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HandbookPolyanin

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A published reference handbook by Andrei D. Polyanin (Chapman & Hall/CRC, 2002) giving exact solutions of more than 2000 linear equations and problems of mathematical physics. It covers parabolic, hyperbolic and elliptic equations with constant and variable coefficients, plus higher-order equations, Green's function formulas, and supplements on special functions and on separation of variables for nonlinear equations. This is a downloaded copy of someone else's book, not Phil's own work.

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Andrei D. PolyaninHANDBOOK OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New Y ork Washington, D.C. This book contains information obtained from authentic and highly regarded sources. Reprinted materialis quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonableefforts have been made to publish reliable data and information, but the author and the publisher cannotassume responsibility for the validity of all materials or for the consequences of their use.Apart from any fair dealing for the purpose of research or private study, or criticism or review, as permittedunder the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, storedor transmitted, in any form or by any means, electronic or mechanical, including photocopying, micro-ﬁlming, and recording, or by any information storage or retrieval system, without the prior permissionin writing of the publishers, or in the case of reprographic reproduction only in accordance with theterms of the licenses issued by the Copyright Licensing Agency in the UK, or in accordance with theterms of the license issued by the appropriate Reproduction Rights Organization outside the UK.All rights reserved. Authorization to photocopy items for internal or personal use, or the personal orinternal use of speciﬁc clients, may be granted by CRC Press LLC, provided that $1.50 per pagephotocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923USA. The fee code for users of the Transactional Reporting Service is ISBN 1-58488-299-9/02/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granteda photocopy license by the CCC, a separate system of payment has been arranged.The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, forcreating new works, or for resale. Speciﬁc permission must be obtained in writing from CRC Press LLCfor such copying.Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identi ﬁcation and explanation, without intent to infringe. Visit the CRC P ress Web site at www.crcpress.com © 2002 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-299-9 Library of Congress Card Number 2001052427 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Library of Congress Cataloging-in-Publication Data Polianin, A. D. (Andrei Dmitrievich) Handbook of linear partial differential equations for engineers and scientists / by Andrei D. Polyanin p. cm. Includes bibliographical references and index.ISBN 1-58488-299-91. Differential equations, Linear--Numerical solution--Handbooks, manuals, etc. I. Title.QA377 .P568 2001515 ′ .354—dc21 2001052427 CIP FOREW ORD Linear partial differential equations arise invarious ®elds ofscience andnumerous applications, e.g., heat andmass transfer theory ,wavetheory ,hydrodynamics, aerodynamics, elasticity ,acous- tics, electrostatics, electrodynamics, electrical engineering, diffraction theory ,quantum mechanics, control theory ,chemical engineering sciences, andbiomechanics. This book presents brief statements andexact solutions ofmore than 2000 linear equations andproblems ofmathematical physics. Nonstationary andstationary equations with constant and variable coef®cients ofparabolic, hyperbolic, andelliptic types areconsidered. Anumber ofnew solutions tolinear equations andboundary value problems aredescribed. Special attention ispaid toequations andproblems ofgeneral form thatdepend onarbitrary functions. Formulas forthe effectiveconstruction ofsolutions tononhomogeneous boundary valueproblems ofvarious types are given.Weconsider second-order andhigher -order equations aswell asthecorresponding boundary value problems. Allinall,thehandbook presents more equations andproblems ofmathematical physics than anyother book currently available. Forthereader' sconvenience, theintroduction outlines some de®nitions andbasic equations, problems, andmethods ofmathematical physics. Italso givesuseful formulas thatenable oneto express solutions tostationary andnonstationary boundary value problems ofgeneral form interms oftheGreen' sfunction. Twosupplements aregivenattheendofthebook. Supplement Alists properties ofthemost common special functions (thegamma function, Bessel functions, degenerate hyper geometric func- tions, Mathieu functions, etc.). Supplement Bdescribes themethods ofgeneralized andfunctional separation ofvariables fornonlinear partial differential equations. Wegivespeci®c examples and anovervie wapplication ofthese methods toconstruct exactsolutions forvarious classes ofsecond-, third-, fourth-, andhigher -order equations (intotal, about 150nonlinear equations with solutions are described). Special attention ispaid toequations ofheat andmass transfer theory ,wavetheory ,and hydrodynamics aswell astomathematical physics equations ofgeneral form thatinvolvearbitrary functions. Theequations inallchapters areinascending order ofcomple xity.Manysections canberead independently ,which facilitates working with thematerial. Anextended table ofcontents willhelp thereader ®ndthedesired equations andboundary value problems. Werefer tospeci®c equations using notation likeª1.8.5.2, ºwhich means ªEquation 2inSubsection 1.8.5. º Toextend therange ofpotential readers with diverse mathematical backgrounds, theauthor strovetoavoidtheuseofspecial terminology where verpossible. Forthisreason, some results are presented schematically ,inasimpli®ed manner (without details), which ishoweverquite suf®cient inmost applications. Separate sections ofthebook canserveasabasis forpractical courses andlectures onequations ofmathematical physics. Theauthor thanks AlexeiZhuro vforuseful remarks onthemanuscript. The author hopes thatthehandbook will beuseful forawide range ofscientists, university teachers, engineers, andstudents invarious areas ofmathematics, physics, mechanics, control, and engineering sciences. Andr eiD.Polyanin Pageiii BASIC NOTATION Latin Character s fundamental solution Im[ ]imaginary partofacomple xquantity Green' sfunction -dimensional Euclidean space, ={- < < ; =1, , } Re[ ]realpartofacomple xquantity , , cylindrical coordinates, = 2+ 2and = cos , = sin , , spherical coordinates, = 2+ 2+ 2and = sin cos , =sin sin , = cos time ( ³0)unkno wnfunction (dependent variable), , space (Cartesian) coordinates1, , Cartesian coordinates in -dimensional space x -dimensional vector ,x={ 1, , } |x|magnitude (length) of -dimensional vector ,|x|= 2 1+ 2 2+ + 2 y -dimensional vector ,y={ 1, , } Greek Character sLaplace operator 2two-dimensional Laplace operator , 2= 2 2+ 2 2 3three-dimensional Laplace operator , 3= 2 2+ 2 2+ 2 2 -dimensional Laplace operator , = =1 2 2 ( )Dirac delta function; - ( ) ( - ) = ( ),where ( )isanycontinuous function,!>0 " Kroneck erdelta, "= #1if = $, 0if ¹ $%( )Heaviside unitstepfunction, %( )= #1if ³0, 0if <0 Brief Notation forDeriv atives& '= &&, & = &&, & '('= &2 &2, & = &2 &2(partial derivatives) )= , )*) = 2 2, )*)*) = 3 3, ( )= (derivativesfor = ( )) Special Functions (See Also Supplement A) Ai( )=1+ , 0cos -1 3 3+ /. Airy function; Ai( )=10 11 3 21 33 -2 3 3 32 . Ce2 + 4( , 5)= , =0 2 + 4 2 + 4cosh[(2 + 6) ] evenmodi®ed Mathieu functions, where 6=0,1; Ce2 + 4( , 5)=ce2 + 4( 78, 5) Pagev ce2 ( , 5)= , =0 2 2 cos2 even +-periodic Mathieu functions; these satisfy the equation )*)+( !- 2 5cos2 ) = 0, where != ! 2 ( 5) are eigenvalues ce2 +1( , 5)= , =0 2 +1 2 +1cos[( 2 +1) ] even 2 +-periodic Mathieu functions; these satisfy the equation )*)+( !- 2 5cos2 ) = 0, where != ! 2 +1( 5) are eigenvalues9 := 9 :( ) parabolic cylinder function (see Paragraph 7.3.4-1); it satis®es the equation )*)+ -8;+1 2-1 4 2 .= 0 erf =2<+ 0exp -- =2 . = error function erfc =2<+ ,exp -- =2 . = complementary error function>( )=(-1) ?2 - ?-2.Hermite polynomial>(1) :( )= @ :( )+ 7BA :( ) Hankel function of ®rst kind, 72= -1>(2) :( )= @ :( )- 7BA :( ) Hankel function of second kind, 72= -1C( !, D, E; )= 1 + , =1( !) ( D) ( E) !hypergeometric function, ( !) = !( !+ 1) ( !+ - 1)F:( )= , =0( G2) : +2 ! H( ;+ + 1)modi®ed Bessel function of ®rst kind@ :( )= , =0(-1) ( G2) : +2 ! H( ;+ + 1)Bessel function of ®rst kind2 :( )= + 2 F - :( )- FI:( ) sin( +;)modi®ed Bessel function of second kindJ K( )=1! - K ? -L + K ?- .generalized Laguerre polynomialM( )=1!2 ( 2- 1) Legendre polynomialM "( )=(1 - 2) "32 " " M( ) associated Legendre functions Se2 + 4( , 5)= , =0 N2 + 4 2 + 4sinh[( 2 + 6) ] odd modi®ed Mathieu functions, where 6= 0,1; Se2 + 4( , 5)= - 7se2 + 4( 78, 5) se2 ( , 5)= , =0 N2 2 sin2 odd +-periodic Mathieu functions; these satisfy the equation )*)+( !- 2 5cos2 ) = 0, where != D2 ( 5) are eigenvalues se2 +1( , 5)= , =0 N2 +1 2 +1sin[(2 +1) ] odd 2 +-periodic Mathieu functions; these satisfy the equation )*)+( !- 2 5cos2 ) = 0, where != D2 +1( 5) are eigenvaluesA :( )= @ :( ) cos( +;)- @- :( ) sin( +;)Bessel function of second kindO( P, )= 0 ?- Q= R-1 = incomplete gamma functionH( P)= , 0 ?- Q=R-1 = gamma functionS( !, D; )= 1 + , =1( !) ( D) !degenerate hypergeometric function, ( !) = !( !+ 1) ( !+ - 1) Page vi AUTHOR Andr eiD.Polyanin, D.Sc., Ph.D .,isanoted scientist ofbroad interests, who works invarious areas ofmathematics, mechanics, andchemical engineering sciences. A.D.Polyanin graduated from theDepartment ofMechanics andMathematics oftheMosco wState University in1974. He recei vedhisPh.D. degree in1981 andD.Sc. degree in1986 at theInstitute forProblems inMechanics oftheRussian (former USSR) Academy ofSciences. Since 1975, A.D.Polyanin has been amember ofthestaffoftheInstitute forProblems inMe- chanics oftheRussian Academy ofSciences. Professor Polyanin hasmade important contrib utions tode- veloping newexact andapproximate analytical methods ofthe theory ofdifferential equations, mathematical physics, integral equations, engineering mathematics, nonlinear mechanics, theory ofheat andmass transfer ,andchemical hydrodynamics. Heob- tained exact solutions forseveralthousand ordinary differential, partial differential, mathematical physics, andintegralequations. Professor Polyanin isanauthor of27books inEnglish, Russian, German, andBulgarian, aswell asover120research papers andthree patents. Hehaswritten anumber offundamental handbooks, including A.D.Polyanin andV.F.Zaitse v,Handbook ofExact Solutions forOrdinary Differential Equations ,CRC Press, 1995; A.D.Polyanin andA.V.Manzhiro v,Handbook ofIntegralEquations , CRC Press, 1998; andA.D.Polyanin, V.F.Zaitse v,andA.Moussiaux, Handbook ofFirstOrder Partial Differential Equations ,Gordon andBreach, 2001. In1991, A.D.Polyanin wasawarded aChaplygin Prize oftheUSSR Academy ofSciences for hisresearch inmechanics. Addr ess: Institute forProblems inMechanics, RAS, 101Vernadsk yAvenue, Building 1,117526 Mosco w,Russia E-mail: [email protected] Pagevii CONTENTS Foreword BasicNotatio nandRemark s Autho r Introduction .Som eDe®nitions ,Formulas ,Methods ,andSolution s 0.1.Classi®catio nofSecond-Orde rPartialDifferentia lEquation s 0.1.1 .Equation swithTwoIndependen tVariable s 0.1.2 .Equation swithManyIndependen tVariable s 0.2.BasicProblem sofMathematica lPhysic s 0.2.1 .Initia landBoundar yConditions .Cauch yProblem .Boundar yValueProblem s 0.2.2 .First,Second ,Third ,andMixedBoundar yValueProblem s 0.3.Propertie sandParticula rSolution sofLinea rEquation s 0.3.1 .Homogeneou sLinea rEquation s 0.3.2 .Nonhomogeneou sLinea rEquation s 0.4.Separatio nofVariable sMetho d 0.4.1 .Genera lDescriptio noftheSeparatio nofVariable sMetho d 0.4.2 .Solutio nofBoundar yValueProblem sforParaboli candHyperboli cEquation s 0.5.IntegralTransform sMetho d 0.5.1 .MainIntegralTransform s 0.5.2 .Laplac eTransfor mandItsApplicatio ninMathematica lPhysic s 0.5.3 .Fourie rTransfor mandItsApplicatio ninMathematica lPhysic s 0.6.Representatio noftheSolutio noftheCauch yProble mviatheFundamenta lSolutio n 0.6.1 .Cauch yProble mforParaboli cEquation s 0.6.2 .Cauch yProble mforHyperboli cEquation s 0.7.Nonhomogeneo husBoundar yValueProblem swithOneSpac eVariable .Representation ofSolution sviatheGreen 'sFunctio n 0.7.1 .Problem sforParaboli cEquation s 0.7.2 .Problem sforHyperboli cEquation s 0.8.Nonhomogeneou sBoundar yValueProblem swithManySpac eVariables .Representa- tionofSolution sviatheGreen 'sFunctio n 0.8.1 .Problem sforParaboli cEquation s 0.8.2 .Problem sforHyperboli cEquation s 0.8.3 .Problem sforEllipti cEquation s 0.8.4 .Compariso noftheSolutio nStructure sforBoundar yValueProblem sfor Equation sofVariou sTypes 0.9.Constructio noftheGreen 'sFunctions .Genera lFormula sandRelation s 0.9.1 .Green 'sFunction sofBoundar yValueProblem sforEquation sofVariou sTypes inBounde dDomain s 0.9.2 .Green 'sFunction sAdmittin gIncomplet eSeparatio nofVariable s 0.9.3 .Constructio nofGreen 'sFunction sviaFundamenta lSolution s Page ix 0.10.Duhamel 'sPrinciple sinNonstationar yProblem s 0.10.1 .Problem sforHomogeneou sLinea rEquation s 0.10.2 .Problem sforNonhomogeneou sLinea rEquation s 0.11.Transformation sSimplifyin gInitia landBoundar yCondition s 0.11.1 .Transformation sThatLeadtoHomogeneou sBoundar yCondition s 0.11.2 .Transformation sThatLeadtoHomogeneou sInitia landBoundar yCondition s 1.Paraboli cEquation swithOneSpac eVariabl e 1.1.Constan tCoef®cien tEquation s 1.1.1 .HeatEquatio n T '= !2T 2 1.1.2 .Equatio noftheForm T '= !2T 2+ S( , ) 1.1.3 .Equatio noftheForm T '= !2T 2+ D + S( , ) 1.1.4 .Equatio noftheForm T '= !2T 2+ D T + S( , ) 1.1.5 .Equatio noftheForm T '= !2T 2+ D T + E + S( , ) 1.2.HeatEquatio nwithAxia lorCentra lSymmetr yandRelate dEquation s 1.2.1 .Equatio noftheForm T '= !- 2T U2+1U T U . 1.2.2 .Equatio noftheForm T '= !- 2T U2+1U T U .+ S( , ) 1.2.3 .Equatio noftheForm T '= !- 2T U2+2U T U . 1.2.4 .Equatio noftheForm T '= !- 2T U2+2U T U .+ S( , ) 1.2.5 .Equatio noftheForm T '= 2T 2+1-2 V T 1.2.6 .Equatio noftheForm T '= 2T 2+1-2 V T + S( , ) 1.3.Equation sContainin gPowerFunction sandArbitrar yParameter s 1.3.1 .Equation softheForm T '= !2T 2+ ( , ) 1.3.2 .Equation softheForm T '= !2T 2+ ( , ) T 1.3.3 .Equation softheForm T '= !2T 2+ ( , ) T + W( , ) + X( , ) 1.3.4 .Equation softheForm T '=( !+ D) 2T 2+ ( , ) T + W( , ) 1.3.5 .Equation softheForm T '=( !2+ DY+ E) 2T 2+ ( , ) T + W( , ) 1.3.6 .Equation softheForm T '= ( ) 2T 2+ W( , ) T + X( , ) 1.3.7 .Equation softheForm T '= ( , ) 2T 2+ W( , ) T + X( , ) 1.3.8 .Liquid-Fil mMassTransfe rEquatio n(1- 2) T = !2T 2 1.3.9 .Equation softheForm ( , ) T + W( , ) T = 2T 2+ X( , ) 1.4.Equation sContainin gExponentia lFunction sandArbitrar yParameter s 1.4.1 .Equation softheForm T '= !2T 2+ ( , ) 1.4.2 .Equation softheForm T '= !2T 2+ ( , ) T 1.4.3 .Equation softheForm T '= !2T 2+ ( , ) T + W( , ) 1.4.4 .Equation softheForm T '= ! 2T 2+ ( , ) T + W( , ) 1.4.5 .Equation softheForm T '= ! ?V 2T 2+ ( , ) T + W( , ) 1.4.6 .Othe rEquation s 1.5.Equation sContainin gHyperboli cFunction sandArbitrar yParameter s 1.5.1 .Equation sContainin gaHyperboli cCosin e 1.5.2 .Equation sContainin gaHyperboli cSine 1.5.3 .Equation sContainin gaHyperboli cTangen t 1.5.4 .Equation sContainin gaHyperboli cCotangen t Page x 1.6.Equation sContainin gLogarithmi cFunction sandArbitrar yParameter s 1.6.1 .Equation softheForm T '= !2T 2+ ( , ) T + W( , ) 1.6.2 .Equation softheForm T '= ! 2T 2+ ( , ) T + W( , ) 1.7.Equation sContainin gTrigonometri cFunction sandArbitrar yParameter s 1.7.1 .Equation sContainin gaCosin e 1.7.2 .Equation sContainin gaSine 1.7.3 .Equation sContainin gaTangen t 1.7.4 .Equation sContainin gaCotangen t 1.8.Equation sContainin gArbitrar yFunction s 1.8.1 .Equation softheForm T '= !2T 2+ ( , ) 1.8.2 .Equation softheForm T '= !2T 2+ ( , ) T 1.8.3 .Equation softheForm T '= !2T 2+ ( , ) T + W( , ) 1.8.4 .Equation softheForm T '= ! 2T 2+ ( , ) T + W( , ) 1.8.5 .Equation softheForm T '= ! ?V 2T 2+ ( , ) T + W( , ) 1.8.6 .Equation softheForm T '= ( ) 2T 2+ W( , ) T + X( , ) 1.8.7 .Equation softheForm T '= ( ) 2T 2+ W( , ) T + X( , ) 1.8.8 .Equation softheForm T '= ( , ) 2T 2+ W( , ) T + X( , ) 1.8.9 .Equation softheForm Z( ) T '= [ 6( ) T \- 5( ) + S( , ) 1.9.Equation sofSpecia lForm 1.9.1 .Equation softheDiffusio n(Thermal )Boundar yLaye r 1.9.2 .One-Dimensiona lSchrÈodinge rEquatio n 7^ ] X T '= - _ `2 2 " 2T 2+ a( ) 2.Paraboli cEquation swithTwoSpac eVariable s 2.1.HeatEquatio n T '= ! 2 2.1.1 .Boundar yValueProblem sinCartesia nCoordinate s 2.1.2 .Problem sinPolarCoordinate s 2.1.3 .Axisymmetri cProblem s 2.2.HeatEquatio nwithaSourc e T '= ! 2 + S( , , ) 2.2.1 .Problem sinCartesia nCoordinate s 2.2.2 .Problem sinPolarCoordinate s 2.2.3 .Axisymmetri cProblem s 2.3.Othe rEquation s 2.3.1 .Equation sContainin gArbitrar yParameter s 2.3.2 .Equation sContainin gArbitrar yFunction s 3.Paraboli cEquation swithThreeorMoreSpac eVariable s 3.1.HeatEquatio n T '= ! 3 3.1.1 .Problem sinCartesia nCoordinate s 3.1.2 .Problem sinCylindrica lCoordinate s 3.1.3 .Problem sinSpherica lCoordinate s 3.2.HeatEquatio nwithSourc e T '= ! 3 + S( , , , ) 3.2.1 .Problem sinCartesia nCoordinate s 3.2.2 .Problem sinCylindrica lCoordinate s 3.2.3 .Problem sinSpherica lCoordinate s 3.3.Othe rEquation swithThre eSpac eVariable s 3.3.1 .Equation sContainin gArbitrar yParameter s 3.3.2 .Equation sContainin gArbitrar yFunction s 3.3.3 .Equation softheForm b( , , ) T '=div[ !( , , )Ñ ]- 5( , , ) + S( , , , ) Page xi 3.4.Equation swith Spac eVariable s 3.4.1 .Equation softheForm T '= ! + S( 1, , , ) 3.4.2 .Othe rEquation sContainin gArbitrar yParameter s 3.4.3 .Equation sContainin gArbitrar yFunction s 4.Hyperboli cEquation swithOneSpac eVariabl e 4.1.Constan tCoef®cien tEquation s 4.1.1 .WaveEquatio n 2T ' 2= !22T 2 4.1.2 .Equation softheForm 2T ' 2= !22T 2+ S( , ) 4.1.3 .Equatio noftheForm 2T ' 2= !22T 2- D + S( , ) 4.1.4 .Equatio noftheForm 2T ' 2= !22T 2- D T + S( , ) 4.1.5 .Equatio noftheForm 2T ' 2= !22T 2+ D T + E + S( , ) 4.2.WaveEquatio nwithAxia lorCentra lSymmetr y 4.2.1 .Equation softheForm 2T ' 2= !2- 2T U2+1U T U . 4.2.2 .Equatio noftheForm 2T ' 2= !2- 2T U2+1U T U .+ S( , ) 4.2.3 .Equatio noftheForm 2T ' 2= !2- 2T U2+2U T U . 4.2.4 .Equatio noftheForm 2T ' 2= !2- 2T U2+2U T U .+ S( , ) 4.2.5 .Equatio noftheForm 2T ' 2= !2- 2T U2+1U T U .- D + S( , ) 4.2.6 .Equatio noftheForm 2T ' 2= !2- 2T U2+2U T U .- D + S( , ) 4.3.Equation sContainin gPowerFunction sandArbitrar yParameter s 4.3.1 .Equation softheForm 2T ' 2=( !+ D) 2T 2+ E T + + S( , ) 4.3.2 .Equation softheForm 2T ' 2=( !2+ D) 2T 2+ Ec T + + S( , ) 4.3.3 .Othe rEquation s 4.4.Equation sContainin gtheFirstTimeDerivative 4.4.1 .Equation softheForm 2T ' 2+ T '= !22T 2+ D T + E + S( , ) 4.4.2 .Equation softheForm 2T ' 2+ T '= ( ) 2T 2+ W( ) T + X( ) + S( , ) 4.4.3 .Othe rEquation s 4.5.Equation sContainin gArbitrar yFunction s 4.5.1 .Equation softheForm Z( ) 2T ' 2= [ 6( ) T \- 5( ) + S( , ) 4.5.2 .Equation softheForm 2T ' 2+ !( ) T '= D( ) d [ 6( ) T \- 5( ) e+ S( , ) 4.5.3 .Othe rEquation s 5.Hyperboli cEquation swithTwoSpac eVariable s 5.1.WaveEquatio n 2T ' 2= !2 2 5.1.1 .Problem sinCartesia nCoordinate s 5.1.2 .Problem sinPolarCoordinate s 5.1.3 .Axisymmetri cProblem s 5.2.Nonhomogeneou sWaveEquatio n 2T ' 2= !2 2 + S( , , ) 5.2.1 .Problem sinCartesia nCoordinate s 5.2.2 .Problem sinPolarCoordinate s 5.2.3 .Axisymmetri cProblem s 5.3.Equation softheForm 2T ' 2= !2 2 - D + S( , , ) 5.3.1 .Problem sinCartesia nCoordinate s 5.3.2 .Problem sinPolarCoordinate s 5.3.3 .Axisymmetri cProblem s Page xii 5.4.TelegraphEquatio n 2T ' 2+ T '= !2 2 - D + S( , , ) 5.4.1 .Problem sinCartesia nCoordinate s 5.4.2 .Problem sinPolarCoordinate s 5.4.3 .Axisymmetri cProblem s 5.5.Othe rEquation swithTwoSpac eVariable s 6.Hyperboli cEquation swithThreeorMoreSpac eVariable s 6.1.WaveEquatio n 2T ' 2= !2 3 6.1.1 .Problem sinCartesia nCoordinate s 6.1.2 .Problem sinCylindrica lCoordinate s 6.1.3 .Problem sinSpherica lCoordinate s 6.2.Nonhomogeneou sWaveEquatio n 2T ' 2= !2 3 + S( , , , ) 6.2.1 .Problem sinCartesia nCoordinate s 6.2.2 .Problem sinCylindrica lCoordinate s 6.2.3 .Problem sinSpherica lCoordinate s 6.3.Equation softheForm 2T ' 2= !2 3 - D + S( , , , ) 6.3.1 .Problem sinCartesia nCoordinate s 6.3.2 .Problem sinCylindrica lCoordinate s 6.3.3 .Problem sinSpherica lCoordinate s 6.4.TelegraphEquatio n 2T ' 2+ T '= !2 3 - D + S( , , , ) 6.4.1 .Problem sinCartesia nCoordinate s 6.4.2 .Problem sinCylindrica lCoordinate s 6.4.3 .Problem sinSpherica lCoordinate s 6.5.Othe rEquation swithThre eSpac eVariable s 6.5.1 .Equation sContainin gArbitrar yParameter s 6.5.2 .Equatio noftheForm b( , , ) 2T ' 2=div[ !( , , )Ñ \- 5( , , ) + S( , , , ) 6.6.Equation swith Spac eVariable s 6.6.1 .WaveEquatio n 2T ' 2= !2 6.6.2 .Nonhomogeneou sWaveEquatio n 2T ' 2= !2 + S( 1, , , ) 6.6.3 .Equation softheForm 2T ' 2= !2 - D + S( 1, , , ) 6.6.4 .Equation sContainin gtheFirstTimeDerivative 7.Ellipti cEquation swithTwoSpac eVariable s 7.1.Laplac eEquatio n 2 =0 7.1.1 .Problem sinCartesia nCoordinat eSyste m 7.1.2 .Problem sinPolarCoordinat eSyste m 7.1.3 .Othe rCoordinat eSystems .Conforma lMapping sMetho d 7.2.Poisso nEquatio n 2 =- S(x) 7.2.1 .Preliminar yRemarks .Solutio nStructur e 7.2.2 .Problem sinCartesia nCoordinat eSyste m 7.2.3 .Problem sinPolarCoordinat eSyste m 7.2.4 .Arbitrar yShap eDomain .Conforma lMapping sMetho d 7.3.Helmholt zEquatio n 2 + f =- S(x) 7.3.1 .Genera lRemarks ,Results ,andFormula s 7.3.2 .Problem sinCartesia nCoordinat eSyste m 7.3.3 .Problem sinPolarCoordinat eSyste m 7.3.4 .Othe rOrthogona lCoordinat eSystems .Ellipti cDomai n Page xiii 7.4.Othe rEquation s 7.4.1 .Stationar ySchrÈodinge rEquatio n 2 = ( , ) 7.4.2 .ConvectiveHeatandMassTransfe rEquation s 7.4.3 .Equation sofHeatandMassTransfe rinAnisotropi cMedi a 7.4.4 .Othe rEquation sArisin ginApplication s 7.4.5 .Equation softheForm !( ) 2T 2+ 2T 2+ D( ) T + E( ) =- S( , ) 8.Ellipti cEquation swithThreeorMoreSpac eVariable s 8.1.Laplac eEquatio n 3 =0 8.1.1 .Problem sinCartesia nCoordinate s 8.1.2 .Problem sinCylindrica lCoordinate s 8.1.3 .Problem sinSpherica lCoordinate s 8.1.4 .Othe rOrthogona lCurvilinea rSystem sofCoordinate s 8.2.Poisso nEquatio n 3 + S(x)=0 8.2.1 .Preliminar yRemarks .Solutio nStructur e 8.2.2 .Problem sinCartesia nCoordinate s 8.2.3 .Problem sinCylindrica lCoordinate s 8.2.4 .Problem sinSpherica lCoordinate s 8.3.Helmholt zEquatio n 3 + f =- S(x) 8.3.1 .Genera lRemarks ,Results ,andFormula s 8.3.2 .Problem sinCartesia nCoordinate s 8.3.3 .Problem sinCylindrica lCoordinate s 8.3.4 .Problem sinSpherica lCoordinate s 8.3.5 .Othe rOrthogona lCurvilinea rCoordinate s 8.4.Othe rEquation swithThre eSpac eVariable s 8.4.1 .Equation sContainin gArbitrar yFunction s 8.4.2 .Equation softheFormdiv[ !( , , )Ñ ]- 5( , , ) =- S( , , ) 8.5.Equation swith Spac eVariable s 8.5.1 .Laplac eEquatio n =0 8.5.2 .Othe rEquation s 9.Highe r-Orde rPartia lDiffe rentia lEquation s 9.1.Third-Orde rPartialDifferentia lEquation s 9.2.Fourth-Orde rOne-Dimensiona lNonstationar yEquation s 9.2.1 .Equation softheForm T '+ !24T 4= S( , ) 9.2.2 .Equation softheForm 2T ' 2+ !24T 4=0 9.2.3 .Equation softheForm 2T ' 2+ !24T 4= S( , ) 9.2.4 .Equation softheForm 2T ' 2+ !24T 4+ = S( , ) 9.2.5 .Othe rEquation s 9.3.Two-Dimensiona lNonstationar yFourth-Orde rEquation s 9.3.1 .Equation softheForm T '+ !2- 4T 4+ 4T 4 .= S( , , ) 9.3.2 .Two-Dimensiona lEquation softheForm 2T ' 2+ !2 =0 9.3.3 .Three -and -Dimensiona lEquation softheForm 2T ' 2+ !2 =0 9.3.4 .Equation softheForm 2T ' 2+ !2 + = S( , , ) 9.3.5 .Equation softheForm 2T ' 2+ !2- 4T 4+ 4T 4 .+ = S( , , ) 9.4.Fourth-Orde rStationar yEquation s 9.4.1 .Biharmoni cEquatio n =0 9.4.2 .Equation softheForm = S( , ) Page xiv 9.4.3 .Equation softheForm - f = S( , ) 9.4.4 .Equation softheForm 4T 4+ 4T 4= S( , ) 9.4.5 .Equation softheForm 4T 4+ 4T 4+ = S( , ) 9.4.6 .StokesEquatio n(Axisymmetri cFlowsofViscou sFluids ) 9.5.Highe r-Orde rLinea rEquation swithConstan tCoef®cient s 9.5.1 .Fundamenta lSolutions .Cauch yProble m 9.5.2 .Ellipti cEquation s 9.5.3 .Hyperboli cEquation s 9.5.4 .RegularEquations .Numbe rofInitia lCondition sintheCauch yProble m 9.5.5 .SomeSpecial- TypeEquation s 9.6.Highe r-Orde rLinea rEquation swithVariabl eCoef®cient s 9.6.1 .Equation sContainin gtheFirstTimeDerivative 9.6.2 .Equation sContainin gtheSecon dTimeDerivative 9.6.3 .Nonstationar yProblem swithManySpac eVariable s 9.6.4 .SomeSpecial- TypeEquation s Supplemen tA.Specia lFunction sandThei rPropertie s A.1.SomeSymbol sandCoef®cient s A.1.1 .Factorial s A.1.2 .Binomia lCoef®cient s A.1.3 .Pochhamme rSymbo l A.1.4 .Bernoull iNumber s A.2.ErrorFunction sandExponentia lIntegral A.2.1 .ErrorFunctio nandComplementar yErrorFunctio n A.2.2 .Exponentia lIntegral A.2.3 .Logarithmi cIntegral A.3.SineIntegralandCosin eIntegral.Fresne lIntegrals A.3.1 .SineIntegral A.3.2 .Cosin eIntegral A.3.3 .Fresne lIntegrals A.4.Gamm aandBetaFunction s A.4.1 .Gamm aFunctio n A.4.2 .BetaFunctio n A.5.Incomplet eGamm aandBetaFunction s A.5.1 .Incomplet eGamm aFunctio n A.5.2 .Incomplet eBetaFunctio n A.6.Besse lFunction s A.6.1 .De®nition sandBasicFormula s A.6.2 .IntegralRepresentation sandAsymptoti cExpansion s A.6.3 .Zero sandOrthogonalit yPropertie sofBesse lFunction s A.6.4 .HankelFunction s(Besse lFunction softheThirdKind ) A.7.Modi®e dBesse lFunction s A.7.1 .De®nitions .BasicFormula s A.7.2 .IntegralRepresentation sandAsymptoti cExpansion s A.8.AiryFunction s A.8.1 .De®nitio nandBasicFormula s A.8.2 .PowerSerie sandAsymptoti cExpansion s Page xv A.9.Degenerat eHype rgeometri cFunction s A.9.1 .De®nition sandBasicFormula s A.9.2 .IntegralRepresentation sandAsymptoti cExpansion s A.10 .Hype rgeometri cFunction s A.10.1 .De®nitio nandSomeFormula s A.10.2 .BasicPropertie sandIntegralRepresentation s A.11 .Whitta kerFunction s A.12 .Legendr ePolynomial sandLegendr eFunction s A.12.1 .De®nitions .BasicFormula s A.12.2 .Zero sofLegendr ePolynomial sandtheGeneratin gFunctio n A.12.3 .Associate dLegendr eFunction s A.13 .Paraboli cCylinde rFunction s A.13.1 .De®nitions .BasicFormula s A.13.2 .IntegralRepresentation sandAsymptoti cExpansion s A.14 .Mathie uFunction s A.14.1 .De®nition sandBasicFormula s A.15 .Modi®e dMathie uFunction s A.16 .Orthogona lPolynomial s A.16.1 .Laguerr ePolynomial sandGeneralize dLaguerr ePolynomial s A.16.2 .Chebysh evPolynomial sandFunction s A.16.3 .Hermit ePolynomia l A.16.4 .Jacob iPolynomial s Supplemen tB.Method sofGeneralize dandFunctiona lSeparatio nofVariable sin Nonlinea rEquation sofMathematica lPhysics B.1.Introductio n B.1.1 .Preliminar yRemark s B.1.2 .Simpl eCase sofVariabl eSeparatio ninNonlinea rEquation s B.1.3 .Example sofNontr ivialVariabl eSeparatio ninNonlinea rEquation s B.2.Method sofGeneralize dSeparatio nofVariable s B.2.1 .Structur eofGeneralize dSeparabl eSolution s B.2.2 .Solutio nofFunctiona lDifferentia lEquation sbyDifferentiatio n B.2.3 .Solutio nofFunctiona lDifferentia lEquation sbySplittin g B.2.4 .Simpli®e dSchem eforConstructin gExac tSolution sofEquation swithQuadratic Nonlinearitie s B.3.Method sofFunctiona lSeparatio nofVariable s B.3.1 .Structur eofFunctiona lSeparabl eSolution s B.3.2 .Specia lFunctiona lSeparabl eSolution s B.3.3 .Differentiatio nMetho d B.3.4 .Splittin gMethod .Reductio ntoaFunctiona lEquatio nwithTwoVariable s B.3.5 .SomeFunctiona lEquation sandTheirSolutions .Exac tSolution sofHeatand WaveEquation s B.4.First-Orde rNonlinea rEquation s B.4.1 .Preliminar yRemark s B.4.2 .Individua lEquation s B.5.Second-Orde rNonlinea rEquation s B.5.1 .Paraboli cEquation s B.5.2 .Hyperboli cEquation s B.5.3 .Ellipti cEquation s B.5.4 .Equation sContainin gMixedDerivatives Page xvi B.5.5 .Genera lFormEquation s B.6.Third-Orde rNonlinea rEquation s B.6.1 .Stationar yHydrodynami cBoundar yLaye rEquation s B.6.2 .Nonstationar yHydrodynami cBoundar yLaye rEquation s B.7.Fourth-Orde rNonlinea rEquation s B.7.1 .Stationar yHydrodynami cEquation s(Navier±StokesEquations ) B.7.2 .Nonstationar yHydrodynami cEquation s B.8.Highe r-Orde rNonlinea rEquation s B.8.1 .Equation softheForm T '= C-L, , , T , , g T g . B.8.2 .Equation softheForm 2T ' 2= C-L, , , T , , g T g . B.8.3 .Othe rEquation s Refe rence s Page xvii Introduction Some De®nitions,Formulas, Methods, and Solutions 0.1. Classi®cation of Second-Order Partial Differential Equations 0.1.1. Equations with Two Independent Variables 0.1.1-1. Examples of equations encountered in applications. Three basic types of partial differential equations are distinguishedÐ parabolic ,hyperbolic , and elliptic . The solutions of the equations pertaining to each of the types have their own characteristic qualitative differences. The simplest example of a parabolic equation is the heat equation - 2 2= 0, ( 1) where the variables and play the role of time and the spatial coordinate, respectively. Note that equation (1) contains only one highest derivative term. The simplest example of a hyperbolic equation is the wave equation2 2- 2 2= 0, ( 2) where the variables and play the role of time and the spatial coordinate, respectively. Note that the highest derivative terms in equation (2) differ in sign. The simplest example of an elliptic equation is the Laplace equation2 2+ 2 2= 0, ( 3) where and play the role of the spatial coordinates. Note that the highest derivative terms in equation (3) have like signs. Any linear partial differential equation of the second-order with two independent variables can be reduced, by appropriate manipulations, to a simpler equation which has one of the three highest derivative combinations speci®ed above in examples (1), (2), and (3). 0.1.1-2. Types of equations. Characteristic equations. Consider a second-order partial differential equation with two independent variables which has the general form( , ) 2 2+ 2 ( , ) 2 + ( , ) 2 2= , , , , , ( 4) where , , are some functions of and that have continuous derivatives up to the second-order inclusive.* Given a point ( , ), equation (4) is said to be parabolic if 2- = 0, hyperbolic if 2- >0, elliptic if 2- <0 at this point. In order to reduce equation (4) to a canonical form, one should ®rst write out the characteristic equation 2- 2 + 2= 0, which splits into two equations - + 2- = 0, ( 5) and - - 2- = 0, ( 6) and ®nd their general integrals. 0.1.1-3. Canonical form of parabolic equations (case 2- = 0). In this case, equations (5) and (6) coincide and have a common general integral,( , )= . By passing from , to new independent variables , in accordance with the relations= ( , ), = ( , ), where = ( , ) is any twice differentiable function that satis®es the condition of nondegeneracy of the Jacobian ( , )( , )in the given domain, we reduce equation (4) to the canonical form2 2= 1 , , , , . ( 7) As , one can take = or = . It is apparent that, just as the heat equation (1), the transformed equation (7) has only one highest-derivative term. !In the degenerate case where the function 1does not depend on the derivative , equation (7) is an ordinary differential equation for the variable , in which serves as a parameter. 0.1.1-4. Canonical form of hyperbolic equations (case 2- >0). The general integrals( , )= 1, "( , )= 2 of equations (5) and (6) are real and different. These integrals determine two different families of real characteristics. * The right-hand side of equation (4) may be nonlinear. The classi®cation and the procedure of reducing such equations to a canonical form are only determined by the left-hand side of the equation. Page 2 By passing from , to new independent variables , in accordance with the relations= ( , ), = "( , ), we reduce equation (4) to2 = 2 , , , , . This is the so-called ®rst canonical form of a hyperbolic equation. The transformation= + #, = - # brings the above equation to another canonical form,2 2- 2 #2= 3 , #, , , # , where 3= 4 2. This is the so-called second canonical form of a hyperbolic equation. Apart from notation, the left-hand side of the last equation coincides with that of the wave equation (2). 0.1.1-5. Canonical form of elliptic equations (case 2- <0). In this case the general integrals of equations (5) and (6) are complex conjugate; these determine two families of complex characteristics. Let the general integral of equation (5) have the form( , )+ $ "( , )= , $2= -1 , where ( , ) and "( , ) are real-valued functions. By passing from , to new independent variables , in accordance with the relations= ( , ), = "( , ), we reduce equation (4) to the canonical form2 2+ 2 2= 4 , , , , . Apart from notation, the left-hand side of the last equation coincides with that of the Laplaceequation (3). 0.1.2. Equations with Many Independent Variables Consider a second-order partial differential equation with %independent variables 1, &'&'&, (that has the form ()*, +=1 *+(x) 2 * += x, , 1, &'&'&, ( , ( 8) where *+are some functions that have continuous derivatives with respect to all variables to the second-order inclusive, and x= { 1, &'&'&, (}. [The right-hand side of equation (8) may be nonlinear. The left-hand side only is required for the classi®cation of this equation.] At a point x=x0, the following quadratic form is assigned to equation (8):,= ()*, +=1 *+(x0) *+. ( 9) Page 3 TABLE 1 Classi®cation of equations with many independent variables Type of equation (8) at a point x=x0 Coef®cients of the canonical form (11) Parabolic (in the broad sense) At least one coef®cient of the *is zero Hyperbolic (in the broad sense) All *are nonzero and some *differ in sign Elliptic All *are nonzero and have like signs By an appropriate linear nondegenerate transformation *= ().- =1 / * - - ( $= 1, &'&'&, %) ( 10) the quadratic form (9) can be reduced to the canonical form,= ()*=1 *2*, ( 11) where the coef®cients *assume the values 1,-1, and 0. The number of negative and zero coef®cients in (11) does not depend on the way in which the quadratic form is reduced to the canonical form. Table1present sthebasiccriteri aaccordin gtowhic htheequation swithmanyindependent variables are classi®ed. Suppose all coef®cients of the highest derivatives in (8) are constant, *+=const. By introducing the new independent variables 1, &'&'&, (in accordance with the formulas *= (0 - =1/ * - - , where the/ * - are the coef®cients of the linear transformation (10), we reduce equation (8) to the canonical form ()*=1 * 2 2*= 1 y, , 1, &'&'&, ( . ( 12) Here, the coef®cients *are the same as in the quadratic form (11), and y= { 1, &'&'&, (}. 1 !Among the parabolic equations, it is conventional to distinguish the parabolic equations in the narrow sense, i.e., the equations for which only one of the coef®cients, - , is zero, while the other *is the same, and in this case the right-hand side of equation (12) must contain the ®rst-order partial derivative with respect to - . 2 !In turn, the hyperbolic equations are divided into normal hyperbolic equationsÐ for which all *but one have like signsÐand ultrahyperbolic equationsÐfor which there are two or more positive *and two or more negative *. Speci®c equations of parabolic, elliptic, and hyperbolic types will be discussed further in Subsection 0.2.354 References for Section 0.1: V . M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), S. J. Farlow (1982), D. Colton (1988), E. Zauderer (1989), A. N. Tikhonov and A. A. Samarskii (1990), I. G. Petrovsky (1991), W. A. Strauss (1992),R. B. Guenther and J. W. Lee (1996), D. Zwillinger (1998). 0.2. Basic Problems of Mathematical Physics 0.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems Every equation of mathematical physics governs in®nitely many qualitatively similar phenomenaor processes. This follows from the fact that differential equations have in®nitely many particular Page 4 solutions. The speci®c solution that describes the physical phenomenon under study is separatedfrom the set of particular solutions of the given differential equation by means of the initial and boundary conditions. Throughout this section, we consider linear equations in the%-dimensional Euclidean space6 ( or in an open domain 7 8 6 ( (exclusive of the boundary) with a suf®ciently smooth boundary9= 7. 0.2.1-1. Parabolic equations. Initial and boundary conditions. In general, a linear second-order partial differential equation of the parabolic type with %independent variables can be written as - :x, ;[ ]= <(x, ), ( 1) where:x, ;[ ]º ()*, +=1 *+(x, ) 2 * ++ ()*=1 *(x, ) *+ (x, ) , ( 2) x= { 1, &'&'&, (}, ()*, +=1 *+(x, ) *+³ = ()*=1 2*, =>0. Parabolic equations govern unsteady thermal, diffusion, and other phenomena dependent on time . Equation (1) is called homogeneous if <(x, )º 0. Cauchy problem ( ³ 0,x 8 6 ( ). Find a function that satis®es equation (1) for >0and the initial condition= >(x) at = 0. ( 3) Boundary value problem * ( ³ 0,x 8 7). Find a function that satis®es equation (1) for >0, the initial condition (3), and the boundary condition? x, ;[ ]= @(x, ) at x 8 9( >0). ( 4) In general, ? x, ;is a ®rst-order linear differential operator in the space variables xwith coef®cient de- pendent on xand . The basic types of boundary conditions are described below in Subsection 0.2.2. The initial condition (3) is called homogeneous if >(x)º 0. The boundary condition (4) is called homogeneous if @(x, )º 0. 0.2.1-2. Hyperbolic equations. Initial and boundary conditions. Consider a second-order linear partial differential equation of the hyperbolic type with %independent variables of the general form2 2+ (x, ) - :x, ;[ ]= <(x, ), ( 5) where the linear differential operator :x, ;is de®ned by (2). Hyperbolic equations govern unsteady wave processes, which depend on time . Equation (5) is said to be homogeneous if <(x, )º 0. Cauchy problem ( ³ 0,x 8 6 ( ). Find a function that satis®es equation (5) for >0and the initial conditions= >0(x) at = 0,; = >1(x) at = 0.(6) *Boundary value problems for parabolic and hyperbolic equations are sometimes called mixed orinitial-boundary value problems . Page 5 Boundary value problem ( ³ 0,x 8 7). Find a function that satis®es equation (5) for >0, the initial conditions (6), and boundary condition (4). The initial conditions (6) are called homogeneous if >0(x)º 0and >1(x)º 0. Goursat problem. On the characteristics of a hyperbolic equation with two independent variables, the values of the unknown function are prescribed. 0.2.1-3. Elliptic equations. Boundary conditions. In general, a second-order linear partial differential equation of elliptic type with %independent variables can be written as - :x[ ]= <(x), ( 7) where:x[ ]º ()*, +=1 *+(x) 2 * ++ ()*=1 *(x) *+ (x) , ( 8)()*, +=1 *+(x) *+³ = ()*=1 2*, =>0. Elliptic equations govern steady-state thermal,diffusion, and other phenomena independent of time . Equation (7) is said to be homogeneous if <(x)º 0. Boundary value problem . Find a function that satis®es equation (7) and the boundary condition? x[ ]= @(x) at x 8 9. ( 9) In general, ? xis a ®rst-order linear differential operator in the space variables x. The basic types of boundary conditions are described below in Subsection 0.2.2. The boundary condition (9) is called homogeneous if @(x)º 0. The boundary value problem (7)±(9) is said to be homogeneous if <º 0and @º 0. 0.2.2. First, Second, Third, and Mixed Boundary Value Problems For any (parabolic, hyperbolic, and elliptic) second-order partial differential equations, it is con- ventional to distinguish four basic types of boundary value problems, depending on the form of the boundary conditions (4) [see also the analogous equation (9)]. For simplicity, here we con®ne ourselves to the case where the coef®cients *+of equations (1) and (5) have the special form*+(x, )= (x, ) A *+, A *+= B1if $= C, 0if $¹ C. This situation is rather frequent in applications; such coef®cients are used to describe variousphenomena (processes) in isotropic media. First boundary value problem. The function(x, ) takes prescribed values at the boundary 9 of the domain,(x, )= @1(x, ) for x 8 9. ( 10) Second boundary value problem. The derivative along the (outward) normal is prescribed at the boundary 9of the domain, D= @2(x, ) for x 8 9. ( 11) In heat transfer problems, where is temperature, the left-hand side of the boundary condition (11) is proportional to the heat ¯ux per unit area of the surface 9. Page 6 Third boundary value problem. A linear relationship between the unknown function and its normal derivative is prescribed at the boundary 9of the domain, D+ E(x, ) = @3(x, ) for x 8 9. ( 12) Usually, it is assumed that E(x, )=const. In mass transfer problems, where is concentration, the boundary condition (12) with @3º 0describes a surface chemical reaction of the ®rst order. Mixed boundary value problems. Conditions of various types, listed above, are set at different portions of the boundary 9. If @1º 0, @2º 0, or @3º 0, the respective boundary conditions (10), (11), (12) are said to be homogeneous.354 References for Section 0.2: V . M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), M. A. Pinsky (1984), R. Leis (1986), R. Haberman (1987), A. A. Dezin (1987), A. G. Mackie (1989), A. N. Tikhonov and A. A. Samarskii (1990), I. Stakgold (2000). 0.3. Properties and Particular Solutions of Linear Equations 0.3.1. Homogeneous Linear Equations 0.3.1-1. Preliminary remarks. For brevity, in this paragraph a homogeneous linear partial differential equation will be written asF[ ]= 0. ( 1) For second-order linear parabolic and hyperbolic equations, the linear differential operator F[ ] is de®ned by the left-hand side of equations (1) and (5) from Subsection 0.2.1, respectively. It is assumed that equation (1) is an arbitrary homogeneous linear partial differential equation of anyorder in the variables, 1, &'&'&, (with suf®ciently smooth coef®cients. A linear operator Fpossesses the propertiesF[ 1+ 2]= F[ 1]+ F[ 2],F[ G ]= G F[ ], G=const. An arbitrary homogeneous linear equation (1) has a trivial solution, º 0. A function is called a classical solution of equation (1) if , when substituted into (1), turns the equation into an identity and if all partial derivatives of that occur in (1) are continuous; the notion of a classical solution is directly linked to the range of the independent variables. In what follows, we usually write ªsolutionº instead of ªclassical solutionº for brevity. 0.3.1-2. Usage of particular solutions for the construction of other particular solutions. Below are some properties of particular solutions of homogeneous linear equations. 1 H. Let 1= 1(x, ), 2= 2(x, ), &'&'&, - = - (x, ) be any particular solutions of the homogeneous equation (1). Then the linear combination= G1 1+ G2 2+ I'I'I+ G - - (2) with arbitrary constants G1, G2, &'&'&, G - is also a solution of equation (1); in physics, this property is known as the principle of linear superposition . Suppose { - }is an in®nite sequence of solutions of equation (1). Then the series J 0 - =1 - , irrespective of its convergence, is called a formal solution of (1). If the solutions - are classical, the series is uniformly convergent, and the sum of the series has all the necessary particular derivatives, then the sum of the series is a classical solution of equation (1). Page 7 2H. Let the coef®cients of the differential operator Fbe independent of time . If equation (1) has a particular solution K = K (x, ), then the partial derivatives of K with respect to time,*K , 2K 2, &'&'&, -K - , &'&'&, are also solutions of equation (1)3H. Let the coef®cients of the differential operator Fbe independent of the space variables 1, &'&'&, (. If equation (1) has a particular solution K = K (x, ), then the partial derivatives of K with respect to the space coordinates,K 1, K 2, K 3, &'&'&, 2K 21, 2K 1 2, &'&'&, - + LK - 2 L 3, &'&'&, are also solutions of equation (1) If the coef®cients of Fare independent of only one space coordinate, say 1, and equation (1) has a particular solution K = K (x, ), then the partial derivativesK 1, 2K 21, &'&'&, -K - 1, &'&'& are also solutions of equation (1).4H. Let the coef®cients of Fbe constant and let equation (1) have a particular solution K = K (x, ). Then any particular derivatives of K with respect to time and the space coordinates (inclusive mixed derivatives),K , K 1, &'&'&, 2K 22, 2K M 1, &'&'&, -K - 3, &'&'&, are solutions of equation (1).5H. Suppose equation (1) has a particular solution dependent on a parameter N, K = K (x, ; N), and the coef®cients of Fare independent of N(but can depend on time and the space coordinates). Then, by differentiating K with respect to N, one obtains other solutions of equation (1),K N, 2K N2, &'&'&, -K N - , &'&'& Let some constants N1, &'&'&, N - belong to the range of the parameter N. Then the sum= G1 K (x, ; N1)+ I'I'I+ G -K (x, ; N - ), ( 3) where G1, &'&'&, G - are arbitrary constants, is also a solution of the homogeneous linear equation (1). The number of terms in sum (3) can be both ®nite and in®nite. 6H. Another effective way of constructing solutions involves the following. The particular solutionK (x, ; N), which depends on the parameter N(as before, it is assumed that the coef®cients of Fare independent of N), is ®rst multiplied by an arbitrary function ( N). Then the resulting expression is integrated with respect to Nover some interval [ O,/]. Thus, one obtains a new function,P QR K (x, ; N) ( N) N, which is also a solution of the original homogeneous linear equation. The properties listed in Items 1H±6Henable one to use known particular solutions to construct other particular solutions of homogeneous linear equations of mathematical physics. * Here and in what follows, it is assumed that the particular solution S Tis differentiable suf®ciently many times with respect to Uand V1, WMWMW, V X(or the parameters). Page 8 TABLE 2 Homogeneous linear partial differential equations that admit separable solutions No Form of equation (1) Form of particular solutions 1Equation coef®cients are constant (x, )= Gexp( Y +/1 1+ I'I'I+/ ( (),Y,/1, &'&'&,/ (are related by an algebraic equation 2Equation coef®cients are independent of time (x, )= Z [ ;"(x),Yis an arbitrary constant, x= { 1, &'&'&, (} 3Equation coef®cients are independent of the coordinates 1, &'&'&, ( (x, )=exp(/1 1+ I'I'I+/ ( () "( ),/1, &'&'&,/ (are arbitrary constants 4Equation coef®cients are independent of the coordinates 1, &'&'&, -(x, )=exp(/1 1+ I'I'I+/ - - ) "( , - +1, &'&'&, (),/1, &'&'&,/ - are arbitrary constants 5 F[ ]= :;[ ]+ :x[ ], operator :;depends on only , operator :xdepends on only x (x, )= ( ) "(x),( ) satis®es the equation :;[ ]+ Y = 0,"(x) satis®es the equation :x[ "]- Y "= 0 6 F[ ]= :;[ ]+ :1[ ]+ I'I'I+ : ([ ], operator :;depends on only , operator : - depends on only - (x, )= ( ) "1( 1) &'&'&M" (( (),( ) satis®es the equation :;[ ]+ Y = 0," - ( - ) satis®es the equation : - [ " - ]+/ -" - = 0,Y+/1+ I'I'I+/ (= 0 7 F[ ]= >0( 1) :;[ ]+ (0 - =1 > - ( 1) : - [ ], operator :;depends on only , operator : - depends on only - (x, )= ( ) "1( 1) &'&'&M" (( (),:;[ ]+ Y = 0,: - [ " - ]+/ -" - = 0, E= 2, &'&'&, %,>1( 1) :1[ "1]- \]Y >0( ^1)+ (0 - =2/ -> - ( ^1) _ "1= 0 8 F[ `]= a `a b+ :1, ;[ `]+ I'I'I+ : ( , ;[ `], where : - , ;[ `]= L c0d=0 > -d( ^ - ,b) a d`a ^ d - `(x,b)= "1( ^1,b) "2( ^2,b) &'&'&e" (( ^ (,b),a " -a b+ : - , ;[ " - ]= Y - (b) " - , E= 1, &'&'&, %,Y1(b)+ Y2(b)+ I'I'I+ Y ((b)= 0 0.3.1-3. Separable solutions. Many homogeneous linear partial differential equations have solutions that can be represented as theproduct of functions depending on different arguments. Such solutions are referred to as separablesolutions. Table2present sthemostcommonl yencountere dtypesofhomogeneou slinea rdifferentia lequa- tions with many independent variables that admit exact separable solutions. Linear combinations ofparticular solutions that correspond to different values of the separation parameters,Y,/1, &'&'&,/ (, are also solutions of the equations in question. For brevity, the word ªoperatorº is used to denoteªlinear differential operator.