z)=–1 2πie–πνi/ 2H(2) ν(ze–πi/2)(π/2<a r g z≤π). 11.7-2. Integral Representations and Asymptotic Expansions. The functions Iν(x)a n dKν(x) can be represented in terms of deﬁnite integrals: Iν(x)=xν π1/22νΓ(ν+1 2)⎝integraldisplay1 –1exp(–xt)(1 –t2)ν–1/2dt (x>0 ,ν>–1 2), Kν(x)=⎝integraldisplay∞ 0exp(–xcosht)c o s h ( νt)dt (x>0 ) , Kν(x)=1 cos⎝parenleftbig1 2πν⎝parenrightbig⎝integraldisplay∞ 0cos(x sinht)c o s h ( νt)dt (x>0 , – 1< ν<1 ) , Kν(x)=1 sin⎝parenleftbig1 2πν⎝parenrightbig⎝integraldisplay∞ 0sin(xsinht)s i n h ( νt)dt (x>0 , – 1< ν<1 ) . For integer ν=n, In(x)=1 π⎝integraldisplayπ 0exp(xcost)c o s ( nt)dt (n=0 ,1 ,2 ,... ), K0(x)=⎝integraldisplay∞ 0cos(x sinht)dt=⎝integraldisplay∞ 0cos(xt ) √ t2+1dt (x>0 ) . Asymptotic expansions as x→∞ : Iν(x)=ex √ 2πx⎝braceleftbigg 1+M⎝summationdisplay m=1(–1)m(4ν2– 1)(4ν2–32)...[4ν2–( 2m–1 )2] m!( 8x)m⎝bracerightbigg , Kν(x)=⎝radicalbigg π 2xe–x⎝braceleftbigg 1+M⎝summationdisplay m=1(4ν2– 1)(4ν2–32)...[4ν2–( 2m–1 )2] m!( 8x)m⎝bracerightbigg . The terms of the order of O(x–M–1) are omitted in the braces. Integrals with modiﬁed Bessel functions: ⎝integraldisplayx 0xλIν(x)dx=xλ+ν+1 2ν(λ+ν+1 )Γ(ν+1 )F⎝parenleftbiggλ+ν+1 2,λ+ν+3 2,ν+1;x2 4⎝parenrightbigg ,R e ( λ+ν) > –1, where F(a,b,c;x) is the hypergeometric series (see Supplement 11.10-1), ⎝integraldisplayx 0xλKν(x)dx=2ν–1Γ(ν) λ–ν+1xλ–ν+1F⎝parenleftbiggλ–ν+1 2,1 –ν,λ–ν+3 2;x2 4⎝parenrightbigg +2–ν–1Γ(–ν) λ+ν+1xλ+ν+1F⎝parenleftbiggλ+ν+1 2,1 +ν,λ+ν+3 2;x2 4⎝parenrightbigg ,R e λ>|Reν|–1 . 11.8. A IRYFUNCTIONS 1023 11.8. Airy Functions 11.8-1. Deﬁnition and Basic Formulas. The Airy function of the ﬁrst kind ,A i (x), and the Airy function of the second kind ,B i (x), are solutions of the Airy equation y/prime/prime xx–xy=0 and are deﬁned by the formulas Ai(x)=1 π⎝integraldisplay∞ 0cos⎝parenleftbig1 3t3+xt⎝parenrightbig dt, Bi(x)=1 π⎝integraldisplay∞ 0⎝bracketleftbig exp⎝parenleftbig –1 3t3+xt⎝parenrightbig +s i n⎝parenleftbig1 3t3+xt⎝parenrightbig⎝bracketrightbig dt. Wronskian: W{Ai(x), Bi(x )}=1/π. Relation to the Bessel functions and the modiﬁed Bessel functions ( x>0 ) : Ai(x)=1 3√ x⎝bracketleftbig I–1/3(z)–I1/3(z)⎝bracketrightbig =π–1⎝radicalBig 1 3xK 1/3(z),z=2 3x3/2, Ai(–x)=1 3√ x⎝bracketleftbig J–1/3(z)+J1/3(z)⎝bracketrightbig , Bi(x)=⎝radicalBig 1 3x⎝bracketleftbig I–1/3(z)+I1/3(z)⎝bracketrightbig , Bi(–x)=⎝radicalBig 1 3x⎝bracketleftbig J–1/3(z)–J1/3(z)⎝bracketrightbig . 11.8-2. Power Series and Asymptotic Expansions. Power series expansions as x→0: Ai(x)=c1f(x)–c2g(x), Bi(x)=√ 3[c1f(x)+c2g(x)], f(x)=1+1 3!x3+1×4 6!x6+1×4×7 9!x9+···=∞⎝summationdisplay k=03k⎝parenleftbig1 3⎝parenrightbig kx3k (3k)!, g(x)=x+2 4!x4+2×5 7!x7+2×5×8 10!x10+···=∞⎝summationdisplay k=03k⎝parenleftbig2 3⎝parenrightbig kx3k+1 (3k+1 ) !, where c1=3–2/3/Γ(2/3)≈0.3550 and c2=3–1/3/Γ(1/3)≈0.2588. For large values of x, the leading terms of asymptotic expansions of the Airy functions are Ai(x)/similarequal1 2π–1/2x–1/4exp(–z),z=2 3x3/2, Ai(–x)/similarequalπ–1/2x–1/4sin⎝parenleftbig z+π 4⎝parenrightbig , Bi(x)/similarequalπ–1/2x–1/4exp(z), Bi(–x)/similarequalπ–1/2x–1/4cos⎝parenleftbig z+π 4⎝parenrightbig , where x>0 . 1024 SPECIAL FUNCTIONS AND THEIR PROPERTIES TABLE 1 Special cases of the Kummer conﬂuent hypergeometric function Φ(a,b;z) a b z Φ Conventional notation a a x ex 1 2 2x 1 xexsinhx a a+1 –x ax–aγ(a,x) Incomplete gamma function γ(a,x)=⎝integraldisplayx 0e–tta–1dt 1 2 3 2 –x2 √ π 2erfx Error function erfx=2 √ π⎝integraldisplayx 0exp(–t2)dt –n 1 2 x2 2 n! (2n)!⎝parenleftBig –1 2⎝parenrightBig–n H2n(x) Hermite polynomial Hn(x) = (–1)nex2dn dxn⎝parenleftbig e–x2⎝parenrightbig , n=0 ,1 ,2 , ... –n 3 2 x2 2 n! (2n+1)!⎝parenleftBig –1 2⎝parenrightBig–n H2n+1(x) –n b x n! (b)nL(b–1) n(x) Laguerre polynomial L(α) n(x)=exx–α n!dn dxn⎝parenleftbig e–xxn+α⎝parenrightbig , α=b–1, (b)n=b(b+1)...(b+n–1) ν+1 2 2ν+1 2x Γ(1+ν)ex⎝parenleftBigx 2⎝parenrightBig–ν Iν(x) Modiﬁed Bessel function Iν(x) n+1 2n+2 2x Γ⎝parenleftBig n+3 2⎝parenrightBig ex⎝parenleftBigx 2⎝parenrightBig–n–1 2In+1 2(x) 11.9. Conﬂuent Hypergeometric Functions 11.9-1. Kummer and Tricomi Conﬂuent Hypergeometric Functions. The conﬂuent hypergeometric functions Φ(a,b;x)a n dΨ(a,b;x) are solutions of the degenerate hypergeometric equation (orconﬂuent hypergeometric equation ) xy/prime/prime xx+(b–x)y/prime x–ay=0 . In the case b≠0, –1, –2, –3, ...,t h e Kummer conﬂuent hypergeometric function Φ(a,b;x) can be represented as Kummer’s series: Φ(a,b;x)=1+∞⎝summationdisplay k=1(a)k (b)kxk k!, where ( a)k=a(a+1 )...(a+k–1 ) ,(a)0=1 . Table 1 presents some special cases where Φcan be expressed in terms of simpler functions. 11.9. C ONFLUENT HYPERGEOMETRIC FUNCTIONS 1025 TABLE 2 Special cases of the Tricomi conﬂuent hypergeometric function Ψ(a,b;z) a b z Ψ Conventional notation 1–a 1–a x exΓ(a,x) Incomplete gamma function Γ(a,x)=⎝integraldisplay∞ xe–tta–1dt 1 2 1 2 x2 √ πexp(x2) erfcx Complementary error function erfcx=2 √ π⎝integraldisplay∞ xexp(–t2)dt 1 1 –x –e–xEi(x) Exponential integral Ei(x)=⎝integraldisplayx –∞et tdt 1 1 –l nx –x–1lix Logarithmic integral lix=⎝integraldisplayx 0dt t 1–n 2 3 2 x2 2–nx–1Hn(x) Hermite polynomial Hn(x) = (–1)nex2dn dxn⎝parenleftbig e–x2⎝parenrightbig , n=0 ,1 ,2 , ... ν+1 2 2ν+1 2x π–1/2(2x)–νexKν(x) Modiﬁed Bessel function Kν(x) –ν 2 1 2 1 2x2 2–ν/2ex2/4Dν(x) Weber parabolic cylinder function Dν(x) 1–ν 2 3 2 1 2x2 2(1–ν)/2x–1ex2/4Dν(x) The Tricomi conﬂuent hypergeometric function Ψ(a,b;x) is deﬁned as follows: Ψ(a,b;x)=Γ(1 –b) Γ(a–b+1 )Φ(a,b;x)+Γ(b–1 ) Γ(a)x1–bΦ(a–b+1 , 2– b;x). Table 2 presents some special cases where Ψcan be expressed in terms of simpler functions. Kummer transformation: Φ(a,b;x)=exΦ(b–a,b;–x),Ψ(a,b;x)=x1–bΨ(1 +a–b,2–b;x). Linear relations for Φ: (b–a)Φ(a–1 ,b;x)+( 2a–b+x)Φ(a,b;x)–aΦ(a+1 ,b;x)=0 , b(b–1 )Φ(a,b–1 ;x)–b(b–1+x)Φ(a,b;x)+(b–a)xΦ(a,b+1 ;x)=0 , (a–b+1 )Φ(a,b;x)–aΦ(a+1 ,b;x)+(b–1 )Φ(a,b–1 ;x)=0 , bΦ(a,b;x)–bΦ(a–1 ,b;x)–xΦ(a,b+1 ;x)=0 , b(a+x)Φ(a,b;x)–(b–a)xΦ(a,b+1 ;x)–abΦ(a+1 ,b;x)=0 , (a–1+x)Φ(a,b;x)+(b–a)Φ(a–1,b;x)–(b–1 )Φ(a,b–1 ;x)=0 . 1026 SPECIAL FUNCTIONS AND THEIR PROPERTIES Linear relations for Ψ: Ψ(a–1 ,b;x)–( 2a–b+x)Ψ(a,b;x)+a(a–b+1 )Ψ(a+1 ,b;x)=0 , (b–a–1 )Ψ(a,b–1 ;x)–(b–1+x)Ψ(a,b;x)+xΨ(a,b+1 ;x)=0 , Ψ(a,b;x)–aΨ(a+1 ,b;x)–Ψ(a,b–1 ;x)=0 , (b–a)Ψ(a,b;x)–xΨ(a,b+1;x)+Ψ(a–1,b;x)=0 , (a+x)Ψ(a,b;x)+a(b–a–1 )Ψ(a+1 ,b;x)–xΨ(a,b+1 ;x)=0 , (a–1+x)Ψ(a,b;x)–Ψ(a–1 ,b;x)+(a–c+1 )Ψ(a,b–1 ;x)=0 . Differentiation formulas: d dxΦ(a,b;x)=a bΦ(a+1 ,b+1 ;x), d dxΨ(a,b;x)=–aΨ(a+1 ,b+1 ;x),dn dxnΦ(a,b;x)=(a)n (b)nΦ(a+n,b+n;x), dn dxnΨ(a,b;x) = (–1)n(a)nΨ(a+n,b+n;x). Wronskian: W(Φ,Ψ)=ΦΨ/prime x–Φ/prime xΨ=–Γ(b) Γ(a)x–bex. The Tricomi conﬂuent hypergeometric function for b=n+1(n=0 ,1 ,2 , ...): Ψ(a,n+1 ;x)=(–1)n–1 n!Γ(a–n)⎝braceleftbigg Φ(a,n+1;x)l nx +∞⎝summationdisplay r=0(a)r (n+1 )r⎝bracketleftbig ψ(a+r)–ψ(1 +r)–ψ(1 +n+r)⎝bracketrightbigxr r!⎝bracerightbigg +(n–1 ) ! Γ(a)n–1⎝summationdisplay r=0(a–n)r (1 –n)rxr–n r!. Here the last sum is dropped for n=0 ,ψ(z)=[ l nΓ(z)]/prime zis the logarithmic derivative of the gamma function, ψ(1) = –C,ψ(n)=–C+n–1⎝summationdisplay k=1k–1, whereC= 0.5772 ...is the Euler constant. Ifb< 0, then the formula Ψ(a,b;x)=x1–bΨ(a–b+1 , 2– b;x) is valid for any x. Forb≠0, –1, –2, –3, ..., the general solution of the degenerate hypergeometric equation can be represented in the form y=C1Φ(a,b;x)+C2Ψ(a,b;x), and for b= 0, –1, –2, –3, ..., in the form y=x1–b⎝bracketleftbig C1Φ(a–b+1 , 2– b;x)+C2Ψ(a–b+1 , 2– b;x)⎝bracketrightbig . 11.9. C ONFLUENT HYPERGEOMETRIC FUNCTIONS 1027 11.9-2. Integral Representations and Asymptotic Expansions. Integral representations: Φ(a,b;x)=Γ(b) Γ(a)Γ(b–a)⎝integraldisplay1 0extta–1(1 –t)b–a–1dt (forb>a>0 ) , Ψ(a,b;x)=1 Γ(a)⎝integraldisplay∞ 0e–xtta–1(1 +t)b–a–1dt (fora>0 ,x>0 ) , whereΓ(a) is the gamma function. Asymptotic expansion as |x|→∞ : Φ(a,b;x)=Γ(b) Γ(a)exxa–b⎝bracketleftbiggN⎝summationdisplay n=0(b–a)n(1 –a)n n!x–n+ε⎝bracketrightbigg ,x>0 , Φ(a,b;x)=Γ(b) Γ(b–a)(–x)–a⎝bracketleftbiggN⎝summationdisplay n=0(a)n(a–b+1 )n n!(–x)–n+ε⎝bracketrightbigg ,x<0 , Ψ(a,b;x)=x–a⎝bracketleftbiggN⎝summationdisplay n=0(–1)n(a)n(a–b+1 )n n!x–n+ε⎝bracketrightbigg ,–∞<x<∞, where ε=O(x–N–1). Integrals with conﬂuent hypergeometric functions: ⎝integraldisplay Φ(a,b;x)dx=b–1 a–1Ψ(a–1 ,b–1 ;x)+C, ⎝integraldisplay Ψ(a,b;x)dx=1 1–aΨ(a–1 ,b–1 ;x)+C, ⎝integraldisplay xnΦ(a,b;x)dx=n!n+1⎝summationdisplay k=1(–1)k+1(1 –b)kxn–k+1 (1 –a)k(n–k+1 ) !Φ(a–k,b–k;x)+C, ⎝integraldisplay xnΨ(a,b;x)dx=n!n+1⎝summationdisplay k=1(–1)k+1xn–k+1 (1 –a)k(n–k+1 ) !Ψ(a–k,b–k;x)+C. 11.9-3. Whittaker Conﬂuent Hypergeometric Functions. The Whittaker conﬂuent hypergeometric functions (or Whittaker functions )Mk,µ(x)a n dWk,µ(x) are linearly independent solutions of the Whittaker equation : y/prime/prime xx+⎝bracketleftbig –1 4+1 2k+⎝parenleftbig1 4–µ2⎝parenrightbig x–2⎝bracketrightbig y=0 . The Whittaker functions are expressed in terms of the Kummer and Tricomi conﬂuent hyperge- ometric functions as Mk,µ(x)=xµ+1/2e–x/2Φ⎝parenleftbig1 2+µ–k,1+2 µ;x⎝parenrightbig , Wk,µ(x)=xµ+1/2e–x/2Ψ⎝parenleftbig1 2+µ–k,1+2 µ;x⎝parenrightbig . 1028 SPECIAL FUNCTIONS AND THEIR PROPERTIES 11.10. Gauss Hypergeometric Functions 11.10-1. Various Representations of the Gauss Hypergeometric Function. The Gauss hypergeometric function (or hypergeometric function )F(α,β,γ;x) is a solution of the Gaussian hypergeometric equation x(x–1 )y/prime/prime xx+[ (α+β+1 )x–γ]y/prime x+αβy =0 . Forγ≠0, –1, –2, –3, ..., the function F(α,β,γ;x) can be expressed in terms of the hypergeo- metric series: F(α,β,γ;x)=1+∞⎝summationdisplay k=1(α)k(β)k (γ)kxk k!,(α)k=α(α+1 )...(α+k–1 ) , which certainly converges for |x|<1 . Ifγis not an integer, then the general solution of the hypergeometric equation can be written in the form y=C1F(α,β,γ;x)+C2x1–γF(α–γ+1 ,β–γ+1 , 2– γ;x). Table 3 shows some special cases where Fcan be expressed in term of elementary functions. Forγ>β> 0, the hypergeometric function can be expressed in terms of a deﬁnite integral: F(α,β,γ;x)=Γ(γ) Γ(β)Γ(γ–β)⎝integraldisplay1 0tβ–1(1 –t)γ–β–1(1 –tx)–αdt, whereΓ(β) is the gamma function. 11.10-2. Basic Properties. Linear transformation formulas: F(α,β,γ;x)=F(β,α,γ;x), F(α,β,γ;x)=( 1– x)γ–α–βF(γ–α,γ–β,γ;x), F(α,β,γ;x)=( 1– x)–αF⎝parenleftBig α,γ–β,γ;x x–1⎝parenrightBig , F(α,β,γ;x)=( 1– x)–βF⎝parenleftBig β,γ–α,γ;x x–1⎝parenrightBig . Gauss’s linear relations for contiguous functions: (β–α)F(α,β,γ;x)+αF(α+1 ,β,γ;x)–βF(α,β+1 ,γ;x)=0 , (γ–α–1 )F(α,β,γ;x)+αF(α+1 ,β,γ;x)–(γ–1 )F(α,β,γ–1 ;x)=0 , (γ–β–1 )F(α,β,γ;x)+βF(α,β+1 ,γ;x)–(γ–1)F(α,β,γ–1;x)=0 , (γ–α–β)F(α,β,γ;x)+α(1 –x)F(α+1 ,β,γ;x)–(γ–β)F(α,β–1 ,γ;x)=0 , (γ–α–β)F(α,β,γ;x)–(γ–α)F(α–1 ,β,γ;x)+β(1 –x)F(α,β+1 ,γ;x)=0 . Differentiation formulas: d dxF(α,β,γ;x)=αβ γF(α+1 ,β+1 ,γ+1 ;x), dn dxnF(α,β,γ;x)=(α)n(β)n (γ)nF(α+n,β+n,γ+n;x), dn dxn⎝bracketleftbig xγ–1F(α,β,γ;x)⎝bracketrightbig =(γ–n)nxγ–n–1F(α,β,γ–n;x), dn dxn⎝bracketleftbig xα+n–1F(α,β,γ;x)⎝bracketrightbig =(α)nxα–1F(α+n,β,γ;x), where ( α)n=α(α+1 )...(α+n–1 ) . See Abramowitz and Stegun (1964) and Bateman and Erd ´elyi (1953, V ol. 1) for more detailed information about hypergeometric functions. 11.10. G AUSS HYPERGEOMETRIC FUNCTIONS 1029 TABLE 3 Some special cases where the Gauss hypergeometric function F(α,β,γ;z) can be expressed in terms of elementary functions α β γ z F –n β γ x n⎝summationdisplay k=0(–n)k(β)k (γ)kxk k!,w h e r e n=1 ,2 , ... –n β –n–m x n⎝summationdisplay k=0(–n)k(β)k (–n–m)kxk k!,w h e r e n=1 ,2 , ... α β β x (1 –x)–α α α+1 2 2α+1 x ⎝parenleftBig1+√ 1–x 2⎝parenrightBig–2α α α+1 2 2α x 1 √ 1–x⎝parenleftBig1+√ 1–x 2⎝parenrightBig1–2α α α+1 2 3 2 x2 (1 +x)1–2α–( 1–x )1–2α 2x( 1–2α) α α+1 2 1 2 x2 1 2⎝bracketleftbig (1 +x)–2α+( 1–x)–2α⎝bracketrightbig α α+1 2 1 2 –t a n2x cos2αxcos(2αx) α α–1 2 2α x 22α–1⎝parenleftbig 1+√ 1–x⎝parenrightbig1–2α α 1 2α+1 1 2α x (1 +x)(1 –x)–α–1 α 2–α 3 2 sin2x sin[(2α–2 )x] (α–1 )s i n ( 2 x) α 1–α 3 2 sin2x sin[(2α–1 )x] (α–1 )s i n ( 2 x) α 1–α 1 2 –x2 ⎝parenleftbig√ 1+x2+x⎝parenrightbig2α–1+⎝parenleftbig√ 1+x2–x⎝parenrightbig2α–1 2√ 1+x2 α 1–α 1 2 sin2x cos[(2 α–1 )x] cosx α –α 1 2 –x2 1 2⎝bracketleftbig⎝parenleftbig√ 1+x2+x⎝parenrightbig2α+⎝parenleftbig√ 1+x2–x⎝parenrightbig2α⎝bracketrightbig α –α 1 2 sin2x cos(2αx) 1 1 2 –x 1 xln(x+1 ) 1 2 1 3 2 x2 1 2xln1+x 1–x 1 2 1 3 2 –x2 1 xarctan x 1 2 1 2 3 2 x2 1 xarcsin x 1 2 1 2 3 2 –x2 1 xarcsinh x n+1 n+m+1 n+m+l+2 x (–1)m(n+m+l+1 ) ! n!l!(n+m)! (m+l)!dn+m dxn+m⎝braceleftBig (1 –x)m+ldlF dxl⎝bracerightBig , F=–ln(1 –x) x,n,m,l=0 ,1 ,2 ,... 1030 SPECIAL FUNCTIONS AND THEIR PROPERTIES 11.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions 11.11-1. Legendre Polynomials and Legendre Functions. The Legendre polynomials Pn(x)a n dt h e Legendre functions Qn(x) are solutions of the second-order linear ordinary differential equation (1 –x2)y/prime/prime xx–2xy/prime x+n(n+1 )y=0 . The Legendre polynomials Pn(x) and the Legendre functions Qn(x) are deﬁned by the formulas Pn(x)=1 n!2ndn dxn(x2–1 )n, Qn(x)=1 2Pn(x)l n1+x 1–x–n⎝summationdisplay m=11 mPm–1(x)Pn–m(x). The polynomials Pn=Pn(x) can be calculated using the formulas P0(x)=1 , P1(x)=x,P2(x)=1 2(3x2–1 ) , P3(x)=1 2(5x3–3x),P4(x)=1 8(35x4–3 0x2+3 ) , Pn+1(x)=2n+1 n+1xPn(x)–n n+1Pn–1(x). The ﬁrst ﬁve functions Qn=Qn(x) have the form Q0(x)=1 2ln1+x 1–x,Q1(x)=x 2ln1+x 1–x–1 , Q2(x)=1 4(3x2–1 )l n1+x 1–x–3 2x,Q3(x)=1 4(5x3–3x)l n1+x 1–x–5 2x2+2 3, Q4(x)=1 16(35x4–3 0x2+3 )l n1+x 1–x–35 8x3+55 24x. The polynomials Pn(x) have the explicit representation Pn(x)=2–n[n/2]⎝summationdisplay m=0(–1)mCm nCn 2n–2mxn–2m, where [ A] stands for the integer part of a number A. Integral representation of the Legendre polynomials ( Laplace integral ): Pn(x)=1 π⎝integraldisplayπ 0⎝parenleftbig x±√ x2–1 c o s t⎝parenrightbigndt,x>1 . Integral representation of the Legendre polynomials ( Dirichlet–Mehler integral ): Pn(cosθ)=√ 2 π⎝integraldisplayθ 0cos⎝bracketleftbig (n+1 2⎝parenrightbig ψ⎝bracketrightbig dψ √ cosψ–c o sθ,0 < θ<π,n=0 ,1 , ... 11.11. L EGENDRE POLYNOMIALS ,LEGENDRE FUNCTIONS ,AND ASSOCIA TED LEGENDRE FUNCTIONS 1031 Integral representation of the Legendre functions: Qn(x)=2n⎝integraldisplay∞ x(t–x)n (t2–1 )n+1dt,x>1 . Properties: Pn(–x) = (–1)nPn(x),Qn(–x) = (–1)n+1Qn(x). Recurrence relations: (n+1 )Pn+1(x)–( 2n+1 )xPn(x)+nPn–1(x)=0 , (x2–1 )d dxPn(x)=n⎝bracketleftbig xPn(x)–Pn–1(x)⎝bracketrightbig =n(n+1 ) 2n+1⎝bracketleftbig Pn+1(x)–Pn–1(x)⎝bracketrightbig . Values of the Legendre polynomials and their derivatives at x=0 : P2m(0) = (–1)m(2m– 1)!! 2mm!,P2m+1(0) = 0, P/prime 2m(0) = 0, P/prime 2m+1(0) = (–1)m(2m+ 1)!! 2mm!. Asymptotic formula as n→∞ : Pn(cosθ)≈⎝parenleftbigg2 πnsinθ⎝parenrightbigg1/2 sin⎝bracketleftbigg⎝parenleftBig n+1 2⎝parenrightBig θ+π 4⎝bracketrightbigg ,0 < θ<π. The polynomials Pn(x) (with natural n) have exactly nreal distinct zeros; all zeros lie on the interval –1 < x< 1. The zeros of Pn(x)a n dPn+1(x) alternate with each other. The function Qn(x) has exactly n+ 1 zeros, which lie on the interval –1 < x<1 . The functions Pn(x) form an orthogonal system on the interval –1 ≤x≤1, with ⎝integraldisplay1 –1Pn(x)Pm(x)dx=⎝braceleftBigg0i f n≠m, 2 2n+1ifn=m. The generating function for Legendre polynomials is 1 √ 1–2sx+s2=∞⎝summationdisplay n=0Pn(x)sn(|s|<1 ) . The generating function for Legendre functions is 1 √ 1–2sx+s2ln⎝bracketleftbiggx–s+√ 1–2sx+s2 √ 1–x2⎝bracketrightbigg =∞⎝summationdisplay n=0Qn(x)sn(|s|<1 ,x>1 ) . 11.11-2. Associated Legendre Functions with Integer Indices and Real Argument. The associated Legendre functions Pm n(x)o fo r d e r mare deﬁned by the formulas Pm n(x)=( 1– x2)m/2dm dxmPn(x), n=1 ,2 ,3 , ...,m=0 ,1 ,2 , ... It is assumed by deﬁnition that P0 n(x)=Pn(x). Properties: Pm n(x)=0 i f m>n,Pm n(–x) = (–1)n–mPm n(x). 1032 SPECIAL FUNCTIONS AND THEIR PROPERTIES The associated Legendre functions Pm n(x) have exactly n–mreal zeros, which lie on the interval –1 <x<1 . The associated Legendre functions Pm n(x) with low indices: P1 1(x)=( 1– x2)1/2,P1 2(x)=3x(1 –x2)1/2,P2 2(x)=3 ( 1– x2), P1 3(x)=3 2(5x2– 1)(1 – x2)1/2,P2 3(x)=1 5x(1 –x2),P3 3(x) = 15(1 – x2)3/2. The associated Legendre functions Pm n(x) with n>mare solutions of the linear ordinary differential equation (1 –x2)y/prime/prime xx–2xy/prime x+⎝bracketleftbigg n(n+1 )–m2 1–x2⎝bracketrightbigg y=0 . The functions Pm n(x) form an orthogonal system on the interval –1 ≤x≤1, with ⎝integraldisplay1 –1Pm n(x)Pm k(x)dx=⎧ ⎨ ⎩0i fn≠k, 2 2n+1(n+m)! (n–m)!ifn=k. The functions Pm n(x) (with m≠0) are orthogonal on the interval –1 ≤x≤1 with weight (1– x2)–1, that is, ⎝integraldisplay1 –1Pm n(x)Pk n(x) 1–x2dx=⎧ ⎨ ⎩0i f m≠k, (n+m)! m(n–m)!ifm=k. 11.11-3. Associated Legendre Functions. General Case. In the general case, the associated Legendre functions of the ﬁrst and the second kind, Pµ ν(z) andQµ ν(z), are linearly independent solutions of the Legendre equation (1 –z2)y/prime/prime zz–2zy/prime z+⎝bracketleftbigg ν(ν+1 )–µ2 1–z2⎝bracketrightbigg y=0 , where the parameters νandµand the variable zcan assume arbitrary real or complex values. For|1–z|< 2, the formulas Pµ ν(z)=1 Γ(1 –µ)⎝parenleftBigz+1 z–1⎝parenrightBigµ/2 F⎝parenleftBig –ν,1 +ν,1 –µ;1–z 2⎝parenrightBig , Qµ ν(z)=A⎝parenleftBigz–1 z+1⎝parenrightBigµ 2F⎝parenleftBig –ν,1 +ν,1 +µ;1–z 2⎝parenrightBig +B⎝parenleftBigz+1 z–1⎝parenrightBigµ 2F⎝parenleftBig –ν,1 +ν,1 –µ;1–z 2⎝parenrightBig , A=eiµπΓ(–µ)Γ(1 +ν+µ) 2Γ(1 +ν–µ),B=eiµπΓ(µ) 2,i2= –1, are valid, where F(a,b,c;z) is the hypergeometric series (see Supplement 11.10). For|z|>1 , Pµ ν(z)=2–ν–1Γ(–1 2–ν) √ πΓ(–ν–µ)z–ν+µ–1(z2–1 )–µ/2F⎝parenleftBig1+ν–µ 2,2+ν–µ 2,2ν+3 2;1 z2⎝parenrightBig +2νΓ(1 2+ν) Γ(1 +ν–µ)zν+µ(z2–1 )–µ/2F⎝parenleftBig –ν+µ 2,1–ν–µ 2,1–2ν 2;1 z2⎝parenrightBig , Qµ ν(z)=eiπµ√ πΓ(ν+µ+1 ) 2ν+1Γ(ν+3 2)z–ν–µ–1(z2–1 )µ/2F⎝parenleftBig2+ν+µ 2,1+ν+µ 2,2ν+3 2;1 z2⎝parenrightBig . 11.11. L EGENDRE POLYNOMIALS ,LEGENDRE FUNCTIONS ,AND ASSOCIA TED LEGENDRE FUNCTIONS 1033 The functions Pν(z)≡P0 ν(z)a n dQν(z)≡Q0 ν(z) are called the Legendre functions . Forn=1 ,2 , ..., Pn ν(z)=(z2–1 )n/2dn dznPν(z),Qn ν(z)=(z2–1 )n/2dn dznQν(z). Relations between associated Legendre functions: Pµ ν(z)=Pµ –ν–1(z),Pn ν(z)=Γ(ν+n+1 ) Γ(ν–n+1 )P–n ν(z),n=0 ,1 ,2 ,... , Pµ ν+1(z)=2ν+1 ν–µ+1zPµ ν(z)–ν+µ ν–µ+1Pµ ν–1(z), Pµ ν+1(z)=Pµ ν–1(z)+( 2ν+1 ) (z2–1 )1/2Pµ–1 ν(z), (z2–1 )d dzPµ ν(z)=νzPµ ν(z)–(ν+m)Pµ ν–1(z), Qµ ν(z)=π 2s i n (µπ)eiπµ⎝bracketleftbigg Pµ ν(z)–Γ(1 +ν+µ Γ(1 +ν–µ)P–µ ν(z)⎝bracketrightbigg , Qµ ν(z)=eiπµ⎝parenleftBigπ 2⎝parenrightBig1/2 Γ(ν+µ+1 ) (z2–1 )–1/4P–ν–1/2 –µ–1/2⎝parenleftbiggz √ z2–1⎝parenrightbigg ,R e z>0 . Integral representation for Re(– µ)>R e ν> –1: Pµ ν(z)=2–ν(z2–1 )–µ/2 Γ(ν+1 )Γ(–µ–ν)⎝integraldisplay∞ 0(z+c o s h t)µ–ν–1(sinht)2ν+1dt, where zdoes not lie on the real axis between –1 and ∞. Integral representation for µ<1/2: Pµ ν(z)=2µ(z2–1 )–µ/2 √ πΓ(1 2–µ)⎝integraldisplayπ 0⎝parenleftbig z+√ z2–1c o s t⎝parenrightbigν+µ(sint)–2µdt, where zdoes not lie on the real axis between –1 and 1. Integral representation for Re ν>– 1a n dR e ( ν+µ+1 )>0 : Qµ ν(z)=eπµiΓ(ν+µ+1 ) (z2–1 )–µ/2 2ν+1Γ(ν+1 )⎝integraldisplayπ 0⎝parenleftbig z+c o st⎝parenrightbigµ–ν–1(sint)2ν+1dt, where zdoes not lie on the real axis between –1 and 1. Forn=0 ,1 ,2 ,... , Pn ν(z)=Γ(ν+n+1 ) πΓ(ν+1 )⎝integraldisplayπ 0⎝parenleftbig z+√ z2–1c o s t⎝parenrightbigνcos(nt )dt,R e z>0 ; Qn ν(z)=( – 1 )nΓ(ν+n+1 ) 2ν+1Γ(ν+1 )(z2–1 )–n/2⎝integraldisplayπ 0(z+c o st)n–ν–1(sint)2ν+1dt,R e ν> –1. Note that z≠x,– 1< x< 1, in the latter formula. The modiﬁed associated Legendre functions ,o nt h ec u t z=x,– 1< x< 1, of the real axis are deﬁned by the formulas Pµ ν(x)=1 2⎝bracketleftbig e1 2iµπPµ ν(x+i0) +e–1 2iµπPµ ν(x–i0)⎝bracketrightbig =1 Γ(1 –µ)⎝parenleftBig1+x 1–x⎝parenrightBigµ/2 F⎝parenleftBig –ν,1 +ν,1 –µ;1–x 2⎝parenrightBig , Qµ ν(x)=1 2e–iµπ⎝bracketleftbig e–1 2iµπQµ ν(x+i0) +e1 2iµπQµ ν(x–i0)⎝bracketrightbig =π 2s i n (πµ)⎝bracketleftbigg cos(πµ )Pµ ν(x)–Γ(ν+µ+1 ) Γ(ν–µ+1 )P–µ ν(x)⎝bracketrightbigg . 1034 SPECIAL FUNCTIONS AND THEIR PROPERTIES Notation: Pν(x)= P0 ν(x), Qν(x)= Q0 ν(x). For –1 < x< 1, the modiﬁed associated Legendre functions can be represented in the form of the trigonometric series: Pµ ν(cosθ)=2µ+1 √ πΓ(ν+µ+1 ) Γ(ν+3 2)(sinθ)µ∞⎝summationdisplay k=0(1 2+µ)k(1 +ν+µ)k k!(ν+3 2)ksin[(2k+ν+µ+1 )θ], Qµ ν(cosθ)=√ π2µΓ(ν+µ+1 ) Γ(ν+3 2)(sinθ)µ∞⎝summationdisplay k=0(1 2+µ)k(1 +ν+µ)k k!(ν+3 2)kcos[(2 k+ν+µ+1 )θ], where 0 < θ<π. For 0 < x<1 , Pµ ν(–x)= Pµ ν(x)c o s [π(ν+µ)] – 2π–1Qµ ν(x)s i n [π(ν+µ)], Qµ ν(–x)=– Qµ ν(x)c o s [π(ν+µ)] –1 2πPµ ν(x)s i n [π(ν+µ)]. For –1 < x<1 , Pµ ν+1(x)=2ν+1 ν–µ+1xPµ ν(x)–ν+µ ν–µ+1Pµ ν–1(x), Pµ ν+1(x)= Pµ ν–1(x)–( 2ν+ 1)(1 – x2)1/2Pµ–1 ν(x), Pµ ν+1(x)=xPµ ν(x)–(ν+µ)(1 –x2)1/2Pµ–1 ν(x), d dxPµ ν(x)=νx x2–1Pµ ν(x)–ν+µ x2–1Pµ ν–1(x). Wronskian: Pµ ν(x)d dxQµ ν(x)–Qµ ν(x)d dxPµ ν(x)=k 1–x2,k=22µΓ⎝parenleftbigν+µ+1 2⎝parenrightbig Γ⎝parenleftbigν+µ+2 2⎝parenrightbig Γ⎝parenleftbigν–µ+1 2⎝parenrightbig Γ⎝parenleftbigν–µ+2 2⎝parenrightbig. Forn=1 ,2 , ..., Pn ν(x)=( – 1 )n(1 –x2)n/2dn dxnPν(x), Qn ν(x)=( – 1 )n(1 –x2)n/2dn dxnQν(x). 11.12. Parabolic Cylinder Functions 11.12-1. Deﬁnitions. Basic Formulas. The W eber parabolic cylinder function Dν(z) is a solution of the linear ordinary differential equation: y/prime/prime zz+⎝parenleftbig –1 4z2+ν+1 2⎝parenrightbig y=0 , where the parameter νand the variable zcan assume arbitrary real or complex values. Another linearly independent solution of this equation is the function D–ν–1(iz); ifνis noninteger, then Dν(–z) can also be taken as a linearly independent solution. The parabolic cylinder functions can be expressed in terms of conﬂuent hypergeometric functions as Dν(z)=21/2exp⎝parenleftbig –1 4z2⎝parenrightbig⎝bracketleftbiggΓ⎝parenleftbig1 2⎝parenrightbig Γ⎝parenleftbig1 2–ν 2⎝parenrightbigΦ⎝parenleftbig –ν 2,1 2;1 2z2⎝parenrightbig +2–1/2Γ⎝parenleftbig –1 2⎝parenrightbig Γ⎝parenleftbig –ν 2⎝parenrightbigzΦ⎝parenleftbig1 2–ν 2,3 2;1 2z2⎝parenrightbig⎝bracketrightbigg . 11.13. E LLIPTIC INTEGRALS 1035 For nonnegative integer ν=n,w eh a v e Dn(z)=1 2n/2exp⎝parenleftbigg –z2 4⎝parenrightbigg Hn⎝parenleftbiggz √ 2⎝parenrightbigg ,n=0 ,1 ,2 , ...; Hn(z) = (–1)nexp⎝parenleftbig z2⎝parenrightbigdn dznexp⎝parenleftbig –z2⎝parenrightbig , where Hn(z) is the Hermitian polynomial of order n. Connection with the error function: D–1(z)=⎝radicalbigg π 2exp⎝parenleftbiggz2 4⎝parenrightbigg erfc⎝parenleftbiggz √ 2⎝parenrightbigg , D–2(z)=⎝radicalbigg π 2zexp⎝parenleftbiggz2 4⎝parenrightbigg erfc⎝parenleftbiggz √ 2⎝parenrightbigg –e x p⎝parenleftbigg –z2 4⎝parenrightbigg . 11.12-2. Integral Representations, Asymptotic Expansions, and Linear Relations. Integral representations: Dν(z)=⎝radicalbig 2/πexp⎝parenleftbig1 4z2⎝parenrightbig⎝integraldisplay∞ 0tνexp⎝parenleftbig –1 2t2⎝parenrightbig cos⎝parenleftbig zt–1 2πν⎝parenrightbig dt for Re ν> –1, Dν(z)=1 Γ(–ν)exp⎝parenleftbig –1 4z2⎝parenrightbig⎝integraldisplay∞ 0t–ν–1exp⎝parenleftbig –zt–1 2t2⎝parenrightbig dt for Re ν<0 . Asymptotic expansion as |z|→∞ : Dν(z)=zνexp⎝parenleftbig –1 4z2⎝parenrightbig⎝bracketleftbiggN⎝summationdisplay n=0(–2)n⎝parenleftbig –ν 2⎝parenrightbig n⎝parenleftbig1 2–ν 2⎝parenrightbig n n!1 z2n+O⎝parenleftbig |z|–2N –2⎝parenrightbig⎝bracketrightbigg for |argz|<3π 4, where ( a)0=1 ,(a)n=a(a+1 )...(a+n–1 )f o r n=1 ,2 ,3 , ... Recurrence relations: Dν+1(z)–zDν(z)+νDν–1(z)=0 , d dzDν(z)+1 2zDν(z)–νDν–1(z)=0 , d dzDν(z)–1 2zDν(z)+Dν+1(z)=0 . 11.13. Elliptic Integrals 11.13-1. Complete Elliptic Integrals. Complete elliptic integral of the ﬁrst kind : K(k)=⎝integraldisplayπ/2 0dα √ 1–k2sin2α=⎝integraldisplay1 0dx ⎝radicalbig (1 –x2)(1 –k2x2). Complete elliptic integral of the second kind : E(k)=⎝integraldisplayπ/2 0√ 1–k2sin2αd α =⎝integraldisplay1 0√ 1–k2x2 √ 1–x2dx. The argument kis called the elliptic modulus (k2<1 ) . 1036 SPECIAL FUNCTIONS AND THEIR PROPERTIES Notation: k/prime=√ 1–k2, K/prime(k)= K(k/prime), E/prime(k)= E(k/prime), where k/primeis the complementary modulus . Properties: K(–k)= K(k), E(–k)= E(k); K(k)= K/prime(k/prime), E(k)= E/prime(k/prime); E(k)K/prime(k)+ E/prime(k)K(k)–K(k)K/prime(k)=π 2. Conversion formulas for complete elliptic integrals: K⎝parenleftbigg1–k/prime 1+k/prime⎝parenrightbigg =1+k/prime 2K(k), E⎝parenleftbigg1–k/prime 1+k/prime⎝parenrightbigg =1 1+k/prime⎝bracketleftbig E(k)+k/primeK(k)⎝bracketrightbig , K⎝parenleftbigg2√ k 1+k⎝parenrightbigg =( 1+ k)K(k), E⎝parenleftbigg2√ k 1+k⎝parenrightbigg =1 1+k⎝bracketleftbig 2E(k)–(k/prime)2K(k)⎝bracketrightbig . Representation of complete elliptic integrals in the form of series in powers of the modulus k: K(k)=π 2⎝braceleftbigg 1+⎝parenleftbigg1 2⎝parenrightbigg2 k2+⎝parenleftbigg1×3 2×4⎝parenrightbigg2 k4+···+⎝bracketleftbigg(2n– 1)!! (2n)!!⎝bracketrightbigg2 k2n+···⎝bracerightbigg , E(k)=π 2⎝braceleftbigg 1–⎝parenleftbigg1 2⎝parenrightbigg2k2 1–⎝parenleftbigg1×3 2×4⎝parenrightbigg2k4 3–···–⎝bracketleftbigg(2n– 1)!! (2n)!!⎝bracketrightbigg2k2n 2n–1–···⎝bracerightbigg . Representation of complete elliptic integrals in the form of series in powers of the complementary modulus k/prime=√ 1–k2: K(k)=π 1+k/prime⎝braceleftbigg 1+⎝parenleftbigg1 2⎝parenrightbigg2⎝parenleftbigg1–k/prime 1+k/prime⎝parenrightbigg2 +⎝parenleftbigg1×3 2×4⎝parenrightbigg2⎝parenleftbigg1–k/prime 1+k/prime⎝parenrightbigg4 +···+⎝bracketleftbigg(2n– 1)!! (2n)!!⎝bracketrightbigg2⎝parenleftbigg1–k/prime 1+k/prime⎝parenrightbigg2n +···⎝bracerightbigg , K(k)=l n4 k/prime+⎝parenleftbigg1 2⎝parenrightbigg2⎝parenleftbigg ln4 k/prime–2 1×2⎝parenrightbigg (k/prime)2+⎝parenleftbigg1×3 2×4⎝parenrightbigg2⎝parenleftbigg ln4 k/prime–2 1×2–2 3×4⎝parenrightbigg (k/prime)4 +⎝parenleftbigg1×3×5 2×4×6⎝parenrightbigg2⎝parenleftbigg ln4 k/prime–2 1×2–2 3×4–2 5×6⎝parenrightbigg (k/prime)6+···; E(k)=π(1 +k/prime) 4⎝braceleftbigg 1+1 22–⎝parenleftbigg1–k/prime 1+k/prime⎝parenrightbigg2 +12 (2×4)2⎝parenleftbigg1–k/prime 1+k/prime⎝parenrightbigg4 +···+⎝bracketleftbigg(2n– 3)!! (2n)!!⎝bracketrightbigg2⎝parenleftbigg1–k/prime 1+k/prime⎝parenrightbigg2n +···⎝bracerightbigg , E(k)=1+1 2⎝parenleftbigg ln4 k/prime–1 1×2⎝parenrightbigg (k/prime)2+12×3 22×4⎝parenleftbigg ln4 k/prime–2 1×2–1 3×4⎝parenrightbigg (k/prime)4 +12×32×5 22×42×6⎝parenleftbigg ln4 k/prime–2 1×2–2 3×4–1 5×6⎝parenrightbigg (k/prime)6+···. Differentiation formulas: dK(k) dk=E(k) k(k/prime)2–K(k) k,dE(k) dk=E(k)–K(k) k. 11.13. E LLIPTIC INTEGRALS 1037 The functions K(k)a n d K/prime(k) satisfy the second-order linear ordinary differential equation d dk⎝bracketleftbigg k(1 –k2)dK dk⎝bracketrightbigg –kK=0 . The functions E(k)a n d E/prime(k)–K/prime(k) satisfy the second-order linear ordinary differential equation (1 –k2)d dk⎝parenleftbigg kdE dk⎝parenrightbigg +kE=0 . 11.13-2. Incomplete Elliptic Integrals (Elliptic Integrals). Elliptic integral of the ﬁrst kind : F(ϕ,k)=⎝integraldisplayϕ 0dα √ 1–k2sin2α=⎝integraldisplaysinϕ 0dx ⎝radicalbig (1 –x2)(1 –k2x2). Elliptic integral of the second kind : E(ϕ,k)=⎝integraldisplayϕ 0√ 1–k2sin2αd α =⎝integraldisplaysinϕ 0√ 1–k2x2 √ 1–x2dx. Elliptic integral of the third kind : Π(ϕ,n,k)=⎝integraldisplayϕ 0dα (1 –nsin2α)√ 1–k2sin2α=⎝integraldisplaysinϕ 0dx (1 –nx2)⎝radicalbig (1 –x2)(1 –k2x2). The quantity kis called the elliptic modulus (k2<1 ) ,k/prime=√ 1–k2is the complementary modulus , andnis the characteristic parameter . Complete elliptic integrals: K(k)=F⎝parenleftBigπ 2,k⎝parenrightBig , E(k)=E⎝parenleftBigπ 2,k⎝parenrightBig , K/prime(k)=F⎝parenleftBigπ 2,k/prime⎝parenrightBig , E/prime(k)=E⎝parenleftBigπ 2,k/prime⎝parenrightBig . Properties of elliptic integrals: F(–ϕ,k)=–F(ϕ,k), F(nπ±ϕ,k)=2nK(k)±F(ϕ,k); E(–ϕ,k)=–E(ϕ,k), E(nπ±ϕ,k)=2nE(k)±E(ϕ,k). Conversion formulas for elliptic integrals (ﬁrst set): F⎝parenleftbigg ψ,1 k⎝parenrightbigg =kF(ϕ,k), E⎝parenleftbigg ψ,1 k⎝parenrightbigg =1 k⎝bracketleftbig E(ϕ,k)–(k/prime)2F(ϕ,k)⎝bracketrightbig , where the angles ϕandψare related by sin ψ=ksinϕ,c o sψ=⎝radicalbig 1–k2sin2ϕ. 1038 SPECIAL FUNCTIONS AND THEIR PROPERTIES Conversion formulas for elliptic integrals (second set): F⎝parenleftbigg ψ,1–k/prime 1+k/prime⎝parenrightbigg =( 1+ k/prime)F(ϕ,k), E⎝parenleftbigg ψ,1–k/prime 1+k/prime⎝parenrightbigg =2 1+k/prime⎝bracketleftbig E(ϕ,k)+k/primeF(ϕ,k)⎝bracketrightbig –1–k/prime 1+k/primesinψ, where the angles ϕandψare related by tan( ψ–ϕ)=k/primetanϕ. Transformation formulas for elliptic integrals (third set): F⎝parenleftbigg ψ,2√ k 1+k⎝parenrightbigg =( 1+ k)F(ϕ,k), E⎝parenleftbigg ψ,2√ k 1+k⎝parenrightbigg =1 1+k⎝bracketleftbigg 2E(ϕ,k)–(k/prime)2F(ϕ,k)+2ksinϕcosϕ 1+ksin2ϕ⎝radicalbig 1–k2sin2ϕ⎝bracketrightbigg , where the angles ϕandψare related by sin ψ=(1 +k)s i nϕ 1+ksin2ϕ. Trigonometric expansions for small kandϕ: F(ϕ,k)=2 πK(k)ϕ–s i nϕcosϕ⎝parenleftbigg a0+2 3a1sin2ϕ+2×4 3×5a2sin4ϕ+···⎝parenrightbigg , a0=2 πK(k)–1 , an=an–1–⎝bracketleftbigg(2n– 1)!! (2n)!!⎝bracketrightbigg2 k2n; E(ϕ,k)=2 πE(k)ϕ–s i nϕcosϕ⎝parenleftbigg b0+2 3b1sin2ϕ+2×4 3×5b2sin4ϕ+···⎝parenrightbigg , b0=1–2 πE(k),bn=bn–1–⎝bracketleftbigg(2n– 1)!! (2n)!!⎝bracketrightbigg2k2n 2n–1. Trigonometric expansions for k→1: F(ϕ,k)=2 πK/prime(k)l nt a n⎝parenleftbiggϕ 2+π 4⎝parenrightbigg –tanϕ cosϕ⎝parenleftbigg a/prime 0–2 3a/prime 1tan2ϕ+2×4 3×5a/prime 2tan4ϕ–···⎝parenrightbigg , a/prime 0=2 πK/prime(k)–1 , a/prime n=a/prime n–1–⎝bracketleftbigg(2n– 1)!! (2n)!!⎝bracketrightbigg2 (k/prime)2n; E(ϕ,k)=2 πE/prime(k)l nt a n⎝parenleftbiggϕ 2+π 4⎝parenrightbigg +tanϕ cosϕ⎝parenleftbigg b/prime 0–2 3b/prime 1tan2ϕ+2×4 3×5b/prime 2tan4ϕ–···⎝parenrightbigg , b/prime 0=2 πE/prime(k)–1 , b/prime n=b/prime n–1–⎝bracketleftbigg(2n– 1)!! (2n)!!⎝bracketrightbigg2(k/prime)2n 2n–1. 11.14. Elliptic Functions An elliptic function is a function that is the inverse of an elliptic integral. An elliptic function is a doubly periodic meromorphic function of a complex variable. All its periods can be written in theform 2 mω 1+2nω2with integer mandn,w h e r e ω1andω2are a pair of (primitive) half-periods. The ratio τ=ω2/ω1is a complex quantity that may be considered to have a positive imaginary part, Imτ>0 . Throughout the rest of this section, the following brief notation will be used: K=K(k)a n d K/prime=K(k/prime) are complete elliptic integrals with k/prime=√ 1–k2. 11.14. E LLIPTIC FUNCTIONS 1039 11.14-1. Jacobi Elliptic Functions. When the upper limit ϕof the incomplete elliptic integral of the ﬁrst kind u=⎝integraldisplayϕ 0dα √ 1–k2sin2α=F(ϕ,k) is treated as a function of u, the following notation is used: u=a mϕ. Naming: ϕis the amplitude anduis the argument . Jacobi elliptic functions : snu=s i nϕ=s i na m u (sine amplitude ), cnu=c o sϕ=c o s a m u (cosine amplitude ), dnu=⎝radicalbig 1–k2sin2ϕ=dϕ du(delta amlplitude ). Along with the brief notations sn u,c nu,d nu, the respective full notations are also used: sn( u,k), cn(u,k), dn(u,k). Simple properties: sn(–u)=–s n u,c n ( – u)=c n u,d n ( – u)=d n u; sn2u+c n2u=1 , k2sn2u+d n2u=1 , d n2u–k2cn2u=1–k2, where i2= –1. Jacobi functions for special values of the modulus ( k=0a n d k=1 ) : sn(u,0 )=s i n u,c n ( u,0 )=c o s u,d n ( u,0 )=1 ; sn(u,1 )=t a n h u,c n ( u,1 )=1 coshu,d n ( u,1 )=1 coshu. Jacobi functions for special values of the argument: sn(1 2K,k)=1 √ 1+k/prime,c n (1 2K,k)=⎝radicalbigg k/prime 1+k/prime,d n (1 2K,k)=√ k/prime; sn(K,k)=1 , c n (K,k) = 0, dn( K,k)=k/prime. Reduction formulas: sn(u±K)=±cnu dnu,c n ( u±K)=∓k/primesnu dnu,d n ( u±K)=k/prime dnu; sn(u±2K)=–s n u,c n ( u±2K)=–c n u,d n ( u±2K)=d n u; sn(u+iK/prime)=1 ksnu,c n ( u+iK/prime)=–i kdnu snu,d n ( u+iK/prime)=–icnu snu; sn(u+2iK/prime)=s n u,c n ( u+2iK/prime)=–c n u,d n ( u+2iK/prime)=–d n u; sn(u+K+iK/prime)=dnu kcnu,c n ( u+K+iK/prime)=–ik/prime kcnu,d n ( u+K+iK/prime)=ik/primesnu cnu; sn(u+2 K+2iK/prime)=–s n u,c n ( u+2 K+2iK/prime)=c n u,d n ( u+2 K+2iK/prime)=–d n u. 1040 SPECIAL FUNCTIONS AND THEIR PROPERTIES Periods, zeros, poses, and residues (see Table 4). TABLE 4 Periods, zeros, poles, and residues of the Jacobian elliptic functions (m,n=0 ,±1,±2,...;i2= –1) Functions Periods Zeros Poles Residues snu 4mK+2nK/primei 2mK+2nK/primei 2mK+(2n+1) K/primei (–1)m1 k cnu (4m+2n)K+2nK/primei (2m+1) K+2nK/primei 2mK+(2n+1) K/primei (–1)m–1i k dnu 2mK+4nK/primei (2m+1) K+(2n+1) K/primei 2mK+(2n+1) K/primei (–1)n–1i Double-argument formulas: sn(2u)=2s nucnudnu 1–k2sn4u=2s nucnudnu cn2u+s n2udn2u, cn(2u)=cn2u–s n2udn2u 1–k2sn4u=cn2u–s n2udn2u cn2u+s n2udn2u, dn(2u )=dn2u–k2sn2ucn2u 1–k2sn4u=dn2u+c n2u(dn2u–1 ) dn2u–c n2u(dn2u–1 ). Half-argument formulas: sn2u 2=1 k21–d n u 1+c n u=1–c n u 1+d n u, cn2u 2=cnu+d nu 1+d n u=1–k2 k21–d n u dnu–c nu, dn2u 2=cnu+d nu 1+c n u=( 1– k2)1–c n u dnu–c nu. Argument addition formulas: sn(u±v)=snucnvdnv±snvcnudnu 1–k2sn2usn2v, cn(u±v)=cnucnv∓snusnvdnudnv 1–k2sn2usn2v, dn(u±v)=dnudnv∓k2snusnvcnucnv 1–k2sn2usn2v. Table 5 presents conversion formulas for Jacobi elliptic functions. If k>1 ,t h e n k1=1/k<1 . Elliptic functions with real modulus can be reduced, using the ﬁrst set of conversion formulas, to elliptic functions with a modulus lying between 0 and 1. Descending Landen transformations (Gauss’s transformations): sn(u,k)=(1 +µ)s n (v,µ2) 1+µsn2(v,µ2),c n ( u,k)=cn(v,µ2)d n (v,µ2) 1+µsn2(v,µ2),d n ( u,k)=dn2(v,µ2)+µ–1 1+µ–d n2(v,µ2), where µ=⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle1–k /prime 1+k/prime⎝vextendsingle⎝vextendsingle⎝vextendsingle⎝vextendsingle,v=u 1+µ. 11.14. E LLIPTIC FUNCTIONS 1041 TABLE 5 Conversion formulas for Jacobi elliptic functions. Full notation is used: sn( u,k), cn(u,k), dn(u,k) u1 k1 sn(u1,k1) cn(u1,k1) dn(u1,k1) ku 1 k ksn(u,k) dn(u,k) cn(u,k) iu k/prime isn(u,k) cn(u,k) 1 cn(u,k) dn(u,k) cn(u,k) k/primeu ik k/prime k/primesn(u,k) dn(u,k) cn(u,k) dn(u,k) 1 dn(u,k) iku ik/prime k iksn(u,k) dn(u,k) 1 dn(u,k) cn(u,k) dn(u,k) ik/primeu 1 k/prime ik/primesn(u,k) cn(u,k) dn(u,k) cn(u,k) 1 cn(u,k) (1+k)u 2√ k 1+k (1+k)s n (u,k) 1+ksn2(u,k) cn(u,k)d n (u,k) 1+ksn2(u,k) 1–k sn2(u,k) 1+ksn2(u,k) (1+k/prime)u 1–k/prime 1+k/prime (1+k/prime)s n (u,k)c n (u,k) dn(u,k) 1–(1+ k/prime)s n2(u,k) dn(u,k) 1–(1–k/prime)s n2(u,k) dn(u,k) Ascending Landen transformations: sn(u,k)=( 1+ σ)sn(v,µ)c n (v,µ) dn(v,µ),c n (u,k)=1+σ µdn2(v,µ)–σ dn(v,µ),d n (u,k)=1–σ µdn2(v,µ)+σ dn(v,µ), where µ=4k (1 +k)2,σ=⎝vextendsingle⎝vextendsingle⎝vextendsingle1–k 1+k⎝vextendsingle⎝vextendsingle⎝vextendsingle,v=u 1+σ. Representation Jacobi functions in the form of power series in u: snu=u–1 3!(1 +k2)u3+1 5!(1 + 14 k2+k4)u5–1 7!(1 + 135 k2+ 135k4+k6)u7+···, cnu=1–1 2!u2+1 4!(1 + 4k2)u4–1 6!(1 + 44 k2+1 6k4)u6+···, dnu=1–1 2!k2u2+1 4!k2(4 +k2)u4–1 6!k2(16 + 44 k2+k4)u6+···, amu=u–1 3!k2u3+1 5!k2(4 +k2)u5–1 7!k2(16 + 44k2+k4)u7+···. These functions converge for |u|<|K(k/prime)|. Representation Jacobi functions in the form of trigonometric series: snu=2π kK√ q∞⎝summationdisplay n=1qn 1–q2n–1sin⎝bracketleftbigg (2n–1 )πu 2K⎝bracketrightbigg , cnu=2π kK√ q∞⎝summationdisplay n=1qn 1+q2n–1cos⎝bracketleftbigg (2n–1 )πu 2K⎝bracketrightbigg , dnu=π 2K+2π K∞⎝summationdisplay n=1qn 1+q2ncos⎝parenleftbiggnπu K⎝parenrightbigg , 1042 SPECIAL FUNCTIONS AND THEIR PROPERTIES amu=πu 2K+2∞⎝summationdisplay n=11 nqn 1+q2nsin⎝parenleftbiggnπu K⎝parenrightbigg , where q=e x p ( – πK/prime/K),K=K(k),K/prime=K(k/prime), and k/prime=√ 1–k2. Derivatives: d dusnu=c nudnu,d ducnu=–s n udnu,d dudnu=–k2snucnu. Integrals:⎝integraldisplay snud u =1 kln(dnu–kcnu)=–1 kln(dnu+kcnu), ⎝integraldisplay cnud u =1 karccos(dn u)=1 karcsin( ksnu), ⎝integraldisplay dnud u = arcsin(sn u)=a m u. The arbitrary additive constant Cin the integrals is omitted. 11.14-2. Weierstrass Elliptic Function. The Weierstrass elliptic function (orW eierstrass ℘-function )i sd e ﬁ n e da s ℘(z)=℘(z|ω1,ω2)=1 z2+⎝summationdisplay m,n⎝bracketleftbigg1 (z–2mω 1–2nω2)2–1 (2mω 1+2nω2)2⎝bracketrightbigg , where the summation is assumed over all integer mandn, except for m=n= 0. This function is a complex, double periodic function of a complex variable zwith periods 2 ω1and 2ω1: ℘(–z)=℘(z), ℘(z+2mω 1+2nω2)=℘(z), where m,n=0 ,±1,±2,...and Im(ω 2/ω1)≠0. The series deﬁning the Weierstrass ℘-function converges everywhere except for second-order poles located at zmn=2mω 1+2nω2. Argument addition formula: ℘(z1+z2)=–℘(z1)–℘(z2)+1 4⎝bracketleftbigg℘/prime(z1)–℘/prime(z2) ℘(z1)–℘(z2)⎝bracketrightbigg2 . The Weierstrass function ℘=℘(z,g2,g3)=℘(z|ω1,ω2) is deﬁned implicitly by the elliptic integral: z=⎝integraldisplay∞ ℘dt ⎝radicalbig 4t3–g2t–g3=⎝integraldisplay∞ ℘dt 2√ (t–e1)(t–e2)(t–e3). The parameters g2andg3are known as the invariants . The parameters e1,e2,e3, which are the roots of the cubic equation 4 z3–g2z–g3=0 ,a r er e l a t e d to the half-periods ω1,ω2and invariants g2,g3by e1=℘(ω1),e2=℘(ω1+ω2),e1=℘(ω2), e1+e2+e3=0 , e1e2+e1e3+e2e3=–1 4g2,e1e2e3=1 4g3. 11.15. J ACOBI THETA FUNCTIONS 1043 Homogeneity property: ℘(z,g2,g3)=λ2℘(λz,λ–4g2,λ–6g3). The Weierstrass ℘-function can be expanded into a Laurent series: ℘(z)=1 z2+g2 20z2+g3 28z4+g2 2 1200z6+3g2g3 6160z8+···=1 z2+∞⎝summationdisplay k=2akz2k–2, ak=3 (k– 3)(2k+1 )k–2⎝summationdisplay m=2amak–mfork≥4, 0 < |z|<m i n ( |ω1|,|ω2|). The Weierstrass ℘-function satisﬁes the ﬁrst-order and second-order nonlinear differential equa- tions: (℘/prime z)2=4℘3–g2℘–g3, ℘/prime/prime zz=6℘2–1 2g2. Direct and inverse representations of the Weierstrass elliptic function via Jacobi elliptic functions: ℘(z)=e1+(e1–e3)cn2w sn2w=e2+(e1–e3)dn2w sn2w=e3+e1–e3 sn2w; snw=⎝radicalbigg e1–e3 ℘(z)–e3,c n w=⎝radicalBigg ℘(z)–e1 ℘(z)–e3,d n w=⎝radicalBigg ℘(z)–e2 ℘(z)–e3; w=z√ e1–e3=Kz/ω 1. The parameters are related by k=⎝radicalbigg e2–e3 e1–e3,k/prime=⎝radicalbigg e1–e2 e1–e3, K=ω1√ e1–e3,iK/prime=ω2√ e1–e3. 11.15. Jacobi Theta Functions 11.15-1. Series Representation of the Jacobi Theta Functions. Simplest Properties. The Jacobi theta functions (orelliptic theta functions ) are deﬁned by the following series: ϑ1(v)=ϑ1(v,q)=ϑ1(v|τ)=2∞⎝summationdisplay n=0(–1)nq(n+1/2)2sin[(2n+1 )πv]=i∞⎝summationdisplay n=–∞(–1)nq(n–1/2)2eiπ(2n–1)v, ϑ2(v)=ϑ2(v,q)=ϑ2(v|τ)=2∞⎝summationdisplay n=0q(n+1/2)2cos[(2 n+1 )πv]=∞⎝summationdisplay n=–∞q(n–1/2)2eiπ(2n–1)v, ϑ3(v)=ϑ3(v,q)=ϑ3(v|τ)=1+2∞⎝summationdisplay n=0qn2cos(2nπv)=∞⎝summationdisplay n=–∞qn2e2iπnv, ϑ4(v)=ϑ4(v,q)=ϑ4(v|τ)=1+2∞⎝summationdisplay n=0(–1)nqn2cos(2nπv)=∞⎝summationdisplay n=–∞(–1)nqn2e2iπnv, where vis a complex variable and q=eiπτis a complex parameter ( τhas a positive imaginary part). The Jacobi theta functions are periodic entire functions that possess the following properties: ϑ1(v) odd, has period 2, vanishes at v=m+nτ; ϑ2(v) even, has period 2, vanishes at v=m+nτ+1 2; ϑ3(v) even, has period 1, vanishes at v=m+(n+1 2)τ+1 2; ϑ4(v) even, has period 1, vanishes at v=m+(n+1 2)τ. Herem,n=0 ,±1,±2,... 1044 SPECIAL FUNCTIONS AND THEIR PROPERTIES Remark. The theta functions are not elliptic functions. The very good convergence of their series allows the computation of various elliptic integrals and elliptic functions using the relationsgiven above in Supplement 11.15-1. 11.15-2. Various Relations and Formulas. Connection w ith Jacobi Elliptic Functions. Linear relations (ﬁrst set): ϑ1⎝parenleftBig v+1 2⎝parenrightBig =ϑ2(v), ϑ2⎝parenleftBig v+1 2⎝parenrightBig =–ϑ1(v), ϑ3⎝parenleftBig v+1 2⎝parenrightBig =ϑ4(v), ϑ4⎝parenleftBig v+1 2⎝parenrightBig =ϑ3(v), ϑ1⎝parenleftBig v+τ 2⎝parenrightBig =ie–iπ⎝parenleftbig v+τ 4⎝parenrightbig ϑ4(v), ϑ2⎝parenleftBig v+τ 2⎝parenrightBig =e–iπ⎝parenleftbig v+τ 4⎝parenrightbig ϑ3(v), ϑ3⎝parenleftBig v+τ 2⎝parenrightBig =e–iπ⎝parenleftbig v+τ 4⎝parenrightbig ϑ2(v), ϑ4⎝parenleftBig v+τ 2⎝parenrightBig =ie–iπ⎝parenleftbig v+τ 4⎝parenrightbig ϑ1(v). Linear relations (second set): ϑ1(v|τ+1 )= eiπ/4ϑ1(v|τ), ϑ2(v|τ+1 )= eiπ/4ϑ2(v|τ), ϑ3(v|τ+1 )= ϑ4(v|τ), ϑ4(v|τ+1 )= ϑ3(v|τ), ϑ1⎝parenleftBigv τ⎝vextendsingle⎝vextendsingle⎝vextendsingle–1 τ⎝parenrightBig =1 i⎝radicalbigg τ ieiπv2/τϑ1(v|τ), ϑ2⎝parenleftBigv τ⎝vextendsingle⎝vextendsingle⎝vextendsingle–1 τ⎝parenrightBig =⎝radicalbigg τ ieiπv2/τϑ4(v|τ), ϑ3⎝parenleftBigv τ⎝vextendsingle⎝vextendsingle⎝vextendsingle–1 τ⎝parenrightBig =⎝radicalbigg τ ieiπv2/τϑ3(v|τ), ϑ4⎝parenleftBigv τ⎝vextendsingle⎝vextendsingle⎝vextendsingle–1 τ⎝parenrightBig =⎝radicalbigg τ ieiπv2/τϑ2(v|τ). Quadratic relations: ϑ2 1(v)ϑ2 2(0) =ϑ2 4(v)ϑ2 3(0) –ϑ2 3(v)ϑ2 4(0), ϑ2 1(v)ϑ2 3(0) =ϑ2 4(v)ϑ2 2(0) –ϑ2 2(v)ϑ2 4(0), ϑ2 1(v)ϑ2 4(0) =ϑ2 3(v)ϑ2 2(0) –ϑ2 2(v)ϑ2 3(0), ϑ2 4(v)ϑ2 4(0) =ϑ2 3(v)ϑ2 3(0) –ϑ2 2(v)ϑ2 2(0). Representation of the theta functions in the form of inﬁnite products: ϑ1(v)=2q0q1/4sin(πv)∞⎝productdisplay n=1⎝bracketleftbig 1–2q2ncos(2πv)+q4n⎝bracketrightbig , ϑ2(v)=2q0q1/4cos(πv )∞⎝productdisplay n=1⎝bracketleftbig 1+2q2ncos(2πv)+q4n⎝bracketrightbig , ϑ3(v)=q0∞⎝productdisplay n=1⎝bracketleftbig 1+2q2n–1cos(2πv)+q4n–2⎝bracketrightbig , ϑ4(v)=q0∞⎝productdisplay n=1⎝bracketleftbig 1–2q2n–1cos(2πv)+q4n–2⎝bracketrightbig , where q0=∞⎝producttext n=1(1 –q2n). Representations of Jacobi elliptic functions in terms of the theta functions: snw=ϑ3(0) ϑ2(0)ϑ1(v) ϑ4(v),c n w=ϑ4(0) ϑ2(0)ϑ2(v) ϑ4(v),d n w=ϑ4(0) ϑ3(0)ϑ3(v) ϑ4(v),w=2 Kv. The parameters are related by k=ϑ2 2(0) ϑ23(0),k/prime=ϑ2 4(0) ϑ23(0), K=π 2ϑ2 3(0), K/prime=–iτK. 11.16. M ATHI EU FUNCTIONS AND MODIFIED MATHI EU FUNCTIONS 1045 TABLE 6 The Mathieu functions ce n=c e n(x,q)a n ds e n=s en(x,q) (for odd n, functions cenand se nare 2π-periodic, and for even n,t h e ya r e π-periodic); deﬁnite eigenvalues a=an(q)a n da=bn(q) correspond to each value of parameter q Mathieu functions Recurrence relations for coefﬁcients Normalization conditions ce2n=∞⎝summationdisplay m=0A2n 2mcos 2mx qA2n 2=a2nA2n 0; qA2n 4=(a2n–4)A2n 2–2qA2n 0; qA2n 2m+2=(a2n–4m2)A2n 2m –qA2n 2m–2,m≥2 (A2n 0)2+∞⎝summationdisplay m=0(A2n 2m)2 =⎝braceleftBig 2i fn=0 1i fn≥1 ce2n+1=∞⎝summationdisplay m=0A2n+1 2m+1cos(2m+1)x qA2n+1 3=(a2n+1–1–q )A2n+1 1; qA2n+1 2m+3=[a2n+1–(2m+1)2]A2n+1 2m+1 –qA2n+1 2m–1,m≥1 ∞⎝summationdisplay m=0(A2n+1 2m+1)2=1 se2n=∞⎝summationdisplay m=0B2n 2msin 2mx, se0=0 qB2n 4=(b2n–4)B2n 2; qB2n 2m+2=(b2n–4m2)B2n 2m –qB2n 2m–2,m≥2 ∞⎝summationdisplay m=0(B2n 2m)2=1 se2n+1=∞⎝summationdisplay m=0B2n+1 2m+1sin(2m+1)x qB2n+1 3=(b2n+1–1–q )B2n+1 1; qB2n+1 2m+3=[b2n+1–(2m+1)2]B2n+1 2m+1 –qB2n+1 2m–1,m≥1 ∞⎝summationdisplay m=0(B2n+1 2m+1)2=1 11.16. Mathieu Functions and Modiﬁed Mathieu Functions 11.16-1. Mathieu Functions. The Mathieu functions ce n(x,q)a n ds e n(x,q) are periodical solutions of the Mathieu equation y/prime/prime xx+(a–2qcos 2x)y=0 . Such solutions exist for deﬁnite values of parameters aandq(those values of aare referred to as eigenvalues). The Mathieu f unctions are listed in Table 6. The Mathieu functions possess the following properties: ce2n(x,–q) = (–1)nce2n⎝parenleftBigπ 2–x,q⎝parenrightBig ,c e 2n+1(x,–q)=( – 1 )nse2n+1⎝parenleftBigπ 2–x,q⎝parenrightBig , se2n(x,–q)=( – 1 )n–1se2n⎝parenleftBigπ 2–x,q⎝parenrightBig ,s e 2n+1(x,–q)=( – 1 )nce2n+1⎝parenleftBigπ 2–x,q⎝parenrightBig . Selecting sufﬁciently large number mand omitting the term with the maximum number in the recur- rence relations (indicated in Table 6), we can obt ain approximate relations for eigenvalues an(orbn) with respect to parameter q. Then, equating the determinant of the corresponding homogeneous linear system of equations for coefﬁcients An m(orBn m) to zero, we obtain an algebraic equation for ﬁnding an(q)( o rb n(q)). For ﬁxed real q≠0, eigenvalues anandbnare all real and different, while ifq>0 t h e n a0<b1<a1<b2<a2<···; ifq<0 t h e n a0<a1<b1<b2<a2<a3<b3<b4<···. 1046 SPECIAL FUNCTIONS AND THEIR PROPERTIES The eigenvalues possess the properties a2n(–q)=a2n(q),b2n(–q)=b2n(q),a2n+1(–q)=b2n+1(q). Tables of the eigenvalues an=an(q)a n dbn=bn(q) can be found in Abramowitz and Stegun (1964, chap. 20). The solution of the Mathieu equation corresponding to eigenvalue an(orbn)h a snzeros on the interval 0 ≤x<π(qis a real number). Listed below are two leading terms of asymptotic expansions of the Mathieu functions ce n(x,q) and se n(x,q), as well as of the corresponding eigenvalues an(q)a n dbn(q), asq→0: ce0(x,q)=1 √ 2⎝parenleftBig 1–q 2cos 2x⎝parenrightBig ,a0(q)=–q2 2+7q4 128; ce1(x,q)=c o s x–q 8cos 3x,a1(q)=1+ q; ce2(x,q)=c o s 2 x+q 4⎝parenleftBig 1–cos 4x 3⎝parenrightBig ,a2(q)=4+5q2 12; cen(x,q)=c o s nx+q 4⎝bracketleftbiggcos(n +2 )x n+1–cos(n –2 )x n–1⎝bracketrightbigg ,an(q)=n2+q2 2(n2–1 )(n≥3); se1(x,q)=s i n x–q 8sin 3x,b1(q)=1– q; se2(x,q)=s i n2 x–qsin 4x 12,b2(q)=4–q2 12; sen(x,q)=s i n nx–q 4⎝bracketleftbiggsin(n+2 )x n+1–sin(n–2 )x n–1⎝bracketrightbigg ,bn(q)=n2+q2 2(n2–1 )(n≥3). Asymptotic results as q→∞ (–π/2<x<π/2): an(q)≈–2q+2 ( 2n+1 )√ q+1 4(2n2+2n+1 ) , bn+1(q)≈–2q+2 ( 2n+1 )√ q+1 4(2n2+2n+1 ) , cen(x,q)≈λnq–1/4cos–n–1x⎝bracketleftbig cos2n+1ξexp(2√ qsinx)+s i n2n+1ξexp(–2√ qsinx)⎝bracketrightbig , sen+1(x,q)≈µn+1q–1/4cos–n–1x⎝bracketleftbig cos2n+1ξexp(2√ qsinx)–s i n2n+1ξexp(–2√ qsinx)⎝bracketrightbig , where λnandµnare some constants independent of the parameter q,a n dξ=1 2x+π 4. 11.16-2. Modiﬁed Mathieu Functions. The modiﬁed Mathieu functions Ce n(x,q)a n dS e n(x,q) are solutions of the modiﬁed Mathieu equation y/prime/prime xx–(a–2qcosh 2 x)y=0 , witha=an(q)a n da=bn(q) being the eigenvalues of the Mathieu equation (see Supplement 11.16-1). The modiﬁed Mathieu functions are deﬁned as Ce2n+p(x,q)=c e 2n+p(ix,q)=∞⎝summationdisplay k=0A2n+p 2k+pcosh[(2 k+p)x], Se2n+p(x,q)=–ise2n+p(ix,q)=∞⎝summationdisplay k=0B2n+p 2k+psinh[(2 k+p)x], where pmay be equal to 0 and 1, and coefﬁcients A2n+p 2k+pandB2n+p 2k+pare indicated in Supplement 11.16-1. 11.17. O RTHOGONAL POLYNOMIALS 1047 11.17. Orthogonal Polynomials All zeros of each of the orthogonal polynomials Pn(x) considered in this section are real and simple. The zeros of the polynomials Pn(x)a n dPn+1(x) are alternating. For Legendre polynomials see Supplement 11.11-1. 11.17-1. Laguerre Polynomials and Generalized Laguerre Polynomials. The Laguerre polynomials Ln=Ln(x) satisfy the second-order linear ordinary differential equation xy/prime/prime xx+( 1– x)y/prime x+ny=0 and are deﬁned by the formulas Ln(x)=1 n!exdn dxn⎝parenleftbig xne–x⎝parenrightbig =(–1)n n!⎝bracketleftbigg xn–n2xn–1+n2(n–1 )2 2!xn–2+···⎝bracketrightbigg . The ﬁrst four polynomials have the form L0(x)=1 , L1(x)=–x+1 , L2(x)=1 2(x2–4x+2 ) , L3(x)=1 6(–x3+9x2–1 8x+6 ) . To calculate Ln(x)f o rn ≥2, one can use the recurrence formulas Ln+1(x)=1 n+1⎝bracketleftbig (2n+1–x)Ln(x)–nLn–1(x)⎝bracketrightbig . The functions Ln(x) form an orthonormal system on the interval 0 < x<∞with weight e–x: ⎝integraldisplay∞ 0e–xLn(x)Lm(x)dx=⎝braceleftBig0i f n≠m, 1i f n=m. The generating function is 1 1–sexp⎝parenleftBig –sx 1–s⎝parenrightBig =∞⎝summationdisplay n=0Ln(x)sn, |s|<1 . The generalized Laguerre polynomials Lα n=Lα n(x)(α> –1) satisfy the equation xy/prime/prime xx+(α+1–x)y/prime x+ny=0 and are deﬁned by the formulas Lα n(x)=1 n!x–αexdn dxn⎝parenleftbig xn+αe–x⎝parenrightbig =n⎝summationdisplay m=0Cn–m n+α(–x)m m!=n⎝summationdisplay m=0Γ(n+α+1 ) Γ(m+α+1 )(–x)m m!(n–m)!. Notation: L0 n(x)=Ln(x). Special cases: Lα 0(x)=1 , Lα 1(x)=α+1–x,L–n n(x)=( – 1 )nxn n!. To calculate Lα n(x)f o rn ≥2, one can use the recurrence formulas Lα n+1(x)=1 n+1⎝bracketleftbig (2n+α+1–x)Lα n(x)–(n+α)Lα n–1(x)⎝bracketrightbig . Other recurrence formulas: Lα n(x)=Lα n–1(x)+Lα–1 n(x),d dxLα n(x)=–Lα+1 n–1(x),xd dxLα n(x)=nLα n(x)–(n+α)Lα n–1(x). 1048 SPECIAL FUNCTIONS AND THEIR PROPERTIES The functions Lα n(x) form an orthogonal system on the interval 0 < x<∞with weight xαe–x: ⎝integraldisplay∞ 0xαe–xLα n(x)Lα m(x)dx=⎝braceleftbigg0i f n≠m, Γ(α+n+1) n!ifn=m. The generating function is (1 –s)–α–1exp⎝parenleftBig –sx 1–s⎝parenrightBig =∞⎝summationdisplay n=0Lα n(x)sn, |s|<1 . 11.17-2. Chebyshev Polynomials and Functions. The Chebyshev polynomials of the ﬁrst kind Tn=Tn(x) satisfy the second-order linear ordinary differential equation (1 –x2)y/prime/prime xx–xy/prime x+n2y=0 ( 1 ) and are deﬁned by the formulas Tn(x)=c o s ( narccos x)=(–2)nn! (2n)!√ 1–x2dn dxn⎝bracketleftbig (1 –x2)n–1 2⎝bracketrightbig =n 2[n/2]⎝summationdisplay m=0(–1)m(n–m–1 ) ! m!(n–2m)!(2x)n–2m(n=0 ,1 ,2 ,... ), where [ A] stands for the integer part of a number A. An alternative representation of the Chebyshev polynomials: Tn(x)=(–1)n (2n– 1)!!(1 –x2)1/2dn dxn(1 –x2)n–1/2. The ﬁrst ﬁve Chebyshev polynomials of the ﬁrst kind are T0(x)=1 , T1(x)=x,T2(x)=2x2–1 , T3(x)=4x3–3x,T4(x)=8x4–8x2+1 . The recurrence formulas: Tn+1(x)=2xTn(x)–Tn–1(x), n≥2. The functions Tn(x) form an orthogonal system on the interval –1 < x< 1 with weight (1– x2)–1/2: ⎝integraldisplay1 –1Tn(x)Tm(x) √ 1–x2dx=⎝braceleftBigg0i f n≠m, 1 2πifn=m≠0, π ifn=m=0 . The generating function is 1–sx 1–2sx+s2=∞⎝summationdisplay n=0Tn(x)sn(|s|<1 ) . The functions Tn(x) have only real simple zeros, all lying on the interval –1 < x<1 . The normalized Chebyshev polynomials of the ﬁrst kind, 21–nTn(x), deviate from zero least of all. This means that among all polynomials of degree nwith the leading coefﬁcient 1, it is the maximum of the modulus max –1≤x≤1|21–nTn(x)|that has the least value, the maximum being equal to 21–n. 11.17. O RTHOGONAL POLYNOMIALS 1049 The Chebyshev polynomials of the second kind Un=Un(x) satisfy the second-order linear ordinary differential equation (1 –x2)y/prime/prime xx–3xy/prime x+n(n+2 )y=0 and are deﬁned by the formulas Un(x)=sin[(n+ 1) arccos x] √ 1–x2=2n(n+1 ) ! (2n+1 ) !1 √ 1–x2dn dxn(1 –x2)n+1/2 =[n/2]⎝summationdisplay m=0(–1)m(n–m)! m!