º Foraconstan tcoef®cien tequatio n(seethe®rstrowinTable2),theseparatio nparameter smust satisfy the algebraic equationf ( Y,/1, &'&'&,/ ()= 0, ( 4) which results from substituting the solution into the equation (1). In physical applications, equa-tion (4) is usually referred to as a variance equation. Any%of the %+ 1separation parameters in (4) can be treated as arbitrary. Page 9 Note that constant coef®cient equations also admit more sophisticated solutions; see the second and third rows, the last column. Theeight hrowofTable2present sthecaseofincomplet eseparatio nofvariable swher ethe solution is separated with respect to the space variables ^1, &'&'&, ^ (but is not separated with respect to timeb. !For stationary equations, which do not depend onb, one should set Y= 0, :;[ `]º 0, and (b)º1inrows1,6,and7ofTable2. 0.3.1-4. Solutions in the form of in®nite series inb. 1H. The equationa `a b= g[ `], where gis an arbitrary linear differential operator of the second (or any) order that only depends on the space variables, has the formal series solution`(x,b)= >(x)+ J ).- =1 b -E! g - [ >(x)], g - [ >]= g h5g - -1[ >] i, where >(x) is an arbitrary in®nitely differentiable function. This solution satis®es the initial condition`(x,0)= >(x). 2 H. The equationa2`a b2= g[ `], where gis a linear differential operator, just as in Item 1H, has a formal solution represented by the sum of two series as`(x,b)= J ) - =0 b2 - (2 E)! g - [ >(x)]+ J ) - =0 b2 - +1 (2 E+ 1)! g - [ @(x)], where >(x) and @(x) are arbitrary in®nitely differentiable functions. This solution satis®es the initial conditions `(x,0)= >(x) anda ; `(x,0)= @(x). 0.3.2. Nonhomogeneous Linear Equations 0.3.2-1. Simplest properties of nonhomogeneous linear equations. For brevity, we write a nonhomogeneous linear partial differential equation in the formF[ `]= <(x,b), ( 5) where the linear differential operator Fis de®ned above, see the beginning of Paragraph 0.3.1-1. Below are the simplest properties of particular solutions of the nonhomogeneous equation (5). 1H. If K ` j(x,b) is a particular solution of the nonhomogeneousequation (5) and K `0(x,b) is a particular solution of the corresponding homogeneous equation (1), then the sumG K `0(x,b)+ K ` j(x,b), where Gis an arbitrary constant, is also a solution of the nonhomogeneous equation (5). The following, more general statement holds: The general solution of the nonhomogeneous equation (5)is the sum of the general solution of the corresponding homogeneous equation (1) and any particularsolution of the nonhomogeneous equation (5).2H. Suppose `1and `2are solutions of nonhomogeneous linear equations with the same left-hand side and different right-hand sides, i.e.,F[ `1]= <1(x,b), F[ `2]= <2(x,b). Then the function `= `1+ `2is a solution of the equationF[ `]= <1(x,b)+ <2(x,b). Page 10 0.3.2-2. Fundamental and particular solutions of stationary equations. Consider the second-order linear stationary (time-independent) nonhomogeneous equation:x[ `]= <(x). ( 6) Here, :xis a linear differential operator of the second (or any) order of general form whose coef®cients are dependent on x, where x 8 6 ( . A distribution k k= k k(x,y) that satis®es the equation with a special right-hand side:x[ k k]= A(x-y) ( 7) is called a fundamental solution corresponding to the operator :x. In (7), A(x) is an l-dimensional Dirac delta function and the vector quantity y= { m1, n'n'n, m o}appears in equation (7) as an l-dimensional free parameter. It is assumed that y 8 6o. The l-dimensional Dirac delta function possesses the following basic properties: 1. A(x)= A( ^1) A( ^2) n'n'npA( ^ o), 2. P q r<(y) A(x-y) sy= <(x), where A( ^ - ) is the one-dimensional Dirac delta function, <(x) is an arbitrary continuous function, and sy= s m1 n'n'n s m o. For constant coef®cient equations, a fundamental solution always exists; it can be found by means of the l-dimensional Fourier transform (see Paragraph 0.5.3-2). The fundamental solution k k= k k(x,y) can be used to construct a particular solution of the linear stationary nonhomogeneous equation (6) for arbitrary continuous <(x); this particular solution is expressed as follows:`(x)= Pq r<(y) k k(x,y) sy. ( 8)t uv w x y z {The fundamental solution k kis not unique; it is de®ned up to an additive term`0= `0(x) which is an arbitrary solution of the homogeneous equation :x[ `0]= 0.t uv w x y | {For constant coef®cient differential equations, the fundamental solution possesses the property k k(x,y)= k k(x-y).t uv w x y } {The right-hand sides of equations (6) and (7) are often pre®xed with the minus sign. In this case, formula (8) remains valid.t uv w x y ~ {Particular solutions of linear nonstationary nonhomogeneous equations can be ex- pressed in terms of the fundamental solution of the Cauchy problem; see Section 0.6.5 References for Section 0.3: G. A. Korn and T. M. Korn (1968), W. Miller, Jr. (1977), R. P. Kanwal (1983), L. H Èormander (1983, 1990), V . S. Vladimirov (1988), A. N. Tikhonov and A. A. Samarskii (1990), D. Zwillinger (1998), A. D. Polyanin,A. V . Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998). 0.4. Separation of Variables Method 0.4.1. General Description of the Separation of Variables Method 0.4.1-1. Scheme of solving linear boundary value problems by separation of variables. Manylinea rproblem sofmathematica lphysic scanbesolvedbyseparatio nofvariables .Figur e1 depicts the scheme of application of this method to solve boundary value problems for second-orderhomogeneous linear equations of the parabolic and hyperbolic type* with homogeneous boundary * The separation of variables method is also used to solve linear boundary value problems for elliptic equations. Page 11 conditions and nonhomogeneous initial conditions. For simplicity, problems with two independentvariables^andb, are considered, with ^1£ ^£ ^2andb³ 0. Theschem epresente dinFig.1applie stoboundar yvalueproblem sforsecond-orde rlinear homogeneous partial differential equations of the formO(b) a2`a b2+ (b) a `a b= ( ^) a2`a ^2+ ( ^) a `a ^+ h5( ^)+ (b) i` (1) with homogeneous linear boundary conditions, 1a `+ 1 `= 0 at ^= ^1, 2a `+ 2 `= 0 at ^= ^2,(2) and arbitrary initial conditions,`= 0( ^) atb= 0, ( 3)a `= 1( ^) atb= 0. ( 4) For parabolic equations, which correspond to (b)º 0in (1), only the initial condition (3) is set. Below we consider the basic stages of the method of separation of variables in more detail. We assume that the coef®cients of equation (1) and boundary conditions (2) meet the followingrequirements:(b), (b), (b), ( ^), ( ^), ( ^) are continuous functions,(b)³ 0,0< ( ^) < ,| 1| + | 1|>0,| 2| + | 2|>0. 0.4.1-2. Search for particular solutions. Derivation of equations and boundary conditions. The approach is based on searching for particular solutions of equation (1) in the product form`( ^,b)= ( ^) (b). ( 5) After separation of the variables and elementary manipulations, one arrives at the following linearordinary differential equations for the functions= ( ^) and = (b):( ^) + ( ^) +[ + ( ^)] = 0, ( 6)(b) + (b) +[ - (b)] = 0. ( 7) These equations contain a free parameter called the separation constant. With the notation adopte dinFig.1,equation s(6)and(7)canberewritte nasfollows: 1( ^, , , )+ = 0 and 2(b, , , )+ = 0. Substituting (5) into (2) yields the boundary conditions for = ( ^): 1 + 1 = 0 at ^= ^1, 2 + 2 = 0 at ^= ^2.(8) The homogeneous linear ordinary differential equation (6) in conjunction with the homogeneouslinear boundary conditions (8) make up an eigenvalue problem. Page 12 Figure 1. Scheme of solving linear boundary value problems by separation of variables (for parabolic equations, the function 2does not depend on , and = 0). Page 13 0.4.1-3. Solution of eigenvalue problems. Orthogonality of eigenfunctions. Suppose 1= 1( , ) and 2= 2( , ) are linearly independent particular solutions of equation (6). Then the general solution of this equation can be represented as the linear combination= 1 1( , )+ 2 2( , ), ( 9) where 1and 2are arbitrary constants. Substituting solution (9) into the boundary conditions (8) yields the following homogeneous linear algebraic system of equations for 1and 2: 11( ) 1+ 12( ) 2= 0, 21( ) 1+ 22( ) 2= 0,(10) where ¢¡( )= h ( ¡) + ¡i= £. For system (10) to have nontrivial solutions, its determinant must be zero; we have 11( ) 22( )- 12( ) 21( )= 0. ( 11) Solving the transcendental equation (11) for ,one obtains the eigenvalues = o, where l=1,2, n'n'n For these values of , there are nontrivial solutions of equation (6), o( )= 12( o) 1( , o)- 11( o) 2( , o), ( 12) which are called eigenfunctions (these functions are de®ned up to a constant multiplier). To facilitate the further analysis, we represent equation (6) in the form [ ¤( ) ] +[ ¥( )- ¦( )] = 0, ( 13) where¤( )=exp §©¨ ª( )( ) « ¬, ¦( )= - ®( )( )exp §¯¨ ª( )( ) « ¬, ¥( )=1( )exp §©¨ ª( )( ) « ¬. (14) It follows from the adopted assumptions (see the end of Paragraph 0.4.1-1) that ¤( ), ¤ °±( ), ¦( ), and ¥( ) are continuous functions, with ¤( ) >0and ¥( ) >0. The eigenvalue problem (13), (8) is known to possess the following properties: 1. All eigenvalues ²1, ²2, ³'³'³are real, and ² ´ µ ¶as · µ ¶; consequently, the number of negative eigenvalues is ®nite. 2. The system of eigenfunctions ¸1( ), ¸2( ), ³'³'³is orthogonal on the interval 1£ £ 2with weight ¹( ), i.e.,¨ ±2±1 ¹( ) ¸ ´( ) ¸ º( )« = 0 for ·¹ ». ( 15) 3. If ¼ ( )³ 0, ½1 ¾1£ 0, ½2 ¾2³ 0, ( 16) there are no negative eigenvalues. If ¼ º 0and¾1=¾2= 0, the least eigenvalue is ²1= 0and the corresponding eigenfunction is ¸1=const. Otherwise, all eigenvalues are positive, provided that conditions (16) are satis®ed; the ®rst inequality in (16) is satis®ed if®( )£ 0. Subsection 1.8.9 presents some estimates for the eigenvalues ² ´and eigenfunctions ¸ ´( ). Page 14 0.4.2. Solution of Boundary Value Problems for Parabolic and Hyperbolic Equations 0.4.2-1. Solution of boundary value problems for parabolic equations. For parabolic equations, one should set ¿( À)º 0in (1) and (7). In addition, we assume that ( À) >0 and Á( À) < min ² ´. First we search for the solutions of equation (7) corresponding to the eigenvalues ²= ² ´and satisfying the normalizing conditions Â ´(0)= 1to obtainÂ ´( À)=exp Ã©¨ Ä 0 Á( Å)- ² ´( Å) « Å ¬. ( 17) Then the solution of the original nonstationary boundary value problem (1)±(3) for the parabolic equation is sought in the form Æ ( , À)= Ç È´=1 É ´ ¸ ´( ) Â ´( À), ( 18) where theÉ ´are arbitrary constants and the functions Æ´( , À)= ¸ ´( ) Â ´( À) are particular solu- tions (5) satisfying the boundary conditions (2). By the principle of linear superposition, series (18) is also a solution of the original partial differential equation which satis®es the boundary conditions. To determine the coef®cientsÉ ´, we substitute series (18) into the initial condition (3), thus obtainingÇ È´=1 É ´ ¸ ´( )= Ê0( ). Multiplying this equation by ¹( ) ¸ ´( ), integrating the resulting relation with respect to over the interval 1£ £ 2, and taking into account the properties (15), we ®ndÉ ´=1Ë¸ ´ Ë2 ¨ ±2±1 ¹( ) ¸ ´( ) Ê0( )« , Ë¸ ´ Ë2= ¨ ±2±1 ¹( ) ¸2´( )« . ( 19) The weight function ¹( ) is de®ned in (14). Relations (18), (12), (17), and (19) give a formal solution of the nonstationary boundary value problem (1)±(3) if ¿( À)º 0. Example 1. Let Ì( Í)= 1and Î( Í)= 0. Substituting these values into (17) yieldsÏ Ð( Í)=exp(- Ñ ÐÍ). ( 20) If the function Ò0( Ó) is twice continuously differentiable and the compatibility conditions (see Paragraph 0.4.2-3) are satis®ed, then series (18) is convergent and admits termwise differentiation, once with respect to Íand twice with respect to Ó. In this case, relations (18), (12), (19), and (20) give the classical smooth solution of problem (1)±(3). [If Ò0( Ó) is not as smooth as indicated or if the compatibility conditions are not met, then series (18) may converge to a discontinuous function, thus giving only a generalized solution.] 0.4.2-2. Solution of boundary value problems for hyperbolic equations. For hyperbolic equations, the solution of the boundary value problem (1)±(4) is sought in the form Æ ( , À)=Ç È´=1 ¸ ´( ) ÔÉ ´ Â ´1( À)+ Õ ´ Â ´2( À) Ö. ( 21) Here,É ´and Õ ´are arbitrary constants. The functions Â ´1( À) and Â ´2( À) are particular solutions of the linear equation (7) for Â(with ²= ² ´) which satisfy the conditionsÂ ´1(0)= 1, Â °´1(0)= 0; Â ´2(0)= 0, Â °´2(0)= 1. ( 22) Page 15 Substituting solution (21) into the initial conditions (3)±(4) yieldsÇ È´=1 É ´ ¸ ´( )= Ê0( ), Ç È´=1 Õ ´ ¸ ´( )= Ê1( ). Multiplying these equations by ¹( ) ¸ ´( ), integrating the resulting relations with respect to on the interval 1£ £ 2, and taking into account the properties (15), we obtain the coef®cients of series (21) in the formÉ ´=1Ë¸ ´ Ë2 ¨ ±2±1 ¹( ) ¸ ´( ) Ê0( )« , Õ ´=1Ë¸ ´ Ë2 ¨ ±2±1 ¹( ) ¸ ´( ) Ê1( )« . ( 23) The quantity Ë¸ ´ Ëis de®ned in (19). Relations (21), (12), and (23) give a formal solution of the nonstationary boundary value problem (1)±(4) for ¿( À) >0. Example 2. Let ×( Í)= 1, Ì( Í)= Î( Í)= 0, and Ñ Ð>0. The solutions of (7) satisfying conditions (22) are expressed asÏÐ 1( Í)=cos ØeÙ Ñ ÐÍ]Ú, ÏÐ 2( Í)=1ÛÑ Ðsin Ø Ù Ñ ÐÍ]Ú. ( 24) If Ò0( Ó) and Ò1( Ó) have three and two continuous derivatives, respectively, and the compatibility conditions are met (see Paragraph 0.4.2-3), then series (21) is convergent and admits double termwise differentiation. In this case, formulas (21), (12), (23), and (24) give the classical smooth solution of problem (1)±(4). 0.4.2-3. Conditions of compatibility of initial and boundary conditions. Parabolic equations , ¿( À)º 0. Suppose the function Æ has a continuous derivative with respect to Àand two continuous derivatives with respect to and is a solution of problem (1)±(3). Then the boundary conditions (2) and the initial condition (3) must be consistent; namely, the following compatibility conditions must hold: [ ½1 Ê °0+¾1 Ê0] ±= ±1= 0, [ ½2 Ê °0+¾2 Ê0] ±= ±2= 0. ( 25) If ½1= 0or ½2= 0, then the additional compatibility conditions [ ( ) Ê °°0+ª( ) Ê °0] ±= ±1= 0 if ½1= 0, [ ( ) Ê °°0+ª( ) Ê °0] ±= ±2= 0 if ½2= 0(26) must also hold; the primes denote the derivatives with respect to . Hyperbolic equations. Suppose Æ is a twice continuously differentiable solution of prob- lem (1)±(4). Then conditions (25) and (26) must hold. In addition, the following conditions of compatibility of the boundary conditions (2) and initial condition (4) must be satis®ed: [ ½1 Ê °1+¾1 Ê1] ±= ±1= 0, [ ½2 Ê °1+¾2 Ê1] ±= ±2= 0. 0.4.2-4. Linear nonhomogeneous equations with nonhomogeneous boundary conditions. Parabolic equations , ¿( À)º 0. The solution of the boundary value problem for the parabolic linear homogeneous equation (1) subject to the homogeneous linear boundary conditions (2) and nonhomogeneous initial condition (3) is given by relations (18), (12), (17), and (19). This solution can be rewritten in the form Æ ( , À)= ¨ ±2±1 Ü( , Ý, À,0) Ê0( Ý)« Ý. Page 16 Here,Ü( , Ý, À, Þ) is the Green's function, which is expressed asÜ( , Ý, À, Þ)= ¹( Ý)Ç È´=1 ¸ ´( ) ¸ ´( Ý)Ë¸ ´ Ë2 Â ´( À, Þ), ( 27) where Â ´= Â ´( À, Þ) is the solution of equation (7) with ¿( À)º 0and ²= ² ´which satis®es the initial conditionÂ ´= 1 at À= Þ. The function Â ´( À, Þ) can be calculated by formula (17) with the lower limit of integration equal to Þ(rather than zero). The simplest way to obtain the solutions of more general boundary value problems for the corre- sponding nonhomogeneous linear equations with nonhomogeneous boundary and initial conditionsis to take advantage of formula (6) from Subsection 0.7.1 and use the Green's function (27). Hyperbolic equations. The solution of the boundary value problem for the hyperbolic linear homogeneous equation (1) subject to the homogeneous linear boundary conditions (2) and semiho-mogeneous initial conditions (3)±(4) withÊ0( )º 0is given by relations (21), (12), and (23) withÉ ´= 0( ·= 1,2, ³'³'³). This solution can be rewritten in the form Æ ( , À)= ¨ ±2±1Ü( , Ý, À,0) Ê1( Ý)« Ý. Here,Ü( , Ý, À, Þ) is the Green's function de®ned by relation (27), where Â ´= Â ´( À, Þ) is the solution of equation (7) for Âwith ²= ² ´which satis®es the initial conditionsÂ ´= 0 at À= Þ, Â °´= 1 at À= Þ. In the special case ¿( À)= 1, ( À)= Á( À)= 0, we have Â ´( À, Þ)= ²-1 ß2´sin Ôà²1 ß2´( À- Þ) Ö. The simplest way to obtain the solutions of more general boundary value problems for the corre- sponding nonhomogeneous linear equations with nonhomogeneous boundary and initial conditionsis to take advantage of formula (14) from Subsection 0.7.2 and use the Green's function (27).á5â References for Section 0.4: V . M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), E. Butkov (1968), E. C. Zach- manoglou and D. W. Thoe (1986), T. U.-Myint and L. Debnath (1987), A. N. Tikhonov and A. A. Samarskii (1990),R. B. Guenther and J. W. Lee (1996), D. Zwillinger (1998), I. Stakgold (2000), A. D. Polyanin (2001a). 0.5. Integral Transforms Method 0.5.1. Main Integral Transforms Various integral transforms are widely used to solve linear problems of mathematical physics. An integral transform is de®ned as ãÊ( ²)= ¨ äå ¸( , ²) Ê( )« . The function ãÊ( ²) is called the transform of the function Ê( ) and ¸( , ²) is called the kernel of the integral transform. The function Ê( ) is called the inverse transform of ãÊ( ²). The limits of integration andªare real numbers (usually, = 0,ª= ¶or = - ¶,ª= ¶). Corresponding inversion formulas, which have the formÊ( )= ¨ æ Â( , ²) ãÊ( ²)« ² make it possible to recover Ê( ) if ãÊ( ²) is given. The integration path çcan lie either on the real axis or in the complex plane. Themostcommonl yusedintegraltransform sarelistedinTable3(fortheconstraint simposed on the functions and parameters occurring in the integrand, see the references given at the end ofSection 0.5). Page 17 TABLE 3 Main integral transforms IntegralTransformDe®nition Inversion Formula Laplacetransformè Ò( é)= ê ë 0 ì- íïîÒ( Í) ðïÍÒ( Í)=1 2 ñ ò ê ó+ ôëó- ôë ì íïîè Ò( é) ðMé Fouriertransformè Ò( õ)=1Û 2 ñ ê ë -ë ì- ô5ö'÷Ò( Ó) ðïÓ Ò( Ó)=1Û 2 ñ ê ë -ë ì ô5ö'÷è Ò( õ) ðïõ Fourier sinetransformè Òs( õ)= ø2ñ ê ë 0sin( Ó õ) Ò( Ó) ðïÓ Ò( Ó)= ø2ñ ê ë 0sin( Ó õ)è Òs( õ) ðïõ Fourier cosinetransformè Òc( õ)= ø2ñ ê ë 0cos( Ó õ) Ò( Ó) ðïÓ Ò( Ó)= ø2ñ ê ë 0cos( Ó õ)è Òc( õ) ðïõ Mellintransformù Ò( ú)=ê ë 0 Ó û-1Ò( Ó) ðïÓÒ( Ó)=1 2 ñ ò ê ó+ ôëó- ôë Ó-ûù Ò( ú) ðïú Hankeltransformù Ò ü( õ)=ê ë 0 Ó ý ü( Ó õ) Ò( Ó) ðïÓ Ò( Ó)=ê ë 0 õ ý ü( Ó õ)ù Ò ü( õ) ðïõ Meijertransformù Ò( ú)= ø2ñ ê ë 0 ÛúMÓ þ ü( úMÓ) Ò( Ó) ðïÓ Ò( Ó)=1ò Û 2 ñ ê ó+ ôëó- ôë ÛúMÓ ÿpü( úMÓ)ù Ò( ú) ðïú Notation : ò2=-1, ý( Ó) and ( Ó) are the Bessel functions of the ®rst and the second kind, respectively; ÿ( Ó) and þ ( Ó) are the modi®ed Bessel functions of the ®rst and the second kind. The Laplace transform and the Fourier transform are in most common use. These integral transforms are brie¯y described below. 0.5.2. Laplace Transform and Its Application in Mathematical Physics 0.5.2-1. The Laplace transform. The inverse Laplace transform. The Laplace transform of an arbitrary (complex-valued) function Ê( À) of a real variable À( À³ 0) is de®ned by ãÊ( )= Ç 0 - Ä Ê( À) À, ( 1) where = ½+ is a complex variable, 2= -1 . The Laplace transform exists for any continuous or piecewise-continuous function satisfying the condition | Ê( À)|< 0Äwith some >0and 0³ 0. In what follows, 0often means the greatest lower bound of the possible values of 0in this condition. For any Ê( À), the transform ãÊ( ) is de®ned in the half-plane Re > 0and is analytic there. Given the transform ãÊ( ), the function Ê( À) can be found by means of the inverse Laplace transformÊ( À)=1 2 + Ç- Ç ãÊ( ) Ä , ( 2) where the integration path is parallel to the imaginary axis and lies to the right of all singularities of ãÊ( ), which corresponds to > 0. The integral in (2) is understood in the sense of the Cauchy principal value:+ Ç- Ç ãÊ( ) Ä =lim Ç + - ãÊ( ) Ä . In the domain À<0, formula (2) gives Ê( À)º 0. Page 18 TABLE 4 Main properties of the Laplace transform No Function Laplace transform Operation 1 Ê1( À)+ Ê2( ) ãÊ1( )+ ãÊ2( ) Linearity 2 Ê( ), >0 ãÊ( ) Scaling 3 Ê( ); = 1,2, (-1) ãÊ()( )Differentiation of the transform 4 åÊ( ) ãÊ( - )Shift in the complex plane 5 Ê ( ) ãÊ( )- Ê(+0) Differentiation 6Ê()( ) ãÊ( )- =1 - Ê( -1)(+0) Differentiation 7! Ê()( ), "³ (-1) ! Ô ãÊ( ) Ö(!)Differentiation 8 0 Ê( Þ) Þ1 ãÊ( ) Integration 9 0 Ê1( Þ) Ê2( - Þ) Þ ãÊ1( ) ãÊ2( ) Convolution Formula (2) holds for continuous functions. If Ê( ) has a (®nite) jump discontinuity at a point= 0>0, then the left-hand side of (2) is equal to1 2[ Ê( 0- 0)+ Ê( 0+ 0)] at this point (for 0= 0, the ®rst term in the square brackets must be omitted). We will brie¯y denote the Laplace transform (1) and the inverse Laplace transform (2) as ãÊ( )= ç{ Ê( )}, Ê( )= ç-1{ ãÊ( )}. 0.5.2-2. Main properties of the Laplace transform. The main properties of the correspondence between functions and their Laplace transforms aregathere dinTable4.TheLaplac etransform sofsomefunction sarelistedinTable5. There are a number of books that contain detailed tables of direct and inverse Laplace transforms elsewhere; see the references at the end of Section 0.5. Such tables are convenient to use in solvinglinear differential equations. Note the important case in which the transform is a rational function of the formãÊ( )= #( )$( ), where $( ) and#( ) are polynomials in the variable and the degree of $( ) exceeds that of#( ). Assume that the zeros of the denominator are simple, i.e., $( )ºconst ( - %1)( - %2) ( - %). Then the inverse transform can be determined by the formula&( )= ' =1 #( % )$( % )exp( % ), where the primes denote the derivatives. Page 19 TABLE 5 The Laplace transforms of some functions No Function, &( ) Laplace transform, ( &( )) Remarks 1 11) 2 !)+1 = 1,2, 3+* ,( + 1) )-*-1>-1 4 --* ( )+ )-1 5+* -- . ,( + 1)( )+ )-*-1>-1 6 sinh( ) )2- 2 7 cosh( ) ))2- 2 8 ln -1)(ln )+ /) /= 0.5772 is the Euler constant 9 sin( ) )2+ 2 10 cos( ) ))2+ 2 11 erfc 0 2 1 2 31)exp 4- 1 ) 5³ 0 12 60( 2)17)2+ 2 60( 8) is the Bessel function 0.5.2-3. Solving linear problems of mathematical physics by the Laplace transform. Figur e2showsschematicall yhowonecanutiliz etheLaplac etransform stosolveboundar yvalue problems for linear parabolic or hyperbolic equations with two independent variables in the casewhere the equation coef®cients are independent of2. It is signi®cant that with the Laplace transform, the original problems for a partial differential equation is reduced to a simpler problem for an ordinary differential equation with parameter ); the derivatives with respect to 2are replaced by appropriate algebraic expressions taking into account theinitia lcondition s(seepropert y5or6inTable4). Example 1. Consider the following problem for the heat equation:9:<;= 9=>= ;, ( ?>0, @>0),;= 0 at @= 0 (initial condition),;= ; 0at ?= 0 (boundary condition),; A0at ? A B(boundary condition). We apply the Laplace transform with respect to @. Settingè ;= C{ ;}and taking into account the relationsC{ 9:<;} = éè ;- ;| : =0= éè ;(usedarepropert y5ofTable4andtheinitia lconditio n),C{ ; 0} = ; 0 C{1} = ; 0 D é (usedarepropert y1ofTable4andtherelatio n C{1}=1D é), Page 20 Figure 2. Scheme for solving linear boundary value problems by the Laplace transform. we arrive at the following problem for a second-order linear ordinary differential equation with parameter : - = 0,= 0 at = 0 (boundary condition), 0 at (boundary condition). Integrating the equation yields the general solution = 1( ) - + 2( ) . Using the boundary conditions, we determine the constants, 1( )= 0 and 2( )= 0. Thus, we have= 0 - . LetusapplytheinverseLaplac etransfor mtobothsidesofthisrelation .WerefertoTable5,row11with = to®ndthe inverse transform of the right-hand side. Finally, we obtain the solution of the original problem in the form= 0erfc 2 . 0.5.3. Fourier Transform and Its Application in Mathematical Physics 0.5.3-1. The Fourier transform and its properties. The Fourier transform is de®ned as follows:( )=1 2 - ( ) - ! ", #2= -1. ( 3) This relation is meaningful for any function ( ) absolutely integrable on the interval ( $,+ $). We will brie¯y write ( )= %{ ( )}to denote the Fourier transform (3). Given ( ), the function ( ) can be found by means of the inverse Fourier transform( )=1 2 - ( ) ! ", ( 4) Page 21 TABLE 6 Main properties of the Fourier transform No Function Fourier transform Operation 1 & 1( )+ ' 2( )& 1( )+ ' 2( ) Linearity 2 ( (&),&>0& (& ) Scaling 3 ) ( ); *= 1,2, +,+,+#-) ())( )Differentiation of the transform 4 ./.! !( ) - 2 ( ) Differentiation 5 ())!( ) ( #0)) ( ) Differentiation 6- 1( 1) 2( - 1) "1 1( ) 2( ) Convolution where the integral is understood in the sense of the Cauchy principal value. We will brie¯y write( )= %-1{ ( )}to denote the inverse Fourier transform (4). The inversion formula (4) holds for continuous functions. If ( ) has a (®nite) jump discontinuity at a point = 0, then the left-hand side of (4) is equal to1 2 2 ( 0- 0)+ ( 0+ 0) 3at this point. The main properties of the correspondence between functions and their Fourier transforms are gathere dinTable6. 0.5.3-2. Solving linear problems of mathematical physics by the Fourier transform. The Fourier transform is usually employed to solve boundary value problems for linear partialdifferential equations whose coef®cients are independent of the space variable,- $< < $. The scheme for solving linear boundary value problems with the help of the Fourier transform is similar to that used in solving problems with help of the Laplace transform. With the Fouriertransform, the derivatives with respect toin the equation are replaced by appropriate algebraic expressions ;seepropert y4or5inTable6.Inthecaseoftwoindependen tvariables ,theproble mfor a partial differential equation is reduced to a simpler problem for an ordinary differential equationwith parameter. On solving the latter problem, one determines the transform. After that, by applying the inverse Fourier transform, one obtains the solution of the original boundary valueproblem. Example 2. Consider the following Cauchy problem for the heat equation:4,5= 4 (- < < ),= 6( ) at = 0 (initial condition). We apply the Fourier transform with respect to the space variable . Setting = 7{ }and taking into account the relation7{ 4 } = - 82 (seepropert y4ofTable6),wearriveatthefollowingproble mfora®rst-orde rlinea rordinar ydifferential equation with parameter 8: 5+ 82 = 0,= 6( 8) at = 0, where 6( 8) is de®ned by (3). On solving this problem for the transform , we ®nd= 6( 8) - 92 5 . Let us apply the inversion formula to both sides of this equation. After some calculations, we obtain the solution of theoriginal problem in the form=12 : ; <-< 6( 8) - 92 5>= 9 ?8=1 2 :; <-< @; <-< 6( A) -= 9B ?A>C - 92 5 += 9 ?8 =1 2 :; <-< 6( A) ?A; <-< - 92 5 += 9( - B) ?8=12 : ; <-< 6( A) exp @ -( - A)2 4 C ?A. Page 22 At the last stage we used the relation; <-<exp D- 282+ E08 F ?8= : | |exp E2 4 2. The Fourier transform admits *-dimensional generalization:(u)=1 (2 )) G2 H I (x) - (u×x) "x, ( u×x)= 1 1+ J,J,J+ ) ), ( 5) where (x)= ( 1, +,+,+, )), (u)= ( 1, +,+,+, )), and "x= "1 +,+,+ "). The corresponding inversion formula is(x)=1 (2 )) G2 H I (u) (u×x) "u, "u= "1 +,+,+ "). The Fourier transform (5) is frequently used in the theory of linear partial differential equations with constant coef®cients ( x K L)).MON References for Section 0.5: H. Bateman and A. Erd Âelyi (1954), V . A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), B. Davis (1978), Yu. A. Brychkov and A. P. Prudnikov (1989), L. H Èormander (1990), J. R. Hanna and J. H. Rowland (1990), W. H. Beyer (1991), D. Zwillinger (1998), A. D. Polyanin and A. V . Manzhirov (1998). 0.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution 0.6.1. Cauchy Problem for Parabolic Equations 0.6.1-1. General formula for the solution of the Cauchy problem. Letx= { 1, +,+,+, )}andy= { P1, +,+,+, P)}, where x K L)andy K L). Consider a nonhomogeneous linear equation of the parabolic type with an arbitrary right-hand side, Q RQ S- Tx, U[ R ]= V(x, S ), ( 1) where the second-order linear differential operator Tx, Uis de®ned by relation (2) from Subsection 0.2.1. The solution of the Cauchy problem for equation (1) with an arbitrary initial condition,R = (x) at S = 0, can be represented as the sum of two integrals,R (x, S )= U 0 H I V(y, W) X X(x,y, S , W) "y "W+ H I (y) X X(x,y, S ,0) "y, "y= "P1 +,+,+ "P). Here, X X= X X(x,y, S , W) is the fundamental solution of the Cauchy problem that satis®es, for S > W³ 0, the homogeneous linear equationQX XQ S- Tx, U[ X X]= 0 (2) with the nonhomogeneous initial condition of special formX X= Y(x-y) at S = W. ( 3) The quantities Wandyappear in problem (2)±(3) as free parameters, and Y(x)= Y( 1) +,+,+ZY( )) is the *-dimensional Dirac delta function.[ \^] _ `>a b cIf the coef®cients of the differential operator Tx, Uin (2) are independent of time S , then the fundamental solution of the Cauchy problem depends on only three arguments, X X(x,y, S , W)=X X(x,y, S - W).[ \^] _ `>a d cIf the differential operator Tx, Uhas constant coef®cients, then the fundamental solution of the Cauchy problem depends on only two arguments, X X(x,y, S , W)= X X(x-y, S - W). Page 23 0.6.1-2. The fundamental solution allowing incomplete separation of variables. Consider the special case where the differential operator Tx, Uin equation (1) can be represented as the sumTx, U[ R ]= T1, U[ R ]+ J,J,J+ T), U[ R ], ( 4) where each term depends on a single space coordinate and time,T e, U[ R ]º& e( e, S ) Q 2 RQ2e+ 'e( e, S ) Q RQ e+ f,e( e, S ) R , g= 1, +,+,+, *. Equations of this form are often encountered in applications. The fundamental solution of the Cauchy problem for the *-dimensional equation (1) with operator (4) can be represented in the product formX X(x,y, S , W)= ) he=1 X Xe( e, P e, S , W), ( 5) where X Xe= X Xe( e, P e, S , W) are the fundamental solutions satisfying the one-dimensional equationsQX XeQ S- T e, U[ X Xe]= 0 ( g= 1, +,+,+, *) with the initial conditionsX Xe= Y( e- P e) at S = W. In this case, the fundamental solution of the Cauchy problem (5) admits incomplete separation of variables; the fundamental solution is separated in the space variables 1, +,+,+, )but not in time S . 0.6.2. Cauchy Problem for Hyperbolic Equations Consider a nonhomogeneous linear equation of the hyperbolic type with an arbitrary right-hand side,Q 2 RQ S 2+ i(x, S ) Q RQ S- Tx, U[ R ]= V(x, S ), ( 6) where the second-order linear differential operator Tx, Uis de®ned by relation (2) from Subsection 0.2.1, with x K L). The solution of the Cauchy problem for equation (6) with general initial conditions,R = 0(x) at S = 0,QU R = 1(x) at S = 0, can be represented as the sumR (x, S )= U 0 HI V(y, W) X X(x,y, S , W) "y "W- HI 0(y) j QQW X X(x,y, S , W) k l =0 "y + HI 2 1(y)+ 0(y) i(y,0) 3X X(x,y, S ,0) "y, "y= "P1 +,+,+ "P). Here, X X= X X(x,y, S , W) is the fundamental solution of the Cauchy problem that satis®es, for S > W³ 0, the homogeneous linear equationQ2X XQ S 2+ i(x, S ) QX XQ S- Tx, U[ X X]= 0 (7) with the semihomogeneous initial conditions of special formX X= 0 at S = W,QU X X= Y(x-y) at S = W.(8) Page 24 The quantities Wandyappear in problem (7)±(8) as free parameters ( y K L)).[ \^] _ `>a b cIf the coef®cients of the differential operator Tx, Uin (7) are independent of time S , then the fundamental solution of the Cauchy problem depends on only three arguments, X X(x,y, S , W)=X X(x,y, S - W). Here, mm lX X(x,y, S , W) n n l =0= - mm U X X(x,y, S ).[ \^] _ `>a d cIf the differential operator Tx, Uhas constant coef®cients, then the fundamental solution of the Cauchy problem depends on only two arguments, X X(x,y, S , W)= X X(x-y, S - W).MON References for Section 0.6: V . M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), G. E. Shilov (1965), A. D. Polyanin (2000a, 2000b, 2000c, 2001a). 0.7. Nonhomogeneous Boundary Value Problems with One Space Variable. Representation of Solutions via the Green's Function 0.7.1. Problems for Parabolic Equations 0.7.1-1. Statement of the problem ( S ³ 0, 1£ £ 2). In general, a nonhomogeneous linear differential equation of the parabolic type with variable coef-®cients in one dimension can be written asQ RQ S- T!, U[ R ]= V( , S ), ( 1) whereT!, U[ R ]º&( , S ) Q 2 RQ2+ '( , S ) Q RQ+ f( , S ) R ,&( , S ) >0. ( 2) Consider the nonstationary boundary value problem for equation (1) with an initial condition of general form,R = ( ) at S = 0, ( 3) and arbitrary nonhomogeneous linear boundary conditions,o 1 Q RQ+ g1( S ) R = p1( S ) at = 1, ( 4)o 2 Q RQ+ g2( S ) R = p2( S ) at = 2. ( 5) By appropriately choosing the coef®cients o 1, o 2and the functions g1= g1( S ), g2= g2( S ) in (4) and (5), we obtain the ®rst, second, third, and mixed boundary value problems for equation (1). 0.7.1-2. Representation of the problem solution in terms of the Green's function. The solution of the nonhomogeneous linear boundary value problem (1)±(5) can be represented asR ( , S )= U 0 !2!1 V( P, W) q( , P, S , W) "P "W+ !2!1 ( P) q( , P, S ,0) "P + U 0 p1( W)&( 1, W) r1( , S , W) "W+ U 0 p2( W)&( 2, W) r2( , S , W) "W. ( 6) Here, q( , P, S , W) is the Green's function that satis®es, for S > W³ 0, the homogeneous equationQqQ S- T!, U[ q]= 0 (7) Page 25 TABLE 7 Expressions of the functions r1( , S , W) and r2( , S , W) involved in the integrands of the last two terms in solution (6) Type of problem Form of boundary conditions Functions r s( , S , W) First boundary value problem ( o 1= o 2= 0, g1= g2= 1) R = p1( S ) at = 1R = p2( S ) at = 2 r1( , S , W)= Q tq( , P, S , W) n n t = !1r2( , S , W)= - Q tq( , P, S , W) n n t = !2 Second boundary value problem ( o 1= o 2= 1, g1= g2= 0) Q! R = p1( S ) at = 1Q! R = p2( S ) at = 2 r1( , S , W)= - q( , 1, S , W)r2( , S , W)= q( , 2, S , W) Third boundary value problem ( o 1= o 2= 1, g1<0, g2>0) Q! R + g1 R = p1( S ) at = 1Q! R + g2 R = p2( S ) at = 2 r1( , S , W)= - q( , 1, S , W)r2( , S , W)= q( , 2, S , W) Mixed boundary value problem ( o 1= g2= 0, o 2= g1= 1) R = p1( S ) at = 1Q! R = p2( S ) at = 2 r1( , S , W)= Q tq( , P, S , W) n n t = !1r2( , S , W)= q( , 2, S , W) Mixed boundary value problem ( o 1= g2= 1, o 2= g1= 0) Q! R = p1( S ) at = 1R = p2( S ) at = 2 r1( , S , W)= - q( , 1, S , W)r2( , S , W)= - Q tq( , P, S , W) n n t = !2 with the nonhomogeneous initial condition of special formq= Y( - P) at S = W (8) and the homogeneous boundary conditionso 1 QqQ+ g1( S ) q= 0 at = 1, ( 9)o 2 QqQ+ g2( S ) q= 0 at = 2. ( 10) The quantities Pand Wappear in problem (7)±(10) as free parameters ( 1£ P£ 2), and Y( ) is the Dirac delta function. The initial condition (8) implies the limit relation( )=limUvu l !2!1 ( P) q( , P, S , W) "P for any continuous function = ( ). The functions r1( , S , W) and r2( , S , W) involved in the integrands of the last two terms in solution (6) can be expressed in terms of the Green's function q( , P, S , W). The corresponding formulas for r s( , S , W)aregiveninTable7forthebasictypesofboundar yvalueproblems. It is signi®cant that the Green's function qand the functions r1, r2are independent of the functions V, , p1, and p2that characterize various nonhomogeneities of the boundary value problem. If the coef®cients of equation (1)±(2) and the coef®cients g1, g2in the boundary conditions (4) and (5) are independent of time S , i.e., the conditions&=&( ), '= '( ), f= f( ), g1=const, g2=const ( 11) hold, then the Green's function depends on only three arguments,q( , P, S , W)= q( , P, S - W). Page 26 In this case, the functions r sdepend on only two arguments, r s= r s( , S - W), w= 1,2. Formula (6) also remains valid for the problem with boundary conditions of the third kind ifg1= g1( S ) and g2= g2( S ). Here, the relation between r s( w= 1,2) and the Green's function qis the same as that in the case of constants g1and g2; the Green's function itself is now different. The condition that the solution must vanish at in®nity, R x 0as x$, is often set for the ®rst, second, and third boundary value problems that are considered on the interval 1£ < $. In this case, the solution is calculated by formula (6) with r2= 0and r1speci®e dinTable7. 0.7.2. Problems for Hyperbolic Equations 0.7.1-2. Statement of the problem ( S ³ 0, 1£ £ 2). In general, a one-dimensional nonhomogeneous linear differential equation of hyperbolic type withvariable coef®cients is written asQ 2 RQ S 2+ i( , S ) Q RQ S- T!, U[ R ]= V( , S ), ( 12) where the operator T!, U[ R ] is de®ned by (2). Consider the nonstationary boundary value problem for equation (12) with the initial conditionsR = y0( ) at S = 0,QU R = y1( ) at S = 0(13) and arbitrary nonhomogeneous linear boundary conditions (4)±(5). 0.7.2-2. Representation of the problem solution in terms of the Green's function. The solution of problem (12), (13), (4), (5) can be represented as the sumR ( , S )= U 0 !2!1 V( P, W) q( , P, S , W) "P "W - !2!1 y0( P)j QQW q( , P, S , W)k l =0 "P+ !2!1 2 y1( P)+ y0( P) i( P,0) 3^q( , P, S ,0) "P + U 0 p1( W)&( 1, W) r1( , S , W) "W+ U 0 p2( W)&( 2, W) r2( , S , W) "W. ( 14) Here, the Green's function q( , P, S , W) is determined by solving the homogeneous equationQ 2qQ S 2+ i( , S ) QqQ S- T!, U[ q]= 0 (15) with the semihomogeneous initial conditionsq= 0 at S = W,QU q= Y( - P) at S = W,(16) (17) and the homogeneous boundary conditions (9) and (10). The quantities Pand Wappear in problem (15)±(17), (9), (10) as free parameters ( 1£ P£ 2), and Y( ) is the Dirac delta function. The functions r1( , S , W) and r2( , S , W) involved in the integrands of the last two terms in solution (14) can be expressed via the Green's function q( , P, S , W). The corresponding formulas for r s( , S , W)aregiveninTable7forthebasictypesofboundar yvalueproblems. It is signi®cant that the Green's function qand r1, r2are independent of the functions V, y0,y1, p1, and p2that characterize various nonhomogeneities of the boundary value problem. If the coef®cients of equation (12) and the coef®cients g1, g2in the boundary conditions (4) and (5) are independent of time S , then the Green's function depends on only three arguments,q( , P, S , W)= q( , P, S - W). In this case, one can set mm lq( , P, S , W) n n l =0= - mm U q( , P, S ) in solu- tion (14).MON References for Section 0.7: V . M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), E. Butkov (1968), A. G. Butkovskiy (1982), E. Zauderer (1989), A. D. Polyanin (2000a, 2000b, 2000c, 2001a). Page 27 0.8. Nonhomogeneous Boundary Value Problems with Many Space Variables. Representation of Solutions via the Green's Function 0.8.1. Problems for Parabolic Equations 0.8.1-1. Statement of the problem. In general, a nonhomogeneous linear differential equation of the parabolic type in *space variables has the form Q RQ S- Tx, U[ R ]= V(x, S ), ( 1) whereTx, U[ R ]º ) z, {=1 & |{(x, S ) Q 2 RQ }| Q }{+ ) z|=1 '|(x, S ) Q RQ }|+ f(x, S ) R , x= { } 1, +,+,+, })}, ) z|, {=1 & |{(x, S ) 1| 1{³ ~ ) z|=1 12|, ~>0.(2) Let be some simply connected domain in L with a suf®ciently smooth boundary = Q. We consider the nonstationary boundary value problem for equation (1) in the domain with an arbitrary initial condition,R = y(x) at S = 0, ( 3) and nonhomogeneous linear boundary conditions, x, U[ R ]= p(x, S ) for x K . ( 4) In the general case, x, Uis a ®rst-order linear differential operator in the space coordinates with coef®cients dependent on xand S . 0.8.1-2. Representation of the problem solution in terms of the Green's function. The solution of the nonhomogeneous linear boundary value problem (1)±(4) can be represented as the sumR (x, S )= 0 (y, ) (x,y, , ) + y(y) (x,y, ,0) + 0 p(y, ) (x,y, , ) , ( 5) where (x,y, , ) is the Green's function; for > ³ 0, it satis®es the homogeneous equation- x,[ ]= 0 (6) with the nonhomogeneous initial condition of special form= Y(x-y) at = (7) and the homogeneous boundary condition x,[ ]= 0 for x . ( 8) The vector y= { 1, ,,, }appears in problem (6)±(8) as an -dimensional free parameter ( y ), and Y(x-y)= Y( } 1- 1) ,,ZY( }- ) is the -dimensional Dirac delta function. The Green's Page 28 TABLE 8 The form of the function (x,y, , ) for the basic types of nonstationary boundary value problems Type of problem Form of boundary condition (4) Function (x,y, , ) 1st boundary value problem = p(x, ) for x (x,y, , )= -m m (x,y, , ) 2nd boundary value problem m m = p(x, ) for x (x,y, , )= (x,y, , ) 3rd boundary value problem m m + = p(x, ) for x (x,y, , )= (x,y, , ) function is independent of the functions , , and pthat characterize various nonhomogeneities of the boundary value problem. In (5), the integration is everywhere performed with respect to y, with = 1 ,,> . The function (x,y, , ) involved in the integrand of the last term in solution (5) can be expressed via the Green's function (x,y, , ). The corresponding formulas for (x,y, , ) are giveninTable8forthethreebasictypesofboundar yvalueproblems ;inthethirdboundar yvalue problem, the coef®cient can depend on xand . The boundary conditions of the second and third kind, as well as the solution of the ®rst boundary value problem, involve operators of differentiationalong the conormal of operator (2); these operators act as follows: º , =1 ¡ (x, ) ¢ £, º , =1 ¡ (y, ) ¢ , ( 9) where N= { ¢1, ,,, ¢}is the unit outward normal to the surface . In the special case where¡ (x, )= 1and¡ (x, )= 0for ¤¹ ¥, operator (9) coincides with the ordinary operator of differen- tiation along the outward normal to ¦. If the coef®cient of equation (6) and the boundary condition (8) are independent of , then the Green's function depends on only three arguments, (x,y, , )= (x,y, - ).[ \^§ ¨ ©>ª «Let ¦ ( ¤= 1, ,,, ¬) be different portions of the surface ¦such that ¦= ® =1 ¦ and let boundary conditions of various types be set on the ¦ ,¯( ) x,[ ]= p (x, ) for x ¦ , ¤= 1, ,,, ¬. ( 10) Then formula (5) remains valid but the last term in (5) must be replaced by the sum =1 0 ° p (y, ) (x,y, , ) ¦ . ( 11) 0.8.2. Problems for Hyperbolic Equations 0.8.2-1. Statement of the problem. The general nonhomogeneous linear differential hyperbolic equation in space variables can be written as22+ ±(x, ) - x,[ ]= (x, ), ( 12) where the operator x,[ ] is explicitly de®ned in (2). We consider the nonstationary boundary value problem for equation (12) in the domain ²with arbitrary initial conditions,= 0(x) at = 0, = 1(x) at = 0,(13) (14) and the nonhomogeneous linear boundary condition (4). Page 29 0.8.2-2. Representation of the problem solution in terms of the Green's function. The solution of the nonhomogeneous linear boundary value problem (12)±(14),(4) can be representedas the sum(x, )= 0 (y, ) (x,y, , ) ² - 0(y) ³ (x,y, , ) ´ l =0 ² + µ 1(y)+ 0(y) ±(y,0) ¶^(x,y, ,0) ² + 0 p(y, ) (x,y, , ) ¦ . (15) Here, (x,y, , ) is the Green's function; for > ³ 0it satis®es the homogeneous equation22+ ±(x, ) - x,[ ]= 0 (16) with the semihomogeneous initial conditions= 0 at = , = Y(x-y) at = , and the homogeneous boundary condition (8). If the coef®cients of equation (16) and the boundary condition (8) are independent of time , then the Green's function depends on only three arguments, (x,y, , )= (x,y, - ). In this case, one can setmm l(x,y, , ) n n l =0= -mm (x,y, ) in solution (15). The function (x,y, , ) involved in the integrand of the last term in solution (15) can be expressed via the Green's function (x,y, , ). The corresponding formulas for are given in Table8forthethreebasictypesofboundar yvalueproblems ;inthethirdboundar yvalueproblem, the coef®cient can depend on xand .[ \^§ ¨ ©>ª «Let ¦ ( ¤= 1, ,,, ¬) be different portions of the surface ¦such that ¦= ® =1 ¦ and let boundary conditions of various types (10) be set on the ¦ . Then formula (15) remains valid but the last term in (15) must be replaced by the sum (11). 0.8.3. Problems for Elliptic Equations 0.8.3-1. Statement of the problem. In general, a nonhomogeneous linear elliptic equation can be written as - x[ ]= (x), ( 17) wherex[ ]º · , =1 ¡ (x) 2 £ £ + · =1 ¸ (x) £+ ¹(x) . ( 18) Two-dimensional problems correspond to = 2and three-dimensional problems, to = 3. We consider equation (17)±(18) in a domain ²and assume that the equation is subject to the general linear boundary condition¯ x[ ]= p(x) for x ¦. ( 19) The solution of the stationary problem (17)±(19) can be obtained by passing in (5) to the limit as º ». To this end, one should start with equation (1) whose coef®cients are independent of and take the homogeneous initial condition (3), with (x)= 0, and the stationary boundary condition (4). Page 30 TABLE 9 The form of the function (x,y) involved in the integrand of the last term in solution (20) for the basic types of stationary boundary value problems Type of problem Form of boundary condition (19) Function (x,y) 1st boundary value problem = p(x) for x ¦ (x,y)= - m m (x,y) 2nd boundary value problem m m = p(x) for x ¦(x,y)= (x,y) 3rd boundary value problem m m + = p(x) for x ¦(x,y)= (x,y) 0.8.3-2. Representation of the problem solution in terms of the Green's function. The solution of the linear boundary value problem (17)±(19) can be represented as the sum(x)= (y) (x,y) ² + p(y) (x,y) ¦ . ( 20) Here, the Green's function (x,y) satis®es the nonhomogeneous equation of special form - x[ ]= ¼(x-y) ( 21) with the homogeneous boundary condition¯ x[ ]= 0 for x ¦. ( 22) The vector y= { 1, ,,, ·}appears in problem (21), (22) as an -dimensional free parameter ( y ²). Note that is independent of the functions and pcharacterizing various nonhomogeneities of the original boundary value problem. The function (x,y) involved in the integrand of the second term in solution (20) can be expresse dviatheGreen 'sfunctio n (x,y).Thecorrespondin gformula sfor aregiveninTable9 for the three basic types of boundary value problems. The boundary conditions of the secondand third kind, as well as the solution of the ®rst boundary value problem, involve operators ofdifferentiation along the conormal of operator (18); these operators are de®ned by (9); in this case,the coef®cients¡ depend on only x.½ ¾^§ ¨ ©>ª «For the second boundary value problem with ¹(x)º 0, the thus de®ned Green's function must not necessarily exist; see Remark 2 in Paragraph 8.2.1-2. 0.8.4. Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types Table10listsbriefformulation sofboundar yvalueproblem sforsecond-orde requation sofelliptic, parabolic, and hyperbolic types. The coef®cients of the differential operators xand ¯ xin the space variables £ 1, ,,, £·are assumed to be independent of time ; these operators are the same for the problems under consideration. Below are the respective general formulas de®ning the solutions of these problems with zero initial conditions ( = 0= 1= 0):0(x)= (y) 0(x,y) ² + p(y) ¿µ 0(x,y) ¶ ¦ ,1(x, )= 0 (y, ) 1(x,y, - ) ² + 0 p(y, ) ¿µ 1(x,y, - ) ¶ ¦ ,2(x, )= 0 (y, ) 2(x,y, - ) ² + 0 p(y, ) ¿µ 2(x,y, - ) ¶ ¦ , Page 31 TABLE 10 Formulations of boundary value problems for equations of various types Type of equation Form of equation Initial conditions Boundary conditions Elliptic - x[ ]= (x) not set ¯ x[ ]= p(x) for x ¦ Parabolic - x[ ]= (x, ) = (x) at = 0 ¯ x[ ]= p(x, ) for x ¦ Hyperbolic À - x[ ]= (x, ) = 0(x) at = 0, = 1(x) at = 0 ¯ x[ ]= p(x, ) for x ¦ where the ·are the Green's functions, the subscripts 0,1, and 2refer to the elliptic, parabolic, and hyperbolic problem, respectively. All solutions involve the same operator ¿[ ]; it is explicitly de®ned in Subsections 0.8.1±0.8.3 (see also Section 0.7) for different boundary conditions. It is apparent that the solutions of the parabolic and hyperbolic problems with zero initial conditions have the same structure. The structure of the solution to the problem for a parabolicequation differs from that for an elliptic equation by the additional integration with respect to.ÁOÂ References for Section 0.8: P. M. Morse and H. Feshbach (1953), V . M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), E. Butkov (1968), A. G. Butkovskiy (1979, 1982), E. Zauderer (1989), A. N. Tikhonov and A. A. Samarskii (1990),A. D. Polyanin (2000a, 2000c, 2001a). 0.9. Construction of the Green's Functions. General Formulas and Relations 0.9.1. Green's Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains 0.9.1-1. Expressions of the Green's function in terms of in®nite series. Table11liststheGreen 'sfunction sofboundar yvalueproblem sforsecond-orde requation sof various types in a bounded domain ². It is assumed that xis a second-order linear self-adjoint differential operator (e.g., see Zwillinger, 1998) in the space variables £ 1, ,,, £·, and ¯ xis a zeroth- or ®rst-order linear boundary operator that can de®ne a boundary condition of the ®rst, second, orthird kind; the coef®cients of the operatorsxand ¯ xcan depend on the space variables but are independent of time . The coef®cients Ã Äand the functions Å Ä(x) are determined by solving the homogeneous eigenvalue problemx[ Å]+ Ã Å= 0, ( 1)¯ x[ Å]= 0 for x ¦. ( 2) Itisapparen tfromTable11that,giventheGreen 'sfunctio nintheproble mforaparaboli c(or hyperbolic) equation, one can easily construct the Green's functions of the corresponding problemsfor elliptic and hyperbolic (or parabolic) equations. In particular, the Green's function of the problemfor an elliptic equation can be expressed via the Green's function of the problem for a parabolicequation as follows:0(x,y)= Æ 0 1(x,y, ) . ( 3) Here, the fact that all Ã Äare positive is taken into account; for the second boundary value problem, it is assumed that Ã= 0is not an eigenvalue of problem (1)±(2). Page 32 TABLE 11 The Green's functions of boundary value problems for equations of various types in bounded domains. In all problems, the operators xand ¯ xare the same; x= { £ 1, ,,, £·} EquationInitial and boundary conditionsGreen's function Elliptic equation - x[ ]= (x) ¯ x[ ]= p(x) for x ¦ (no initial condition required) (x,y)= Æ Ä=1 Å Ä(x) Å Ä(y)ÇÅ Ä Ç2Ã Ä, Ã Ä¹ 0 Parabolic equation - x[ ]= (x, ) = (x) at = 0¯ x[ ]= p(x, ) for x ¦ (x,y, )= Æ Ä=1 Å Ä(x) Å Ä(y)ÇÅ Ä Ç2exp È- Ã Ä ZÉ Hyperbolic equationÀ - x[ ]= (x, ) = 0(x) at = 0= 1(x) at = 0¯ x[ ]= p(x, ) for x ¦ (x,y, )= Æ Ä=1 Å Ä(x) Å Ä(y)ÇÅ Ä Ç2 ÊÃ ÄsinÈ Ë Ã ÄÉ 0.9.1-2. Some remarks and generalizations.½ ¾^§ ¨ ©>ª Ì «Formula (3) can also be used if the domain ²is in®nite. In this case, one should make sure that the integral on the right-hand side is convergent.½ ¾^§ ¨ ©>ª Í «Suppos etheequation sgiveninthe®rstcolum nofTable11contai n- Îx[ Ï]- Ð Ï instead of - Îx[ Ï], with Ðbeing a free parameter. Then the Ã Äin the expressions of the Green's functio ninthethirdcolum nofTable11mustbereplace dby Ã Ä- Ð;justaspreviousl y,the Ã ÄandÅ Ä(x) were determined by solving the eigenvalue problem (1)±(2).½ ¾^§ ¨ ©>ª Ñ «Theformula sfortheGreen 'sfunction spresente dinTable11willalsoholdfor boundary value problems described by equations of the fourth or higher order in the space variables;provided that the eigenvalue problem for equation (1) subject to appropriate boundary conditions isself-adjoint. 0.9.2. Green's Functions Admitting Incomplete Separation of Variables 0.9.2-1. Boundary value problems for rectangular domains. 1 Ò. Consider the parabolic equation ÓÏ Ó Ô= Î1, Õ[ Ï]+ Ö,Ö,Ö+ Î·, Õ[ Ï]+ ×(x, Ô ), ( 4) where each term Î Ø, Õ[ Ï] depends on only one space variable, £Ø, and time Ô :Î Ø, Õ[ Ï]º Ù Ø( Ú Ø, Ô ) Ó 2Ï ÓÚ2Ø+¸ Ø( Ú Ø, Ô ) ÓÏ ÓÚ Ø+ ¹Ø( Ú Ø, Ô ) Ï, Û= 1, Ü,Ü,Ü, Ý. For equation (4) we set the initial condition of general formÏ= Þ(x) at Ô = 0. ( 5) Consider the domain ²= { ß Ø£ Ú Ø£ Ð Ø, Û= 1, Ü,Ü,Ü, Ý}which is an Ý-dimensional paral- lelepiped. We set the following boundary conditions at the faces of the parallelepiped:o(1)Ø ÓÏ ÓÚ Ø+ à(1)Ø( Ô ) Ï= p(1)Ø(x, Ô ) at Ú Ø= ß Ø,o(2)Ø ÓÏ ÓÚ Ø+ à(2)Ø( Ô ) Ï= p(2)Ø(x, Ô ) at Ú Ø= Ð Ø.(6) Page 33 By appropriately choosing the coef®cients o(1)Ø, o(2)Øand functions à(1)Ø= à(1)Ø( Ô ), à(2)Ø= à(2)Ø( Ô ), we can obtain the boundary conditions of the ®rst, second, or third kind. For in®nite domains, the boundary conditions corresponding to ß Ø= - »or Ð Ø= »are omitted. 2Ò. The Green's function of the nonstationary Ý-dimensional boundary value problem (4)±(6) can be represented in the product form á (x,y, Ô , â)= · ãØ=1 áØ( Ú Ø, ä Ø, Ô , â), ( 7) where the Green's functions áØ= áØ( Ú Ø, ä Ø, Ô , â) satisfy the one-dimensional equationsÓ áØ Ó Ô- Î Ø, Õ[ áØ]= 0 ( Û= 1, Ü,Ü,Ü, Ý) with the initial conditions áØ= å( Ú Ø- ä Ø) at Ô = â and the homogeneous boundary conditionso(1)Ø Ó áØ ÓÚ Ø+ à(1)Ø( Ô ) áØ= 0 at Ú Ø= ß Ø,o(2)Ø Ó áØ ÓÚ Ø+ à(2)Ø( Ô ) áØ= 0 at Ú Ø= Ð Ø. Here, ä Øand âare free parameters ( ß Ø£ ä Ø£ Ð Øand Ô ³ â³ 0), and å( Ú) is the Dirac delta function. It can be seen that the Green's function (7) admits incomplete separation of variables; it separates in the space variables Ú1, Ü,Ü,Ü, Ú æbut not in time Ô . 0.9.2-2. Boundary value problems for a cylindrical domain with arbitrary cross-section. 1Ò. Consider the parabolic equationÓÏ Ó Ô= Îx, Õ[ Ï]+ ç è, Õ[ Ï]+ ×(x, é, Ô ), ( 8) where Îx, Õis an arbitrary second-order linear differential operator in Ú1, Ü,Ü,Ü, Ú æwith coef®cients dependent on xand Ô , and ç è, Õis an arbitrary second-order linear differential operator in éwith coef®cients dependent on éand Ô . For equation (8) we set the general initial condition (5), where Þ(x) must be replaced by Þ(x, é). We assume that the space variables belong to a cylindrical domain ê= {x ë ì, é1£ é£ é2} with arbitrary cross-section ì. We set the boundary conditions*í 1[ Ï]= p1(x, Ô ) at é= é1 (x ë ì),í 2[ Ï]= p2(x, Ô ) at é= é2 (x ë ì),í 3[ Ï]= p3(x, é, Ô ) for x ë Óì( é1£ é£ é2),(9) where the linear boundary operators í î( à= 1,2,3) can de®ne boundary conditions of the ®rst, second, or third kind; in the last case, the coef®cients of the differential operators í îcan be dependent on Ô . * If ï1= - ðor ï2= ð, the corresponding boundary condition is to be omitted. Page 34 2Ò. The Green's function of problem (8)±(9), (5) can be represented in the product form á (x,y, é, ñ, Ô , â)= á ò (x,y, Ô , â) á ó ( é, ñ, Ô , â), ( 10) where á ò = á ò (x,y, Ô , â) and á ó = á ó ( é, ñ, Ô , â) are auxiliary Green's functions; these can be determined from the following two simpler problems with fewer independent variables: Problem on the cross-section ì: Problem on the interval é1£ é£ é2:ôõ õöõ õ÷ Ó á òÓ Ô= Îx, Õ[ á ò ] for x ë ì, á ò = å(x-y) at Ô = â,í 3[ á ò ]= 0 for x ë Óì, ôõ õöõ õ÷ Ó á óÓ Ô= ç è, Õ[ á ó ] for é1< é< é2, á ó = å( é- ñ) at Ô = â,í î[ á ó ]= 0 at é= é î( à= 1,2). Here, y, ñ, and âare free parameters ( y ë ì, é1£ ñ£ é2, Ô ³ â³ 0). It can be seen that the Green's function (10) admits incomplete separation of variables; it separates in the space variables xand ébut not in time Ô . 0.9.3. Construction of Green's Functions via Fundamental Solutions 0.9.3-1. Elliptic equations. Fundamental solution. Consider the elliptic equationÎx[ Ï]+ Ó 2Ï Óé2= ×(x, é), ( 11) where x= { Ú1, Ü,Ü,Ü, Ú æ} ë ø æ, é ë ø1, and Îx[ Ï] is a linear differential operator that depends onÚ1, Ü,Ü,Ü, Ú æbut is independent of é. For subsequent analysis it is signi®cant that the homogeneous equation (with ×º 0) does not change under the replacement of éby- éand éby é+const. Let ù ù= ù ù(x,y, é- ñ) be a fundamental solution of equation (11), which means thatÎx[ ù ù]+ Ó 2ù ù Óé2= å(x-y) å( é- ñ). Here, y= { ä1, Ü,Ü,Ü, ä æ} ë ø æand ñ ë ø1are free parameters. The fundamental solution of equation (11) is an even function in the last argument, i.e.,ù ù(x,y, é)= ù ù(x,y,- é). Below, Paragraphs 0.9.3-2 and 0.9.3-3 present relations that permit one to express the Green's functions of some boundary value problems for equation (11) via its fundamental solution. 0.9.3-2. Domain: x ë ø æ,0 £ é< ú. Boundary value problems for elliptic equations. 1Ò.First boundary value problem. The boundary condition:Ï= Þ(x) at é= 0. Green's function: á (x,y, é, ñ)= ù ù(x,y, é- ñ)- ù ù(x,y, é+ ñ). Domain of the free parameters: y ë ø æand0 £ ñ< ú. Page 35 2Ò.Second boundary value problem. The boundary condition:Óè,Ï= Þ(x) at é= 0. Green's function: á (x,y, é, ñ)= ù ù(x,y, é- ñ)+ ù ù(x,y, é+ ñ). 3Ò.Third boundary value problem. The boundary condition:Óè,Ï- à Ï= Þ(x) at é= 0. Green's function: á (x,y, é, ñ)= ù ù(x,y, é- ñ)+ ù ù(x,y, é+ ñ)- 2 à û ü 0 ý- î^þù ù(x,y, é+ ñ+ o) ÿ o = ù ù(x,y, é- ñ)+ ù ù(x,y, é+ ñ)- 2 û üè+ ý- î ( - è- )ù ù(x,y, ) ÿ . 0.9.3-3. Domain: x ë ø æ,0 £ é£ . Boundary value problems for elliptic equations. 1 .First boundary value problem. Boundary conditions:Ï= Þ1(x) at é= 0, Ï= Þ2(x) at é= . Green's function:(x,y, é, ñ)= ü æ=-ü ù ù(x,y, é- ñ+ 2 )- ù ù(x,y, é+ ñ+ 2 ) . ( 12) Domain of the free parameters: y ë ø æand0 £ ñ£ . 2.Second boundary value problem. Boundary conditions:è,Ï= Þ1(x) at é= 0, è Ï= Þ2(x) at é= . Green's function:(x,y, é, ñ)= ü æ=-ü ù ù(x,y, é- ñ+ 2 )+ ù ù(x,y, é+ ñ+ 2 ) . ( 13) 3.Mixed boundary value problem. The unknown function and its derivative are prescribed at the left and right end, respectively:Ï= Þ1(x) at é= 0, è,Ï= Þ2(x) at é= . Green's function:(x,y, é, ñ)= ü æ=-ü(-1) æ ù ù(x,y, é- ñ+ 2 )- ù ù(x,y, é+ ñ+ 2 ) . ( 14) 4.Mixed boundary value problem. The derivative and the unknown function itself are prescribed at the left and right end, respectively:è,Ï= Þ1(x) at é= 0, Ï= Þ2(x) at é= . Green's function:(x,y, é, ñ)= ü æ=-ü(-1) æ ù ù(x,y, é- ñ+ 2 )+ ù ù(x,y, é+ ñ+ 2 ) . ( 15) One should make sure that series (12)±(15) are convergent; in particular, for the three-dimensional Laplace equation, series (12), (14), and (15) are convergent and series (13) is divergent. Page 36 TABLE 12 Representation of the Green's functions of some nonstationary boundary value problems in terms of the fundamental solution of the Cauchy problem Boundary value problemsBoundary conditions Green's functions First problem x ë ø æ, é ë ø1 = 0 at é= 0 (x,y, é, ñ, , )= ù ù(x,y, é- ñ, , )- ù ù(x,y, é+ ñ, , ) Second problem x ë ø æ, é ë ø1 è = 0 at é= 0 (x,y, é, ñ, , )= ù ù(x,y, é- ñ, , )+ ù ù(x,y, é+ ñ, , ) Third problem x ë ø æ, é ë ø1 è - = 0at é= 0 (x,y, é, ñ, , )= ùù(x,y, é- ñ, , )+ ù ù(x,y, é+ ñ, , ) -2 û ü 0 ý- î^þù ù(x,y, é+ ñ+ o, , ) ÿ o First problem x ë ø æ,0 £ é£ = 0at é= 0,= 0at é= (x,y, é, ñ, , )=ü æ=-ü ù ù(x,y, é- ñ+2 , , ) - ù ù(x,y, é+ ñ+2 , , ) Second problem x ë ø æ,0 £ é£ è = 0at é= 0,è = 0at é= (x,y, é, ñ, , )=ü æ=-ü ù ù(x,y, é- ñ+2 , , ) + ù ù(x,y, é+ ñ+2 , , ) Mixed problem x ë ø æ,0 £ é£ = 0at é= 0,è = 0at é= (x,y, é, ñ, , )=ü æ=-ü(-1) æ ù ù(x,y, é- ñ+2 , , ) - ù ù(x,y, é+ ñ+2 , , ) Mixed problem x ë ø æ,0 £ é£ è = 0at é= 0,= 0at é= (x,y, é, ñ, , )=ü æ=-ü(-1) æ ù ù(x,y, é- ñ+2 , , ) + ù ù(x,y, é+ ñ+2 , , ) 0.9.3-4. Boundary value problems for parabolic equations. Letx ë ø æ, é ë ø1, and ³ 0. Consider the parabolic equationÏ= Îx, [ Ï]+ 2Ïé2+ (x, é, ), ( 16) where Îx, [ Ï] is a linear differential operator that depends on 1, , and but is independent of . Let ù ù= ù ù(x,y, - ñ, , ) be a fundamental solution of the Cauchy problem for equation (16), i.e.,ù ù= Îx, [ ù ù]+ 2ù ù2for > ,ù ù= (x-y) ( - ñ) at = . Here, y ø , ñ ø1, and ³ 0are free parameters. The fundamental solution of the Cauchy problem possesses the propertyù ù(x,y, , , )= ù ù(x,y,- , , ). Table12present sformula sthatpermi tonetoexpresstheGree n'sfunction sofsomenonstation ary boundary value problems for equation (16) via the fundamental solution of the Cauchy problem.!#" References for Section 0.9: V . M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), A. D. Polyanin (2000b, 2001a). Page 37 0.10. Duhamel's Principles in Nonstationary Problems 0.10.1. Problems for Homogeneous Linear Equations 0.10.1-1. Parabolic equations with two independent variables. Consider the problem for the homogeneous linear equation of parabolic typeÏ= $( ) 2Ï2+ %( ) Ï+ &( ) Ï (1) with the homogeneous initial conditionÏ= 0 at = 0 (2) and the boundary conditions ' 1 (Ï+ 1 Ï= p( ) at = 1, ( 3) ' 2 (Ï+ 2 Ï= 0 at = 2. ( 4) By appropriately choosing the values of the coef®cients ' 1, ' 2, 1, and 2in (3) and (4), one can obtain the ®rst, second, third, and mixed boundary value problems for equation (1). The solution of problem (1)±(4) with the nonstationary boundary condition (3) at = 1can be expressed by the formula (Duhamel's ®rst principle)Ï( , )= û 0 )( , - ) p( ) ÿ = û 0 ) ( , - ) p( ) ÿ (5) in terms of the solution)( , ) of the auxiliary problem for equation (1) with the initial and boundary conditions (2) and (4), for)instead of Ï, and the following simpler stationary boundary condition at = 1: ' 1 ()+ 1)= 1 at = 1. ( 6) A similar formula also holds for the homogeneous boundary condition at = 1and a nonhomogeneous nonstationary boundary condition at = 2. 0.10.1-2. Hyperbolic equations with two independent variables. Consider the problem for the homogeneous linear hyperbolic equation2Ï2+ *( ) Ï= $( ) 2Ï2+ %( ) Ï+ &( ) Ï (7) with the homogeneous initial conditionsÏ= 0 at = 0, Ï= 0 at = 0,(8) and the boundary conditions (3) and (4). The solution of problem (7), (8), (3), (4) with the nonstationary boundary condition (3) at= 1can be expressed by formula (5) in terms of the solution)( , ) of the auxiliary problem for equation (7) with the initial conditions (8) and boundary condition (4), for)instead of Ï, and the simpler stationary boundary condition (6) at = 1. In this case, the remark made in Paragraph 0.10.1-1 remains valid. Page 38 0.10.1-3. Second-order equations with several independent variables. Duhamel's ®rst principle can also be used to solve homogeneous linear equations of the parabolic or hyperbolic type with many space variables, +Ï += ,, -=1 $ ,-(x) 2Ï , -+ ,=1 % ,(x) Ï ,+ &(x) Ï, ( 9) where = 1,2andx= { 1, , }. Let .be some bounded domain in / with a suf®ciently smooth surface 0= .. The solution of the boundary value problem for equation (9) in .with the homogeneous initial conditions (2) if= 1or (8) if = 2, and the nonhomogeneous linear boundary condition1 x[ Ï]= p( ) for x 0, ( 10) is given byÏ(x, )= û 0 )(x, - ) p( ) ÿ = û 0 ) (x, - ) p( ) ÿ . Here,)(x, ) is the solution of the auxiliary problem for equation (9) with the same initial conditions, (2) or (8), for)instead of Ï, and the simpler stationary boundary condition1 x[)]= 1 for x 0. Note that (10) can represent a boundary condition of the ®rst, second, or third kind; the coef®cients of the operator 1 xare assumed to be independent of . 0.10.2. Problems for Nonhomogeneous Linear Equations 0.10.2-1. Parabolic equations. The solution of the nonhomogeneous linear equationÏ= ,, -=1 $ ,-(x) 2Ï , -+ ,=1 % ,(x) Ï ,+ &(x) Ï+ (x, ) with the homogeneous initial condition (2) and the homogeneous boundary condition1 x[ Ï]= 0 for x 0 (11) can be represented in the form (Duhamel's second principle)Ï(x, )= û 0 2(x, - , ) ÿ . ( 12) Here,2(x, , ) is the solution of the auxiliary problem for the homogeneous equation2 = ,, -=1 $ ,-(x) 22 , -+ ,=1 % ,(x) 2 ,+ &(x)2 with the boundary condition (11), in which Ïmust be substituted by2, and the nonhomogeneous initial condition2= (x, ) at = 0, where is a parameter. Note that (11) can represent a boundary condition of the ®rst, second, or third kind; the coef®cients of the operator 1 xare assumed to be independent of . Page 39 0.10.2-2. Hyperbolic equations. The solution of the nonhomogeneous linear equation2Ï2+ *(x) Ï= ,, -=1 $ ,-(x) 2Ï , -+ ,=1 % ,(x) Ï ,+ &(x) Ï+ (x, ) with the homogeneous initial conditions (8) and homogeneous boundary condition (11) can beexpressed by formula (12) in terms of the solution2=2(x, , ) of the auxiliary problem for the homogeneous equation22 2+ *(x) 2 = ,, -=1 $ ,-(x) 22 , -+ ,=1 % ,(x) 2 ,+ &(x)2 with the homogeneous initial and boundary conditions, (2) and (11), where Ïmust be replaced by2, and the nonhomogeneous initial condition2= (x, ) at = 0, where is a parameter. Note that (11) can represent a boundary condition of the ®rst, second, or third kind.!#" References for Section 0.10: E. Butkov (1968), S. J. Farlow (1982), E. Zauderer (1989), R. Courant and D. Hilbert (1989), D. Zwillinger (1998). 0.11. Transformations Simplifying Initial and Boundary Conditions 0.11.1. Transformations That Lead to Homogeneous Boundary Conditions A linear problem with arbitrary nonhomogeneous boundary conditions,1( +) x, [ Ï]= p +(x, ) for x 0 +, ( 1) can be reduced to a linear problem with homogeneous boundary conditions. To this end, one shouldperform the change of variableÏ(x, )= 3(x, )+)(x, ), ( 2) where)is a new unknown function and 3is any function that satis®es the nonhomogeneous boundary conditions (1),1( +) x, [ 3]= p +(x, ) for x 0 +. ( 3) Table13givesexample sofsuchtransformation sforlinea rboundar yvalueproblem swithone space variable for parabolic and hyperbolic equations. In the third boundary value problem, it isassumed that1<0and 2>0. Note that the selection of the function 3is of a purely algebraic nature and is not connected with the equation in question; there are in®nitely many suitable functions 3that satisfy condition (3). Transformations of the form (2) can often be used at the ®rst stage of solving boundary valueproblems. Page 40 TABLE 13 Simple transformations of the form ( , )= ( , )+ ( , ) that lead to homogeneous boundary conditions in problems with one space variables ( 0 £ £ ) No Problems Boundary conditions Function ( , ) 1First boundary value problem = 1( ) at = 0= 2( ) at = ( , )= 1( )+ 2( )- 1( ) 2Second boundary value problem = 1( ) at = 0 = 2( ) at = ( , )= 1( )+ 2 2 2( )- 1( ) 3Third boundary value problem + 1 = 1( ) at = 0 + 2 = 2( ) at = ( , )=( 2 -1- 2 ) 1( )+(1- 1 ) 2( )2- 1- 1 2 4Mixed boundary value problem = 1( ) at = 0 = 2( ) at = ( , )= 1( )+ 2( ) 5Mixed boundary value problem = 1( ) at = 0= 2( ) at = ( , )=( - ) 1( )+ 2( ) 0.11.2. Transformations That Lead to Homogeneous Initial and Boundary Conditions A linear problem with nonhomogeneous initial and boundary conditions can be reduced to a linearproblem with homogeneous initial and boundary conditions. To this end, one should introduce anew dependent variableby formula (2), where the function must satisfy nonhomogeneous initial and boundary conditions. Below we specify some simple functions that can be used in transformation (2) to obtain boundary value problems with homogeneous initial and boundary conditions. To be speci®c, weconsider a parabolic equation with one space variable and the general initial condition= ( ) at = 0. ( 4) 1. First boundary value problem : the initial condition is (4) and the boundary conditions are giveninrow1ofTable13.Suppos ethattheinitia landboundar ycondition sarecompatible ,i.e.,(0)= 1(0) and ( )= 2(0). Then, in transformation (2), one can take( , )= ( )+ 1( )- 1(0)+ 2( )- 1( )+ 1(0)- 2(0) . 2. Second boundary value problem : the initial condition is (4) and the boundary conditions are giveninrow2ofTable13.Suppos ethattheinitia landboundar ycondition sarecompatible ,i.e., (0)= 1(0) and ( )= 2(0). Then, in transformation (2), one can set( , )= ( )+ 1( )- 1(0)+ 2 2 2( )- 1( )+ 1(0)- 2(0). 3. Third boundary value problem : the initial condition is (4) and the boundary conditions aregiveninrow3ofTable13.Iftheinitia landboundar ycondition sarecompatible ,then,in transformation (2), one can take( , )= ( )+( 2 - 1 - 2 )[ 1( )- 1(0)]+(1 - 1 )[ 2( )- 2(0)]2- 1- 1 2 ( 1<0, 2>0). Page 41 4. Mixed boundary value problem : the initial condition is (4) and the boundary conditions are giveninrow4ofTable13.Suppos ethattheinitia landboundar ycondition sarecompatible ,i.e.,(0)= 1(0) and ( )= 2(0). Then, in transformation (2), one can set( , )= ( )+ 1( )- 1(0)+ 2( )- 2(0). 5. Mixed boundary value problem : the initial condition is (4) and the boundary conditions are giveninrow5ofTable13.Suppos ethattheinitia landboundar ycondition sarecompatible ,i.e., (0)= 1(0) and ( )= 2(0). Then, in transformation (2), one can take( , )= ( )+( - ) 1( )- 1(0) + 2( )- 2(0). References for Section 0.11: V . M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), A. D. Polyanin, A. V . Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998). Page 42 Chapter 2 Parabolic Equations with TwoSpace Variab les 2.1. Heat Equation = 2 2.1.1. Boundar yValue Problems inCartesian Coor dinates Inrectangular Cartesian coordinates, thetwo-dimensional sourceless heat equation hastheform = 2 2+ 2 2 . Itgoverns two-dimensional unsteady heat transfer processes inquiescent media orsolid bodies with constant thermal diffusivity .Asimilar equation isused tostudy analogous two-dimensional unsteady mass transfer phenomena with constant diffusivity; inthiscase theequation iscalled a diffusion equation. 2.1.1-1. Particular solutions:( , )= + 1 + 2 + 3,( , , )= 2+ 2+2 ( + ) ,( , , )= ( 2+2 )( 2+2 )+ ,( , , )= exp 1 + 2 +( 2 1+ 2 2) + ,( , , )= cos( 1 + 1)cos( 2 + 2)exp -( 2 1+ 2 2) ,( , , )= cos( 1 + 1)sinh( 2 + 2)exp -( 2 1- 2 2) ,( , , )= cos( 1 + 1)cosh( 2 + 2)exp -( 2 1- 2 2) ,( , , )= exp(- - )cos( -2 2 + 1)cos( -2 2 + 2),( , , )= - 0exp -( - 0)2+( - 0)2 4 ( - 0) ,( , , )= erf - 0 2 erf - 0 2 + , where , , 1, 2, 3, 1, 2, 0, 0,and 0arearbitrary constants. Fundamental solution: ( , , )=1 4 exp - 2+ 2 4 . 2.1.1-2. Formulas toconstruct particular solutions. Remarks ontheGreen' sfunctions. 1 .Apart from usual separable solutions ( , , )= 1( ) 2( ) 3( ),theequation inquestion has more sophisticated solutions intheproduct form( , , )= ( , ) ( , ), Page161 where = ( , ) and = ( , ) are solutions of the one-dimensional heat equations = 2 2, = 2 2, considered in Subsection 1.1.1.2. Suppose = ( , , ) is a solution of the heat equation. Then the functions 1= ( + 1, + 2, 2 + 3), 2= ( cos !- sin !+ 1, sin !+ cos !+ 2, + 3), 3= exp 1 + 2 + ( 21+ 22) "( + 2 1 + 1, + 2 2 + 2, + 3), 4= #+ ! exp - !( 2+ 2) 4 ( #+ ! ) #+ ! , #+ ! , $+ #+ ! , #- !$= 1, where , 1, 2, 3, !, #, , 1and 2are arbitrary constants, are also solutions of this equation. The signs at 's in the formula for 1are taken arbitrarily, independently of each other.%& Reference : W. Miller, Jr. (1977). 3 . For all two-dimensional boundary value problems discussed in Subsection 2.1.1, the Green's function can be represented in the product form'( , , (, ), )= ' 1( , (, ) ' 2( , ), ), where ' 1( , (, ) and ' 2( , ), ) are the Green's functions of the corresponding one-dimensional boundary value problems (these functions are speci®ed in Subsections 1.1.1 and 1.1.2). Example 1. The Green's function of the ®rst boundary value problem for a semiin®nite strip ( 0 £ *£ +,0 £ ,< -), considered in Subsection 2.1.1-12, is the product of two one-dimensional Green's functions. The ®rst Green's function isthat of the ®rst boundary value problem on a closed interval ( 0 £*£ +) presented in Subsection 1.1.2-5. The second Green's function is that of the ®rst boundary value problem on a semiin®nite interval ( 0 £ ,< -) presented in Subsection 1.1.2-2, where *and .must be renamed ,and /, respectively. 2.1.1-3. Transformations that allow separation of variables. Table18listspossibl etransformation sthatallowreductio nofthetwo-dimensiona lheatequatio nto a separable equation. All transformations of the independent variables have the form ( , , ) 01 2 ( (, ), ). The transformations that can be obtained by interchange of independent variables, 3 , are omitted. The anharmonic oscillator functions are solutions of the second-order ordinary differential equation 4 565 787+( 94+ :"92+ ;) 4= 0. The Ince polynomials are the 2 -periodic solutions of the Whittaker±Hill equation 4 565 787+ sin2 9 4 5 7+( - :<cos2 9) 4= 0; see Arscott (1964, 1967).%& Reference : W. Miller, Jr. (1977). 2.1.1-4. Domain: - =< < =,- =< < =. Cauchy problem. An initial condition is prescribed:= ( , ) at = 0. Solution:( , , )=1 4 > ? -? > ? -? ( (, )) exp -( - ()2+( - ))2 4 @ (@ ). Example 2. The initial temperature is piecewise-constant and equal to A1in the domain | *|< *0,| ,|< ,0and A2in the domain | *|> *0,| ,|> ,0, speci®cally, B ( *, ,)= C A1for| *|< *0,| ,|< ,0,A2for| *|> *0,| ,|> ,0. Page 162 DFG2 A TABLE 18 Transformations ( , , ) 01 2( (, ), ) that allow solutions with J-separated variables, =exp[ J( (, ), )] ( () K( )) L( ), for the two-dimensional heat equation M= N N + O<O . Everywhere, the function L( ) is exponential No Transformations Factor exp J Function ( () Function K( )) 1 = (,= ) J= 0Exponential functionExponential function 2 = (,= )| | J= 0Exponential functionHermite function 3 = (| |,= )| | J= 0 Hermite function Hermite function 4 =1 2( (2- )2),= ( ) J= 0Parabolic cylinder functionParabolic cylinder function 5 = (cos ),= (sin ) J= 0 Bessel functionExponential function 6 =cosh (cos ),=sinh (sin ) J= 0Modi®ed Mathieu functionMathieufunction 7 =| | (cos ),=| | (sin ) J= 0 Laguerre functionExponential function 8 =| |cosh (cos ),=| |sinh (sin ) J= 0 Ince polynomial Ince polynomial 9 = (,= )+ 2 J= - ) Exponential functionAiry function 10 = (,= ) + :FP J= -1 4 )2 +1 2 :8) P Exponential functionAiry function 11 = (,= ) 1 + 2 J= -1 4 )2 Exponential functionParabolic cylinder function 12 = (,= ) Q|1 - 2| J= -1 4 R )2 ,R=sign( 1 - 2)Exponential functionHermite function 13 = ( ,= ) J= -1 4( (2+ )2) Exponential functionExponential function 14 = (+ 2,= )+ : 2 J= -( (+ :8)) Airy function Airy function 15 = ( + P ,= ) + :FP J= -1 4( (2+ )2) +1 2( (+ :8)) P Airy function Airy function 16 =1 2( (2- )2) ,= ( ) J= -1 16( (2+ )2)2 Parabolic cylinder functionParabolic cylinder function 17 = ( 1 + 2,= ) 1 + 2 J= -1 4( (2+ )2) Parabolic cylinder functionParabolic cylinder function 18 = ( Q|1 - 2|,= ) Q|1 - 2| J= -1 4 R( (2+ )2) ,R=sign( 1 - 2)Hermite function Hermite function Page 163 TABLE 18 (continued ) No Transformations Factor exp J Function ( () Function K( )) 19 =1 2( (2- )2)+ 2,= ( ) J= -1 2 ( (2- )2) Anharmonic oscillator functionAnharmonic oscillator function 20 =1 2( (2- )2) + P ,= ( ) J= -1 16( (2+ )2)2 +1 4 ( (2- )2) P Anharmonic oscillator functionAnharmonic oscillator function 21 = ( cos ),= ( sin ) J= -1 4 (2 Bessel functionExponential function 22 = cosh (cos ),= sinh (sin ) J= -1 4(sinh2(+cos2)) Modi®ed Mathieu functionMathieu function 23 = 1 + 2(cos ),= 1 + 2(sin ) J= -1 4 (2 Whittaker functionExponential function 24 =Q|1 - 2| (cos ),=Q|1 - 2| (sin ) J= -1 4 R (2 ,R=sign( 1 - 2)Laguerre functionExponential function 25 = 1 + 2cosh (cos ),= 1 + 2sinh (sin ) J= -1 4(sinh2(+cos2)) Ince polynomial Ince polynomial 26 =Q|1 - 2|cosh (cos ),= Q|1 - 2|sinh (sin ) J= -1 4 R(sinh2(+cos2)) ,R=sign( 1 - 2)Ince polynomial Ince polynomial Solution:A=1 4( A1- A2) Serf T *0- * 2 U H<V W+erf T *0+ * 2 U H<V W X Serf T ,0- , 2 U H<V W+erf T ,0+ , 2 U H<V W X+ A2. If the initial temperature distribution ( Y, Z) is an in®nitely differentiable function in both arguments, then the solution can be represented in the series form[( Y, Z, \)= ( Y, Z)+ ? ]_^ =1( ` \) ^a! b ^ ced ( Y, Z) f,bº g2g Y2+ g2g Z2. Such a representation is useful for small \.hi Reference : H. S. Carslaw and J. C. Jaeger (1984). 2.1.1-5. Domain: 0 £ Y< j,- j< Z< j. First boundary value problem. A half-plane is considered. The following conditions are prescribed:[= d ( Y, Z) at \= 0 (initial condition),[= k( Z, \) at Y= 0 (boundary condition). Solution:[( Y, Z, \)= l m 0 l m -m d ( (, )) '( Y, Z, (, ), \) n ) n ( + ` l o 0 l m -m k( ), p) q gg ( '( Y, Z, (, ), \- p) r s =0 n ) n p, Page 164 tFv2 y where'( Y, Z, (, ), \)=1 4 z ` \ {exp q-( Y- ()2+( Z- ))2 4 ` \ r-exp q-( Y+ ()2+( Z- ))2 4 ` \ r |.hi Reference : H. S. Carslaw and J. C. Jaeger (1984). 2.1.1-6. Domain: 0 £ Y< j,- j< Z< j. Second boundary value problem. A half-plane is considered. The following conditions are prescribed:[= d ( Y, Z) at \= 0 (initial condition),g } [= k( Z, \) at Y= 0 (boundary condition). Solution:[( Y, Z, \)= l m 0 l m -m d ( (, )) '( Y, Z, (, ), \) n ) n (- ` l o 0 l m -m k( ), p) '( Y, Z,0, ), \- p) n ) n p, where'( Y, Z, (, ), \)=1 4 z ` \ {exp q-( Y- ()2+( Z- ))2 4 ` \ r+exp q-( Y+ ()2+( Z- ))2 4 ` \ r |. 2.1.1-7. Domain: 0 £ Y< j,- j< Z< j. Third boundary value problem. A half-plane is considered. The following conditions are prescribed:[= d ( Y, Z) at \= 0 (initial condition),g } [- ~ [= k( Z, \) at Y= 0 (boundary condition). The solution [( Y, Z, \) is determined by the formula in Paragraph 2.1.1-6 where'( Y, Z, (, ), \)=1 4 z ` \exp q-( Z- ))2 4 ` \ r{exp q-( Y- ()2 4 ` \ r+exp q-( Y+ ()2 4 ` \ r - 2 ~ l m 0exp q-( Y+ (+ )2 4 ` \- ~ 8r n |. 2.1.1-8. Domain: 0 £ Y< j,0 £ Z< j. First boundary value problem. A quadrant of the plane is considered. The following conditions are prescribed:[= d ( Y, Z) at \= 0 (initial condition),[= k1( Z, \) at Y= 0 (boundary condition),[= k2( Y, \) at Z= 0 (boundary condition). Solution:[( Y, Z, \)= l m 0 l m 0 d ( (, )) '( Y, Z, (, ), \) n ( n ) + ` lo 0 l m 0 k1( ), p) q gg ( '( Y, Z, (, ), \- p) r s =0 n ) n p + ` l o 0 l m 0 k2( (, p) q gg ) '( Y, Z, (, ), \- p) r =0 n ( n p, Page 165 where'( Y, Z, (, ), \)=1 4 z ` \ {exp q-( Y- ()2 4 ` \ r-exp q-( Y+ ()2 4 ` \ r |{exp q-( Z- ))2 4 ` \ r-exp q-( Z+ ))2 4 ` \ r |. Example 3. The initial temperature is uniform, ( *, )=y0. The boundary is maintained at zero temperature, 1( , )= 2( *, )= 0. Solution:y=y0erf * 2 w< erf 2 w< .hi References : A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984). 2.1.1-9. Domain: 0 £ < j,0 £ < j. Second boundary value problem. A quadrant of the plane is considered. The following conditions are prescribed:= d ( , ) at = 0 (initial condition),g } = k1( , ) at = 0 (boundary condition),g = k2( , ) at = 0 (boundary condition). Solution:( , , )= l m 0 l m 0 d ( (, )) '( , , (, ), ) n ( n ) - ` lo 0 l m 0 k1( ), p) '( , ,0, ), - p) n ) n p- ` lo 0 l m 0 k2( (, p) '( , , (,0, - p) n ( n p, where'( , , (, ), )=1 4 z ` {exp q-( - ()2 4 ` r+exp q-( + ()2 4 ` r |{exp q-( - ))2 4 ` r+exp q-( + ))2 4 ` r |. 2.1.1-10. Domain: 0 £ < j,0 £ < j. Third boundary value problem. A quadrant of the plane is considered. The following conditions are prescribed:= d ( , ) at = 0 (initial condition),g } - ~1 = k1( , ) at = 0 (boundary condition),g - ~2 = k2( , ) at = 0 (boundary condition). The solution ( , , ) is determined by the formula in Paragraph 2.1.1-9 where'( , , (, ), )=1 4 z ` {exp q-( - ()2 4 ` r+exp q-( + ()2 4 ` r - 2 ~1 z ` exp c` ~2 1 + ~1( + ) ferfc + 2 ` + ~1 ` | ´{exp q-( - )2 4 ` r+exp q-( + )2 4 ` r - 2 ~2 z ` exp c` ~2 2 + ~2( + ) ferfc + 2 ` + ~2 ` |. Example 4. The initial temperature is constant, ( *, )=y0. The temperature of the environment is zero, 1( , )= 2( *, )= 0. Solution:y=y0 erf * 2 w< +exp( 1 *+ wF2 1 ) erfc * 2 w<+ 1 w< ´erf 2 w< +exp( 2 + wF2 2 ) erfc 2 w<+ 2 w< .hi Reference : H. S. Carslaw and J. C. Jaeger (1984). Page 166 tFv2 y 2.1.1-11. Domain: 0 £ < j,0 £ < j. Mixed boundary value problems. 1 . A quadrant of the plane is considered. The following conditions are prescribed:= d ( , ) at = 0 (initial condition),= k1( , ) at = 0 (boundary condition),g = k2( , ) at = 0 (boundary condition). Solution:( , , )= l m 0 l m 0 d ( , ) ( , , , , ) n n + l o 0 l m 0 k1( , p) q gg ( , , , , - p) r s =0 n n p - lo 0 l m 0 k2( , p) ( , , ,0, - p) n n p, where( , , , , )=1 4 z {exp q-( - )2 4 r-exp q-( + )2 4 r |{exp q-( - )2 4 r+exp q-( + )2 4 r |. 2. A quadrant of the plane is considered. The following conditions are prescribed:= d ( , ) at = 0 (initial condition),g } - ~ = k1( , ) at = 0 (boundary condition),= k2( , ) at = 0 (boundary condition). Solution:( , , )= l m 0 l m 0 d ( , ) ( , , , , ) n n - lo 0 l m 0 k1( , p) ( , ,0, , - p) n n p + lo 0 l m 0 k2( , p) q gg ( , , , , - p) r =0 n n p, where( , , , , )=1 4 z {exp q( - )2 4 r-exp q-( + )2 4 r |{exp q-( - )2 4 r+exp q-( + )2 4 r - 2 ~ z exp ~2+ ~( + ) ferfc + 2 + ~ |. Example 5. The initial temperature is uniform, ( *, )=y0. Heat exchange with the environment of zero temperature occurs at one side and the other side is maintained at zero temperature: 1( , )= 2( *, )= 0. Solution:y=y0 erf * 2 w< +exp( *+ wF2) erfc * 2 w<+ w< erf 2 w< . Page 167 2.1.1-12. Domain: 0 £ £ ,0 £ < j. First boundary value problem. A semiin®nite strip is considered. The following conditions are prescribed:= d ( , ) at = 0 (initial condition),= k1( , ) at = 0 (boundary condition),= k2( , ) at = (boundary condition),= k3( , ) at = 0 (boundary condition). Solution:( , , )= l m 0 l 0 d ( , ) ( , , , , ) n n + l o 0 l m 0 k1( , p) q gg ( , , , , - p) r s =0 n n p - l o 0 l m 0 k2( , p) q gg ( , , , , - p) rs = n n p + l o 0 l 0 k3( , p) q gg ( , , , , - p) r =0 n n p, where( , , , , )= 1( , , ) 2( , , ),1( , , )=2 m _ =1sin az sin az exp - a2z22 ,2( , , )=1 2 z {exp q-( - )2 4 r-exp q-( + )2 4 r |. Example 6. The initial temperature is uniform, ( *, )=y0. The boundary is maintained at zero temperature, 1( , )= 2( , )= 3( *, )= 0. Solution:y=4y0erf 2 w< =01 2 ¡+ 1sin(2 ¡+ 1) *+exp- 2(2 ¡+ 1)2 ¢<£+2.¤¥ Reference : H. S. Carslaw and J. C. Jaeger (1984). 2.1.1-13. Domain: 0 £ ¦£ ,0 £ §< ¨. Second boundary value problem. A semiin®nite strip is considered. The following conditions are prescribed:©= ª( ¦, §) at «= 0 (initial condition),¬ ©= ®1( §, «) at ¦= 0 (boundary condition),¬ ©= ®2( §, «) at ¦= (boundary condition),¬ ¯©= ®3( ¦, «) at §= 0 (boundary condition). Solution:©( ¦, §, «)= ° ± 0 ° ² 0 ª( ³, ´) µ( ¦, §, ³, ´, «) ¶ ³ ¶ ´ - · ° ¸ 0 ° ± 0 ®1( ´, ¹) µ( ¦, §,0, ´, «- ¹) ¶ ´ ¶ ¹ + · °¸ 0 °± 0 ®2( ´, ¹) µ( ¦, §, º, ´, «- ¹) ¶ ´ ¶ ¹ - · ° ¸ 0 ° ² 0 ®3( ³, ¹) µ( ¦, §, ³,0, «- ¹) ¶ ³ ¶ ¹, Page 168 »F½2 ¿ whereµ( ¦, §, ³, ´, «)= µ1( ¦, ³, «) µ2( §, ´, «),µ1( ¦, ³, «)=1º+2º ± À_Á =1cos Â a Ã¦º Äcos Â a Ã³º Äexp Â- · a2 Ã2«º2Ä,µ2( §, ´, «)=1 2 Å Ã· « Æexp q-( §- ´)2 4 · « Ç+exp q-( §+ ´)2 4 · « Ç È. 2.1.1-14. Domain: 0 £ ¦£ º,0 £ §< ¨. Third boundary value problem. A semiin®nite strip is considered. The following conditions are prescribed:©= ª( ¦, §) at «= 0 (initial condition),¬ ©- É1 ©= ®1( §, «) at ¦= 0 (boundary condition),¬ ©+ É2 ©= ®2( §, «) at ¦= º(boundary condition),¬ ¯©- É3 ©= ®3( ¦, «) at §= 0 (boundary condition). The solution ©( ¦, §, «) is determined by the formula in Paragraph 2.1.1-13 where the Green's function µ( ¦, §, ³, ´, «) is the product of the Green's function of Subsection 1.1.1-11 and that of Subsection 1.1.1-8; one should replace ¦, ³, and Éby §, ´and É3, respectively, in the last Green's function. 2.1.1-15. Domain: 0 £ ¦£ º,0 £ §< ¨. Mixed boundary value problems. 1 Ê. A semiin®nite strip is considered. The following conditions are prescribed:©= ª( ¦, §) at «= 0 (initial condition),©= ®1( §, «) at ¦= 0 (boundary condition),©= ®2( §, «) at ¦= º(boundary condition),¬ ¯©= ®3( ¦, «) at §= 0 (boundary condition). Solution:©( ¦, §, «)= °± 0 °² 0 ª( ³, ´) µ( ¦, §, ³, ´, «) ¶ ³ ¶ ´ + · ° ¸ 0 ° ± 0 ®1( ´, ¹) q ¬¬³ µ( ¦, §, ³, ´, «- ¹)Ç Ë=0 ¶ ´ ¶ ¹ - · ° ¸ 0 ° ± 0 ®2( ´, ¹) q ¬¬³ µ( ¦, §, ³, ´, «- ¹)Ç Ë=² ¶ ´ ¶ ¹ - · °¸ 0 °² 0 ®3( ³, ¹) µ( ¦, §, ³,0, «- ¹) ¶ ³ ¶ ¹, whereµ( ¦, §, ³, ´, «)= µ1( ¦, ³, «) µ2( §, ´, «),µ1( ¦, ³, «)=2º ± À_Á =1sin Â a Ã¦º Äsin Â a Ã³º Äexp Â- · a2 Ã2«º2Ä,µ2( §, ´, «)=1 2 Å Ã· « Æexp q-( §- ´)2 4 · « Ç+exp q-( §+ ´)2 4 · « Ç È. Page 169 2 Ê. A semiin®nite strip is considered. The following conditions are prescribed:©= ª( ¦, §) at «= 0 (initial condition),¬ ©= ®1( §, «) at ¦= 0 (boundary condition),¬ ©= ®2( §, «) at ¦= º(boundary condition),©= ®3( ¦, «) at §= 0 (boundary condition). Solution:©( ¦, §, «)= °± 0 °² 0 ª( ³, ´) µ( ¦, §, ³, ´, «) ¶ ³ ¶ ´ - · ° ¸ 0 ° ± 0 ®1( ´, ¹) µ( ¦, §,0, ´, «- ¹) ¶ ´ ¶ ¹ + · ° ¸ 0 ° ± 0 ®2( ´, ¹) µ( ¦, §, º, ´, «- ¹) ¶ ´ ¶ ¹ + · °¸ 0 °² 0 ®3( ³, ¹) q ¬¬´ µ( ¦, §, ³, ´, «- ¹)Ç Ì=0 ¶ ³ ¶ ¹, whereµ( ¦, §, ³, ´, «)= µ1( ¦, ³, «) µ2( §, ´, «),µ1( ¦, ³, «)=1º+2º ± À Á =1cos Â a Ã¦º Äcos Â a Ã³º Äexp Â- · a2 Ã2«º2Ä,µ2( §, ´, «)=1 2 Å Ã· « Æexp q-( §- ´)2 4 · « Ç-exp q-( §+ ´)2 4 · « Ç È. 2.1.1-16. Domain: 0 £ ¦£ º1,0 £ §£ º2. First boundary value problem. A rectangle is considered. The following conditions are prescribed:©= ª( ¦, §) at «= 0 (initial condition),©= ®1( §, «) at ¦= 0 (boundary condition),©= ®2( §, «) at ¦= º1(boundary condition),©= ®3( ¦, «) at §= 0 (boundary condition),©= ®4( ¦, «) at §= º2(boundary condition). Solution:©( ¦, §, «)= °²1 0 °²2 0 ª( ³, ´) µ( ¦, §, ³, ´, «) ¶ ´ ¶ ³ + · ° ¸ 0 ° ²2 0 ®1( ´, ¹) q ¬¬³ µ( ¦, §, ³, ´, «- ¹)Ç Ë=0 ¶ ´ ¶ ¹ - · °¸ 0 °²2 0 ®2( ´, ¹) q ¬¬³ µ( ¦, §, ³, ´, «- ¹)Ç Ë=²1 ¶ ´ ¶ ¹ + · ° ¸ 0 ° ²1 0 ®3( ³, ¹) q ¬¬´ µ( ¦, §, ³, ´, «- ¹)ÇÌ=0 ¶ ³ ¶ ¹ - · ° ¸ 0 ° ²1 0 ®4( ³, ¹) q ¬¬´ µ( ¦, §, ³, ´, «- ¹)Ç Ì=²2 ¶ ³ ¶ ¹, Page 170 »F½2 ¿ whereµ( ¦, §, ³, ´, «)=4º1 º2 ± À_Á =1 ± ÀÍ=1sin a Ã¦º1sin a Ã³º1sin Î Ã§º2sin Î Ã´º2exp q- Ã2Â a2º2 1+ Î2º2 2 Ä · «Ç. Example 7. The initial temperature is uniform, ( *, Ï)=¿0. The boundary is maintained at zero temperature,Ð 1( Ï, £)= Ð 2( Ï, £)= Ð 3( *, £)= Ð 4( *, £)= 0. Solution:¿=16¿0Ñ2 Ò Ó Ô Õ =01 2 Ö+ 1sin ×(2 Ö+ 1) Ñ*+1 Øexp ×- Ñ2(2 Ö+ 1)2 Ù<Ú+2 1 Ø Û ´Ò Ó ÔÜ=01 2 Ý+ 1sin ×(2 Ý+ 1) ÑÏ+2 Øexp ×- Ñ2(2 Ý+ 1)2Ù<Ú+2 2 Ø Û.Þß Reference : H. S. Carslaw and J. C. Jaeger (1984). 2.1.1-17. Domain: 0 £ à£ á1,0 £ â£ á2. Second boundary value problem. A rectangle is considered. The following conditions are prescribed:ã= ä( à, â) at å= 0 (initial condition),æ çã= è1( â, å) at à= 0 (boundary condition),æ çã= è2( â, å) at à= á1 (boundary condition),æ éã= è3( à, å) at â= 0 (boundary condition),æ éã= è4( à, å) at â= á2 (boundary condition). Solution:ã( à, â, å)= ê ë1 0 ê ë2 0 ä( ì, í) î( à, â, ì, í, å) ï í ï ì - ð ê ñ 0 ê ë2 0 è1( í, ò) î( à, â,0, í, å- ò) ï í ï ò + ð êñ 0 ê ë2 0 è2( í, ò) î( à, â, á1, í, å- ò) ï í ï ò - ð ê ñ 0 ê ë1 0 è3( ì, ò) î( à, â, ì,0, å- ò) ï ì ï ò + ð êñ 0 ê ë1 0 è4( ì, ò) î( à, â, ì, á2, å- ò) ï ì ï ò, whereî( à, â, ì, í, å)=1á1 á2 ó1 + 2 ô õ_ö =1exp ÷- ø2 ù2ð åá2 1 úcos ùø û ü 1cos ùø ìü 1 ý ´ó1 + 2 ô õþ=1exp ÷- ø2 ÿ2ð ü2 2 úcos ÿø ü 2cos ÿø íü 2 ý. Reference : H. S. Carslaw and J. C. Jaeger (1984). 2.1.1-18. Domain: 0 £û£ ü 1,0 £ £ ü 2. Third boundary value problem. A rectangle is considered. The following conditions are prescribed:= (û, ) at = 0 (initial condition), - 1 = è1( , ) atû= 0 (boundary condition), + 2 = è2( , ) atû= ü 1(boundary condition), é- 3 = è3(û, ) at = 0 (boundary condition), é+ 4 = è4(û, ) at = ü 2(boundary condition). Page 171 The solution (û, , ) is determined by the formula in Paragraph 2.1.1-17 whereî(û, , ì, í, )= ô õ_ö =1 ö (û) ö ( ì) ö2exp(- ð 2 ö) ô õþ=1 þ( ) þ( í) þ 2exp(- ð 2þ) , ö (û)=cos( öû)+ 1 ö sin( öû), ö2= 2 2 2 ö 2 ö + 2 1 2 ö + 2 2+ 1 2 2 ö + ü 1 2 ÷1 + 2 1 2 öú, þ( )=cos( þ )+ 3 þsin( þ ), þ 2= 4 2 2þ 2þ+ 2 32þ+ 2 4+ 3 2 2þ+ ü 2 2 ÷1 + 2 32þú. Here, the ö and þare positive roots of the transcendental equations tan( ü 1) = 1+ 2 2- 1 2,tan( ü 2)= 3+ 42- 3 4. 2.1.1-19. Domain: 0 £û£ ü 1,0 £ £ ü 2. Mixed boundary value problems. 1 . A rectangle is considered. The following conditions are prescribed:= (û, ) at = 0 (initial condition),= è1( , ) atû= 0 (boundary condition),= è2( , ) atû= ü 1(boundary condition), é= è3(û, ) at = 0 (boundary condition), é= è4(û, ) at = ü 2(boundary condition). Solution:(û, , )= ê ë1 0 ê ë2 0 ( ì, í) î(û, , ì, í, ) ï í ï ì + ð ê ñ 0 ê ë2 0 è1( í, ò)ó ì î(û, , ì, í, - ò)ý =0 ï í ï ò - ð ê ñ 0 ê ë2 0 è2( í, ò)ó ì î(û, , ì, í, - ò)ý =ë1 ï í ï ò - ð êñ 0 ê ë1 0 è3( ì, ò) î(û, , ì,0, - ò) ï ì ï ò + ð ê ñ 0 ê ë1 0 è4( ì, ò) î(û, , ì, ü 2, - ò) ï ì ï ò, whereî(û, , ì, í, )=4ü 1 ü 2 ó ô õ ö =1sin ùø û ü 1sin ùø ìü 1exp ÷- ø2 ù2ð ü2 1 ú ý ´ó1 2+ ô õþ=1cos ÿø ü 2cos ÿø íü 2exp ÷- ø2 ÿ2ð ü2 2 ú ý. 2 . A rectangle is considered. The following conditions are prescribed:= (û, ) at = 0 (initial condition),= è1( , ) atû= 0 (boundary condition), = è2( , ) atû= ü 1(boundary condition),= è3(û, ) at = 0 (boundary condition), é= è4(û, ) at = ü 2(boundary condition). Page 172 2 Solution:(û, , )= ê ë1 0 ê ë2 0 ( ì, í) î(û, , ì, í, ) ï í ï ì + ð ê ñ 0 ê ë2 0 è1( í, ò)ó ì î(û, , ì, í, - ò)ý =0 ï í ï ò + ð êñ 0 ê ë2 0 è2( í, ò) î(û, , ü 1, í, - ò) ï í ï ò + ð ê ñ 0 ê ë1 0 è3( ì, ò)ó í î(û, , ì, í, - ò)ý =0 ï ì ï ò + ð êñ 0 ê ë1 0 è4( ì, ò) î(û, , ì, ü 2, - ò) ï ì ï ò, whereî(û, , ì, í, )=4ü 1 ü 2 ô õ ö =0sinó ø(2 ù+ 1)û2 ü 1 ýsinó ø(2 ù+ 1) ì 2 ü 1 ýexpó- ðø2(2 ù+ 1)2 4 ü2 1 ý ´ ô õþ=0sinó ø(2 ÿ+ 1) 2 ü 2 ýsinó ø(2 ÿ+ 1) í 2 ü 2 ýexpó- ðø2(2 ÿ+ 1)2 4 ü2 2 ý . 2.1.2. Problems in Polar Coordinates The sourceless heat equation with two space variables in the polar coordinate system ,has the form= ð ÷ 2 2+1 +12 2 2ú, = û2+ 2. One-dimensional problems with axial symmetry that have solutions of the form = ( , ) are considered in Subsection 1.2.1. 2.1.2-1. Domain: 0 £ < ,0 ££ 2ø. Cauchy problem. An initial condition is prescribed:= ( ,) at = 0. Solution:( ,, )=1 4ø ð 2 0 ô 0 exp !- 2+ 2- 2 cos(- ") 4 # ý ( , ") $ $ ". 2.1.2-2. Domain: 0 £ £ %,0 ££ 2ø. First boundary value problem. A circle is considered. The following conditions are prescribed:= ( ,) at = 0 (initial condition),= è(, ) at = %(boundary condition). Solution:( ,, )=2 0 &0 ( , ") '( ,, , ", ) $ $ " - # % (0 2 0 è( ", )) ! '( ,, , ", - ))ý =& $ " $ ). Page 173 Here,'( ,, , ", )=1ø %2 ô * ö =0 ô *þ=1 + ö [ , - ö ( öþ%)]2 , ö ( öþ) , ö ( öþ ) cos[ .(- ")] exp( - 2 öþ# ),+0= 1,+ ö = 2 ( .= 1,2, ///), where , ö ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument), and öþare positive roots of the transcendental equation , ö ( %)= 0. Reference : H. S. Carslaw and J. C. Jaeger (1984). 2.1.2-3. Domain: 0 £ £ %,0 ££ 2 0. Second boundary value problem. A circle is considered. The following conditions are prescribed:= ( ,) at = 0 (initial condition), 1= è(, ) at = %(boundary condition). Solution:( ,, )=2 0 &0 ( , ") '( ,, , ", ) $ $ "+ # % (0 2 0 è( ", )) '( ,, %, ", - )) $ " $ ). Here,'( ,, , ", )=10 %2+10 ô *_ö =0 ô *þ=1 + ö 2 öþ, ö ( öþ) , ö ( öþ ) ( 2 öþ%2- .2)[ , ö ( öþ%)]2cos[ .(- ")] exp( - 2 öþ# ),+0= 1,+ ö = 2 ( .= 1,2, ///), where , ö ( ) are the Bessel functions, and öþare positive roots of the transcendental equation, - ö ( %)= 0. 2.1.2-4. Domain: 0 £ £ %,0 ££ 2 0. Third boundary value problem. A circle is considered. The following conditions are prescribed:= ( ,) at = 0 (initial condition), 1+ = è(, ) at = %(boundary condition). The solution ( ,, ) is determined by the formula in Paragraph 2.1.2-3 where'( ,, , ", )=10 ô * ö =0 ô *þ=1 + ö 2 öþ, ö ( öþ) , ö ( öþ ) ( 2 öþ%2+ 2%2- .2)[ , ö ( öþ%)]2cos[ .(- ")] exp( - 2 öþ# ),+0= 1,+ ö = 2 ( .= 1,2, ///). Here, , ö ( ) are the Bessel functions, and öþare positive roots of the transcendental equation , - ö ( %)+ , ö ( %)= 0. Reference : H. S. Carslaw and J. C. Jaeger (1984). Page 174 2 2.1.2-5. Domain: %1£ £ %2,0 ££ 2 0. First boundary value problem. An annular domain is considered. The following conditions are prescribed:= ( ,) at = 0 (initial condition),= è1(, ) at = %1(boundary condition),= è2(, ) at = %2(boundary condition). Solution:( ,, )=2 0 &2&1 ( , ") '( ,, , ", ) $ $ " + # %1 (0 2 0 è1( ", )) ! '( ,, , ", - ))ý =&1 $ " $ ) - # %2 (0 2 0 è2( ", )) ! '( ,, , ", - ))ý =&2 $ " $ ). Here,'( ,, , ", )= 0 2 ô * ö =0 ô *þ=1+ ö 2 öþ 3 ö ( öþ) 3 ö ( öþ ) cos[ .(- ")] exp( - 2 öþ# ),+ ö = 1 42if .= 0, 1 if .¹ 0, 2 öþ= 2 öþ,2 ö ( öþ%2),2 ö ( öþ%1)- ,2 ö ( öþ%2),3 ö ( öþ)= , ö ( öþ%1) 5 ö ( öþ)- 5 ö ( öþ%1) , ö ( öþ), where , ö ( ) and 5 ö ( ) are the Bessel functions, and öþare positive roots of the transcendental equation, ö ( %1) 5 ö ( %2)- 5 ö ( %1) , ö ( %2)= 0. Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 2.1.2-6. Domain: %1£ £ %2,0 ££ 2 0. Second boundary value problem. An annular domain is considered. The following conditions are prescribed:= ( ,) at = 0 (initial condition), 1= 61(, ) at = %1(boundary condition), 1= 62(, ) at = %2(boundary condition). Solution:( ,, )=2 0 &2&1 ( , ") '( ,, , ", ) $ $ " - # %1(0 2 0 61( ", )) '( ,, %1, ", - )) $ " $ ) + # %2(0 2 0 62( ", )) '( ,, %2, ", - )) $ " $ ). Here,'( ,, , ", )=10( %2 2- %2 1)+10 ô * ö =0 ô *þ=1 + ö 2 öþ 3 ö ( öþ) 3 ö ( öþ ) cos[ .(- ")] exp( - 2 öþ# ) ( 2 öþ%2 2- .2) 32 ö ( öþ%2)-( 2 öþ%2 1- .2) 32 ö ( öþ%1),3 ö ( öþ)= , - ö ( öþ%1) 5 ö ( öþ)- 5 - ö ( öþ%1) , ö ( öþ), where+0= 1and+ ö = 2for .= 1,2, ///; , ö ( ) and 5 ö ( ) are the Bessel functions, and öþare positive roots of the transcendental equation, - ö ( %1) 5 - ö ( %2)- 5 - ö ( %1) , - ö ( %2)= 0. Page 175 2.1.2-7. Domain: %1£ £ %2,0 ££ 2 0. Third boundary value problem. An annular domain is considered. The following conditions are prescribed:= ( ,) at = 0 (initial condition), 1- 1 = 61(, ) at = %1(boundary condition), 1+ 2 = 62(, ) at = %2(boundary condition). Solution:( ,, )=2 0 &2&1 ( , ") '( ,, , ", ) $ $ " - # %1 (0 2 0 61( ", )) '( ,, %1, ", - )) $ " $ ) + # %2 (0 2 0 62( ", )) '( ,, %2, ", - )) $ " $ ). Here,'( ,, , ", )=10 ô * ö =0 ô *þ=1 + ö 2 öþ 3 ö ( öþ) 3 ö ( öþ ) cos[ .(- ")] exp( - 2 öþ# ) ( 2 2 %2 2+ 2 öþ%2 2- .2) 32 ö ( öþ%2)-( 2 1 %2 1+ 2 öþ%2 1- .2) 32 ö ( öþ%1),3 ö ( öþ)= 78 öþ, - ö ( öþ%1)- 1 , ö ( öþ%1) 9:5 ö ( ö ;) - 7< ö ;5 - ö ( ö ;%1)- 1 5 ö ( ö ;%1) 9 , ö ( ö ;), where+0= 1and+ ö = 2for .= 1,2, ///; , ö ( ) and 5 ö ( ) are the Bessel functions, and ö ; are positive roots of the transcendental equation7 , - ö ( %1)- 1 , ö ( %1) 9 7 5 - ö ( %2)+ 2 5 ö ( %2) 9 = 7< 5 - ö ( %1)- 1 5 ö ( %1) 9 7< , - ö ( %2)+ 2 , ö ( %2) 9.=> Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 2.1.2-8. Domain: 0 £ < ,0 ££0. First boundary value problem. A wedge domain is considered. The following conditions are prescribed:?= @( , A) at B= 0 (initial condition),?= 61( , B) at A= 0 (boundary condition),?= 62( , B) at A= A0(boundary condition). Solution:?( , A, B)= C0 0 ô 0 @( , ") '( , A, , ", B) $ $ " + #(0 ô 0 61( , ))1 ! DD " '( E, A, , ", B- )) F G =0 $ $ ) - #(0 ô 0 62( , ))1 ! DD " '( E, A, , ", B- )) F G =C0 $ $ ). Here,'( E, A, , ", B)=1# A0 Bexp H- E2+ 2 4 # B I ô * ö =1 J ö KC0 L E 2 # B MsinL . 0 AA0 MsinL . 0 "A0 M, whereJON( E) are the modi®ed Bessel functions.=> Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 176 PR2 U 2.1.2-9. Domain: 0 £ E< V,0 £ A£ A0. Second boundary value problem. A wedge domain is considered. The following conditions are prescribed:?= @( E, A) at B= 0 (initial condition),E-1DC ?= 61( E, B) at A= 0 (boundary condition),E-1DC ?= 62( E, B) at A= A0(boundary condition). Solution:?( E, A, B)= WC0 0 W X 0 @( Y, Z) '( E, A, Y, Z, B) Y [ Y [ Z - \ W(0 W X 0 ]1( Y, )) '( E, A, Y,0, B- )) [ Y [ ) + \ W(0 W X 0]2( Y, )) '( E, A, Y, A0, B- )) [ Y [ ). Here,'( E, A, Y, Z, B)=1\ A0 Bexp H- E2+ Y2 4 \ B I ^1 2J0L E Y 2 \ _ M+ X `_ö =1 J ö aKcb0 L E Y 2 \ _ McosL d e ff0 McosL d e Zf0 M F, whereJON( g) are the modi®ed Bessel functions.hi Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 2.1.2-10. Domain: 0 £ g£ j,0 £f£f0. First boundary value problem. A circular sector is considered. The following conditions are prescribed:k= l( g,f) at _= 0 (initial condition),k=]1(f, _) at g= j (boundary condition),k=]2( g, _) atf= 0 (boundary condition),k=]3( g, _) atf=f0(boundary condition). Solution:k( g,f, _)= W b0 0 W&0 l( Y, Z) '( g,f, Y, Z, _) Y [ Y [ Z - \ j W(0 W b0 0]1( Z, ))^ mm Y '( g,f, Y, Z, _- )) n o =& [ Z [ ) + \ W(0 W&0 ]2( Y, ))1Y ^ mm Z '( g,f, Y, Z, _- )) n p =0 [ Y [ ) - \ W(0 W&0 ]3( Y, ))1Y ^ mm Z '( g,f, Y, Z, _- )) n p = b0 [ Y [ ). Here,'( g,f, Y, Z, _)=4j2f0 X ` ö =1 X `q=1 r ö a sb0( t öqg)r ö a sb0( t öqY) [r u ö a sb0( t öqj)]2sin vd e ff0 wsin vd e Zf0 wexp(- t2 öq\ _), where ther ö a sb0( g) are the Bessel functions, and the t öqare positive roots of the transcendental equationr ö a sb0( t j)= 0. Page 177 Example. The initial temperature is uniform, x( y, z)= {0. The boundary is maintained at zero temperature, |1( z, })=|2( y, })= |3( y, })= 0. Solution:{=8 {0~ 2 =01 2 + 1sin( z) =1exp(- 2 }) :<( y) [ <( )]2 0 :<( ) , =(2 + 1) ~z0, where the are positive roots of the transcendental equation<( )= 0.hi References : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), H. S. Carslaw and J. C. Jaeger (1984). 2.1.2-11. Domain: 0 £ g£ j,0 £f£f0. Second boundary value problem. A circular sector is considered. The following conditions are prescribed:k= l( g,f) at _= 0 (initial condition),m k=]1(f, _) at g= j (boundary condition),g-1m b k=]2( g, _) atf= 0 (boundary condition),g-1m b k=]3( g, _) atf=f0(boundary condition). Solution:k( g,f, _)= W b0 0 W 0 l( Y, Z) ( g,f, Y, Z, _) Y [ Y [ Z + \ j W 0 W b0 0 ]1( Z, ) ( g,f, j, Z, _- ) [ Z [ - \ W 0 W 0 ]2( Y, ) ( g,f, Y,0, _- ) [ Y [ + \ W 0 W 0]3( Y, ) ( g,f, Y,f0, _- ) [ Y [ . Here,( g,f, Y, Z, _)=2j2f0+ 4f0 X ` =0 X `q=1 t2 qr a sb0( t qg)r a sb0( t qY) ( j2f20t2 q-d2e2)[r a sb0( t qj)]2 ´cos vd e ff0 wcos vd e Zf0 wexp(- t2 q\ _), where ther a sb0( g) are the Bessel functions, and the t qare positive roots of the transcendental equationr u a sb0( t j)= 0.hi Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 2.1.2-12. Domain: 0 £ g£ j,0 £f£f0. Mixed boundary value problem. A circular sector is considered. The following conditions are prescribed:k= l( g,f) at _= 0 (initial condition),m k- k=](f, _) at g= j (boundary condition),m b k= 0 atf= 0 (boundary condition),m b k= 0 atf=f0(boundary condition). Solution:k( g,f, _)= W b0 0 W 0 l( Y, Z) ( g,f, Y, Z, _) Y [ Y [ Z + \ j W 0 W b0 0]( Z, ) ( g,f, j, Z, _- ) [ Z [ . Page 178 2 { Here,( g,f, Y, Z, _)= X ` =0 X `q=1 qr ( t qg)r ( t qY) cos( f) cos( Z) exp( - t2 q\ _), =d ef0, q=4 t2 qf0( t2 qj2+ 2j2- 2 ) ¡r ( t qj) ¢2, wherer ( g) are the Bessel functions, and t qare positive roots of the transcendental equationtr u ( t j)+ r ( t j)= 0.hi Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 2.1.2-13. Domain: j1£ g£ j2,0 £f£f0. Different boundary value problems. Some problems for this domain were studied in Budak, Samarskii, and Tikhonov (1980). 2.1.3. Axisymmetric Problems In the case of angular symmetry, the two-dimensional sourceless heat equation in the cylindricalcoordinate system has the formm km _= \ £m2 km g2+1g m km g+m2 km ¤2 ¥, g= ¦ §2+ ¨2. This equation governs two-dimensional unsteady thermal processes in quiescent media or solidbodies (bounded by coordinate surfaces of the cylindrical system) in the case where the initial andboundary conditions are independent of the angular coordinate. A similar equation is used to studyanalogous two-dimensional unsteady mass transfer phenomena. 2.1.3-1. Particular solutions. Remarks on the Green's functions. 1 ©. Apart from usual separable solutions k( g,¤, ª)= l1( g) l2(¤) l3( ª), the equation in question has more sophisticated solutions in the product formk( g,¤, ª)= «( g, ª) ¬(¤, ª), where «= «( g, ª) and ¬= ¬(¤, ª) are solutions of the simpler one-dimensional equationsm «m ª= £m2«m ®2+1® m «m ® ¥(see Subsection 1.2.1-1 for particular solutions of this equation),m ¬m ª= m2¬m ¤2(see Subsection 1.1.1-1 for particular solutions of this equation). 2 ©. For all two-dimensional boundary value problems considered in Subsection 2.1.3, the Green's function can be represented in the product form(®,¤, Y, ¯, ª)= 1(®, Y, ª) 2(¤, ¯, ª), where 1(®, Y, ª) and 2(¤, ¯, ª) are the Green's functions of appropriate one-dimensional boundary value problems. Page 179 2.1.3-2. Domain: 0 £®£ j,0 £¤< °. First boundary value problem. A semiin®nite circular cylinder is considered. The following conditions are prescribed:k= l(®,¤) at ª= 0 (initial condition),k= ±1(¤, ª) at®= j(boundary condition),k= ±2(®, ª) at¤= 0 (boundary condition). Solution:k(®,¤, ª)= 2 ² ³ X 0 ³ 0 Y ´( Y, ¯) (®, µ, Y, ¯, ª) ¶ Y ¶ ¯ - 2 ² · ³ 0 ³ X 0 ±1( ¸, ) ¹mm Y ( º, µ, Y, ¸, »- ) n o = ¶ ¸ ¶ + 2 ¼ ½ ³ 0 ³ 0 Y ¾2( Y, ) ¹mm ¸ ( º, µ, Y, ¸, »- ) n ¿ =0 ¶ Y ¶ . Here,( º, µ, Y, ¸, »)= 1( º, Y, ») 2( µ, ¸, »),1( º, Y, »)=1¼ ·2 X À =11Á2 1( Â ) Á 0 Ã Â º· Ä Á 0 Ã Â Y· ÄexpÃ- ½ Â2 »·2Ä,2( µ, ¸, »)=1 2 Å ¼ ½ » Æexp ¹-( µ- ¸)2 4 ½ » n-exp ¹-( µ+ ¸)2 4 ½ » n Ç, where the Â are positive zeros of the Bessel function, Á 0( Â )= 0. Example 1. The initial temperature is the same at every point of the cylinder, È( É, Ê)= Ë0. The lateral surface and the end face are maintained at zero temperature, Ì1( É, })= Ì2( Ê, })= 0. Solution:Ë( É, Ê, })=2 Ë0erf Í Ê 2 Î Ï:} Ð Ñ Ò Ó =1 Ô0( Õ ÓÉ)Õ ÓÔ1( Õ Ó)exp(- Õ2 ÓÏ:}), Õ Ó = Ö Ó. Example 2. The initial temperature of the cylinder is everywhere zero, È( É, Ê)= 0. The lateral surface É= is maintained at a constant temperature Ë0, and the end face Ê= 0at zero temperature. Solution:Ë( É, Ê, })= Ë0- Ë0Ñ Ò Ó =1 Ô0( Õ ÓÉ)Õ ÓÔ1( Õ Ó) ×2exp(- Õ2 ÓÏ:}) erf Í Ê 2Î Ï:} Ð +exp( Õ ÓÊ) erfc Í Ê 2 Î Ï:}+ Õ ÓÎÏ:}Ð+exp(- Õ ÓÊ) erfc Í Ê 2 Î Ï:}- Õ ÓÎÏ:}Ð Ø, where the Õ Ó are positive zeros of the Bessel function,Ô0( Õ )= 0.ÙÚ References : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), H. S. Carslaw and J. C. Jaeger (1984). 2.1.3-3. Domain: 0 £ º£ ·,0 £ µ< Û. Second boundary value problem. A semiin®nite circular cylinder is considered. The following conditions are prescribed:Ü= ´( º, µ) at »= 0 (initial condition),m Ý Ü= ¾1( µ, ») at º= ·(boundary condition),m Þ Ü= ¾2( º, ») at µ= 0 (boundary condition). Page 180 Solution:( , , )= 2 0 0 (, ) ( , ,, , ) + 2 0 0 1( , ) ( , , , , - ) - 2 0 0 2(, ) ( , ,,0, - ) . Here,( , ,, , )= 1( ,, ) 2( , , ),1( ,, )=1 2+1 2 =112 0( ) 0 0 exp- 2 2,2( , , )=1 2 !exp "-( - )2 4 #+exp "-( + )2 4 # $, where the are positive zeros of the ®rst-order Bessel function, 1( )= 0. 2.1.3-4. Domain: 0 £ £ ,0 £ < %. Third boundary value problem. A semiin®nite circular cylinder is considered. The following conditions are prescribed:=( , ) at = 0 (initial condition),& '+ (1 =1( , ) at = (boundary condition),& )- (2 =2( , ) at = 0 (boundary condition). The solution ( , , ) is determined by the formula in Paragraph 2.1.3-3 where( , ,, , )= 1( ,, ) 2( , , ),1( ,, )=1 2 =1 2 ( (2 1 2+ 2 ) 2 0( ) 0 0 exp- 2 2,2( , , )=1 2 !exp "-( - )2 4 #+exp "-( + )2 4 #- 2 (2 0exp "-( + + *)2 4 - (2 *# *$. Here, 0( ) is the zeroth Bessel function and the are positive roots of the transcendental equation 1( )- (1 0( )= 0. Example 3. The initial temperature is the same at every point of the cylinder, +( ,, -)=0. At the lateral surface and the end face, heat exchange of the cylinder with the zero temperature environment occurs, .1( -, /)= .2( ,, /)= 0. Solution:( ,, -, /)=20 011 2erf 3 - 2 4 5/ 6+exp(02 -+02 2 5/) erfc 3 - 2 4 5/+02 45/6 7 8 9 : =1 ;0( < :,) exp( - <2 :5/) (02 1+ <2 : );0( < :1), where the < : are positive roots of the transcendental equation <;1( < 1)-01;0( < 1)= 0.=?> Reference : H. S. Carslaw and J. C. Jaeger (1984). Page 181 2.1.3-5. Domain: 0 £ £ ,0 £ < %. Mixed boundary value problems. 1 @. A semiin®nite circular cylinder is considered. The following conditions are prescribed:=( , ) at = 0 (initial condition),=1( , ) at = (boundary condition),& )=2( , ) at = 0 (boundary condition). Solution:( , , )= 2 0 0 (, ) ( , ,, , ) - 2 0 01( , ) " && ( , ,, , - )# A= - 2 0 0 2(, ) ( , ,,0, - ) . Here,( , ,, , )= 1( ,, ) 2( , , ),1( ,, )=1 2 =112 1( ) 0 0 exp- 2 2,2( , , )=1 2 !exp "-( - )2 4 #+exp "-( + )2 4 # $, where the are positive zeros of the Bessel function, 0( )= 0. 2 @. A semiin®nite circular cylinder is considered. The following conditions are prescribed:=( , ) at = 0 (initial condition),& '=1( , ) at = (boundary condition),=2( , ) at = 0 (boundary condition). Solution:( , , )= 2 0 0 (, ) ( , ,, , ) + 2 0 01( , ) ( , , , , - ) + 2 0 0 2(, ) " && ( , ,, , - )# B=0 . Here,( , ,, , )= 1( ,, ) 2( , , ),1( ,, )=1 2+1 2 =112 0( ) 0 0 exp- 2 2,2( , , )=1 2 !exp "-( - )2 4 #-exp "-( + )2 4 # $, where the are positive zeros of the ®rst-order Bessel function, 1( )= 0. Page 182 2 3 @. A semiin®nite circular cylinder is considered. The following conditions are prescribed:=( , ) at = 0 (initial condition),& '+ ( =1( , ) at = (boundary condition),=2( , ) at = 0 (boundary condition). The solution ( , , ) is determined by the formula in Paragraph 2.1.3-5, Item 2 @where( , ,, , )= 1( ,, ) 2( , , ),1( ,, )=1 2 =1 2 ( (22+ 2 ) 2 0( ) 0 0 exp- 2 2,2( , , )=1 2 !exp "-( - )2 4 #-exp "-( + )2 4 # $, where the are positive roots of the transcendental equation 1( )- ( 0( )= 0. Example 4. The initial temperature is the same at every point of the cylinder, +( ,, -)=0. Heat exchange of the cylinder with the zero temperature environment occurs at the lateral surface, .1( -, /)= 0. The end face is maintained at zero temperature, .2( ,, /)= 0. Solution:( ,, -, /)=20 01 erf 3 - 2 4 5/ 6 8 9 : =1 ;0( C :,) (02+ C2 : );0( C :1)exp(- C2 :5/), C : = D :1. Example 5. The initial temperature of the cylinder is everywhere zero, +( ,, -)= 0. Heat exchange of the cylinder with the zero temperature environment occurs at the lateral surface, .1( -, /)= 0. The end face is maintained at a constant temperature, .2( ,, /)=0. Solution:( ,, -, /)= 0 018 9 : =1 ;0( C :,) ( C2 : +02);0( C :1) 22exp(- C :-) +exp( C :-) erfc 3 C :45/+ - 24 5/ 6-exp(- C :-) erfc 3 C :45/- - 24 5/ 6 7, C : = D :1.=?> Reference : H. S. Carslaw and J. C. Jaeger (1984). 2.1.3-6. Domain: 0 £ £ ,0 £ £ E. First boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:=( , ) at = 0 (initial condition),=1( , ) at = (boundary condition),=2( , ) at = 0 (boundary condition),=3( , ) at = E(boundary condition). Solution:( , , )= 2 F 0 0 (, ) ( , ,, , ) - 2 0 F 0 1( , ) " && ( , ,, , - )# A= + 2 0 0 2(, ) " && ( , ,, , - )# B=0 - 2 0 0 3(, ) " && ( , ,, , - )# B= F . Page 183 Here,( , ,, , )= 1( ,, ) 2( , , ),1( ,, )=1 2 =112 1( ) 0 0 exp- 2 2,2( , , )=2E =1sin G E sin G E exp- G22 E2, where the are positive zeros of the Bessel function, 0( )= 0. Example 6. The initial temperature is the same at every point of the cylinder, +( ,, -)=0. The lateral surface and the end faces are maintained at zero temperature, .1( -, /)= .2( ,, /)= .3( ,, /)= 0. Solution:=80H I8 9 : =01 2 J+ 1sin2(2 J+ 1) H-K7exp2- (2 J+ 1)2 H2/K27 L I8 9 : =11D :;1(D : );0 3 D :,16exp 3- D2 :5/126 L, whereD : are positive zeros of the Bessel function,;0(D : )= 0.=?> Reference : H. S. Carslaw and J. C. Jaeger (1984). 2.1.3-7. Domain: 0 £ £ ,0 £ £ E. Second boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:=( , ) at = 0 (initial condition),& '=1( , ) at = (boundary condition),& )=2( , ) at = 0 (boundary condition),& )=3( , ) at = E(boundary condition). Solution:( , , )= 2 F 0 0 (, ) ( , ,, , ) + 2 0 F 0 1( , ) ( , , , , - ) - 2 0 0 2(, ) ( , ,,0, - ) + 2 0 0 3(, ) ( , ,, E, - ) . Here,( , ,, , )= 1( ,, ) 2( , , ),1( ,, )=1 2+1 2 =112 0( ) 0 0 exp- 2 2,2( , , )=1E+2E =1cos G E cos G E exp- G22 E2, where the are positive zeros of the ®rst-order Bessel function, 1( )= 0. Page 184 2 2.1.3-8. Domain: 0 £ £ ,0 £ £ E. Third boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:=( , ) at = 0 (initial condition),& '+ (1 =1( , ) at = (boundary condition),& )- (2 =2( , ) at = 0 (boundary condition),& )+ (3 =3( , ) at = E(boundary condition). The solution ( , , ) is determined by the formula in Paragraph 2.1.3-7 where( , ,, , )= 1( ,, ) 2( , , ),1( ,, )=1 2 =1 2 ( (2 1 2+ 2 ) 2 0( ) 0 0 exp- 2 2,2( , , )= M=1 N M( O)N M( )PN M P2exp(- Q2M R),N M( O)=cos( Q MO)+ (2Q Msin( Q MO),PN M P2= (3 2 Q2M Q2M+ (2 2Q2M+ (2 3+ (2 2 Q2M+ E 2 1 + (2 2Q2M, and the and Q Mare positive roots of the transcendental equations 1( )- (1 0( )= 0,tan( Q E)Q= (2+ (3Q2- (2 (3. 2.1.3-9. Domain: 0 £ S£ ,0 £ O£ E. Mixed boundary value problems. 1 @. A circular cylinder of ®nite length is considered. The following conditions are prescribed:T=( S, O) at R= 0 (initial condition),T=1( O, R) at S= (boundary condition),& )T=2( S, R) at O= 0 (boundary condition),& )T=3( S, R) at O= E(boundary condition). Solution:T( S, O, R)= 2 U V F 0 V W 0 X Y(X, ) ( S, O,X, , R) ZX Z - 2 U [ \ V ] 0 V F 0 ^1( , _) ` &&X ( S, O,X, , R- _) aA=W Z Z _ - 2 U [ V ] 0 V W 0 X^2(X, _) ( S, O,X,0, R- _) ZX Z _ + 2 U [ V] 0 VW 0 X^3(X, _) ( S, O,X, E, R- _) ZX Z _. Here,( S, O,X, , R)= 1( S,X, R) 2( O, , R),1( S,X, R)=1U \2 b cd =11e2 1( f d ) e 0 g f d h\ i e 0 g f dX\ iexpg- [ f2 d j\2i,2( k, l, j )=1 m+2 mb cd =1cosg n o k micosg n o l miexpg- [n2o2 jm 2i, where the f d are positive zeros of the Bessel function, e 0( f d )= 0. Page 185 2 p. A circular cylinder of ®nite length is considered. The following conditions are prescribed:q=Y( h , k) at j = 0 (initial condition),r 'q=^1( k, j ) at h = \(boundary condition),q=^2( h , j ) at k= 0 (boundary condition),q=^3( h , j ) at k= m (boundary condition). Solution:q( h , k, j )= 2o s t0 s W 0 u v(u, l) ( h , k,u, l, j ) wu w l + 2o x y s z0 s t0 {1( l, |) ( h , k,y, l, j - |) w l w | + 2o x sz0 s W 0 u {2(u, |) } rrl ( h , k,u, l, j - |) ~ =0 wu w | - 2o x sz0 s W 0 u {3(u, |) } rrl ( h , k,u, l, j - |) ~ =t wu w |. Here,( h , k,u, l, j )= 1( h ,u, j ) 2( k, l, j ),1( h ,u, j )=1o y2+1o y2 cd =11e2 0( f d ) e 0 g f d hy i e 0 g f duy iexpg-x f2 d jy2i,2( k, l, j )=2 m cd =1singn o k misingn o l miexpg-x n2o2 jm 2i, where the f d are positive zeros of the ®rst-order Bessel function, e 1( f d )= 0. 2.1.3-10. Domain:y1£ h £y2,0 £ k£ m . First boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:q=v( h , k) at j = 0 (initial condition),q={1( k, j ) at h =y1(boundary condition),q={2( k, j ) at h =y2(boundary condition),q={3( h , j ) at k= 0 (boundary condition),q={4( h , j ) at k= m (boundary condition). Solution:q( h , k, j )= 2o s t0 s W2W1 u v(u, l) ( h , k,u, l, j ) wu w l + 2o x y1sz0 s t0 {1( l, |) } rru ( h , k,u, l, j - |) ~ =W1 w l w | - 2o x y2sz0 s t0 {2( l, |) } rru ( h , k,u, l, j - |) ~ =W2 w l w | + 2o x sz0 s W2W1 u {3(u, |) } rrl ( h , k,u, l, j - |) ~ =0 wu w | - 2o x s z0 s W2W1 u {4(u, |) } rrl ( h , k,u, l, j - |) ~ =t wu w |. Page 186 2 Here,( h , k,u, l, j )= 1( k, l, j ) 2( h ,u, j ),1( k, l, j )=2 m cd =1sing n o k mising n o l miexpg-x n2o2 jm 2i,2( h ,u, j )=o4y2 1 cd =1 f2 de2 0( f d )e2 0( f d )- e2 0( f d ) d ( h ) d (u) expg-x f2 d jy2 1 i, d ( h )= 0( f d ) e 0 g f d hy1 i- e 0( f d ) 0 g f d hy1 i, =y2y1, where e 0( f) and 0( f) are the Bessel functions, the f d are positive roots of the transcendental equatione 0( f) 0( f)- e 0( f) 0( f)= 0. 2.1.3-11. Domain:y1£ h £y2,0 £ k£ m . Second boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:q=v( h , k) at j = 0 (initial condition),r q={1( k, j ) at h =y1(boundary condition),r q={2( k, j ) at h =y2(boundary condition),r q={3( h , j ) at k= 0 (boundary condition),r q={4( h , j ) at k= m (boundary condition). Solution:q( h , k, j )= 2o s t0 s W2W1 u v(u, l) ( h , k,u, l, j ) wu w l - 2o x y1s z0 s t0 {1( l, |) ( h , k,y1, l, j - |) w l w | + 2o x y2s z0 s t0 {2( l, |) ( h , k,y2, l, j - |) w l w | - 2o x s z0 s W2W1 u {3(u, |) ( h , k,u,0, j - |) wu w | + 2o x sz0 s W2W1 u {4(u, |) ( h , k,u, m , j - |) wu w |. Here,( h , k,u, l, j )= 1( k, l, j ) 2( h ,u, j ),1( k, l, j )=1 m+2 m cd =1cosgn o k micosgn o l miexpg-x n2o2 jm 2i,2( h ,u, j )=1o(y2 2-y2 1)+o4y2 1 cd =1 f2 de2 1( f d )e2 1( f d )- e2 1( f d ) d ( h ) d (u) expg-x f2 d jy2 1 i, d ( h )= 1( f d ) e 0 g f d hy1 i- e 1( f d ) 0 g f d hy1 i, =y2y1, where e ( f) and ( f) are the Bessel functions of order = 0,1and the f d are positive roots of the transcendental equatione 1( f) 1( f)- e 1( f) 1( f)= 0. Page 187 2.1.3-12. Domain:y1£ h £y2,0 £ k£ m . Third boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:q=v( h , k) at j = 0 (initial condition),r q- 1 q={1( k, j ) at h =y1(boundary condition),r q+ 2 q={2( k, j ) at h =y2(boundary condition),r q- 3 q={3( h , j ) at k= 0 (boundary condition),r q+ 4 q={4( h , j ) at k= m (boundary condition). For the solution of this problem, see Subsection 2.2.3-4 with º 0. 2.2. Heat Equation with a Source = 2+ ( , ,) 2.2.1. Problems in Cartesian Coordinates In the rectangular Cartesian coordinate system, the heat equation has the form =x 2 2+ 2 2 + ( , , ). It governs two-dimensional unsteady thermal processes in quiescent media or solids with constantthermal diffusivity in the cases where there are volume thermal sources or sinks. 2.2.1-1. Domain: - ¡< < ¡,- ¡< < ¡. Cauchy problem. An initial condition is prescribed:=v( , ) at = 0. Solution:( , , )=s - s - v(u, ¢) £( , ,u, ¢, ) wu w ¢ +sz0 s - s - (u, ¢, |) £( , ,u, ¢, - |) wu w ¢ w |, where£( , ,u, ¢, )=1 4 ¤x exp }-( -u)2+( - ¢)2 4x ~.¥?¦ Reference : A. G. Butkovskiy (1979). 2.2.1-2. Domain: 0 £ < ¡,- ¡< < ¡. First boundary value problem. A half-plane is considered. The following conditions are prescribed:=v( , ) at = 0 (initial condition),={( , ) at = 0 (boundary condition). Solution:( , , )=s 0 s - v(u, ¢) £( , ,u, ¢, ) w ¢ wu +x sz0 s - {( ¢, |) § u £( , ,u, ¢, - |) ¨ =0 w ¢ w | +s z0 s 0 s - (u, ¢, |) £( , ,u, ¢, - |) w ¢ wu w |, Page 188 2 where£( , ,u, ¢, )=1 4 ¤x exp }-( -u)2+( - ¢)2 4x ~-exp }-( +u)2+( - ¢)2 4x ~ ®.¥?¦ References : A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984). 2.2.1-3. Domain: 0 £ < ¡,- ¡< < ¡. Second boundary value problem. A half-plane is considered. The following conditions are prescribed:= ¯( , ) at = 0 (initial condition), ° = ±( , ) at = 0 (boundary condition). Solution:( , , )= ² ³ 0 ² ³ -³ ¯( ´, ¢) £( , , ´, ¢, ) µ ¢ µ ´- ¶ ² · 0 ² ³ -³ ±( ¢, ¸) £( , ,0, ¢, - ¸) µ ¢ µ ¸ + ² · 0 ² ³ 0 ² ³ -³ ( ´, ¢, ¸) £( , , ´, ¢, - ¸) µ ¢ µ ´ µ ¸, where£( , , ´, ¢, )=1 4 ¤ ¶ exp ¹-( - ´)2+( - ¢)2 4 ¶ º+exp ¹-( + ´)2+( - ¢)2 4 ¶ º®. 2.2.1-4. Domain: 0 £ < ¡,- ¡< < ¡. Third boundary value problem. A half-plane is considered. The following conditions are prescribed:= ¯( , ) at = 0 (initial condition), ° - » = ±( , ) at = 0 (boundary condition). The solution ( , , ) is determined by the formula in Paragraph 2.2.1-3 where£( , , ´, ¢, )=1 4 ¤ ¶ exp ¹-( - ¢)2 4 ¶ º exp ¹-( - ´)2 4 ¶ º+exp ¹-( + ´)2 4 ¶ º - 2 » ² ³ 0exp ¹-( + ´+ ¼)2 4 ¶ - » ¼ ºµ ¼ ®. 2.2.1-5. Domain: 0 £ < ¡,0 £ < ¡. First boundary value problem. A quadrant of the plane is considered. The following conditions are prescribed:= ¯( , ) at = 0 (initial condition),= ±1( , ) at = 0 (boundary condition),= ±2( , ) at = 0 (boundary condition). Solution:( , , )= ² ³ 0 ² ³ 0 ¯( ´, ¢) £( , , ´, ¢, ) µ ´ µ ¢ + ¶ ² · 0 ² ³ 0 ±1( ¢, ¸) § ´ £( , , ´, ¢, - ¸) ¨ ½ =0 µ ¢ µ ¸ + ¶ ² · 0 ² ³ 0 ±2( ´, ¸) § ¢ £( , , ´, ¢, - ¸) ¨ ¾ =0 µ ´ µ ¸ + ² · 0 ² ³ 0 ² ³ 0 ( ´, ¢, ¸) £( , , ´, ¢, - ¸) µ ´ µ ¢ µ ¸, Page 189 where£( , , ´, ¢, )=1 4 ¤ ¶ exp ¹-( - ´)2 4 ¶ º-exp ¹-( + ´)2 4 ¶ º® exp ¹-( - ¢)2 4 ¶ º-exp ¹-( + ¢)2 4 ¶ º®.¥?¦ Reference : H. S. Carslaw and J. C. Jaeger (1984). 2.2.1-6. Domain: 0 £ < ¡,0 £ < ¡. Second boundary value problem. A quadrant of the plane is considered. The following conditions are prescribed:= ¯( , ) at = 0 (initial condition), ° = ±1( , ) at = 0 (boundary condition), ¿ = ±2( , ) at = 0 (boundary condition). Solution:( , , )= ² ³ 0 ² ³ 0 ¯( ´, ¢) £( , , ´, ¢, ) µ ´ µ ¢ - ¶ ² · 0 ² ³ 0 ±1( ¢, ¸) £( , ,0, ¢, - ¸) µ ¢ µ ¸ - ¶ ² · 0 ² ³ 0 ±2( ´, ¸) £( , , ´,0, - ¸) µ ´ µ ¸ + ² · 0 ² ³ 0 ² ³ 0 ( ´, ¢, ¸) £( , , ´, ¢, - ¸) µ ´ µ ¢ µ ¸, where£( , , ´, ¢, )=1 4 ¤ ¶ exp ¹-( - ´)2 4 ¶ º+exp ¹-( + ´)2 4 ¶ º® exp ¹-( - ¢)2 4 ¶ º+exp ¹-( + ¢)2 4 ¶ º®. 2.2.1-7. Domain: 0 £ < ¡,0 £ < ¡. Third boundary value problem. A quadrant of the plane is considered. The following conditions are prescribed:= ¯( , ) at = 0 (initial condition), ° - »1 = ±1( , ) at = 0 (boundary condition), ¿ - »2 = ±2( , ) at = 0 (boundary condition). The solution ( , , ) is determined by the formula in Paragraph 2.2.1-6 where£( , , ´, ¢, )=1 4 ¤ ¶ exp ¹-( - ´)2 4 ¶ º+exp ¹-( + ´)2 4 ¶ º - 2 »1 À ¤ ¶ exp Á?¶ »2 1 + »1( + ´) Âerfc Ã Ä+ ´ 2À ¶ + »1 À ¶ Å® ´ Æexp ¹-( Ç- È)2 4 ¶ º+exp ¹-( Ç+ È)2 4 ¶ º - 2 »2 À É ¶ exp Á?¶ »2 2 + »2( Ç+ È) Âerfc Ã Ç+ È 2À ¶ + »2 À ¶ Å Ê. Page 190 ËÍ2 Ð 2.2.1-8. Domain: 0 £Ä< Ô,0 £ Ç< Ô. Mixed boundary value problem. A quadrant of the plane is considered. The following conditions are prescribed:Õ= Ö(Ä, Ç) at = 0 (initial condition),Õ= ±1( Ç, ) atÄ= 0 (boundary condition),× ¿Õ= ±2(Ä, ) at Ç= 0 (boundary condition). Solution:Õ(Ä, Ç, )= ² ³ 0 ² ³ 0 Ö( ´, È) Ø(Ä, Ç, ´, È, ) µ ´ µ È + ¶ ² · 0 ² ³ 0 ±1( È, ¸) § ××´ Ø(Ä, Ç, ´, È, - ¸) ¨ ½ =0 µ È µ ¸ - ¶ ² · 0 ² ³ 0 ±2( ´, ¸) Ø(Ä, Ç, ´,0, - ¸) µ ´ µ ¸ + ² · 0 ² ³ 0 ² ³ 0 Ù( ´, È, ¸) Ø(Ä, Ç, ´, È, - ¸) µ ´ µ È µ ¸, whereØ(Ä, Ç, ´, È, )=1 4É ¶ Æexp ¹-(Ä- ´)2 4 ¶ º-exp ¹-(Ä+ ´)2 4 ¶ º ÊÆexp ¹-( Ç- È)2 4 ¶ º+exp ¹-( Ç+ È)2 4 ¶ º Ê. 2.2.1-9. Domain: 0 £Ä£ Ú,0 £ Ç< Ô. First boundary value problem. A semiin®nite strip is considered. The following conditions are prescribed:Õ= Ö(Ä, Ç) at = 0 (initial condition),Õ= ±1( Ç, ) atÄ= 0 (boundary condition),Õ= ±2( Ç, ) atÄ= Ú(boundary condition),Õ= ±3(Ä, ) at Ç= 0 (boundary condition). The solution is given by the formula of Subsection 2.1.1-12 with the additional term² · 0 ² Û 0 ² ³ 0 Ù( ´, È, ¸) Ø(Ä, Ç, ´, È, - ¸) µ È µ ´ µ ¸, which takes into account the equation's nonhomogeneity. 2.2.1-10. Domain: 0 £Ä£ Ú,0 £ Ç< Ô. Second boundary value problem. A semiin®nite strip is considered. The following conditions are prescribed:Õ= Ö(Ä, Ç) at = 0 (initial condition),× °Õ= ±1( Ç, ) atÄ= 0 (boundary condition),× °Õ= ±2( Ç, ) atÄ= Ú(boundary condition),× ¿Õ= ±3(Ä, ) at Ç= 0 (boundary condition). Solution:Õ(Ä, Ç, )= ² ³ 0 ² Û 0 Ö( ´, È) Ø(Ä, Ç, ´, È, ) µ ´ µ È+ ² · 0 ² ³ 0 ² Û 0 Ù( ´, È, ¸) Ø(Ä, Ç, ´, È, - ¸) µ ´ µ È µ ¸ - ¶ ² · 0 ² ³ 0 ±1( È, ¸) Ø(Ä, Ç,0, È, - ¸) µ È µ ¸+ ¶ ² · 0 ² ³ 0 ±2( È, ¸) Ø(Ä, Ç, Ú, È, - ¸) µ È µ ¸ - ¶ ² · 0 ² Û 0 ±3( ´, ¸) Ø(Ä, Ç, ´,0, - ¸) µ ´ µ ¸, Page 191 whereØ(Ä, Ç, ´, È, )= Ø1(Ä, ´, ) Ø2( Ç, È, ),Ø1(Ä, ´, )=1Ú+2Ú ³ ÜÝ =1cos Þ ß ÉÄÚ àcos Þ ß É ´Ú àexp Ã- ¶ß2É2 Ú2 Å,Ø2( Ç, È, )=1 2À É ¶ Æexp ¹-( Ç- È)2 4 ¶ º+exp ¹-( Ç+ È)2 4 ¶ º Ê. 2.2.1-11. Domain: 0 £Ä£ Ú,0 £ Ç< Ô. Third boundary value problem. A semiin®nite strip is considered. The following conditions are prescribed:Õ= Ö(Ä, Ç) at = 0 (initial condition),× °Õ- »1 Õ= ±1( Ç, ) atÄ= 0 (boundary condition),× °Õ+ »2 Õ= ±2( Ç, ) atÄ= Ú(boundary condition),× ¿Õ- »3 Õ= ±3(Ä, ) at Ç= 0 (boundary condition). The solution Õ(Ä, Ç, ) is determined by the formula in Paragraph 2.2.1-10 where the Green's function Ø(Ä, Ç, ´, È, ) is the product of the Green's function of Subsection 1.1.1-11 and that of Subsection 1.1.1-8;Ä, ´, and »in the last Green's function must be replaced by Ç, È, and »3, respectively. 2.2.1-12. Domain: 0 £Ä£ Ú,0 £ Ç< Ô. Mixed boundary value problem. A semiin®nite strip is considered. The following conditions are prescribed:Õ= Ö(Ä, Ç) at = 0 (initial condition),Õ= ±1( Ç, ) atÄ= 0 (boundary condition),Õ= ±2( Ç, ) atÄ= Ú(boundary condition),× ¿Õ= ±3(Ä, ) at Ç= 0 (boundary condition). The solution is given by the formula of Subsection 2.1.1-15 (Item 1 á) with the additional termâ ã 0 âÛ 0 â³ 0 Ù( ä, È, å) Ø(Ä, Ç, ä, È, æ- å) ç È ç ä ç å, which takes into account the equation's nonhomogeneity. 2.2.1-13. Domain: 0 £Ä£ Ú1,0 £ Ç£ Ú2. First boundary value problem. A rectangle is considered. The following conditions are prescribed:Õ= Ö(Ä, Ç) at æ= 0 (initial condition),Õ= ±1( Ç, æ) atÄ= 0 (boundary condition),Õ= ±2( Ç, æ) atÄ= Ú1(boundary condition),Õ= ±3(Ä, æ) at Ç= 0 (boundary condition),Õ= ±4(Ä, æ) at Ç= Ú2(boundary condition). The solution is given by the formula of Subsection 2.1.1-16 with the additional termâ ã 0 âÛ1 0 âÛ2 0 Ù( ä, È, å) Ø(Ä, Ç, ä, È, æ- å) ç È ç ä ç å, which takes into account the equation's nonhomogeneity.è?é Reference : H. S. Carslaw and J. C. Jaeger (1984). Page 192 ËÍ2 Ð 2.2.1-14. Domain: 0 £Ä£ Ú1,0 £ Ç£ Ú2. Second boundary value problem. A rectangle is considered. The following conditions are prescribed:Õ= Ö(Ä, Ç) at æ= 0 (initial condition),× °Õ= ±1( Ç, æ) atÄ= 0 (boundary condition),× °Õ= ±2( Ç, æ) atÄ= Ú1(boundary condition),× êÕ= ±3(Ä, æ) at Ç= 0 (boundary condition),× êÕ= ±4(Ä, æ) at Ç= Ú2(boundary condition). Solution:Õ(Ä, Ç, æ)= âÛ1 0 âÛ2 0 Ö( ä, È) Ø(Ä, Ç, ä, È, æ) ç È ç ä+ âã 0 âÛ1 0 âÛ2 0 Ù( ä, È, å) Ø(Ä, Ç, ä, È, æ- å) ç È ç ä ç å - ë â ã 0 âÛ2 0 ±1( È, å) Ø(Ä, Ç,0, È, æ- å) ç È ç å+ ë â ã 0 âÛ2 0 ±2( È, å) Ø(Ä, Ç, Ú1, È, æ- å) ç È ç å - ë âã 0 âÛ1 0 ±3( ä, å) Ø(Ä, Ç, ä,0, æ- å) ç ä ç å+ ë âã 0 âÛ1 0 ±4( ä, å) Ø(Ä, Ç, ä, Ú2, æ- å) ç ä ç å, whereØ(Ä, Ç, ä, È, æ)=1Ú1 Ú2 ì1 + 2 ³ Ü Ý =1exp Ã- É2ß2ë æÚ2 1 Åcos ß ÉÄÚ1cos ß É äÚ1 í ´ì1 + 2 ³ Üî=1exp Ã- É2 ï2ë æÚ2 2 Åcos ïÉ ÇÚ2cos ïÉ ÈÚ2 í.è?é Reference : H. S. Carslaw and J. C. Jaeger (1984). 2.2.1-15. Domain: 0 £Ä£ Ú1,0 £ Ç£ Ú2. Third boundary value problem. A rectangle is considered. The following conditions are prescribed:Õ= Ö(Ä, Ç) at æ= 0 (initial condition),× °Õ- ð1 Õ= ±1( Ç, æ) atÄ= 0 (boundary condition),× °Õ+ ð2 Õ= ±2( Ç, æ) atÄ= Ú1(boundary condition),× êÕ- ð3 Õ= ±3(Ä, æ) at Ç= 0 (boundary condition),× êÕ+ ð4 Õ= ±4(Ä, æ) at Ç= Ú2(boundary condition). The solution Õ(Ä, Ç, æ) is determined by the formula in Paragraph 2.2.1-14 whereØ(Ä, Ç, ä, È, æ)= Æ ³ Ü Ý =1 ñ Ý (Ä)ñ Ý ( ä)òñ Ýò2exp(- ë ó2 Ýæ) ÊÆ ³ Üî=1 ô î( Ç)ô î( È)òô î ò2exp(- ë õ2îæ) Ê,ñ Ý (Ä)=cos( ó ÝÄ)+ ð1ó Ý sin( ó ÝÄ), òñ Ýò2= ð2 2 ó2 Ýó2 Ý + ð2 1ó2 Ý + ð2 2+ ð1 2 ó2 Ý + Ú1 2 ö1 + ð2 1ó2 Ý ÷ ,ô î( Ç)=cos( õ îÇ)+ ð3õ îsin( õ îÇ), òô î ò2= ð4 2 õ2î õ2î+ ð2 3õ2î+ ð2 4+ ð3 2 õ2î+ Ú2 2 ö1 + ð2 3õ2î ÷ . Here, the ó Ý and õ îare positive roots of the transcendental equations tan( ó Ú1)ó= ð1+ ð2ó2- ð1 ð2,tan( õ Ú2)õ= ð3+ ð4õ2- ð3 ð4. Page 193 2.2.1-16. Domain: 0 £ ø£ Ú1,0 £ Ç£ Ú2. Mixed boundary value problem. A rectangle is considered. The following conditions are prescribed:Õ= Ö( ø, Ç) at æ= 0 (initial condition),Õ= ±1( Ç, æ) at ø= 0 (boundary condition),Õ= ±2( Ç, æ) at ø= Ú1(boundary condition),× êÕ= ±3( ø, æ) at Ç= 0 (boundary condition),× êÕ= ±4( ø, æ) at Ç= Ú2(boundary condition). The solution is given by the formula of Subsection 2.1.1-19 (Item 1 á) with the additional termâã 0 âÛ1 0 âÛ2 0 Ù( ä, È, å) Ø( ø, Ç, ä, È, æ- å) ç È ç ä ç å, which takes into account the equation's nonhomogeneity. 2.2.2. Problems in Polar Coordinates The heat equation with a volume source in the polar coordinate system ù,ñis written as×Õ×æ= ëö ×2 Õ×ù2+1ù ×Õ×ù+1ù2 ×2 Õ×ñ2 ÷ +Ù( ù,ñ, æ). Solutions of the form Õ= Õ( ù, æ) that are independent of the angular coordinateñand govern plane thermal processes with central symmetry, are presented in Subsection 1.2.2. 2.2.2-1. Domain: 0 £ ù< Ô,0 £ñ£ 2É. Cauchy problem. An initial condition is prescribed:Õ= Ö( ù,ñ) at æ= 0. Solution:Õ( ù,ñ, æ)= â2 ú 0 â³ 0 Ö( ä, È) Ø( ù,ñ, ä, È, æ) ä ç ä ç È + â ã 0 â2 ú 0 â³ 0 Ù( ä, È, å) Ø( ù,ñ, ä, È, æ- å) ä ç ä ç È ç å, whereØ( ù,ñ, ä, È, æ)=1 4É ë æexpì- ù2+ ä2- 2 ù äcos(ñ- È) 4 ë æí. 2.2.2-2. Domain: 0 £ ù£ û,0 £ñ£ 2É. Different boundary value problems. 1 á. The solution of the ®rst boundary value problem for a circle of radius ûis given by the formula from Subsection 2.1.2-2 with the additional termâã 0 â2 ú 0 â ü 0 Ù( ä, È, å) Ø( ù,ñ, ä, È, æ- å) ä ç ä ç È ç å, ( 1) which allows for the equation's nonhomogeneity.2á. The solution of the second boundary value problem for a circle is given by the formula in Paragraph 2.1.2-3 with the additional term (1).3á. The solution of the third boundary value problem for a circle is given by the formula in Paragraph 2.1.2-4 with the additional term (1). Page 194 ËÍ2 Ð 2.2.2-3. Domain: û1£ ù£ û2,0 £ñ£ 2É. Different boundary value problems. 1 á. The solution of the ®rst boundary value problem for an annular domain is given by the formula in Paragraph 2.1.2-5 with the additional termâã 0 â2 ú 0 â ü2ü1 Ù( ä, È, å) Ø( ù,ñ, ä, È, æ- å) ä ç ä ç È ç å, ( 2) which allows for the equation's nonhomogeneity.2á. The solution of the third boundary value problem for an annular domain is given by the formula in Paragraph 2.1.2-7 with the additional term (2). 2.2.2-4. Domain: 0 £ ù< Ô,0 £ñ£ñ0. Different boundary value problems. 1 á. The solution of the ®rst boundary value problem for a wedge domain is given by the formula in Paragraph 2.1.2-8 with the additional termâã 0 â þ0 0 â³ 0 Ù( ä, È, å) Ø( ù,ñ, ä, È, æ- å) ä ç ä ç È ç å, ( 3) which allows for the equation's nonhomogeneity.2á. The solution of the second boundary value problem for a wedge domain is given by the formula in Paragraph 2.1.2-9 with the additional term (3). 2.2.2-5. Domain: 0 £ ù£ û,0 £ñ£ñ0. Different boundary value problems. 1 á. The solution of the ®rst boundary value problem for a sector of a circle is given by the formula of Paragraph 2.1.2-10 with the additional termâ ã 0 â þ0 0 â ü 0 Ù( ä, È, å) Ø( ù,ñ, ä, È, æ- å) ä ç ä ç È ç å, ( 4) which allows for the equation's nonhomogeneity.2á. The solution of the mixed boundary value problem for a sector of a circle is given by the formula of Paragraph 2.1.2-11 with the additional term (4). 2.2.3. Axisymmetric Problems In the case of axial symmetry, the heat equation in the cylindrical coordinate system is written as×Õ×æ= ëö ×2 Õ×ù2+1ù ×Õ×ù+ ×2 Õ× ÿ2 ÷ +Ù( ù, ÿ, æ), provided there are heat sources or sinks. One-dimensional axisymmetric problems that have solutions of the form Õ= Õ( ù, æ) can be found in Subsection 1.2.2. 2.2.3-1. Domain: 0 £ ù£ û,0 £ ÿ< Ô. Different boundary value problems. 1 á. The solution to the ®rst boundary value problem for a semiin®nite circular cylinder of radius û is given by the formula of Subsection 2.1.3-2 with the term 2É â ã 0 â³ 0 â ü 0 äÙ( ä, È, å) Ø( ù, ÿ, ä, È, æ- å) ç ä ç È ç å (1) added; this term takes into account the nonhomogeneity of the equation. Page 195 2 á. The solution to the second boundary value problem for a semiin®nite circular cylinder is given by the formula of Subsection 2.1.3-3 with the additional term (1).3á. The solution to the third boundary value problem for a semiin®nite circular cylinder is given by the formula of Subsection 2.1.3-4 with the additional term (1).4á. The solutions to various mixed boundary value problems for a semiin®nite circular cylinder are de®ned by formulas of Subsection 2.1.3-5 with additional terms of the form (1). 2.2.3-2. Domain: 0 £ ù£ û,0 £ ÿ£ Ú. Different boundary value problems. 1 á. The solution to the ®rst boundary value problem for a circular cylinder of radius ûand length Ú is given by the formula of Subsection 2.1.3-6 with the term 2É â ã 0 âÛ 0 â ü 0 äÙ( ä, È, å) Ø( ù, ÿ, ä, È, æ- å) ç ä ç È ç å, ( 2) added; this term takes into account the nonhomogeneity of the equation.2á. The solution to the second boundary value problem for a ®nite circular cylinder is given by the formula of Subsection 2.1.3-7 with the additional term (2).3á. The solution to the third boundary value problem for a ®nite circular cylinder is given by the formula of Subsection 2.1.3-8 with the additional term (2).4á. The solutions to various mixed boundary value problems for a ®nite circular cylinder are de®ned by formulas of Subsection 2.1.3-9 with additional terms of the form (2). 2.2.3-3. Domain: û1£ ù£ û2,0 £ ÿ£ Ú. First and second boundary value problems. 1 á. The solution to the ®rst boundary value problem for a hollow circular cylinder of interior radius û1, exterior radius û2, and length Úis given by the formula of Subsection 2.1.3-10 with the term 2É â ã 0 âÛ 0 â ü2ü1 äÙ( ä, È, å) Ø( ù, ÿ, ä, È, æ- å) ç ä ç È ç å (3) added; this term takes into account the equation's nonhomogeneity.2á. The solution to the second boundary value problem for a ®nite hollow circular cylinder is given by the formula of Subsection 2.1.3-11 with the additional term (3). 2.2.3-4. Domain: û1£ ù£ û2,0 £ ÿ£ Ú. Third boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:Õ= Ö( ù, ÿ) at æ= 0 (initial condition),× Õ- ð1 Õ= 1( ÿ, æ) at ù= û1(boundary condition),× Õ+ ð2 Õ= 2( ÿ, æ) at ù= û2(boundary condition),× Õ- ð3 Õ= 3( ù, æ) at ÿ= 0 (boundary condition),× Õ+ ð4 Õ= 4( ù, æ) at ÿ= Ú (boundary condition). Page 196 ËÍ2 Ð Solution:Õ( ù, ÿ, æ)= 2 âÛ 0 â ü2ü1 ä Ö( ä, ) Ø( ù, ÿ, ä, , æ) ç ä ç - 2 ë û1 â ã 0 âÛ 0 1( , å) Ø( ù, ÿ, û1, , æ- å) ç ç å + 2 ë û2 âã 0 âÛ 0 2( , å) Ø( ù, ÿ, û2, , æ- å) ç ç å - 2 ë âã 0 â ü2ü1 ä 3( ä, å) Ø( ù, ÿ, ä,0, æ- å) ç ä ç å + 2 ë â ã 0 â ü2ü1 ä 4( ä, å) Ø( ù, ÿ, ä, Ú, æ- å) ç ä ç å + 2 â ã 0 âÛ 0 â ü2ü1 äÙ( ä, , å) Ø( ù, ä, ÿ, , æ- å) ç ä ç ç å. Here, the Green's function is given byØ( ù, ÿ, ä, , æ)= Ø1( ù, ä, æ) Ø2( ÿ, , æ),Ø1( ù, ä, æ)= 4 ³ Ü Ý =1 õ2 Ý Ý ð2 0( õ Ýû2)- õ Ý1( õ Ýû2) 2 Ý ( ù) Ý ( ä) exp( - õ2 Ýë æ), 2( ÿ, , æ)= ³ î=1 ñ î( ÿ)ñ î( )òñ î ò2exp - ó2îë æ , where =( õ2 + ð2 2) ð1 0( õ û1)+ õ 1( õ û1)2-( õ2 + ð2 1) ð2 0( õ û2)- õ 1( õ û2)2, ( ù)= ð1 0( õ û1)+ õ 1( õ û1)0( õ ù)- ð1 0( õ û1)+ õ 1( õ û1)0( õ ù),ñ î( ÿ)= ó îcos( ó î ÿ)+ ð3sin( ó î ÿ), òñ î ò2= ð4 2 ó2î+ ð2 3ó2î+ ð2 4+ ð3 2+ 2 ó2î+ ð2 3 ,0( õ),1( õ),0( õ), and1( õ) are the Bessel functions, the õ are positive roots of the transcendental equationð1 0( õ û1)+ õ1( õ û1) ð2 0( õ û2)- õ1( õ û2) - ð2 0( õ û2)- õ1( õ û2) ð1 0( õ û1)+ õ1( õ û1)= 0, and the ó îare positive roots of the transcendental equation tan óó= ð3+ ð4ó2- ð3 ð4.è?é Reference : A. G. Butkovskiy (1979). 2.2.3-5. Domain: û1£ ù£ û2,0 £ ÿ£. Mixed boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= ( ù, ÿ) at æ= 0 (initial condition), - ð1 = 1( ÿ, æ) at ù= û1(boundary condition), + ð2 = 2( ÿ, æ) at ù= û2(boundary condition),= 3( ù, æ) at ÿ= 0 (boundary condition),= 4( ù, æ) at ÿ=(boundary condition). Page 197 Solution:( ù, ÿ, æ)= 2 0 ü2ü1 ä ( ä, ) ( ù, , ä, , ) ä - 2 û1 ã 0 0 1( , å) ( ù, , û1, , - å) å + 2 û2 ã 0 0 2( , å) ( ù, , û2, , - å) å + 2 ã 0 ü2ü1 ä3( ä, å) ( ù, , ä, , - å) ! " =0 # $ - 2 % 0 ü2ü1 #4( #, $) ( ù, , #, , - $) !" = # $ + 2 % 0 0 ü2ü1 # &( #, , $) ( ù, #, , , - $) # $. Here,( ù, #, , , )= 1( ù, #, )2 ' ( )*=1sin + , ' -sin + , ' -exp .- 2,2 ' 2 /, where the expression of 1( ù, #, ) is speci®ed in Subsection 2.2.3-4.0Subsection 3.2.2 presents solutions of other boundary value problems; a more general, three- dimensional equation is discussed there. 2.3. Other Equations 2.3.1. Equations Containing Arbitrary Parameters 1. 1 21 3= 4 . 1221 52+ 1221 62 /+ ( 75+ 86+ 9)2. The transformation : ( ;, <, )= =( #, , ) exp ( >?;+ @A<+ B) +1 3 ( >2+ @2) 3, #= ;+ >?2, = <+ @A2 leads to the two-dimensional heat equation % == C DD =+ "E"= F. See also Niederer (1973) and Boyer (1974). 2. 1 21 3= 4 . 1221 52+ 1221 62 /±C 752+ 762+ 9F2, 7> 0. The transformation ( Gis an arbitrary constant): ( ;, <, )= =( #, , $) exp H1 2 I >( ;2+ <2)+C2 J >- BF LK,#= ;exp C2 J > F, = <exp C2 J > F, $=1 4J >exp C4 J > F+ G leads to the two-dimensional heat equation M == C DD =+ "E"=F. See also Niederer (1973) and Boyer (1974). Page 198 3. 1 21 3= 4 . 1221 52+ 1221 62 /+ CL752+ 762± 9 F2, 7> 0. The transformation: ( ;, <, )=1 cosC2J > Fexp H J > 2J tanC2 J > F C ;2+ <2F- B LK =( #, , $),#= ; cos C2J > F, = < cos C2J > F, $= J 2J >tanC2 J > F leads to the two-dimensional heat equation M == DD =+ "E"=. See also Niederer (1973) and Boyer (1974). 4. 1 21 3= 1221 52+ 1221 62+C 45±2+ 76±2F2. This is a special case of equation 2.3.2.7. Boyer (1976) showed that this equation admits the separation of variables into 25 systems of coordinates for >= 0and 15 systems of coordinates for >¹ 0. 5. 1 21 3= 4 . 1221 52+ 1221 62 /+ ( 73ON 5+ 83?P 6+s3RQ)2. This is a special case of equation 2.3.2.2 with S( T)= >?TVU, W( T)= @AT *, and X( T)= YAT[Z. 6. 1 21 3= 4 . 1221 52+ 1221 62 /+ \± 7(52+62) + 813?N15+ 823?N26+s3RQ 2. This is a special case of equation 2.3.2.3 with S( T)= @1 T]U1, W( T)= @2 T]U2, and X( T)= YAT[Z. 7. 1 21 3= 4 . 1221 52+ 1221 62 /+ 71 1 21 5+ 72 1 21 6+ 82. This equation describes an unsteady temperature (concentration) ®eld in a medium moving with aconstant velocity, provided there is volume release (absorption) of heat proportional to temperature. The substitution: ( ;, <, T)=expC_^1 ;+^2 <+ ` TF a( ;, <, T),^1= - >1 2 b,^2= - >2 2 b, `= @- >21+ >22 4 b, leads to the two-dimensional heat equation c% a= b d2 athat is considered in Subsection 2.1.1. 8. 1 21 3= 4 .1221 52+ 1221 62 /+ 71 1 21 5+ 72 1 21 6+ ( 815+ 826+ 9)2. The transformation : ( ;, <, T)=exp \( @1 ;+ @2 <) T+1 3 b( @21+ @22) T3+1 2( >1 @1+ >2 @2) T2+ B T a( #, e, T),#= ;+ b @1 T2+ >1 T, e= <+ b @2 T2+ >2 T leads to the two-dimensional heat equation c% a= b( cDD a+ c "E"a) that is considered in Subsec- tion 2.1.1.9.1 21 3= 4 . 1221 52+ 1221 62 /+ 713?N11 21 5+ 723?N21 21 6+ ( 813?P15+ 823?P26+s3RQ)2. This is a special case of equation 2.3.2.5. The equation can be reduced to the two-dimensional heat equation treated in Subsection 2.1.1. Page 199 10. f g h1 21 3+ g h2 2 i . 1221 52+ 1221 62 /= 0. Two-dimensional Schr Èodinger equation, j2= -1 . Fundamental solution: k k ( ;, <, T)= - jL, 2 l g X2Texp H jL, 2 g X T( ;2+ <2)- j l 2 K.mon Reference : V . S. Vladimirov, V . P. Mikhailov, A. A. Vasharin, et al. (1974). 2.3.2. Equations Containing Arbitrary Functions 1. 1 21 3= 4 . 1221 52+ 1221 62 /+ p(3)2. This equation describes two-dimensional thermal phenomena in quiescent media or solids with constant thermal diffusivities in the case of unsteady volume heat release proportional to temperature. The substitution : ( ;, <, T)=exp \rq S( T) s Tra( ;, <, T) leads to the two-dimensional heat equationc ta= b( c u ua+ c vEva) treated in Subsection 2.1.1. 2. 1 21 3= 4 w x2 yx z2+ x2 yx {2 |+ [z p( }) +{ ~( }) + h( })] y. The transformation ( , , T)= ( , e, T) exp ( T)+ ( T)+ ( )+ b 2( ) + b 2( ) ,= + 2 b ( ) , e= + 2 b ( ) , where( )= ( ) , ( )= ( ) , ( )= ( ) , leads to the two-dimensional heat equation c = b _c E+ c E . 3. x yx }= w x2 yx z2+ x2 yx {2|+ [± (z2+{2) + ( })z+~( }){+ ( })] y. 1 . Case >0. The transformation ( , , )= ( , e, ) exp 1 2 b( 2+ 2) ,¡= exp 2 ¢ b £, e= exp 2 ¢ ¤ £, =1 4¢ ¤ exp 4 ¢ ¤ £ leads to an equation of the form 2.3.2.2:¥ ¦¥£= ¤ § ¥2 ¦¥¡2+ ¥2 ¦¥ ¨2 ©+ ª«( ) ¡+ ¬( ) ¨+ ( ) ® ¦,( )=1 ( ¯A)3 °2 §ln( ¯A)¯ ©, ¬( )=1 ( ¯A)3 °2 §ln( ¯A)¯ ©, ( )=1¯A §ln( ¯A)¯ ©+1 2 , ¯= 4 ¢¤ . 2. Case <0. The transformation ± ( , , £)= ²(Å ¡, Å ¨, Å ) exp ³ ¢- 2¢ ¤tan 2 ¢- ¤ £( 2+ 2) , Å ¡= cos 2¢- ¤ £, Å ¨= cos 2¢- ¤ £, Å =1 2¢- ¤ tan 2 ¢- ¤ £ also leads to an equation of the form 2.3.2.2 (the transformed equation is not written out here). Page 200 4. = 1( ) 2 2+ 2( ) 2 2+ ( , , ). This is a special case of equation 2.3.2.8. Let 0< 1( ) < and0< 2( ) < . For the ®rst, second, third, and mixed boundary value problems treated in rectangular, ®nite, or in®nite domains ( 1£ £ 2, 1£ £ 2), the Green's function can be represented in the product form( , , , , , )= 1( , , 1) 2( , , 2),1= 1( ) , 2= 2( ) . Here, 1= 1( , , ) is the auxiliary Green's function that corresponds to the one-dimensional heat equation for 1( )= 1, 2( )= 0, and ( , , )= 0with homogeneous boundary conditions at = 1and = 2(the 1's for various boundary value problems can be found in Subsections 1.1.1 and 1.1.2). Similarly, 2= 2( , , ) is the auxiliary Green's function that corresponds to the one-dimensional heat equation for 1( )= 0, 2( )= 1, and ( , , )= 0with homogeneous boundary conditions at = 1and = 2. Note that the Green's functions 1and 2are introduced for = 0. See Subsection 0.8.1 for solution of various boundary value problems with the help of the Green's function. Example 1. Domain: - < < ,- < < . Cauchy problem. An initial condition is prescribed: = ( , ) at = 0. Solution: ( , , )= 0 - - ( , !, ") #( , , , !, , ") $% $%! $%"+ - - ( , !) #( , , , !, ,0) $% $%!, where#( , , , !, , ")=1 4 & ' (1 (2exp )-( - )2 4 (1-( - !)2 4 (2 *, (1= + ,1( !) $%!, (2= + ,2( !) $%!. Example 2. Domain: 0 £ < ,0 £ < . Second boundary value problem. The following conditions are prescribed: = ( , ) at = 0 (initial condition),-/. = 01( , ) at = 0 (boundary condition),-/1 = 02( , ) at = 0 (boundary condition). Solution: ( , , )= 0 0 0( , !, ") #( , , , !, , ") $% $%! $%"+ 0 0 ( , !) #( , , , !, ,0) $% $%! - 0 0 ,1( ") 01( !, ") #( , ,0, !, , ") $%! $%"- 0 0 ,2( ") 02( , ") #( , , ,0, , ") $% $%", where#( , , , !, , ")= #1( , , (1) #2( , !, (2),#1( , , (1)=1 2 ' & (1 2exp )-( - )2 4 (1 *+exp )-( + )2 4 (1 * 3, (1= +,1( !) $%!,#2( , !, (2)=1 2 '& (2 2exp )-( - !)2 4 (2 *+exp )-( + !)2 4 (2 * 3, (2= +,2( !) $%!. Example 3. Domain: 0 £ £ 41,0 £ £ 42. First boundary value problem. Page 201 The following conditions are prescribed: = ( , ) at = 0 (initial condition), = 01( , ) at = 0 (boundary condition), = 02( , ) at = 41(boundary condition), = 51( , ) at = 0 (boundary condition), = 52( , ) at = 42(boundary condition). Solution: ( , , )= 0 61 0 62 0( , !, ") #( , , , !, , ") $%! $% $%"+ 61 0 62 0 ( , !) #( , , , !, ,0) $%! $% + 0 62 0 ,1( ") 01( !, ") ) -- #( , , , !, , ")* 7=0 $%! $%"- 0 62 0 ,1( ") 02( !, ") ) -- #( , , , !, , ")* 7=61 $%! $%" + 0 61 0 ,2( ") 51( , ") ) --! #( , , , !, , ")* 8=0 $% $%"- 0 61 0 ,2( ") 52( , ") ) --! #( , , , !, , ")* 8=62 $% $%", where#( , , , !, , ")= #1( , , (1) #2( , !, (2),#1( , , (1)=241 9 : =1sin ; < & 41 =sin ; < & 41 =exp ;- <2&2(1412=, (1= +,1( !) $%!,#2( , !, (2)=242 9 : =1sin ; < & 42 =sin ; < & !42 =exp ;- <2&2(2422=, (2= +,2( !) $%!. 5. = 1( ) 2 2+ 2( ) 2 2+ [ >1( ) + ?1( )] + [ >2( ) + ?2( )] + [s1( ) +s2( ) + @( )] . The transformationA( , , )=exp BDC1( ) + C2( ) + E( ) F%G( , , ), = H1( ) + I1( ), = H2( ) + I2( ), whereH J( )= K Jexp L MJ( ) ON,C J( )= H J( ) PJ( )H J( ) + Q J H J( ),I J( )= B2 J( ) C J( )+ RSJ( )F H J( ) + T J,E( )= B 1( ) C2 1( )+ 2( ) C2 2( )+ R1( ) C1( )+ R2( ) C2( )+ U( )F + V, ( W= 1,2; K J, Q J, T J, and Vare arbitrary constants), leads to an equation of the form 2.3.2.4:XGX= 1( ) H21( ) X2GX 2+ 2( ) H22( ) X2GX2. 6. = 1( ) 2 2+ 2( ) 2 2+ [ >1( ) + ?1( )] + [ >2( ) + ?2( )] + [s1( ) 2+s2( ) 2+ @1( ) + @2( ) + Y( )] . The substitutionA( , , )=expB C1( ) 2+ C2( ) 2F G( , , ), where the functions C1= C1( ) and C2= C2( ) are solutions of the Riccati equationsC Z1= 4 1( ) C2 1+ 2 M 1( ) C1+ P 1( ),C Z2= 4 2( ) C2 2+ 2 M 2( ) C2+ P 2( ), leads to an equation of the form 2.3.2.5 for G= G( , , ). Page 202 7. = 1( ) 2 2+ 2( ) 2 2+ >1( ) + >2( ) + [ ?1( ) + ?2( )] + ( , , ). Domain: 1£ £ 2, 1£ £ 2. Different boundary value problems:A= C( , ) at = 0 (initial condition),P 1 X [A- W1 A= E1( , ) at = 1(boundary condition),P 2 X [A+ W2 A= E2( , ) at = 2(boundary condition),P 3 X \A- W3 A= E3( , ) at = 1(boundary condition),P 4 X \A+ W4 A= E4( , ) at = 2(boundary condition). By choosing appropriate parameters P%], W ]( ^= 1,2,3,4), one obtains the ®rst, second, third, or mixed boundary value problem. If the domain is in®nite, say, 2= , the corresponding boundary condition should be omitted; this is also valid for 1= - , 1= - , or 2= . The Green's function admits incomplete separation of variables; speci®cally, it can be repre- sented in the product form( , , , , )= 1( , , ) 2( , , ). Here, 1= 1( , , ) and 2= 2( , , ) are auxiliary Green's functions that are determined by solving the following simpler one-dimensional problems with homogeneous boundary conditions:X 1X= 1( ) X2 1X 2+ M 1( ) X 1X + R1( ) 1, X 2X= 2( ) X2 2X2+ M 2( ) X 2X+ R2( ) 2, 1= _( - ) at = 0,P 1 X [ 1- W1 1= 0 at = 1,P 2 X [ 1+ W2 1= 0 at = 2, 2= _( - ) at = 0,P 3 X \ 2- W3 2= 0 at = 1,P 4 X \ 2+ W4 2= 0 at = 2, where and are free parameters, and _( ) is the Dirac delta function. The equation for 1coincides with equation 1.8.6.5, which is reduced to the equation of Subsection 1.8.9 (where the expression of the Green's function can also be found). In the general case, the equation for 2differs from the equation for 1in only notation. 