(n–2m)!(2x)n–2m(n=0 ,1 ,2 , ...). The ﬁrst ﬁve Chebyshev polynomials of the second kind are U0(x)=1 , U1(x)=2x,U2(x)=4x2–1 , U3(x)=8x3–4x,U4(x)=1 6x4–1 2x2+1 . The recurrence formulas: Un+1(x)=2xUn(x)–Un–1(x),n≥2. The generating function is 1 1–2sx+s2=∞⎝summationdisplay n=0Un(x)sn(|s|<1 ) . The Chebyshev polynomials of the ﬁrst and second kind are related by Un(x)=1 n+1d dxTn+1(x). The Chebyshev functions of the second kind , U0(x)=a r c s i n x, Un(x)=s i n ( narccos x)=√ 1–x2 ndTn(x) dx(n=1 ,2 , ...), just as the Chebyshev polynomials, also satisfy the differential equation (1). The ﬁrst ﬁve Chebyshev functions are U0(x)=0 , U1(x)=√ 1–x2, U2(x)=2x√ 1–x2, U3(x)=( 4x2–1 )√ 1–x2, U5(x)=( 8x3–4x)√ 1–x2. The recurrence formulas: Un+1(x)=2xUn(x)–Un–1(x), n≥2. The functions Un(x) form an orthogonal system on the interval –1 < x< 1 with weight (1– x2)–1/2: ⎝integraldisplay1 –1Un(x)Um(x) √ 1–x2dx=⎝braceleftbigg0i f n≠morn=m=0 , 1 2πifn=m≠0. The generating function is √ 1–x2 1–2sx+s2=∞⎝summationdisplay n=0Un+1(x)sn(|s|<1 ) . 1050 SPECIAL FUNCTIONS AND THEIR PROPERTIES 11.17-3. Hermite Polynomials and Functions. The Hermite polynomials Hn=Hn(x) satisfy the second-order linear ordinary differential equation y/prime/prime xx–2xy/prime x+2ny=0 and are deﬁned by the formulas Hn(x)=( – 1 )nexp⎝parenleftbig x2⎝parenrightbigdn dxnexp⎝parenleftbig –x2⎝parenrightbig =[n/2]⎝summationdisplay m=0(–1)m n! m!(n–2m)!(2x)n–2m, where [ A] stands for the integer part of a number A. The ﬁrst ﬁve polynomials are H0(x)=1 , H1(x)=2x,H2(x)=4x2–2 , H3(x)=8x3–1 2x,H4(x)=1 6x4–4 8x2+ 12. Recurrence formulas: Hn+1(x)=2xHn(x)–2nHn–1(x), n≥2; d dxHn(x)=2nHn–1(x). Integral representation: H2n(x)=(–1)n22n+1 √ πexp⎝parenleftbig x2⎝parenrightbig⎝integraldisplay∞ 0exp⎝parenleftbig –t2⎝parenrightbig t2ncos(2xt)dt, H2n+1(x)=(–1)n22n+2 √ πexp⎝parenleftbig x2⎝parenrightbig⎝integraldisplay∞ 0exp⎝parenleftbig –t2⎝parenrightbig t2n+1sin(2xt)dt, where n=0 ,1 ,2 , ... The functions Hn(x) form an orthogonal system on the interval – ∞<x<∞with weight e–x2: ⎝integraldisplay∞ –∞exp⎝parenleftbig –x2⎝parenrightbig Hn(x)Hm(x)dx=⎝braceleftbigg0i f n≠m,√ π2nn!i fn=m. Generating function: exp⎝parenleftbig –s2+2sx⎝parenrightbig =∞⎝summationdisplay n=0Hn(x)sn n!. Asymptotic formula as n→∞ : Hn(x)≈2n+1 2nn 2e–n 2exp⎝parenleftbig x2⎝parenrightbig cos⎝parenleftBig√ 2n+1x–1 2πn⎝parenrightBig . The Hermite functions hn(x) are introduced by the formula hn(x)=e x p⎝parenleftBig –1 2x2⎝parenrightBig Hn(x)=( – 1 )nexp⎝parenleftBig1 2x2⎝parenrightBigdn dxnexp⎝parenleftbig –x2⎝parenrightbig ,n=0 ,1 ,2 ,... The Hermite functions satisfy the second-order linear ordinary differential equation h/prime/prime xx+( 2n+1–x2)h=0 . The functions hn(x) form an orthogonal system on the interval – ∞<x<∞, with ⎝integraldisplay∞ –∞hn(x)hm(x)dx=⎝braceleftbigg0i f n≠m,√ π2nn!i fn=m. 11.17. O RTHOGONAL POLYNOMIALS 1051 11.17-4. Jacobi Polynomials. The Jacobi polynomials ,P(α,β) n(x), are solutions of the second-order linear ordinary differential equation (1 –x2)y/prime/prime xx+⎝bracketleftbig β–α–(α+β+2 )x⎝bracketrightbig y/prime x+n(n+α+β+1 )y=0 and are deﬁned by the formulas P(α,β) n(x)=(–1)n 2nn!(1 –x)–α(1 +x)–βdn dxn⎝bracketleftBig (1 –x)α+n(1 +x)β+n⎝bracketrightBig =2–nn⎝summationdisplay m=0Cm n+αCn–m n+β(x–1 )n–m(x+1 )m, where the Ca bare binomial coefﬁcients. The generating function: 2α+βR–1(1 –s+R)–α(1 +s+R)–β=∞⎝summationdisplay n=0P(α,β) n(x)sn,R=√ 1–2xs+s2,|s|<1 . The Jacobi polynomials are orthogonal on the interval –1 ≤x≤1 with weight (1 – x)α(1 +x)β: ⎝integraldisplay1 –1(1 –x)α(1 +x)βP(α,β) n(x)Pα,β m(x)dx=⎧ ⎨ ⎩0i fn≠m, 2α+β+1 α+β+2n+1Γ(α+n+1 )Γ(β+n+1 ) n!Γ(α+β+n+1 )ifn=m. Forα>– 1a n d β> –1, all zeros of the polynomial P(α,β) n(x) are simple and lie on the interval –1 <x<1 . 11.17-5. Gegenbauer Polynomials. The Gegenbauer polynomials (also called ultraspherical polynomials ),C(λ) n(x), are solutions of the second-order linear ordinary differential equation (1 –x2)y/prime/prime xx–( 2λ+1 )xy/prime x+n(n+2λ)y=0 and are deﬁned by the formulas C(λ) n(x)=(–2)n n!Γ(n+λ)Γ(n+2λ) Γ(λ)Γ(2n+2λ)(1 –x2)–λ+1/2dn dxn(1 –x2)n+λ–1/2 =[n/2]⎝summationdisplay m=0(–1)mΓ(n–m+λ) Γ(λ)m!(n–2m)!(2x)n–2m. Recurrence formulas: C(λ) n+1(x)=2(n+λ) n+1xC(λ) n(x)–n+2λ–1 n+1C(λ) n–1(x); C(λ) n(–x) = (–1)nC(λ) n(x),d dxC(λ) n(x)=2λC(λ+1) n–1(x). The generating function: 1 (1 – 2xs+s2)λ=∞⎝summationdisplay n=0C(λ) n(x)sn. The Gegenbauer polynomials are orthogonal on the interval –1 ≤x≤1 with weight (1 – x2)λ–1/2: ⎝integraldisplay1 –1(1 –x2)λ–1/2C(λ) n(x)C(λ) m(x)dx=⎧ ⎨ ⎩0i fn≠m, πΓ(2λ+n) 22λ–1(λ+n)n!Γ2(λ)ifn=m. 1052 SPECIAL FUNCTIONS AND THEIR PROPERTIES 11.18. Nonorthogonal Polynomials 11.18-1. Bernoulli Polynomials. The Bernoulli polynomials Bn(x) are introduced by the formula Bn(x)=n⎝summationdisplay k=0Ck nBkxn–k(n=0 ,1 ,2 , ...), where Ck nare the binomial coefﬁcients and Bnare Bernoulli numbers (see Supplement 11.1-3). The Bernoulli polynomials can be deﬁned using the recurrence relation B0(x)=1 ,n–1⎝summationdisplay k=0Ck nBk(x)=nxn–1,n=2 ,3 , ... The ﬁrst six Bernoulli polynomials are given by B0(x)=1 , B1(x)=x–1 2,B2(x)=x2–x+1 6,B3(x)=x3–3 2x2+1 2x, B4(x)=x4–2x3+x2–1 30,B5(x)=x5–5 2x4+5 3x3–1 6x. Basic properties: Bn(x+1 )– Bn(x)=nxn–1,B/prime n+1(x)=(n+1 )Bn(x), Bn(1 –x)=( – 1 )nBn(x), (–1)nEn(–x)=En(x)+nxn–1, where the prime denotes a derivative with respect to x,a n dn=0 ,1 , ... Multiplication and addition formulas: Bn(mx)=mn–1m–1⎝summationdisplay k=0Bn⎝parenleftBig x+k m⎝parenrightBig , Bn(x+y)=n⎝summationdisplay k=0Ck nBk(x)yn–k, where n=0 ,1 , ...andm=1 ,2 , ... The generating function is expressed as text et–1≡∞⎝summationdisplay n=0Bn(x)tn n!(|t|<2π). This relation may be used as a deﬁnition of the Bernoulli polynomials. Fourier series expansions: Bn(x)=– 2n! (2π)n∞⎝summationdisplay k=1cos(2πkx –1 2πn) kn(n=1 , 0<x <1 ;n>1 ,0 ≤x≤1); B2n–1(x) = 2(–1)n(2n–1 ) ! (2π)2n–1∞⎝summationdisplay k=1sin(2kπx) k2n–1(n=1 , 0<x <1 ;n>1 ,0 ≤x≤1); B2n(x) = 2(–1)n(2n)! (2π)2n∞⎝summationdisplay k=1cos(2kπx) k2n(n=1 ,2 , ...,0≤x≤1). Integrals:⎝integraldisplayx aBn(t)dt=Bn+1(x)–Bn+1(a) n+1, ⎝integraldisplay1 0Bm(t)Bn(t)dt= (–1)n–1m!n! (m+n)!Bm+n, where mandnare positive integers and Bnare Bernoulli numbers. 11.18. N ONORTHOGONAL POLYNOMIALS 1053 11.18-2. Euler Polynomials. Deﬁnition: En(x)=n⎝summationdisplay k=0Ck nEk 2n⎝parenleftBig x–1 2⎝parenrightBign–k (n=0 ,1 ,2 , ...), where Ck nare the binomial coefﬁcients and Enare Euler numbers (see Supplement 11.1-4). The ﬁrst six Euler polynomials are given by E0(x)=1 , E1(x)=x–1 2,E2(x)=x2–x,E3(x)=x3–3 2x2+1 4, E4(x)=x4–2x3+x,E5(x)=x5–5 2x4+5 2x2–1 2. Basic properties: En(x+1 )+ En(x)=2xn,E/prime n+1=(n+1 )En(x), En(1 –x)=( – 1 )nEn(x), (–1)n+1En(–x)=En(x)–2xn, where the prime denotes a derivative with respect to x,a n dn=0 ,1 , ... Multiplication and addition formulas: En(mx)=mnm–1⎝summationdisplay k=0(–1)kEn⎝parenleftBig x+k m⎝parenrightBig ,n=0 ,1 , ...,m=1 ,3 , ...; En(mx)=–2 n+1mnm–1⎝summationdisplay k=0(–1)kEn+1⎝parenleftBig x+k m⎝parenrightBig ,n=0 ,1 , ...,m=2 ,4 , ...; En(x+y)=n⎝summationdisplay k=0Ck nEk(x)yn–k,n=0 ,1 , ... The generating function is expressed as 2ext et+1≡∞⎝summationdisplay n=0En(x)tn n!(|t|<π). This relation may be used as a deﬁnition of the Euler polynomials. Fourier series expansions: En(x)=4n! πn+1∞⎝summationdisplay k=0sin⎝parenleftbig (2k+1 )πx–1 2πn⎝parenrightbig (2k+1 )n+1(n=0 ,0<x <1 ;n>0 , 0 ≤x≤1); E2n(x)=4 ( – 1 )n(2n)! π2n+1∞⎝summationdisplay k=0sin⎝parenleftbig (2k+1 )πx⎝parenrightbig (2k+1 )2n+1(n=0 , 0<x <1 ;n>0 , 0 ≤x≤1); E2n–1(x)=4 ( – 1 )n(2n–1 ) ! π2n∞⎝summationdisplay k=0cos⎝parenleftbig (2k+1 )πx⎝parenrightbig (2k+1 )2n(n=1 ,2 , ...,0≤x≤1). Integrals: ⎝integraldisplayx aEn(t)dt=En+1(x)–En+1(a) n+1, ⎝integraldisplay1 0Em(t)En(t)dt= 4(–1)n(2m+n+2–1 )m!n! (m+n+2 ) !Bm+n+2, 1054 SPECIAL FUNCTIONS AND THEIR PROPERTIES where m,n=0 ,1 , ...andBnare Bernoulli numbers. The Euler polynomials are orthogonal for evenn+m. Connection with the Bernoulli polynomials: En–1(x)=2n n⎝bracketleftbigg Bn⎝parenleftBigx+1 2⎝parenrightBig –Bn⎝parenleftBigx 2⎝parenrightBig⎝bracketrightbigg =2 n⎝bracketleftbigg Bn(x)–2nBn⎝parenleftBigx 2⎝parenrightBig⎝bracketrightbigg , where n=1 ,2 , ... References for Supplement 11: H. Bateman and A. Erd ´elyi (1953, 1955), N. W. McLachlan (1955), M. Abramowitz and I. A. Stegun (1964), W. Magnus, F. Oberhettinger, and R. P. Soni (1966), H. Buchholz (1969), S. Yu. Slavyanov and W. Lay(2000), D. Zwillinger (2002), A. D. Polyanin and V . F. Zaitsev (2003), E. W. Weisstein (2003). Supplement 12 Some Notions of Functional Analysis 12.1. Functions of Bounded Variation 12.1-1. Deﬁnition of a Function of Bounded Variation. 1◦.L e tf(x) be a function deﬁned on a ﬁnite segment [ a,b]. Consider an arbitrary partition of the segment by the points a=x0<x1<x2<···<xn–1<xn=b and construct the sum v=n–1⎝summationdisplay k=0⎝vextendsingle⎝vextendsinglef(xk+1)–f(xk)⎝vextendsingle⎝vextendsingle (1) whose terms are absolute values of the increments of f(x) on each segment of the partition. If, for all partitions, the sums (1) are bounded by a constant independent of the partition, one says that the function f(x)h a s bounded variation on the segment [ a,b]. The supremum of all such sums over all partitions is called the total variation of the function f(x) on the segment [ a,b]. The total variation is denoted by bV af(x)=s u p {v}. A function f(x) is said to have bounded variation on the inﬁnite interval [ a,∞) if it is a function of bounded variation on any ﬁnite segment [ a,b] and its total variation on [ a,b] is bounded by a constant independent of b. By deﬁnition, ∞V af(x)=s u p b>a⎝braceleftBigbV af(x)⎝bracerightBig . 2◦. In the above deﬁnitions, the continuity of the function f(x) is not mentioned. A continuous function (without additional conditions) may have bounded or unbounded variation. Example. Consider the continuous function f(x)=⎝braceleftBigg xcosπ 2xifx≠0, 0i f x=0 and the partition of the segment [0, 1] by the points 0<1 2n<1 2n–1<···<1 3<1 2<1 . Then the sums (1) corresponding to this partition have the form vn=1+1 2+···+1 n→∞ asn→∞ . Therefore,1V 0f(x)=∞. 1055 1056 SOME NOTIONS OF FUNCTIONAL ANALYSIS 12.1-2. Classes of Functions of Bounded Variation. Next, we list some common classes of functions of bounded variation. 1. Any bounded monotone function has bounded variation. Its total variation on the segment [a,b]i sd e ﬁ n e db ybV af(x)=|f(b)–f(a)|. Remark. The last statement is true for inﬁnite intervals (– ∞,a]a n d[ a,∞); in the latter case, the total variation is equal to∞V af(x)=|f(∞)–f(a)|. 2. Suppose that f(x) is a bounded function on [ a,b] and this segment can be divided into ﬁnitely many parts [ak,ak+1](k=0 ,1 , ...,m–1 ; a0=a,am=b), so that the function f(x) is monotone on each part. Then f(x) has bounded variation on [ a,b]. Remark. This statement is also true for inﬁnite segments. 3. Let f(x) be a function on a ﬁnite segment [ a,b] satisfying the Lipschitz condition ⎝vextendsingle⎝vextendsinglef(x1)–f(x2)⎝vextendsingle⎝vextendsingle≤L|x1–x2|, for any x1andx2in [a,b], where Lis a constant. Then f(x) has bounded variation andbV af(x)≤ L(b–a). 4. Let f(x) be a function on a ﬁnite segment [ a,b] with a bounded derivative |f/prime(x)|≤L ,w h e r e L= const. Then, f(x) is of bounded variation andbV af(x)≤L(b–a). 5. Let f(x)b eaf u n c t i o no n[ a,b]o r[a,∞) and suppose that f(x) can be represented as an integral with variable upper limit, f(x)=c+⎝integraldisplayx aϕ(t)dt, where ϕ(t) is an absolutely continuous function on the interval under consideration. Then f(x)h a s bounded variation and bV af(x)=⎝integraldisplayb a|ϕ(x)|dx. Corollary . Suppose that ϕ(t) on a ﬁnite segment [ a,b]o r[a,∞) is integrable, but not absolutely integrable. Then the total variation of f(x)i si n ﬁ n i t e . 12.1-3. Properties of Functions of Bounded Variation. Here, all functions are considered on a ﬁnite segment [ a,b]. 1. Any function of bounded variation is bounded. 2. The sum, difference, or product of ﬁnitely many functions of bounded variation is a function of bounded variation. 3. Let f(x)a n dg(x) be two functions of bounded variation and |g(x)|≥K > 0. Then the ratio f(x)/g(x) is a function of bounded variation. 4. Let a<c<b.I ff(x) has bounded variation on the segment [ a,b], then it has bounded variation on each segment [ a,c]a n d[ c,b]; and the converse statement is true. In this case, the following additivity condition holds: bV af(x)=cV af(x)+bV cf(x). 12.2. S TIELTJES INTEGRAL 1057 5. Let f(x) be a function of bounded variation of the segment [ a,b]. Then, for a≤x≤b,t h e variation of f(x) with variable upper limit F(x)=xV af(x) is a monotonically increasing bounded function of x. 6. Any function f(x) of bounded variation on the segment [ a,b] has a left-hand limit lim x→x0–0f(x) and a right-hand limit lim x→x0+0f(x) at any point x0∈[a,b]. 12.1-4. Criteria for Functions to Have Bounded Variation. 1. A function f(x) has bounded variation on a ﬁnite segment [ a,b] if and only if there is a monotonically increasing bounded function Φ(x)s u c ht h a tf o ra l l x1,x2∈[a,b](x1<x2), the following inequality holds: |f(x2)–f(x1)|≤Φ(x2)–Φ(x1). 2. A function f(x) has bounded variation on a ﬁnite segment [a ,b] if and only if f(x) can be represented as the difference of two monotonically increasing bounded functions on that segment:f(x)=g 2(x)–g1(x). Remark. The above criteria are valid also for inﬁnite intervals (– ∞,a], [a,∞), and (–∞, ∞). 12.1-5. Properties of Continuous Functions of Bounded Variation. 1. Let f(x) be a function of bounded variation on the segment [a ,b]. Iff(x) is continuous at a pointx0(a<x0<b), then the function F(x)=xV af(x) is also continuous at that point. 2. A continuous function of bounded variation can be represented as the difference of two continuous increasing functions. 3. Let f(x) be a continuous function on the segment [ a,b]. Consider a partition of the segment a=x0<x1<x2<···<xn–1<xn=b and the sum v=n–1⎝summationtext k=0⎝vextendsingle⎝vextendsinglef(xk+1)–f(xk)⎝vextendsingle⎝vextendsingle. Letting λ=m a x |xk+1–xk|and passing to the limit as λ→0, we get lim λ→0v=bV af(x). 12.2. Stieltjes Integral 12.2-1. Basic Deﬁnitions. Letf(x)a n d ϕ(x) be functions deﬁned on an interval [ a,b]. Let us partition this interval into nelementary subintervals deﬁned by a set of points {x0,x1,...,xn}such that a=x0<x1<···< xn=b. Each subinterval [ xk–1,xk] will be characterized by its length ∆xk=xk–xk–1and an arbitrarily chosen point ξk∈[xk–1,xk]. Let us make up a Stieltjes integral sum sn=n⎝summationdisplay k=1f(ξk)∆kϕ(x), 1058 SOME NOTIONS OF FUNCTIONAL ANALYSIS where ∆kϕ(x)=ϕ(xk)–ϕ(xk–1) is the increment of the function ϕ(x)o nt h e kth elementary subinterval. If there exists a limit of the integral sums sn, as the number of subintervals nincreases indeﬁnitely so that the length of every subinterval ∆xkvanishes, and this limit depends on neither the way the interval [ a,b] was partitioned nor the way the points ξkwere selected, then this limit is called the Stieltjes integral of the function f(x) with respect to the function ϕ(x) over the interval [ a,b]: ⎝integraldisplayb af(x)dϕ(x) = lim λ→0sn⎝parenleftBig max 1≤k≤n∆xk→0a sn→∞⎝parenrightBig . Thenf(x)i sc a l l e da n integrable function with respect to ϕ(x), and ϕ(x) is called an integrating function . The Stieltjes integral is a generalization of the Riemann integral; the latter corresponds to the special case ϕ(x)=x+ const. 12.2-2. Properties of the Stieltjes Integral. The Stieltjes integral has properties analogous to those of the deﬁnite Riemann integral: 1)⎝integraldisplayb adϕ(x)=ϕ(b)–ϕ(a); 2)⎝integraldisplayb a⎝bracketleftbig Af(x)±Bg(x)⎝bracketrightbig dϕ(x)=A⎝integraldisplayb af(x)dϕ(x)±B⎝integraldisplayb ag(x)dϕ(x); 3)⎝integraldisplayb af(x)d[Aϕ(x)±Bψ(x)] =A⎝integraldisplayb af(x)dϕ(x)±B⎝integraldisplayb af(x)dψ(x); 4)⎝integraldisplayb af(x)dϕ(x)=⎝integraldisplayc af(x)dϕ(x)+⎝integraldisplayb cf(x)dϕ(x)(a<c<b). It is assumed that all integrals on the left- and right-hand sides exist. THEOREM (MEAN V ALUE ).If a function f(x)satisﬁes inequalities m≤f(x)≤Mon an interval [a,b]and is integrable with respect to an increasing function ϕ(x),t h e n ⎝integraldisplayb af(x)dϕ(x)=µ[ϕ(b)–ϕ(a)], where m<µ<M. 12.2-3. Existence Theorems for the Stieltjes Integral. The existence of the Stieltjes integral and its reduction to the Riemann integral is established by the following theorem. THEOREM 1.Iff(x)is continuous on [a,b]andϕ(x)has a bounded variation* on [a,b],t h e n the integral⎝integraldisplayb af(x)dϕ(x)exists. * A function ϕ(x)i ss a i dt oh a v ea bounded variation on an interval [ a,b] if there exists a number M> 0 such that for any set of points a=x0<x1<···<xn=bthe inequalityn⎝summationtext k=1|ϕ(xk+1)–ϕ(xk)|<Mholds (see also Supplement 12.1). 12.3. L EBESGUE INTEGRAL 1059 THEOREM 2.Letf(x)be integrable on [a,b]in the sense of Riemann and let ϕ(x)satisfy the Lipschitz condition |ϕ(x2)–ϕ(x1)|<K|x2–x1|, where x1andx2are arbitrary points of the interval [a,b]andKis a ﬁxed positive constant. Then the function f(x)is integrable with respect to the function ϕ(x). THEOREM 3.Letf(x)be integrable on [a,b]in the sense of Riemann and let ϕ(x)be differen- tiable and have an integrable derivative on [a,b]. Then the function f(x)is integrable with respect to the function ϕ(x)and, moreover, ⎝integraldisplayb af(x)dϕ(x)=⎝integraldisplayb af(x)ϕ/prime(x)dx, where the integral on the right-hand side is understood in the sense of Riemann. Remark. If a function f(x) is integrable on an interval [ a,b] with respect to a function ϕ(x), then, vice versa, the function ϕ(x) is also integrable with respect to the function f(x)o n[a,b]. Owing to this property, the functions f(x)a n dϕ(x) are interchangeable in Theorems 1 and 2. THEOREM 4.Letf(x)be continuous on [a,b]and let ϕ(x)have an absolutely integrable deriva- tiveϕ/prime(x)everywhere on [a,b], except, perhaps, ﬁnitely many points. Let, in addition, the function ϕ(x)undergo a jump discontinuity at ﬁnitely many points a=c0<c1<···<cm=b. Then the Stieltjes integral exists and is calculated as ⎝integraldisplayb af(x)dϕ(x)=⎝integraldisplayb af(x)ϕ/prime(x)dx+f(a)[ϕ(a+0 )– ϕ(a)] +m–1⎝summationdisplay k=1f(ck)[ϕ(ck+0 )– ϕ(ck–0 ) ]+ f(b)[ϕ(b)–ϕ(b– 0)], where the right-hand side contains a Riemann integral. Note the presence of terms outside the integral on the right-hand side, where, apart from the ordinary jumps of the function ϕ(x)at the internal points of discontinuity, there are terms with one-sided jumps at the endpoints (if there is no jump at either endpoint, the corresponding term vanishes). The Stieltjes integral is useful for ﬁnding static moments, moments of inertia, and some other distributed quantities on an interval [ a,b], where, apart from continuous distributions, there are concentrated quantities like point masses that correspond to a discontinuous function ϕ(x) with ﬁnite jumps. 12.3. Lebesgue Integral∗ 12.3-1. Riemann Integral and the Lebesgue Integral. The space C[a,b] of continuous functions on a ﬁnite interval [ a,b] is a metric space with the metric ρ(f,g)=⎝integraldisplayb a|f(x)–g(x)|dx, where the integral is understood in the sense of Riemann. It is well known that this metric space is incomplete, in the sense that there is a Cauchy sequence (with respect to this metric) that does not converge to any element of C[a,b]. One can consider a formal completion L[a,b] of the space C[a,b] in this metric. The space L[a,b] is wider than C[a,b] and the problem is to describe the structure of its elements. It turns out that L[a,b] consists of the so-called summable orLebesgue integrable functions. Below, we brieﬂy describe a version of the Lebesgue integration theory. * Supplement 12.3 was written by G. A. Yosiﬁan. 1060 SOME NOTIONS OF FUNCTIONAL ANALYSIS 12.3-2. Sets of Zero Measure. Notion of “Almost Everywhere”. Let [a,b] be a ﬁnite interval on the real axis x. As e tA⊂[a,b] is called a set of zero measure if for any εit can be covered by ﬁnitely many or countably many intervals whose joint length is less than ε. In particular, any ﬁnite or countable set of points on [ a,b] is a set of zero measure on [ a,b]. The union of ﬁnitely many (or countably many) sets of zero measure is a set of zero measure. As e tB⊂[a,b] is called a s e to ff u l lm e a s u r e on [a,b] if its complement [ a,b]\Bis a set of zero measure on [ a,b]. If some property holds for all points of a segment [ a,b] except points of some set of zero measure, one says that this property holds almost everywhere on [a,b], or holds for almost all x∈[a,b], or holds on a set of full measure. A function is said to be deﬁned almost everywhere on [a,b] if it is deﬁned at all points of [ a,b] except points forming a set of zero measure on [ a,b]. Letfn(x) be a sequence of functions deﬁned almost everywhere on [ a,b]. One says that the sequence fn(x)converges to a function f(x)almost everywhere on [a,b]a sn→∞ if there is pointwise convergence fn(x)→f(x) for almost all x∈[a,b]; in other words, if there is pointwise convergence on a set of full measure. 12.3-3. Step Functions and Measurable Functions. Apartition of a segment [ a,b] is a system of intervals ( xi,xi+1),i=0 , 1 , ...k, such that a=x0< x1<···<xk=b. Astep function on [a,b] is a function that takes a constant value on every interval ( xi,xi+1)o f some partition of [ a,b]. Ameasurable function f(x)o n[a,b] is a function that is deﬁned and ﬁnite almost everywhere on [a,b] and can be represented as the pointwise limit (almost everywhere) of a sequence of step functions; in other words, there is a sequence of step functions fn(x)s u c ht h a t fn(x)c o n v e r g e st o f(x)a l m o s te v e r y w h e r eo n[ a,b]a sn→∞ . Since measurable functions are deﬁned almost eve rywhere, two such functions are identiﬁed if they coincide on a set of full measure. Obviously, any step function is measurable. Many properties of step functions can be transferred to measurable functions. In particular: (i) All step functions on [ a,b] form a linear space, i.e., if f,gare step functions, then their linear combination αf+βgis a step function. It follows that all measurable functions on [ a,b]f o r ma linear space. (ii) The product of two step functions is a step function, and accordingly, the product of two measurable functions is a measurable function. (iii) The ratio of two step functions is a step function, provided that the denominator is different from zero. The ratio of two measurable functions is a measurable function, provided that thedenominator differs from zero almost everywhere on [ a,b]. (iv) The absolute value |h(x)|of a step function h(x) is a step function. The absolute value of any measurable function is also a measurable function. (v) Let f(x),g(x) be measurable functions, then the functions h 1(x)=m a x {f(x),g(x)},h2(x)=m i n {f(x),g(x)} are measurable. In particular, for any measurable function f(x), the functions f+(x)=m a x {f(x), 0},f–(x)=m a x {0, –f (x)} are measurable. The functions f+andf–are called the positive part and the negative part off, respectively. Any continuous function on [ a,b] (or even a piecewise continuous function) is measurable. 12.3. L EBESGUE INTEGRAL 1061 12.3-4. Deﬁnition and Properties of the Lebesgue Integral. Leth(x) be a step function on the interval [a ,b] taking constant values h1,...,hkon mutually disjoint segments ∆1,...,∆kinto which [ a,b] is divided by points a=x0<x1<···<xk=b.T h e integral of such a step function h(x)i sd e ﬁ n e db y Ih=⎝integraldisplay [a,b]h(x)dx=k⎝summationdisplay j=1hj|∆j|, where |∆j|is the length of the interval ∆j. For a sequence of function gn(x)o n[a,b], we write gn/arrownortheastgifgnconverge to a function galmost everywhere on [ a,b] and the numerical sequence gn(x) is monotonically increasing for almost all x∈[a,b]. DEFINITION 1.A function f(x)on[a,b]is said to belong to the classL+if it can be represented as the limit (in the sense of convergence almost everywhere) of a monotonically increasing sequence of step functions hn/arrownortheastfand the integrals of these step functions are bounded by the same constant: Ihn≤C. Any function of class L+is measurable. Continuous functions belong to L+. The integral off∈L+is deﬁned by the formula If= lim n→∞Ihn, where hn/arrownortheastfis the sequence from Deﬁnition 1 of the class L+. The value Ifforf∈L+does not depend on the sequence of step functions hn/arrownortheastf. DEFINITION 2.A function φ(x)on[a,b]is called summable orLebesgue integrable on[a,b], (or simply, integrable ) if it can be represented in the form φ=f–g,for some f,g∈L+. The set of all summable functions is denoted by L. Properties of summable functions: (i) iff,g∈L, then any linear combination αg+βgbelongs to L;i no t h e rw o r d s , Lis a linear space; (ii) iff∈L,t h e n |f|∈L; (iii) if f,g∈Landh1(x)=m a x {f(x),g(x)},h2(x)=m i n {f(x),g(x)},t h e nh1,h2∈L. DEFINITION 3.The integral of a summable function φ∈Lis deﬁned by Iφ=If–Ig,where φ=f–g,f,g∈L+. The value Iφdoes not depend on the representation φ=f–g. Properties of the integral of summable functions: (i)I(φ1+φ2)=Iφ1+Iφ2for any φ1,φ2∈L; (ii)I(αφ)=αIφ for any φ∈Land any scalar α; (iii) if f,g∈Landf(x)≥g(x) almost everywhere, then If≥Ig. 1062 SOME NOTIONS OF FUNCTIONAL ANALYSIS THEOREM 1.Any Riemann integrable function on [a,b](in particular, any continuous function on[a,b]) is Lebesgue integrable, and its Riemann integral coincides with its Lebesgue integral. For a sequence φn∈Lsuch that φn→φalmost everywhere, it cannot be claimed, in general, thatIφn→Iφ. For example, consider the sequence φn(x)=⎝braceleftbiggnsinnx for 0 ≤x≤π n, 0f o rπ n≤x≤π. It is easy to verify that φn(x)→0f o ra n y x∈[0,π], butIφn=2 . An important result with regard to integrating pointwise convergent sequences is the following theorem. THEOREM 2( L EBESGUE THEOREM ON DOMINATED CONVERGENCE ).Letφnbe a sequence of summable functions that converges to a function φalmost everywhere a nd satisﬁes the condition |φn(x)|≤φ0(x)∈L. Thenφis a summable function and Iφ= lim n→∞Iφn. In particular, Iφ= lim n→∞Iφnif the functions φn are uniformly bounded. Some important properties of measurable and summable functions: (i) Ifφis a measurable function that satisﬁes (almost everywhere) the inequality –φ0≤φ≤φ0∈L. Thenφ∈L. (ii) The limit of a sequence of measurable functions that converges almost everywhere to a ﬁnite limit is a measurable function. (iii) (Fatou lemma.) If φn≥0 is a sequence of summable functions, φn→φalmost everywhere, andIφn≤C,t h e nφis a summable function and 0 ≤Iφ≤C. (iv) If φ0(x)≥0 is a summable function such that Iφ0=0 ,t h e n φ0= 0 almost everywhere. THEOREM 3( F ISCHER –RIESZ).The space Lendowed with the norm /bardblφ/bardbl=I(|φ|) is a Banach space. THEOREM 4.The space Lis the completion of the space C[a,b]with respect to the norm /bardblf/bardbl=⎝integraldisplayb a|f(x)|dx. In other words, continuous functions form a dense set in L. 12.3-5. Measurable Sets. As e tA⊂[a,b] is called measurable if its characteristic function χA(x)=⎝braceleftbigg1f o r x∈A, 0f o r x∈[a,b]\A is measurable. 12.3. L EBESGUE INTEGRAL 1063 The integral of the characteristic function of a measurable set A⊂[a,b] is called the measure ofAand is denoted by µ(A), i.e., µ(A)=⎝integraldisplay [a,b]χA(x)dx. In particular, for a set Bof zero measure, we have µ(B)=0 . Measurable sets have the following properties: (i) the union A=⎝uniontextAjof ﬁnitely many or countably many measurable sets A1,...,An,...is a measurable set; moreover, if the sets Ajare mutually disjoint, i.e., Aj∩Ai=∅for all i≠j,t h e n µ(A)=µ(A1)+···+µ(An)+···; (ii) the intersection A=⎝intersectiontextAjof ﬁnitely many or countably many measurable sets A1,...,An,... is a measurable set; (iii) the difference A=B\Cof measurable sets B,Cis a measurable set, in particular, the complement of B,i . e . ,[ a,b]\B, is a measurable set; (iv) any interval [α ,β], (α,β], (α,β), [α,β) is a measurable set and its measure is equal to its length β–α; (v) any open and any closed set on [ a,b] is measurable. 12.3-6. Integration Over Measurable Sets. So far, the domain of integration has been the interval [ a,b]. It is easy to extend the notion of integral to any measurable set E⊂[a,b]. A function φis called summable (orintegrable )o n a measurable set Eif the function χE(x)φ(x) is summable on [ a,b], where χEis the characteristic function of E.T h e integral ofφoverEis deﬁned by⎝integraldisplay Eφdx =⎝integraldisplay [a,b]χE(x)φ(x)dx=I(χEφ). This integral has the following additive property :i fφis summable on a set E=E1∪E2∪··· , where E1,E2,...are mutually disjoint measurable sets, then φis summable on each Ejand ⎝integraldisplay Eφdx =⎝integraldisplay E1φdx +⎝integraldisplay E2φdx +···. 12.3-7. Case of an Inﬁnite Interval. The above considerations pertain to functions deﬁned on a ﬁnite interval [ a,b]. It is not very difﬁcult to extend the above theory to the cases of intervals [ a,∞), –(∞,b], or (–∞,∞). In all these cases, a step function is deﬁned as a function taking constant values on ﬁnitely many ﬁnite intervals ∆j=(xj,xj+1)(xj<xj+1) and on the rest of the inﬁnite interval, it is supposed to be equal to zero. A measurable function is a function φ(x) that is the limit (almost everywhere on every ﬁnite segment) of a sequence of step functions. The integral of a step function h(x)t a k i n g values hjon an interval ∆jof length |∆j|(j=1 ,...,k) is naturally deﬁned by the formula Ih=k⎝summationdisplay j=1hj|∆j|. The classL+consists of all functions f(x) that can be represented as the limit of an increasing sequence of step functions fn(x) with bounded integrals. The classLis deﬁned as the set of differences φ=f–g,f,g∈L+. The results formulated above for a ﬁnite interval can be easily extended to the case inﬁnite intervals. 1064 SOME NOTIONS OF FUNCTIONAL ANALYSIS 12.3-8. Case of Several Variables. We limit ourselves to functions of two variables φ(x,y) deﬁned on a rectangle D={a1≤x≤b1,a2≤ y≤b2}. As e tA⊂Dis called a set of zero measure inDif for any εthe set Dcan be covered by a ﬁnite or countable system of rectangles Dj=⎝braceleftbig a(j) 1≤x≤b(j) 1,a(j) 2≤y≤b(j) 2⎝bracerightbig whose joint area does not exceed ε. A partition of Dis a system of mutually disjoint open rectangles D1,...,Dk⊂Dsuch that D= D1∪···∪ Dk,w h e r e Djis the closure of Dj. Astep function onDis a function that takes constant values on each rectangle Djof some partition of D,D= D1∪···∪ Dk. The integral of a step function h(x) with values hjon the rectangles Djof some partition is deﬁned as Ih=k⎝summationdisplay j=1hj|Dj|, where |Dj|is the area the rectangle of Dj. As in the one-dimensional case, the classL+is the set of functions fsuch that fis a limit (almost everywhere on D) of a sequence of step functions fnwith uniformly bounded integrals. The classLof summable functions is again deﬁned as the set of differences φ=f–g,f,g∈L+. The properties formulated above for the one-dimensional case are obviously modiﬁed in the case of two dimensions. However, in the two-dimensional case, there is the question of the reduction ofan integral over a two-dimensional domain Dto a double integral over linear segments, and also the question of changing the order of double integration. The answers to these questions are given by the following theorem. T HEOREM 5( F UBINI THEOREM ).Letφ(x,y)be a summable function on a rectangle D={a1≤ x≤b1,a2≤y≤b2}. Then: (i)regarded as a function of the argument xfor a ﬁxed y, this function is integrable in xfor almost ally; (ii)its integral over the interval a1≤x≤b1, denoted by Ixφ(x,y), is a summable function of yon the interval a2≤y≤b2; (iii) the integral over Dcan be reduced to a double integral in which the order of integration can be changed: Iφ=Iy{Ixφ(x,y)}=Ix{Iyφ(x,y)}. As in the one-dimensional case, a set G⊂Dis called measurable if its characteristic function χG(x,y) is measurable and the integral of φoverGis deﬁned by the formula ⎝integraldisplay Gφd xd y =⎝integraldisplay DχG(x,y)φ(x,y)dx dy . 12.3-9. Spaces Lp. For a measurable set Gandp>0 ,t h e classLp(G) consists of all measurable functions f(x)o nG for which |f|pis summable on G, i.e., ⎝integraldisplay G|f|pdx<∞. For any p> 0, this class of functions is a linear space. 12.4. L INEAR NORMED SPACES 1065 Forp≥1, the class Lp(G) is a Banach space (complete normed space) with the norm /bardblf/bardblp=⎝parenleftbigg⎝integraldisplay G|f|p⎝parenrightbigg1/p . The set of continuous functions is dense in the Banach space Lp(G), i.e., for any f∈Lp,t h e r ei sa sequence of continuous functions fnsuch that /bardblf–fn/bardblp→0a sn→∞ . Letp>1 ,q> 1 be real numbers such that p–1+q–1=1 .F o r f∈Lp,g∈Lq, the product fgis summable on Gand the H¨older inequality holds: ⎝integraldisplay Gfgdx ≤/bardblf/bardblp/bardblg/bardblq. 12.4. Linear Normed Spaces 12.4-1. Linear Spaces. Alinear space or a vector space Lover the ﬁeld of real or complex numbers (called the ﬁeld of scalars ) is a nonempty set of elements (also called vectors ) for which two operations are deﬁned: addition of elements and their multiplication by scalars. To be more precise: for any two elements x,y∈L, there is a unique element z∈L, called their sum and denoted by z=x+y∈L,a n df o r any scalar α(real or complex) and any element x∈Lthere is a unique element y, called the product ofαandxand denoted by y=αx, so that for these two operations the following axioms hold: I. Axioms for addition of vectors:1)x+y=y+x(commutative property); 2)x+(y+z)=(x+y)+z(associative property); 3) there is an element 0 ∈Lsuch that x+0=xfor all x∈L(existence of zero); 4) for any x∈L, equation x+y= 0 is solvable; the element yis called the opposite ofxand is denoted by – x,s ot h a t x+(–x)= 0 (existence of an opposite element). II. Axioms relating addition of vectors with their multiplication by scalars: 5)α(βx)=(αβ)xfor any vector x∈Land any scalars α,β; 6) 1 ⋅x=xfor any x∈L; 7) (α+β)x=αx+βxfor any x∈Land any scalars α,β; 8)α(x+y)=αx+βyfor any scalar αand any vectors x,y∈L. If the ﬁeld of scalars is the set of real numbers, then Lis called a real linear space .I ft h e ﬁ e l d of scalars is the set of all complex numbers, then Lis called a complex linear space . Elements (vectors) y 1,y2,...,ynof a linear space Lare called linearly dependent if there exist scalar coefﬁcients α1,α2,...,αnsuch that at least one of them is different from zero and α1y1+α2y2+···+αnyn= 0. Otherwise, vectors y1,y2,...,ynare called linearly independent . A nonempty subset ¯Lof a linear space Lis called its subspace if for any x,y∈¯Land any scalars α,β,w eh a v e αx+βy∈¯L. 12.4-2. Linear Normed Spaces. A linear space Lis called a normed space if any element y∈Lis associated with a real number /bardbly/bardbl≥0, called the norm ofy, so that the following properties (axioms of a linear normed space) hold: 1)/bardbly/bardbl= 0 if and only if y=0 ; 2)/bardblλy/bardbl=|λ|/bardbly/bardblfor any scalar λ(homogeneity of the norm); 3)/bardbly1+y2/bardbl≤/bardbly1/bardbl+/bardbly2/bardbl(triangle inequality). A sequence {yn}of elements of a normed space Lis called convergent to an element y0if /bardbly0–yn/bardbl→ 0a sn→∞ . 1066 SOME NOTIONS OF FUNCTIONAL ANALYSIS 12.4-3. Space of Continuous Functions C(a,b). The linear normed space C(a,b) consists of all continuous functions y(x) on the interval [ a,b], with the norm deﬁned by /bardbly/bardbl=m a x a≤x≤b|y(x)|. The distance between two functions in this space has the form ρ(y1,y2)= m a x a≤x≤b|y1(x)–y2(x)|. The convergence of a sequence of functions {yn}in the space C(a,b)t oa ne l e m e n t y0(x) means uniform convergence of the functions yn(x)t oy0(x). 12.4-4. Lebesgue Space Lp(a,b). The linear normed space Lp(a,b)(p≥1) consists of all measurable functions y(x)o n(a,b)s u c h that|y(x)|pis integrable (has ﬁnite integral) on [ a,b], and the norm in Lp(a,b)i sd e ﬁ n e db y /bardbly/bardbl=⎝parenleftbigg⎝integraldisplayb a|y(x)|pdx⎝parenrightbigg1/p . Convergence yn→y0inLp(a,b) means that ⎝integraldisplayb a|yn(x)–y0(x)|pdx→0. Remark 1. Functions y1(x)a n d y2(x)i nLp(a,b) that coincide almost ev erywhere (i.e., may differ only on a set of zero measure) are identiﬁed. Remark 2. With regard to the space L2(a,b), see also Subsection 9.1-1. 12.4-5. H ¨older Space Cα(0, 1). The normed linear space Cα(0, 1) is the set of all functions y(x) deﬁned on the interval [0, 1] and satisfying the H¨older condition with exponent α(0 <α≤1): |y(x1)–y(x2)|≤A|x1–x2|α(0≤x1,x2≤1). The norm of a function y(x)i nCα(0, 1) is introduced by the formula /bardbly/bardbl=|y(0)|+s u p 0≤x1,x2≤1|y(x1)–y(x2)| |x1–x2|α. 12.4-6. Space of Functions of Bounded Variation V(0, 1). The normed linear space V(0, 1) is the set of all functions of bounded variation (see Supplement 12.1) on the interval [0, 1]. The norm of y(x)i nV(0, 1) is introduced by /bardbly/bardbl=|y(0)|+1V 0y(x). 12.5. E UCLIDEAN AND HILBERT SPACES .LINEAR OPERATORS IN HILBERT SPACES 1067 12.5. Euclidean and Hilbert Spaces. Linear Operators in Hilbert Spaces 12.5-1. Preliminary Remarks. The mathematical concept of a Hilbert space generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to inﬁnite-dimensional spaces. In more formal terms, a Hilbert space is an inner product space—an abstract vector space in which distances and angles can be measured—which is “complete,” meaning that if a sequence of vectors approaches a limit, then that limit is guaranteed to be in the space as well. Geometric intuition plays an important role in many aspects of Hilbert space theory. An element of a Hilbert space can be uniquely speciﬁed by its coordinates with respect to an orthonormal basis, in analogy with cartesian coordinates in the plane. This means that Hilbert space can also usefully be thought of in terms of inﬁnite sequences that are square-summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases , they are simply transformations that stretch the space by different factors in mutually perpendicular directions. 12.5-2. Euclidean and Hilbert Spaces. AEuclidean space Eis a (real or complex) linear space endowed with a scalar product {x,y}/mapsto→ (x,y), i.e., a mapping of E×Einto the ﬁeld of real or complex numbers satisfying the following conditions: (x,y)= (y,x)f o r a l l x,y∈E, (x+y,z)=(x,z)+(y,z)f o r a l l x,y,z∈E, (λx,y)=λ(x,y)f o r a l l x,y∈Eand all (real or complex) λ, (x,x)≥0f o r a l l x∈E,a n d x=0⇐⇒ (x,x)=0 . Here the bar over a complex number denotes its complex conjugate. For a Euclidean space E, the formula /bardblx/bardbl=⎝radicalbig (x,x),x∈E, deﬁnes a norm on E. Therefore, any Euclidean space can be regarded as a normed space. Vectors x,y∈Eare called orthogonal if (x,y) = 0. A set of nonzero vectors {ei,i∈I}⊂E(hereIis a set of indices) is called an orthogonal system in Eifeiandejare orthogonal for all i≠j,i,j∈I.A n orthogonal system {ei,i∈I}is called an orthonormal system if/bardblei/bardbl=1f o ra n y i∈I. A system of vectors {ei,i∈I},ei∈E, is called complete if any x∈Ecan be approximated in the norm of E(with any given accuracy) by ﬁnite linear combinations of the vectors ei, i.e., for any ε>0t h e r e is a ﬁnite linear combination⎝summationtext icieisuch that⎝vextenddouble⎝vextenddouble⎝vextenddoublex–⎝summationtext iciei⎝vextenddouble⎝vextenddouble⎝vextenddouble<ε. A normed linear space is called a complete space or a Banach space if the Cauchy criterion holds for that space, namely, for any sequence {x n,n∈N},xn∈G(here Nis the set of all positive integers) the following conditions are equivalent: a) there exists an x0∈Gsuch that lim n→∞/bardblxn–x0/bardbl=0 ; b) for any ε> 0, there exists an N∈Nsuch that /bardblxn–xm/bardbl<εfor all m,n>N. A complete Euclidean space is called a Hilbert space . An orthogonal system {ei,i∈I}in a Hilbert space Eis complete if and only if the only vector in Eorthogonal to every vector of the system {ei,i∈I}is the zero-vector. For a closed linear subspace Lin a Hilbert space E,t h e symbol L⊥denotes the set of all vectors y∈Esuch that ( x,y)=0f o ra l l x∈L.T h e s e t L⊥is a closed linear subspace of Ecalled the orthogonal complement ofL. Any vector x∈Ecan be uniquely represented as a sum x=y+z,w h e r e y∈Landz∈L⊥. In particular, the orthogonal complement of L⊥coincides with L. 1068 SOME NOTIONS OF FUNCTIONAL ANALYSIS THEOREM 1.Any closed subspace of a Hilbert space is either ﬁnite-dimensional or is itself a Hilbert space. A Hilbert space His said to be represented as a direct sum of its orthogonal subspaces M1,M2,...,Mn, H=M1⊕M2⊕···⊕Mn, if for any f∈Hthere exist h1∈M1,...,hn∈Mnsuch that f=h1+···+hn, and any element ofMiis orthogonal to any element of Mkfori≠k. THEOREM 2.Any element f∈Hcan be uniquely represented in the form f=h1+h2+···+hn, where hj∈Mj. COROLLARY .If{ϕi n}are complete orthonormal systems in the subspaces Mi, then the union of all{ϕi n}is a complete orthonormal system in H. 12.5-3. Linear Operators in Hilbert Spaces. Given two linear spaces LandL1any mapping y=Ax(x∈L,y∈L1) of subset of L(possibly Litself) int L1is called operator (from LtoL1). The operator Ais said to belinear if A(αx+βy)=αAx+βAy. LetDAbe the set of all x∈Lfor which Ais deﬁned. Then DAis called the domain (of deﬁnition ) of operator A. Although in general DAneed not equal L, we will always assume that DAis a linear subspace of L, i.e., that x,y∈DAimplies αx+βy∈DAfor all αandβ. The operator Ais said to be continuous at the point x0∈Dif, given any neighborhood Vof the pointy0=Ax0, there is a neighborhood Uof the point x0such that Ax∈Vfor all x∈U∩DA. We say that the operator Aiscontinuous if it is continuous at every point x0∈DA. Suppose LandL1are normed linear spaces. Then it is easy to see that Ais continuous if and only if, given any ε>0 ,t h e r ei sa δ> 0 such that /bardblx–y/bardbl<δ(x,y∈DA) implies /bardblAx–Ay/bardbl<ε. Given a bounded linear operator mapping a normed linear space Linto another linear space L1, the number /bardblA/bardbl=s u p /bardblx/bardbl<1/bardblAx/bardbl, equal to the least upper bound of /bardblAx/bardblon the closed unit sphere /bardblx/bardbl< 1, is called the norm ofA. The norm /bardblA/bardblhas the following properties: /bardblA/bardbl=s u p x≠0/bardblAx/bardbl /bardblx/bardbl, /bardblAx/bardbl≤/bardblA/bardbl/bardblx/bardblfor all x∈L. An eigenvalue of a linear operator AinHis deﬁned as a scalar µfor which there is x≠0 such that Ax=µx. The element xin this relation is called an eigenvector or an eigenfunction ofA corresponding to the eigenvalue µ. The set of all eigenvalues of Ais called spectrum ofA,a n da l l other values of µare said to be regular (points ). 12.5. E UCLIDEAN AND HILBERT SPACES .LINEAR OPERATORS IN HILBERT SPACES 1069 IfHis Hilbert space, then by the adjoint of an operator Amapping HintoH, we mean the operator A∗deﬁned by (A x,y)=(x,A∗y)f o ra l l x,y∈H. A bounded linear operator Amapping a Hilbert space Hinto itself is said to be self-adjoint if A=A∗, i.e., if ( Ax,y)=(x,Ay)f o ra l l x,y∈H. An operator AinHispositive deﬁnite if for all nonzero x,(Ax,x)>0 . LetAbe a linear operator mapping a Hilbert space Hinto itself. Then Ais completely continuous if and only if: 1)Amaps every relatively compact set in the weak topology into a relatively compact set in the strong topology; 2)Amaps every weakly convergent sequence into a strongly convergent sequence. THEOREM 3.All eigenvalues of a self-adjoint operator in Hare real, and eigenvectors corre- sponding to different eigenvalues are orthogonal. THEOREM 4.The set of all eigenvalues of a compact operator in His no more than countable. Zero is the only possible limit point of this set. THEOREM 5.All eigenvalues of a compact self-adjoint positive deﬁnite operator in Hare positive. THEOREM 6( H ILBERT –SCHMIDT ).Let Abe a compact self-adjoint linear operator in a Hilbert spaceH. Then there is an orthonormal system of eigenvectors {φn}corresponding to eigenvalues {µn}(µn≠0)such that each element ξ∈Hcan be uniquely represented in the form ξ=⎝summationdisplay kckφk+ξ/prime, where ξ/prime∈Ker A, i.e., Aξ/prime=0.M o r e o v e r , Aξ=⎝summationdisplay kµkckφk, and if the system {φn}is inﬁnite, then lim n→∞µn=0. COROLLARY .If zero is not an eigenvalue of the operator A, then the system {φn}is complete inH. In particular, for a compact self-adjoint positi ve deﬁnite operator, this system forms a basis inH. Suppose that a Hilbert space His represented as a direct sum of its two orthogonal closed subspaces: H=H1⊕H2. Thus each element h∈Hcan be uniquely represented in the form h=h1+h2(hi∈Hi,i=1 ,2 ) . An operator Pi:H→Hideﬁned by the relation Pih=hiis called the orthogonal projector ofHontoHi(i= 1, 2). Obviously P2=I–P1,w h e r e Iis the identity operator. Any orthogonal projector is a linear continuous self-adjoint operator in H. Orthogonal projectors have the following properties: Pihi=hi,P1h2=P2h1=0 , P1P2h=0 ,/bardblPi/bardbl=1 . THEOREM 7.In a Hilbert space Han operator of orthogonal projection onto a subspace is compact if and only if this subspace has a ﬁnite dimension. THEOREM 8.A linear operator PonHis an orthogonal projector if and only if Pis self-adjoint and satisﬁes the condition P(Px)=Pxfor any x∈H(i.e., P2=P). THEOREM 9.A linear combination of compact operators is a compact operator. THEOREM 10.IfAis a compact operator and Bis a bounded linear operator, then the operators ABand BAare compact. References for Supplement 12: L. V . Kantorovich and G. P. Akilov (1964), R. Edwards (1965), M. G. Krein (1972), M. Reed and B. Simon (1972), W. Rudin (1973), K. 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V ., Integral Equation of the First Kind , World Scientiﬁc, Singapore, 1995. Bochner, S., Lectures on F ourier Integrals , Princeton Univ. Press, Princeton, 1959. Bochner, S. and Chandrasekharan, K. C., F ourier Transforms , Princeton Univ. Press, Princeton, 1949. Bracewell, R., The F ourier Transform and Its Applications, 3rd Edition , McGraw-Hill, New York, 1999. Brakhage, H., Nickel, K., and Rieder, P ., Auﬂ¨osung der Abelschen Integralgleichung 2. Art, ZAMP , V ol. 16, Fasc. 2, S. 295–298, 1965. Bronshtein, I. N. and Semendyayev, K. A., Handbook of Mathematics, 4th Edition , Springer- Verlag, Berlin, 2004. Brunner, H., Collocation Methods for V olterra Integral and Related Functional Differential Equations , Cambridge University Press, Cambridge, 2004. Brychkov, Yu. A. and Prudnikov, A. P ., Integral Transforms, In: Mathematical Encyclopedia, V ol. 2, pp. 589–590 [in Russian], Sovetskaya Entsiklopediya, Moscow, 1979. Brychkov, Yu. A. and Prudnikov, A. 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B., T ables of Integrals and Other Mathematical Data , Macmillan, New York, 1961. Dzhuraev, A., Methods of Singular Integral Equations , J. Wiley, New York, 1992. Edwards, C. H. and Penney, D., Calculus, 6th Edition , Pearson Education, Toronto, 2002. Erdogan, F . E., Complex Function T echnique, In: Continuum Physics (Ed. A. C. Eringen), V ol. 2, pp. 523–603, Academic Press, New York, 1975. Erdogan, F . E., Gupta, G. D., and Cook, T. S., The Numerical Solutions of Singular Integral Equations, Mechanics of Fracture. V ol. 1. Methods of Analysis and Solutions of Crack Problems , pp. 368–425, Noordhoff Intern. Publ., Leyden, 1973. Estrada, R. and Kanwal, R. P ., Singular Integral Equations ,B i r k h ¨auser, Boston, 1999. Feny ¨o, S. and Stolle H. W., Theorie und Praxis der Linearen Integralgleichungen, Bd. 3 ,B i r k h ¨auser Verlag, Basel, 1984. Fock, V . A., Some integral equations of mathematical physics, Doklady AN SSSR, V ol. 26, No. 4–5, pp. 147–151, 1942. Frank-Kamenetskii, D. A., Diffusion and Heat Transfer in Chemical Kinetics [in Russian], Nauka, Moscow, 1987. Gakhov, F. D., Boundary V alue Problems [in Russian], Nauka, Moscow, 1977. Gakhov, F. D., Boundary V alue Problems , Dover Publ., New York, 1990. 