8. ` a` b= c1( d,b) `2a` d2+ c2( e,b) `2a` e2+ >1( d,b) ` a` d + >2( e,b) ` a` e+ [ ?1( d,b) + ?2( e,b)]a+ f( d, e,b). Suppose this equation is subject to the same initial and boundary conditions as equation 2.3.2.7. Then the Green's function for this problem can be represented in the product formg( h, i, j, k, l, m)= g 1( h, j, l, m) g 2( i, k, l, m). Here, g 1= g 1( h, j, l, m) and g 2= g 2( i, k, l, m) are auxiliary Green's functions that are determined by solving the following simpler boundary value problems with homogeneous boundary conditions:-#1- n=,1( o, n) -2#1-o2+ p1( o, n) -#1-o+ q1( o, n) #1, -#2- n=,2( r, n) -2 s2-r2+ p2( r, n) -s2-r+ q2( r, n) s2,s1= t( o- u) at n= v,w 1 -/.s1- x1 s1= 0 at o= o1,w 2 -/.s1+ x2 s1= 0 at o= o2, s2= t( r- y) at n= v,w 3 z/{ s2- x3 s2= 0 at r= r1,w 4 z/{ s2+ x4 s2= 0 at r= r2, where j, k, and mare free parameters, and _( h) is the Dirac delta function, l³ m. See Subsection 0.8.1 for the solution of boundary value problems with the help of the Green's function. Page 203 Chapter 3 Parabolic Equations with Three orMore Space Variab les 3.1. Heat Equation | }| ~= 3} 3.1.1. Problems inCartesian Coor dinates Thethree-dimensional sourceless heat equation intherectangular Cartesian system ofcoordinates hastheform l= 2 h2+ 2 i2+ 2 2 . Itgovernsthree-dimensional thermal phenomena inquiescent media orsolids with constant thermal diffusivity.Asimilar equation isused tostudy thecorresponding three-dimensional unsteady mass-e xchange processes with constant diffusivity. 3.1.1-1. Particular solutions: ( , , , l)= 2+ 2+ 2+2 ( + + ) , ( , , , )= ( 2+2 )( 2+2 )( 2+2 )+ , ( , , , )= exp 1 + 2 + 3 +( 2 1+ 2 2+ 2 3) + , ( , , , )= cos( 1 + 1)cos( 2 + 2)cos( 3 + 3)exp -( 2 1+ 2 2+ 2 3) , ( , , , )= cos( 1 + 1)cos( 2 + 2)sinh( 3 + 3)exp -( 2 1+ 2 2- 2 3) , ( , , , )= cos( 1 + 1)cos( 2 + 2)cosh( 3 + 3)exp -( 2 1+ 2 2- 2 3) , ( , , , )= exp(- 1 - 2 - 3 )cos( 1 -2 2 1 )cos( 2 -2 2 2 )cos( 3 -2 2 3 ), ( , , , )= ( - 0)3 2exp -( - 0)2+( - 0)2+( - 0)2 4 ( - 0) , ( , , , )= erf - 0 2 erf - 0 2 erf - 0 2 + , where , , , 1, 2, 3, 1, 2, 3, 0, 0, 0,and 0arearbitrary constants. Fundamental solution: ( , , , )=1 8( )3 2exp - 2+ 2+ 2 4 . 3.1.1-2. Formulas toconstruct particular solutions. Remarks ontheGreen' sfunctions. 1 .Apart from usual solutions with separated variables, ( , , , )= 1( ) 2( ) 3( ) 4( ), Page205 the equation in question admits more sophisticated solutions in the product form ( , , , )= 1( , ) 2( , ) 3( , ), where the functions 1= 1( , ), 2= 2( , ), and 3= 3( , ) are solutions of the one-dimensional heat equations1= 212, 2= 222, 3= 23 2, treated in Subsection 1.1.1.2. Suppose = ( , , , ) is a solution of the three-dimensional heat equation. Then the functions 1= ( + 1, + 2, + 3, 2+ 4), 2= exp 1 + 2 + 3 +( 21+ 22+ 23) ( + 2 1 , + 2 2 , + 2 3 , ), 3= | + |3 2exp - ( 2+ 2+ 2) 4 ( + ) + , + , + , + + , - = 1, where , 1, 2, 3, 4, , 1, 2, 3, , and are arbitrary constants, are also solutions of this equation. The signs at in the formula for 1can be taken independently of one another. 3 . For the three-dimensional boundary value problems considered in Subsection 3.1.1, the Green's function can be represented in the product form ( , , , ¡, ¢, £, )= 1( , ¡, ) 2( , ¢, ) 3( , £, ), where 1( , ¡, ), 2( , ¢, ), 3( , £, ) are the Green's functions of the corresponding one- dimensional boundary value problems; these functions can be found in Subsections 1.1.1 and1.1.2. Example 1. The Green's function of the mixed boundary value problem for a semiin®nite layer ( - ¤< ¥< ¤, 0 £ ¦< ¤,0 £ §< ¨) presented in Paragraph 3.1.1-14 is the product of three one-dimensional Green's functions from Paragraph 1.1.2-1 (Cauchy problem for - ¤< ¥< ¤), Paragraph 1.1.2-2 (®rst boundary value problem for 0 £ ¦< ¤), and Paragraph 1.1.2-6 (second boundary value problem for 0 £ §< ¨), in which one needs to carry out obvious renaming of variables. 3.1.1-3. Domain: - ©< < ©,- ©< < ©,- ©< < ©. Cauchy problem. An initial condition is prescribed: = ( , , ) at = 0. Solution: ( , , , )=1 8( )3 2 ª «-« ª «-« ª «-« ( ¡, ¢, £) exp -( - ¡)2+( - ¢)2+( - £)2 4 ¬ ¡¬ ¢¬ £. Example 2. The initial temperature is constant and is equal to 1in the domain | ¥|< ¥0,| ¦|< ¦0,| §|< §0and is equal to 2in the domain | ¥|> ¥0,| ¦|> ¦0,| §|> §0; speci®cally,®( ¥, ¦, §)= ¯ 1for| ¥|< ¥0,| ¦|< ¦0,| §|< §0,2for| ¥|> ¥0,| ¦|> ¦0,| §|> §0. Solution:=1 8( 1- 2) °erf ± ¥0- ¥ 2 ² ³µ´ ¶+erf ± ¥0+ ¥ 2 ² ³µ´ ¶ · ´ °erf ± ¦0- ¦ 2 ² ³µ´ ¶+erf ± ¦0+ ¦ 2 ² ³µ´ ¶ · °erf ± §0- § 2 ² ³µ´ ¶+erf ± §0+ § 2 ² ³µ´ ¶ ·+ 2.¸D¹ Reference : H. S. Carslaw and J. C. Jaeger (1984). Page 206 º/¼ 3.1.1-4. Domain: 0 £ ¾< ©,- ©< ¿< ©,- ©< À< ©. First boundary value problem. A half-space is considered. The following conditions are prescribed:Á= ( ¾, ¿, À) at = 0 (initial condition),Á= Â( ¿, À, ) at ¾= 0 (boundary condition). Solution:Á( ¾, ¿, À, )=ª «-« ª «-« ª «0 ( ¡, ¢, £) ( ¾, ¿, À, ¡, ¢, £, )¬ ¡¬ ¢¬ £ + Ãª Ä0 ª «-« ª «-« Â( ¢, £, Å) ÆÆ ¡ ( ¾, ¿, À, ¡, ¢, £, Ç- Å) È=0 ¬ ¢¬ £¬ Å, where ( ¾, ¿, À, ¡, ¢, £, Ç)=1 8( Ã Ç)3 É2 Êexp Ë-( ¾- ¡)2 4 Ã Ç Ì-exp Ë-( ¾+ ¡)2 4 Ã Ç Ì Íexp Ë-( ¿- ¢)2+( À- £)2 4 Ã Ç Ì.¸D¹ References : A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984). 3.1.1-5. Domain: 0 £ ¾< ©,- ©< ¿< ©,- ©< À< ©. Second boundary value problem. A half-space is considered. The following conditions are prescribed:Á= ( ¾, ¿, À) at Ç= 0 (initial condition),Æ Î Á= Â( ¿, À, Ç) at ¾= 0 (boundary condition). Solution:Á( ¾, ¿, À, Ç)=ª «-« ª «-« ª «0 ( ¡, ¢, £) ( ¾, ¿, À, ¡, ¢, £, Ç)¬ ¡¬ ¢¬ £ - ÃªÄ0 ª «-« ª «-« Â( ¢, £, Å) ( ¾, ¿, À,0, ¢, £, Ç- Å)¬ ¢¬ £¬ Å, where ( ¾, ¿, À, ¡, ¢, £, Ç)=1 8( Ã Ç)3 É2 Êexp Ë-( ¾- ¡)2 4 Ã Ç Ì+exp Ë-( ¾+ ¡)2 4 Ã Ç Ì Íexp Ë-( ¿- ¢)2+( À- £)2 4 Ã Ç Ì.¸D¹ Reference : A. G. Butkovskiy (1979). 3.1.1-6. Domain: 0 £ ¾< ©,- ©< ¿< ©,- ©< À< ©. Third boundary value problem. A half-space is considered. The following conditions are prescribed:Á= ( ¾, ¿, À) at Ç= 0 (initial condition),Æ Î Á- Ï Á= Â( ¿, À, Ç) at ¾= 0 (boundary condition). The solution Á( ¾, ¿, À, Ç) is determined by the formula in Paragraph 3.1.1-5 where ( ¾, ¿, À, ¡, ¢, £, Ç)=1 8( Ã Ç)3 É2exp Ë-( ¿- ¢)2+( À- £)2 4 Ã Ç Ì Êexp Ë-( ¾- ¡)2 4 Ã Ç Ì+exp Ë-( ¾+ ¡)2 4 Ã Ç Ì - 2 Ï Ð Ñ Ã Çexp ÒÏ2Ã Ç+ Ï( ¾+ Ó) Ôerfc Õ ¾+ Ó 2Ð Ã Ç+ Ï Ð Ã Ç ÖÍ.×DØ Reference : H. S. Carslaw and J. C. Jaeger (1984). Page 207 3.1.1-7. Domain: - Ù< Ú< Ù,- Ù< Û< Ù,0 £ Ü£ Ý. First boundary value problem. An in®nite layer is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),Þ= á1( Ú, Û, à) at Ü= 0 (boundary condition),Þ= á2( Ú, Û, à) at Ü= Ý(boundary condition). Solution:Þ( Ú, Û, Ü, à)= â ã 0 â ä -ä â ä -ä ß( Ó, å, æ) ç( Ú, Û, Ü, Ó, å, æ, à) è Ó è å è æ + Ã âÄ0 â ä -ä â ä -ä á1( Ó, å, Å) é êê æ ç( Ú, Û, Ü, Ó, å, æ, à- Å) ë ì =0 è í è å è Å - Ã âÄ0 âä -ä âä -ä á2( í, å, Å)é êê æ ç( Ú, Û, Ü, í, å, æ, à- Å)ë ì =ã è í è å è Å, whereç( Ú, Û, Ü, í, å, æ, à)=1 2 î Ã Ýïàexpé-( Ú- í)2+( Û- å)2 4 Ã à ë ä ðòñ =1sin ^ î ÜÝsin ^ î æÝexp ó- ^2î2Ã àÝ2 Ö, orç( Ú, Û, Ü, í, å, æ, à)=1 8( î Ã à)3 ô2exp é-( Ú- í)2+( Û- å)2 4 Ã à ë ´ ä ð ñ =-ä õ expé-(2 ^ Ý+ Ü- æ)2 4 Ã à ë-expé-(2 ^ Ý+ Ü+ æ)2 4 Ã à ë ö.×DØ Reference : H. S. Carslaw and J. C. Jaeger (1984). 3.1.1-8. Domain: - Ù< Ú< Ù,- Ù< Û< Ù,0 £ Ü£ Ý. Second boundary value problem. An in®nite layer is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),ê ÷ Þ= á1( Ú, Û, à) at Ü= 0 (boundary condition),ê ÷ Þ= á2( Ú, Û, à) at Ü= Ý(boundary condition). Solution:Þ( Ú, Û, Ü, à)= â ã 0 â ä -ä â ä -ä ß( í, å, æ) ç( Ú, Û, Ü, í, å, æ, à) è í è å è æ - Ã âÄ0 â ä -ä â ä -ä á1( í, å, Å) ç( Ú, Û, Ü, í, å,0, à- Å) è í è å è Å + Ã âÄ0 âä -ä âä -ä á2( í, å, Å) ç( Ú, Û, Ü, í, å, Ý, à- Å) è í è å è Å, whereç( Ú, Û, Ü, í, å, æ, à)=1 4 î Ã Ýïàexpé-( Ú- í)2+( Û- å)2 4 Ã à ë ´é1 + 2 ä ðøñ =1cos ^ î ÜÝcos ^ î æÝexp ó- ^2î2Ã àÝ2 Öë, Page 208 ù/û orç( Ú, Û, Ü, í, å, æ, à)=1 (2 ÿ î Ã à)3expé-( Ú- í)2+( Û- å)2 4 Ã à ë ´ ä ð ñ =-ä õ expé-( Ü- æ+ 2 ^ Ý)2 4 Ã à ë+expé-( Ü+ æ+ 2 ^ Ý)2 4 Ã à ë ö.×DØ Reference : H. S. Carslaw and J. C. Jaeger (1984). 3.1.1-9. Domain: - Ù< Ú< Ù,- Ù< Û< Ù,0 £ Ü£ Ý. Third boundary value problem. An in®nite layer is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),ê ÷ Þ- 1 Þ= á1( Ú, Û, à) at Ü= 0 (boundary condition),ê ÷ Þ+ 2 Þ= á2( Ú, Û, à) at Ü= Ý(boundary condition). The solution Þ( Ú, Û, Ü, à) is determined by the formula in Paragraph 3.1.1-8 whereç( Ú, Û, Ü, í, å, æ, à)=1 4 î Ã àexpé-( Ú- í)2+( Û- å)2 4 Ã à ë ä ðòñ =1 ñ ( Ü) ñ ( æ) ñ2exp(- Ã 2 ñà), ñ ( Ü)=cos( ñÜ)+ 1 ñ sin( ñÜ), ñ2= 2 2 2 ñ2 ñ + 2 12 ñ + 2 2+ 1 2 2 ñ + Ý 2 ó1 + 2 12 ñÖ. Here, the ñ are positive roots of the transcendental equationtan( Ý)= 1+ 22- 1 2. 3.1.1-10. Domain: - Ù< Ú< Ù,- Ù< Û< Ù,0 £ Ü£ Ý. Mixed boundary value problem. An in®nite layer is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),Þ= á1( Ú, Û, à) at Ü= 0 (boundary condition),ê ÷ Þ= á2( Ú, Û, à) at Ü= Ý(boundary condition). Solution:Þ( Ú, Û, Ü, à)= â ã 0 â ä -ä â ä -ä ß( í, å, æ) ç( Ú, Û, Ü, í, å, æ, à) è í è å è æ + Ã âÄ0 â ä -ä â ä -ä á1( í, å, Å)é êê æ ç( Ú, Û, Ü, í, å, æ, à- Å)ë ì =0 è í è å è Å + Ã âÄ0 âä -ä âä -ä á2( í, å, Å) ç( Ú, Û, Ü, í, å, Ý, à- Å) è í è å è Å, whereç( Ú, Û, Ü, í, å, æ, à)=1 2 î Ã Ýïàexpé-( Ú- í)2+( Û- å)2 4 Ã à ë ´ ä ðòñ =0siné(2 ^+ 1) î Ü 2 Ý ësiné(2 ^+ 1) î æ 2 Ý ëexpé-(2 ^+ 1)2î2Ã à 4 Ý2 ë, orç( Ú, Û, Ü, í, å, æ, à)=1 (2 ÿ î Ã à)3expé-( Ú- í)2+( Û- å)2 4 Ã à ë ´ ä ð ñ =-ä(-1) ñõ expé-( Ü- æ+ 2 ^ Ý)2 4 Ã à ë-expé-( Ü+ æ+ 2 ^ Ý)2 4 Ã à ë ö.×DØ Reference : A. G. Butkovskiy (1979). Page 209 3.1.1-11. Domain: - Ù< Ú< Ù,0 £ Û< Ù,0 £ Ü£ Ý. First boundary value problem. A semiin®nite layer is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),Þ= á1( Ú, Ü, à) at Û= 0 (boundary condition),Þ= á2( Ú, Û, à) at Ü= 0 (boundary condition),Þ= á3( Ú, Û, à) at Ü= Ý(boundary condition). Solution:Þ( Ú, Û, Ü, à)= â ã 0 â ä 0 â ä -ä ß( í, å, æ) ç( Ú, Û, Ü, í, å, æ, à) è í è å è æ + Ã âÄ0 âã 0 âä -ä á1( í, æ, Å)é êê å ç( Ú, Û, Ü, í, å, æ, à- Å)ë =0 è í è æ è Å + Ã âÄ0 âä 0 âä -ä á2( í, å, Å)é êê æ ç( Ú, Û, Ü, í, å, æ, à- Å)ë ì =0 è í è å è Å - Ã âÄ0 â ä 0 â ä -ä á3( í, å, Å)é êê æ ç( Ú, Û, Ü, í, å, æ, à- Å)ë ì =ã è í è å è Å, whereç( Ú, Û, Ü, í, å, æ, à)=1 2 î Ã Ýïàexpé-( Ú- í)2 4 Ã à ë õ expé-( Û- å)2 4 Ã à ë-expé-( Û+ å)2 4 Ã à ë ö ´ ä ðòñ =1sin ^ î ÜÝsin ^ î æÝexp ó- ^2î2Ã àÝ2 Ö. 3.1.1-12. Domain: - Ù< Ú< Ù,0 £ Û< Ù,0 £ Ü£ Ý. Second boundary value problem. A semiin®nite layer is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),ê Þ= á1( Ú, Ü, à) at Û= 0 (boundary condition),ê ÷ Þ= á2( Ú, Û, à) at Ü= 0 (boundary condition),ê ÷ Þ= á3( Ú, Û, à) at Ü= Ý(boundary condition). Solution:Þ( Ú, Û, Ü, à)= â ã 0 â ä 0 â ä -ä ß( í, å, æ) ç( Ú, Û, Ü, í, å, æ, à) è í è å è æ - Ã âÄ0 âã 0 âä -ä á1( í, æ, Å) ç( Ú, Û, Ü, í,0, æ, à- Å) è í è æ è Å - Ã âÄ0 âä 0 âä -ä á2( í, å, Å) ç( Ú, Û, Ü, í, å,0, à- Å) è í è å è Å + Ã âÄ0 âä 0 âä -ä á3( í, å, Å) ç( Ú, Û, Ü, í, å, Ý, à- Å) è í è å è Å, whereç( Ú, Û, Ü, í, å, æ, à)=1 4 î Ã Ýïàexpé-( Ú- í)2 4 Ã à ë õ expé-( Û- å)2 4 Ã à ë+expé-( Û+ å)2 4 Ã à ë ö ´é1 + 2 ä ðñ =1cos ^ î ÜÝcos ^ î æÝexp ó- ^2î2Ã àÝ2 Öë. Page 210 ù/û 3.1.1-13. Domain: - Ù< Ú< Ù,0 £ Û< Ù,0 £ Ü£ Ý. Third boundary value problem. A semiin®nite layer is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),ê Þ- 1 Þ= á1( Ú, Ü, à) at Û= 0 (boundary condition),ê ÷ Þ- 2 Þ= á2( Ú, Û, à) at Ü= 0 (boundary condition),ê ÷ Þ+ 3 Þ= á3( Ú, Û, à) at Ü= Ý(boundary condition). The solution Þ( Ú, Û, Ü, à) is determined by the formula in Paragraph 3.1.1-12 whereç( Ú, Û, Ü, í, å, æ, à)=1 4 î Ã àexpé-( Ú- í)2 4 Ã à ë ( Û, å, à) ä ðòñ =1 ñ ( Ü) ñ ( æ) ñ2exp(- Ã 2 ñà),( Û, å, à)=exp é-( Û- å)2 4 Ã à ë+exp é-( Û+ å)2 4 Ã à ë- 2 1 â ä 0exp é-( Û+ å+ )2 4 Ã à- 1 ë è . Here, ñ ( Ü)=cos( ñÜ)+ 2 ñ sin( ñÜ), ñ2= 3 2 2 ñ2 ñ + 2 22 ñ + 2 3+ 2 2 2 ñ + Ý 2 ó1 + 2 22 ñÖ; the ñ are positive roots of the transcendental equationtan( Ý)= 2+ 32- 2 3. 3.1.1-14. Domain: - Ù< Ú< Ù,0 £ Û< Ù,0 £ Ü£ Ý. Mixed boundary value problems. 1 . A semiin®nite layer is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),Þ= á1( Ú, Ü, à) at Û= 0 (boundary condition),ê ÷ Þ= á2( Ú, Û, à) at Ü= 0 (boundary condition),ê ÷ Þ= á3( Ú, Û, à) at Ü= Ý(boundary condition). Solution:Þ( Ú, Û, Ü, à)= âã 0 âä 0 âä -ä ß( í, å, æ) ç( Ú, Û, Ü, í, å, æ, à) è í è å è æ + Ã âÄ0 â ã 0 â ä -ä á1( í, æ, Å) é êê å ç( Ú, Û, Ü, í, å, æ, à- Å) ë =0 è í è æ è Å - Ã âÄ0 âä 0 âä -ä á2( í, å, Å) ç( Ú, Û, Ü, í, å,0, à- Å) è í è å è Å + Ã âÄ0 âä 0 âä -ä á3( í, å, Å) ç( Ú, Û, Ü, í, å, Ý, à- Å) è í è å è Å, whereç( Ú, Û, Ü, í, å, æ, à)=1 4 î Ã Ýïàexpé-( Ú- í)2 4 Ã à ë õ expé-( Û- å)2 4 Ã à ë-expé-( Û+ å)2 4 Ã à ë ö ´é1 + 2 ä ðòñ =1cos ^ î ÜÝcos ^ î æÝexp ó- ^2î2Ã àÝ2 Öë. Page 211 2 . A semiin®nite layer is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),ê Þ= á1( Ú, Ü, à) at Û= 0 (boundary condition),Þ= á2( Ú, Û, à) at Ü= 0 (boundary condition),Þ= á3( Ú, Û, à) at Ü= Ý(boundary condition). Solution:Þ( Ú, Û, Ü, à)= â ã 0 â ä 0 â ä -ä ß( í, å, æ) ç( Ú, Û, Ü, í, å, æ, à) è í è å è æ - Ã âÄ0 âã 0 âä -ä á1( í, æ, Å) ç( Ú, Û, Ü, í,0, æ, à- Å) è í è æ è Å + Ã âÄ0 âä 0 âä -ä á2( í, å, Å)é êê æ ç( Ú, Û, Ü, í, å, æ, à- Å)ë ì =0 è í è å è Å - Ã âÄ0 â ä 0 â ä -ä á3( í, å, Å)é êê æ ç( Ú, Û, Ü, í, å, æ, à- Å)ë ì =ã è í è å è Å, whereç( Ú, Û, Ü, í, å, æ, à)=1 2 î Ã Ýïàexpé-( Ú- í)2 4 Ã à ë õ expé-( Û- å)2 4 Ã à ë+expé-( Û+ å)2 4 Ã à ë ö ´ ä ðòñ =1sin ^ î ÜÝsin ^ î æÝexp ó- ^2î2Ã àÝ2 Ö. 3.1.1-15. Domain: 0 £ Ú< Ù,0 £ Û< Ù,0 £ Ü< Ù. First boundary value problem. An octant is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),Þ= á1( Û, Ü, à) at Ú= 0 (boundary condition),Þ= á2( Ú, Ü, à) at Û= 0 (boundary condition),Þ= á3( Ú, Û, à) at Ü= 0 (boundary condition). Solution:Þ( Ú, Û, Ü, à)= â ä 0 â ä 0 â ä 0 ç( Ú, Û, Ü, í, å, æ, à) ß( í, å, æ) è í è å è æ + Ã âÄ0 â ä 0 â ä 0 á1( å, æ, Å)é êê í ç( Ú, Û, Ü, í, å, æ, à- Å)ë =0 è å è æ è Å + Ã âÄ0 âä 0 âä 0 á2( í, æ, Å)é êê å ç( Ú, Û, Ü, í, å, æ, à- Å)ë =0 è í è æ è Å + Ã âÄ0 âä 0 âä 0 á3( í, å, Å)é êê æ ç( Ú, Û, Ü, í, å, æ, à- Å)ë ì =0 è í è å è Å, whereç( Ú, Û, Ü, í, å, æ, à)=1 2 ÿî Ã à 3 ( Ú, í, à)( Û, å, à)( Ü, æ, à),( Ú, í, à)=expé-( Ú- í)2 4 Ã à ë-expé-( Ú+ í)2 4 Ã à ë. Page 212 ÄExample 3. The initial temperature is uniform, ( , , )=þ0. The faces are maintained at zero temperature, 1= 2= 3= 0. Solution:þ=þ0erf 2 ü erf 2 ü erf 2 ü .×DØ Reference : H. S. Carslaw and J. C. Jaeger (1984). 3.1.1-16. Domain: 0 £ Ú< Ù,0 £ Û< Ù,0 £ Ü< Ù. Second boundary value problem. An octant is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),ê Þ= á1( Û, Ü, à) at Ú= 0 (boundary condition),ê Þ= á2( Ú, Ü, à) at Û= 0 (boundary condition),ê ÷ Þ= á3( Ú, Û, à) at Ü= 0 (boundary condition). Solution:Þ( Ú, Û, Ü, à)= âä 0 âä 0 âä 0 ç( Ú, Û, Ü, í, å, æ, à) ß( í, å, æ) è í è å è æ - â 0 â ä 0 â ä 0 á1( å, æ, ) ç( Ú, Û, Ü,0, å, æ, à- ) è å è æ è - â 0 â ä 0 â ä 0 á2( í, æ, ) ç( Ú, Û, Ü, í,0, æ, à- ) è í è æ è - â 0 â ä 0 â ä 0 á3( í, å, ) ç( Ú, Û, Ü, í, å,0, à- ) è í è å è , whereç( Ú, Û, Ü, í, å, æ, à)=1 2 ÿî à 3 ( Ú, í, à)( Û, å, à)( Ü, æ, à),( Ú, í, à)=expé-( Ú- í)2 4 à ë+expé-( Ú+ í)2 4 à ë. 3.1.1-17. Domain: 0 £ Ú< Ù,0 £ Û< Ù,0 £ Ü< Ù. Third boundary value problem. An octant is considered. The following conditions are prescribed:Þ= ß( Ú, Û, Ü) at à= 0 (initial condition),ê Þ- 1 Þ= á1( Û, Ü, à) at Ú= 0 (boundary condition),ê Þ- 2 Þ= á2( Ú, Ü, à) at Û= 0 (boundary condition),ê ÷ Þ- 3 Þ= á3( Ú, Û, à) at Ü= 0 (boundary condition). The solution Þ( Ú, Û, Ü, à) is determined by the formula in Paragraph 3.1.1-16 whereç( Ú, Û, Ü, í, å, æ, à)=1 2 ÿ î à 3 ( Ú, í, à; 1)( Û, å, à; 2)( Ü, æ, à; 3),( Ú, í, à; )=expé-( Ú- í)2 4 à ë+expé-( Ú+ í)2 4 à ë - 2 ÿî àexp 2à+ ( Ú+ í) erfc ó Ú+ í 2 ÿ à+ ÿ à . Page 213 Example 4. The initial temperature is uniform, ( , , )=þ0. The temperature of the contacting media is zero, 1= 2= 3= 0. Solution:þ=þ0 !erf 2 ü +exp( "1 + "2 1 ü) erfc 2 ü+ "1 ü # ´!erf 2 ü +exp( "2 + "2 2 ü) erfc 2 ü+ "2 ü # ´!erf 2 ü +exp( "3 + "2 3 ü) erfc 2 ü+ "3 ü #.$% Reference : H. S. Carslaw and J. C. Jaeger (1984). 3.1.1-18. Domain: 0 £ &< ',0 £ (< ',0 £ )< '. Mixed boundary value problems. 1 . An octant is considered. The following conditions are prescribed:*= +( &, (, )) at ,= 0 (initial condition),*= -1( (, ), ,) at &= 0 (boundary condition),. *= -2( &, ), ,) at (= 0 (boundary condition),.÷ *= -3( &, (, ,) at )= 0 (boundary condition). Solution:*( &, (, ), ,)= / 0 0 / 0 0 / 0 0 1( &, (, ), 2, 3, 4, ,) +( 2, 3, 4) 5 2 5 3 5 4 + 6 / 7 0 / 0 0 / 0 0 -1( 3, 4, 8) 9 ..21( &, (, ), 2, 3, 4, ,- 8) : ; =0 5 3 5 4 5 8 - 6 / 7 0 / 0 0 / 0 0 -2( 2, 4, 8)1( &, (, ), 2,0, 4, ,- 8) 5 2 5 4 5 8 - 6 / 7 0 / 0 0 / 0 0 -3( 2, 3, 8)1( &, (, ), 2, 3,0, ,- 8) 5 2 5 3 5 8, where1( &, (, ), 2, 3, 4, ,)=1< 2 = > 6 ? @3 Aexp 9-( B- 2)2 4 6 ? :-exp 9-( B+ 2)2 4 6 ? : C D( E, 3, ?) D( F, 4, ?),D( E, 3, ?)=exp 9-( E- 3)2 4 6 ? :+exp 9-( E+ 3)2 4 6 ? :. 2 G. An octant is considered. The following conditions are prescribed:H= I( B, E, F) at ?= 0 (initial condition),H= J1( E, F, ?) at B= 0 (boundary condition),H= J2( B, F, ?) at E= 0 (boundary condition),K LH= J3( B, E, ?) at F= 0 (boundary condition). Solution:H( B, E, F, ?)= / 0 0 / 0 0 / 0 0 1( B, E, F, 2, 3, 4, ?) I( 2, 3, 4) 5 2 5 3 5 4 + 6 / 7 0 / 0 0 / 0 0 J1( 3, 4, 8) 9 KK21( B, E, F, 2, 3, 4, ?- 8) : ; =0 5 3 5 4 5 8 + 6 / 7 0 / 0 0 / 0 0 J2( 2, 4, 8) 9 KK31( B, E, F, 2, 3, 4, ?- 8) : M =0 5 2 5 4 5 8 - 6 / 7 0 / 0 0 / 0 0 J3( 2, 3, 8)1( B, E, F, 2, 3,0, ?- 8) 5 2 5 3 5 8, Page 214 NPQ where1( B, E, F, 2, 3, 4, ?)=1< 2 = > 6 ? @3 D( B, 2, ?) D( E, 3, ?)Aexp 9-( F- 4)2 4 6 ? :+exp 9-( F+ 4)2 4 6 ? : C,D( B, 2, ?)=exp 9-( B- 2)2 4 6 ? :-exp 9-( B+ 2)2 4 6 ? :. 3.1.1-19. Domain: 0 £ B£ U1,0 £ E£ U2,- V< F< V. First boundary value problem. An in®nite cylindrical domain of a rectangular cross-section is considered. The following conditionsare prescribed:H= I( B, E, F) at ?= 0 (initial condition),H= J1( E, F, ?) at B= 0 (boundary condition),H= J2( E, F, ?) at B= U1(boundary condition),H= J3( B, F, ?) at E= 0 (boundary condition),H= J4( B, F, ?) at E= U2(boundary condition). Solution:H( B, E, F, ?)= / 0 -0 / W2 0 / W1 0 I( 2, 3, 4)1( B, E, F, 2, 3, 4, ?) 5 2 5 3 5 4 + 6 / 7 0 / 0 -0 / W2 0 J1( 3, 4, 8) 9 KK21( B, E, F, 2, 3, 4, ?- 8) : ; =0 5 3 5 4 5 8 - 6 / 7 0 / 0 -0 / W2 0 J2( 3, 4, 8) 9 KK21( B, E, F, 2, 3, 4, ?- 8) : ; =W1 5 3 5 4 5 8 + 6 / 7 0 / 0 -0 / W1 0 J3( 2, 4, 8) 9 KK31( B, E, F, 2, 3, 4, ?- 8) : M =0 5 2 5 4 5 8 - 6 / 7 0 / 0 -0 / W1 0 J4( 2, 4, 8) 9 KK31( B, E, F, 2, 3, 4, ?- 8) : M =W2 5 2 5 4 5 8, where1( B, E, F, 2, 3, 4, ?)=1 2 = > 6 ?exp 9-( F- 4)2 4 6 ? : D1( B, 2, ?) D2( E, 3, ?),D1( B, 2, ?)=2U1 0 XZY =1sin [ > \ BU1 ]sin [ > \ 2U1 ]exp [- >2\26 ?U2 1 ],D2( E, 3, ?)=2U2 0 XZY =1sin [ > \ EU2 ]sin [ > \ 3U2 ]exp [- >2\26 ?U2 2 ]. 3.1.1-20. Domain: 0 £ B£ U1,0 £ E£ U2,- V< F< V. Second boundary value problem. An in®nite cylindrical domain of a rectangular cross-section is considered. The following conditionsare prescribed:H= I( B, E, F) at ?= 0 (initial condition),K ^H= J1( E, F, ?) at B= 0 (boundary condition),K ^H= J2( E, F, ?) at B= U1(boundary condition),K _H= J3( B, F, ?) at E= 0 (boundary condition),K _H= J4( B, F, ?) at E= U2(boundary condition). Page 215 Solution:H( B, E, F, ?)= / 0 -0 / W2 0 / W1 0 I( 2, 3, 4)1( B, E, F, 2, 3, 4, ?) 5 2 5 3 5 4 - 6 / 7 0 / 0 -0 / W2 0 J1( 3, 4, 8)1( B, E, F,0, 3, 4, ?- 8) 5 3 5 4 5 8 + 6 / 7 0 / 0 -0 / W2 0 J2( 3, 4, 8)1( B, E, F, U1, 3, 4, ?- 8) 5 3 5 4 5 8 - 6 / 7 0 / 0 -0 / W1 0 J3( 2, 4, 8)1( B, E, F, 2,0, 4, ?- 8) 5 2 5 4 5 8 + 6 / 7 0 / 0 -0 / W1 0 J4( 2, 4, 8)1( B, E, F, 2, U2, 4, ?- 8) 5 2 5 4 5 8, where1( B, E, F, 2, 3, 4, ?)=1 2 = > 6 ?exp 9-( F- 4)2 4 6 ? : D1( B, 2, ?) D2( E, 3, ?),D1( B, 2, ?)=1U1 91 + 2 0 XZY =1cos [ > \ BU1 ]cos [ > \ 2U1 ]exp [- >2\26 ?U2 1 ] :,D2( E, 3, ?)=1U2 91 + 2 0 XZY =1cos [ > \ EU2 ]cos [ > \ 3U2 ]exp [- >2\26 ?U2 2 ] :. 3.1.1-21. Domain: 0 £ B£ U1,0 £ E£ U2,- V< F< V. Third boundary value problem. An in®nite cylindrical domain of a rectangular cross-section is considered. The following conditionsare prescribed:H= I( B, E, F) at ?= 0 (initial condition),K ^H- `1 H= J1( E, F, ?) at B= 0 (boundary condition),K ^H+ `2 H= J2( E, F, ?) at B= U1(boundary condition),K _H- `3 H= J3( B, F, ?) at E= 0 (boundary condition),K _H+ `4 H= J4( B, F, ?) at E= U2(boundary condition). The solution H( B, E, F, ?) is determined by the formula in Paragraph 3.1.1-20 where1( B, E, F, 2, 3, 4, ?)=1 2 = > 6 ?exp 9-( F- 4)2 4 6 ? : D1( B, 2, ?) D2( E, 3, ?),D1( B, 2, ?)= 0 XZY =1 a Y ( B)a Y ( 2)ba Yb2exp(- 6 c2 Y?), D2( E, 3, ?)= 0 Xd=1 e d( E)e d( 3)be d b2exp(- 6 f2d?). Here,a Y ( B)=cos( c YB)+ `1c Y sin( c YB), ba Yb2= `2 2 c2 Yc2 Y + `2 1c2 Y + `2 2+ `1 2 c2 Y + U1 2 [1 + `2 1c2 Y],e d( E)=cos( f dE)+ `3f dsin( f dE), be d b2= `4 2 f2d f2d+ `2 3f2d+ `2 4+ `3 2 f2d+ U2 2 [1 + `2 3f2d]; the c Y and f dare positive roots of the transcendental equations tan( c U1)c= `1+ `2c2- `1 `2,tan( f U2)f= `3+ `4f2- `3 `4. Page 216 NPQ 3.1.1-22. Domain: 0 £ B£ U1,0 £ E£ U2,- V< F< V. Mixed boundary value problem. An in®nite cylindrical domain of a rectangular cross-section is considered. The following conditionsare prescribed:H= I( B, E, F) at ?= 0 (initial condition),H= J1( E, F, ?) at B= 0 (boundary condition),H= J2( E, F, ?) at B= U1(boundary condition),K _H= J3( B, F, ?) at E= 0 (boundary condition),K _H= J4( B, F, ?) at E= U2(boundary condition). Solution:H( B, E, F, ?)= g h -h g W2 0 g W1 0 I( i, j, k) l( B, E, F, i, j, k, ?) m i m j m k + n g 7 0 gh -h g W2 0 J1( j, k, 8) 9 KKi l( B, E, F, i, j, k, ?- 8) o p =0 m j m k m 8 - n g 7 0 gh -h g W2 0 J2( j, k, 8) 9 KKi l( B, E, F, i, j, k, ?- 8) o p =W1 m j m k m 8 - n g 7 0 g h -h g W1 0 J3( i, k, 8) l( B, E, F, i,0, k, ?- 8) m i m k m 8 + n g 7 0 g h -h g W1 0 J4( i, k, 8) l( B, E, F, i, U2, k, ?- 8) m i m k m 8, wherel( B, E, F, i, j, k, ?)=2U1 U2 => n ?exp 9-( F- k)2 4 n ? o 9 h XZY =1sin [ > \ BU1 ]sin [ > \ iU1 ]exp [- >2\2n ?U2 1 ] o ´ 91 2+ h Xd=1cos [ > q BU2 ]cos [ > q iU2 ]exp [- >2q2n ?U2 2 ] o. 3.1.1-23. Domain: 0 £ B£ U1,0 £ E£ U2,0 £ F< V. First boundary value problem. A semiin®nite cylindrical domain of a rectangular cross-section is considered. The followingconditions are prescribed:H= I( B, E, F) at ?= 0 (initial condition),H= J1( E, F, ?) at B= 0 (boundary condition),H= J2( E, F, ?) at B= U1(boundary condition),H= J3( B, F, ?) at E= 0 (boundary condition),H= J4( B, F, ?) at E= U2(boundary condition),H= J5( B, E, ?) at F= 0 (boundary condition). Page 217 Solution:H( B, E, F, ?)= g h 0 g W2 0 g W1 0 I( i, j, k) l( B, E, F, i, j, k, ?) m i m j m k + n g 7 0 g h 0 g W2 0 J1( j, k, 8) 9 KKi l( B, E, F, i, j, k, ?- 8) o p =0 m j m k m 8 - n g 7 0 g h 0 g W2 0 J2( j, k, 8) 9 KKi l( B, E, F, i, j, k, ?- 8) o p =W1 m j m k m 8 + n g 7 0 g h 0 g W1 0 J3( i, k, 8) 9 KKj l( B, E, F, i, j, k, ?- 8) o M =0 m i m k m 8 - n g 7 0 g h 0 g W1 0 J4( i, k, 8) 9 KKj l( B, E, F, i, j, k, ?- 8) o M =W2 m i m k m 8 + n g 7 0 g W2 0 g W1 0 J5( i, j, 8) 9 KKk l( B, E, F, i, j, k, ?- 8) o r =0 m i m j m 8, wherel( B, E, F, i, j, k, ?)= l1( B, i, ?; U1) l1( E, j, ?; U2) l2( F, k, ?),l1( B, i, ?; U)=2U h XZY =1sin [ > \ BU]sin [ > \ iU]exp [- >2\2n ?U2],l2( F, k, ?)=1 2 s t n u vexp 9-( w- k)2 4 n u o-exp 9-( w+ k)2 4 n u o x. 3.1.1-24. Domain: 0 £ y£ z1,0 £ {£ z2,0 £ w< |. Second boundary value problem. A semiin®nite cylindrical domain of a rectangular cross-section is considered. The followingconditions are prescribed:} = ~( y, {, w) at u= 0 (initial condition), ^ } = 1( {, w, u) at y= 0 (boundary condition), ^ } = 2( {, w, u) at y= z1(boundary condition), _ } = 3( y, w, u) at {= 0 (boundary condition), _ } = 4( y, w, u) at {= z2(boundary condition), } = 5( y, {, u) at w= 0 (boundary condition). Solution:} ( y, {, w, u)= gh 0 g 2 0 g 1 0 ~( i, j, k) l( y, {, w, i, j, k, u) m i m j m k - n g 7 0 gh 0 g 2 0 1( j, k, 8) l( y, {, w,0, j, k, u- 8) m j m k m 8 + n g 7 0 gh 0 g 2 0 2( j, k, 8) l( y, {, w, z1, j, k, u- 8) m j m k m 8 - n g 7 0 g h 0 g 1 0 3( i, k, 8) l( y, {, w, i,0, k, u- 8) m i m k m 8 + n g 7 0 g h 0 g 1 0 4( i, k, 8) l( y, {, w, i, z2, k, u- 8) m i m k m 8 - n g 7 0 g 2 0 g 1 0 5( i, j, 8) l( y, {, w, i, j,0, u- 8) m i m j m 8, Page 218 P wherel( y, {, w, i, j, k, u)=1 2 s t n u vexp 9-( w- k)2 4 n u o+exp 9-( w+ k)2 4 n u o x l1( y, i, u) l2( {, j, u),l1( y, i, u)=1z1 91 + 2 h Z =1cos t yz1 cos t iz1 exp - t22n uz2 1 o,l2( {, j, u)=1z2 1 + 2 Z =1cos t {z2 cos t z2 exp - t22 uz2 2 . 3.1.1-25. Domain: 0 £ y£ z1,0 £ {£ z2,0 £ w< |. Third boundary value problem. A semiin®nite cylindrical domain of a rectangular cross-section is considered. The followingconditions are prescribed:} = ~( y, {, w) at = 0 (initial condition), } - 1 } = 1( {, w, ) at y= 0 (boundary condition), } + 2 } = 2( {, w, ) at y= z1(boundary condition), } - 3 } = 3( y, w, ) at {= 0 (boundary condition), } + 4 } = 4( y, w, ) at {= z2(boundary condition), } - 5 } = 5( y, {, ) at w= 0 (boundary condition). The solution } ( y, {, w, ) is determined by the formula in Paragraph 3.1.1-24 where( y, {, w, , , , )= 1( y, , ) 2( {, , ) 3( , , ),3( , , )=1 2 exp-( - )2 4 +exp-( + )2 4 - 5exp ¡2 5 + 5( + ) ¢erfc + 2 + 5 , and the functions 1( £, , ) and 2( ¤, , ) can be found in Paragraph 3.1.1-21. 3.1.1-26. Domain: 0 £ ££ ¥1,0 £ ¤£ ¥2,0 £ < ¦. Mixed boundary value problems. 1 §. A semiin®nite cylindrical domain of a rectangular cross-section is considered. The following conditions are prescribed: ¨ = ©( £, ¤, ) at = 0 (initial condition), ¨ = ª1( ¤, , ) at £= 0 (boundary condition), ¨ = ª2( ¤, , ) at £= ¥1(boundary condition), ¨ = ª3( £, , ) at ¤= 0 (boundary condition), ¨ = ª4( £, , ) at ¤= ¥2(boundary condition),« ¬ ¨ = ª5( £, ¤, ) at = 0 (boundary condition). Page 219 Solution:¨ ( £, ¤, , )= 0 ®2 0 ®1 0 ©( , , ) ( £, ¤, , , , , ) ¯ ¯ ¯ + ° 0 0 ®2 0 ª1( , , ±) «« ( £, ¤, , , , , - ±) ²=0 ¯ ¯ ¯ ± - ° 0 0 ®2 0 ª2( , , ±) «« ( £, ¤, , , , , - ±) ²=®1 ¯ ¯ ¯ ± + ° 0 0 ®1 0 ª3( , , ±) «« ( £, ¤, , , , , - ±) ³=0 ¯ ¯ ¯ ± - ° 0 0 ®1 0 ª4( , , ±) «« ( £, ¤, , , , , - ±) ³=®2 ¯ ¯ ¯ ± - ° 0 ®2 0 ®1 0 ª5( , , ±) ( £, ¤, , , ,0, - ±) ¯ ¯ ¯ ±, where( £, ¤, , , , , )=1 2 exp-( - )2 4 +exp-( + )2 4 ( £, , ; ¥1) ( ¤, , ; ¥2),( £, , ; ¥)=2¥ ´Zµ =1sin ¶ · £¥ ¸sin ¶ · ¥ ¸exp ¶- 2·2 ¥2¸. 2 §. A semiin®nite cylindrical domain of a rectangular cross-section is considered. The following conditions are prescribed: ¨ = ©( £, ¤, ) at = 0 (initial condition),« ¨ = ª1( ¤, , ) at £= 0 (boundary condition),« ¨ = ª2( ¤, , ) at £= ¥1(boundary condition),« ¨ = ª3( £, , ) at ¤= 0 (boundary condition),« ¨ = ª4( £, , ) at ¤= ¥2(boundary condition), ¨ = ª5( £, ¤, ) at = 0 (boundary condition). Solution:¨ ( £, ¤, , )= 0 ®2 0 ®1 0 ©( , , ) ( £, ¤, , , , , ) ¯ ¯ ¯ - ° 0 0 ®2 0 ª1( , , ±) ( £, ¤, ,0, , , - ±) ¯ ¯ ¯ ± + ° 0 0 ®2 0 ª2( , , ±) ( £, ¤, , ¥1, , , - ±) ¯ ¯ ¯ ± - ° 0 0 ®1 0 ª3( , , ±) ( £, ¤, , ,0, , - ±) ¯ ¯ ¯ ± + ° 0 0 ®1 0 ª4( , , ±) ( £, ¤, , , ¥2, , - ±) ¯ ¯ ¯ ± + ° 0 ®2 0 ®1 0 ª5( , , ±) «« ( £, ¤, , , , , - ±) ¹=0 ¯ ¯ ¯ ±, Page 220 where( , , , , , , )=1 2 exp -( - )2 4 -exp -( + )2 4 ( , , ; 1)( , , ; 2),( , , ; )=1 1 + 2 =1cos cos exp - 22 2 . 3.1.1-27. Domain: 0 £ £ 1,0 £ £ 2,0 £ £ 3. First boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:= ( , , ) at = 0 (initial condition),= !1( , , ) at = 0 (boundary condition),= !2( , , ) at = 1(boundary condition),= !3( , , ) at = 0 (boundary condition),= !4( , , ) at = 2(boundary condition),= !5( , , ) at = 0 (boundary condition),= !6( , , ) at = 3(boundary condition). Solution:( , , , )= " #3 0 " #2 0 " #1 0 ( , , ) ( , , , , , , ) $ $ $ + " % 0 "#3 0 "#2 0 !1( , , &) '' ( , , , , , , - &) (=0 $ $ $ & - " % 0 "#3 0 "#2 0 !2( , , &) '' ( , , , , , , - &) (=#1 $ $ $ & + " % 0 " #3 0 " #1 0 !3( , , &) '' ( , , , , , , - &) )=0 $ $ $ & - " % 0 "#3 0 "#1 0 !4( , , &) '' ( , , , , , , - &) )=#2 $ $ $ & + " % 0 " #2 0 " #1 0 !5( , , &) '' ( , , , , , , - &) *=0 $ $ $ & - " % 0 " #2 0 " #1 0 !6( , , &) '' ( , , , , , , - &) *=#3 $ $ $ &, where( , , , , , , )= 1( , , ) 2( , , ) 3( , , ), 1( , , )=21 =1sin 1 sin 1 exp - 22 2 1 , 2( , , )=22 =1sin 2 sin 2 exp - 22 2 2 , 3( , , )=23 =1sin 3 sin 3 exp - 22 2 3 .+-, Reference : H. S. Carslaw and J. C. Jaeger (1984). Page 221 3.1.1-28. Domain: 0 £ £ 1,0 £ £ 2,0 £ £ 3. Second boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:= ( , , ) at = 0 (initial condition),' . = !1( , , ) at = 0 (boundary condition),' . = !2( , , ) at = 1(boundary condition),' / = !3( , , ) at = 0 (boundary condition),' / = !4( , , ) at = 2(boundary condition),' 0 = !5( , , ) at = 0 (boundary condition),' 0 = !6( , , ) at = 3(boundary condition). Solution:( , , , )= " #3 0 " #2 0 " #1 0 ( , , ) ( , , , , , , ) $ $ $ - " % 0 " #3 0 " #2 0 !1( , , &) ( , , ,0, , , - &) $ $ $ & + " % 0 " #3 0 " #2 0 !2( , , &) ( , , , 1, , , - &) $ $ $ & - " % 0 " #3 0 " #1 0 !3( , , &) ( , , , ,0, , - &) $ $ $ & + " % 0 " #3 0 " #1 0 !4( , , &) ( , , , , 2, , - &) $ $ $ & - " % 0 " #2 0 " #1 0 !5( , , &) ( , , , , ,0, - &) $ $ $ & + " % 0 " #2 0 " #1 0 !6( , , &) ( , , , , , 3, - &) $ $ $ &, where( , , , , , , )= 1( , , ) 2( , , ) 3( , , ), 1( , , )=11 1 + 2 =1cos 1 cos 1 exp - 22 2 1 , 2( , , )=12 1 + 2 =1cos 2 cos 2 exp - 22 2 2 , 3( , , )=13 1 + 2 =1cos 3 cos 3 exp - 22 2 3 . 3.1.1-29. Domain: 0 £ £ 1,0 £ £ 2,0 £ £ 3. Third boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:= ( , , ) at = 0 (initial condition),' . - 11 = !1( , , ) at = 0 (boundary condition),' . + 12 = !2( , , ) at = 1(boundary condition),' / - 13 = !3( , , ) at = 0 (boundary condition),' / + 14 = !4( , , ) at = 2(boundary condition),' 0 - 15 = !5( , , ) at = 0 (boundary condition),' 0 + 16 = !6( , , ) at = 3(boundary condition). Page 222 The solution ( , , , ) is determined by the formula in Paragraph 3.1.1-28 where( , , , , , , )=1( , , )2( , , )3( , , ). The functions1( , , ) and2( , , ) can be found in Paragraph 3.1.1-21, and the function3( , , ) is given by3( , , )= =1 2 ( )2 ( )32 32exp(- 42 ),2 ( )=cos( 4 )+ 154 sin( 4 ), 32 32= 16 2 42 42 + 12 542 + 12 6+ 15 2 42 + 3 2 1 + 12 542 , where the 4 are positive roots of the transcendental equationtan( 4 3)4= 15+ 1642- 15 16. 3.1.1-30. Domain: 0 £ £ 1,0 £ £ 2,0 £ £ 3. Mixed boundary value problems. 1 5. A rectangular parallelepiped is considered. The following conditions are prescribed:= ( , , ) at = 0 (initial condition),= !1( , , ) at = 0 (boundary condition),= !2( , , ) at = 1(boundary condition),= !3( , , ) at = 0 (boundary condition),= !4( , , ) at = 2(boundary condition),' 0 = !5( , , ) at = 0 (boundary condition),' 0 = !6( , , ) at = 3(boundary condition). Solution:( , , , )= "#3 0 "#2 0 "#1 0 ( , , ) ( , , , , , , ) $ $ $ + " % 0 " #3 0 " #2 0 !1( , , &) '' ( , , , , , , - &) (=0 $ $ $ & - " % 0 "#3 0 "#2 0 !2( , , &) '' ( , , , , , , - &) (=#1 $ $ $ & + " % 0 " #3 0 " #1 0 !3( , , &) '' ( , , , , , , - &) )=0 $ $ $ & - " % 0 "#3 0 "#1 0 !4( , , &) '' ( , , , , , , - &) )=#2 $ $ $ & - " % 0 "#2 0 "#1 0 !5( , , &) ( , , , , ,0, - &) $ $ $ & + " % 0 " #2 0 " #1 0 !6( , , &) ( , , , , , 3, - &) $ $ $ &, Page 223 where( , , , , , , )= 1( , , ) 2( , , ) 3( , , ), 1( , , )=21 =1sin 1 sin 1 exp - 22 2 1 , 2( , , )=22 76 =1sin 1 2 sin 1 2 exp - 212 2 2 , 3( , , )=13+23 8=1cos 9 3 cos 9 3 exp - 292 2 3 . 2 5. A rectangular parallelepiped is considered. The following conditions are prescribed:= ( , , ) at = 0 (initial condition),= !1( , , ) at = 0 (boundary condition),= !2( , , ) at = 1(boundary condition),' / = !3( , , ) at = 0 (boundary condition),' / = !4( , , ) at = 2(boundary condition),' 0 = !5( , , ) at = 0 (boundary condition),' 0 = !6( , , ) at = 3(boundary condition). Solution:( , , , )= " #3 0 " #2 0 " #1 0 ( , , ) ( , , , , , , ) $ $ $ + " % 0 "#3 0 "#2 0 !1( , , &) '' ( , , , , , , - &) (=0 $ $ $ & - " % 0 "#3 0 "#2 0 !2( , , &) '' ( , , , , , , - &) (=#1 $ $ $ & - " % 0 " #3 0 " #2 0 !3( , , &) ( , , , ,0, , - &) $ $ $ & + " % 0 "#3 0 "#1 0 !4( , , &) ( , , , , 2, , - &) $ $ $ & - " % 0 " #2 0 " #1 0 !5( , , &) ( , , , , ,0, - &) $ $ $ & + " % 0 "#2 0 "#1 0 !6( , , &) ( , , , , , 3, - &) $ $ $ &, where( , , , , , , )= 1( , , ) 2( , , ) 3( , , ), 1( , , )=21 =1sin 1 sin 1 exp - 22 2 1 , 2( , , )=12+22 76 =1cos 1 2 cos 1 2 exp - 212 2 2 , 3( , , )=13+23 8=1cos 9 3 cos 9 3 exp - 292 2 3 . Page 224 3.1.2. Problems in Cylindrical Coordinates The three-dimensional sourceless heat equation in the cylindrical coordinate system has the form' ' = 1 :'' : :' ' :+1 : 2 '2 ' ;2+ '2 ' 2, : = < 2+ 2. It is used to describe nonsymmetric unsteady processes in moving media or solids with cylindrical orplane boundaries. A similar equation is used to study the corresponding three-dimensional unsteadymass-exchange processes with constant diffusivity. One-dimensional problems with axial symmetry that have solutions of the form= ( : , ) are discussed in Subsection 1.2.1. Two-dimensional problems whose solutions have the form= ( : ,;, ) or = ( : , , ) are considered in Subsections 2.1.2 and 2.1.3. 3.1.2-1. Remarks on the Green's functions. For the three-dimensional problems dealt with in Subsection 3.1.2, the Green's function can berepresented in the product form( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), where 1( : ,;, , , ) is the Green's function of the two-dimensional boundary value problem (such functions are presented in Subsection 2.1.2), and 2( , , ) is the Green's function of the correspond- ing one-dimensional boundary value problem (such functions can be found in Subsections 1.1.1 and1.1.2). Example. The Green's function of the ®rst boundary value problem for a semiin®nite circular cylinder ( 0 £ =£ >, 0 £ ?£ 2 @,0 £ A< B) of Paragraph 3.1.2-5 is the product of the two-dimensional Green's function of the ®rst boundary value problem of Paragraph 2.1.2-2 ( 0 £ =£ >,0 £ ?£ 2 @) and the one-dimensional Green's function of the ®rst boundary value problem of Paragraph 1.1.2-2 ( 0 £ A< B), in which one should perform obvious renaming of variables. General formulas that enable one to obtain solutions of basic boundary value problems with the help of the Green's function can be found in Subsection 0.8.1. 3.1.2-2. Domain: 0 £ : £ C,0 £;£ 2 ,- D< < D. First boundary value problem. An in®nite circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),= !(;, , ) at : = C(boundary condition). Solution:( : ,;, , )= " - "2 E 0 " F 0 ( , , ) ( : ,;, , , , , ) $ $ $ - C " % 0 " - "2 E 0 !( , , &) '' ( : ,;, , , , , - &) (=F $ $ $ &. Here,( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 C2 =0 8=1 G [ H I ( J 8C)]2 H ( J 8 : ) H ( J 8) cos[ (;- )] exp( - J2 8 ), 2( , , )=1 2 exp -( - )2 4 ,G = 1for = 0, 2for = 1,2, KKK, where the H ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the J 8are positive roots of the transcendental equation H ( J C)= 0.+-, Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 225 3.1.2-3. Domain: 0 £ : £ C,0 £;£ 2 ,- D< < D. Second boundary value problem. An in®nite circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),' L = !(;, , ) at : = C(boundary condition). Solution:( : ,;, , )= " - "2 E 0 " F 0 ( , , ) ( : ,;, , , , , ) $ $ $ + C " % 0 " - "2 E 0 !( , , &) ( : ,;, , C, , , - &) $ $ $ &. Here,( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 C2+1 =0 8=1 G J2 8H ( J 8 : ) H ( J 8) ( J2 8C2- 2)[ H ( J 8C)]2cos[ (;- )] exp( - J2 8 ), 2( , , )=1 2 exp -( - )2 4 ,G = 1for = 0, 2for = 1,2, KKK, where the H ( ) are the Bessel functions and the J 8are positive roots of the transcendental equationH I ( J C)= 0.+-, Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 3.1.2-4. Domain: 0 £ : £ C,0 £;£ 2 ,- D< < D. Third boundary value problem. An in®nite circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),' L + 1 = !(;, , ) at : = C(boundary condition). The solution ( : ,;, , ) is determined by the formula in Paragraph 3.1.2-3 where( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 =0 8=1G J2 8H ( J 8 : ) H ( J 8) ( J2 8C2+ 12C2- 2)[ H ( J 8C)]2cos[ (;- )] exp( - J2 8 ), 2( , , )=1 2 exp -( - )2 4 ,G = 1for = 0, 2for = 1,2, KKK Here, the H ( ) are the Bessel functions and the J 8are positive roots of the transcendental equationJ H I ( J C)+ 1 H ( J C)= 0.+-, Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 226 3.1.2-5. Domain: 0 £ : £ C,0 £;£ 2 ,0 £ < D. First boundary value problem. A semiin®nite circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),= !1(;, , ) at : = C(boundary condition),= !2( : ,;, ) at = 0 (boundary condition). Solution:( : ,;, , )= " 0 "2 E 0 " F 0 ( , , ) ( : ,;, , , , , ) $ $ $ - C " % 0 " 0 "2 E 0 !1( , , &) '' ( : ,;, , , , , - &) (=F $ $ $ & + " % 0 "2 E 0 " F 0 !2( , , &) '' ( : ,;, , , , , - &) *=0 $ $ $ &. Here,( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 C2 =0 8=1G [ H I ( J 8C)]2 H ( J 8 : ) H ( J 8) cos[ (;- )] exp( - J2 8 ), 2( , , )=1 2 exp -( - )2 4 -exp -( + )2 4 ,G = 1for = 0, 2for = 1,2, KKK, where the H ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the J 8are positive roots of the transcendental equation H ( J C)= 0.+-, Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 3.1.2-6. Domain: 0 £ : £ C,0 £;£ 2 ,0 £ < D. Second boundary value problem. A semiin®nite circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),' L = !1(;, , ) at : = C(boundary condition),' 0 = !2( : ,;, ) at = 0 (boundary condition). Solution:( : ,;, , )= " 0 "2 E 0 " F 0 ( , , ) ( : ,;, , , , , ) $ $ $ + C " % 0 " 0 "2 E 0 !1( , , &) ( : ,;, , C, , , - &) $ $ $ & - " % 0 "2 E 0 " F 0 !2( , , &) ( : ,;, , , ,0, - &) $ $ $ &. Here,( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 C2+1 =0 8=1G J2 8H ( J 8 : ) H ( J 8) ( J2 8C2- 2)[ H ( J 8C)]2cos[ (;- )] exp( - J2 8 ), 2( , , )=1 2 exp -( - )2 4 +exp -( + )2 4 ,G = 1for = 0, 2for = 1,2, KKK, where the H ( ) are the Bessel functions and the J 8are positive roots of the transcendental equationH I ( J C)= 0.+-, Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 227 3.1.2-7. Domain: 0 £ : £ C,0 £;£ 2 ,0 £ < D. Third boundary value problem. A semiin®nite circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),' L + 11 = !(;, , ) at : = C(boundary condition),' 0 - 12 = !2( : ,;, ) at = 0 (boundary condition). The solution ( : ,;, , ) is determined by the formula in Paragraph 3.1.2-6 where( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 =0 8=1 G J2 8H ( J 8 : ) H ( J 8) ( J2 8C2+ 12 1 C2- 2)[ H ( J 8C)]2cos[ (;- )] exp( - J2 8 ), 2( , , )=1 2 exp -( - )2 4 +exp -( + )2 4 -2 12 " 0exp -( + + M)2 4 - 12 M $ M. Here,G0= 1andG = 2for = 1,2, KKK; the H ( ) are the Bessel functions and the J 8are positive roots of the transcendental equationJ H I ( J C)+ 11 H ( J C)= 0. 3.1.2-8. Domain: 0 £ : £ C,0 £;£ 2 ,0 £ < D. Mixed boundary value problems. 1 5. A semiin®nite circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),= !1(;, , ) at : = C(boundary condition),' 0 = !2( : ,;, ) at = 0 (boundary condition). Solution:( : ,;, , )= " 0 "2 E 0 " F 0 ( , , ) ( : ,;, , , , , ) $ $ $ - C " % 0 " 0 "2 E 0 !1( , , &) '' ( : ,;, , , , , - &) (=F $ $ $ & - " % 0 "2 E 0 " F 0 !2( , , &) ( : ,;, , , ,0, - &) $ $ $ &. Here,( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 C2 =0 8=1G [ H I ( J 8C)]2 H ( J 8 : ) H ( J 8) cos[ (;- )] exp( - J2 8 ), 2( , , )=1 2 exp -( - )2 4 +exp -( + )2 4 ,G = 1for = 0, 2for = 1,2, KKK, where the H ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the J 8are positive roots of the transcendental equation H ( J C)= 0. Page 228 2 5. A semiin®nite circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),' L = !1(;, , ) at : = C(boundary condition),= !2( : ,;, ) at = 0 (boundary condition). Solution:( : ,;, , )= " 0 "2 E 0 " F 0 ( , , ) ( : ,;, , , , , ) $ $ $ + C " % 0 " 0 "2 E 0 !1( , , &) ( : ,;, , C, , , - &) $ $ $ & + " % 0 "2 E 0 " F 0 !2( , , &) '' ( : ,;, , , , , - &) *=0 $ $ $ &. Here,( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 C2+1 =0 8=1G J2 8H ( J 8 : ) H ( J 8) ( J2 8C2- 2)[ H ( J 8C)]2cos[ (;- )] exp( - J2 8 ), 2( , , )=1 2 exp -( - )2 4 -exp -( + )2 4 ,G = 1for = 0, 2for = 1,2, KKK, where the H ( ) are the Bessel functions and the J 8are positive roots of the transcendental equationH I ( J C)= 0. 3.1.2-9. Domain: 0 £ : £ C,0 £;£ 2 ,0 £ £ . First boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),= !1(;, , ) at : = C(boundary condition),= !2( : ,;, ) at = 0 (boundary condition),= !3( : ,;, ) at = (boundary condition). Solution:( : ,;, , )= " # 0 "2 E 0 " F 0 ( , , ) ( : ,;, , , , , ) $ $ $ - C " % 0 "# 0 "2 E 0 !1( , , &) '' ( : ,;, , , , , - &) (=F $ $ $ & + " % 0 "2 E 0 " F 0 !2( , , &) '' ( : ,;, , , , , - &) *=0 $ $ $ & - " % 0 "2 E 0 " F 0 !3( , , &) '' ( : ,;, , , , , - &) *=# $ $ $ &. Here,( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 C2 =0 8=1G [ H I ( J 8C)]2 H ( J 8 : ) H ( J 8) cos[ (;- )] exp( - J2 8 ), 2( , , )=2 =1sin sin exp - 222,G = 1for = 0, 2for = 1,2, KKK, Page 229 where the H ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the J 8are positive roots of the transcendental equation H ( J C)= 0. 3.1.2-10. Domain: 0 £ : £ C,0 £;£ 2 ,0 £ £ . Second boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),' L = !1(;, , ) at : = C(boundary condition),' 0 = !2( : ,;, ) at = 0 (boundary condition),' 0 = !3( : ,;, ) at = (boundary condition). Solution:( : ,;, , )= "# 0 "2 E 0 " F 0 ( , , ) ( : ,;, , , , , ) $ $ $ + C " % 0 " # 0 "2 E 0 !1( , , &) ( : ,;, , C, , , - &) $ $ $ & - " % 0 "2 E 0 " F 0 !2( , , &) ( : ,;, , , ,0, - &) $ $ $ & + " % 0 "2 E 0 " F 0 !3( , , &) ( : ,;, , , , , - &) $ $ $ &. Here,( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 C2+1 =0 8=1 G J2 8H ( J 8 : ) H ( J 8) ( J2 8C2- 2)[ H ( J 8C)]2cos[ (;- )] exp( - J2 8 ), 2( , , )=1+2 =1cos cos exp - 222,G = 1for = 0, 2for = 1,2, KKK, where the H ( ) are the Bessel functions and the J 8are positive roots of the transcendental equationH I ( J C)= 0. 3.1.2-11. Domain: 0 £ : £ C,0 £;£ 2 ,0 £ £ . Third boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),' L + 11 = !(;, , ) at : = C(boundary condition),' 0 - 12 = !2( : ,;, ) at = 0 (boundary condition),' 0 + 13 = !3( : ,;, ) at = (boundary condition). The solution ( : ,;, , ) is determined by the formula in Paragraph 3.1.2-10 where( : ,;, , , , , )= 1( : ,;, , , ) ON =1 P N ( )P N ( )3P N32exp(- Q2 N), 1( : ,;, , , )=1 =0 8=1 G J2 8H ( J 8 : ) H ( J 8) ( J2 8C2+ 12 1 C2- 2)[ H ( J 8C)]2cos[ (;- )] exp( - J2 8 ),P N ( )=cos( Q N )+ 12Q N sin( Q N ), 3P N32= 13 2 Q2 NQ2 N + 12 2Q2 N + 12 3+ 12 2 Q2 N + 2 1 + 12 2Q2 N. Page 230 Here,G0= 1andG = 2for = 1,2, KKK; the H ( ) are the Bessel functions; and the J 8and Q N are positive roots of the transcendental equationsJ H I ( J C)+ 11 H ( J C)= 0,tan( Q )Q= 12+ 13Q2- 12 13. 3.1.2-12. Domain: 0 £ : £ C,0 £;£ 2 ,0 £ £ . Mixed boundary value problems. 1 5. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),= !1(;, , ) at : = C(boundary condition),' 0 = !2( : ,;, ) at = 0 (boundary condition),' 0 = !3( : ,;, ) at = (boundary condition). Solution:( : ,;, , )= "# 0 "2 E 0 " F 0 ( , , ) ( : ,;, , , , , ) $ $ $ - C " % 0 " # 0 "2 E 0 !1( , , &) '' ( : ,;, , , , , - &) (=F $ $ $ & - " % 0 "2 E 0 " F 0 !2( , , &) ( : ,;, , , ,0, - &) $ $ $ & + " % 0 "2 E 0 " F 0 !3( , , &) ( : ,;, , , , , - &) $ $ $ &. Here,( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 C2 =0 8=1G [ H I ( J 8C)]2 H ( J 8 : ) H ( J 8) cos[ (;- )] exp( - J2 8 ), 2( , , )=1+2 =1cos cos exp - 222,G = 1for = 0, 2for = 1,2, KKK, where the H ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the J 8are positive roots of the transcendental equation H ( J C)= 0. 2 5. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),' L = !1(;, , ) at : = C(boundary condition),= !2( : ,;, ) at = 0 (boundary condition),= !3( : ,;, ) at = (boundary condition). Solution:( : ,;, , )= " # 0 "2 E 0 " F 0 ( , , ) ( : ,;, , , , , ) $ $ $ + C " % 0 " # 0 "2 E 0 !1( , , &) ( : ,;, , C, , , - &) $ $ $ & + " % 0 "2 E 0 " F 0 !2( , , &) '' ( : ,;, , , , , - &) *=0 $ $ $ & - " % 0 "2 E 0 " F 0 !3( , , &) '' ( : ,;, , , , , - &) *=# $ $ $ &. Page 231 Here,( : ,;, , , , , )= 1( : ,;, , , ) 2( , , ), 1( : ,;, , , )=1 C2+1 =0 8=1 G J2 8H ( J 8 : ) H ( J 8) ( J2 8C2- 2)[ H ( J 8C)]2cos[ (;- )] exp( - J2 8 ), 2( , , )=2 =1sin sin exp - 222,G = 1for = 0, 2for = 1,2, KKK, where the H ( ) are the Bessel functions and the J 8are positive roots of the transcendental equationH I ( J C)= 0. 3.1.2-13. Domain: C1£ : £ C2,0 £;£ 2 ,- D< < D. First boundary value problem. An in®nite hollow circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),= !1(;, , ) at : = C1(boundary condition),= !2(;, , ) at : = C2(boundary condition). Solution:( : ,;, , )= " - "2 E 0 " F2F1 ( , , ) ( : ,;, , , , , ) $ $ $ + C1 " % 0 " - "2 E 0 !1( , , &) '' ( : ,;, , , , , - &) (=F1 $ $ $ & - C2 " % 0 " - "2 E 0 !2( , , &) '' ( : ,;, , , , , - &) (=F2 $ $ $ &. Here,( : ,;, , , , , )=1 2 exp -( - )2 4 1( : ,;, , , ), 1( : ,;, , , )= 2 =0 8=1G R 8 S ( J 8 : ) S ( J 8) cos[ (;- )] exp( - J2 8 ),G = 1 T2for = 0, 1 for ¹ 0, R 8= J2 8H2 ( J 8C2)H2 ( J 8C1)- H2 ( J 8C2),S ( J 8 : )= H ( J 8C1) U ( J 8 : )- U ( J 8C1) H ( J 8 : ), where the H ( : ) and U ( : ) are the Bessel functions and the J 8are positive roots of the transcen- dental equationH ( J C1) U ( J C2)- U ( J C1) H ( J C2)= 0. 3.1.2-14. Domain: C1£ : £ C2,0 £;£ 2 ,- D< < D. Second boundary value problem. An in®nite hollow circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),' L = !1(;, , ) at : = C1(boundary condition),' L = !2(;, , ) at : = C2(boundary condition). Page 232 Solution:( : ,;, , )= " - "2 E 0 " F2F1 ( , , ) ( : ,;, , , , , ) $ $ $ - C1 " % 0 " - "2 E 0 !1( , , &) ( : ,;, , C1, , , , - &) $ $ $ & + C2 " % 0 " - "2 E 0 !2( , , &) ( : ,;, , C2, , , - &) $ $ $ &. Here,( : ,;, , , , , )=1 2 exp -( - )2 4 1( : ,;, , , ), 1( : ,;, , , )=1( C2 2- C2 1)+1 =0 8=1 G J2 8 S ( J 8 : ) S ( J 8) cos[ (;- )] exp( - J2 8 ) ( J2 8C2 2- 2) S2 ( J 8C2)-( J2 8C2 1- 2) S2 ( J 8C1),S ( J 8 : )= H I ( J 8C1) U ( J 8 : )- U I ( J 8C1) H ( J 8 : ), whereG0= 1andG = 2for = 1,2, KKK; the H ( : ) and U ( : ) are the Bessel functions (the prime denotes the derivative with respect to the argument); and the J 8are positive roots of the transcendental equationH I ( J C1) U I ( J C2)- U I ( J C1) H I ( J C2)= 0. 