1074 REFERENCES Gakhov, F . D. and Cherskii, Yu. I., Equations of Convolution Type [in Russian], Nauka, Moscow, 1978. Gantmakher, F . R., Non-symmetric Kellogg Kernels [in Russian], Doklady RAN, V ol. 1, No. 3, 1936. Gantmakher, F . R. and Krein, M. G., Oscillating Matrices and Small Vibrations of Mechanical Systems [in Russian], Gostekhizdat, Moscow, 1950. Gohberg, I. C. and Krein, M. G., The Theory of V olterra Operators in a Hilbert Space and Its Applications [in Russian], Nauka, Moscow, 1967. Golberg, A. (Editor), Numerical Solution of Integral Equations , Plenum Press, New York, 1990. Gorenﬂo, R. and Vessella, S., Abel Integral Equations: Analysis and Applications , Springer-Verlag, Berlin–New York, 1991. Goursat, E., Cours d’Analyse Math ´ematique, III,3me´ed., Gauthier–Villars, Paris, 1923. Gradshteyn, I. S. and Ryzhik, I. M., T ables of Integrals, Series, and Products, 6th Edition , Academic Press, New York, 2000. Grifﬁth, J. L., On the Hankel J-,YandHtransforms, Proc. Amer . Math. Soc., Vo l . 9 , N o . 5 , pp. 738–741, 1958. Gripenberg, G., Londen, S.-O., and Staffans, O., V olterra Integral and Functional Equations , Cambridge Univ. Press, Cambridge–New York, 1990. Guenther, R. B. and Lee, J. W., Partial Differential Equations of Mathematical Physics and Integral Equations , Dover Publications, New York, 1996. Gupalo, Yu. P ., Polyanin, A. D., and Ryazantsev, Yu. S., Mass and Heat Transfer of Reacting Particles with the Flow [in Russian], Nauka, Moscow, 1985. Hackbusch, W., Integral Equations: Theory and Numerical Treatment ,B i r k h ¨auser Verlag, Boston, 1995. Hansen, E. R., A T able of Series and Products , Prentice Hall, Englewood Cliffs, London, 1975. Hirschman, I. I. and Widder, D. V ., The Convolution Transform , Princeton Univ. Press, Princeton, New Jersey, 1955. Iovane, G., Lifanov, I.K., and Sumbatyan, M.A., On direct numerical treat ment of hypersingular integral equations arising in mechanics and acoustics, arXiv:math-ph/0301034 v1 26 Jan 2003, pp. 1–19, 2003. Jentzch, R., ¨Uber Integralgleichungen mit positivem Kern, J. Math., V ol. 141, p. 235, 1912. Jerry, A. J., Introduction to Integral Equations With Applications , Marcel Dekker, New York–Basel, 1985. Joseph, D. D., Stability of Fluid Motions , Springer-Verlag, Berlin, 1976. Kalandiya, A. I., Mathematical Methods of Two-Dimensional Elasticity, Mir Publ., Moscow, 1973. Kamke, E., Differentialgleichungen: L ¨osungsmethoden und L ¨osungen, Bd. 1, B. G. Teubner, Leipzig, 1977. Kantorovich, L. V . and Akilov, G. P., Functional Analysis in Normed Spaces , Macmillan, New York, 1964. Kantorovich, L. V . and Krylov, V . I., Approximate Methods of Higher Analysis , Interscience Publ., New York, 1958. Kanwal, R. P ., Linear Integral Equations ,B i r k h ¨auser Verlag, Boston, 1996. Karlin, S., T otal Positivity, V ol. 1 , Stanford Univ. Press, Stanford, 1968. Karon, J. M., The sign-regularity properties of a class of Green’s functions for ordinary differential equations, J. Diff. Equations , V ol. 6, p. 484, 1969. Kellogg, O. D., Orthogonal functions sets arising from integral equations, Amer . J. Math., V ol. 40, p. 145, 1918. Kellogg, O. D., The oscillation on functions of an orthogonal set, Amer . J. Math., V ol. 38, p. 1, 1916. Kline, M., Calculus: An Intuitive and Physical Approach, 2nd Edition , Dover Publications, New York, 1998. REFERENCES 1075 Kolmogorov, A. N. and Fomin, S. V ., Elements of the Theory of Functions and Functional Analysis , Dover Publications, New York, 1999. Kolmogorov, A. N. and Fomin, S. V ., Introductory Real Analysis , Prentice Hall, Englewood Cliffs, 1970. Kondo, J., Integral Equations , Clarendon Press, Oxford, 1991. K o r n ,G .A .a n dK o r n ,T .M . , Mathematical Handbook for Scientists and Engineers, 2nd Edition , Dover Publications, New York, 2000. Krantz, S. G., Handbook of Complex V ariables ,B i r k h ¨auser, Boston, 1999. Krasnosel’skii, M. A., T opological Methods in the Theory of Nonlinear Integral Equations , Macmillan, New York, 1964. Krasnov, M. L., Kiselev, A. I., and Makarenko, G. I., Problems and Exercises in Integral Equations , Mir Publ., Moscow, 1971. Krasnov, M. L., Integral Equations: Introduction to the Theory [in Russian], Nauka, Moscow, 1975. Krein, M. G. (Editor), Functional Analysis, 2nd Edition [in Russian], Nauka, Moscow, 1972. Krein, M. G., Integral equations on a half-line with kernels depending upon the difference of the arguments [in Russian], Uspekhi Mat. Nauk, V ol. 13, No. 5 (83), pp. 3–120, 1958. Krein, M. G., Non-symmetric oscillating Green’s functions of ordinary differential operators [in Russian], Doklady AN SSSR , V ol. 25, p. 643, 1939. Kress, R., Linear Integral Equations, 2nd Edition , Springer-Verlag, New York, 1999. Kress, R., Numerical Analysis , Springer-Verlag, New York, 1998. Krylov, V . I., Bobkov, V . V ., and Monastyrnyi, P . I., Introduction to the Theory of Numerical Methods. Integral Equations, Ill-Posed Problems, and Improvement of Convergence [in Russian], Nauka i Tekhnika, Minsk, 1984. Ky t he , P. K . a nd P ur i , P. , Computational Methods for Linear Integral Equations ,B i r k h ¨auser Verlag, Basel, 2002. Ladopoulos, E. G., Singular Integral Equations: Linear and Non-Linear Theory and Its Applications in Science and Engineering , Springer-Verlag, New York, 2000. Lavrentiev, M. M., Some Improperly Posed Problems of Mathematical Physics , Springer-Verlag, New York, 1967. Lavrentiev, M. M., Romanov, V . G., and Shishatskii, S. 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Index A Abel equation ﬁrst kind, 10generalized, 519, 527 generalized, ﬁrst kind, 531 generalized, second kind, 141, 548second kind, 138 Abel problem, 520Abel type two-dimensional equation, 15absolutely continuous function, 529abstract Hilbert space, 873 Airy equation, 1023 Airy function, 1023 asymptotic expansions, 1023deﬁnition, 1023ﬁrst kind, 1023power series, 1023second kind, 1023 algebraic equations linear, inﬁnite system, 858, 861, 864, 868, 971linear, inﬁnite system, symmetric matrix, 850, 853 alternating sums of powers of natural numbers, 920 alternative, Fredholm, 637, 638, 643 symmetric equations, 643 alternative Fourier transform, 512 amplitude, 1039 analysis, functional, 1055analytic continuation theorem, 595, 714application of integral equations to differential equations, 875 approach, Carleman–Vekua, 778approximate methods nonlinear equations, constant integration limits, 826 nonlinear equations, variable integration limit, 811 approximate solution, 688, 693approximate values of eigenvalues, Hilbert– Schmidt kernel, 845 approximating a kernel, 687approximation characteristic values, 646eigenfunctions, Hilbert–Schmidt operator, 868, 872 eigenvalues, Hilbert–Schmidt operator, 868, 872 kernel, 687Lanczos, 798approximation ( continued ) method, successive, 566, 579, 632, 633, 811, 826, 876 solution, 854 arbitrary functions, 111, 191, 278, 357, 406, 410, 413, 437, 444, 456 arbitrary parameters, 408, 411, 433, 453 arbitrary powers, 12, 139, 223, 317, 939, 977 arccosine, 176, 344 arccotangent, 178, 347arcsine, 177, 345 arctangent, 178, 346 argument, complicated, 227, 346, 254 Arutyunyan equation, 198 associated Legendre functions, 107, 271, 1031, 1032, 1033 ﬁrst kind, 1032 general case, 1032 integer indices, real argument, 1031 modiﬁed, 1033 second kind, 1032 asymmetric form Fourier cosine transform, 514Fourier sine transform, 515 Fourier transform, 512 asymptotic expansions, 509, 1017, 1022–1024, 1035 Airy functions, 1023 Bessel functions, 1017modiﬁed Bessel functions, 1022 parabolic cylinder functions, 1035 Tricomi conﬂuent hypergeometric functions, 1024 asymptotic methods, 618 equations with logarithmic singularity, 618 auxiliary conditions, 843, 845, 851, 856, 862, 869, 870 auxiliary equation, 546, 550, 551, 527 application, 527 ﬁrst kind, 550 second kind, 551 auxiliary integral conditions, 841–843 auxiliary results, 784 axioms for addition, vectors, 1065 axioms for addition and multiplication by scalars, vectors, 1065 axis, real, 575, 713 H¨older condition, 575 Sokhotski–Plemelj formulas, 713 1081 1082 INDEX B Banach space, 1062, 1065, 1067 base, Napierian, 906 base of Napierian logarithm, 905 base of natural logarithm, 905, 906basis abstract space, 844, 863 Euclidean space, 857, 869Hilbert space, 857, 867, 869 Hilbert space, special, 869 orthonormal, 855, 856 Bateman method, 689 general scheme, 689 special cases, 690 Bernoulli numbers, 1008 Bernoulli polynomials, 1052 Bessel’s formula, 1018Bessel equation, 1016 modiﬁed, 1021 Bessel function, 88, 187, 264, 269, 353, 958, 1016 asymptotic expansions, 1017 deﬁnitions, 1016 ﬁrst kind, 261, 297, 1016integral representations, 1017 modiﬁed, 97, 189, 269, 355, 1021 modiﬁed, ﬁrst kind, 266, 1021modiﬁed, second kind, 266, 1021 orthogonality properties, 1019 second kind, 264, 299, 1016 third kind, 1020 zeros, 1019 beta function, 1012, 1014 incomplete, 1014, 1015 bifurcation point, nonlinear integral equations, 834, 835 bilinear series, 640 iterated kernels, 642 binomial coefﬁcients, 909, 920, 1007Boas transform, 250 boundary conditions, 887 boundary value problem ﬁrst, 895, 896 Hilbert, 742 linear, representation, 892nth-order differential equations, 882 ODEs, 881 ODEs, reduction to Fredholm equations, 881ODEs, reduction to V olterra equations, 877 Riemann, 595, 714 second, 895, 897second-order differential equations, 883 bounded closed domain, 839 bounded set, 866 closed, 842 bounded variation function, 1055, 1058 classes, 1056 criteria, 1057 deﬁnition, 1055properties, 1056, 1057Boussinesq equation, 900 Bubnov–Galerkin method, 697Buchholz transform, 274 Bueckner equation, 801 C C(a,b), space of continuous functions, 1066 Cα(0, 1), H ¨older space, 1066 calculation of eigenvalues, 877 canonical factorization, 680canonical form, 805–807 Hammerstein equation, 807 canonical function, nonhomogeneous Riemann problem, 605 Carleman equation, 243, 590Carleman method characteristic equations, 761 equation, convolution type, ﬁrst kind, 606equation, convolution type, second kind, 660 equation, difference kernels, 610 Carleman–Vekua regularization, 778Cauchy criterion, 1067Cauchy integral, 708 Cauchy kernel, 707, 757 characteristic equation, 761complete singular equation, 757 equation on real axis, 743 general singular equation, ﬁrst kind, 745generalized, 783integral equation, 743, 757 Cauchy principal value, 709 Cauchy problem ﬁrst-order ODEs, 875, 876 ODEs, reduction to integral equations, 875 second-order ODEs, 876special nth-order linear ODE, 876 Cauchy residue theorem, 504 Cauchy–Schwarz–Bunyakovsky inequality, 501 Cauchy type and Fourier integrals, 592Cauchy type integral, 708 Cauchy-type kernel, 751, 753 characteristic equation, 758, 761 Cauchy kernel, 761exceptional case, 767 Hilbert kernel, 769 real axis, 765transposed, 758, 764 characteristic operator, 758 transposed, 758 characteristic value, 301, 625, 637, 639, 645, 697 approximation, 646 extremal properties, 644 system, 640 Chebyshev formula, 535 Chebyshev functions, 1049 Chebyshev nodes, 748Chebyshev polynomial ﬁrst kind, 109, 1048 second kind, 750, 1049 INDEX 1083 closed-form solution case of constant coefﬁcients, 770 general case, 771 closed bounded set, 842 closed domain, bounded, 839 closed kernel, 578coefﬁcient binomial, 909, 920, 1007 discontinuous, 739rational, 601, 723 Riemann problem, 596, 718 undetermined, 692 collocation method, 692, 693, 815 hypersingular integral equation, 755 collocation points, 693 combination elementary functions, 73, 255 hyperbolic functions, 39 trigonometric functions, 63, 252 compact operator, 842, 843, 1069 self-adjoint, 843 self-adjoint positive, 873self-adjoint positive deﬁnite, 1069 compact self-adjoint operator, 843 compact self-adjoint positive deﬁnite operator, 1069 eigenvalues, 1069 compact self-adjoint positive operator, 873 compactness of integral operator, sufﬁcient condition, 842 compatibility condition, 896, 897 complementary error function, 1009, 1025 complementary modulus, 1036, 1037complete elliptic integral ﬁrst kind, 1035 second kind, 1035 complete equation generalized Cauchy kernel, 783 Hilbert kernel, 780 complete kernel, 578 complete orthonormal system of functions, 844, 855 complete singular integral equation, 757, 770, 772 Cauchy kernel, 757 Hilbert kernel, 759, 780 regularization method, 772solution methods, 757 complete space, 1067 complete system, 1067complete system of eigenfunctions, 640 complex linear space, 1065 complicated argument, 227, 246, 254concentration, 890 integral equation, 890 integral equation, numerical method, 891 condition auxiliary integral, 841–843, 845, 851, 856, 862, 869, 870 boundary, 887compatibility, 896, 897condition ( continued ) H¨older, 709, 1066 H¨older, real axis, 575 Lipschitz, 709, 1056, 1059normality, 596sufﬁcient for compactness of integral operator, 842 conﬂuent hypergeometric equation, 1024conﬂuent hypergeometric function, 107, 1024 Kummer, 1024Tricomi, 1024, 1025Tricomi, asymptotic expansions, 1024 Tricomi, integral representations, 1024 Whittaker, 1027Wronskian, 1026 conjugate kernels, 582connected domain, 731constant Euler, 533, 1013, 1017, 1026H¨older, 709 eigenfunctions, 696, 699resolvent, 633 continuation analytic, 714 continuity, principle, 714 continuous function of real argument values in Banach space, 840values in Hilbert space, 840values in space of functions square integrable over a closed bounded set, 842 values in space of functions square integrable over a ring-shaped domain, 841 values in space of square integrable functions, 840 continuous operator, 1068contour, smooth, 708convergence almost everywhere, 1060mean-square, 501 convergent series, 509convolution theorem, 507, 513convolution type, 574, 606, 660, 669coordinate functions, 693, 697 cosine, 46, 166, 246, 335, 558, 928 hyperbolic, 22, 154, 238, 327 cosine integral, 87, 258, 1011cosine transform, 514cotangent, 62, 175, 252, 343 hyperbolic, 38, 162, 242, 333 criterion, Cauchy, 1067Crum transform, 268curves, open, Riemann problem, 734cuspidal point, 708cylinder function, 1016cylindrical function, 1016 deﬁnitions, 1016 D De Moivre formulas, 911deﬁnite integrals, tables, 951 1084 INDEX deﬁnition Cauchy type integral, 708 hyperbolic functions, 913 degenerate hypergeometric equation, 1024 degenerate kernel, 111, 191, 278, 357, 519, 522, 539, 540–543, 569, 573, 589, 625, 627, 631,810, 817 general, 523, 628simplest, 627 density, potential, 893 derivative fractional, deﬁnition, 529 fractional, left-sided, 529 fractional, properties, 530fractional, right-sided, 529 integrable, fractional, 531 logarithmic of gamma function, 1017, 1021Riemann–Liouville, 529 determinant, Fredholm, 636 method, 635 difference kernel, 114, 203, 283, 372, 519, 524, 539, 544, 573, 574, 586, 625, 610, 626, 655,683 entire axis, 655 ﬁnite interval, 683weak singularity, 588 differential equation nth-order, boundary value problems, 882 ordinary, 527, 547, 686, 875, 877 ordinary, linear, 881 second-order, boundary value problems, 883 differential equation and V olterra integral equations, 877 differentiating, method for integral equations, 820 differentiation fractional, method, 529 method, 564, 583, 810 differentiation formulas, 910, 913, 916, 917diffusion ﬂux, integral equations, 890 digamma function, 1013, 1017 direct sum of orthogonal subspaces, 845, 863, 869 Dirichlet–Mehler integral, 1030 Dirichlet problem exterior, 896interior, 895 reduction to integral equations, 895, 896 discontinuous coefﬁcient, 739divisor transform, 269 Dixon equation, 136 domain bounded closed, 839 circular, 841 multidimensional, 839one-dimensional, 839 ring-shaped, 841, 855, 862 double layer potential, 893 Gauss formula, 894 dual integral equation ﬁrst kind, 295, 575, 610ﬁrst kind, exact solutions, 613dual integral equation ( continued ) reduction to Fredholm equation, 615 second kind, 627 second kind, convolution type, 669 E eigenfunctions, 301, 625, 639, 834, 867 construction, 696, 699 extremal properties, 644Fredholm equation, second kind, 694Hilbert–Schmidt kernel, 854, 856, 858, 861, 864, 865 Hilbert–Schmidt operator, 871 kernel, 844 linear operator, 1068nonlinear equation, 834nonlinear operator, 834 system, 640 system, complete, 640system, incomplete, 640 eigenvalues, 301, 625, 834 calculation, 877 compact self-adjoint positive deﬁnite operator, 1069 Hilbert–Schmidt kernel, 854, 856, 858, 861, 864, 865 Hilbert–Schmidt operator, 871kernel, 844 linear operator, 1068 matrix, 845, 848, 856, 859, 861, 868, 872operator, 867positive, 648self-adjoint operator, 1069 eigenvectors of matrix, orthonormal, 845, 848, 856, 859, 868, 872 eigenvectors of self-adjoint operator, 1069electrostatic problem, Roben, 897elementary functions, 73, 255, 257, 348, 905 combinations, 179 properties, 905 elements linearly dependent, 1065linearly independent, 843, 1065 elliptic function, 1038 Jacobi, 1039Weierstrass, 1042 elliptic integral, 1035, 1036 complete, 1035 complete, ﬁrst kind, 1035 complete, second kind, 1035ﬁrst kind, 1037incomplete, 1037second kind, 1037 third kind, 1037 elliptic modulus, 1037elliptic theta functions, 1043 entire axis, equation, 574, 586, 587, 626, 655 equation Abel, ﬁrst kind, 10Abel, generalized, 519, 527Abel, generalized, ﬁrst kind, 531 INDEX 1085 equation ( continued ) Abel, generalized, second kind, 141, 548 Abel, second kind, 138Abel type, ﬁrst kind, 15 Abel type, two-dimensional, 15 Airy, 1023Arutyunyan, 198 auxiliary, 546 auxiliary, application, 527auxiliary, ﬁrst kind, 550 auxiliary, second kind, 551 Bessel, 1016Bessel, modiﬁed, 1021 Boussinesq, 900 Bueckner, 801Carleman, 243, 590 Cauchy kernel, complete, 757 Cauchy kernel, ﬁrst kind, 707 Cauchy kernel, ﬁrst kind, real axis, 743 Cauchy kernel, general of ﬁrst kind, 745Cauchy kernel, simplest of ﬁrst kind, 707, 743 Cauchy kernel, simplest of ﬁrst kind, real axis, 743 characteristic, 758, 761characteristic, Cauchy kernel, 761 characteristic, exceptional case, 767 characteristic, Hilbert kernel, 769characteristic, real axis, 765 characteristic, transposed, 758, 764 compact self-adjoint and positive deﬁnite operator, 843 complete, generalized Cauchy kernels, 783 complete, Hilbert kernel, 780 complete singular, 757, 770, 772complete singular, Cauchy kernel, 757 complete singular, regularization method, 772 conﬂuent hypergeometric, 1024contain arbitrary functions, 410, 413 contain arbitrary parameters, 408, 411 contain modulus, 278, 583 contain unknown function of complicated argument, 227, 254 convolution type, ﬁrst kind, 574convolution type, ﬁrst kind, Carleman method, 606 convolution type, second kind, 626, 655, 657 convolution type, second kind, Carleman method, 660 degenerate kernel, 111, 191, 278, 357, 522, 540–543 degenerate kernel, nonlinear, method of differentiation, 810 difference kernel, 114, 203, 283, 372, 524, 544, 574, 586, 626, 685 difference kernel, Carleman method, 610 difference kernel, entire axis, 655 difference kernel, ﬁnite interval, 683, 685difference kernel, weak singularity, 588 differential, 875, 877 differential, nth-order, boundary value problems, 882equation ( continued ) differential, ordinary, 527, 547, 686 differential, ordinary, linear, 881differential, second-order, boundary value problems, 883 diffusion ﬂux, 890 Dixon, 136dual, ﬁrst kind, 295, 575, 610 dual, ﬁrst kind, exact solutions, 613 dual, reduction to Fredholm equation, 615dual, second kind, 627 dual, second kind, convolution type, 669 eigenfunctions, Fredholm equation, second kind, 694 elasticity, 621 entire axis, 574, 586, 587, 626, 655 exact methods, 588–592exact solutions, 3–500 exponential nonlinearity, 411, 467 ﬁnite interval, 683, 685ﬁnite interval, ﬁrst kind, 744 ﬁrst kind, 3, 519, 591, 624 ﬁrst kind, reduction to equations of second kind, 591 ﬁrst kind, weak singularity, 574 Fredholm, degenerate kernel, second kind, 627 Fredholm, ﬁrst kind, 573, 623Fredholm, second kind, 625, 685, 698, 701 Fredholm, second kind, system, 701 Fredholm, second kind on contour, 759Fredholm, spectrum, 760 Fredholm, symmetric kernel, second kind, 639 Fredholm and dual equations, 615Fredholm and Green’s function, 881 function of complicated argument, 246 Gaussian hypergeometric, 1028Gelfand–Levitan–Marchenko, 900 Gelfand–Levitan–Marchenko type, 898 general degenerate kernel, 523generalized Abel, 519, 527 generalized Abel, ﬁrst kind, 531 generalized Abel, second kind, 141, 548generalized Cauchy kernel, complete, 783 generalized Schlomilch, equation, generalized Schl¨omilch 254 Hammerstein, canonical form, 807Hammerstein, ﬁrst kind, 807 Hammerstein, second kind, 807 Hammerstein, second kind, degenerate kernel, 817 Hammerstein type, 807 Hilbert kernel, complete, 759, 780 Hilbert kernel, ﬁrst kind, 707, 746Hilbert kernel, general of ﬁrst kind, 708, 747 Hilbert kernel, simplest of ﬁrst kind, 707, 746 Hilbert kernel, simplest of ﬁrst kind, complete, 759 Hilbert–Plessner, 255 homogeneous, 301, 502, 539, 625, 627, 637, 708, 751 hyperbolic nonlinearity, 414, 468 1086 INDEX equation ( continued ) hypergeometric, 1028 hypergeometric, conﬂuent, 1024 hypergeometric, degenerate, 1024 hypersingular, Cauchy-type kernel, ﬁrst kind, 751 hypersingular, Cauchy-type kernel, general of ﬁrst kind, 751 hypersingular, Cauchy-type kernel, simplest of ﬁrst kind, 231, 751, 753 hypersingular, collocation method, 755 hypersingular, Hilbert-type kernel, ﬁrst kind, 751 hypersingular, Hilbert-type kernel, general of ﬁrst kind, 751 hypersingular, Hilbert-type kernel, simplest of ﬁrst kind, 255, 754 hypersingular, numerical methods, 754 inﬁnite integration limit, ﬁrst kind, 537 inﬁnite limits of integration, second kind, 702 Kadomtsev–Petviashvili, 901 kernel contains arbitrary functions, 111, 191, 278, 357 kernel contains arbitrary powers, 12 kernel contains combinations of elementary functions, 73, 179, 255, 348 kernel contains combinations of various functions, 565 kernel contains exponential functions, 15, 144, 231, 320 kernel contains higher-order polynomials in arguments, 6 kernel contains hyperbolic functions, 22, 154, 238, 327 kernel contains inverse trigonometric functions, 66, 176, 344 kernel contains logarithmic functions, 42, 45, 164, 242, 334 kernel contains power-law functions, 4, 45, 127, 217, 301 kernel contains rational functions, 7 kernel contains special functions, 86, 187, 258, 353 kernel contains square roots, 9 kernel contains sum of exponential functions, 564 kernel contains sum of hyperbolic functions, 564 kernel contains sum of trigonometric functions, 564 kernel contains trigonometric functions, 46, 166, 246, 335 kernel cubic in arguments, 5 kernel linear in arguments, 4 kernel quadratic in arguments, 4 Korteweg–de Vries, 899Korteweg–de Vries, modiﬁed, 900 Krein’s method, 588 Lalesco–Picard, 323 Laplace, 893 Laplace, potentials, properties, 892equation ( continued ) Laplace, potentials, types, 892Legendre, 1032 linear, constant integration limits, 502 linear, constant integration limits, ﬁrst kind, 217, 502, 573 linear, constant integration limits, second kind, 301, 502, 625 linear, ﬁrst kind, 502 linear, operator methods, 549 linear, second kind, 502 linear, solution methods, 519, 539, 573, 625linear, structure of solutions, 502 linear, variable integration limit, ﬁrst kind, 3, 502 linear, variable integration limit, second kind, 127, 502 linear and nonlinear PDEs, 898 logarithmic nonlinearity, 419, 472 logarithmic singularity, 618logarithmic singularity, asymptotic methods, 618 Mathieu, 1045 Mathieu, modiﬁed, 1046method of differentiating, 564, 583, 820 mixed multidimensional, bounded set, projection method, 866 mixed multidimensional, closed bounded set, 842 mixed multidimensional, Fredholm operator, 842 mixed multidimensional, Hilbert–Schmidt operator, 869 mixed multidimensional, integral operators of V olterra and Hilbert–Schmidt types, 866 mixed multidimensional, integral operators of V olterra and Schmidt types, 866 mixed multidimensional, methods of solving, 839–874 mixed multidimensional, Schmidt operator, 843mixed multidimensional, Schmidt operator, equivalent form, 843 mixed multidimensional, symmetric Fredholm kernel, 842 mixed operator, 866, 869mixed operator, auxiliary conditions, 869 mixed two-dimensional, circular domain, 841 mixed two-dimensional, ﬁnite interval, 840 mixed two-dimensional, ﬁnite interval, methods of solving, 843–854 mixed two-dimensional, Hilbert–Schmidt kernel and auxiliary conditions, ﬁnite interval, 845 mixed two-dimensional, Hilbert–Schmidt kernel and auxiliary conditions, ring-shapeddomain, 856 mixed two-dimensional, Hilbert–Schmidt kernel and given right-hand side, ﬁnite interval, 843 mixed two-dimensional, Hilbert–Schmidt kernel and given right-hand side, ring-shapeddomain, 