3.1.2-15. Domain: C1£ : £ C2,0 £;£ 2 ,- D< < D. Third boundary value problem. An in®nite hollow circular cylinder is considered. The following conditions are prescribed:= ( : ,;, ) at = 0 (initial condition),' L - 11 = !1(;, , ) at : = C1(boundary condition),' L + 12 = !2(;, , ) at : = C2(boundary condition). The solution ( : ,;, , ) is given by relations in Paragraph 3.1.2-14 in which( : ,;, , , , , )=1 2 exp -( - )2 4 1( : ,;, , , ), 1( : ,;, , , )=1 =0 8=1 G J2 8 S ( J 8 : ) S ( J 8) cos[ (;- )] exp( - J2 8 ) ( 12 2 C2 2+ J2 8C2 2- 2) S2 ( J 8C2)-( 12 1 C2 1+ J2 8C2 1- 2) S2 ( J 8C1),S ( J 8 : )= VWJ 8H I ( J 8C1)- 11 H ( J 8C1) XYU ( J 8 : ) - VZJ 8U I ( J 8C1)- 11 U ( J 8C1) X H ( J 8 : ). Here,G0= 1andG = 2for = 1,2, KKK; H ( : ) and U ( : ) are the Bessel functions; and the J 8 are positive roots of the transcendental equationVZJ H I ( J C1)- 11 H ( J C1) X VWJ U I ( J C2)+ 12 U ( J C2) X = VZJ U I ( J C1)- 11 U ( J C1) X VZJ H I ( J C2)+ 12 H ( J C2) X. Page 233 3.1.2-16. Domain: C1£ : £ C2,0 £;£ 2 ,0 £ [< D. First boundary value problem. A semiin®nite hollow circular cylinder is considered. The following conditions are prescribed:= ( : ,;, [) at \= 0 (initial condition),= !1(;, [, \) at : = C1(boundary condition),= !2(;, [, \) at : = C2(boundary condition),= !3( : ,;, \) at [= 0 (boundary condition). Solution:( : ,;, [, \)= " 0 "2 E 0 " F2F1 ( ], ^, _) `( : ,;, [, ], ^, _, \) ] $ ] $ ^ $ _ + a C1 " % 0 " 0 "2 E 0 !1( ^, _, &) b '' ] `( : ,;, [, ], ^, _, \- &) c(=F1 $ ^ $ _ $ & - a C2 " % 0 " d 0 "2 E 0 !2( ^, _, &) b '' ] `( : ,;, [, ], ^, _, \- &) c(=F2 $ ^ $ _ $ & + a " % 0 "2 E 0 " F2F1 !3( ], ^, &) b '' _ `( : ,;, [, ], ^, _, \- &) c*=0 ] $ ] $ ^ $ &. Here,`( : ,;, [, ], ^, _, \)=1 2 e f a \ gexp b-( [- _)2 4 a \ c-exp b-( [+ _)2 4 a \ c h`1( i, j, ], ^, \),`1( i, j, ], ^, \)= f 2 d kl =0 d km=1 n l o lm p l ( q lmi) p l ( q lm]) cos[ r( j- ^)] exp( - q2 lm s t),n l =g1 u2for r= 0, 1 for r¹ 0, o lm= q2 lm v2 l ( q lm w2)v2 l ( q lm w1)- v2 l ( q lm w2),p l ( q lmi)= v l ( q lm w1) U l ( q lmi)- U l ( q lm w1) v l ( q lmi), where the v l ( i) and U l ( i) are the Bessel functions and the q lmare positive roots of the transcen- dental equationv l ( q w1) U l ( q w2)- U l ( q w1) v l ( q w2)= 0. 3.1.2-17. Domain: w1£ i£ w2,0 £ j£ 2 x,0 £ y< z. Second boundary value problem. A semiin®nite hollow circular cylinder is considered. The following conditions are prescribed:{= |( i, j, y) at t= 0 (initial condition),} ~{= 1( j, y, t) at i= w1(boundary condition),} ~{= 2( j, y, t) at i= w2(boundary condition),} {= 3( i, j, t) at y= 0 (boundary condition). Solution:{( i, j, y, t)= 0 2 0 21 |( ], ^, _) ( i, j, y, ], ^, _, t) ] ] ^ _ - s w1 % 0 0 2 0 1( ^, _, ) ( i, j, y, w1, ^, _, t- ) ^ _ + s w2 % 0 0 2 0 2( ^, _, ) ( i, j, y, w2, ^, _, t- ) ^ _ - s % 0 2 0 21 3( ], ^, ) ( i, j, y, ], ^,0, t- ) ] ] ^ . Page 234 Here,( i, j, y, ], ^, _, t)=1 2 x s t exp -( y- _)2 4 s t +exp -( y+ _)2 4 s t h 1( i, j, ], ^, t),1( i, j, ], ^, t)=1x( w2 2- w2 1)+1x kl =0 km=1 n lq2 lm p l ( q lmi) p l ( q lm]) cos[ r( j- ^)] exp( - q2 lm s t) ( q2 lm w2 2- r2) p2 l ( q lm w2)-( q2 lm w2 1- r2) p2 l ( q lm w1),p l ( q lmi)= v l ( q lm w1) U l ( q lmi)- U l ( q lm w1) v l ( q lmi), wheren0= 1andn l = 2for r= 1,2, ; the v l ( i) and U l ( i) are the Bessel functions; and theq lmare positive roots of the transcendental equationv l ( q w1) U l ( q w2)- U l ( q w1) v l ( q w2)= 0. 3.1.2-18. Domain: w1£ i£ w2,0 £ j£ 2 x,0 £ y< z. Third boundary value problem. A semiin®nite hollow circular cylinder is considered. The following conditions are prescribed:{= |( i, j, y) at t= 0 (initial condition),} ~{- 1 {= 1( j, y, t) at i= w1(boundary condition),} ~{+ 2 {= 2( j, y, t) at i= w2(boundary condition),} {- 3 {= 3( i, j, t) at y= 0 (boundary condition). The solution {( i, j, y, t) is determined by the formula in Paragraph 3.1.2-17 where( i, j, y, ], ^, _, t)= 1( y, _, t) 2( i, j, ], ^, t),1( y, _, t)=1 2 x s t exp -( y- _)2 4 s t +exp -( y+ _)2 4 s t -2 3 0exp -( y+ _+ )2 4 s t- 3 h,2( i, j, ], ^, t)=1x kl =0 km=1 n lq2 lm p l ( q lmi) p l ( q lm]) cos[ r( j- ^)] exp( - q2 lm s t) ( 2 2 w2 2+ q2 lm w2 2- r2) p2 l ( q lm w2)-( 2 1 w2 1+ q2 lm w2 1- r2) p2 l ( q lm w1),p l ( q lmi)= q lmv l ( q lm w1)- 1 v l ( q lm w1) YU l ( q lmi) - Wq lmU l ( q lm w1)- 1 U l ( q lm w1) v l ( q lmi). Here,n0= 1andn l = 2for r= 1,2, ; the v l ( i) and U l ( i) are the Bessel functions; and theq lmare positive roots of the transcendental equationZq v l ( q w1)- 1 v l ( q w1) Wq U l ( q w2)+ 2 U l ( q w2) = Zq U l ( q w1)- 1 U l ( q w1) Zq v l ( q w2)+ 2 v l ( q w2) . 3.1.2-19. Domain: w1£ i£ w2,0 £ j£ 2 x,0 £ y< z. Mixed boundary value problems. 1 . A semiin®nite hollow circular cylinder is considered. The following conditions are prescribed:{= |( i, j, y) at t= 0 (initial condition),{= 1( j, y, t) at i= w1(boundary condition),{= 2( j, y, t) at i= w2(boundary condition),} {= 3( i, j, t) at y= 0 (boundary condition). Page 235 Solution:{( i, j, y, t)= 0 2 0 21 |( ], ^, _) ( i, j, y, ], ^, _, t) ] ] ^ _ + s w1 % 0 0 2 0 1( ^, _, ) }}] ( i, j, y, ], ^, _, t- ) =1 ^ _ - s w2 % 0 0 2 0 2( ^, _, ) }}] ( i, j, y, ], ^, _, t- ) =2 ^ _ - s % 0 2 0 21 3( ], ^, ) ( i, j, y, ], ^,0, t- ) ] ] ^ . Here,( i, j, y, ], ^, _, t)=1 2 x s t exp -( y- _)2 4 s t +exp -( y+ _)2 4 s t h 1( i, j, ], ^, t),1( i, j, ], ^, t)= x 2 kl =0 km=1 n l o lm p l ( q lmi) p l ( q lm]) cos[ r( j- ^)] exp( - q2 lm s t),n l =1 u2for r= 0, 1 for r¹ 0, o lm= q2 lm v2 l ( q lm w2)v2 l ( q lm w1)- v2 l ( q lm w2),p l ( q lmi)= v l ( q lm w1) U l ( q lmi)- U l ( q lm w1) v l ( q lmi), where the v l ( i) and U l ( i) are the Bessel functions and the q lmare positive roots of the transcen- dental equationv l ( q w1) U l ( q w2)- U l ( q w1) v l ( q w2)= 0. 2 . A semiin®nite hollow circular cylinder is considered. The following conditions are prescribed:{= |( i, j, y) at t= 0 (initial condition),} ~{= 1( j, y, t) at i= w1(boundary condition),} ~{= 2( j, y, t) at i= w2(boundary condition),{= 3( i, j, t) at y= 0 (boundary condition). Solution:{( i, j, y, t)= 0 2 0 21 |( ], ^, _) ( i, j, y, ], ^, _, t) ] ] ^ _ - s w1 % 0 0 2 0 1( ^, _, ) ( i, j, y, w1, ^, _, t- ) ^ _ + s w2 % 0 0 2 0 2( ^, _, ) ( i, j, y, w2, ^, _, t- ) ^ _ + s % 0 2 0 21 3( ], ^, ) }}_ ( i, j, y, ], ^, _, t- ) =0 ] ] ^ . Here,( i, j, y, ], ^, _, t)=1 2 x s t exp -( y- _)2 4 s t -exp -( y+ _)2 4 s t h 1( i, j, ], ^, t),1( i, j, ], ^, t)=1x( w2 2- w2 1)+1x kl =0 km=1 n lq2 lm p l ( q lmi) p l ( q lm]) cos[ r( j- ^)] exp( - q2 lm s t) ( q2 lm w2 2- r2) p2 l ( q lm w2)-( q2 lm w2 1- r2) p2 l ( q lm w1),p l ( q lmi)= v l ( q lm w1) U l ( q lmi)- U l ( q lm w1) v l ( q lmi), Page 236 wheren0= 1andn l = 2for r= 1,2, ; the v l ( i) and U l ( i) are the Bessel functions (the prime denotes the derivative with respect to the argument); and the q lmare positive roots of the transcendental equationv l ( q w1) U l ( q w2)- U l ( q w1) v l ( q w2)= 0. 3.1.2-20. Domain: w1£ i£ w2,0 £ j£ 2 x,0 £ y£ . First boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:{= |( i, j, y) at t= 0 (initial condition),{= 1( j, y, t) at i= w1(boundary condition),{= 2( j, y, t) at i= w2(boundary condition),{= 3( i, j, t) at y= 0 (boundary condition),{= 4( i, j, t) at y= (boundary condition). Solution:{( i, j, y, t)= 0 2 0 21 |( ], ^, _) ( i, j, y, ], ^, _, t) ] ] ^ _ + s w1 % 0 0 2 0 1( ^, _, ) }}] ( i, j, y, ], ^, _, t- ) =1 ^ _ - s w2 % 0 0 2 0 2( ^, _, ) }}] ( i, j, y, ], ^, _, t- ) =2 ^ _ + s % 0 2 0 21 3( ], ^, ) }}_ ( i, j, y, ], ^, _, t- ) =0 ] ] ^ - s % 0 2 0 21 4( ], ^, ) }}_ ( i, j, y, ], ^, _, t- ) = ] ] ^ . Here,( i, j, y, ], ^, _, t)= 1( i, j, ], ^, t) 2 kl =1sin r x y sin r x _ exp - sr2x2t2 ,1( , ¡, ], ^, t)= x 2 kl =0 km=1 n l o lm p l ( q lm ) p l ( q lm]) cos[ r( ¡- ^)] exp( - q2 lm s t),n l =1 u2for r= 0, 1 for r¹ 0, o lm= q2 lm v2 l ( q lm w2)v2 l ( q lm w1)- v2 l ( q lm w2),p l ( q lm )= v l ( q lm w1) U l ( q lm )- U l ( q lm w1) v l ( q lm ), where the v l ( ) and U l ( ) are the Bessel functions and the q lmare positive roots of the transcen- dental equationv l ( q w1) U l ( q w2)- U l ( q w1) v l ( q w2)= 0. 3.1.2-21. Domain: w1£ £ w2,0 £ ¡£ 2 x,0 £ y£ . Second boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:{= |( , ¡, y) at t= 0 (initial condition),} ~{= 1( ¡, y, t) at = w1(boundary condition),} ~{= 2( ¡, y, t) at = w2(boundary condition),} {= 3( , ¡, t) at y= 0 (boundary condition),} {= 4( , ¡, t) at y= (boundary condition). Page 237 Solution:{( , ¡, y, t)= 0 2 0 21 |( ], ^, _) ( , ¡, y, ], ^, _, t) ] ] ^ _ - s w1 % 0 0 2 0 1( ^, _, ) ( , ¡, y, w1, ^, _, t- ) ^ _ + s w2 % 0 0 2 0 2( ^, _, ) ( , ¡, y, w2, ^, _, t- ) ^ _ - s % 0 2 0 21 3( ], ^, ) ( , ¡, y, ], ^,0, t- ) ] ] ^ + s % 0 2 0 21 4( ], ^, ) ( , ¡, y, ], ^, , t- ) ] ] ^ . Here,( , ¡, y, ], ^, _, t)= 1( , ¡, ], ^, t) 1+2 kl =1cos r x y cos r x _ exp - sr2x2t2 ,1( , ¡, ], ^, t)=1x( w2 2- w2 1)+1x kl =0 km=1 n lq2 lm p l ( q lm ) p l ( q lm]) cos[ r( ¡- ^)] exp( - q2 lm s t) ( q2 lm w2 2- r2) p2 l ( q lm w2)-( q2 lm w2 1- r2) p2 l ( q lm w1),p l ( q lm )= v l ( q lm w1) U l ( q lm )- U l ( q lm w1) v l ( q lm ), wheren0= 1andn l = 2for r= 1,2, ; the v l ( ) and U l ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument); and the q lmare positive roots of the transcendental equationv l ( q w1) U l ( q w2)- U l ( q w1) v l ( q w2)= 0. 3.1.2-22. Domain: w1£ £ w2,0 £ ¡£ 2 x,0 £ y£ . Third boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:{= |( , ¡, y) at t= 0 (initial condition),} ~{- 1 {= 1( ¡, y, t) at = w1(boundary condition),} ~{+ 2 {= 2( ¡, y, t) at = w2(boundary condition),} {- 3 {= 3( , ¡, t) at y= 0 (boundary condition),} {+ 4 {= 4( , ¡, t) at y= (boundary condition). The solution {( , ¡, y, t) is determined by the formula in Paragraph 3.1.2-21 where( , ¡, y, ], ^, _, t)= 1( , ¡, ], ^, t) 2( y, _, t). Here, the ®rst factor has the form1( , ¡, ], ^, t)=1x kl =0 km=1 n lq2 lm p l ( q lm ) p l ( q lm]) cos[ r( ¡- ^)] exp( - q2 lm s t) ( 2 2 w2 2+ q2 lm w2 2- r2) p2 l ( q lm w2)-( 2 1 w2 1+ q2 lm w2 1- r2) p2 l ( q lm w1),p l ( q lm )= Wq lmv l ( q lm w1)- 1 v l ( q lm w1) YU l ( q lm ) - Zq lmU l ( q lm w1)- 1 U l ( q lm w1) v l ( q lm ), Page 238 wheren0= 1andn l = 2for r= 1,2, ; the v l ( ) and U l ( ) are the Bessel functions; and theq lmare positive roots of the transcendental equationZq v l ( q w1)- 1 v l ( q w1) Wq U l ( q w2)+ 2 U l ( q w2) = Zq U l ( q w1)- 1 U l ( q w1) Zq v l ( q w2)+ 2 v l ( q w2) . The second factor is given by2( y, _, t)= kO¢ =1 £ ¢ ( y)£ ¢ ( _)¤£ ¢¤2exp(- s ¥2 ¢t),£ ¢ ( y)=cos( ¥ ¢y)+ 3¥ ¢ sin( ¥ ¢y), ¤£ ¢¤2= 4 2 ¥2 ¢¥2 ¢ + 2 3¥2 ¢ + 2 4+ 3 2 ¥2 ¢ + 2 1 + 2 3¥2 ¢, where the ¥ ¢ are positive roots of the transcendental equationtan( ¥)¥= 3+ 4¥2- 3 4. 3.1.2-23. Domain: w1£ £ w2,0 £ ¡£ 2 x,0 £ y£ . Mixed boundary value problems. 1 . A circular cylinder of ®nite length is considered. The following conditions are prescribed:{= |( , ¡, y) at t= 0 (initial condition),{= 1( ¡, y, t) at = w1(boundary condition),{= 2( ¡, y, t) at = w2(boundary condition),} {= 3( , ¡, t) at y= 0 (boundary condition),} {= 4( , ¡, t) at y= (boundary condition). Solution:{( , ¡, y, t)= 0 2 0 21 |( ¦, §, ¨) ( , ¡, y, ¦, §, ¨, t) ¦ ¦ § ¨ + s w1 % 0 0 2 0 1( §, ¨, ) }}¦ ( , ¡, y, ¦, §, ¨, t- ) =1 § ¨ - s w2 % 0 0 2 0 2( §, ¨, ) }}¦ ( , ¡, y, ¦, §, ¨, t- ) =2 § ¨ - s % 0 2 0 21 3( ¦, §, ) ( , ¡, y, ¦, §,0, t- ) ¦ ¦ § + s % 0 2 0 21 4( ¦, §, ) ( , ¡, y, ¦, §, , t- ) ¦ ¦ § . Here,( , ¡, y, ¦, §, ¨, t)= 1( , ¡, ¦, §, t) 1+2 kl =1cos r x y cos r x ¨ exp - sr2x2t2 ,1( , ¡, ¦, §, t)= x 2 kl =0 km=1 n l o lm p l ( q lm ) p l ( q lm¦) cos[ r( ¡- §)] exp( - q2 lm s t),n l =1 u2for r= 0, 1 for r¹ 0, o lm= q2 lm v2 l ( q lm w2)v2 l ( q lm w1)- v2 l ( q lm w2),p l ( q lm )= v l ( q lm w1) U l ( q lm )- U l ( q lm w1) v l ( q lm ), Page 239 where the v l ( ) and U l ( ) are the Bessel functions and the q lmare positive roots of the transcen- dental equationv l ( q w1) U l ( q w2)- U l ( q w1) v l ( q w2)= 0. 2 . A circular cylinder of ®nite length is considered. The following conditions are prescribed:{= |( , ¡, y) at t= 0 (initial condition),} ~{= 1( ¡, y, t) at = w1(boundary condition),} ~{= 2( ¡, y, t) at = w2(boundary condition),{= 3( , ¡, t) at y= 0 (boundary condition),{= 4( , ¡, t) at y= (boundary condition). Solution:{( , ¡, y, t)= 0 2 0 21 |( ¦, §, ¨) ( , ¡, y, ¦, §, ¨, t) ¦ ¦ § ¨ - s w1 % 0 0 2 0 1( §, ¨, ) ( , ¡, y, w1, §, ¨, t- ) § ¨ + s w2 % 0 0 2 0 2( §, ¨, ) ( , ¡, y, w2, §, ¨, t- ) § ¨ + s % 0 2 0 21 3( ¦, §, ) }}¨ ( , ¡, y, ¦, §, ¨, t- ) =0 ¦ ¦ § - s % 0 2 0 21 4( ¦, §, ) }}¨ ( , ¡, y, ¦, §, ¨, t- ) = ¦ ¦ § . Here,( , ¡, y, ¦, §, ¨, t)= 1( , ¡, ¦, §, t) 2 kl =1sin r x y sin r x ¨ exp - sr2x2t2 ,1( , ¡, ¦, §, t)=1x( w2 2- w2 1)+1x kl =0 km=1 n lq2 lm p l ( q lm ) p l ( q lm¦) cos[ r( ¡- §)] exp( - q2 lm s t) ( q2 lm w2 2- r2) p2 l ( q lm w2)-( q2 lm w2 1- r2) p2 l ( q lm w1),p l ( q lm )= v l ( q lm w1) U l ( q lm )- U l ( q lm w1) v l ( q lm ), wheren0= 1andn l = 2for r= 1,2, ; the v l ( ) and U l ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument); and the q lmare positive roots of the transcendental equationv l ( q w1) U l ( q w2)- U l ( q w1) v l ( q w2)= 0. 3.1.2-24. Domain: 0 £ < z,0 £ ¡£ ¡0,- z< y< z. First boundary value problem. A dihedral angle is considered. The following conditions are prescribed:{= |( , ¡, y) at t= 0 (initial condition),{= 1( , y, t) at ¡= 0 (boundary condition),{= 2( , y, t) at ¡= ¡0(boundary condition). Solution:{( , ¡, y, t)= - ©0 0 0 |( ¦, §, ¨) ( , ¡, y, ¦, §, ¨, t) ¦ ¦ § ¨ + s % 0 - 0 1( ¦, ¨, )1¦ }}§ ( , ¡, y, ¦, §, ¨, t- ) ª=0 ¦ ¨ - s % 0 - 0 2( ¦, ¨, )1¦ }}§ ( , ¡, y, ¦, §, ¨, t- ) ª=©0 ¦ ¨ . Page 240 Here,( , , , , , , )=1 2 exp -( - )2 4 1( , , , , ), 1( , , , , )=1 0 exp - 2+ 2 4 =1 0 2 sin 0 sin 0 , where the! ( ) are the modi®ed Bessel functions. 3.1.2-25. Domain: 0 £ < ",0 £ £ 0,- "< < ". Second boundary value problem. A dihedral angle is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),-1 % #= &1( , , ) at = 0 (boundary condition),-1 % #= &2( , , ) at = 0(boundary condition). Solution:#( , , , )= '- ' 0 0 '0 $( , , ) ( , , , , , , ) ( ( ( - ' ) 0 '- '0 &1( , , *) ( , , , ,0, , - *) ( ( ( * + ' ) 0 '- '0 &2( , , *) ( , , , , 0, , - *) ( ( ( *. Here,( , , , , , , )=1 2 exp -( - )2 4 1( , , , , ), 1( , , , , )=1 0 exp - 2+ 2 4 1 20 2 + =1 0 2 cos 0 cos 0 , where the! ( ) are the modi®ed Bessel functions. 3.1.2-26. Domain: 0 £ < ",0 £ £ 0,0 £ < ". First boundary value problem. The upper half of a dihedral angle is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),#= &1( , , ) at = 0 (boundary condition),#= &2( , , ) at = 0(boundary condition),#= &3( , , ) at = 0 (boundary condition). Solution:#( , , , )= '0 ' 0 0 '0 $( , , ) ( , , , , , , ) ( ( ( + ') 0 '0 '0 &1( , , *)1 %% ( , , , , , , - *) +=0 ( ( ( * - ') 0 '0 '0 &2( , , *)1 %% ( , , , , , , - *) += 0 ( ( ( * + ' ) 0 ' 0 0 '0 &3( , , *) %% ( , , , , , , - *) ,=0 ( ( ( *. Page 241 Here,( , , , , , , )=1 2 -exp -( - )2 4 -exp -( + )2 4 . 1( , , , , ), 1( , , , , )=1 0 exp - 2+ 2 4 =1 0 2 sin 0 sin 0 , where the! ( ) are the modi®ed Bessel functions. 3.1.2-27. Domain: 0 £ < ",0 £ £ 0,0 £ < ". Second boundary value problem. The upper half of a dihedral angle is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),-1 % #= &1( , , ) at = 0 (boundary condition),-1 % #= &2( , , ) at = 0(boundary condition),% / #= &3( , , ) at = 0 (boundary condition). Solution:#( , , , )= '0 ' 0 0 '0 $( , , ) ( , , , , , , ) ( ( ( - ' ) 0 '0 '0 &1( , , *) ( , , , ,0, , - *) ( ( ( * + ') 0 '0 '0 &2( , , *) ( , , , , 0, , - *) ( ( ( * - ') 0 ' 0 0 '0 &3( , , *) ( , , , , ,0, - *) ( ( ( *. Here,( , , , , , , )=1 2 -exp -( - )2 4 +exp -( + )2 4 . 1( , , , , ), 1( , , , , )=1 0 exp - 2+ 2 4 1 20 2 + =1 0 2 cos 0 cos 0 , where the! ( ) are the modi®ed Bessel functions. 3.1.2-28. Domain: 0 £ < ",0 £ £ 0,0 £ < ". Mixed boundary value problems. 1 0. The upper half of a dihedral angle is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),#= &1( , , ) at = 0 (boundary condition),#= &2( , , ) at = 0(boundary condition),% / #= &3( , , ) at = 0 (boundary condition). Page 242 Solution:#( , , , )= '0 ' 0 0 '0 $( , , ) ( , , , , , , ) ( ( ( + ') 0 '0 '0 &1( , , *)1 %% ( , , , , , , - *) +=0 ( ( ( * - ') 0 '0 '0 &2( , , *)1 %% ( , , , , , , - *) += 0 ( ( ( * - ' ) 0 ' 0 0 '0 &3( , , *) ( , , , , ,0, - *) ( ( ( *. Here,( , , , , , , )=1 2 -exp -( - )2 4 +exp -( + )2 4 . 1( , , , , ), 1( , , , , )=1 0 exp - 2+ 2 4 =1 0 2 sin 0 sin 0 , where the! ( ) are the modi®ed Bessel functions. 2 0. The upper half of a dihedral angle is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),-1% #= &1( , , ) at = 0 (boundary condition),-1 % #= &2( , , ) at = 0(boundary condition),#= &3( , , ) at = 0 (boundary condition). Solution:#( , , , )= '0 ' 0 0 '0 $( , , ) ( , , , , , , ) ( ( ( - ' ) 0 '0 '0 &1( , , *) ( , , , ,0, , - *) ( ( ( * + ' ) 0 '0 '0 &2( , , *) ( , , , , 0, , - *) ( ( ( * + ') 0 ' 0 0 '0 &3( , , *) %% ( , , , , , , - *) ,=0 ( ( ( *. Here,( , , , , , , )=1 2 -exp -( - )2 4 -exp -( + )2 4 . 1( , , , , ), 1( , , , , )=1 0 exp - 2+ 2 4 1 20 2 + =1 0 2 cos 0 cos 0 , where the! ( ) are the modi®ed Bessel functions. Page 243 3.1.2-29. Domain: 0 £ < ",0 £ £ 0,0 £ £ 1. First boundary value problem. A wedge domain of ®nite thickness is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),#= &1( , , ) at = 0 (boundary condition),#= &2( , , ) at = 0(boundary condition),#= &3( , , ) at = 0 (boundary condition),#= &4( , , ) at = 1 (boundary condition). Solution:#( , , , )= ' 2 0 ' 0 0 '0 $( , , ) ( , , , , , , ) ( ( ( + ' ) 0 ' 2 0 '0 &1( , , *)1 %% ( , , , , , , - *) +=0 ( ( ( * - ') 0 ' 2 0 '0 &2( , , *)1 %% ( , , , , , , - *)+= 0 ( ( ( * + ') 0 ' 0 0 '0 &3( , , *) %% ( , , , , , , - *) ,=0 ( ( ( * - ') 0 ' 0 0 '0 &4( , , *) %% ( , , , , , , - *) ,=2 ( ( ( *. Here,( , , , , , , )= 1( , , , , ) 21 =1sin 1 sin 1 exp - 2212 , 1( , , , , )=1 0 exp - 2+ 2 4 =1 0 2 sin 0 sin 0 , where the! ( ) are the modi®ed Bessel functions. 3.1.2-30. Domain: 0 £ < ",0 £ £ 0,0 £ £ 1. Second boundary value problem. A wedge domain of ®nite thickness is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),-1% #= &1( , , ) at = 0 (boundary condition),-1 % #= &2( , , ) at = 0(boundary condition),% / #= &3( , , ) at = 0 (boundary condition),% / #= &4( , , ) at = 1 (boundary condition). Page 244 Solution:#( , , , )= ' 2 0 ' 0 0 '0 $( , , ) ( , , , , , , ) ( ( ( - ') 0 ' 2 0 '0 &1( , , *) ( , , , ,0, , - *) ( ( ( * + ') 0 ' 2 0 '0 &2( , , *) ( , , , , 0, , - *) ( ( ( * - ') 0 ' 0 0 '0 &3( , , *) ( , , , , ,0, - *) ( ( ( * + ') 0 ' 0 0 '0 &4( , , *) ( , , , , , 1, - *) ( ( ( *. Here,( , , , , , , )= 1( , , , , ) 2( , , ), 1( , , , , )=1 0 exp - 2+ 2 4 1 20 2 + =1 0 2 cos 0 cos 0 , 2( , , )=11+21 =1cos 1 cos 1 exp - 2212, where the! ( ) are the modi®ed Bessel functions. 3.1.2-31. Domain: 0 £ < ",0 £ £ 0,0 £ £ 1. Mixed boundary value problems. 1 0. A wedge domain of ®nite thickness is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),#= &1( , , ) at = 0 (boundary condition),#= &2( , , ) at = 0(boundary condition),% / #= &3( , , ) at = 0 (boundary condition),% / #= &4( , , ) at = 1 (boundary condition). Solution:#( , , , )= ' 2 0 ' 0 0 '0 $( , , ) ( , , , , , , ) ( ( ( + ' ) 0 ' 2 0 '0 &1( , , *)1 %% ( , , , , , , - *) +=0 ( ( ( * - ') 0 ' 2 0 '0 &2( , , *)1 %% ( , , , , , , - *) += 0 ( ( ( * - ') 0 ' 0 0 '0 &3( , , *) ( , , , , ,0, - *) ( ( ( * + ' ) 0 ' 0 0 '0 &4( , , *) ( , , , , , 1, - *) ( ( ( *. Here,( , , , , , , )= 1( , , , , ) 2( , , ), Page 245 1( , , , , )=1 0 exp - 2+ 2 4 =1 0 2 sin 0 sin 0 , 2( , , )=11+21 =1cos 1 cos 1 exp - 2212, where the! ( ) are the modi®ed Bessel functions. 2 0. A wedge domain of ®nite thickness is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),-1% #= &1( , , ) at = 0 (boundary condition),-1 % #= &2( , , ) at = 0(boundary condition),#= &3( , , ) at = 0 (boundary condition),#= &4( , , ) at = 1 (boundary condition). Solution:#( , , , )= ' 2 0 ' 0 0 '0 $( , , ) ( , , , , , , ) ( ( ( - ') 0 ' 2 0 '0 &1( , , *) ( , , , ,0, , - *) ( ( ( * + ' ) 0 ' 2 0 '0 &2( , , *) ( , , , , 0, , - *) ( ( ( * + ' ) 0 ' 0 0 '0 &3( , , *) %% ( , , , , , , - *) ,=0 ( ( ( * - ' ) 0 ' 0 0 '0 &4( , , *) %% ( , , , , , , - *),=2 ( ( ( *. Here,( , , , , , , )= 1( , , , , ) 21 =1sin 1 sin 1 exp - 2212 , 1( , , , , )=1 0 exp - 2+ 2 4 1 20 2 + =1 0 2 cos 0 cos 0 , where the! ( ) are the modi®ed Bessel functions. 3.1.2-32. Domain: 0 £ £ 3,0 £ £ 0,- "< < ". First boundary value problem. An in®nite cylindrical sector is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),#= &1( , , ) at = 3 (boundary condition),#= &2( , , ) at = 0 (boundary condition),#= &3( , , ) at = 0(boundary condition). Page 246 Solution:#( , , , )= '- ' 0 0 ' 4 0 $( , , ) ( , , , , , , ) ( ( ( - 3 ') 0 '- ' 0 0 &1( , , *) %% ( , , , , , , - *) 5=4 ( ( ( * + ' ) 0 '- ' 4 0 &2( , , *)1 %% ( , , , , , , - *) +=0 ( ( ( * - ' ) 0 '- ' 4 0 &3( , , *)1 %% ( , , , , , , - *) += 0 ( ( ( *. Here,( , , , , , , )=1 2 exp -( - )2 4 1( , , , , ), 1( , , , , )=432 0 =1 6=1 7 0( 8 6)7 0( 8 6) [7 9 0( 8 63)]2sin 0 sin 0 exp(- 82 6 ), where the7 0( ) are the Bessel functions and the 8 6are positive roots of the transcendental equation7 0( 8 3)= 0. 3.1.2-33. Domain: 0 £ £ 3,0 £ £ 0,0 £ < ". First boundary value problem. A semiin®nite cylindrical sector is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),#= &1( , , ) at = 3 (boundary condition),#= &2( , , ) at = 0 (boundary condition),#= &3( , , ) at = 0(boundary condition),#= &4( , , ) at = 0 (boundary condition). Solution:#( , , , )= '0 ' 0 0 ' 4 0 $( , , ) ( , , , , , , ) ( ( ( - 3 ') 0 '0 ' 0 0 &1( , , *) %% ( , , , , , , - *) 5=4 ( ( ( * + ' ) 0 '0 ' 4 0 &2( , , *)1 %% ( , , , , , , - *) +=0 ( ( ( * - ' ) 0 '0 ' 4 0 &3( , , *)1 %% ( , , , , , , - *) += 0 ( ( ( * + ') 0 ' 0 0 ' 4 0 &4( , , *) %% ( , , , , , , - *) ,=0 ( ( ( *. Here,( , , , , , , )=1 2 -exp -( - )2 4 -exp -( + )2 4 . 1( , , , , ), 1( , , , , )=432 0 =1 6=1 7 0( 8 6)7 0( 8 6) [7 9 0( 8 63)]2sin 0 sin 0 exp(- 82 6 ), where the7 0( ) are the Bessel functions and the 8 6are positive roots of the transcendental equation7 0( 8 3)= 0. Page 247 3.1.2-34. Domain: 0 £ £ 3,0 £ £ 0,0 £ < ". Mixed boundary value problem. A semiin®nite cylindrical sector is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),#= &1( , , ) at = 3 (boundary condition),#= &2( , , ) at = 0 (boundary condition),#= &3( , , ) at = 0(boundary condition),% / #= &4( , , ) at = 0 (boundary condition). Solution:#( , , , )= '0 ' 0 0 ' 4 0 $( , , ) ( , , , , , , ) ( ( ( - 3 ') 0 '0 ' 0 0 &1( , , *) %% ( , , , , , , - *) 5=4 ( ( ( * + ') 0 '0 ' 4 0 &2( , , *)1 %% ( , , , , , , - *) +=0 ( ( ( * - ') 0 '0 ' 4 0 &3( , , *)1 %% ( , , , , , , - *) += 0 ( ( ( * - ') 0 ' 0 0 ' 4 0 &4( , , *) ( , , , , ,0, - *) ( ( ( *. Here,( , , , , , , )=1 2 -exp -( - )2 4 +exp -( + )2 4 . 1( , , , , ), 1( , , , , )=432 0 =1 6=1 7 0( 8 6)7 0( 8 6) [79 0( 8 63)]2sin 0 sin 0 exp(- 82 6 ), where the7 0( ) are the Bessel functions and the 8 6are positive roots of the transcendental equation7 0( 8 3)= 0. 3.1.2-35. Domain: 0 £ £ 3,0 £ £ 0,0 £ £ 1. First boundary value problem. A cylindrical sector of ®nite thickness is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),#= &1( , , ) at = 3 (boundary condition),#= &2( , , ) at = 0 (boundary condition),#= &3( , , ) at = 0(boundary condition),#= &4( , , ) at = 0 (boundary condition),#= &5( , , ) at = 1 (boundary condition). Page 248 Solution:#( , , , )= ' 2 0 ' 0 0 ' 4 0 $( , , ) ( , , , , , , ) ( ( ( - 3 ' ) 0 ' 2 0 ' 0 0 &1( , , *) %% ( , , , , , , - *) 5=4 ( ( ( * + ' ) 0 ' 2 0 ' 4 0 &2( , , *)1 %% ( , , , , , , - *) +=0 ( ( ( * - ') 0 ' 2 0 ' 4 0 &3( , , *)1 %% ( , , , , , , - *) += 0 ( ( ( * + ' ) 0 ' 0 0 ' 4 0 &4( , , *) %% ( , , , , , , - *) ,=0 ( ( ( * - ' ) 0 ' 0 0 ' 4 0 &5( , , *) %% ( , , , , , , - *) ,=2 ( ( ( *. Here,( , , , , , , )= 1( , , , , ) 2( , , ), 1( , , , , )=432 0 =1 6=1 7 0( 8 6)7 0( 8 6) [7 9 0( 8 63)]2sin 0 sin 0 exp(- 82 6 ), 2( , , )=21 =1sin 1 sin 1 exp - 2212, where the7 0( ) are the Bessel functions and the 8 6are positive roots of the transcendental equation7 0( 8 3)= 0. 3.1.2-36. Domain: 0 £ £ 3,0 £ £ 0,0 £ £ 1. Mixed boundary value problem. A cylindrical sector of ®nite thickness is considered. The following conditions are prescribed:#= $( , , ) at = 0 (initial condition),#= &1( , , ) at = 3 (boundary condition),#= &2( , , ) at = 0 (boundary condition),#= &3( , , ) at = 0(boundary condition),% / #= &4( , , ) at = 0 (boundary condition),% / #= &5( , , ) at = 1 (boundary condition). Solution:#( , , , )= ' 2 0 ' 0 0 ' 4 0 $( , , ) ( , , , , , , ) ( ( ( - 3 ') 0 ' 2 0 ' 0 0 &1( , , *) %% ( , , , , , , - *) 5=4 ( ( ( * + ' ) 0 ' 2 0 ' 4 0 &2( , , *)1 %% ( , , , , , , - *) +=0 ( ( ( * - ' ) 0 ' 2 0 ' 4 0 &3( , , *)1 %% ( , , , , , , - *)+= 0 ( ( ( * - ' ) 0 ' 0 0 ' 4 0 &4( , , *) ( , , , , ,0, - *) ( ( ( * + ' ) 0 ' 0 0 ' 4 0 &5( , , *) ( , , , , , 1, - *) ( ( ( *. Page 249 Here,( , , , , , , )= 1( , , , , ) 11+21 =1cos 1 cos 1 exp - 2212 , 1( , , , , )=432 0 =1 6=1 7 0( 8 6)7 0( 8 6) [79 0( 8 63)]2sin 0 sin 0 exp(- 82 6 ), where the7 0( ) are the Bessel functions and the 8 6are positive roots of the transcendental equation7 0( 8 3)= 0. 3.1.3. Problems in Spherical Coordinates The heat equation in the spherical coordinate system has the form% #%= 12 %% 2 % #% +12sin : %%: sin : % #%: +12sin2: %2 #% 2, = ; <2+ =2+ 2. This representation is convenient to describe three-dimensional heat and mass exchange phenomenain domains bounded by coordinate surfaces of the spherical coordinate system. One-dimensional problems with central symmetry that have solutions of the form#= #( , ) are discussed in Subsection 1.2.3. 3.1.3-1. Domain: 0 £ £ 3,0 £ :£ ,0 £ £ 2 . First boundary value problem. A spherical domain is considered. The following conditions are prescribed:#= $( , :, ) at = 0 (initial condition),#= &( :, , ) at = 3(boundary condition). Solution:#( , :, , )= '2 0 ' 0 ' 4 0 $( , , ) ( , :, , , , , ) 2sin ( ( ( - 32' ) 0 '2 0 ' 0 &( , , *) %% ( , :, , , , , - *) 5=4sin ( ( ( *, where( , :, , , , , )=1 2 32 =0 6=1 ?> =0 @ > A 6 >7 +1 2( B 6)7 +1 2( B 6) ´ C > (cos :) C > (cos ) cos[ D( - )] exp( - B2 6 ),@ > =-1for D= 0, 2for D¹ 0, A 6 > =(2+ 1)(- D)! (+ D)! E7 9 +1 2( B 63) F2. Here, the7 +1 2( ) are the Bessel functions, the C > ( 8) are the associated Legendre functions expressed in terms of the Legendre polynomials C ( 8) as follows:C > ( 8)=(1 - 82) >G 2 ( >( 8 >C ( 8), C ( 8)=1!2 ( ( 8 ( 82- 1) ; and the B 6are positive roots of the transcendental equation7 +1 2( B 3)= 0.HJI References : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), H. S. Carslaw and J. C. Jaeger (1984). Page 250 3.1.3-2. Domain: 0 £ £ 3,0 £ :£ ,0 £ £ 2 . Second boundary value problem. A spherical domain is considered. The following conditions are prescribed:#= $( , :, ) at = 0 (initial condition),% K #= &( :, , ) at = 3(boundary condition). Solution:#( , :, , )= '2 0 ' 0 ' 4 0 $( , , ) ( , :, , , , , ) 2sin ( ( ( + 32') 0 '2 0 ' 0 &( , , *) ( , :, , 3, , , - *) sin ( ( ( *, where( , :, , , , , )=3 4 33+1 2 =0 6=1 ?> =0 @ > A 6 >7 +1 2( B 6)7 +1 2( B 6) ´ C > (cos :) C > (cos ) cos[ D( - )] exp( - B2 6 ),@ > =-1for D= 0, 2for D¹ 0, A 6 > = B2 6(2+ 1)(- D)! (+ D)!E 32B2 6-(+ 1)F E7 +1 2( B 63)F2. Here, the7 +1 2( ) are the Bessel functions, the C > ( 8) are the associated Legendre functions (see Paragraph 3.1.3-1), and the B 6are positive roots of the transcendental equation 2 B 37 9 +1 2( B 3)-7 +1 2( B 3)= 0. 3.1.3-3. Domain: 0 £ £ 3,0 £ :£ ,0 £ £ 2 . Third boundary value problem. A spherical domain is considered. The following conditions are prescribed:#= $( , :, ) at = 0 (initial condition),% K #+ D #= &( :, , ) at = 3(boundary condition). The solution #( , :, , ) is determined by the formula in Paragraph 3.1.3-2 where( , :, , , , , )=1 2 =0 6=1 ML =0@ L A 6 L7 +1 2( B 6)7 +1 2( B 6) ´ C L (cos :) C L (cos ) cos[ N( - )] exp( - B2 6 ),@ L =-1for N= 0, 2for N¹ 0, A 6 L = B2 6(2+ 1)(- N)! (+ N)!E 32B2 6+( D 3+)( D 3-- 1)F E7 +1 2( B 63)F2. Here, the7 +1 2( ) are the Bessel functions, the C L ( 8) are the associated Legendre functions (see Paragraph 3.1.3-1), and the B 6are positive roots of the transcendental equationB 37 9 +1 2( B 3)+ OPD 3-1 2 Q7 +1 2( B 3)= 0.HJI Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 251 3.1.3-4. Domain: 31£ £ 32,0 £ :£ ,0 £ £ 2 . First boundary value problem. A spherical layer is considered. The following conditions are prescribed:#= $( , :, ) at = 0 (initial condition),#= &1( :, , ) at = 31(boundary condition),#= &2( :, , ) at = 32(boundary condition). Solution:#( , :, , )= '2 0 ' 0 ' 4241 $( , , ) ( , :, , , , , ) 2sin ( ( ( + 32 1 ') 0 '2 0 ' 0 &1( , , *) %% ( , :, , , , , - *) 5=41sin ( ( ( * - 32 2 ' ) 0 '2 0 ' 0 &2( , , *) %% ( , :, , , , , - *) 5=42sin ( ( ( *, where( , :, , , , , )= 8 =0 6=1 R> =0 @ > A 6 > S +1 2( B 6) S +1 2( B 6) ´ C > (cos :) C > (cos ) cos[ D( - )] exp( - B2 6 ). Here, S +1 2( B 6)=7 +1 2( B 631) T +1 2( B 6)- T +1 2( B 631)7 +1 2( B 6),@ > =-1for D= 0, 2for D¹ 0, A 6 > = B2 6(2+ 1)(- D)!72 +1 2( B 632) (+ D)! E72 +1 2( B 631)-72 +1 2( B 632) F, where the7 +1 2( ) are the Bessel functions, the C > ( 8) are the associated Legendre functions expressed in terms of the Legendre polynomials C ( 8) as follows:C > ( 8)=(1 - 82) >G 2 ( >( 8 >C ( 8), C ( 8)=1!2 ( ( 8 ( 82- 1) ; and the B 6are positive roots of the transcendental equation S +1 2( B 32)= 0.HJI Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 3.1.3-5. Domain: 31£ £ 32,0 £ :£ ,0 £ £ 2 . Second boundary value problem. A spherical layer is considered. The following conditions are prescribed:#= $( , :, ) at = 0 (initial condition),% K #= &1( :, , ) at = 31(boundary condition),% K #= &2( :, , ) at = 32(boundary condition). Solution:#( , :, , )= '2 0 ' 0 ' 4241 $( , , ) ( , :, , , , , ) 2sin ( ( ( - 32 1 ' ) 0 '2 0 ' 0 &1( , , *) ( , :, , 31, , , - *) sin ( ( ( * + 32 2 ' ) 0 '2 0 ' 0 &2( , , *) ( , :, , 32, , , - *) sin ( ( ( *, Page 252 where( , :, , , , , )=3 4 ( 33 2- 33 1)+1 4 =0 6=1 ?> =0 @ >A 6 > S +1 2( B 6) S +1 2( B 6) ´ C > (cos :) C > (cos ) cos[ D( - )] exp( - B2 6 ). Here,@ > =-1for D= 0, 2for D¹ 0, A 6 > =(+ D)! (2+ 1)(- D)! ' 4241 S 2 +1 2( B 6) ( ,S +1 2( B )= B7 9 +1 2( B 31)-1 2 317 +1 2( B 31) T +1 2( B ) - B T9 +1 2( B 31)-1 2 31 T +1 2( B 31)7 +1 2( B ), where the7 +1 2( ) and T +1 2( ) are the Bessel functions, the C > ( 8) are the associated Legendre functions (see Paragraph 3.1.3-4), and the B 6are positive roots of the transcendental equationB S9 +1 2( B 32)-1 2 32 S +1 2( B 32)= 0. The integrals that determine the coef®cients A 6 > can be expressed in terms of the Bessel functions and their derivatives; see Budak, Samarskii, and Tikhonov (1980). 3.1.3-6. Domain: 31£ £ 32,0 £ :£ ,0 £ £ 2 . Third boundary value problem. A spherical layer is considered. The following conditions are prescribed:#= $( , :, ) at = 0 (initial condition),% K #- D1 #= &1( :, , ) at = 31(boundary condition),% K #+ D2 #= &2( :, , ) at = 32(boundary condition). The solution #( , :, , ) is determined by the formula in Paragraph 3.1.3-5 where( , :, , , , , )=1 4 =0 6=1 ML =0 @ LA 6 L S +1 2( B 6) S +1 2( B 6) ´ C L (cos :) C L (cos ) cos[ N( - )] exp( - B2 6 ). Here,@ L =-1for N= 0, 2for N¹ 0, A 6 L =(+ N)! (2+ 1)(- N)! ' 4241 S 2 +1 2( B 6) ( ,S +1 2( B )= B7 9 +1 2( B 31)- D1+1 2 31 7 +1 2( B 31) T +1 2( B ) - B T9 +1 2( B 31)- D1+1 2 31 T +1 2( B 31)7 +1 2( B ), where the7 +1 2( ) and T +1 2( ) are the Bessel functions, the C L ( 8) are the associated Legendre functions (see Paragraph 3.1.3-4), and the B 6are positive roots of the transcendental equationB S9 +1 2( B 32)+ D2-1 2 32 S +1 2( B 32)= 0. The integrals that determine the coef®cients A 6 L can be expressed in terms of the Bessel functions and their derivatives.HJI Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 253 3.1.3-7. Domain: 0 £ < ",0 £ :£ :0,0 £ £ 2 . First boundary value problem. A cone is considered. The following conditions are prescribed:#= $( , :, ) at = 0 (initial condition),#= &( , , ) at := :0(boundary condition). Solution:#( , :, , )= '2 0 ' U0 0 '0 $( , , ) ( , :, , , , , ) 2sin ( ( ( - ') 0 '2 0 '0 &( , , *) sin %% ( , :, , , , , - *) +=U0 ( ( ( *, where( , :, , , , , )= -1 4 6=0 @ 6(2 V+ 1) A6 exp - 2+ 2 4 +1 2 2 ´ C- 6 (cos :) C- 6 (cos ) cos[ W( - )],@ 6=-1for W= 0, 2for W¹ 0, A6 = (1 - 8)2 (( 8 C- 6 ( 8) (( V C- 6 ( 8) X=cosU0. Here, C- 6 ( 8) is the modi®ed Legendre function expressed asC- 6 ( 8)=1Y(1 + W) 1 - 8 1 + 8 6 2 ZO- V, V+ 1,1 + W;1 2-1 2 8Q, where Z ( , [, \; 8) is the Gaussian hypergeometric function and Y( ]) is the gamma function. The summation with respect to Vis performed over all roots of the equation C- 6 (cos :0)= 0that are greater than -1 ^2.HJI Reference : H. S. Carslaw and J. C. Jaeger (1984). 3.2. Heat Equation with Source _ `_ a= b c3`+ d( e, f, g,a) 3.2.1. Problems in Cartesian Coordinates In the Cartesian coordinate system, the three-dimensional heat equation with a volume source hasthe formh ih j= k l h 2 ih m 2+ h 2 ih n 2+ h 2 ih]2 o+ p( m , n , ], j ). It describes three-dimensional unsteady thermal phenomena in quiescent media or solids withconstant thermal diffusivity. A similar equation is used to study the corresponding three-dimensionalmass transfer processes with constant diffusivity. 3.2.1-1. Domain: - "< m < ",- "< n < ",- "< ]< ". Cauchy problem. An initial condition is prescribed:i = q( m , n , ]) at j = 0. Page 254 rt Solution:i ( m , n , }, j )= ~ - ~ - ~ - q( , , ) ( m , n , }, , , , j ) + ~ 0 ~ - ~ - ~ - p( , , , ) ( m , n , }, , , , j - ) , where( m , n , }, , , , j )=1 8( k j )3 2exp -( m - )2+( n - )2+( }- )2 4 k j .J References : A. G. Butkovskiy (1979). 3.2.1-2. Domain: 0 £ m < ,- < n < ,- < }< . Different boundary value problems. 1 . The solution of the ®rst boundary value problem for a half-space is given by the formula in Paragraph 3.1.1-4 with the additional term~ 0 ~ - ~ - ~ 0 p( , , , ) ( m , n , }, , , , j - ) , ( 1) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for a half-space is given by the formula in Paragraph 3.1.1-5 with the additional term (1).3. The solution of the third boundary value problem for a half-space is given by the formula in Paragraph 3.1.1-6 with the additional term (1).J References : A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984). 3.2.1-3. Domain: - < m < ,- < n < ,0 £ }£ . Different boundary value problems. 1. The solution of the ®rst boundary value problem for an in®nite layer is given by the formula in Paragraph 3.1.1-7 with the additional term~ 0 ~ 0 ~ - ~ - p( , , , ) ( m , n , }, , , , j - ) , ( 2) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for an in®nite layer is given by the formula in Paragraph 3.1.1-8 with the additional term (2).3. The solution of the third boundary value problem for an in®nite layer is given by the formula in Paragraph 3.1.1-9 with the additional term (2).4. The solution of a mixed boundary value problem for an in®nite layer is given by the formula in Paragraph 3.1.1-10 with the additional term (2). 3.2.1-4. Domain: - < m < ,0 £ n < ,0 £ }£ . Different boundary value problems. 1. The solution of the ®rst boundary value problem for a semiin®nite layer is given by the formula in Paragraph 3.1.1-11 with the additional term~ 0 ~ 0 ~ 0 ~ - p( , , , ) ( m , n , }, , , , j - ) , ( 3) which allows for the equation's nonhomogeneity. Page 255 2. The solution of the second boundary value problem for a semiin®nite layer is given by the formula in Paragraph 3.1.1-12 with the additional term (3).3. The solution of the third boundary value problem for a semiin®nite layer is given by the formula in Paragraph 3.1.1-13 with the additional term (3).4. The solutions of mixed boundary value problems for a semiin®nite layer are given by the formulas in Paragraph 3.1.1-14 with additional terms of the form (3).J References : A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984). 3.2.1-5. Domain: 0 £ m < ,0 £ n < ,0 £ }< . Different boundary value problems. 1. The solution of the ®rst boundary value problem for the ®rst octant is given by the formula in Paragraph 3.1.1-15 with the additional term~ 0 ~ 0 ~ 0 ~ 0 p( , , , ) ( m , n , }, , , , j - ) , ( 4) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for the ®rst octant is given by the formula in Paragraph 3.1.1-16 with the additional term (4).3. The solution of the third boundary value problem for the ®rst octant is given by the formula in Paragraph 3.1.1-17 with the additional term (4).4. The solutions of mixed boundary value problems for the ®rst octant are given by the formulas in Paragraph 3.1.1-18 with additional terms of the form (4). 3.2.1-6. Domain: 0 £ m £ 1,0 £ n £ 2,- < }< . Different boundary value problems. 1. The solution of the ®rst boundary value problem in an in®nite rectangular domain is given by the formula in Paragraph 3.1.1-19 with the additional term~ 0 ~ - ~ 2 0 ~ 1 0 p( , , , ) ( m , n , }, , , , j - ) , ( 5) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem in an in®nite rectangular domain is given by the formula in Paragraph 3.1.1-20 with the additional term (5).3. The solution of the third boundary value problem in an in®nite rectangular domain is given by the formula in Paragraph 3.1.1-21 with the additional term (5).4. The solution of a mixed boundary value problem in an in®nite rectangular domain is given by the formula in Paragraph 3.1.1-22 with the additional term (5). 3.2.1-7. Domain: 0 £ m £ 1,0 £ n £ 2,0 £ }< . Different boundary value problems. 1. The solution of the ®rst boundary value problem in a semiin®nite rectangular domain is given by the formula of Paragraph 3.1.1-23 with the additional term~ 0 ~ 0 ~ 2 0 ~ 1 0 p( , , , ) ( m , n , }, , , , j - ) , ( 6) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem in a semiin®nite rectangular domain is given by the formula in Paragraph 3.1.1-24 with the additional term (6).3. The solution of the third boundary value problem in a semiin®nite rectangular domain is given by the formula in Paragraph 3.1.1-25 with the additional term (6).4. The solutions of mixed boundary value problems in a semiin®nite rectangular domain are given by the formulas in Paragraph 3.1.1-26 with additional terms of the form (6). Page 256 rt 3.2.1-8. Domain: 0 £ m £ 1,0 £ n £ 2,0 £ }£ 3. Different boundary value problems. 1. The solution of the ®rst boundary value problem for a rectangular parallelepiped is given by the formula in Paragraph 3.1.1-27 with the additional term~ 0 ~ 3 0 ~ 2 0 ~ 1 0 p( , , , ) ( m , n , }, , , , j - ) , ( 7) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for a rectangular parallelepiped is given by the formula in Paragraph 3.1.1-28 with the additional term (7).3. The solution of the third boundary value problem for a rectangular parallelepiped is given by the formula in Paragraph 3.1.1-29 with the additional term (7).4. The solutions of mixed boundary value problems for a rectangular parallelepiped are given by the formulas in Paragraph 3.1.1-30 with additional terms of the form (7).J References : A. G. Butkovskiy (1979), H. S. Carslaw and J. C. Jaeger (1984). 3.2.2. Problems in Cylindrical Coordinates In the cylindrical coordinate system, the heat equation with a volume source is written ash ih j= k 1 hh l h ih o+1 2 h 2 ih 2+ h 2 ih}2 + p( , , }, j ). This representation is used to describe nonsymmetric unsteady thermal (diffusion) processes inquiescent media or solids bounded by cylindrical surfaces and planes. 3.2.2-1. Domain: 0 £ £ ,0 £ £ 2 ,- < }< . Different boundary value problems. 1. The solution of the ®rst boundary value problem for an in®nite circular cylinder is given by the formula in Paragraph 3.1.2-2 with the additional term~ 0 ~ - ~2 0 ~ 0 p( , , , ) ( , , }, , , , j - ) , ( 1) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for an in®nite circular cylinder is given by the formula in Paragraph 3.1.2-3 with the additional term (1).3. The solution of the third boundary value problem for an in®nite circular cylinder is the sum of the solution presented in Paragraph 3.1.2-4 and expression (1). 3.2.2-2. Domain: 0 £ £ ,0 £ £ 2 ,0 £ }< . Different boundary value problems. 1. The solution of the ®rst boundary value problem for a semiin®nite circular cylinder is given by the formula in Paragraph 3.1.2-5 with the additional term~ 0 ~ 0 ~2 0 ~ 0 p( , , , ) ( , , }, , , , j - ) , ( 2) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for a semiin®nite circular cylinder is given by the formula in Paragraph 3.1.2-6 with the additional term (2).3. The solution of the third boundary value problem for a semiin®nite circular cylinder is the sum of the solution presented in Paragraph 3.1.2-7 and expression (2).4. The solutions of mixed boundary value problems for a semiin®nite circular cylinder are given by the formulas in Paragraph 3.1.2-8 with additional terms of the form (2). Page 257 3.2.2-3. Domain: 0 £ £ ,0 £ £ 2 ,0 £ }£ . Different boundary value problems. 1. The solution of the ®rst boundary value problem for a circular cylinder of ®nite length is given by the formula in Paragraph 3.1.2-9 with the additional term~ 0 ~ 0 ~2 0 ~ 0 p( , , , ) ( , , }, , , , j - ) , ( 3) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for a circular cylinder of ®nite length is given by the formula in Paragraph 3.1.2-10 with the additional term (3).3. The solution of the third boundary value problem for a circular cylinder of ®nite length is the sum of the solution presented in Paragraph 3.1.2-11 and expression (3).4. The solutions of mixed boundary value problems for a circular cylinder of ®nite length are given by the formulas in Paragraph 3.1.2-12 with additional terms of the form (3). 3.2.2-4. Domain: 1£ £ 2,0 £ £ 2 ,- < }< . Different boundary value problems. 1. The solution of the ®rst boundary value problem for an in®nite hollow cylinder is given by the formula in Paragraph 3.1.2-13 with the additional term~ 0 ~ - ~2 0 ~ 21 p( , , , ) ( , , }, , , , j - ) , ( 4) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for an in®nite hollow cylinder is given by the formula in Paragraph 3.1.2-14 with the additional term (4).3. The solution of the third boundary value problem for an in®nite hollow cylinder is the sum of the solution presented in Paragraph 3.1.2-15 and expression (4). 3.2.2-5. Domain: 1£ £ 2,0 £ £ 2 ,0 £ }< . Different boundary value problems. 1. The solution of the ®rst boundary value problem for a semiin®nite hollow cylinder is given by the formula in Paragraph 3.1.2-16 with the additional term~ 0 ~ 0 ~2 0 ~ 21 p( , , , ) ( , , }, , , , j - ) , ( 5) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for a semiin®nite hollow cylinder is given by the formula in Paragraph 3.1.2-17 with the additional term (5).3. The solution of the third boundary value problem for a semiin®nite hollow cylinder is the sum of the solution presented in Paragraph 3.1.2-18 and expression (5).4. The solutions of mixed boundary value problems for a semiin®nite hollow cylinder are given by the formulas in Paragraph 3.1.2-19 with additional terms of the form (5). Page 258 rt 3.2.2-6. Domain: 1£ £ 2,0 £ £ 2 ,0 £ }£ . Different boundary value problems. 1. The solution of the ®rst boundary value problem for a hollow cylinder of ®nite length is given by the formula in Paragraph 3.1.2-20 with the additional term~ 0 ~ 0 ~2 0 ~ 21 p( , , , ) ( , , }, , , , j - ) , ( 6) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for a hollow cylinder of ®nite length is given by the formula in Paragraph 3.1.2-21 with the additional term (6).3. The solution of the third boundary value problem for a hollow cylinder of ®nite length is the sum of the solution speci®ed in Paragraph 3.1.2-22 and expression (6).4. The solutions of mixed boundary value problems for a hollow cylinder of ®nite length are given by the formulas in Paragraph 3.1.2-23 with additional terms of the form (6). 3.2.2-7. Domain: 0 £ < ,0 £ £ 0,- < }< . Different boundary value problems. 1. The solution of the ®rst boundary value problem for an in®nite wedge domain is given by the formula in Paragraph 3.1.2-24 with the additional term~ 0 ~ - ~ 0 0 ~ 0 p( , , , ) ( , , }, , , , j - ) , ( 7) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for an in®nite wedge domain is given by the formula in Paragraph 3.1.2-25 with the additional term (7). 3.2.2-8. Domain: 0 £ < ,0 £ £ 0,0 £ }< . Different boundary value problems. 1 . The solution of the ®rst boundary value problem for a semiin®nite wedge domain is given by the formula in Paragraph 3.1.2-26 with the additional term~ 0 ~ 0 ~ 0 0 ~ 0 p( , , , ) ( , , }, , , , j - ) , ( 8) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for a semiin®nite wedge domain is given by the formula in Paragraph 3.1.2-27 with the additional term (8).3. The solutions of mixed boundary value problems for a semiin®nite wedge domain are given by the formulas in Paragraph 3.1.2-28 with additional terms of the form (8). 3.2.2-9. Domain: 0 £ < ,0 £ £ 0,0 £ }£ . Different boundary value problems. 1. The solution of the ®rst boundary value problem for a wedge domain of ®nite height is given by the formula in Paragraph 3.1.2-29 with the additional term~ 0 ~ 0 ~ 0 0 ~ 0 p( , , , ) ( , , }, , , , j - ) , ( 9) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for a wedge domain of ®nite height is given by the formula in Paragraph 3.1.2-30 with the additional term (9).3. The solutions of mixed boundary value problems for a wedge domain of ®nite height are given by the formulas in Paragraph 3.1.2-31 with additional terms of the form (9). Page 259 3.2.2-10. Different boundary value problems for a cylindrical sector. 1. The solution of the ®rst boundary value problem for an unbounded cylindrical sector ( 0 £ £ , 0 £ £ 0,- < }< ) is given by the formula in Paragraph 3.1.2-32 with the additional term~ 0 ~ - ~ 0 0 ~ 0 p( , , , ) ( , , }, , , , j - ) , which allows for the equation's nonhomogeneity.2. The solution of the ®rst boundary value problem for a semibounded cylindrical sector ( 0 £ £ , 0 £ £ 0,0 £ }< ) is given by the formula in Paragraph 3.1.2-33 with the additional term~ 0 ~ 0 ~ 0 0 ~ 0 p( , , , ) ( , , }, , , , j - ) , ( 10) which allows for the equation's nonhomogeneity.3. The solution of the mixed boundary value problem for a semibounded cylindrical sector (0 £ £ ,0 £ £ 0,0 £ }< ) is given by the formula in Paragraph 3.1.2-34 with the additional term (10).4. The solution of the ®rst boundary value problem for a cylindrical sector of ®nite height ( 0 £ £ , 0 £ £ 0,0 £ }£ ) is given by the formula in Paragraph 3.1.2-35 with the additional term~ 0 ~ 0 ~ 0 0 ~ 0 p( , , , ) ( , , }, , , , j - ) , ( 11) which allows for the equation's nonhomogeneity.3. The solution of a mixed boundary value problem for a cylindrical sector of ®nite height is given by the formula in Paragraph 3.1.2-36 with the additional term (11). 