855 INDEX 1087 equation ( continued ) mixed two-dimensional, ring-shaped domain, 841 mixed two-dimensional, ring-shaped domain, methods of solving, 855–866 mixed two-dimensional, Schmidt kernel, 841 mixed two-dimensional, Schmidt kernel, equivalent form, 842 mixed two-dimensional, Schmidt kernel and auxiliary conditions, ring-shapeddomain, 862 mixed two-dimensional, Schmidt kernel and given right-hand side, ﬁnite interval, 848 modiﬁed Bessel, 1021 modiﬁed Korteweg–de Vries, 900 modiﬁed Mathieu, 1046Nekrasov, 836 nonhomogeneous, 502, 539, 627, 708, 751 nonhomogeneous, positive solutions, 649 nonhomogeneous, solution, 642 nonlinear, 805, 807, 834, 899nonlinear, bifurcation points, 834, 835 nonlinear, constant integration limits, 806, 829 nonlinear, constant integration limits, approxi- mate methods, 826 nonlinear, constant int egration limits, exact methods, 817 nonlinear, constant integration limits, ﬁrst kind, 433 nonlinear, constant integration limits, numerical methods, 826 nonlinear, constant int egration limits, second kind, 453 nonlinear, degenerate kernels, 817 nonlinear, eigenfunctions, 834 nonlinear, existence theorems, 830 nonlinear, uniqueness theorems, 830 nonlinear, variable integration limit, 805nonlinear, variable integration limit, approxi- mate methods, 811 nonlinear, variable integration limit, exact methods, 809 nonlinear, variable integration limit, ﬁrst kind, 393 nonlinear, variable integration limit, numerical methods, 811 nonlinear, variable integration limit, second kind, 403 nonlinear, V olterra, 805 nonlinear, with parameter, local solutions, 835 nonlinearity, general form, 399, 425, 447, 477 nonnegative kernel, 648 nonsymmetric kernel, ﬁrst kind, 580one-sided, ﬁrst kind, 574 one-sided, second kind, 626 operator, general projection problem, 873 operator, mixed, 866, 869 operator, mixed with auxiliary conditions, 869operator, “quadratic”, 552 operator, solution, 553 ordinary differential, 527, 547, 686equation ( continued ) parameter, 625 Picard–Goursat, 134Poisson, 894 power-law nonlinearity, 408, 464 power-law nonlinearity that contains arbitrary functions, 444 quadratic nonlinearity, 819 quadratic nonlinearity that contains arbitrary functions, 397, 406, 437, 456 quadratic nonlinearity that contains arbitrary parameters, 393, 403, 433, 453 “quadratic” operator, 552 reducible to symmetric equation, 647 renewal, 203 right-hand side, 519, 539, 573, 625 right-hand side, special, 555Schl¨omilch, 254, 452, 825 Schl¨omilch, generalized, 254 Schmidt integral operator, 843Schmidt kernel, 843, 859, 863 Schmidt kernel and auxiliary conditions, ﬁnite interval, 851 Schmidt kernel and auxiliary conditions, ring-shaped domain, 862 Schmidt kernel and given right-hand side, ﬁnite interval, 848 Schmidt kernel and given right-hand side, ring-shaped domain, 859 Schmidt operator, 869second kind, 591 second kind, operator method, 654 semiaxis, 574, 587, 626, 657simplest hypersingular, Cauchy-type kernel, ﬁrst kind, 231, 753 simplest hypersingular, Hilbert-type kernel, ﬁrst kind, 255, 754 single kernel, ﬁrst kind, 574, 626 singular, 228, 255, 319, 344, 707 singular, Bueckner type, 801singular, complete, 757, 770, 772 singular, ﬁrst kind, 707, 743 singular, generalized kernel, 792singular, numerical solution, 799 singular, transposed, 758 singular, two-dimensional, 231skew-symmetric, 647 solution methods, 501–901 special right-hand side, 555surface concentration, 890 surface concentration, numerical method, 891 symmetric, 639, 647symmetric, Fredholm alternative, 643 symmetric kernel, 639 symmetric kernel, ﬁrst kind, 577system, 701 transposed, 573, 575, 625, 627, 637 transposed of characteristic equation, 764 Tricomi, 319, 769, 769 Tricomi–Gellerstedt, 320trigonometric nonlinearity, 420, 473 1088 INDEX equation ( continued ) “truncated” ﬁrst kind, 549 two kernels, ﬁrst kind, 574, 607two kernels, second kind, 626, 664 Urysohn, 806, 832 Urysohn, ﬁrst kind, 806, 829Urysohn, ﬁrst kind, special, method, 821 Urysohn, second kind, 806 Urysohn, second kind, degenerate kernel, 818Urysohn, second kind, special, method, 822 Urysohn type, 806 variable integration limit, 3variable lower integration limit, ﬁrst kind, 537 variable lower integration limit, second kind, 570 V olterra, 549, 805, 877V olterra, ﬁrst kind, 519, 524, 565 V olterra, ﬁrst kind, connection with V olterra equations of second kind, 524 V olterra, ﬁrst kind, existence of solution, 519V olterra, ﬁrst kind, Hammerstein form, 806 V olterra, ﬁrst kind, problems, 520 V olterra, ﬁrst kind, uni queness of solution, 519 V olterra, ﬁrst kind, Urysohn form, 805, 815 V olterra, Hammerstein form, 806 V olterra, nonlinear, 805 V olterra, quadratic nonlinearity, 809 V olterra, reduction to Wiener–Hopf equation, 528 V olterra, second kind, 524, 539, 565 V olterra, second kind, connection with V olterra equations of ﬁrst kind, 524 V olterra, second kind, Hammerstein form, 816V olterra, second kind, sequence, 855 V olterra, second kind, sequence of independent, 853, 865, 872 V olterra, second kind, Urysohn form, 805V olterra, sequence, 844, 850, 862 V olterra, sequence of independent, 847, 858 V olterra, Urysohn form, 805, 811, 814, 816weak singularity, 519 weak singularity, ﬁrst kind, 532, 574 weak singularity, second kind, 625weakly singular kernel, 532 Whittaker, 1027 Wiener–Hopf, 574, 626, 679Wiener–Hopf, ﬁrst kind, 285, 574, 538, 606 Wiener–Hopf, Krein’s method, 679 Wiener–Hopf, second kind, 373, 547, 571, 626, 660, 679 Wiener–Hopf, second kind, exceptional case, 678 Wiener–Hopf, second kind, homogeneous, 672 Wiener–Hopf, second kind, index, 661Wiener–Hopf, second kind, nonhomogeneous, 677 Wiener–Hopf, second kind, solution, 681 Wiener–Hopf, V olterra equation, 528 equidistant surface, method, 891 equilibrium potential, 897equivalent regularization, problem, 776Erd´elyi–Kober operators, 532 error function, 86, 258, 549, 1009, 1024 complementary, 1009, 1025 estimates for spectral radius, 649Euclidean space, 845, 857, 863, 869, 1067 basis, 857, 869 Euler constant, 533, 1013, 1017, 1026 Euler formula, 911, 1013 Euler numbers, 1008Euler polynomials, 1053exceptional case characteristic equation, 767regularization, 779Riemann problem, 605, 727 Wiener–Hopf equation, second kind, 678 existence theorems, 875 nonlinear equations, 830Stieltjes integral, 1058 expansion, asymptotic, 509 Airy functions, 1023Bessel functions, 1017modiﬁed Bessel functions, 1022 parabolic cylinder functions, 1034 Tricomi conﬂuent hypergeometric functions, 1024 expansion in power series, 910, 913, 916, 918exponent, growth, 505exponential form, 555exponential function, 15, 73, 77, 78, 144, 151, 179–181, 231, 234, 236, 257, 320, 326, 348,349, 419, 564, 905, 940, 954, 963, 978, 984,990, 998, 1002 properties, 905 exponential integral, 86, 258, 1009, 1010, 1025exponential nonlinearity, 411, 467 exponents, singularity, 787, 789 expressions with arbitrary powers, 977exponential functions, 963, 978, 984, 990, 998, 1002 hyperbolic functions, 964, 979, 985, 991logarithmic functions, 965, 980, 985, 992, 999, 1002 power-law functions, 963, 983, 989, 998, 1001rational functions, 971 special functions, 967, 981, 987, 993, 1000, 1004 square roots, 975trigonometric functions, 966, 981, 986, 992, 999, 1003 exterior Dirichlet problem, 896 reduction to integral equations, 896 exterior Neumann problem, 897 reduction to integral equations, 896 F factorization, 597, 674, 676, 677, 679, 720, 723 canonical, 680 factorization problem, 676, 679Feller potential, 226 INDEX 1089 Feller transform, 226 ﬁeld of scalars, 1065 ﬁnite functional sums, 922ﬁnite interval, 683, 840, 843 equation, 683, 685 integrals, 951, 956mixed equations, 840 ﬁnite numerical sums, 919 ﬁnite sums, 919ﬁnitely many singular points, 507 ﬁrst-order ODEs, 875, 876 ﬁrst boundary value problem, 895, 896Fischer–Riesz, theorem, 1062 ﬂow ﬂuid, 888nonisothermal in plane channel, 884 ﬂuid ﬂow, 888 ﬂux, diffusion integral equations, 890form canonical, 805–807 canonical of Hammerstein equation, 807equivalent of mixed multidimensional equation with Schmidt operator, 843 equivalent of mixed two-dimensional equation with Schmidt kernel, 842 exponential, 555Hammerstein, for V olterra equation, 806 Hammerstein, for V olterra equation of ﬁrst kind, 806 Hammerstein, for V olterra equation of second kind, 816 polynomial, 553 quadratic, 644 Urysohn, for V olterra equation, 805, 811, 814, 816 Urysohn, for V olterra equation of ﬁrst kind, 805, 815 Urysohn, for V olterra equation of second kind, 805 form of inﬁnite products, representation, 910, 916formula Bessel’s, 1018 Chebyshev, 535Euler, 1013 Fourier inversion, 512 Gauss, 535 Gauss, for double layer potential, 894 Gauss, for volume potential, 894Green’s, 895 Hilbert inversion, 746 Hopf–Fock, 683Kontorovich–Lebedev inversion, 516 Meijer inversion, 516 Poincar ´e–Bertrand, 714 Poisson’s, 1018 Post–Widder, 510 quadrature, 534, 815Sokhotski–Plemelj, 713, 785 Stirling, 1013 formulas addition, 909, 915formulas ( continued ) calculation, 504 De Moivre, 911differentiation, 910, 913, 916, 917Euler, 911 integration, 910, 913, 916, 918 quadrature, 534, 793reduction, 907, 939, 947 Sokhotski–Plemelj, for real axis, 713 Fourier cosine transform, 514, 518 asymmetric form, 514 Parseval’s relation, 514 tables, 983 Fourier integral left, 594 one-sided, 593, 594 relationships with Cauchy type integral, 592right, 594 Fourier inversion formula, 512 Fourier sine transform, 514, 518 asymmetric form, 515 Parseval’s relation, 515 tables, 989 Fourier transform, 235, 511, 512, 518, 658 alternative, 512 asymmetric form, 512 deﬁnition, 512inverse, 512 inversion formula, 512 properties, 513rational, 685 fractional derivative, 529 deﬁnition, 529integrable, 531left-sided, 529 properties, 530 right-sided, 529 fractional differentiation, method, 529 fractional integral deﬁnition, 529left-sided, 529 properties, 530 Riemann–Liouville, 529right-sided, 529 fractional integration, 548 by parts, 529 operator, 529semigroup property, 529 fractional order, integral, 529 fractional powers, 138fracture mechanics, 791 Fredholm alternative, 637, 638 symmetric equations, 643 Fredholm determinant, 636 method, 635 Fredholm equation, 615, 881 degenerate kernel, second kind, 627ﬁrst kind, 573, 623 second kind, 625, 685, 698, 701 second kind, on contour, 759second kind, system, 701 1090 INDEX Fredholm equation ( continued ) spectrum, 760 symmetric kernel, second kind, 639 Fredholm kernel, 573, 625, 839–841 positive deﬁnite, 840 positive deﬁnite, symmetric, 866symmetric deﬁnite, 840symmetric positive, 841 symmetric positive deﬁnite, 866 Fredholm minor, 636Fredholm operator, 758, 842 symmetric kernel, generalization, 843 Fredholm theorems, 637, 702, 777Fresnel cosine integral, 1012 generalized, 1012 Fresnel integrals, 87, 258, 1011, 1012 generalized, 1012 Fresnel sine integral, 1012 generalized, 1012 Fubini theorem, 1064full measure, set, 1060 function absolutely continuous, 529Airy, 1023 arccosine, 66 arccotangent, 71arcsine, 68 arctangent, 70 associated Legendre, 107, 271, 1030–1033associated Legendre, ﬁrst kind, 1032 associated Legendre, general case, 1032 associated Legendre, integer indices and real argument, 1031 associated Legendre, second kind, 1032Bessel, 88, 187, 264, 269, 353, 958, 1016 Bessel, asymptotic expansions, 1017 Bessel, deﬁnitions, 1016Bessel, ﬁrst kind, 261, 297, 1016 Bessel, integral representations, 1017 Bessel, modiﬁed, 97, 189, 269, 355, 1021Bessel, modiﬁed, ﬁrst kind, 266, 1021 Bessel, modiﬁed, second kind, 266, 1021 Bessel, orthogonality properties, 1019Bessel, second kind, 264, 299, 1016 Bessel, third kind, 1020 Bessel, zeros, 1019beta, 1012, 1014beta, incomplete, 1014, 1015 canonical of nonhomogeneous Riemann problem, 605 Chebyshev, 1049complementary error, 1009, 1025 conﬂuent hypergeometric, 107, 1024 conﬂuent hypergeometric, Kummer, 1024conﬂuent hypergeometric, Tricomi, 1024 conﬂuent hypergeometric, Whittaker, 1027 conﬂuent hypergeometric, Wronskian, 1026cosine, 46 cotangent, 62 cylinder, 1016cylindrical, 1016function ( continued ) digamma, 1013, 1017 elementary, 73, 179, 255, 257, 348elementary, properties, 905 elliptic, 1038 elliptic, Jacobi, 1039elliptic, Weierstrass, 1042 elliptic theta, 1043 error, 86, 258, 549, 1009, 1024error, complementary, 1009, 1025 exponential, 15, 73, 77, 78, 144, 151, 179–181, 213, 234, 236, 257, 320, 326, 348, 349, 419,564, 905, 940, 954, 963, 978, 984, 990, 998,1002 exponential, properties, 905gamma, 260, 1012 gamma, incomplete, 88, 260, 1014, 1024, 1025 gamma, logarithmic derivative, 1017, 1021Gauss hypergeometric, 275, 1028 generalized Riemann zeta, 277 generating, 555, 580generating, power-law, 557 generating contain cosines, 558 generating contain sines, 558generating of exponential form, 555 Green’s, 881–883 Hankel, 1020Hankel, ﬁrst kind, 265 Hankel, second kind, 265 harmonic, 893Hermite, 1050 hyperbolic, 22, 73, 83, 84, 154, 164, 179, 185, 186, 238, 255, 327, 334, 348, 351, 352, 564,911, 913, 922, 940, 955, 964, 979, 985, 991 hyperbolic, inverse, 917hyperbolic, of half argument, 915 hyperbolic, of multiple argument, 915 hypergeometric, 1028hypergeometric, conﬂuent, 107, 1024 hypergeometric, conﬂuent, Wronskian, 1026 hypergeometric, Gauss, 275, 1028hypergeometric, Kummer conﬂuent, 272 hypergeometric, Tricomi conﬂuent, 273, 1025 hypergeometric, Whittaker conﬂuent, 274, 1027incomplete beta, 1014, 1015 incomplete gamma, 88, 260, 1014, 1024, 1025 index, 595inﬂuence, 577, 882 integrable, 501, 502, 1058 integrable, Lebesgue, 1059, 1061inverse hyperbolic, 917 inverse trigonometric, 66, 176, 344, 911, 948 irrational, 937Jacobi elliptic, 1039 Jacobi elliptic, connection with Jacobi theta functions, 1044 Jacobi theta, 110, 1043 Jacobi theta, connection with Jacobi elliptic functions, 1044 Jacobi weight, 793Kummer conﬂuent hypergeometric, 272, 1024 INDEX 1091 function ( continued ) Lebesgue integrable, 1059, 1061 left, 594Legendre, 270, 1030 Legendre, associated, 107, 271, 1030–1033 Legendre, associated, ﬁrst kind, 1032Legendre, associated, second kind, 1032 Legendre, modiﬁed associated, 1033 Legendre, spherical of ﬁrst kind, 299Legendre, Wronskians, 1034 logarithmic, 42, 45, 77, 83, 85, 164, 165, 180, 185, 187, 242, 244, 255, 256, 334, 335, 349,351, 353, 905, 943, 955, 965, 980, 985, 992,999, 1002 logarithmic, properties, 906MacDonald, 266, 1021 Mathieu, 1045, 1046 Mathieu, modiﬁed, 1046measurable, 1060 modiﬁed associated Legendre, 1033 modiﬁed Bessel, 97, 189, 269, 355, 1021modiﬁed Bessel, asymptotic expansions, 1022 modiﬁed Bessel, deﬁnitions, 1021 modiﬁed Bessel, ﬁrst kind, 266, 1021 modiﬁed Bessel, integral representations, 1022 modiﬁed Bessel, second kind, 266, 1021modiﬁed Mathieu, 1046 multivalued, 711 Neumann, 1016of complicated argument, 227, 346, 254 one-sided, 594 parabolic cylinder, 276, 1034parabolic cylinder, asymptotic expansions, 1035 parabolic cylinder, basic formulas, 1034 parabolic cylinder, deﬁnitions, 1034parabolic cylinder, integral representations, 1035 parabolic cylinder, linear relations, 1035 parabolic cylinder, Weber, 1034power, properties, 905 power-law, 4, 45, 127, 151, 165, 217, 236, 244, 301, 326, 335, 419, 951, 963, 983, 989, 998,1001 power-law generating, 557psi, 1012, 1013 rational, 7, 136, 220, 314, 933, 971 rational, inverse transforms, 506Riemann zeta, generalized, 277 special, 86, 111, 187, 258, 277, 353, 967, 981, 987, 993, 1000, 1004 special, properties, 1007spherical, Legendre of ﬁrst kind, 299 square integrable, 501, 502 Struve, 264, 299, 516, 518summable, 1059, 1061 summable, integral, 1061 tangent, 60theta, Jacobi, 1043 total variation, 1055 Tricomi conﬂuent hypergeometric, 273, 1024, 1025Tricomi conﬂuent hypergeometric, asymptotic expansions, 1024 Tricomi conﬂuent hypergeometric, integral representations, 1024 trigonometric, 78, 84, 85, 166, 176, 181, 186, 187, 246, 252, 256, 295, 335, 344, 349, 352,353, 564, 907, 922, 944, 956, 966, 981, 986,992, 999, 1003 trigonometric, inverse, 176, 344, 911, 948trigonometric, of half argument, 909trigonometric, of multiple arguments, 909 trigonometric, of single argument, relations, 908 trigonometric, powers, 908Weber, 88Weber parabolic cylinder, 1034Weierstrass elliptic, 1042 weight, Jacobi, 793 Whittaker, 1027 Whittaker conﬂuent hypergeometric, 274, 1027 function of bounded variation, 1055function of real argument values in Banach space, continuous, 840 values in Hilbert space, continuous, 840values in space of functions square integrable functions, continuous, 841 values in space of functions square integrable over closed bounded set, continuous, 842 values in space of functions square integrable over ring-shaped domain, continuous, 841 function of several variables, 839 functional analysis, some notions, 1055functional series, inﬁnite, 925functional sums, ﬁnite, 922functions coordinate, 693, 697 measurable, 1060of bounded variation, 1055, 1058, 1066orthogonal, 582power, 905 real-valued, multidimensional, 839 with ﬁnitely many singular points, 507 fundamental solution, 881 G Galerkin method, 582 gamma function, 260, 1012 incomplete, 88, 260, 1014, 1024, 1025logarithmic derivative, 1017, 1021 Gauss formula, 535 for double layer potential, 894 for volume potential, 894 Gauss hypergeometric functions, 275, 1028Gauss transform, 237Gaussian hypergeometric equation, 1028Gegenbauer polynomials, 1051 Gelfand–Levitan–Marchenko equation, 900 general degenerate kernel, 523general equation of ﬁrst kind with Cauchy kernel, 745 general hypersingular equation of ﬁrst kind with Cauchy-type kernel, ﬁnite interval, 751 1092 INDEX general hypersingular equation of ﬁrst kind with Hilbert-type kernel, 751 general projection problem, 873 special case, 846, 852, 857, 870 general scheme Bateman method, 689method of quadratures, 568quadrature method for Fredholm equations of second kind, 698 solving of dual integral equations, 611 successive approximation method, 566 general singular equation of ﬁrst kind with Hilbert kernel, 708, 747 generalization of Fredholm integral operator with symmetric kernel, 843 generalized Abel equation, 519, 527 ﬁrst kind, 531second kind, 141, 548 generalized Cauchy kernel, 783generalized Fresnel cosine integral, 1012 generalized Fresnel integral, 1012 generalized Fresnel sine integral, 1012generalized Jentzch theorem, 648generalized kernel of integral equation, 783generalized Laguerre polynomials, 1047generalized Liouville theorem, 595, 714generalized Mehler–Fock transform, 271generalized Riemann zeta function, 277 generalized Schl ¨omilch equation, 254 generating function, 555, 580 containing cosines, 558containing sines, 558exponential form, 555power-law, 557 Green’s formula, 895Green’s function, 881–883 growth exponent, 505 H Hammerstein equation, 807, 817, 830 canonical form, 807 degenerate kernel, second kind, 817 ﬁrst kind, 807second kind, 807 Hammerstein form, V olterra equation, 806 ﬁrst kind, 806second kind, 816 Hankel function, 1020 ﬁrst kind, 265 second kind, 265 Hankel transform, 261, 515, 518 Parseval’s relation, 515, 516 Hardy transform, 264harmonic function, 893Hartley transform, 252, 518Hermite functions, 1050Hermite interpolation polynomial, 716 Hermite polynomial, 108, 1024, 1025, 1050 Hilbert boundary value problem, 742Hilbert inversion formula, 746Hilbert kernel, 707, 780 characteristic equation, 769 complete singular equation, 759, 780equation, 759 equations of ﬁrst kind, 746 Hilbert–Plessner equation, 255Hilbert problem, 742 Hilbert–Schmidt kernel, 841, 843, 845, 853, 855, 856, 860 approximate values of eigenvalues, 845eigenfunctions, 854, 856, 858, 861, 864, 865 eigenvalues, 854, 856, 858, 861, 864, 865 Hilbert–Schmidt operator, 842, 843, 866, 871 approximation for eigenfunctions, 868, 872 approximation for eigenvalues, 868, 872 eigenfunctions, 871eigenvalues, 871 Hilbert–Schmidt theorem, 641, 1069 Hilbert–Schmidt theory, 843Hilbert space, 839, 845, 857, 863, 867, 869, 1067 abstract, 873 basis, 857, 867, 869linear operators, 1067, 1068 special basis, 869 Hilbert transform, 228, 255, 518, 743Hilbert transform on semiaxis, 229 Hilbert-type kernel, 751, 754 H¨older condition, 709, 1066 H¨older condition on real axis, 575 H¨older constant, 709 H¨older inequality, 1065 H¨older space C α(0, 1), 1066 homogeneous integral equation, 301, 502, 539, 625, 627, 637, 708, 751 homogeneous problem, 596, 602, 742homogeneous problem solution, 720 homogeneous Wiener–Hopf equation, second kind, 672 Hopf–Fock formula, 683hyperbolic cosine, 22, 154, 238, 327 hyperbolic cotangent, 38, 162, 242, 333 hyperbolic function, 22, 73, 83, 84, 154, 179, 185, 186, 238, 255, 327, 334, 348, 351, 352, 564,911, 913, 922, 940, 955, 964, 979, 985, 991 combinations, 164 half argument, 915 multiple argument, 915 hyperbolic nonlinearity, 414, 468 hyperbolic sine, 28, 156, 238, 329 hyperbolic tangent, 36, 161, 241, 332hypergeometric equation, 1028 conﬂuent, 1024 degenerate, 1024 hypergeometric function, 1028 conﬂuent, 107, 1024 conﬂuent, Kummer, 272, 1024conﬂuent, Tricomi, 1024, 1025 conﬂuent, Whittaker, 274, 1027 conﬂuent, Wronskian, 1026Gauss, 275, 1028 INDEX 1093 hypergeometric function ( continued ) Gauss, basic properties, 1028Kummer conﬂuent, 272, 1024Tricomi conﬂuent, 273 Whittaker conﬂuent, 274, 1027 hypergeometric series, 1028hypersingular equation, 751 Cauchy-type kernel, 751, 753collocation method, 755 ﬁrst kind, Cauchy-type kernel on ﬁnite interval, 751 ﬁrst kind, Hilbert-type kernel, 751Hilbert-type kernel, 751, 754numerical methods, 754 simplest of ﬁrst kind, Cauchy-type kernel, 231, 753 simplest of ﬁrst kind, Hilbert-type kernel, 255, 754 hypersingular integral deﬁnition, 751in sense of Hadamard principal value, 752 I identities, integral, 895identity operator, 842, 873ill-posed problem, 623 general notions, 623 incomplete beta function, 1014, 1015incomplete elliptic integrals, 1036incomplete gamma function, 88, 260, 1014, 1024, 1025 incomplete kernel, 578 incomplete system of eigenfunctions, 640 indeﬁnite integrals, tables, 933independent elements, linearly, 843, 1063index, 603, 661, 664 notion, 716 index of function, 595 index of Riemann problem, 596, 731index of Wiener–Hopf equation, 661inequality Cauchy–Schwarz–Bunyakovsky, 501 H¨older, 1063 triangle, 501 inﬁnite functional series, 925inﬁnite numerical series, 924 inﬁnite products, 910, 916 inﬁnite system of linear algebraic equations, 858, 861, 864, 868, 971 inﬁnite system of linear algebraic equations with symmetric matrix, 850, 853 inﬂuence function, 577, 882inner product, 501, 644 integrable fractional derivative, 531 integrable function, 501, 502, 1056 Lebesgue, 1059 integral Cauchy, 708 Cauchy type, 708integral ( continued ) Cauchy type, relationships with Fourier integral, 592 complete elliptic, 1035complete elliptic, ﬁrst kind, 1035 complete elliptic, second kind, 1035 cosine, 87, 258, 1011deﬁnite, tables, 951Dirichlet–Mehler, 1030 elliptic, 1035, 1036 elliptic, complete, 1035elliptic, ﬁrst kind, 1036 elliptic, incomplete, 1036 elliptic, second kind, 1036elliptic, third kind, 1036 exponential, 86, 258, 1009, 1010, 1025 Fourier, left, 594Fourier, one-sided, 593, 594 Fourier, relationships with Cauchy type integral, 592 Fourier, right, 594fractional, deﬁnition, 529 fractional, left-sided, 529 fractional, properties, 530fractional, Riemann–Liouville, 529 fractional, right-sided, 529 fractional order, 529Fresnel, 87, 258, 1011, 1012Fresnel, generalized, 1012 Fresnel cosine, 1012 Fresnel cosine, generalized, 1012Fresnel sine, 1012 Fresnel sine, generalized, 1012 hypersingular, deﬁnition, 751hypersingular, in sense of Hadamard principal value, 752 incomplete elliptic, 1036 indeﬁnite, tables, 933involving arbitrary powers, 939 involving Bessel functions, 958 involving exponential functions, 940, 954involving hyperbolic functions, 940, 955 involving inverse trigonometric functions, 948 involving irrational functions, 937involving logarithmic functions, 943, 955involving power-law functions, 951 involving rational functions, 933 involving trigonometric functions, 944, 956Jacobi weight function, 793 Laplace, 1030 Lebesgue, 1057Lebesgue, deﬁnition, 1059 Lebesgue, properties, 1059 left Fourier, 594logarithmic, 258, 1009, 1010, 1025 Mehler, 299, 615 one-sided Fourier, 593, 594probability, 1009 Riemann, 1057 Riemann–Liouville fractional, 529right-sided fractional, 529 1094 INDEX integral ( continued ) right Fourier, 594sine, 87, 258, 1011singular, 