3.2.3. Problems in Spherical Coordinates In the spherical coordinate system, the heat equation with a volume source has the formh ih j= k 1 2 hh l 2 h ih o+1 2sin hh lsin h ih o+1 2sin2 h 2 ih 2 + p( , , , j ). One-dimensional problems with central symmetry that have solutions of the form i = i ( , j ) are discussed in Subsection 1.2.4. 3.2.3-1. Domain: 0 £ £ ,0 £ £ ,0 £ £ 2 . Different boundary value problems. 1. The solution of the ®rst boundary value problem for a spherical domain is given by the formula in Paragraph 3.1.3-1 with the additional term~ 0 ~2 0 ~ 0 ~ 0 p( , , , ) ( , , , , , , j - ) 2sin , ( 1) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for a spherical domain is given by the formula in Paragraph 3.1.3-2 with the additional term (1).3. The solution of the third boundary value problem for a spherical domain is the sum of the solution speci®ed in Paragraph 3.1.3-3 and expression (1). Page 260 3.2.3-2. Domain: 1£ £ 2,0 £ £ ,0 £ £ 2 . Different boundary value problems. 1 . The solution of the ®rst boundary value problem for a spherical layer is given by the formula in Paragraph 3.1.3-4 with the additional term 0 2 0 0 2 1 ( , , , ) ( , , , , , , - ) 2sin , ( 2) which allows for the equation's nonhomogeneity.2. The solution of the second boundary value problem for a spherical layer is given by the formula in Paragraph 3.1.3-5 with the additional term (2).3. The solution of the third boundary value problem for a spherical layer is the sum of the solution speci®ed in Paragraph 3.1.3-6, and expression (2). 3.2.3-3. Domain: 0 £ < ,0 £ £ 0,0 £ £ 2 . First boundary value problem. The solution of the ®rst boundary value problem for an in®nite cone is given by the formula inParagraph 3.1.3-7 with the additional term 0 2 0 0 0 0 ( , , , ) ( , , , , , , - ) 2sin , which allows for the equation's nonhomogeneity. 3.3. Other Equations with Three Space Variables 3.3.1. Equations Containing Arbitrary Parameters 1. = 2 2+ 2 2+ 2 2 + ( 1+ 2+ 3+ ). The transformation ( !, ", #, )=exp $( %1 !+ %2 "+ %3 #) +1 3 &( %21+ %22+ %23) 3+ '(*)(+( , , , ),= !+& %1 2, = "+& %2 2, = #+& %3 2 leads to the three-dimensional heat equation , +=&( , -.-/++ , 0/0 ++ , 1213+) that is dealt with in Subsection 3.1.1.2. = 2 2+ 2 2+ 2 2 ± $4(2+2+2) + ), > 0. The transformation ( 5is any number) ( !, ", #, )=exp 6 7 & % 2&( !2+ "2+ #2)+ 837 & %- '39 *: +( , , , ),= !exp 827 & % .9, = "exp 827 & % .9, = #exp 827 & % .9, =1 47 & %exp 847 & % .9+ 5 leads to the three-dimensional heat equation , ; +=&( , -.-/++ , 0/0 ++ , 121 +) that is dealt with in Subsection 3.1.1.3. = 2 2+ 2 2+ 2 2 + $± (2+2+2) + 1+ 2+ 3+s ). This is a special case of equation 3.3.2.3 with < =( )= '>=, ?( )= @. Page 261 4. = 2 2+ 2 2+ 2 2+ 1 + 2 + 3 + . This equation governs the nonstationary temperature (concentration) ®eld in a medium moving witha constant velocity, provided there is volume release (absorption) of heat proportional to temperature(concentration). The substitution ( !, ", #, )=exp 8A51 !+ 52 "+ 53 #+ B .9 C( !, ", #, ), where51= - %1 2&, 52= - %2 2&, 53= - %3 2&, B= '-1 4& 8*%21+ %22+ %239, leads to the three-dimensional heat equation , C=& D3 Cthat is considered in Subsection 3.1.1. 5. = 2 2+ 2 2+ 2 2± . This equation is encountered in problems of convective heat and mass transfer in a simple shear¯ow. Fundamental solution:E E( !, ", #, , , , )=1 (4 & )3 F281+1 12 %2291 F2exp G- $H!- -1 2 %2( "+ ) )2 4& 81+1 12 %229-( "- )2+( #- )2 4& I.J4K Reference : E. A. Novikov (1958). 6. = 2 2+ 2 2+ 2 2 + 1 + 2 + 3 + L(,,,). This equation is encountered in problems of convective heat and mass transfer in a linear shear ¯ow. Domain: - < !< ,- < "< ,- < #< . Cauchy problem. An initial condition is prescribed: = <( !, ", #) at = 0. Solution: ( !, ", #, )= - - - <( , , ) ( !, ", #, , , , ) ! " # + 0 - - - ( , , , ) ( !, ", #, , , , - ) ! " # , where( !, ", #, , , , )= M( !, , ; %1) M( ", , ; %2) M( #, , ; %3),M( !, , ; %)= 62 &%( N2 O - 1) :-1 F2 exp 6- % 8P! N O - 92 2&( N2 O - 1) :. 7. = 2 2+ 2 2+ 2 2 + ( 1+ 1) + ( 2+ 2) + ( 3+ 3) + (s1+s2+s3+ Q). This is a special case of equation 3.3.2.5. Page 262 8. R S T + S T2 2 U 2 2+ 2 2+ 2 2= 0. Three-dimensional Schr Èodinger equation, V2= -1 . Fundamental solution:E E( !, ", #, )= - VS W X2 S W 3 F2 exp 6YV X2 S W( !2+ "2+ #2)- V3 4 :.J4K Reference : V . S. Vladimirov, V . P. Mikhailov, A. A. Vasharin, et al. (1974). 9. = Z + [ + \ . This equation describes unsteady heat and mass transfer processes in inhomogeneous (anisotropic)media. It admits separable solutions, as well as solutions with incomplete separation of variables(see Subsection 0.9.2-1). In addition, for]¹ 2,X¹ 2, ^¹ 2there are particular solutions of the form = ( , ), 2= 4 6 !2- _&(2 - ])2+ "2- `%(2 -X)2+ #2- ='(2 - ^)2 :, where the function ( , ) is determined by the one-dimensional nonstationary equation, , = ,2 , 2+ 5 , , , 5= 2 1 2 - ]+1 2 -X+1 2 - ^ - 1. For solutions of this equation, see Subsections 1.2.1, 1.2.3, and 1.2.5. 3.3.2. Equations Containing Arbitrary Functions 1. = 2 2+ 2 2+ 2 2 + a(). This equation describes three-dimensional unsteady thermal phenomena in quiescent media or solidswith constant thermal diffusivity, provided there is unsteady volume heat release proportional totemperature. The substitution ( !, ", #, )=exp $3b <( ) c)3C( !, ", #, ) leads to the usual heat equation, d.C=& D3 Cthat is dealt with in Subsection 3.1.1. 2. = 2 2+ 2 2+ 2 2 + $ a1() + a2() + a3() + e() ). The transformation ( !, ", #, )=exp G! f1( g)+ " f2( g)+ # f3( g)+& h i f2 1( g)+ f2 2( g)+ f2 3( g) j k g+h l( g) k g m n( o, p, q, g),o= r+ 2 sh f1( g) k g, p= t+ 2 sh f2( g) k g, q= u+ 2 sh v3( w) k w,v x( w)=h y x( w) k w, leads to the three-dimensional heat equation z d*n= s {Az |.|/n+ z 0/0 n+ z 1213n }that is dealt with in Subsection 3.1.1. Page 263 3. ~ ~ = ~2~ 2+ ~2~ 2+ ~2~ 2 +i± (2+2+2)+ 1()+ 2()+ 3()+ () j. 1 . Case >0. The transformation ( r, t, u, w)= n( o, p, q, ) exp s 2 s {Pr2+ t2+ u2}(,o= rexp {2 s w.}, p= texp {2 s w.}, q= uexp {2 s w.}, =1 4 s exp {4 s w.}+ 5, where 5is an arbitrary constant, leads to an equation of the form 3.3.2.2:z nz w= s z2nz o2+ z2nz p2+ z2nz q2+i ov1( )+ pv2( )+ qv3( )+ ( ) j/n,v x( )=1 ( ()3 2 y x ln( () , ( )=1( l ln( () +3 4 , = 4 s , = 1,2,3. 2 . Case <0. The transformation ( r, t, u, w)= ( o1, p1, q1, 1) exp - s 2 s( r2+ t2+ u2) tan {2- s w.}(,o1= r cos {2- s w.}, p1= t cos {2- s w.}, q1= u cos {2- s w.}, 1=1 2- s tan {2- s w.} also leads to an equation of the form 3.3.2.2 for = ( o1, p1, q1, 1) (the equation for is not written out here).4.~ ~ = 1() ~2~ 2+ 2() ~2~ 2+ 3() ~2~ 2+ (,,,). Here, 0< sx( w) < ; = 1,2,3. Domain: - < r< ,- < t< ,- < u< . Cauchy problem. An initial condition is prescribed: =y( r, t, u) at w= 0. Solution: ( r, t, u, w)=h d 0 h - h - h - ( o, p, q, ) ( r, t, u, o, p, q, w, ) k o k p k q k +h - h - h - y( o, p, q) ( r, t, u, o, p, q, w,0) k o k p k q, where( r, t, u, o, p, q, w, )=1 8 3 2 123exp -( r- o)2 41-( t- p)2 42-( u- q)2 43 ,1=h d s1( ) k ,2=h d s2( ) k ,3=h d s3( ) k . See also the more general equation 3.4.3.3, where other boundary value problems are considered. Page 264 5. ~ ~ = 1() ~2~ 2+ 2() ~2~ 2+ 3() ~2~ 2+i 1()+ 1() j ~ ~ +i 2()+ 2()j ~ ~ +i 3()+ 3()j ~ ~ +is1()+s2()+s3()+ ()j. The transformation ( r, t, u, w)=expi y1( w) r+y2( w) t+y3( w) u+l( w) j(n( o, p, q, w),o= ¡1( w) r+ ¢1( w), p= ¡2( w) t+ ¢2( w), q= ¡3( w) u+ ¢3( w), where¡x( w)= 5xexp h x( w) k wc,y x( w)= ¡x( w)h £ x( w)¡x( w) k w+ ¤x ¡x( w),¢x( w)=hi2 sx( w)y x( w)+ x( w)j ¡x( w) k w+ ¥x,l( w)=h3¦x=1 i sx( w)y2x( w)+ x( w)y x( w)j k w+ § ¨( ©) k ©+ ª, ( = 1,2,3; « ¬, ¤ ¬, ¥ ¬, ªare arbitrary constants) leads to an equation of the form 3.3.2.4:z z ©= ®1( ©) ¡21( ©) z2z ¯2+ ®2( ©) °22( ©) z2z ±2+ ®3( ©) °23( ©) z2z ²2. 6. ³ ´³ µ= ¶1(µ) ³2´³ ·2+ ¶2(µ) ³2´³ ¸2+ ¶3(µ) ³2´³ ¹2+ º »1(µ)·+ ¼1(µ) ½ ³ ´³ ·+ º4»2(µ)¸+ ¼2(µ) ½ ³ ´³ ¸ + º4»3(µ)¹+ ¼3(µ) ½ ³ ´³ ¹+ ºs1(µ)·2+s2(µ)¸2+s3(µ)¹2+ ¾1(µ)·+ ¾2(µ)¸+ ¾3(µ)¹+ ¿(µ) ½´. The substitution À ( Á, Â, Ã, ©)=exp º Ä1( ©) Á2+ Ä2( ©) Â2+ Ä3( ©) Ã2½/( Á, Â, Ã, ©), where the functions Ä ¬= Ä ¬( ©) are solutions of the respective Riccati equationsÄ Å¬= 4 ® ¬( ©) Ä2¬+ 2 ÆÇ¬( ©) Ä ¬+ È2¬( ©) ( É= 1,2,3), leads to an equation of the form 3.3.2.5 for = ( Á, Â, Ã, ©). 7. ³ ´³ µ+2Ê Ë =1 ºÍÌ Ë (µ) + Î Ë (µ)·3 ½ ³ ´³ · Ë ± ¶ ³ ´³ ·3=2Ê Ë , Ï=1 Ð ËÏ(µ) ³2´³ · Ë³ · Ï+Ð33(µ) ³2´³ ·23. Equation of turbulent diffusion. It describes the diffusion of an admixture in a horizontal stream whose velocity components are linear functions of the height. Fundamental solution:Ñ Ñ( Á1, Á2, Á3, ¯1, ¯2, ¯3)=1 (4 Ò)3 Ó2 Ôdet| Õ|exp Ö-1 43Ê×, Ø=1 Õ-1×Ø( ©) Â ×ÂØ(Ù, Õ ×Ø( ©)= § d 0 Ú ×Ø( Û) Ü Û. Here, the following notation is used ( Ý, Þ= 1,2):Â ß= Á ß- ¯3ß- à ß( ©)- Á3 á ß( ©)- ® § d 0( ©- Û) â ß( Û) Ü Û, Â3= Á3- ¯3+ ® ©,Ú ß ã( ©)= ä ß ã( ©)+ ä33( ©)á ß( ©)á ã( ©),Ú ß3( ©)=Ú3 ß( ©)= - ä33( ©)á ß( ©),Ú33( ©)= ä33( ©), à ß( ©)= § d 0 Ä ß( ©) Ü ©,á ß( ©)= § d 0 â ß( ©) Ü ©, det| Õ|is the determinant of the matrix Twith entries Õ ×Ø( ©), Õ-1×Ø( ©) are the entries of the inverse ofT. The inequalities Õ11( ©) >0, Õ11( ©) Õ22( ©)- Õ2 12( ©) >0, and det | Õ|>0are assumed to hold.å4æ Reference : E. A. Novikov (1958). Page 265 3.3.3. Equations of the Formç( è, é, ê) ë ìë í= div[ î( è, é, ê)Ñì] ± ï( è, é, ê)ì+ ð( è, é, ê,í) Equations of this form are often encountered in the theory of heat and mass transfer. For brevity,the following notation is used: divº4®(r)Ñ À½= ññ Á Öò®(r) ñ Àñ Á Ù+ ññ Â Öò®(r) ñ Àñ Â Ù+ ññ Ã Öò®(r) ñ Àñ Ã Ù, r= { Á, Â, Ã}. The problems presented in this subsection are assumed to refer to a simply connected bounded domain ówith smooth boundaryÚ. It is also assumed that ô(r) >0, ®(r) >0, and õ(r)³ 0. 3.3.3-1. First boundary value problem. The following conditions are prescribed:À = Ä(r) at ö= 0 (initial condition),À = â(r, ö) for r ÷Ú(boundary condition). Solution:À (r, ö)= d 0 ø ù ú( û, Û) ü(r, û, ý- Û) þ ÿ þ Û+ø ù ( û) ( û) ü(r, û, ý) þ ÿ -ø d 0 ø ( û, Û) ®( û) ü(r, û, ý- Û)Ù þÚ þ Û. ( 1) Here, the modi®ed Green's function is given byü(r, û, ý)= =1 (r) ( û) 2exp(- ý), 2=øù (r) 2 (r) þ ÿ, û= { 1, 2, 3}, ( 2) where the and (r) are the eigenvalues and corresponding eigenfunctions of the Sturm±Liouville problem for the following elliptic second-order equation with a homogeneous boundary conditionof the ®rst kind: div4®(r)Ñ - (r) + (r) = 0, ( 3)= 0 for r Ú. ( 4) The integration in solution (1) is carried out with respect to 1, 2, 3; denotes the derivative along the outward normal to the surfaceÚwith respect to 1, 2, and 3. General properties of the Sturm±Liouville problem (3)±(4): 1 . There are countably many eigenvalues. All eigenvalues are real and can be ordered so that 1£ 2£ 3£ , with as ; consequently, there can exist only ®nitely many negative eigenvalues.2. For (r) >0, ®(r) >0, and (r)³ 0, all eigenvalues are positive: >0. 3 . The eigenfunctions are de®ned up to a constant multiplier. Any two eigenfunctions (r) and (r) corresponding to different eigenvalues and are orthogonal with weight (r) in the domain ÿ:ø ù (r) (r) (r) þ ÿ= 0 for ¹ . Page 266 4 . An arbitrary function (r) that is twice continuously differentiable and satis®es the boundary condition of the Sturm±Liouville problem ( = 0forr Ú) can be expanded into an absolutely and uniformly convergent series in the eigenvalues:(r)= =1 (r), =1 2øù (r) (r) (r) þ ÿ, where the formula for 2is given in (2). "! # $&% 'In a three-dimensional problem, to each eigenvalue there generally correspond ®nitely many linearly independent eigenfunctions (1) , (2) , (((, ( ) . These function can always be replaced by their linear combinations Å ( ¬) = « ¬,1 (1) + + « ¬, ¬-1 ( ¬-1) + ( ¬) , )= 1,2, (((, , such that Å (1) , Å (2) , (((, Å ( ) are now pairwise orthogonal. Thus, without loss of generality, we assume that all eigenfunctions are orthogonal. 3.3.3-2. Second boundary value problem. The following conditions are prescribed:*= (r) at ý= 0 (initial condition),* = (r, ý) for r +(boundary condition). Solution:*(r, ý)=ø d 0 øù ú( û, ,) ü(r, û, ý- ,) þ ÿ þ ,+øù ( û) ( û) ü(r, û, ý) þ ÿ +ø d 0 ø ( û, ,) ®( û) ü(r, û, ý- ,) þ + þ ,. ( 5) Here, the modi®ed Green's function is given by (2), where the and (r) are the eigenvalues and corresponding eigenfunctions of the Sturm±Liouville problem for the elliptic second-orderequation (3) with a homogeneous boundary condition of the second kind, = 0 for r +. ( 6) For (r) >0the general properties of the eigenvalue problem (3), (6) are the same as for the ®rst boundary value problem (with all >0). 3.3.3-3. Third boundary value problem. The following conditions are prescribed:*= (r) at ý= 0 (initial condition),* + )(r) *= (r, ý) for r +(boundary condition). The solution of the third boundary value problem is given by formulas (5) and (2), where the and (r) are the eigenvalues and corresponding eigenfunctions of the Sturm±Liouville problem for the second-order elliptic equation (3) with a homogeneous boundary condition of the third kind, + )(r) = 0 for r +. ( 7) For (r)³ 0and )(r) >0the general properties of the eigenvalue problem (3), (7) are the same as for the ®rst boundary value problem (see Paragraph 3.3.3-1). Let )(r)= )=const. Denote the Green's functions of the second and third boundary value problems by -2(r, û, ý) and -3(r, û, ý, )), respectively. If (r) >0, then the following limiting relation holds: -2(r, û, ý)=lim¬/.0 -3(r, û, ý, )).021 References for Subsection 3.3.3: V . S. Vladimirov (1988), A. D. Polyanin (2000a, 2000c). Page 267 3.4. Equations with 3Space Variables 3.4.1. Equations of the Form 4 54 6= 7 8 95+ :( ;1, < < <, ; 9,6) This is an -dimensional nonhomogeneous heat equation. In the Cartesian system of coordinates, it is represented as*ý= ® ¬=1 2 * =2¬+ú(x, ý), x= { = 1, (((, = }. The solutions of various problems for this equation can be constructed on the basis of incompleteseparation of variables (see Paragraphs 0.6.1-2 and 0.9.2-1) taking into account the results ofSubsections 1.1.1 and 1.1.2. Some examples of solving such problems can be found below inParagraphs 3.4.1-2 through 3.4.1-4. 3.4.1-1. Homogeneous equation (úº 0). 1 . Particular solutions:*(x, ý)= «exp > =1 ) =+ ® ý =1 )2 ?,*(x, ý)= «exp >- ® ý =1 )2 ? @=1cos( ) =+ A ),*(x, B)= Cexp >- =1 ) =? @=1cos( ) =- 2 ® )2 B+ A ),*(x, B)= C ( B- B0) D 2exp -1 4 ®( B- B0) =1( =- A )2 E,*(x, B)= C @=1erf > =- A 2 F ® B ?, where x= { = 1, (((, = }; C, ) , A , and B0are arbitrary constants. 2 . Fundamental solution: G G (x, B)=1H 2 F I® B JKexp >-|x|2 4 ® B ?, |x|2= K LNM =1 O2 M . 3 P. Suppose Q= Q(O1, (((,O K, B) is a solution of the homogeneous equation. Then the functionsQ1= C Q( R SO1+ A1, (((, R SO K+ AK, S2B+ AK+1),Q2= Cexp T K LUM =1 S MO M + V B K LNM =1 S2 M WQ(O1+ 2 V S1 B+ A1, (((,O K+ 2 V SK B+ AK, B+ AK+1),Q3= C | X+ Y B|K D 2exp Z- Y 4 V( X+ Y [) K LNM =1 O2 MEQ TO1X+ Y [, (((,O KX+ Y [, \+ S [X+ Y [ W , S X- Y\= 1, where ], ^1, (((, ^K+1, S, S1, (((, SK Y, and Xare arbitrary constants, are also solutions of the equation. The signs at Sin the formula for Q1can be taken independently of one another. Page 268 3.4.1-2. Domain: `K= {- a<O M < a; b= 1, (((, c}. Cauchy problem. An initial condition is prescribed:Q= d(x) at [= 0. Solution:Q(x, [)=1H 2 e IV [ JK f g h d(y) exp T-|x-y|2 4 V [ W i y+f d 0 f g h j(y, k)H 2e IV( [- k) JKexp T-|x-y|2 4 V( [- k) W i y ik, where y= { l1, (((, lK},|x-y| = m(O1- l1)2+ nnn+(O K- lK)2, i y= il1 il2 ((( ilK.o2p Reference : V . S. Vladimirov (1988). 3.4.1-3. Domain: q={0 £O M £ r M ; b= 1, (((, c}. First boundary value problem. The following conditions are prescribed:Q= d(x) at [= 0 (initial condition),Q= s M (x, [) atO M = 0 (boundary conditions),Q= t M (x, [) atO M = r M (boundary conditions). Solution:Q(x, [)=f d 0 f uj(y, k) v(x,y, [- k) i y ik+f u d(y) v(x,y, [) i y + V w L M =1 f d 0 f x( y) Zzs M (y, k) {{ l Mv(x,y, [- k) | }y=0 i ~ ( M ) } ik - V w LNM =1 f d 0 f x( y) Zt M (y, k) {{ l Mv(x,y, [- k) | }y= y i ~ ( M ) } ik, where the following notation is used:i ~ ( M ) }= il1 ((( il M -1 il M +1 ((( ilw, ~ ( M )= {0 £ l £ rfor = 1, (((, b-1, b+1, (((, c}. The Green's function can be represented in the product formv(x,y, [)= w M =1 v M (O M , l M , [), ( 1) where the v M (O M , l M , [) are the Green's functions of the respective one-dimensional boundary value problems (see Paragraph 1.1.2-5):v M (O M , l M , [)=2r M =1sin r sin r exp - 22 r2 . 3.4.1-4. Domain: q= {0 £ £ r; = 1, (((, }. Second boundary value problem. The following conditions are prescribed:Q= (x) at = 0 (initial condition),{ y Q= s (x, ) at = 0 (boundary conditions),{ y Q= t (x, ) at = r(boundary conditions). Page 269 Solution:Q(x, )= d 0 (y, ) (x,y, - ) y + (y) (x,y, ) y - =1 d 0 ( ) (y, ) (x,y, - ) }=0 ¡( ) } + =1 d 0 ( )£¢ (y, ) (x,y, - ) }= ¤ ¡( ) } . ( 2) The Green's function can be represented as the product (1) of the corresponding one-dimensionalGreen's functions of the form (see Paragraph 1.1.2-6) ( , ¥ , )=1¦+2¦ § ¨©=1cos ª ¦ cos ª ¦ exp - ª22 ¦2 . 3.4.2. Other Equations Containing Arbitrary Parameters 1. « ¬« = ® ¯ ¨° =1 «2¬« ±2 ° + ²+ ¯ ¨° =1 ³ °± °¬. This is a special case of equation 3.4.3.1. The transformation´( 1, (((, , )=exp ¨=1 µ +1 3 3 ¨=1 µ2+ ¶ ·( 1, (((, , ), = +µ 2 leads to the -dimensional heat equation ¸ ¹·= º=1 ¸ » »·that is dealt with in Subsection 3.4.1. 2. « ¬« = ® ¯ ¨° =1 «2¬« ±2 ° ± ¼ ²+³ ¯ ¨° =1 ±2 ° ½¬,³> 0. The transformation ( ¾is any number)´( 1, (((, , ¿)=·( À1, (((, À, ) exp Á1 2 Â µ Ã ¨NÄ =1 2 Ä + ÅÆ Ç Ãµ- ¶È ¿ÊÉ,À1= 1expÅ2 Ç Ãµ ¿È, (((, À= expÅ2 Ç Ãµ ¿È, =1 4Ç ÃµexpÅ4 Ç Ãµ ¿È+ ¾ leads to the Æ-dimensional heat equation ¸ Ë·= Ã º Ä =1 ¸ » »·that is dealt with in Subsection 3.4.1. 3. « ¬« = ®¯ ¨ ° =1 «2¬« ±2 ° + ¼±³ ¯ ¨ ° =1 ±2 ° +¯ ¨ ° =1 ² °± ° +s ½¬. This is a special case of equation 3.4.3.2 with Ì Ä ( ¿)= ¶ Ä and( ¿)= Í. 4. « ¬« = ® ¯ ¨° =1 «2¬« ±2 ° + ¯ ¨° =1 ³ °« ¬« ± ° + ²¬. The substitution´( 1, (((, , ¿)=exp ¼ ¾ ¿-1 2 Ã ¨ Ä =1 µ Ä Ä½ Î ( 1, (((, , ¿), where ¾= ¶-1 4 Ã ¨ Ä =1 µ2 Ä , leads to the Æ-dimensional heat equation ¸ ¹ Î = Ã Ï Î that is dealt with in Subsection 3.4.1. Page 270 5. « ¬« = ® ¯ ¨° =1 «2¬« ±2 ° + ¯ ¨° =1 Å³ °± ° + ² °È « ¬« ± ° + ¼ ¯ ¨Ñ° =1s °± ° + Ò ½¬. This is a special case of equation 3.4.3.4.6.Ó Ô Õ« ¬« + Ô Õ2 2 Ö ¯ ¨ ° =1 «2¬« ±2 ° = 0. This is the n-dimensional Schr Èodinger equation, ×2= -1 . Fundamental solution:Ø Ø(x, ¿)= - ×Ô Ù¼ ª2 Ô Ù¿ ½ Ú Û 2 exp ¼ × ª2 Ô Ù¿|x|2- × Æ 4 ½ , |x|2= 21+ ÜÜÜ+ 2Ú.Ý2Þ Reference : V . S. Vladimirov, V . P. Mikhailov, A. A. Vasharin, et al. (1974). 3.4.3. Equations Containing Arbitrary Functions 1. ß àß á= â ã äå =1 ß2àß æ2 å + Á ã äå =1 æ å ç å (á) + è(á) Éà. The transformation é ( ê1, (((, ê Ú, ¿)=exp Á ÚäNÄ =1 ê Ä ë Ä ( ¿)+ Ã ÚäNÄ =1 ì ë 2 Ä ( í) î í+ ï( í) ð ñ( ò1, (((, ò Ú, í),ò Ä = ê Ä + 2 óì ë Ä ( í) î í, ë Ä ( í)=ì ô Ä ( í) î í, ï( í)=ì õ( í) î í, leads to the Æ-dimensional heat equation ö ÷Êñ= ó ø ù Ä =1 ö úüûýúüû ñthat is discussed in Subsection 3.4.1. 2. ß àß á= â ã äå =1 ß2àß æ2 å + þ± ÿ ã äå =1 æ2 å + ã äÑå =1 æ å ç å (á) + è(á) ðà. 1 . Case >0. The transformationé ( ê1, (((, êø, í)= ñ( ò1, (((, òø, ) exp 1 2 ó ø NÄ =1 ê2 Ä ,ò1= ê1expÅ2 ó í , (((, òø= êøexp 2 ó í , =1 4 ó exp 4 ó í + , where is an arbitrary constant, leads to an equation of the form 3.4.3.1:ö ñö í= ó ø =1 ö2ñö ò2 + þ ø =1 ò ë ( )+ ï( ) ð ñ,ë ( )=1 ( )3 2ô ln( ) , ï( )=1õ ln( ) + Æ 4 , = 4 ó . 2 . Case <0. The transformation é ( ê1, (((, êø, í)= ( 1, (((, ø, ) exp þ - 2 ótan 2 - ó í ø =1 ê2 ð,1= ê1 cos 2- ó í , (((, ø= êøcos 2- ó í , =1 2- ó tan 2 - ó í also leads to an equation of the form 3.4.3.1 (this equation is not speci®ed here). Page 271 3. = =1 () 2 2 + (1, ,,). The solutions of various problems for this equation can be constructed on the basis of incompleteseparation of variables (see Paragraphs 0.6.1-2 and 0.9.2-1) taking into account the results ofSubsections 1.1.1 and 1.1.2. Some examples of solving such problems are given below. It isassumed that 0<ó ( í) < , = 1, (((, Æ. 1 . Domain: ø= {- < ê < ; = 1, (((, Æ}. Cauchy problem. An initial condition is prescribed: é =ô(x) at í= 0. Solution:é (x, í)= ! 0 " # $(y, %) &(x,y, ', %) (y ( %+ " #ô(y) &(x,y, ',0) (y, where&(x,y, ', %)=1 2 ) * ) 2 +1 +2 (((+)exp ,- ) =1( ê - - )2 4+ . ,+ = !/ 0 ( 1) ( 1, x= { ê1, (((, ê)},y= { -1, (((, -)}, (y= ( -1 ( -2 ((( ( -). 2 2. Domain: 3={0 £ 4 £ 5 ; = 1, (((, Æ}. First boundary value problem. The following conditions are prescribed:6=ô(x) at '= 0 (initial condition),6=õ (x, ') at 4 = 0 (boundary conditions),6= 7 (x, ') at 4 = 5 (boundary conditions). Solution:6(x, ')= ! 0 8 $(y, %) &(x,y, ', %) (y ( %+ 8ô(y) &(x,y, ') (y + ) =1 ! 0 9 ( :) 0 ( %) ,õ (y, %) ;; - &(x,y, ', %) .}:=0 ( <( ) }( % - )= =1 ! 0 9 ( :) 0 ( %) ,7 (y, %) ;; - &(x,y, ', %) .}:= >: ( <( ) }( %, where the following notation is used:( <( ) }= ( -1 ((( ?( - -1 ( - +1 ((( ?( -), <( )= {0 £ - @£ 5A@for B= 1, (((, -1, +1, (((, C}. The Green's function can be represented in the product form&(x,y, ', %)= )D E =1 & E ( 4 E , - E , ', %), ( 1) where the & E ( 4 E , - E , ', %) are the Green's functions of the respective boundary value problems,& E ( 4 E , - E , ', %)=25 E F=@=1sin G B * 4 E5 E H sin G B * - E5 E H exp G- B2*2+ E52 E H ,+ E = !/0 E ( I) ( I. (2) 3 2. Domain: 3={0 £ 4 E £ 5 E ; J= 1, (((, C}. Second boundary value problem. Page 272 The following conditions are prescribed:6= L(x) at '= 0 (initial condition),; M : 6= N E (x, ') at 4 E = 0 (boundary conditions),; M : 6= 7 E (x, ') at 4 E = 5 E (boundary conditions). Solution:6(x, ')= ! 0 8 $(y, %) &(x,y, ', %) (y ( %+ 8 L(y) &(x,y, ') (y - )=E =1 ! 0 9 ( :) 0 E ( %) OPN E (y, %) &(x,y, ', %) QSR:=0 ( <( E )R( % + )=E =1 ! 0 9 ( :) 0 E ( %)O 7 E (y, %) &(x,y, ', %)Q R:= >: ( <( E )R( %. The Green's function can be represented as the product (1) of the corresponding one-dimensionalGreen's functions& E ( 4 E , - E , ', %)=15 E +25 E F=@=1cos G B * 4 E5 E H cos G B * - E5 E H exp G- B2*2+ E52 E H ,+ E = !/ 0 E ( I) ( I.TVU Reference : A. D. Polyanin (2000a, 2000b). 4. W XW Y= Z =[ =1 \ [ (Y) W2XW ]2 [ + Z =[ =1 OV^ [ (Y)] [ + _ [ (Y) Q W XW ] [ + ,Z =[ =1s [ (Y)] [ + `(Y) .X. Let us perform the transformation6( 41, (((, 4), ')=exp , )=E =1 L E ( ') 4 E + N( ') . a ( b1, (((, b), '), b E = 7 E ( ') 4 E + c E ( '), where the functions L E ( '), N( '), 7 E ( '), and c E ( ') are given by ( d E , e E , f E , and gare arbitrary constants):7 E ( h)= d E exp ikj l E ( h) m hon,L E ( h)= p E ( h) j q E ( h)p E ( h) m h+ e Ep E ( h),c E ( h)= jO2 r E ( h) L E ( h)+ s E ( h)Q p E ( h) m h+ f E ,N( h)= j iut( h)+ v w x =1 r x ( h) L2 x ( h)+ v w x =1 s x ( h) L x ( h) n m h+ g. As a result, we arrive at an equation of the form 3.4.3.3 for the new dependent variable y=y( b1, (((, bv, h): zyzh= v w x =1 r x ( h) p2 x ( h) z 2yzb2 x . 5. { |{ }= ~ w =1 (}) {2|{ 2 + ~ w =1 V (}) + (}) { |{ + i ~ w =1s (})2 + ~ w =1 (}) + (}) n|. The substitution ( 1, , v, )=exp iv w x =1 L x ( ) 2 xny( 1, , v, ), where the functions L x = L x ( ) are solutions of the Riccati equationL x = 4 r x ( ) L2 x + 2 l x ( ) L x +q x ( ) ( = 1, , ), leads to an equation of the form 3.4.3.4 for y= y( 1, , v, ). Page 273 6. { |{ }± ~ w =1 ( ,}) {2|{ 2 + ( ,}) { |{ + ( ,})| n= (1, ,~,}). Here, 0< r x ( x , ) < for all . We introduce the notation x= { 1, , v},y= { 1, , v}and consider the domain = { x £ x £ x , = 1, , }, which is an -dimensional parallelepiped. 1 . First boundary value problem. The following conditions are prescribed: = L(x) at = 0 (initial condition), = x (x, ) at x = x (boundary conditions), = p x (x, ) at x = x (boundary conditions). Solution: (x, )= j 0 j (y, ) (x,y, , ) my m + L(y) (x,y, ,0) y + v w x =1 0 ¡ ( ¢) £ x ( x , ) x (y, ) zz x(x,y, , ) ¤ ¥¢= ¦¢ §( x ) ¥ - v w x =1 0 ¡ ( ¢) £ x ( x , )¨ x (y, ) zz x(x,y, , ) ¤ ¥¢= ©¢ §( x ) ¥ , where y= 1 2 v, §( x ) ¥= 1 ? x -1 x +1 v,§( x )= { ª£ ª£ ªfor «= 1, , -1, +1, , }. The Green's function can be represented in the product form(x,y, , )=v ¬ x =1 x ( x , x , , ). ( 1) Here, the x = x ( x , x , , ) are auxiliary Green's functions that, for > ³ 0, satisfy the one-dimensional linear homogeneous equationsz xz-£ x ( x , ) z 2 xz2 x - x ( x , ) z xz x - ® x ( x , ) x = 0 ( = 1, , ) ( 2) with nonhomogeneous initial conditions of a special form, x = ¯( x - x ) at = , ( 3) and homogeneous boundary conditions of the ®rst kind, x = 0 at x = x , x = 0 at x = x . In determining the function x , the quantities x and play the role of parameters; ¯( ) is the Dirac delta function.2. The second and third boundary value problems. The following conditions are prescribed: = L(x) at = 0 (initial condition),zM ¢ - ° x = x (x, ) at x = x (boundary conditions),zM ¢ + ± x =¨ x (x, ) at x = x (boundary conditions). The second boundary value problem corresponds to ° x = ± x = 0. Page 274 Solution: (x, )= 0 (y, ) (x,y, , ) y + L(y) (x,y, ,0) y - v w³x =1 0 ¡ ( ¢) £ x ( x , ) x (y, ) (x,y, , ) ¥¢= ¦¢ §( x ) ¥ +v w x =1 0 ¡ ( ¢) £ x ( x , )´¨ x (y, ) (x,y, , ) ¥¢= ©¢ §( x ) ¥ . The Green's function can be represented as the product (1) of the corresponding one-dimensionalGreen's functions satisfying the linear equations (2) with the initial conditions (3) and the homogen-eous boundary conditionszM ¢ x - ° x x = 0 at x = x ,zM ¢ x + ± x x = 0 at x = x .µV¶ Reference : A. D. Polyanin (2000a, 2000b). 7. { |{ }=~ w·, ¸=1 {{ ·A ·¸(1, ,~) { |{ ¸ ¤± (1, ,~)|+ (1, ,~,}). The problems considered below are assume to refer to a bounded domain with smooth surface §. We introduce the brief notation x= { 1, , v}and assume that the conditionv w¹, º=1 £ ¹º(x) » ¹»º³ ® v w¹=1 »2¹, ®>0, is satis®ed; this condition imposes the requirement that the differential operator on the right-handside of the equation is elliptic.1. First boundary value problem. The following conditions are prescribed: = L(x) at = 0 (initial condition), = (x, ) for x ¼ §(boundary condition). Solution: (x, )= 0 (y, ) (x,y, - ) ¥ + L(y) (x,y, ) ¥ - 0 ¡ (y, ) zz ½¥(x,y, - ) ¤ § ¥ . ( 1) Here, the Green's function is given by(x,y, )= ¾ ¿v=1 À v(x)À v(y)ÁÀ v Á2exp(- »v Â), ÁÀ v Á2= À2v(x) Ã,y= { Ä1, ÅÅÅ, Äv}, ( 2) where the »vandÀ v(x) are the eigenvalues and corresponding eigenfunctions of the Sturm±Liouville problem for the following elliptic second-order equation with homogeneous boundary condition ofthe ®rst kind:v ¿¹, º=1 ÆÆ Ç ¹ È£ ¹º(x)Æ ÀÆ Ç º ¤- É(x)À+ »À= 0, ( 3)À= 0 for x ¼ §. ( 4) Page 275 The integration in solution (1) is carried out with respect to Ä1, ÅÅÅ, Ä Ê; ËË Ì Íis the differential operator de®ned asÆ ÎÆ ½¥º Ê¿¹, º=1 Ï ¹º(y) ÐºÆ ÎÆ Ä ¹, ( 5) where N= { Ð1, ÅÅÅ, Ð Ê}is the unit outward normal to the surface Ñ. In the special case whereÏ ¹´¹(x)= 1andÏ ¹º(x)= 0for Ò¹ Ó,the operator of (5) coincides with the usual operator of differentiation along the direction of the outward normal to the surface Ñ. General properties of the Sturm±Liouville problem (3)±(4): 1. There are countably many eigenvalues. All eigenvalues are real and can be ordered so that»1£ »2£ »3£ ÔÔÔ, with » Ê Õ as Ö Õ ; consequently, there can exist only ®nitely many negative eigenvalues. 2. For É(x)³ 0all eigenvalues are positive: × Ê>0. 3. The eigenfunctions are de®ned up to a constant multiplier. Any two eigenfunctionsÀ Ê(x) andÀ Ø(x) corresponding to different eigenvalues × Êand ×Øare orthogonal in the domain Ã:ÙÀ Ê(x)À Ø(x) Ú Ã= 0 for Ö¹ Û.Ü ÝSÞ ß à?á âTo each eigenvalue × Êthere generally correspond ®nitely many linearly indepen- dent eigenfunctionsÀ(1)Ê,À(2)Ê, ÅÅÅ,À(Ø)Ê. These functions can always be replaced by their linear combinations ÅÀ( ã)Ê= äã,1À(1)Ê+ ÔÔÔ+ äã, ã-1À( ã-1)Ê+À( ã)Ê, å= 1,2, ÅÅÅ, Û, such that ÅÀ(1)Ê, ÅÀ(2)Ê, ÅÅÅ, ÅÀ(Ø)Êare now pairwise orthogonal. Thus, without loss of generality, we assume that all eigenfunctions are orthogonal. 2 . Second boundary value problem. The following conditions are prescribed:æ= ç(x) atÂ= 0 (initial condition),Æ æÆ è é= (x,Â) for x ê Ñ(boundary condition). Here, the left-hand side of the boundary condition is determined with the help of (5), whereÎ, Ä,y, and Äãmust be replaced by æ,Ç,x, andÇ ã, respectively. Solution:æ(x,Â)= Ù 0 Ù ë(y, ì)Î(x,y,Â- ì) Ú Ã í Ú ì+ Ùç(y)Î(x,y,Â) Ú Ã í + Ù 0 Ù î(y, ì)Î(x,y,Â- ì) Ú Ñ í Ú ì. ( 6) Here, the Green's function is de®ned by (2), where the × ÊandÀ Ê(x) are the eigenvalues and corre- sponding eigenfunctions of the Sturm±Liouville problem for the elliptic second-order equation (3) with a homogeneous boundary condition of the second kind:Æ ÀÆ è é= 0 for x ê Ñ. ( 7) For É(x) >0the general properties of the eigenvalue problem (3), (7) are the same as for the ®rst boundary value problem (see Item 1 ). For É(x)º 0the zero eigenvalue ×0= 0arises which corresponds to the eigenfunctionÀ0=const. It should be noted that the Green's function of the second boundary value problem can be expressed in terms of the Green's function of the third boundary value problem (see Item 3 ). Page 276 3 . Third boundary value problem. The following conditions are prescribed:æ= ç(x) atÂ= 0 (initial condition),Æ æÆ è é+ å(x) æ= (x,Â) for x ê Ñ(boundary condition). The solution of the third boundary value problem is given by relations (6) and (2), where the × Ê andÀ Ê(x) are the eigenvalues and corresponding eigenfunctions of the Sturm±Liouville problem for the second-order elliptic equation (3) with a homogeneous boundary condition of the third kind:Æ ÀÆ è é+ å(x)À= 0 for x ê Ñ. ( 8) For É(x)³ 0and å(x) >0, the general properties of the eigenvalue problem (3), (8) are the same as for the ®rst boundary value problem (see Item 1 ). Let å(x)= å=const. Denote the Green's functions of the second and third boundary value problems byÎ2(x,y,Â) andÎ3(x,y,Â, å), respectively. Then the following relations hold:Î2(x,y,Â)= ð ñPòlimãôó0Î3(x,y,Â, å), if É(x) >0; 1Ã0+limãôó0Î3(x,y,Â, å), if É(x)º 0; where Ã0= ÙÚ Ãis the volume of the domain in question.õVö References : V . M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), A. D. Polyanin (2000a, 2000b). Page 277 Chapter 4 Hyperbolic Equations with One Space Variab le 4.1. Constant Coef®cient Equations 4.1.1. WaveEquation ÷2 ø÷ ù2= ú2÷2 ø÷ û2 This equation isalso knownastheequation ofvibration ofastring .Itisoften encountered in elasticity ,aerodynamics, acoustics, andelectrodynamics. 4.1.1-1. General solution. Some formulas. 1 ü.General solution:æ(Ç, ý)= þ( ÿ+Ï ý)+ ( ÿ-Ï ý), where þ( ÿ)and ( ÿ)arearbitrary functions. Physical interpr etation :Thesolution represents two traveling wavesthatpropagate, respecti vely,totheleftandright along the ÿ-axis ataconstant speedÏ. 2 ü.Fundamental solution: ( ÿ, ý)=1 2Ï Ï ý-| ÿ| , ( )= 0for <0, 1for >0. 3 ü.In®nite series solutions containing arbitrary functions ofthespace variable:æ( ÿ, ý)= ç( ÿ)+ =1( ý)2 (2 Ö)! ç(2 )é( ÿ), ç(Ø)é( ÿ)= ÚØ Ú ÿØ ç( ÿ),æ( ÿ, ý)= ý ( ÿ)+ ý =1( ý)2 (2 Ö+1)! (2 )é( ÿ), where ç( ÿ)and ( ÿ)areanyin®nitely differentiable functions. The®rstsolution satis®es theinitial conditions æ( ÿ,0)= ç( ÿ)and æ( ÿ,0)=0,andthesecond æ( ÿ,0)=0and æ( ÿ,0)= ( ÿ).The sums are®nite if ç( ÿ)and ( ÿ)arepolynomials. 4 ü.In®nite series solutions containing arbitrary functions oftime:æ( ÿ, ý)= ç( ý)+ =112 (2 Ö)! ÿ2 ç(2 )( ý), ç(Ø)( ý)= ÚØ Ú ýØ ç( ý),æ( ÿ, ý)= ÿ ( ý)+ ÿ =112 (2 Ö+1)! ÿ2 (2 )( ý), where ç( ý)and ( ý)areanyin®nitely differentiable functions. Thesums are®nite if ç( ý)and ( ý) arepolynomials. The®rstsolution satis®es theboundary condition ofthe®rstkind æ(0, ý)= ç( ý), andthesecond solution totheboundary condition ofthesecond kind é æ(0, ý)= ( ý). Page279 5 ü. If æ( ÿ, ý) is a solution of the wave equation, then the functionsæ 1= ä æ( × ÿ+ 1, ý+ 2),æ 2= ä æ ÿ- ý 1 -( )2, ý- -2ÿ 1 -( )2 , 3= 2- 2 2, 2- 2 2, are also solutions of the equation everywhere these functions are de®ned ( , 1, 2, , and are arbitrary constants). The signs at 's in the formula for 1are taken arbitrarily, independently of each other. The function 2results from the invariance of the wave equation under the Lorentz transformations. References : G. N. Polozhii (1964), A. V . Bitsadze and D. F. Kalinichenko (1985). 4.1.1-2. Domain: - << . Cauchy problem. Initial conditions are prescribed:= () at = 0, = () at = 0. Solution (D'Alembert's formula):(, )=1 2[ (+ )+ (- )]+1 2 ! "+ #%$"- #%$ &( ') ( '. 4.1.1-3. Domain: 0 £< . First boundary value problem. 1 ). Problem with a homogeneous boundary condition:= () at = 0 (initial condition),*$ =&() at = 0 (initial condition),= 0 at= 0 (boundary condition). Solution:(, )= +, , -, ,. 1 2[ (+ / )+ (- / )]+1 2 / !"+ #%$"- #%$ &( ') ( 'for < /, 1 2[ (+ / )- ( / -)]+1 2 / !"+ #%$#%$-" &( ') ( 'for > /. 2 ). Problem with a nonhomogeneous boundary condition:= () at = 0 (initial condition),*$ =&() at = 0 (initial condition),= 0( ) at= 0 (boundary condition). Solution:(, )= +, , -, ,. 1 2[ (+ / )+ (- / )]+1 2 / ! "+ #%$"- #%$ &( ') ( ' for < /, 1 2[ (+ / )- ( / -)]+1 2 / ! "+ #%$#%$-" &( ') ( '+ 0 1 - / 2for > /. In the domain < 3 /the boundary conditions have no effect on the solution and the expression of (, ) coincides with D'Alembert's solution for an in®nite line (see Paragraph 4.1.1-2). Reference : A. N. Tikhonov and A. A. Samarskii (1990). Page 280 4.1.1-4. Domain: 0 £ < . Second boundary value problem. 1 . Problem with a homogeneous boundary condition:= ( ) at = 0 (initial condition), = ( ) at = 0 (initial condition), = 0 at = 0 (boundary condition). Solution:( , )= 1 2[ ( + )+ ( - )]+1 2 [ ( + )- ( - )] for < , 1 2[ ( + )+ ( - )]+1 2 [ ( + )+ ( - )] for > , where ( )= 0 ( ) . 2 . Problem with a nonhomogeneous boundary condition:= ( ) at = 0 (initial condition), = ( ) at = 0 (initial condition), = ( ) at = 0 (boundary condition). Solution:( , )= 1 2[ ( + )+ ( - )]+1 2 [ ( + )- ( - )] for < , 1 2[ ( + )+ ( - )]+1 2 [ ( + )+ ( - )]- - for > , where ( )= 0 ( ) and ( )= 0 ( ) . In the domain < the boundary conditions have no effect on the solution, and the expression of ( , ) coincides with D'Alembert's solution for an in®nite line (see Paragraph 4.1.1-2). Reference : B. M. Budak, A. N. Tikhonov, and A. A. Samarskii (1980). 4.1.1-5. Domain: 0 £ £ . First boundary value problem. 1 . Vibration of a string with rigidly ®xed ends. The following conditions are prescribed:= ( ) at = 0 (initial condition), = ( ) at = 0 (initial condition),= 0 at = 0 (boundary condition),= 0 at = (boundary condition). Solution:( , )= "! =1 #$ ! cos( % ! )+ & ! sin( % ! ) 'sin( % !), % ! = ( ),$ ! =2 * 0 ( ) sin( % !) , & ! =2( ) * 0 ( ) sin( % !) . Example 1. The initial shape of the string is a triangle with base 0 £ +£ ,and height -at += ., i.e.,/( +)= 01 213 - +.for0 £ +£ .,-( ,- +),- .for .£ +£ ,. Page 281 The initial velocities of the string points are zero, 4( +)= 0. Solution: 5 ( +, 6)=2 - ,272.( ,- .) 8 9 : =11;2sin < ;7., =sin < ;7+, =cos < ;7 >6, =. Example 2. Initially, the string has the shape of a parabola symmetric about the center of the string with elevation -, so that/( +)=4 -,2 +( ,- +). The initial velocities of the string points are zero, 4( +)= 0. Solution: 5 ( +, 6)=32 -738 9 : =01 (2 ;+ 1)3sin ?(2 ;+ 1) 7+, @cos ?(2 ;+ 1) 7 >6, @. 2 . For the solution of the ®rst boundary value problem with a nonhomogeneous boundary condition, see Paragraph 4.1.2-4 with A( , )º 0. References : B. M. Budak, A. N. Tikhonov, and A. A. Samarskii (1980), A. V . Bitsadze and D. F. Kalinichenko (1985). 4.1.1-6. Domain: 0 £ £ . Second boundary value problem. 1 . Longitudinal vibration of an elastic rod with free ends. The following conditions are prescribed:= ( ) at = 0 (initial condition), = ( ) at = 0 (initial condition), = 0 at = 0 (boundary condition), = 0 at = (boundary condition). Solution:( , )=$0+ &0 + "! =1 #$ ! cos( % ! )+ & ! sin( % ! ) 'cos( % !),% ! = ( ),$0=1 * 0 ( ) , &0=1 * 0 ( ) ,$ ! =2 * 0 ( ) cos( % !) , & ! =2( ) * 0 ( ) cos( % !) . 2 . For the solution of the second boundary value problem with a nonhomogeneous boundary condition, see Paragraph 4.1.2-5 with A( , )º 0. Reference : A. V . Bitsadze and D. F. Kalinichenko (1985). 4.1.1-7. Domain: 0 £ £ . Third boundary value problem. 1 . Longitudinal vibration of an elastic rod with clamped ends in the case of equal stiffness coef®- cients. The following conditions are prescribed:= ( ) at = 0 (initial condition), = ( ) at = 0 (initial condition), - B = 0 at = 0 (boundary condition), + B = 0 at = (boundary condition). Solution:( , )= C! =1 #$ ! cos( % ! )+ & ! sin( % ! ) 'sin( % !+ D ! ), Page 282 where$ ! =1EGF !E2 * 0sin( % !+ D ! ) ( ) , & ! =1 % !EGF !E2 * 0sin( % !+ D ! ) ( ) ,D ! =arctan % !B, EGF !E2= * 0sin2( % !+ D ! ) = 2+ BB2+ %2 ! ; the % ! are positive roots of the transcendental equation cot( % )=1 2 H %B- B% I. 2 . Longitudinal vibration of an elastic rod with clamped ends in the case of different stiffness coef®cients. The following conditions are prescribed:= ( ) at = 0 (initial condition), = ( ) at = 0 (initial condition), - B1 = 0 at = 0 (boundary condition), + B2 = 0 at = (boundary condition). Solution:( , )= ! =1 #$ ! cos( % ! )+ & ! sin( % ! ) 'sin( % !+ D ! ), where$ ! =1EGF !E2 * 0sin( % !+ D ! ) ( ) , & ! =1 % !EGF !E2 * 0sin( % !+ D ! ) ( ) ,D ! =arctan % !B1, EGF !E2= * 0sin2( % !+ D ! ) = 2+( %2 ! + B1 B2)( B1+ B2) 2( %2 ! + B2 1)( %2 ! + B2 2); the % ! are positive roots of the transcendental equation cot( % )= %2- B1 B2%( B1+ B2). 3 . For the solution of the third boundary value problem with nonhomogeneous boundary conditions, see Paragraph 4.1.2-6 with A( , )º 0. Reference : B. M. Budak, A. N. Tikhonov, and A. A. Samarskii (1980). 4.1.1-8. Domain: 0 £ £ . Mixed boundary value problem. 1 . Longitudinal vibration of an elastic rod with one end rigidly ®xed and the other free. The following conditions are prescribed:= ( ) at = 0 (initial condition), = ( ) at = 0 (initial condition),= 0 at = 0 (boundary condition), = 0 at = (boundary condition). Solution:( , )= C! =0 #$ ! cos( % ! )+ & ! sin( % ! ) 'sin( % !), % ! = ((2)+ 1) 2 ,$ ! =2 * 0 ( ) sin( % !) , & ! =2 J% !* 0 ( ) sin( % !) . 2 . For the solution of the mixed boundary value problem with nonhomogeneous boundary condi- tions, see Paragraph 4.1.2-7 with A( , )º 0. References : M. M. Smirnov (1975), A. V . Bitsadze and D. F. Kalinichenko (1985). Page 283 4.1.1-9. Goursat problem. The boundary conditions are prescribed to the equation characteristics:= ( ) for - = 0 (0 £ £ K),= ( ) for + = 0 (0 £ £ L), where (0)= (0). Solution:( , )= H + 2 I+ H - 2 I- (0). The solution propagation domain is bounded by four lines:- = 0, + = 0, - = 2 L, + = 2 K. Reference : A. V . Bitsadze and D. F. Kalinichenko (1985). 4.1.2. Equations of the Form M2 NM O2= P2M2 NM Q2+ R(Q,O) 4.1.2-1. Domain: - S< T< S. Cauchy problem. Initial conditions are prescribed: U = V( T) at W= 0,X Y U = Z( T) at W= 0. Solution:U ( T, W)=1 2[ V( T- [ W)+ V( T+ [ W)]+1 2 [ \+ ] Y^\- ] Y Z( _) ` _+1 2 [ Y^ 0 \+ ]( Y - a)^\- ]( Y - a) A( _, b) ` _ ` b. 4.1.2-2. Domain: 0 £ T< S. First boundary value problem. The following conditions are prescribed:U = V( T) at W= 0 (initial condition),X Y U = Z( T) at W= 0 (initial condition),U = c( W) at T= 0 (boundary condition). Solution: U ( T, W)= U 1( T, W)+1 2 [ U 2( T, W), whereU 1( T, W)= de e e e e efe e e e e eg 1 2[ V( T+ [ W)+ V( T- [ W)]+1 2 [ \+ ] Y^\- ] Y Z( _) ` _ for W< T[, 1 2[ V( T+ [ W)- V( [ W- T)]+1 2 [ \+ ] Y^] Y -\ Z( _) ` _+ c hi- j k lfor i> j k,U 2(j, i)= de e e e e e efe e e e e e eg m^ 0 n+ o( m- a)^n- o( m- a) A( _, b) ` _ ` b for i< j k,m- n po^ 0 n+ o( m- a)^o( m- a)- n A( _, b) ` _ ` b+ m^m- n po n+ o( m- a)^n- o( m- a) A( _, b) ` _ ` bfor i> jk. Reference : A. V . Bitsadze and D. F. Kalinichenko (1985). Page 284 4.1.2-3. Domain: 0 £j< q. Second boundary value problem. The following conditions are prescribed:U = r(j) at i= 0 (initial condition),sm U = t(j) at i= 0 (initial condition),sn U = c( i) atj= 0 (boundary condition). Solution: U (j, i)= U 1(j, i)+1 2 k U 2(j, i), whereU 1(j, i)= d e e e e e e e e e e e e efe e e e e e e e e e e e eg 1 2[ r(j+ ki)+ r(j- ki)]+1 2 k n+ o m^n- o m t( _) ` _ for i< jk, 1 2[ r(j+ ki)+ r( ki-j)]+1 2 k n+ o m^ 0 t( _) ` _ +1 2 k o m- n^ 0 t( _) ` _- k m- n po^ 0 c( _) ` _for i> j k,U 2(j, i)= d e e e e e e e e e e e e e efe e e e e e e e e e e e e eg m^ 0 n+ o( m- a)^n- o( m- a) A( _, b) ` _ ` b for i< j k,m- n po^ 0 n+ o( m- a)^ 0 A( _, b) ` _ ` b+ m- n po^ 0 o( m- a)- n^ 0 A( _, b) ` _ ` b + m^m- n po n+ o( m- a)^n- o( m- a) A( _, b) ` _ ` b for i> j k.uv Reference : A. V . Bitsadze and D. F. Kalinichenko (1985). 4.1.2-4. Domain: 0 £j£ w. First boundary value problem. The following conditions are prescribed:U = r0(j) at i= 0 (initial condition),sm U = r1(j) at i= 0 (initial condition),U = t1( i) atj= 0 (boundary condition),U = t2( i) atj= w(boundary condition). Solution:U (j, i)= ssi ^ x 0 r0( _) y(j, _, i) ` _+ ^ x 0 r1( _) y(j, _, i) ` _+ ^ m 0 ^ x 0 A( _, b) y(j, _, i- b) ` _ ` b + k2 ^ m 0 t1( b) z ss_ y(j, _, i- b) { | =0 ` b- k2 ^ m 0 t2( b) z ss_ y(j, _, i- b) { | = x ` b, wherey(j, _, i)=2k } ~ C =11sin h }jw lsin h }_w lsin h } kiw l. Page 285 4.1.2-5. Domain: 0 £j£ w. Second boundary value problem. The following conditions are prescribed: U = r0(j) at i= 0 (initial condition),sm U = r1(j) at i= 0 (initial condition),sn U = t1( i) atj= 0 (boundary condition),sn U = t2( i) atj= w(boundary condition). Solution:U (j, i)= ssi ^x 0 r0( _) y(j, _, i) ` _+ ^x 0 r1( _) y(j, _, i) ` _+ ^ m 0 ^x 0 A( _, b) y(j, _, i- b) ` _ ` b - k2 ^ m 0 t1( b) y(j,0, i- b) ` b+ k2 ^ m 0 t2( b) y(j, w, i- b) ` b, wherey(j, _, i)= iw+2k } ~ =11cos h }jw lcos h }_w lsin h } kiw l. 4.1.2-6. Domain: 0 £j£ w. Third boundary value problem. The following conditions are prescribed: U = r0(j) at i= 0 (initial condition),sm U = r1(j) at i= 0 (initial condition),sn U - B1 U = t1( i) atj= 0 (boundary condition),sn U + B2 U = t2( i) atj= w(boundary condition). The solution U (j, i) is determined by the formula in Paragraph 4.1.2-5 wherey(j, _, i)=1k ~ =11 G 2sin( j+ D ) sin( _+ D ) sin( ki),D =arctan B1, G 2= w 2+( 2 + B1 B2)( B1+ B2) 2( 2 + B2 1)( 2 + B2 2); the are positive roots of the transcendental equation cot( w)= 2- B1 B2( B1+ B2). 4.1.2-7. Domain: 0 £j£ w. Mixed boundary value problem. The following conditions are prescribed: U = r0(j) at i= 0 (initial condition),sm U = r1(j) at i= 0 (initial condition),U = t1( i) atj= 0 (boundary condition),sn U = t2( i) atj= w(boundary condition). Solution:U (j, i)= ssi ^ x 0 r0( _) y(j, _, i) ` _+ ^ x 0 r1( _) y(j, _, i) ` _+ ^ m 0 ^ x 0 A( _, b) y(j, _, i- b) ` _ ` b + k2 ^ m 0 t1( b)z ss_ y(j, _, i- b){ | =0 ` b+ k2 ^ m 0 t2( b) y(j, w, i- b) ` b, wherey(j, _, i)=2kw ~ C =11 sin( j) sin( _) sin( ki), = }(2 + 1) 2 w. Page 286 4.1.3. Equation of the Form 2 2= P22 Q2± + (Q,) This equation with (j, i)º 0and >0is encountered in quantum ®eld theory and a number of applications and is referred to as the Klein±Gordon equation . 4.1.3-1. Solutions of the homogeneous equation ( º 0). 1 . Particular solutions:U (j, i)=exp( i)( j+ ), = - 2,U (j, i)=exp( j)( i+ ), = k2 2,U (j, i)=cos( j)[ cos( i)+ sin( i)], = - k2 2+ 2,U (j, i)=sin( j)[ cos( i)+ sin( i)], = - k2 2+ 2,U (j, i)=exp( i)[ cos( j)+ sin( j)], = - k2 2- 2,U (j, i)=exp( j)[ cos( i)+ sin( i)], = k2 2+ 2,U (j, i)=exp( j)[ exp( i)+ exp(- i)], = k2 2- 2,U (j, i)= 0( _)+ 0( _), _= k k2( i+ 1)2-(j+ 2)2, >0,U (j, i)= 0( )+ 0( ), = - k k2( i+ 1)2-(j+ 2)2, <0, where , , 1, and 2are arbitrary constants, 0( ) and 0( ) are the Bessel functions, and 0( ) and 0( ) are the modi®ed Bessel functions. 2 . Fundamental solutions: (j, i)= ( ki- |j|) 2 k 0 h k k2i2-j2lfor =2>0, (j, i)= ( ki- |j|) 2 k 0 h k k2i2-j2lfor = -2<0, where( ) is the Heaviside unit step function (= 0for <0and= 1for ³ 0), 0( ) is the Bessel function, and 0( ) is the modi®ed Bessel function.uv Reference : V . S. Vladimirov, V . P. Mikhailov, A. A. Vasharin, et al. (1974). 4.1.3-2. Some formulas and transformations of the homogeneous equation ( º 0). 1 . Suppose = (j, i) is a solution of the Klein±Gordon equation. Then the functions1= (j+ 1, i+ 2),2= (-j+ 1, i+ 2),3= - ¡1 -( ¢ £)2, ¡- £-21 -( ¢ £)2 ¤, where , 1, 2, and are arbitrary constants, are also solutions of this equation. 2 .Table19liststransformation softheindependen tvariable sthatallowseparatio nofvariable sin the Klein±Gordon equation. Notation: ¥( ) and ¥( ) are the Bessel functions, ¦¥( ) and ¥( ) are the modi®ed Bessel functions, and § ¨( ) is the parabolic cylinder function.©ª References : E. Kalnins (1975), W. Miller, Jr. (1977). Page 287 TABLE 19 Orthogonal coordinates «= «(, ¡), = (, ¡) admitting separable solutions = ¬( «) ( ) of the Klein±Gordon equation ( £= 1; ®1, ®2, ¯1, ¯2, and °are arbitrary constants) NoRelation between, ¡and «, Function ¬= ¬( «) (differential equation)Function = ( ) (differential equation) 1= «, ¡= ¬= ®1 ± ² ³ ¨+ ´+ ®2 ±-² ³ ¨+ ´= ¯1 ± µ¶³ ¨+ ¯2 ±-µ¶³ ¨ 2= «sinh ,¡= «cosh ¬= · « ¸®1 ¹ ¥ º»« · ¼ ½+ ®2 ¾ ¥ º»« · ¼ ½G¿,À=1 2 ·1+ °2 = ¯1 ± ¨µ+ ¯2 ±- ¨µ 3 = « ,¡=1 2( «2+ 2) ¬= ®1 § ¨( Á «)+ ®2 § ¨(- Á «),Á=(-4 ¼)1 Â4 = ¯1 § ¨( Á )+ ¯2 § ¨(- Á ),Á=(-4 ¼)1 Â4 4=1 2( «2+ 2),¡= « ¬= ®1 § ¨( Á «)+ ®2 § ¨(- Á «),Á=(4 ¼)1 Â4 = ¯1 § ¨( Á )+ ¯2 § ¨(- Á ),Á=(4 ¼)1 Â4 5 =-1 2( «- )2+ «+ ,¡=1 2( «- )2+ «+ ¬= · Ã¸ ®1 ¹1 3( Ä)+ ®2 ¾1 3( Ä) ¿,Ã= «+ °, Ä=2 3 · ¼Ã3 Â2 = · Å¸ ¯1 ¹1 3( Æ)+ ¯2 ¾1 3( Æ) ¿,Å= + °, Æ=2 3 · ¼Å3 Â2 6 ¡+=cosh¸1 2( «- ) ¿,¡-=sinh¸1 2( «+ ) ¿ ¬ ÇÈÇ+( °+ ¼sinh «) ¬= 0 ÇÈÇ+( °+ ¼sinh ) = 0 7 =sinh( «- )-1 2 ± ²+µ,¡=sinh( «- )+1 2 ± ²+µ ¬= ®1 ¹ ¨( Á± ²)+ ®2 ¾ ¨( Á± ²),Á= · ¼ = ¯1 É ¨( Á± µ)+ ¯1 Ê ¨( Á± µ),Á= · ¼ 8 =cosh( «- )-1 2 ± ²+µ,¡=cosh( «- )+1 2 ± ²+µ ¬= ®1 ¹ ¨( Á± ²)+ ®2 ¾ ¨( Á± ²),Á= · ¼ = ¯1 ¹ ¨( Á± µ)+ ¯1 ¾ ¨( Á± µ),Á= · ¼ 9=cosh «sinh ,¡=sinh «cosh ¬ ÇÈÇ+( °+1 2 ¼cosh 2 «) ¬= 0, modi®ed Mathieu equation ÇÈÇ+( °-1 2 ¼cosh 2 ) = 0, modi®ed Mathieu equation 10=sinh «sinh ,¡=cosh «cosh ¬ ÇÈÇ+( °+1 2 ¼cosh 2 «) ¬= 0, modi®ed Mathieu equation ÇÈÇ+( °+1 2 ¼cosh 2 ) = 0, modi®ed Mathieu equation 11 =sin «sin ,¡=cos «cos ¬ ÇÈÇ+( °-1 2 ¼cos2 «) ¬= 0, Mathieu equation ÇÈÇ+( °-1 2 ¼cos2 ) = 0, Mathieu equation 4.1.3-3. Domain: - Ë<< Ë. Cauchy problem. Initial conditions are prescribed: Ì = Í() at ¡= 0,Î Ï Ì = Ð() at ¡= 0. Solution for ¼= - Ñ2<0:Ì (, ¡)=1 2[ Í(+ £ ¡)+ Í(- £ ¡)]+ ÑG¡ 2 £ Ò Ó+ Ô ÏÓ- Ô Ï É1 º Ñ Õ ¡2-(- Ä)2¢ £2½Õ ¡2-(- Ä)2¢ £2 Í( Ä) Ö Ä +1 2 £ Ò Ó+ Ô ÏÓ- Ô ÏÉ0 º Ñ × Ø2-( Ù- Ä)2 Ú Û2½Ð( Ä) Ö Ä +1 2 ÛÒ Ï 0 Ò Ó+ Ô( Ï - Ü)Ó- Ô( Ï - Ü) É0 º Ñ ×( Ø- Ý)2-( Ù- Ä)2 Ú Û2½ Þ( Ä, Ý) Ö Ä Ö Ý, whereÉ0( ß) andÉ1( ß) are the modi®ed Bessel functions of the ®rst kind. Page 288 Solution for ¼= Ñ2>0:Ì ( Ù, Ø)=1 2[ Í( Ù+ ÛØ)+ Í( Ù- ÛØ)]- ÑGØ 2 ÛÒ Ó+ Ô ÏÓ- Ô Ï ¹1 º Ñ Õ Ø2-( Ù- Ä)2 Ú Û2½Õ Ø2-( Ù- Ä)2 Ú Û2 Í( Ä) Ö Ä +1 2 ÛÒ Ó+ Ô ÏÓ- Ô Ï¹0 º Ñ ×Ø2-( Ù- Ä)2 Ú Û2½Ð( Ä) Ö Ä +1 2 ÛÒ Ï 0 Ò Ó+ Ô( Ï - Ü)Ó- Ô( Ï - Ü) ¹0 º Ñ ×( Ø- Ý)2-( Ù- Ä)2 Ú Û2½ Þ( Ä, Ý) Ö Ä Ö Ý, where¹0( ß) and¹1( ß) are the Bessel functions of the ®rst kind.