709 singular, principal value, 709 step-function, 1059Stieltjes, 1055, 1056Stieltjes, basic deﬁnitions, 1055 Stieltjes, existence theorems, 1056 Stieltjes, properties, 1056summable function, 1059 integral conditions, auxiliary, 841–843integral equation, seeequation integral identities, 895 integral operator compactness, sufﬁcient condition, 842Fredholm, 842 Fredholm, symmetric kernel, 843 Hilbert–Schmidt, 842, 843, 866positive deﬁnite, 842positive deﬁnite kernel, 843 Schmidt, 843, 866 self-adjoint, 842, 843spectral radius, 649symmetric kernel, 843V olterra, 842 integral representations Bessel functions, 1017modiﬁed Bessel functions, 1022parabolic cylinder functions, 1034 Tricomi conﬂuent hypergeometric functions, 1024 integral sum, Stieltjes, 1055integral transform, seetransform integrand contain exponential functions, 419 contain power-law functions, 419nonlinearity, 414–416, 418, 420, 422–424, 467–470, 472–475 integration fractional, 548 fractional, by parts, 529 fractional, operator, 529fractional, semigroup property, 529 interior Dirichlet problem, 895 reduction to integral equations, 895 interior Neumann problem, 895 reduction to integral equations, 895 interpolation nodes, 534interpolation polynomial Hermite, 716 Lagrange, 748 inverse Fourier transform, 512inverse hyperbolic functions, 917 inverse Laplace transforms, tables, 969 inverse Mellin transform, 510, 1001inverse transform rational functions, 506 representation as asymptotic expansions, 509 representation as convergent series, 509inverse trigonometric function, 66, 176, 344, 911, 948 inversion formula Hilbert, 746Kontorovich–Lebedev, 516 Meijer, 516 inversion of functions with ﬁnitely many singular points, 507 investigation of differential equations, 875irrational functions, 937 iterated kernel, 566, 632 bilinear series, 642 iteration process, 811, 814 J Jacobi elliptic function, 1038 connection with Jacobi theta functions, 1042 Jacobi polynomials, 1049Jacobi theta function, 110, 1042 connection with Jacobi elliptic functions, 1042 properties, 1042relations and formulas, 1042series representation, 1042 Jacobi weight function, 793 Jentzch theorem, generalized, 648Jordan lemma, 505 jump problem, 596 K K-transform, 518 Kadomtsev–Petviashvili equation, 901Kellog’s method for ﬁnding characteristic values in case of symmetric kernel, 645 kernel approximation, 687Cauchy, 707, 757 Cauchy, characteristic equation, 761 Cauchy, complete singular integral equation, 757 Cauchy, generalized, 783 Cauchy, integral equations, 757 Cauchy-type, 751, 753 closed, 578complete, 578 conjugate, 582 containing arbitrary functions, 111, 191, 278, 357 containing arbitrary powers, 12, 139, 223, 317 containing arccosine, 66, 176, 344 containing arccotangent, 71, 178, 347containing arcsine, 68, 177, 345 containing arctangent, 70, 178, 346 containing associated Legendre functions, 107, 271 containing Bessel functions, 88, 187, 353 containing Bessel functions of ﬁrst kind, 261, 297 containing Bessel functions of second kind, 264, 299 containing Chebyshev polynomials, 109 INDEX 1095 kernel ( continued ) containing combination of Bessel and modiﬁed Bessel functions, 269 containing combination of Bessel functions, 264 containing combination of elementary functions, 179, 255, 348 containing combination of hyperbolic functions, 39, 164, 334 containing combination of trigonometric functions, 63, 176, 252, 344 containing combination of various functions, 565 containing conﬂuent hypergeometric functions, 107 containing cosine, 46, 166, 246, 335containing cosine integral, 87 containing cosine integrals, 258 containing cotangent, 62, 175, 252, 343 containing elementary functions, 257 containing error function, 86, 258 containing exponential function, 15, 19, 73, 77, 78, 144, 151, 179–181, 231, 234, 236, 257, 320, 326, 348, 349 containing exponential integral, 86, 258 containing fractional powers, 138 containing Fresnel integral, 87, 258 containing gamma function, 260 containing Gauss hypergeometric function, 275 containing Hermite polynomial, 108containing higher-order polynomial in arguments, 6, 133, 311 containing hyperbolic cosine, 22, 154, 238, 237 containing hyperbolic cotangent, 38, 162, 242, 333 containing hyperbolic function, 22, 73, 83, 84, 154, 179, 185, 186, 238, 255, 327, 348, 351,352 containing hyperbolic sine, 28, 156, 238, 329containing hyperbolic tangent, 36, 161, 241, 332 containing incomplete gamma function, 88, 260 containing integer powers of arguments, 220 containing inverse trigonometric function, 66, 176, 344 containing Jacobi theta functions, 110 containing Kummer conﬂuent hypergeometric function, 272 containing Laguerre polynomial, 110containing Legendre function, 270 containing Legendre polynomial, 105 containing Legendre spherical function of ﬁrst kind, 299 containing logarithmic function, 42, 45, 77, 83, 85, 164, 165, 180, 185, 187, 242, 244, 255,256, 334, 335, 349, 351, 353 containing logarithmic integral, 258containing modiﬁed Bessel function, 97, 189, 355 containing modiﬁed Bessel function of ﬁrst kind, 266kernel ( continued ) containing modiﬁed Bessel function of second kind, 266 containing other special function, 111, 277containing parabolic cylinder function, 276 containing power-law function, 4, 19, 45, 127, 151, 165, 217, 236, 244, 301, 326, 335 containing rational function, 7, 136, 220, 314 containing sine, 52, 169, 247, 337 containing sine integral, 87, 258containing special function, 86, 187, 258, 353 containing square roots, 9, 222 containing square roots powers, 138 containing sum of exponential functions, 564 containing sum of hyperbolic functions, 564containing sum of trigonometric functions, 564 containing tangent, 60, 174, 251, 342 containing Tricomi conﬂuent hypergeometric function, 273 containing trigonometric function, 46, 78, 84, 85, 166, 181, 186, 187, 246, 256, 295, 335, 349, 352, 353 containing Whittaker conﬂuent hypergeometric function, 274 cubic in arguments, 5, 132, 307 degenerate, 111, 191, 278, 357, 519, 522, 539, 540–543, 569, 573, 589, 625, 627, 631, 810,817 degenerate, general, 523degenerate, general case, 628 degenerate, simplest, 627 difference, 114, 203, 283, 372, 519, 524, 539, 544, 573, 574, 586, 610, 625, 626, 655, 683 difference, on entire axis, 655 difference, with weak singularity, 588eigenfunction, 844 eigenvalue, 844 Fredholm, 573, 625, 839–841 Fredholm, positive deﬁnite, 840 Fredholm, positive deﬁnite, symmetric, 866Fredholm, symmetric deﬁnite, 840, 841 general degenerate, 523 generalized, 783Hilbert, 707, 780 Hilbert, characteristic equation, 769 Hilbert, complete singular integral equation, 759, 780 Hilbert, integral equations, 759 Hilbert–Schmidt, 841, 843, 845, 853, 855, 856, 860 Hilbert–Schmidt, approximate values of eigenvalues, 845 Hilbert–Schmidt, eigenfunctions, 854, 856, 858, 861, 864, 865 Hilbert–Schmidt, eigenvalues, 854, 856, 858, 864, 865 Hilbert-type, 751, 754 incomplete, 578iterated, 566, 632 iterated, bilinear series, 642 linear in arguments, 4, 127, 217, 301 1096 INDEX kernel ( continued ) logarithmic, 519, 588nondegenerate, 589, 631nonnegative, 648nonsymmetric, 580, 647of integral equation, 519, 573, 625of integral transform, 503orthogonal, 634oscillation, 651oscillation, deﬁnition, 651oscillation, theorems, 651 polar, 519, 532, 574, 588 positive deﬁnite, 641quadratic in arguments, 4, 129, 219, 304resolvent, 844Schmidt, 582, 841, 848, 851, 859, 860, 862simplest degenerate, 627singular, weakly, 532spectral radius, 649stochastic, 654symmetric, 573, 577, 625, 639, 645symmetric, resolvent, 644trace, 646transformation, method, 532V olterra, 839 weakly singular, 532 with logarithmic singularity, 533with rational Fourier transforms, 685with weak singularity, 519, 532, 574, 588, 625 Kontorovich–Lebedev inversion formula, 516Kontorovich–Lebedev transform, 267, 516, 518Korteweg–de Vries equation, 899 modiﬁed, 900 Krein’s method, 588, 683 for integral equations, 588for Wiener–Hopf equations, 679 Kummer conﬂuent hypergeometric function, 272, 1024 Kummer series, 1024Kummer transformation, 1025 L L2-norm, 501 Lagrange interpolation polynomial, 748Laguerre polynomial, 110, 1024, 1045 generalized, 1045 Lalesco–Picard equation, 323Lanczos approximation, 798Laplace equation, 893 potentials, properties, 892 Laplace integral, 1030Laplace transform, 235, 505, 511, 518, 524, 544, 658, 809 deﬁnition, 505inverse, tables, 969 inversion formula, 505 properties, 507solution method, 524tables, 961two-side, 234, 518largeλ, solution, 619 Lavrentiev regularization method, 621layer potential, single, 893least squares method, 695 description, 695 normal system, 695 Lebedev transform, 269Lebesgue integrable function, 1059Lebesgue integral, 1057 deﬁnition, 1059properties, 1059 Lebesgue space L p(a,b), 1064 Lebesgue theorem on dominated convergence, 1060 left-sided fractional derivative, 529left-sided fractional integral, 529left Fourier integral, 594 left function, 594 left regularization, 775 method, 775 left regularizer, 703Legendre equation, 1032Legendre functions, 270, 1030 associated, 107, 271, 1030 associated, ﬁrst kind, 1032 associated, modiﬁed, 1033associated, second kind, 1032modiﬁed associated, 1033Wronskians, 1034 Legendre polynomials, 105, 856, 1030 orthonormal, 844 Legendre spherical functions, ﬁrst kind, 299lemma, Jordan, 505limit theorems, 507linear algebraic equations inﬁnite system, 858, 861, 864, 868, 971 inﬁnite system with symmetric matrix, 850, 853 linear boundary value problems, representation, 892 linear equation, 898 constant integ ration limits, 502 ﬁrst kind, 502ﬁrst kind, constant integration limits, 217, 573 ﬁrst kind, variable integration limit, 3 operator methods, 549second kind, 502second kind, constant integration limits, 301, 625 second kind, variable integration limit, 127solution methods, 519, 539, 573, 625structure of solutions, 502 variable integration limit, 502 linear normed spaces, 1063linear operator, 502, 1066 eigenfunction, 1066eigenvalue, 1066 linear operators in Hilbert spaces, 1065, 1066 linear ordinary differential equations, 881 linear relations of parabolic cylinder functions, 1034 INDEX 1097 linear space, 1063 complex, 1063real, 1063 linear superposition principle, 502linearly dependent elements, 1063linearly independent elements, 843, 1063 Liouville theorem, generalized, 714 Lipschitz condition, 709, 1054, 1057local solutions of nonlinear integral equation with parameter, 835 logarithm Napierian, base, 905natural, base, 905 logarithmic derivative of gamma function, 1017, 1021 logarithmic function, 42, 45, 77, 83, 85, 164, 165, 180, 185, 187, 242, 244, 255, 256, 334, 335, 349, 351, 353, 905, 943, 955, 965, 980, 985,992, 999, 1002 properties, 906 logarithmic integral, 258, 1009, 1010, 1025logarithmic kernel, 519, 588logarithmic nonlinearity, 419, 472logarithmic singularity, 533, 618 kernel, 533 L p, spaces, 1062 Lp(a,b), Lebesgue space, 1064 M MacDonald function, 266, 1021 mass transfer to particle in ﬂuid ﬂow complicated by surface reaction, 888 Mathieu equation, 1043 modiﬁed, 1045 Mathieu function, 1043, 1044 modiﬁed, 1043, 1045 matrix eigenvalues, 845, 848, 856, 859, 861, 868, 872eigenvectors, orthonormal, 845, 848, 856, 859, 868, 872 orthonormal eigenvectors, 845, 848, 856, 859, 868, 872 mean-square convergence, 501 measurable function, 1058 measurable set, 1060 integration, 1061 measure full, set, 1058zero, set, 1058 measure of set, 1061 mechanics, fracture, 791 Mehler–Fock transform, 270, 518 generalized, 271 Mehler integral, 299, 615Meijer inversion formula, 516Meijer transform, 266, 516, 517Mellin transform, 510, 511, 518, 587, 657, 658 deﬁnition, 510 inverse, 510inverse, tables, 1001Mellin transform ( continued ) inversion formula, 510 properties, 511 tables, 997 method approximation, successive, 566Bateman, 689Bateman, general scheme, 689Bateman, special cases, 690 Bubnov–Galerkin, 697 Bubnov–Galerkin, description, 697Carleman, for characteristic equations, 761Carleman, for equations of convolution type of ﬁrst kind, 606 Carleman, for equations with difference kernels, 610 Carleman, for integral equations of convolution type of second kind, 660 collocation, 692, 693, 815 collocation, for solving hypersingular integral equation, 755 exact, 588Galerkin, 582Kellog’s, for ﬁnding characteristic values in case of symmetric kernel, 645 Krein’s, 588, 683Krein’s, for integral equations, 588 Krein’s, for Wiener–Hopf equations, 679 Multhopp–Kalandiya, 747Newton–Kantorovich, 813, 814, 827Newton–Kantorovich, modiﬁed, 814, 827nonlinear equations with constant integration limits, exact, 817 nonlinear equations with variable integration limit, exact, 809 operator, 549, 654 operator, for solving integral equations of second kind, 654 Picard, 876projection, for solving mixed equations on bounded set, 866 quadrature, 698, 816 829quadrature, general scheme, 698regularization, 704regularization, for complete singular integral equations, 772 regularization, for equations with inﬁnite limits of integration, 702 regularization, Lavrentiev, 621regularization, Tikhonov, 622, 829solution, Laplace transform, 524successive approximation, 566, 811, 826successive approximation, general scheme, 566successive approximation, resolvent, 566 Tikhonov regularization, 829 trace, for approximation of characteristic values, 646 Wiener–Hopf, 671Wiener–Hopf, scheme, 676Zakharov–Shabat, 898 1098 INDEX method based on solution of auxiliary equation, 546 method for solving “quadratic” operator equations, 552special Urysohn equations of ﬁrst kind, 821 special Urysohn equations of second kind, 822 method of approximating kernel by degenerate one, 687 differentiating, for integral equations, 820, 564, 583 differentiation, 564, 583, 810 differentiation, for nonlinear equations with degenerate kernel, 810 equidistant surface, 891 fractional differentiation, 529 fractional integration, for generalized Abel equation, 548 Fredholm determinants, 635 Fredholm determinants, 635 integral transforms, 586, 655, 809, 819 least squares, 695 least squares, description, 695least squares, normal system, 695 left regularization, 775 model solutions, 559, 655, 659 model solutions, description, 560 numerical integration of equation for surface concentration, 891 quadratures, 534, 568, 698 quadratures, algorithm based on trapezoidal rule, 536 quadratures, general scheme, 535, 568 quadratures, trapezoidal rule, 568 replacing kernel by degenerate kernel, 687right regularization, 775 successive approximations, 579, 632, 633, 811, 876 successive approximations, for ODEs, 876 transformation of kernel, 532 methods approximate, for nonlinear equations with constant integ ration limits, 826 approximate, for nonlinear equations with variable integration limit, 811 asymptotic, 618 asymptotic, for solving equations with logarithmic singularity, 618 exact, for integral equations, 588exact, for nonlinear equations with constant integration limits, 817 exact, for nonlinear equations with variable integration limit, 809 for solving complete singular integral equations, 757 for solving equations with difference kernels on ﬁnite interval, 683 for solving integral equations, 499for solving linear equations, 519, 539, 573, 625 for solving multidimensional mixed integral equations, 839 for solving nonlinear integral equations, 805methods ( continued ) for solving singular integral equations of ﬁrst kind, 707 integral equations of ﬁrst kind, 707numerical, for hypersingular equations, 754 numerical, for nonlinear equations with constant integration limits, 826 numerical, for nonlinear equations with variable integration limit, 811 of solving mixed integral equations on ﬁnite interval, 843 of solving mixed integral equations on ring-shaped domain, 855 operator, for solving linear integral equations, 549 regularization, 621 minor, Fredholm, 636 mixed equation, 839 bounded set, projection method, 866circular domain, 841 closed bounded set, 842 ﬁnite interval, 840 Hilbert–Schmidt kernel, ﬁnite interval, 843 Hilbert–Schmidt kernel, ring-shaped domain and given right-hand side, 855 multidimensional, 839 multidimensional, solution methods, 839 on ﬁnite interval, methods of solving, 843on ring-shaped domain, methods of solving, 855 ring-shaped domain, 841 Schmidt kernel and auxiliary conditions on ring-shaped domain, 862 Schmidt Kernel and given right-hand side on interval, 848 mixed multidimensional equation Fredholm operator, 842 Schmidt operator, 843 Schmidt operator, equivalent form, 843 symmetric Fredholm kernel, 842 V olterra and Hilbert–Schmidt types operators, 866 V olterra and Schmidt types operators, 866 mixed operator equation, 866, 869 with given right-hand side, 866 mixed operator equations with auxiliary condi- tions, 869 mixed two-dimensional equation, Schmidt kernel, 841 Schmidt kernel, equivalent form, 842 model solution cosine-shaped right-hand side, 563exponential right-hand side, 561 power-law right-hand side, 562 sine-shaped right-hand side, 562 method, 559, 655, 659 modiﬁed associated Legendre functions, 1033modiﬁed Bessel equation, 1021 modiﬁed Bessel function, 97, 189, 269, 355, 1021 asymptotic expansions, 1022 INDEX 1099 modiﬁed Bessel function ( continued ) deﬁnitions, 1021ﬁrst kind, 266, 1021integral representations, 1022second kind, 266, 1021 modiﬁed Korteweg–de Vries equation, 900modiﬁed Mathieu function, 1043, 1045modiﬁed Newton–Kantorovich method, 814, 827modulus, 278, 583 complementary, 1036elliptic, 1036 Multhopp–Kalandiya method, 747 multidimensional domain, 839 integration, 839 multidimensional equation, mixed, 839 Fredholm operator, 842integral operators of V olterra and Hilbert– Schmidt types, 866 integral operators of V olterra and Schmidt types, 866 Schmidt operator, 843solution methods, 839symmetric Fredholm kernel, 842 multidimensional real-valued functions, 839multiply connected domain, 731multivalued functions, 711 N Napierian base, 906Napierian logarithms, base, 905natural logarithms, base, 905natural numbers, powers, sums, 919Nekrasov equation, 836Neumann function, 1016Neumann problem exterior, reduction to integral equations, 896interior, 895interior, reduction to integral equations, 895 Neumann series, 567, 633 Newton–Kantorovich method, 813, 814, 827 modiﬁed, 814, 827 nodes Chebyshev, 748interpolation, 534quadrature, 534 nondegenerate kernel, 589, 631nonhomogeneous equation, 502, 539, 627, 708, 751 positive solutions, 649solution, 642 nonhomogeneous problem, 604, 742 solution, 721 nonhomogeneous Riemann problem, canonical function, 605 nonhomogeneous Wiener–Hopf equation of second kind, 677 nonisothermal ﬂow in plane channel, 884 nonlinear equation, 807, 834, 899 bifurcation points, 834, 835constant integration limits, 806, 829nonlinear equation ( continued ) constant integration limits, approximate methods, 826 constant integration limits, exact methods, 817constant integration limits, numerical methods, 826 degenerate kernel, 817 degenerate kernel, method of differentiation, 810 eigenfunctions, 834existence theorems, 830ﬁrst kind with constant limits of integration, 433 parameter, local solutions, 835second kind with variable limit of integration, 403 second kind with constant limits of integration, 453 solution methods, 805uniqueness theorems, 830variable limit of integration, 805variable limit of inte gration, approximate methods, 811 variable limit of integration, exact methods, 809variable limit of integration, numerical methods, 811 nonlinear operator, eigenfunctions, 834nonlinear PDEs, 898 nonlinear problem of nonisothermal ﬂow in plane channel, 884 nonlinear V olterra integral equation, 805nonlinearity, 414–416, 418, 467–470, 472–475 exponential, 411, 467general form, 399, 425, 447, 477hyperbolic, 414, 468logarithmic, 419, 472power-law, 408, 444, 464quadratic, 393, 397, 403, 406, 437, 453, 456trigonometric, 420, 473 nonnegative kernels, 648nonorthogonal polynomials, 1050nonsymmetric kernel, 580, 647norm, 501, 644, 839 L 2, 501 operator, 1066 normal system of method of least squares, 695 normality condition, 596 normed space, 1063 linear, 1063 notion of almost everywhere, 1058notion of index, 716nth-order differential equations, boundary value problems, 882 nth-order linear ODE, 876 number e, 905, 906 numbers, 1007 Bernoulli, 1008Euler, 1008natural, powers, sums, 919 numerical integration, method, 891 1100 INDEX numerical methods for hypersingular equations, 754 numerical methods for nonlinear equations with constant integ ration limits, 826 numerical methods for nonlinear equations with variable limit of integration, 811 numerical series, 924 inﬁnite, 924 numerical solution, singular equations, 799 generalized kernels, 792 numerical sums, 921 ﬁnite, 919 O ODE ﬁrst-order, 875, 876method of successive approximations, 876nth-order, linear, 876 second-order, 876 Olevskii transform, 276one-dimensional domain, 839 integration, 839 one-sided equation, 574, 626one-sided Fourier integrals, 593, 594one-sided function, 594open curves, 734 Riemann problem, 734 operator compact, 842, 843, 1067compact, self-adjoint, 843compact, self-adjoint positive, 873compact, self-adjoint positive deﬁnite, 1067compact, self-adjoint positive deﬁnite, eigenvalues, 1067 Erd´elyi–Kober, 532 Fredholm, 758, 842Fredholm, symmetric kernel, generalization, 843Hilbert–Schmidt, 842, 843, 866, 871Hilbert–Schmidt, approximation for eigenfunc- tions, 868, 872 Hilbert–Schmidt, approximation for eigenvalues, 868, 872 Hilbert–Schmidt, eigenfunction, 871 Hilbert–Schmidt, eigenvalues, 871 identity, 842, 873integral, characteristic, 758integral, characteristic, transposed, 758integral, compactness, sufﬁcient condition, 842integral, continuous, 1066integral, domain, 1066integral, domain of deﬁnition, 1066integral, eigenvalues, 867integral, positive deﬁnite, 842integral, self-adjoint, 842, 843, 1067integral, self-adjoint, eigenvalues, 1067integral, self-adjoint, eigenvectors, 1067 integral, spectral radius, 649 integral, spectrum, 1066integral, transposed, 758integral, transposed characteristic, 758operator ( continued ) integral with positive deﬁnite kernel, 843integral with symmetric kernel, 843linear, 502, 1066linear, eigenfunction, 1066 linear, eigenvalue, 1066 linear in Hilbert spaces, 1065, 1066nonlinear, eigenfunctions, 834norm, 1066orthogonal projection, 1067point, continuous, 1066 positive deﬁnite, 842, 1067 regular, 758regularizing, 703Schmidt, 843, 866singular, 758singular, certain properties, 772 V olterra, 842, 873 operator equation general projection problem, 873general projection problem, 873mixed, 866, 869mixed with auxiliary conditions, 869 “quadratic”, 552 solution, 553 operator method, 549, 654operator method for solving integral equations of second kind, 654 operator of fractional integration, 529 operator of orthogonal projection, 846, 852, 857, 563, 870 order, fractional, integral, 529 ordinary differential equations, 527, 547, 686 linear, 881 orthogonal function, 582orthogonal kernels, 634 orthogonal polynomials, 1045 system, 795 orthogonal projection, operator, 846, 852, 857, 563, 870, 1067 orthogonal projector, 1067orthogonal subspaces, 873 direct sum, 845, 863, 869 orthogonal system, 1065 orthogonal vectors, 1065orthogonality properties of Bessel functions, 1019orthonormal basis, 855, 856orthonormal eigenvectors of matrix, 845, 848, 856, 859, 868, 872 orthonormal Legendre polynomials, 844 orthonormal system, 1065 complete, 844, 855 oscillation kernel, 651 deﬁnition, 651theorems, 651 P ℘-function, Weierstrass, 1041 Paley–Wiener transform, 260 INDEX 1101 parabolic cylinder function, 276, 1034 asymptotic expansions, 1034 basic formulas, 1034deﬁnitions, 1034integral representations, 1034 linear relations, 1034 Weber, 1034 parameter of integral equation, 625 parameters, arbitrary, 408, 411, 433, 453 Parseval’s relation Fourier cosine transform, 514 Fourier sine transform, 515 Hankel transform, 515, 516 particular solutions of PDEs, 887 PDEs, nonlinear, 898 PDEs with boundary conditions third kind, 887third kind, reduction to integral equations, 887 permutator, 654 Picard–Goursat equation, 134Picard method, 876 Pochhammer symbol, 1007 Poincar ´e–Bertrand formula, 714 point bifurcation, 835 bifurcation of nonlinear integral equations, 834, 835 collocation, 693cuspidal, 708 regular, 1066 singular, 507 point operator, continuous, 1066 Poisson’s formula, 1018 Poisson equation, 894polar kernel, 519, 532, 574, 588 polynomial Bernoulli, 1050Chebyshev, 109, 1047 Chebyshev, second kind, 750 Euler, 1051Gegenbauer, 1050generalized Laguerre, 1045 Hermite, 108, 1024, 1025, 1048 higher-order in arguments, 6, 133, 311interpolation, Hermite, 716 interpolation, Lagrange, 748 Jacobi, 1049Lagrange interpolation, 748 Laguerre, 110, 1024, 1045 Laguerre, generalized, 1045Legendre, 105, 856, 1030Legendre, orthonormal, 844 nonorthogonal, 1050 orthogonal, 1045orthogonal, system, 795 orthonormal Legendre, 844 ultraspherical, 1050 polynomial form, 553 positive deﬁnite Fredholm kernel, 840 symmetric, 866 positive deﬁnite integral operator, 842positive deﬁnite kernel, 641 positive deﬁnite operator, 1067 positive eigenvalue, 648positive Fredholm kernel, symmetric, 841 positive solutions of nonhomogeneous integral equation, 649 Post–Widder formula, 510potential density, 893 double layer, 893double layer, Gauss formula, 894 equilibrium, 897 Feller, 226Laplace equation, 892 Laplace equation, properties, 892 layer, single, 893Riesz, 226 Roben, 897 single layer, 893volume, 893 volume, Gauss formula, 894 power-law functions, 4, 45, 127, 151, 165, 217, 236, 244, 301, 326, 335, 419, 951, 963, 983,989, 998, 1001 power-law generating function, 557 power-law nonlinearity, 408, 464 power-law nonlinearity that contain arbitrary functions, 444 power function, 905 properties, 905 power series, 925 expansion, 910, 913, 916, 918 power series in parameter, 632 power series of Airy functions, 1023powers, arbitrary, 139, 223, 317, 939, 977 powers, fractional, 138 powers of natural numbers, sums, 919principal value curvilinear integral, 712 singular curvilinear integral, 712singular integral, 709 principle linear superposition, 502superposition, linear, 502 principle of argument, 714 principle of continuity, 714probability integral, 1009 problem Abel, 520boundary value, ﬁrst, 895, 896 boundary value, for nth-order differential equations, 882 boundary value, for ODEs, 877, 881boundary value, for second-order differential equations, 883 boundary value, linear, representation, 892boundary value, Riemann, 595 boundary value, second, 895, 897 Cauchy, for ODEs, reduction to integral equations, 875 Cauchy, for second-order ODEs, 876 1102 INDEX problem ( continued ) Cauchy, for special nth-order linear ODE, 876 Dirichlet, exterior, reduction to integral equations, 896 Dirichlet, interior, 895 Dirichlet, interior, reduction to integral equations, 895 electrostatic, Roben, 897factorization, 676, 679 general projection, 873 general projection, for operator equation, 873 general projection, special case, 846, 852, 857, 870 Hilbert, 742Hilbert, boundary value, 742 homogeneous, 596, 602, 742 homogeneous, solution, 720ill-posed, 623, 624 ill-posed, general notions, 623 interior Dirichlet, 895interior Dirichlet, reduction to integral equations, 895 interior Neumann, 895 interior Neumann, reduction to integral equations, 895 jump, 596 linear boundary value, representation, 892Neumann, exterior, reduction to integral equations, 896 Neumann, interior, 895 Neumann, interior, reduction to integral equations, 895 nonhomogeneous, 604, 742 nonhomogeneous, solution, 721 nonhomogeneous Riemann, canonical function, 605 nonlinear of nonisothermal ﬂow in plane channel, 884 projection, general, for operator equation, 873projection, general, special case, 846, 852, 857, 870 Riemann, 596, 685, 714 Riemann, boundary value, 595Riemann, coefﬁcient, 596, 718 Riemann, discontinuous coefﬁcient, 739 Riemann, exceptional cases, 727Riemann, for half-plane, 725 Riemann, for open curves, 734 Riemann, for real axis, 592 Riemann, general case, 741 Riemann, index, 596, 731Riemann, multiply connected domain, 731 Riemann, nonhomogeneous, canonical function, 605 Riemann, open curves, 734Riemann, right-hand side, 596, 718 Riemann, statement, 718 Riemann, with discontinuous coefﬁcient, 739Riemann, with rational coefﬁcients, 723 Roben electrostatic, 897 second boundary value, 895, 897problem ( continued ) tautochrone, 520well-posed, 623 well-posed, general notions, 623 problem of equivalent regularization, 776problem with rational coefﬁcients, 601process, iteration, 811, 814 product inﬁnite, 910, 916inner, 501, 644scalar, 839 progressions, 919, 924 projection, orthogonal, operator, 846, 852, 857, 563, 870 projection method for solving mixed equations on bounded set, 866 projection problem general, for operator equation, 873 general, special case, 846, 852, 857, 870 projector, orthogonal, 1067properties basic of Gauss hypergeometric functions, 1028 certain of singular operators, 772orthogonality of Bessel functions, 1019 property, semigroup of fractional integration, 529 psi function, 1012, 1013 Q quadratic form, 644quadratic nonlinearity, 393, 397, 403, 406 containing arbitrary functions, 437, 456containing arbitrary parameters, 433, 453 quadrature formula, 534, 793, 815 quadrature method, 698, 816, 829 general scheme, 698 quadrature nodes, 534quadratures, method, 534, 568, 698 method, algorithm based on trapezoidal rule, 536 method, general scheme, 535 R radius spectral, estimates, 649spectral, of integral operator, 649 spectral, of kernel, 649 rational coefﬁcients, 601, 723rational Fourier transforms, 685rational functions, 7, 136, 220, 314, 933, 971 inverse transforms, 506 reaction, surface, 888real-valued functions, multidimensional, classes, 839 real axis H¨older condition, 575 Sokhotski–Plemelj formulas, 713 real linear space, 1063 rectangle rule, 534 recurrent relations, 636 INDEX 1103 reduction formulas, 907, 939, 947 regular operator, 758 regular points, 1066regular value, 301, 625, 637 regularization, 774 Carleman–Vekua, 778equivalent, problem, 776left, 775 left, method, 775 right, 776right, method, 775 regularization in exceptional cases, 779 regularization method, 621, 704 complete singular integral equations, 772 equations with inﬁnite limits of integration, 702 Lavrentiev, 621Tikhonov, 622, 829 regularizer, 774 left, 703right, 704 regularizing operators, 703 relation linear of parabolic cylinder functions, 1034Parseval’s, Fourier cosine transform, 514 Parseval’s, Fourier sine transform, 515 Parseval’s, Hankel transform, 515, 516recurrent, 636 relations between Mellin, Laplace, and Fourier transforms, 511 remainder, 534renewal equation, 203 representation Bessel functions, 1017form of inﬁnite products, 910, 916 Gauss hypergeometric functions, 1028 inverse transforms as asymptotic expansions, 509 inverse transforms as convergent series, 509 modiﬁed Bessel functions, 1022 parabolic cylinder functions, 1034series of Jacobi theta functions, 1042 Tricomi conﬂuent hypergeometric functions, 1024 residual, 692residue theorem, Cauchy, 504residues, 504 resolvent, 539, 567, 626, 633, 635 construction, 633kernel, 844 symmetric kernel, 644 results, auxiliary, 784Riemann boundary value problem, 595, 714 Riemann integral, 1057 Riemann–Liouville derivatives, 529Riemann–Liouville fractional integrals, 529 Riemann problem, 596, 685, 714 coefﬁcient, 596, 718exceptional cases, 727 for half-plane, 725 for multiply connected domain, 731for open curves, 734Riemann problem ( continued ) for real axis, 592general case, 741index, 596, 731nonhomogeneous, canonical function, 605right-hand side, 596, 718statement, 718 with discontinuous coefﬁcient, 739 with rational coefﬁcients, 723 Riemann zeta function, generalized, 277Riesz potential, 226Riesz–Schauder theory, 843Riesz transform, 226right-hand side, 757 equation, 519, 573, 625 integral equation, 539 Riemann problem, 596, 718special, 555 right-sided fractional derivative, 529right-sided fractional integral, 529right Fourier integral, 594right function, 594right regularization, 776 method, 775 right regularizer, 704 ring-shaped domain, 841, 855, 862Roben electrostatic problem, 897Roben potential, 897roots, square, 138, 222, 975rule rectangle, 534Simpson’s, 534 trapezoidal, 534, 568 S scalar, 1063 scalar product, 839scalars, ﬁeld, 1063 scheme general, Bateman method, 689general, method of quadratures, 568general, successive approximation method, 566 Schl¨omilch equation, 254, 452, 825 generalized, 254 Schmidt integral operator, 843, 866Schmidt kernel, 582, 841, 848, 851, 859, 860, 862 Schmidt operator, 866 second-order differential equations, boundary value problems, 883 second-order ODEs, 876second boundary value problem, 895, 897segment, ﬁnite, equation, 683, 685self-adjoint operator, 842, 843, 1067 eigenvalues, 1067eigenvectors, 1067 semiaxis equation, 574, 587, 626, 657 Hilbert transform, 229 semigroup property of fractional integration, 529 1104 INDEX sequence of independent V olterra equations, 847, 858 sequence of independent V olterra equations of second kind, 853, 865, 872 sequence of V olterra equations, 844, 850, 862sequence of V olterra equations of second kind, 855 series bilinear, 640bilinear, iterated kernels, 642convergent, 509functional, inﬁnite, 925hypergeometric, 1028inﬁnite, 919inﬁnite functional, 925inﬁnite numerical, 924Kummer, 1024 Neumann, 567, 633numerical, 924numerical, inﬁnite, 924power, 913, 925power, expansion, 910, 916, 918power in parameter, 632power of Airy functions, 1023trigonometric, in one variable, involving cosine, 928 trigonometric, in one variable, involving sine, 927 trigonometric, in two variables, 930 series representation of Jacobi theta functions, 1042 set, 866 bounded, closed, 842closed bounded, 842measurable, 1060measure, 1061 set of full measure, 1058set of zero measure, 1058sets, measurable, 1060 measurable, integration, 1061zero measure, 1058 several variables, function, 839side right-hand, 757 right-hand, of equation, 519, 573, 625 right-hand, of integral equation, 539right-hand, of Riemann problem, 596right-hand, of Riemann problem, 718right-hand, special, 555 simple hypersingular equation of ﬁrst kind with Cauchy-type kernel, 231 simple hypersingular equation of ﬁrst kind with Hilbert-type kernel, 255 simplest degenerate kernel, 627simplest equation with Cauchy kernel, 743simplest hypersingular equation for ﬁrst kind with Hilbert-type kernel, 754 simplest singular equation of ﬁrst kind with Hilbert kernel, 707, 746 Simpson’s rule, 534sine, 52, 169, 247, 337, 558, 927 hyperbolic, 28, 156, 238, 329 sine integral, 87, 258, 1011sine transform, Fourier, Parseval’s relation, 515single layer potential, 893singular curvilinear integral, principal value, 712 228, 255, 319, 344 Bueckner type, 801Cauchy kernel, complete, 757Cauchy kernel, ﬁrst kind, 707complete, 757, 770, 772ﬁrst kind, 743generalized kernels, 792generalized kernels, direct numerical solution, 792 Hilbert kernel, 759 Hilbert kernel, complete, 759, 780 numerical solution, 799simplest of ﬁrst kind with Hilbert kernel, 707, 746 transposed, 758two-dimensional, 231 singular equations of ﬁrst kind, 707singular integral, 709 principal value, 709, 712 singular kernel, weakly, 532singular operator, 758singular operators, certain properties, 772singular points, 507singularities, solutions, 783singularity logarithmic, 533, 618 logarithmic, kernel, 533weak, 574, 588, 625weak, kernel, 519, 532, 574, 588, 625 singularity exponents, 787, 789skew-symmetric integral equation, 647small λsolution, 620 smooth contour, 708 Sokhotski–Plemelj formula, 713, 785 Sokhotski–Plemelj formulas for real axis, 713solution approximate, 688, 693approximation, 854convolution representation, 526direct numerical of singular integral equations with generalized kernels, 792 exact of simple hypersingular equation with Cauchy-type kernel, 753 exact of simple hypersingular equation with Hilbert-type kernel, 754 fundamental, 881homogeneous problem, 720integral equations, exact, 1–500model, cosine-shaped right-hand side, 563model, exponential right-hand side, 561model, power-law right-hand side, 562 model, sine-shaped right-hand side, 562 nonhomogeneous problem, 721numerical, of singular integral equations, 799 INDEX 1105 solution ( continued ) simple hypersingular equation with Cauchy-type kernel, exact, 753 simple hypersingular equation with Hilbert-type kernel, exact, 754 stable, 623 trivial, 502 solution method, Laplace transform, 524solution method based on Laplace transform, 544 solution of auxiliary equation, method, 546 solution of generalized Abel equation, 531 solution of operator equations of polynomial form, 553 solutions closed-form, case of constant coefﬁcients, 770 closed-form, general case, 771 fundamental, 881local of nonlinear integral equation with parameter, 835 model, method, 559, 655, 659 particular of PDEs, 887positive of nonhomogeneous integral equation, 649 solutions of dual integral equations, general scheme, 611 solutions of nonlinear PDEs, representation in terms of solutions of linear integralequations, 898 solutions singularities, 783 solving linear equations, methods, 519, 539solving “quadratic” operator equations, 552 Sonine transform, 114 space Banach, 1065 basis, 844, 863 complete, 1065complex linear, 1063 Euclidean, 845, 857, 863, 869, 1065 Euclidean, basis, 857, 869Hilbert, 839, 845, 857, 863, 867, 869, 1065 Hilbert, abstract, 873 Hilbert, basis, 857, 867, 869Hilbert, linear operators, 1065, 1066 Hilbert, special basis, 869 H¨olderC α(0, 1), 1064 Lebesgue Lp(a,b), 1064 linear, 1063linear, complex, 1063 linear, normed, 1063 linear, real, 1063normed, 1063 normed linear, 1063 real linear, 1063vector, 1063 space L p, 1062 space of continuous functions C(a,b), 1064 space of functions of bounded variation V(0, 1), 1064 special basis of Hilbert space, 869 special case of general projection problem, 846, 852, 857, 870special functions, 86, 111, 187, 258, 277, 353, 967, 981, 987, 993, 1000, 1004 calculations, 797 properties, 1007 special right-hand side, 555 special Urysohn equations of ﬁrst kind, method, 821 special Urysohn equations of second kind, method, 822 spectral radius, estimates, 649 spectral radius of integral operator, 649 spectral radius of kernel, 649spectrum of Fredholm integral equation, 760 spectrum of operator, 1066 spherical functions, Legendre of ﬁrst kind, 299square integrable function, 501, 502 square root, 9, 138, 222, 975 stable solution, 623statement of Riemann problem, 718 step-function, 1058 integral, 1059 Stieltjes integral, 1055, 1056 basic deﬁnitions, 1055 existence theorems, 1056properties, 1056 Stieltjes integral sum, 1055 Stieltjes transform, 221Stirling formula, 1013 stochastic kernel, 654 structure of solutions to linear integral equations, 502 Struve function, 264, 299, 516, 518 subspace, 1063 orthogonal, 873orthogonal, direct sum, 845, 863, 869 successive approximation method, 566, 579, 632, 633, 811, 826, 876 for ODEs, 876general scheme, 566 resolvent, 566 sufﬁcient condition for compactness of integral operator, 842 sum contain binomial coefﬁcients, 920 contain integers, 920ﬁnite, 919 ﬁnite functional, 922 ﬁnite numerical, 919functional, ﬁnite, 922 integral, Stieltjes, 1055 involving hyperbolic functions, 922involving trigonometric functions, 922 numerical, 921 numerical, ﬁnite, 919of exponential functions, 564 of hyperbolic functions, 564 of orthogonal subspaces, direct, 845, 863, 869of powers of natural numbers, 919, 920 of powers of natural numbers, alternating, 920 of trigonometric functions, 564Stieltjes integral, 1055 1106 INDEX summable function, 1059 integral, 1059 superposition principle, linear, 502surface, equidistant, method, 891surface concentration equation, method of numerical integration, 891integral equations, 890 surface reaction, 888 symbol, Pochhammer, 1007symbols, 1007symmetric deﬁnite Fredholm kernel, 840symmetric equation, 639, 647 Fredholm alternative, 643 symmetric kernel, 573, 577, 625, 639, 645 resolvent, 644 symmetric positive deﬁnite Fredholm kernel, 866 symmetric positive Fredholm kernel, 841system complete, 1065complete orthonormal, 855Fredholm integral equations of second kind, 701 inﬁnite of linear algebraic equations, 858, 861, 864, 868, 971 inﬁnite of linear algebraic equations with symmetric matrix, 850, 853 normal of method of least squares, 695 orthogonal, 1065 orthonormal, 1065orthonormal, complete, 855V olterra integral equations, 549 system of characteristic values, 640system of eigenfunctions, 640 complete, 640incomplete, 640 system of equations, 701 reduction to single equation, 701 system of Fredholm equations of second kind, 701 system of functions complete orthonormal, 844orthonormal, complete, 844 system of orthogonal polynomials, 795 T tables of deﬁnite integrals, 951tables of Fourier cosine transforms, 983tables of Fourier sine transforms, 989tables of indeﬁnite integrals, 933 tables of inverse Laplace transforms, 969 tables of inverse Mellin transforms, 1001tables of Laplace transforms, 961tables of Mellin transforms, 997tangent, 60, 174, 251, 342 hyperbolic, 36, 161, 241, 332 tautochrone problem, 520terms of potentials, 892 theorem analytic continuation, 595, 714Cauchy residue, 504theorem ( continued ) convolution, 507, 513 existence, 875existence, for nonlinear equations, 830existence, for Stieltjes integral, 1056 Fischer–Riesz, 1060 Fredholm, 637, 702, 777Fubini, 1062 generalized Jentzch, 648 generalized Liouville, 595, 714Hilbert–Schmidt, 641, 1067 Jentzch, generalized, 648 Lebesgue on dominated convergence, 1060limit, 507 residue, Cauchy, 504 uniqueness, 875uniqueness, for nonlinear equations, 830 theory Hilbert–Schmidt, 843 Riesz–Schauder, 843 theta functions, Jacobi, 110, 1042 Tikhonov regularization method, 622, 829 total variation of function, 1053trace method for approximation of characteristic values, 646 trace of kernel, 646 transform alternative Fourier, 512Boas, 250 Bochner, 263, 518 Buchholz, 274cosine, Fourier, Parseval’s relation, 514 Crum, 268 divisor, 269Feller, 226 Fourier, 235, 511, 512, 518, 658 Fourier, alternative, 512Fourier, asymmetric form, 512 Fourier, deﬁnition, 512 Fourier, inverse, 512Fourier, inversion formula, 512Fourier, properties, 513 Fourier, rational, 685 Fourier cosine, 514, 518Fourier cosine, asymmetric form, 514 Fourier cosine, Parseval’s relation, 514 Fourier cosine, tables, 983Fourier sine, 514, 518 Fourier sine, asymmetric form, 515 Fourier sine, Parseval’s relation, 515Fourier sine, tables, 989Gauss, 237 generalized Mehler–Fock, 271 Hankel, 261, 515, 518Hankel, Parseval’s relation, 515, 516 Hardy, 264 Hartley, 252, 518Hilbert, 228, 255, 518, 743 Hilbert, on semiaxis, 229 integral, 503, 515integral, kernel, 503 INDEX 1107 transform (continued ) integral, method, 586, 655, 809, 819 integral, table, 517inverse, 503 inverse, representation as asymptotic expan- sions, 509 inverse, representation as convergent series, 509inverse Fourier, 512 inverse Laplace, tables, 969 inverse Mellin, 510inverse Mellin, tables, 1001inverse of rational functions, 506 kernel, 503, 586, 655, 809, 819 Kontorovich–Lebedev, 267, 516, 518Laplace, 235, 505, 511, 518, 524, 544, 658, 809 Laplace, deﬁnition, 505 Laplace, inverse, tables, 969Laplace, inversion formula, 505 Laplace, properties, 507 Laplace, solution method, 524Laplace, tables, 961 Laplace, two-side, 234, 518 Lebedev, 269Mehler–Fock, 270, 518 Mehler–Fock, generalized, 271 Meijer, 516, 517Mellin, 510, 511, 518, 587, 657, 658 Mellin, deﬁnition, 510 Mellin, inverse, 510Mellin, inverse, tables, 1001 Mellin, inversion formula, 510 Mellin, properties, 511Mellin, tables, 997 Olevskii, 276 Paley–Wiener, 260rational Fourier, 685Riesz, 226 sine, Fourier, Parseval’s relation, 515 Sonine, 114Stieltjes, 221 table, 517 two-side Laplace, 234, 518Weber, 265, 518 Weierstrass, 237, 518 transformation, Kummer, 1025transformation of kernel, method, 532 transposed characteristic equation, 758 transposed characteristic operator, 758transposed equation, 573, 575, 625, 627, 637 transposed equation of characteristic equation, 764 transposed operator, 758transposed singular equation, 758 trapezoidal rule, 534, 568 triangle inequality, 501Tricomi conﬂuent hypergeometric function, 273, 1024, 1025 asymptotic expansions, 1024 integral representations, 1024 Tricomi equation, 319, 769Tricomi–Gellerstedt equation, 320 trigonometric functions, 46, 78, 84, 85, 166, 181, 186, 187, 246, 252, 256, 295, 335, 344, 349,352, 353, 564, 907, 922, 944, 956, 966, 981,986, 992, 999, 1003 addition, 908combinations, 176 inverse, 176, 344, 911, 948 inverse, addition, 912inverse, relations, 912inverse, subtraction, 912 of half argument, 909 of multiple arguments, 909of single argument, relations, 908powers, 908 products, 908 relationship, 916subtraction, 908sum, 564 trigonometric nonlinearity, 420, 473 trigonometric series in one variable, involving cosine, 928in one variable, involving sine, 927 in two variables, 930 trivial solution, 502two-dimensional equation of Abel type, 15two-dimensional integral equation, mixed with Schmidt kernel, 841 two-dimensional singular equation, 231 two-side Laplace transform, 234, 518type, convolution, 574, 606, 660, 669 U ultraspherical polynomials, 1050 undetermined coefﬁcients, 692 uniqueness theorems, 875uniqueness theorems for nonlinear equations, 830unknown function of complicated argument, 227, 246, 254 Urysohn equation, 806, 832 ﬁrst kind, 806, 829second kind, 806second kind with degenerate kernel, 818 special of ﬁrst kind, method, 821 special of second kind, method, 822 Urysohn form V olterra equation, 805, 811, 814, 816 V olterra equation, ﬁrst kind, 805, 815 V olterra equation, second kind, 805 V value approximate of eigenvalues of Hilbert–Schmidt kernel, 845 Cauchy principal, 709characteristic, 301, 625, 637, 639, 645, 697characteristic, approximation, 646 characteristic, extremal properties, 644 characteristic, system, 640 1108 INDEX value ( continued ) in Banach space, continuous function of real argument, 840 in Hilbert space, continuous function of real argument, 840 in space of functions square integrable over closed bounded set, continuous function ofreal argument, 842 in space of functions square integrable over ring-shaped domain, continuous function ofreal argument, 841 in space of square integrable functions, continuous function of real argument, 840 regular, 301, 625, 637 variable integration limit, 3, 805, 809, 811 variable limit of integration, 3, 805, 809, 811 variable lower integration limit, 537, 570 variable lower limit of integration, 537, 570 variables, several, function, 839 variation, total, of function, 1053 variation function, bounded, 1056 vector, 1063 axioms for addition, 1063 axioms relating addition of vectors with their multiplication by scalars, 1063 orthogonal, 1065 vector space, 1063 V olterra equation, 549, 805, 877 ﬁrst kind, 519, 524, 565ﬁrst kind, connection with V olterra equations of second kind, 524 ﬁrst kind, existence of solution, 519 ﬁrst kind, in Hammerstein form, 806 ﬁrst kind, in Urysohn form, 805, 815 ﬁrst kind, problems, 520 ﬁrst kind, uniqueness of solution, 519 Hammerstein form, 806 nonlinear, 805 quadratic nonlinearity, 809 reduction to Wiener–Hopf equation, 528 second kind, 524, 539, 565 second kind, connection with V olterra equations of ﬁrst kind, 524 second kind, in Urysohn form, 805 second kind, of Hammerstein form, 816 second kind, reduction to V olterra equations of ﬁrst kind, 565 second kind, sequence, 855 second kind, sequence of independent, 853, 865, 872 sequence, 844, 850, 862 sequence of independent, 847, 858V olterra equation ( continued ) systems, 549Urysohn form, 805, 811, 814, 816 V olterra integral operator, 842V olterra kernel, 839 V olterra operator, 873 volume potential, 893 Gauss formula, 894 W weak singularity, 574, 588, 625 kernel, 519, 532, 574, 588, 625 weakly singular kernel, 532Weber function, 88 Weber parabolic cylinder function, 1034 Weber transform, 265, 518Weierstrass elliptic function, 1041Weierstrass ℘-function, 1041 Weierstrass transform, 237, 518weight function, Jacobi, 793well-posed problem, 623 general notions, 623 Whittaker conﬂuent hypergeometric function, 274, 1027 Whittaker equation, 1027Wiener–Hopf equation, 574, 626, 679 ﬁrst kind, 285, 538, 574, 606Krein’s method, 679second kind, 373, 547, 571, 626, 660, 679second kind, exceptional case, 678second kind, homogeneous, 672second kind, index, 661second kind, nonhomogeneous, 677second kind, solution, 681V olterra equation, 528 Wiener–Hopf method, 671 scheme, 676 Wronskian, conﬂuent hypergeometric function, 1026 Wronskian, Legendre function, 1034 Y Y-transform, 516, 518 Yν-transform, 264 Z Zakharov–Shabat method, 898zero measure, set, 1058zeros of Bessel functions, 1019