©ª Reference : B. M. Budak, A. N. Tikhonov, and A. A. Samarskii (1980). 4.1.3-4. Domain: 0 £ Ù£ à. First boundary value problem. The following conditions are prescribed: Ì = Í0( Ù) at Ø= 0 (initial condition),Î Ï Ì = Í1( Ù) at Ø= 0 (initial condition), Ì = Ð1( Ø) at Ù= 0 (boundary condition), Ì = Ð2( Ø) at Ù= à(boundary condition). Solution:Ì ( Ù, Ø)= ÎÎØ Ò á0 Í0( Ä) ( Ù, Ä, Ø) Ö Ä+Ò á0 Í1( Ä) ( Ù, Ä, Ø) Ö Ä+Ò Ï 0 Ò á0 Þ( Ä, Ý) ( Ù, Ä, Ø- Ý) Ö Ä Ö Ý + Û2Ò Ï 0 Ð1( Ý) â ÎÎÄ ã( Ù, Ä, Ø- Ý) ä å =0 Ö Ý- Û2Ò Ï 0 Ð2( Ý) â ÎÎÄ ã( Ù, Ä, Ø- Ý) ä å =á Ö Ý, whereã( Ù, Ä, Ø)=2à æ çCè =1sin( é èÙ) sin( é è ê )sinº Øë Û2é2 è + ì íë Û2é2 è + ì, é è = î ïà.ð ñóò ô õ÷ö øLet ì<0and Û2é2 è + ì<0forï= 1, ù¶ù¶ù, úand Û2é2 è + ì>0forï= ú+ 1, ú+ 2, ù¶ù¶ù In this case the Green's function is modi®ed and acquires the formã( Ù, ê , Ø)=2à û çè =1sin( é èÙ) sin( é è ê )sinh º»Øë| Û2é2 è + ì| íë| Û2é2 è + ì| +2à æ ç è =û+1sin( é èÙ) sin( é è ê )sin º»Øë Û2é2 è + ì íë Û2é2 è + ì, é è = î ïà. Analogously, the Green's functions for the second, third, and mixed boundary value problems aremodi®ed in similar cases.üý Reference : A. G. Butkovskiy (1979). 4.1.3-5. Domain: 0 £ Ù£ à. Second boundary value problem. The following conditions are prescribed:Ì = þ0( Ù) at Ø= 0 (initial condition),ÿ Ì = þ1( Ù) at Ø= 0 (initial condition),ÿ Ì = 1( Ø) at Ù= 0 (boundary condition),ÿ Ì = 2( Ø) at Ù= à(boundary condition). Page 289 Solution:Ì ( Ù, Ø)= ÿÿØ 0 þ0( ê ) ( , ê , ) ê + 0 þ1( ê ) ( , ê , ) ê + 0 0 ( ê , ) ( , ê , - ) ê - 2 0 1( ) ( ,0, - ) + 2 0 2( ) ( , , - ) , where( , ê , )=1 ìsin ì í+2 çè =1cos( é è) cos( é è ê )sin ë 2é2 è + ì íë 2é2 è + ì, é è = î ï. 4.1.3-6. Domain: 0 £ £ . Third boundary value problem. The following conditions are prescribed:Ì = þ0( ) at = 0 (initial condition),ÿ Ì = þ1( ) at = 0 (initial condition),ÿ Ì - 1 Ì = 1( ) at = 0 (boundary condition),ÿ Ì + 2 Ì = 2( ) at = (boundary condition). The solution Ì ( , ) is determined by the formula in Paragraph 4.1.3-5 where( , ê , )= çCè =1 è ( ) è ( ê ) sin ë 2é2 è + ì í è2ë 2é2 è + ì, è ( )=cos( é è)+ 1é è sin( é è), è2= 2 2 é2 èé2 è + 2 1é2 è + 2 2+ 1 2 é2 è + 2 1 + 2 1é2 è . Here, the é è are positive roots of the transcendental equationtan( é )é= 1+ 2é2- 1 2. 4.1.3-7. Domain: 0 £ £ . Mixed boundary value problem. The following conditions are prescribed:Ì = þ0( ) at = 0 (initial condition),ÿ Ì = þ1( ) at = 0 (initial condition), Ì = 1( ) at = 0 (boundary condition),ÿ Ì = 2( ) at = (boundary condition). Solution:Ì ( , )= ÿÿ 0 þ0( ê ) ( , ê , ) ê + 0 þ1( ê ) ( , ê , ) ê + 0 0 ( ê , ) ( , ê , - ) ê + 2 0 1( ) ÿÿ ê( , ê , - ) =0 + 2 0 2( ) ( , , - ) , where( , ê , )=2 =0sin( ) sin( )sin 22 + ! " 22 + !, = #(2 $+ 1) 2 . Page 290 4.1.4. Equation of the Form %2 &% '2= (2%2 &% )2± * % &% )+ +(),') 4.1.4-1. Reduction to the nonhomogeneous Klein±Gordon equation. The substitution Ì ( , )=exp 1 2 !, - 2" .( , ) leads the nonhomogeneous Klein±Gordon equation/2./2= 2 /2./2- !2 4 2 .+exp- !, 2 2 ( , ), which is discussed in Subsection 4.1.3. 4.1.4-2. Domain: - 0< < 0. Cauchy problem. Initial conditions are prescribed:Ì = þ( ) at = 0,/ 1 Ì = 2( ) at = 0. Solution:Ì ( , )=1 2 þ( + ) exp- !, 2 +1 2 þ( - ) exp !, 2 - 3 2 exp !, 2 2 4 5+ 6 15- 6 1exp 7- ! 2 82 9 :1 ;3 2-( - )2- 82" 2-( - )2- 82 þ( ) +1 2 8exp7 !, 2 829 4 5+ 6 15- 6 1exp7- ! 2 829:0 ;3 < 2-( - )2- 82"=2( ) +1 2 8 4 1 0 45+ 6( 1 - >)5- 6( 1 - >)exp ? !( - ) 2 82 @:0 ;3<( - A)2-( - )2- 82" B( , A) A, where:0( C) and:1( C) are the Bessel functions of the ®rst kind, and3=1 2| !| - 8. 4.1.4-3. Domain: 0 £ £ D. First boundary value problem. The following conditions are prescribed:Ì = þ0( ) at = 0 (initial condition),/ 1 Ì = þ1( ) at = 0 (initial condition), Ì = 21( ) at = 0 (boundary condition), Ì = 22( ) at = D(boundary condition). Solution:Ì ( , )= // 4 E 0 þ0( ) ( , , ) + 4 E 0 þ1( ) ( , , ) + 4 1 0 4 E 0 B( , A) ( , , - A) A + 82 4 1 0 21( A) ? // ( , , - A)@ F =0 A- 82 4 1 0 22( A) ? // ( , , - A)@ F = E A, where( , , )=2Dexp ? ! 2 82( - )@ G =1sin7 # $ D 9sin7 # $ D 9sin; " , = H 82#2$2D2+ I2 4 82.JLK Reference : A. G. Butkovskiy (1979). Page 291 4.1.4-4. Domain: 0 £ £ D. Second boundary value problem. The following conditions are prescribed:Ì = þ0( ) at = 0 (initial condition),M N Ì = þ1( ) at = 0 (initial condition),M5 Ì = O1( ) at = 0 (boundary condition),M5 Ì = O2( ) at = D(boundary condition). Solution:Ì ( , )= MM P E 0 þ0( Q) R( , Q, ) Q+P E 0 þ1( Q) R( , Q, ) Q+P N 0 P E 0 B( Q, A) R( , Q, - A) Q A - S2P N 0 O1( A) R( ,0, - A) A+ S2P N 0 O2( A) R( , D, - A) A, whereR( , Q, )= I S2 T1 -exp(-I DU S2) Vexp W- I QS2 X+2 Yexp Z I2 S2( - Q) [ \ ]^ =1 _ ^ ( )_ ^ ( Q) sin( ` ^)` ^ (1 + a2 ^ ),_ ^ ( )=cos W b $ YX- I Y 2 S2b $sin W b $ YX, ` ^ =H S2b2$2Y 2+ I2 4 S2, a ^ = I Y 2 S2b $.JLK Reference : A. G. Butkovskiy (1979). 4.1.4-5. Domain: 0 £ £ Y . Third boundary value problem. The following conditions are prescribed:Ì = þ0( ) at = 0 (initial condition),M N Ì = þ1( ) at = 0 (initial condition),M c Ì - d1 Ì = O1( ) at = 0 (boundary condition),M c Ì + d2 Ì = O2( ) at = Y (boundary condition). The solution Ì ( , ) is determined by the formula in Paragraph 4.1.4-4 whereR( , Q, )=exp Z I( - Q) 2 S2 [ \ ]^ =1 _ ^ ( )_ ^ ( Q) sin( S ` ^)S ` ^ e ^ . Here,_ ^ ( )=cos( a ^)+2 S2d1-I2 S2a ^ sin( a ^), ` ^ =H a2 ^ + I2 4 S4,e ^ =2 S2d2+I4 S2a2 ^4 S4a2 ^ +(2 S2d1-I)2 4 S4a2 ^ +(2 S2d2+I)2+2 S2d1-I4 S2a2 ^ + Y 2+ Y (2 S2d1-I)2 8 S4a2 ^ , where the a ^ are positive roots of the transcendental equation tan( a Y )a=4 S4( d1+ d2) 4 S4a2-(2 S2d1-I)(2 S2d2+I).JLK Reference : A. G. Butkovskiy (1979). Page 292 4.1.5. Equation of the Form %2 &% f2= g2%2 &% h2+ i % &% h+ j &+ k(h,f) 4.1.5-1. Reduction to the nonhomogeneous Klein±Gordon equation. The substitution Ì ( l, m)=exp n-1 2 S-2I l o p( l, m) leads to the equationM2pMm2= S2 M2pMl2+ nrq-1 4 S-2 s2o p+exp n1 2 S-2 sl o t( l, m), which is discussed in Subsection 4.1.3. 4.1.5-2. Domain: - u< l< u. Cauchy problem. Initial conditions are prescribed: Ì = þ( l) at m= 0,v wyx= z( l) at m= 0. Solution for q-1 4 S-2 s2= {2>0:x( l, m)=1 2 þ( l+ S m) exp W sm 2 S X+1 2 þ( l- S m) exp W- sm 2 S X + { m 2 Sexp W- sl 2 S2 XP c + | wc - | wexp W sQ 2 S2 X }1 n{ ~ m2-( l- Q)2U S2o~ m2-( l- Q)2U S2 þ( Q) Q +1 2 Sexp W- sl 2 S2 XP c + | wc - | wexp W sQ 2 S2 X}0 n{ m2-( l- Q)2U S2o=z( Q) Q +1 2 S w 0 c + |( w - )c - |( w - )exp s( - l) 2 2 }0 nr{ ( m- )2-( l- )2 2o t( , ) , where}0( ) and}1( ) are the modi®ed Bessel functions of the ®rst kind. Solution for q-1 4 -2 s2= - {2<0:x( l, m)=1 2 ( l+ m) exp ,2 +1 2 ( - ) exp - ,2 - 2 exp - 2 2 + w- wexp 2 2 1 2-( - )2 2 2-( - )2 2( ) +1 2 exp - 2 2 + w- wexp 2 20 2-( - )2 2 =( ) +1 2 w 0 + ( w - )- ( w - )exp ( - ) 2 2 0 (- )2-( - )2 2 ( , ) , where0( ) and1( ) are the Bessel functions of the ®rst kind.L Reference : A. N. Tikhonov and A. A. Samarskii (1990). 4.1.5-3. Domain: 0 £ £ . First boundary value problem. The following conditions are prescribed:=0( ) at= 0 (initial condition), w=1( ) at= 0 (initial condition),= 1() at = 0 (boundary condition),= 2() at = (boundary condition). Page 293 Solution:( ,)= w 0 0 ( , ) ¡( , ,- ) + 0 0( ) ¡( , ,) + 0 1( ) ¡( , ,) + 2 w 0 1( ) ¡( , ,- ) ¢ =0 - 2 w 0 2( ) ¡( , ,- ) ¢ = . Let 2 £2+1 4 -222- ¤ 2>0. Then¡( , ,)=2exp ( - ) 2 2 ¥ ¦§ =1sin £ ¨ sin £ ¨ sinª© « §©« § ,« § = 2£2¨22+ 2 4 2- ¤. Let2£2¨2+1 4 -222- ¤ 2£ 0 at ¨= 1, ¬¬¬, ®;2£2¨2+1 4 -222- ¤ 2>0at ¨= ®+ 1, ®+ 2, ¬¬¬ Then¡( , ,)=2exp ( - ) 2 2 ¯ ¦§ =1sin £ ¨ sin £ ¨ sinh© ° §©° § +2exp ± ( ²- ) 2 ³2 ¥ ¦ § =¯+1sin £ ¨ sin £ ¨² sinª© « §©« § ,° § = ¤- ³2 £2 ¨22- 2 4 ³2,« § = ³2 £2 ¨22+ 2 4 ³2- ¤. For° § = 0the ratio sinh© ° § © ° § must be replaced by.L Reference : A. G. Butkovskiy (1979). 4.1.5-4. Domain: 0 £ £ . Second boundary value problem. The following conditions are prescribed:=0( ) at= 0 (initial condition), w=1( ) at= 0 (initial condition), = 1() at = 0 (boundary condition), = 2() at = (boundary condition). Solution:( ,)= ´ w 0 ´ 0 ( ², ) ¡( , ²,- ) ² + ´ 0 µ0( ²) ¡( , ²,) ²+ ´ 0 µ1( ²) ¡( , ²,) ² - ³2´ w 0 1( ) ¡( ,0,- ) + ³2´ w 0 2( ) ¡( , ,- ) . For ¤<0,¡( , ²,)= ³2r¶ · ¹¸ 2- 1 exp ²³2sin©| ¤| ©| ¤|+2exp ± ( ²- ) 2 ³2 º¥ ¦§ =1 » § ( )» § ( ²) 1 + ¼2 §sin ©« §© « § ,« § = ³2 £2 ¨22+ 2 4 ³2- ¤,» § ( )=cos £ ¨ + ¼ § sin £ ¨ , ¼ § = 2 ³2 £ ¨. For ¤>0,¡( , ²,)= ³2r¶ · ¹¸ 2- 1 exp ²³2sinh© ¤ © ¤+2exp ± ( ²- ) 2 ³2 º ¥ ¦§ =1 » § ( )» § ( ²) 1 + ¼2 §sin ©« §© « § , where the« § ,» § ( ), and ¼ § were speci®ed previously. If the inequality« § <0holds for several ®rst values ¨= 1, ¬¬¬, ®, then the© « § in the corresponding terms of the series should be replaced by©|« § |, and the sines by the hyperbolic sines. Page 294 4.1.5-5. Domain: 0 £ £ . Third boundary value problem. The following conditions are prescribed:=µ0( ) at= 0 (initial condition), w=µ1( ) at= 0 (initial condition), - ½1 = 1() at = 0 (boundary condition), + ½2 = 2() at = (boundary condition). The solution ( ,) is determined by the formula in Paragraph 4.1.5-4 where¡( , ²,)=exp ± ( ²- ) 2 ³2 º¥ ¦§ =1 » § ( )» § ( ²) sinª© « §¾ §©« § . Here,» § ( )=cos( ¼ §)+2 ³2½1+2 ³2¼ § sin( ¼ §),« § = ³2¼2 § + 2 4 ³2- ¤,¾ § =2 ³2½2-4 ³2¼2 §4 ³4¼2 § +(2 ³2½1+)2 4 ³4¼2 § +(2 ³2½2-)2+2 ³2½1+4 ³2¼2 § + 2+ (2 ³2½1+)2 8 ³4¼2 § , where the ¼ § are positive roots of the transcendental equation tan( ¼ )¼=4 ³4( ½1+ ½2) 4 ³4¼2-(2 ³2½1+)(2 ³2½2-). 4.2. Wave Equation with Axial or Central Symmetry 4.2.1. Equations of the Form ¿2 À¿ Á2= Â2 Ã¿2 À¿ Ä2+1Ä ¿ À¿ Ä Å This is the one-dimensional wave equation with axial symmetry, where Æ= 2+»2is the radial coordinate. In the problems considered in Paragraphs 4.2.1-1 through 4.2.1-3, the solutions boundedatÆ= 0are sought (this is not specially stated below). 4.2.1-1. Domain: 0 £ Æ£ Ç. First boundary value problem. The following conditions are prescribed:=µ0( Æ) at È= 0 (initial condition), É=µ1( Æ) at È= 0 (initial condition),= ( È) at Æ= Ç(boundary condition). Solution:( Æ, È)= È ´ Ê 0µ0( ²) ¡( Æ, ², È) ²+ ´ Ê 0µ1( ²) ¡( Æ, ², È) ²- ³2´ É 0 ( ) ± ² ¡( Æ, ², È- )º ¢ =Ê , where¡( Æ, ², È)=2 ²³ Ç ¥ ¦§ =11« § Ë 2 1(« § ) Ë 0 Ì « §ÆÇ Í Ë 0 Ì « §²Ç ÍsinÌ « §³ ÈÇ Í. Here, the« § are positive zeros of the Bessel function, Ë 0(«)= 0. The numerical values of the ®rst ten« § are speci®ed in Paragraph 1.2.1-3.ÎLÏ Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 295 4.2.1-2. Domain: 0 £ Æ£ Ç. Second boundary value problem. The following conditions are prescribed:=µ0( Æ) at È= 0 (initial condition), É=µ1( Æ) at È= 0 (initial condition), Ð= Ñ( È) at Æ= Ç(boundary condition). Solution:( Æ, È)= È ´ Ê 0 µ0( ²) Ò( Æ, ², È) Ó ²+ ´ Ê 0 µ1( ²) Ò( Æ, ², È) Ó ²+ ³2´ É 0 Ñ( ) Ò( Æ, Ç, È- ) Ó , whereÒ( Æ, ², È)=2 Èy²Ç2+2 ²³ Ç Ô ÕÖ =11× Ö Ë 2 0( × Ö ) Ë 0 Ì × ÖÆÇ Í Ë 0 Ì × Ö ØÇ ÍsinÌ × Ö ÙÈÇ Í. Here, the × Ö are positive zeros of the ®rst-order Bessel function, Ë 1( ×)= 0. The numerical values of the ®rst ten roots × Ö are speci®ed in Paragraph 1.2.1-4.ÎLÏ References : M. M. Smirnov (1975), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 4.2.1-3. Domain: 0 £ Æ£ Ç. Third boundary value problem. The following conditions are prescribed:=µ0( Æ) at È= 0 (initial condition), É=µ1( Æ) at È= 0 (initial condition), Ð+ ½ = Ñ( È) at Æ= Ç(boundary condition). The solution ( Æ, È) is determined by the formula in Paragraph 4.2.1-2 whereÒ( Æ, Ø , È)=2 ØÙÇ Ô ÕÖ =1 × Ö ( ½2Ç2+ ×2 Ö ) Ë 2 0( × Ö ) Ë 0 Ì × ÖÆÇ Í Ë 0 Ì × Ö ØÇ ÍsinÌ × Ö ÙÈÇ Í. Here, the × Ö are positive roots of the transcendental equation× Ë 1( ×)- ½ Ç Ë 0( ×)= 0. The numerical values of the ®rst six roots × Ö can be found in Carslaw and Jaeger (1984); see also Abramowitz and Stegun (1964). 4.2.1-4. Domain: Ç1£ Æ£ Ç2. First boundary value problem. The following conditions are prescribed:=µ0( Æ) at È= 0 (initial condition), É=µ1( Æ) at È= 0 (initial condition),= Ñ1( È) at Æ= Ç1(boundary condition),= Ñ2( È) at Æ= Ç2(boundary condition). Solution:( Æ, È)= È ´Ê2Ê1 µ0( Ø ) Ò( Æ, Ø , È) Ó Ø + ´Ê2Ê1 µ1( Ø ) Ò( Æ, Ø , È) Ó Ø + Ù 2´ É 0 Ñ1( Ú) Û ØÒ( Æ, Ø , È- Ú)º Ü =Ê1 Ó Ú- Ù 2 Ý Þ 0 Ñ2( Ú) Û ØÒ( ß, Ø , à- Ú) áÜ = â2 Ó Ú. Page 296 Here,Ò( ß, Ø , à)=Ô ÕÖ =1 ã Ö Ø ä Ö ( ß) ä Ö ( Ø ) sin å × Ö Ùàæ 1 ç,ã Ö = è2 × Ö é 2 0( ê × Ö ) 2 Ùæ 1 ë é 2 0( × Ö )- é 2 0( ê × Ö ) ì,ä Ö ( ß)= í0( î ï) é 0 å î ï ßæ 1 ç- é 0( î ï) í0 å î ï ßæ 1 ç, ê= æ 2æ 1, where é 0( ð) and í0( ð) are the Bessel functions, the î ïare positive roots of the transcendental equation é 0( î) í0( êî)- é 0( êî) í0( î)= 0. The numerical values of the ®rst ®ve roots î ï= î ï( ê) can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984). 4.2.1-5. Domain: æ 1£ ß£ æ 2. Second boundary value problem. The following conditions are prescribed:ñ= ò0( ß) at à= 0 (initial condition),óÞ ñ= ò1( ß) at à= 0 (initial condition),ó Ðñ= Ñ1( à) at ß= æ 1(boundary condition),ó Ðñ= Ñ2( à) at ß= æ 2(boundary condition). Solution:ñ( ß, à)= óóà Ý â2â1 ò0( ô) õ( ß, ô, à) ö ô+ Ý â2â1 ò1( ô) õ( ß, ô, à) ö ô - ÷2 Ý Þ 0 Ñ1( ø) õ( ß, æ 1, à- ø) ö ø+ ÷2 Ý Þ 0 Ñ2( ø) õ( ß, æ 2, à- ø) ö ø. Here,õ( ß, ô, à)=2 àyôæ2 2- æ2 1+ ù úï=1 ã ï ô äï( ß) äï( ô) sin å î ï ÷ àæ 1 ç,ã ï= è2î ï é 2 1( êî ï) 2 ÷ æ 1 ë é 2 1( î ï)- é 2 1( êî ï) ì,äï( ß)= í1( î ï) é 0 å î ï ßæ 1 ç- é 1( î ï) í0 å î ï ßæ 1 ç, ê= æ 2æ 1, where é û ( ð) and í û ( ð) are the Bessel functions ( ü= 0,1); the î ïare positive roots of the transcen- dental equation é 1( î) í1( êî)- é 1( êî) í1( î)= 0. The numerical values of the ®rst ®ve roots î ï= î ï( ê) can be found in Abramowitz and Stegun (1964). 4.2.1-6. Domain: æ 1£ ß£ æ 2. Third boundary value problem. The following conditions are prescribed:ñ= ò0( ß) at à= 0 (initial condition),óÞ ñ= ò1( ß) at à= 0 (initial condition),ó Ðñ- ü1 ñ= Ñ1( à) at ß= æ 1(boundary condition),ó Ðñ+ ü2 ñ= Ñ2( à) at ß= æ 2(boundary condition). Page 297 The solution ñ( ß, à) is determined by the formula in Paragraph 4.2.1-5 whereõ( ß, ô, à)= è2 2 ÷ ù úï=1 ý ïþï ë ü2 é 0(ý ï æ 2)-ý ï é 1(ý ï æ 2) ì2ô ÿ ï( ß) ÿ ï( ô) sin(ý ï ÷ à). Here,þï=(ý2ï+ ü2 2)ë ü1 é 0(ý ï æ 1)+ý ï é 1(ý ï æ 1) ì2-(ý2ï+ ü2 1)ë ü2 é 0(ý ï æ 2)-ý ï é 1(ý ï æ 2) ì2,ÿ ï( ß)=ë ü1 í0(ý ï æ 1)+ý ï í1(ý ï æ 1) ì é 0(ý ï ß)-ë ü1 é 0(ý ï æ 1)+ý ï é 1(ý ï æ 1) ìí0(ý ï ß); é û ( ð) and í û ( ð) are the Bessel functions ( ü= 0,1); and theý ïare positive roots of the transcendental equationë ü1 é 0(ý æ 1)+ý é 1(ý æ 1) ìë ü2 í0(ý æ 2)-ý í1(ý æ 2) ì -ë ü2 é 0(ý æ 2)-ý é 1(ý æ 2) ìë ü1 í0(ý æ 1)+ý í1(ý æ 1) ì= 0. 4.2.2. Equation of the Form ¿2 ¿ 2= 2 ¿2 ¿ 2+1 ¿ ¿ + (, ) 4.2.2-1. Domain: 0 £ ß£ æ. Different boundary value problems. 1 . The solution to the ®rst boundary value problem for a circle of radius æis given by the formula from Paragraph 4.2.1-1 with the additional termÝÞ 0 Ý â 0 ( ô, ø) õ( ß, ô, à- ø) ö ô ö ø, ( 1) which allows for the equation's nonhomogeneity.2. The solution to the second boundary value problem for a circle of radius æis given by the formula from Paragraph 4.2.1-2 with the additional term (1).3. The solution to the third boundary value problem for a circle of radius æis the sum of the solution presented in Paragraph 4.2.1-3 and expression (1). 4.2.2-2. Domain: æ 1£ ß£ æ 2. Different boundary value problems. 1 . The solution to the ®rst boundary value problem for an annular domain is given by the formula from Paragraph 4.2.1-4 with the additional termÝ Þ 0 Ý â2â1 ( ô, ø) õ( ß, ô, à- ø) ö ô ö ø, ( 2) which allows for the equation's nonhomogeneity.2. The solution to the second boundary value problem for an annular domain is given by the formula from Paragraph 4.2.1-5 with the additional term (2).3. The solution to the third boundary value problem for an annular domain is the sum of the solution presented in Paragraph 4.2.1-6 and expression (2). 4.2.3. Equation of the Form ¿2 ¿ 2= 2 ¿2 ¿ 2+2 ¿ ¿ This is the equation of one-dimensional vibration of a gas with central symmetry, where ß= 2+ 2+ ð2is the radial coordinate. In the problems considered in Paragraphs 4.2.3-1 through 4.2.3-3, the solutions bounded at ß= 0are sought; this is not specially stated below. Page 298 4.2.3-1. General solution:ñ( à, ß)= ( + ÷ )+ ( - ÷ ) , where( 1) and ( 2) are arbitrary functions. 4.2.3-2. Reduction to a constant coef®cient equation. The substitution ( , )= ñ( , ) leads to the constant coef®cient equationó2ó2= ÷2 ó2ó 2, which is discussed in Subsection 4.1.1. 4.2.3-3. Domain: 0 £ < . Cauchy problem. Initial conditions are prescribed:ñ= ò( ) at = 0,ó ñ= Ñ( ) at = 0. Solution:ñ( , )=1 2 ë( - ÷ ) ò | - ÷ | +( + ÷ ) ò | + ÷ | ,ì+1 2 ÷ + - Ñ || . Solution at the center = 0: (0, )= !( )+ ( )+ #"( ).$&% Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 4.2.3-4. Domain: 0 £ £ '. First boundary value problem. The following conditions are prescribed: = 0( ) at = 0 (initial condition),( = 1( ) at = 0 (initial condition), = "( ) at = '(boundary condition). Solution: ( , )= (( )0 0() *( ,, ) + )0 1() *( ,, ) - 2 0 "( +) , (( *( ,, - +) - . =) +, where*( ,, )=2/ 0 1ï=112sin 3 2/ ' 4sin 3 2/' 4sin 3 2/' 4. 4.2.3-5. Domain: 0 £ £ '. Second boundary value problem. The following conditions are prescribed: = 0( ) at = 0 (initial condition),( = 1( ) at = 0 (initial condition),( = "( ) at = '(boundary condition). Page 299 Solution: ( , )= (( )0 0() *( ,, ) + )0 1() *( ,, ) + 2 0 "( +) *( , ', - +) +, where*( ,, )=3 2'3+2 0 1ï=1 52ï+ 153ïsin 35 ï ' 4sin 35 ï' 4sin 35 ï ' 4. Here, the5 ïare positive roots of the transcendental equation tan5-5= 0. The numerical values of the ®rst ®ve roots5 ïare speci®ed in Paragraph 1.2.3-5. 4.2.3-6. Domain: 0 £ £ '. Third boundary value problem. The following conditions are prescribed: = 0( ) at = 0 (initial condition),( = 1( ) at = 0 (initial condition),( + 6 = "( ) at = '(boundary condition). The solution ( , ) is determined by the formula in Paragraph 4.2.3-5 where*( ,, )=2 0 1ï=1 52ï+( 6 '- 1)25 ï 752ï+ 6 '( 6 '- 1) 8sin 35 ï ' 4sin 35 ï' 4sin 35 ï ' 4. Here, the5 ïare positive roots of the transcendental equation5cot5+ 6 '- 1 = 0 . The numerical values of the ®rst six roots5 ïcan be found in Carslaw and Jaeger (1984). 4.2.3-7. Domain: '1£ £ '2. First boundary value problem. The following conditions are prescribed: = 0( ) at = 0 (initial condition),( = 1( ) at = 0 (initial condition), = "1( ) at = '1(boundary condition), = "2( ) at = '2(boundary condition). Solution: ( , )= (( )2)1 0() *( ,, ) + )2)1 1() *( ,, ) + 2 0 "1( +) , (( *( ,, - +) - . =)1 +- 2 0 "2( +) , (( *( ,, - +) - . =)2 +, where*( ,, )=2/ 0 1ï=112sin , / 2( - '1)'2- '1 -sin , / 2(- '1)'2- '1 -sin 3 / 2 '2- '1 4. Page 300 4.2.3-8. Domain: 1£ £ 2. Second boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition), = 1( ) at = 1(boundary condition), = 2( ) at = 2(boundary condition). Solution:( , )= 2 1 0( ) ( , , ) + 2 1 1( ) ( , , ) - 2 0 1( ) ( , 1, - ) + 2 0 2( ) ( , 2, - ) , where( , , )=3 23 2- 3 1+2 ( 2- 1) =1(1 + 2 2 2 ) ( ) ( ) sin( )3 2 1+ 2 2+ 1 2(1 + 1 22 ) , ( )=sin[ ( - 1)]+ 1 cos[ ( - 1)]. Here, the are positive roots of the transcendental equation (2 1 2+ 1) tan[( 2- 1)]-( 2- 1)= 0. 4.2.3-9. Domain: 1£ £ 2. Third boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition), - 1 = 1( ) at = 1(boundary condition), + 2 = 2( ) at = 2(boundary condition). The solution ( , ) is determined by the formula in Paragraph 4.2.3-8 where( , , )=2 ( 2- 1) =11 ( 22+ 2 2 2 ) ( ) ( ) sin( ) ( 21+ 2 1 2 )( 22+ 2 2 2 )+( 1 2+ 2 1)( 1 2+ 1 22 ), ( )= 1sin[ ( - 1)]+ 1 cos[ ( - 1)], 1= 1 1+ 1, 2= 2 2- 1. Here, the are positive roots of the transcendental equation ( 1 2- 1 22) sin[( 2- 1)]+( 1 2+ 2 1) cos[( 2- 1)]= 0. 4.2.4. Equation of the Form 2 2= 2 2 !2+2! ! "+ #(!,) 4.2.4-1. Reduction to a nonhomogeneous constant coef®cient equation. The substitution $( , )= ( , ) leads to the nonhomogeneous constant coef®cient equation2$2= 2 2$2+ %( , ), which is discussed in Subsection 4.1.2. Page 301 4.2.4-2. Domain: 0 £ < &. Cauchy problem. Initial conditions are prescribed:= ( ) at = 0, = ( ) at = 0. Solution:( , )=1 2 ( - ) '| - | (+( + ) '| + | ()+1 2 + * - * '| | ( +1 2 0 + *( - +) - *( - +) % '| |, ( .,.- Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 4.2.4-3. Domain: 0 £ £ . Different boundary value problems. 1 /. The solution to the ®rst boundary value problem for a sphere of radius is given by the formula from Paragraph 4.2.3-4 with the additional term 0 0 %( , ) ( , , - ) , ( 1) which allows for the equation's nonhomogeneity.2/. The solution to the second boundary value problem for a sphere of radius is given by the formula from Paragraph 4.2.3-5 with the additional term (1).3/. The solution to the third boundary value problem for a sphere of radius is the sum of the solution presented in Paragraph 4.2.3-6 and expression (1). 4.2.4-4. Domain: 1£ £ 2. Different boundary value problems. 1 /. The solution to the ®rst boundary value problem for a spherical layer is given by the formula from Paragraph 4.2.3-7 with the additional term 0 2 1 %( , ) ( , , - ) , ( 2) which allows for the equation's nonhomogeneity.2/. The solution to the second boundary value problem for a spherical layer is given by the formula from Paragraph 4.2.3-8 with the additional term (2).3/. The solution to the third boundary value problem for a spherical layer is the sum of the solution presented in Paragraph 4.2.3-9 and expression (2). 4.2.5. Equation of the Form 2 2= 2 2 !2+1! ! "± 0 + #(!,) For >0and %º 0, this is the Klein±Gordon equation describing one-dimensional wave phenomena with axial symmetry. In the problems considered in Paragraphs 4.2.5-1 through 4.2.5-3, the solutionsbounded at= 0are sought; this is not specially stated below. Page 302 4.2.5-1. Domain: 0 £ £ . First boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition),= ( ) at = (boundary condition). Solution:( , )= 0 0( ) ( , , ) + 0 1( ) ( , , ) - 2 0 ( ) 1 ( , , - ) 2 3 = + 0 0 %( , ) ( , , - ) . Here,( , , )=2 2 =1142 1( 5 ) 4 0 6 5 7 4 0 6 5 7sin '8:9 (9 , = 252 2+ , where the 5 are positive zeros of the Bessel function, 4 0( 5)= 0. The numerical values of the ®rst ten 5 are speci®ed in Paragraph 1.2.1-3. 4.2.5-2. Domain: 0 £ £ . Second boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition), = ( ) at = (boundary condition). Solution:( , )= 0 0( ) ( , , ) + 0 1( ) ( , , ) + 2 0 ( ) ( , , - ) + 0 0 %( , ) ( , , - ) . Here,( , , )=2 sin '8 9 (29+2 2 =1142 0( 5 ) 4 0 6 5 7 4 0 6 5 7sin '8:9 (9 , = 252 2+ , where the 5 are positive zeros of the ®rst-order Bessel function, 4 1( 5)= 0. The numerical values of the ®rst ten 5 are speci®ed in Paragraph 1.2.1-4. 4.2.5-3. Domain: 0 £ £ . Third boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition), + = ( ) at = (boundary condition). Page 303 The solution ( , ) is determined by the formula in Paragraph 4.2.5-2 where( , , )=22 =1 52 ( 2 2+ 52 ) 42 0( 5 ) 4 0 6 5 7 4 0 6 5 7sin '8:9 (9 , = 252 2+ . Here, the 5 are positive roots of the transcendental equation5 4 1( 5)- 4 0( 5)= 0. The numerical values of the ®rst six roots 5 can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984). 4.2.5-4. Domain: 1£ £ 2. First boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition),= 1( ) at = 1(boundary condition),= 2( ) at = 2(boundary condition). Solution:( , )= 0 2 1 %( , ) ( , , - ) + 2 1 0( ) ( , , ) + 2 1 1( ) ( , , ) + 2 0 1( ) 1 ( , , - ) 2 3 = 1 - 2 0 2( ) 1 ( , , - ) 2 3 = 2 . Here,( , , )= ;2 2 2 1 =1 52 42 0( <)5 ) 42 0( 5 )- 42 0( <)5 ) ( ) ( )sin ':9 (9 , = 252 2 1+ , ( )= =0( 5 ) 4 0 6 5 1 7- 4 0( 5 ) =0 6 5 1 7, <= 2 1, where 4 0( >) and =0( >) are the Bessel functions and the 5 are positive roots of the transcendental equation4 0( 5) =0( <)5)- 4 0( <)5) =0( 5)= 0. The numerical values of the ®rst ®ve roots 5 = 5 ( <) can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984). 4.2.5-5. Domain: 1£ £ 2. Second boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition), = 1( ) at = 1(boundary condition), = 2( ) at = 2(boundary condition). Page 304 Solution:( , )= 0 2 1 %( , ) ( , , - ) + 2 1 0( ) ( , , ) + 2 1 1( ) ( , , ) - 2 0 1( ) ( , 1, - ) + 2 0 2( ) ( , 2, - ) . Here,( , , )=2 sin ' 9 ( ( 2 2- 2 1) 9+ ;2 2 2 1 =1 52 42 1( <)5 ) 42 1( 5 )- 42 1( <)5 ) ( ) ( )sin ':9 (9 , ( )= =1( 5 ) 4 0 6 5 1 7- 4 1( 5 ) =0 6 5 1 7, = 252 2 1+ , <= 2 1, where 4 ?( >) and = ?( >) are the Bessel functions ( = 0,1); the 5 are positive roots of the transcen- dental equation4 1( 5) =1( <)5)- 4 1( <)5) =1( 5)= 0. The numerical values of the ®rst ®ve roots 5 = 5 ( <) can be found in Abramowitz and Stegun (1964). 4.2.5-6. Domain: 1£ £ 2. Third boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition), - 1 = 1( ) at = 1(boundary condition), + 2 = 2( ) at = 2(boundary condition). The solution ( , ) is determined by the formula in Paragraph 4.2.5-5 where( , , )= ;2 2 =1 @2 A B2@2 + 2 4 0(@ 2)-@ 4 1(@ 2) 2 C ( ) C ( ) sin '8ED 2@2 + (. Here,A =(@2 + 2 2) 1 4 0(@ 1)+@ 4 1(@ 1) 2-(@2 + 2 1) 2 4 0(@ 2)-@ 4 1(@ 2) 2,C ( )= 1 =0(@ 1)+@ =1(@ 1) 4 0(@ )- 1 4 0(@ 1)+@ 4 1(@ 1) =0(@ ), where the@ are positive roots of the transcendental equation1 4 0(@ 1)+@ 4 1(@ 1) 2 =0(@ 2)-@ =1(@ 2) - 2 4 0(@ 2)-@ 4 1(@ 2) 1 =0(@ 1)+@ =1(@ 1) = 0. 4.2.6. Equation of the Form 2 2= 2 2 !2+2! ! "± 0 + #(!,) For >0and %º 0, this is the Klein±Gordon equation describing one-dimensional wave phenomena with central symmetry. In the problems considered in Paragraphs 4.2.6-1 through 4.2.6-3, thesolutions bounded at= 0are sought; this is not specially stated below. Page 305 4.2.6-1. Domain: 0 £ £ . First boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition),= ( ) at = (boundary condition). Solution:( , )= 0 0( ) ( , , ) + 0 1( ) ( , , ) - 2 0 ( ) 1 ( , , - ) 2 3 = + 0 0 %( , ) ( , , - ) , where( , , )=2 =1sin6 F ; 7sin6 F ; 7sin '89 (9 , = 2;2F22+ . 4.2.6-2. Domain: 0 £ £ . Second boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition), = ( ) at = (boundary condition). Solution:( , )= 0 0( ) ( , , ) + 0 1( ) ( , , ) + 2 0 ( ) ( , , - ) + 0 0 %( , ) ( , , - ) , where( , , )=3 2sin ' 9 (39+2 =1 52 + 152 9 sin6 5 7sin6 5 7sin '8 B (, = 252 2+ . Here, the 5 are positive roots of the transcendental equation tan 5- 5= 0; for the numerical values of the ®rst ®ve roots 5 , see Paragraph 1.2.3-5. 4.2.6-3. Domain: 0 £ £ . Third boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition), + = ( ) at = (boundary condition). The solution ( , ) is determined by the formula in Paragraph 4.2.6-2 where( , , )=2 =1 52 +( - 1)252 + ( - 1)sin6 5 7sin6 5 7sin '89 (9 , = 252 2+ . Here, the 5 are positive roots of the transcendental equation 5cot 5+ - 1 = 0 . The numerical values of the six ®ve roots 5 can be found in Carslaw and Jaeger (1984). Page 306 4.2.6-4. Domain: 1£ £ 2. First boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition),= 1( ) at = 1(boundary condition),= 2( ) at = 2(boundary condition). Solution:( , )= 0 2 1 %( , ) ( , , - ) + 2 1 0( ) ( , , ) + 2 1 1( ) ( , , ) + 2 0 1( ) 1 ( , , - ) 2 3 = 1 - 2 0 2( ) 1 ( , , - ) 2 3 = 2 , where( , , )=2 ( 2- 1) =1sin 1 ;F( - 1) 2- 1 2sin 1 ;F( - 1) 2- 1 2sin '89 (9 , = 2;2F2 ( 2- 1)2+ . 4.2.6-5. Domain: 1£ £ 2. Second boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition), = 1( ) at = 1(boundary condition), = 2( ) at = 2(boundary condition). Solution:( , )= 0 2 1 %( , ) ( , , - ) + 2 1 0( ) ( , , ) + 2 1 1( ) ( , , ) - 2 0 1( ) ( , 1, - ) + 2 0 2( ) ( , 2, - ) . Here,( , , )=3 2sin '8 9 ( ( 3 2- 3 1) 9+2 ( 2- 1) =1(1 + 2 2 2 ) ( ) ( ) sin '8 B22 + (2 2 1+ 2 2+ 1 2(1 + 1 22 ) B22 + , ( )=sin[ ( - 1)]+ 1 cos[ ( - 1)], where the are positive roots of the transcendental equation (2 1 2+ 1) tan[( 2- 1)]-( 2- 1)= 0. Page 307 4.2.6-6. Domain: 1£ £ 2. Third boundary value problem. The following conditions are prescribed:= 0( ) at = 0 (initial condition), = 1( ) at = 0 (initial condition), - 1 = 1( ) at = 1(boundary condition), + 2 = 2( ) at = 2(boundary condition). The solution ( , ) is determined by the formula in Paragraph 4.2.6-5 where( , , )=2 =1( 22+ 2 2 2 ) ( ) ( ) sin '8 B22 + ( ( 2- 1)( 21+ 2 1 2 )( 22+ 2 2 2 )+( 1 2+ 2 1)( 1 2+ 1 22 ) B22 + , ( )= 1sin[ ( - 1)]+ 1 cos[ ( - 1)], 1= 1 1+1, 2= 2 2-1. Here, the are positive roots of the transcendental equation ( 1 2- 1 22) sin[( 2- 1)]+( 1 2+ 2 1) cos[( 2- 1)]= 0. 4.3. Equations Containing Power Functions and Arbitrary Parameters 4.3.1. Equations of the Form 2 2= ( G+ 0) 2 G2+ H G+ I + #( G,) 1. J2 KJ L2= M2JJ N 6N J KJ N 7+ O(N,L). For %( P, Q)º0,this equation governs small-amplitude free vibration of a hanging heavy homogeneous thread ( 2is the acceleration due to gravity, Rthe de¯ection of the thread from the vertical axis, andPthe vertical coordinate). 1 /. The substitution P=1 4 S2leads to the equationT2RTQ2= 26 T2RTS2+1S TRTS 7+ % '1 4 S2, QU(, which is discussed in Subsections 4.2.1±4.2.2.2/. Domain: 0 £ P£ V. First boundary value problem. The following conditions are prescribed:R= W0( P) at Q= 0 (initial condition),T XR= W1( P) at Q= 0 (initial condition),R= Y( Q) at P= V(boundary condition),R¹ & at P= 0 (boundedness condition). Solution:R( P, Q)= TTQ Z [0 \0( ]) ^( _, ], `) a ]+Z [0 \1( ]) ^( _, ], `) a ] - b2VZ c0 d( e) f gg ] ^( _, ], `- e) h i =[ a e+Z c0 Z [0 j( ], e) ^( _, ], `- e) a ] a e, where^( _, ], `)=2b k V l mn =11o n p 2 1( o n ) p 0 q o n r_V s p 0 q o n r]V ssinq o nb ` 2 k V s. Here, the o n are positive zeros of the Bessel function, p 0( o)= 0. The numerical values of the ®rst ten roots o n are speci®ed in Paragraph 1.2.1-3.t.- Reference : M. M. Smirnov (1975). Page 308 3 /. Domain: 0 £ _£ V. Second boundary value problem. The following conditions are prescribed:u=\0( _) at `= 0 (initial condition),gc u=\1( _) at `= 0 (initial condition),g v u=d( `) at _= V(boundary condition),u¹ w at _= 0 (boundedness condition). Solution:u( _, `)= gg ` Z [0\0( ]) ^( _, ], `) a ]+Z [0\1( ]) ^( _, ], `) a ] + b2VZ c0 d( e) ^( _, V, `- e) a e+Z c0 Z [0 j( ], e) ^( _, ], `- e) a ] a e, where^( _, ], `)= `V+2b k V l mn =11o n p 2 0( o n ) p 0 q o n r_V s p 0 q o n r]V ssinq o nb ` 2 k V s. Here, the o n are positive zeros of the ®rst-order Bessel function, p 1( o)= 0. The numerical values of the ®rst ten roots o n are speci®ed in Paragraph 1.2.1-4. 4 /. Domain: 0 £ _£ V. Third boundary value problem. The following conditions are prescribed:u=\0( _) at `= 0 (initial condition),gc u=\1( _) at `= 0 (initial condition),g v u+ x u=d( `) at _= V(boundary condition),u¹ w at _= 0 (boundedness condition). The solution u( _, `) is given by the formula in Item 3 /with^( _, ], `)=2b k V l mn =1 o n (4 x2V+ o2 n ) p 2 0( o n ) p 0 q o n r_V s p 0 q o n r]V ssinq o nb ` 2 k V s. Here, the o n are positive roots of the transcendental equationo p 1( o)- 2 x kV p 0( o)= 0. The numerical values of the ®rst six roots o n can be found in Carslaw and Jaeger (1984). 2. y2 zy {2= |2yy } q} y zy } s± ~ z+ (},{). For <0andj( _, `)º 0, this equation describes small-amplitude vibration of a heavy homogeneous thread that rotates at a constant angular velocity =k| |about the vertical axis ( b2is the acceleration due to gravity).1/. The substitution _=1 4 2leads to the equationg2 ug `2= b2q g2 ug 2+1 g ug s- u+j 1 42, `U, which is discussed in Subsection 4.2.5. Page 309 2 /. Domain: 0 £ _£ V. First boundary value problem. The following conditions are prescribed:u=\0( _) at `= 0 (initial condition),gc u=\1( _) at `= 0 (initial condition),u=d( `) at _= V(boundary condition),u¹ w at _= 0 (boundedness condition). Solution:u( _, `)= gg ` Z [0\0( ]) ^( _, ], `) a ]+Z [0\1( ]) ^( _, ], `) a ] - b2VZ c0 d( e) f gg ] ^( _, ], `- e) h i =[ a e+Z c0 Z [0 j( ], e) ^( _, ], `- e) a ] a e. Here,^( _, ], `)=1Vl mn =11 p 2 1( o n ) p 0 q o nr_V s p 0 q o n r]V ssin `k nk n , n = b2 o2 n 4 V+ , where the o n are positive zeros of the Bessel function, p 0( o)= 0.t.- Reference : M. M. Smirnov (1975). 3 /. Domain: 0 £ _£ V. Second boundary value problem. The following conditions are prescribed:u=\0( _) at `= 0 (initial condition),gc u=\1( _) at `= 0 (initial condition),g v u=d( `) at _= V(boundary condition),u¹ w at _= 0 (boundedness condition). Solution:u( _, `)= gg ` Z [0 \0( ]) ^( _, ], `) a ]+Z [0 \1( ]) ^( _, ], `) a ] + b2VZ c0 d( e) ^( _, V, `- e) a e+Z c0 Z [0 j( ], e) ^( _, ], `- e) a ] a e. Here,^(, ], `)=sin `:k Vk +1Vl mn =11 p 2 0( o n ) p 0 q o nr_V s p 0 q o n r]V ssin `k nk n , n = b2 o2 n 4 V+ , where the o n are positive zeros of the ®rst-order Bessel function, p 1( o)= 0. The numerical values of the ®rst ten roots o n are speci®ed in Paragraph 1.2.1-4. 4 /. Domain: 0 £ _£ V. Third boundary value problem. The following conditions are prescribed:u=\0( _) at `= 0 (initial condition),gc u=\1( _) at `= 0 (initial condition),g v u+ x u=d( `) at _= V(boundary condition). The solution u( _, `) is given by the formula in Item 3 /with^(, ], `)=1Vl mn =1 o2 n (4 x2V+ o2 n ) p 2 0( o n ) p 0 q o n r_V s p 0 q o n r]V ssin `k nk n , n = b2 o2 n 4 V+ . Here, the o n are positive roots of the transcendental equationo p 1( o)- 2 x kV p 0( o)= 0. The numerical values of the ®rst six roots o n can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984). Page 310 3. y2 zy {2= |2yy } f( ±}) y zy } h. This equation governs small-amplitude free vibration of a heavy homogeneous thread of length V ( b2is the acceleration due to gravity, uthe de¯ection of the thread from the vertical axis, and _the vertical coordinate). The change of variable = V- _leads a special case of equation 4.3.1.1 with= 0andjº 0. 4. y2 zy {2= |2q2 2 + 1 } y2 zy }2+ y zy } s, = 1, 2, General solution:u( _, `)= g n -1g _ n -1 fj(k2(2 + 1) _+ b `)+ (k2(2 + 1) _- b `)k _ h, wherejand are arbitrary functions.t.- Reference : M. M. Smirnov (1975). 5. y2 zy {2= ( |}+ ~) y2 zy }2+ | y zy }+ z+ (},{). The substitution = b _+ leads to an equation of the form 4.3.1.2:g2 ug `2= b2gg q g ug s+ u+j q - b, `s. 6. y2 zy {2= ( |}+ ~) y2 zy }2+1 2 | y zy }+ z+ (},{). The substitution = 2 kb _+ leads to the equationg2 ug `2= b2g2 ug 2+ u+j q 2- 4 4 b, `s, which is considered in Subsection 4.1.3.7.y2 zy {2= ( |2}+ ~2) y2 zy }2+ ( |1}+ ~1) y zy }+ ( |0}+ ~0) z. This is a special case of equation 4.5.3.4 with\( _)= b2 _+ 2,d( _)= b1 _+ 1, ( _)= b0 _+ 0, andjº 0. Particular solutions:u( _, `)=exp( x _) q _+ s .sin `k o+ cos `k o) for o>0,u( _, `)=exp( x _) q _+ s sinh `k- o+ cosh `k- oEfor o<0. Here,, , and oare arbitrary constants; the coef®cients x, , and the function = ( ]) are listedinTable20,where( , ; )= 1 ( , ; )+ 2 ( , ; ), 1, 2are any numbers, is an arbitrary solution of the degenerate hypergeometric equation v v+( - ) v- = 0, and ¡( )= 1 p¡( )+ 2 = ¡( ), 1, 2are any numbers, is an arbitrary solution of the Bessel equation 2 v v+ v+( 2- ¢2) = 0. Page 311 TABLE 20 The coef®cients x, , and the function = ( £) determining the form of particular solutions to equation 4.3.1.7. Notation: ¤( x)= 2 x2+ 1 x+ 0+ o Conditions x = ( £) Parameters¥ 2¹ 0, ¦¹ 0¦º ¥21-4 ¥ 0 ¥ 2 § ¦- ¥ 1 2 ¥ 2- ¥ 2 2 ¥ 2 x+ ¥ 1 2¥ 2 ( , ; £) = ¤( x) ¨(2 ¥ 2 x+ ¥ 1),=( ¥ 2 1- ¥ 1 2) ¥-22¥ 2= 0,¥ 1¹ 0- ¥ 0¥ 112 2 x+ 1¥ 1 ,1 2; © £2 = ¤( x) ¨(2 ¥ 1),©= - ¥ 1 ¨(2 2)¥ 2¹ 0,¥21= 4 ¥ 0 ¥ 2- ¥ 1 2 ¥ 2 ¥ 2 2¥ 2 £ ª 2ª © « £ =1 2-2 2 x+ 1 2 ¥ 2,©= 2 « ¤( x)¥ 2= ¥ 1= 0,¥ 0¹ 0- 1 2 214( 0+ ¬) 2- 21 4 ¥ 0 2 £1 2 1 3 © £3 2 ©=2 3 ® ¥ 02 ¯1 2 For the degenerate hypergeometric functions( ¥, ; ) and ( ¥, ; ), see Supplement A.9 and the books by Abramowitz and Stegun (1964) and Bateman and Erd Âelyi (1953, V ol. 1). For the Bessel functions ° ¡( ) and ± ¡( ), see Supplement A.6 and the books by Abramowitz and Stegun (1964) and Bateman and Erd Âelyi (1953, V ol. 2). 4.3.2. Equations of the Form ²2 ³² ´2= ( µ ¶2+ ·) ²2 ³² ¶2+ ¸ ¶ ² ³² ¶+ ¹ ³+ º( ¶,´) 1. »2 ¼» ½2= ¾2»2 ¼» ¾2+ ¿( ¾,½). This is a special case of equation 4.3.2.2 with ¥= 1and À= Á= 0. 1 Â. Domain: 1 £ £ ¥. First boundary value problem. The following conditions are prescribed:Ã= Ä0( ) at Å= 0 (initial condition),Æ ÇÃ= Ä1( ) at Å= 0 (initial condition),Ã= È1( Å) at = 1 (boundary condition),Ã= È2( Å) at = ¥(boundary condition). Solution:Ã( , Å)= É Ç 0 É Ê 1 Ë( Ì, Í) Î( Ï, Ì, Å- Í) Ð Ì Ð Í + ÆÆÅ É Ê 1 Ä0( Ì) Î( Ï, Ì, Å) Ð Ì+ É Ê 1 Ä1( Ì) Î( Ï, Ì, Å) Ð Ì + É Ç 0 È1( Í) Ñ ÆÆÌ Î( Ï, Ì, Å- Í) Ò Ó =1 Ð Í- Ô2É Ç 0 È2( Í) Ñ ÆÆÌ Î( Ï, Ì, Å- Í) Ò Ó =Ê Ð Í, whereÎ( Ï, Ì, Å)=2 Õ ÏÌ3 2ln Ô Ö ×Ø =11Ù Ø sin( Ú Ø ln Ï) sin( Ú Ø ln Ì) sin( Ù Ø Û ), Ú Ø = Ü Ýln Ô, Ù Ø = Þ Ú2 Ø +1 4. Page 312 2Â. Domain: 1 £ Ï£ Ô. Second boundary value problem. The following conditions are prescribed:Ã= ß0( Ï) at Û = 0 (initial condition),à áÃ= ß1( Ï) at Û = 0 (initial condition),à âÃ= ã1( Û ) at Ï= 1 (boundary condition),à âÃ= ã2( Û ) at Ï= Ô(boundary condition). Solution:Ã( Ï, Û )= É á 0 ÉÊ 1 Ë( Ì, Í) Î( Ï, Ì, Û - Í) Ð Ì Ð Í + àà ÛÉÊ 1 ß0( Ì) Î( Ï, Ì, Û ) Ð Ì+ ÉÊ 1 ß1( Ì) Î( Ï, Ì, Û ) Ð Ì - É á 0 ã1( Í) Î( Ï,1, Û - Í) Ð Í+ Ô2É á 0 ã2( Í) Î( Ï, Ô, Û - Í) Ð Í, whereÎ( Ï, Ì, Û )= Ô Û ( Ô- 1) Ì2+8 Õ ÏÌ3 2ln Ô Ö ×Ø =1 Ú2 ØÙ Ø (1 + Ú2 Ø ) ä Ø ( Ï)ä Ø ( Ì) sin( Ù Ø Û ),ä Ø ( Ï)=cos( Ú Ø ln Ï)-1 2 Ú Ø sin( Ú Ø ln Ï), Ú Ø = Ü Ýln Ô, Ù Ø = Þ Ú2 Ø +1 4.å.æ Reference : A. G. Butkovskiy (1979). 2. ç2 èç é2= ê ë2ç2 èç ë2+ ì ë ç èç ë+ í è+ î( ë,é). The substitution Ï= ï ð ñ( ï¹ 0) leads to the constant coef®cient equation à áòáÃ= Ô àñóñ Ã+( ô- Ô) àñ Ã+Á Ã+Ë( ï ðñ, Û ), which is discussed in Subsection 4.1.5. 3. ç2 èç é2= ( ê ë2+ ì) ç2 èç ë2+ ê ë ç èç ë+ í è. The substitution õ= É Ð ÏÕÔ Ï2+ ôleads to the constant coef®cient equation à áòáÃ= àñóñ Ã+ Á Ã, which is discussed in Subsection 4.1.3.4.ç2 èç é2= ê2çç ë Ñ( ö2± ë2) ç èç ë Ò+ î( ë,é). Domain: 0 £ Ï£ ÷. First boundary value problem. The following conditions are prescribed:Ã= ß0( Ï) at Û = 0 (initial condition),à áÃ= ß1( Ï) at Û = 0 (initial condition),Ã= ã( Û ) at Ï= 0 (boundary condition),Ã¹ ø at Ï= ÷(boundedness condition). Solution:Ã( Ï, Û )= àà ÛÉ ù 0 ß0( Ì) Î( Ï, Ì, Û ) Ð Ì+ É ù 0 ß1( Ì) Î( Ï, Ì, Û ) Ð Ì + Ô2÷2É á 0 ã( Í) Ñ ààÌ Î( Ï, Ì, Û - Í) Ò Ó =0 Ð Í+ É á 0 É ù 0 Ë( Ì, Í) Î( Ï, Ì, Û - Í) Ð Ì Ð Í. Page 313 Here,Î( Ï, Ì, Û )=1Ô ÷Ö ×Ø =14Ý- 1Ù Ø ú 2 Ø -1 û Ï÷ ü ú 2 Ø -1 û Ì÷ üsin( Ù ØÔ Û ), Ù Ø = ý2Ý(2Ý- 1), where ú þ ( ÿ)=1 2 þï! þ ÿ þ ( ÿ2- 1) þ are the Legendre polynomials.å.æ Reference : M. M. Smirnov (1975). 4.3.3. Other Equations 1. ç2 èç é2= ê2 ç2 èç 2+2 ç èç ± (+ 1)2 è ,= 1, 2, 3, General solution: ( , Û )= Øû1 àà ü Ø ( + Ô Û )+ ( - Ô Û ) , where( 1) and ( 2) are arbitrary functions.å.æ Reference : M. M. Smirnov (1975). 2. ç2èç é2= ç2èç ë2+ ë ç èç ë. The hyperbolic Euler±Poisson±Darboux equation.1. For = 1and = 2, see Subsections 4.2.1±4.2.4. For ¹ 1, the substitution õ= ÿ1- leads to an equation of the form 4.5.3.1:à2 à Û 2=(1 - )2õ2 -1 à2 àõ2. 2 . Suppose = ( ÿ, Û ) is a solution of the equation in question for a ®xed value of the parameter . Then the functions de®ned by the relations = à à Û , = ÿ à àÿ+ Ûà à Û , = 2 ÿ Ûà àÿ+( ÿ2+ Û 2) à à Û + Û are also solutions of this equation.3. Suppose = ( ÿ, Û ) is a solution of the equation in question for a ®xed value of the parameter . Using this , one can construct solutions of the equation with other values of the parameter by the formulas 2- = ÿ -1 , -2= ÿ à àÿ+( - 1) , -2= ÿ Ûà àÿ+ ÿ2 à à Û +( - 1) Û , -2= ÿ( ÿ2+ Û 2) à àÿ+ 2 ÿ2 Ûà à Û + ÿ2+( - 1) Û 2 , +2=1ÿ à àÿ, +2= Ûÿ à àÿ+ à à Û , +2= ÿ2+ Û 2ÿ à àÿ+ 2 Ûà à Û + .å.æ The results of Items 2 and3 were obtained by A. V . Aksenov (2001). Page 314 3. ç2 èç é2= ç2 èç ë2+2 êë ç èç ë+ ì2è, 0 < 2 ê< 1. General solution: ( ÿ, Û )= 1 0 + ÿ(2 - 1) [ (1 - )]1- Å -12 óÿ ý (1 - ) + ÿ1-2 1 0 + ÿ(2 - 1) [ (1 - )] Å - 2 óÿ ý (1 - ) , where( 1) and ( 2) are arbitrary functions; Å - ¡( )= (1 - ¢)2- ¡ ¡ - ¡( ); - ¡( ) is the Bessel function.! Reference : M. M. Smirnov (1975). 4. "2 #" é2= ê ë4"2 #" ë2+ $( ë,é). The transformation = 1 % ÿ, &= % ÿleads to the equation'2&'2= Ô '2&'2+ (1,*), which is discussed in Subsection 4.1.2. 5. "2 #" é2= ( ê ë+ ì)4"2 #" ë2. The transformation&= Ô ++ , = Ô+1Ô ++ , ,= - Ô+1Ô ++ leads to the equation ' -/.&= 0. Thus, the general solution of the original equation has the form =( Ô ++ )[ 0( )+ 1( ,)], where 0= 0( ) and 1= 1( ,) are arbitrary functions.! Reference : N. H. Ibragimov (1994). 6. "2 #" é2= ( ê2± ë2)2"2 #" ë2+ $( ë,é). Domain: - 2£ +£ 2. First boundary value problem. The following conditions are prescribed: = 0( +) at= 0 (initial condition),' 3 = 1( +) at= 0 (initial condition), = 0 at += 2(boundary condition), = 0 at += - 2(boundary condition). Solution for 0< 2< Ô: ( +,)= '' 4 -4 0( ) 5( +, ,) 6 + 4 -4 1( ) 5( +, ,) 6 + 3 0 4 -4 7( , 8) 5( +, ,- 8) 6 6 8, where5( +, ,)=2 Ô9( 2- Ô2)2 : ;=< =11> < ? < ( +) ? < ( ) sin( > <),? < ( +)= @ Ô2- +2sin( A B2+A B2 9ln Ô+ +Ô- + ), > < = Ô9@A2B2+ 92, 9=ln Ô+ 2Ô- 2.! Reference : A. G. Butkovskiy (1979). Page 315 7. "2 #" C2= ( D± E1)2( D± E2)2"2 #" D2, E1¹ E2. The transformation ( +,)=( +- Ô2) &( , 8), =ln F F F F + - Ô1+- Ô2 F F F F , 8= | Ô1- Ô2| leads to the constant coef®cient equation ' G G&= ' HIH&- ' H&, which is discussed in Subsection 4.1.4. 8. "2 #" C2= ( E D2+ J D+ í)2"2 #" D2. The transformation ( +,)= &( ,) K| Ô +2+ /++ L|, = M 6 +N+2+ O/++ L leads to the constant coef®cient equation P 3Q3SR= P -/-R+ T NL-1 4 O2 R, which is discussed in Subsec- tion 4.1.3.9.U2 VU C2= E2UU D W D X U VU D )+ Y( D,C). 1 Z. Domain: 0 £ +£ 2. First boundary value problem. The following conditions are prescribed: 1.1. Case 0< [<1: = 00( +) at= 0 (initial condition),P 3 = 01( +) at= 0 (initial condition), = 0 at += 0 (boundary condition), = 1() at += 2(boundary condition). Solution: ( +,)= PP M 4 0 00( ) 5( +, ,) 6 + M 4 0 01( ) 5( +, ,) 6 - N22\ M 3 0 1( 8) ] PP 5( +, ,- 8) ^H =4 6 8+ M 3 0 M 4 07( , 8) 5( +, ,- 8) 6 6 8. ( 1) Here,5( +, ,)=: ;=< =1 , < ( +) , < ( ) sin( > <N)N _, <_2 > < , > < = ` < 2(2 - [) 2 \-2 2, ( 2) where a < ( b)= b1-\2 c dW ` < eb f g2-\2 h, _ a i_2= M j 0 a 2 i ( b) k b, l= F F F F 1 - [ 2 - [ F F F F ; the` i are positive zeros of the Bessel function, c d(`)= 0. 1.2. Case 1 £ [<2: = m0( b) at= 0 (initial condition),P n = m1( b) at= 0 (initial condition), ¹ o at b= 0 (boundedness condition), = p() at b= f (boundary condition). The solution is given by the formulas presented in Item 1.1.q!r Reference : M. M. Smirnov (1975). Page 316 2 Z. Domain: 0 £ b£ f . Mixed boundary value problem. The following conditions are prescribed: = m0( b) at s= 0 (initial condition),P n = m1( b) at s= 0 (initial condition), ( b \ P t )= 0 at b= 0 (boundary condition), = p( s) at b= f (boundary condition). The solution for 0< [<1is given by relations (1) and (2) witha i ( b)= b1-\2 c- dW ` i eb f g2-\2 h, _ a i_2= M j 0 a 2 i ( b) k b, l=1 - [ 2 - [; the` i are positive zeros of the Bessel function, c- d(`)= 0. 3 Z. For uº 0, the change of variable v= b1-\leads to an equation of the form 4.3.3.10:P2 P s2= N2(1 - [)2v \\-1 P2 P v2. 10. U2 VU w2= x2 yX U2 VU y2. 1 Z. Particular solutions ( z1, z2, {1, {2, and`are arbitrary constants): ( b, s)= | b }~z1 c1 2 T` b + z2 1 2 T` b {1sin( N ` s)+ {2cos( N ` s) , ( b, s)= | b }~z1 1 2 T` b + z2 1 2 T` b {1sinh( N ` s)+ {2cosh( N ` s) , where =1 2(2 - [); c ( v) and ( v) are the Bessel functions; ( v) and ( v) are the modi®ed Bessel functions.2Z. Below are discrete transformations that preserve the form of the original equation; what changes is the parameter . 2.1. The point transformationv=1b, R= b(transformation ) leads to a similar equation 2 s2= 2v4-\ 2 v2. The transformation changes the equation parameter in accordance with the rule [ 4- . The double application of the transformation yields the original equation. 2.2. Suppose = ( b, s) is a solution of the original equation. Then the function = ( , ), which is related to the solution = ( b, s) by the B Èacklund transformation( , )= b ( b, s), b= 1 1- , = |1 - | s (transformation ), is a solution of a similar equation 22= 2 -1 22. The transformation changes the equation parameters in accordance with the rule - 1. The double application of the transformation yields the original equation. Page 317 2.3. The composition of transformations = changes the equation parameter as follows: 4 - 3 - 8 - 3 5 - 2 12 - 5 7 - 3 16 - 7 9 - 4 The -fold application of the transformation yields the equation with parameter 4 -(2 - 1) 2 + 1 - . ( 1) 2.4. The composition of transformations = changes the equation parameter as follows: 4 - 3 1 - 8 - 5 3 - 2 12 - 7 5 - 3 16 - 9 7 - 4 The -fold application of the transformation yields the equation with parameter 4 -(2 + 1) 2 - 1 - . ( 2) 2.5. Setting = 0in (1) and (2), we arrive at two families of equations 2 s2= 2 4 i 2 i +1 2 2at = 1,2, ; 2 s2= 2 4 i 2 i -1 2 2at = 1,2, ; whose solutions can be obtained with the aid of the wave equation; for this constant coef®cient waveequation, see Subsection 4.1.1.3¡. Below are some useful transformations that lead to other equations. 3.1. The substitution = 1- leads to an equation of the form 4.3.3.9: 2 s2= 2(1 - )2 ¢ -1 £. 3.2. The transformation =1 2 |2 - | s, = 2- 2leads to an equation of the form 4.3.3.3: 2 2= 2 2+ - 21 . 11. ¤2 ¥¤ w2=w/¦ ¤2 ¥¤ y2. 1 ¡. Domain: - §< < §. Cauchy problem. Initial conditions are prescribed: = ¨( ) at ©= 0, ª = «( ) at ©= 0. Solution for >0: ( , ©)= ¬(2 )¬2( ) ®1 0 ¨¢ ¯+2°+ 2 © ±+2 2(2 ²- 1)£[ ²(1 - ²)] ³-1 ´² + ¬(2 - 2 )¬2(1 - ) ©®1 0 «¢ ¯+2°+ 2 © ±+2 2(2 ²- 1)£[ ²(1 - ²)]-³ ´², where= ° 2( °+ 2),¬( µ)=® ¶0 ·- ¸*¹»º-1´ ¹.¼!½ Reference : M. M. Smirnov (1975). Page 318 2 ¾. Domain: 0 £¯£ ¿. First boundary value problem. The following conditions are prescribed: = ¨(¯) at ©= 0 (initial condition),À ª = «(¯) at ©= 0 (initial condition), = 0 at¯= 0 (boundary condition), = 0 at¯= ¿(boundary condition). Solution for °>-1: (¯, ©)= Á ©¶ Â=Ã =1 Ä~Å Ã Æ - Ç È2 É Ê Ã©1 2 Ç Ë+ Ì Ã ÆÇ È2 É Ê Ã©1 2 Ç Ë Ísin( Ê Ã¯),Å Ã =¬(1 - É)( Ê ÃÉ) Ç2¿ ® Î0 Ï(¯) sin( Ê Ã¯) ´¯, É=1°+ 2,Ì Ã = Ð(1 + É)( Ê ÃÉ)- Ç2¿ ® Î0 Ñ(¯) sin( Ê Ã¯) ´¯, Ê Ã = Ò Ó¿, where Ð( É) is the gamma function.¼!½ Reference : M. M. Smirnov (1975). 12. Ô2 ÕÔ Ö2=Ö/× Ô2 ÕÔ Ø2+ ÙÖ ×±2 2Ô ÕÔ Ø, Ú³2. Domain: - Û<¯< Û. Cauchy problem. Initial conditions are prescribed: =Ï(¯) at Ü= 0,À Ý =Ñ(¯) at Ü= 0. 1 ¾. Solution for | Þ|<1 2 °: (¯, Ü)= Ð( ß+ à)Ð( ß) Ð( à) ®1 0 Ï á ¯+2°+ 2 Ü ±+2 2(2 ²- 1) â ² ³-1(1 - ²) ã-1 ´² + Ð(2 - ß- à)Ð(1 - ß) Ð(1 - à) Ü®1 0 Ñá ¯+2°+ 2 Ü ±+2 2(2 ²- 1) â ²-ã(1 - ²)-³ ´², whereß= °- 2 Þ 2( °+ 2), à= °+ 2 Þ 2( °+ 2), Ð( µ)=®¶0 ·- ¸ ¹º-1 ´ ¹. 2 ¾. Solution for Þ=1 2 °:ä(¯, Ü)=Ï á ¯+2°+ 2 Ü ±+2 2â+2 Ü°+ 2 ®1 0 Ñá ¯+2°+ 2 Ü ±+2 2(2 ²- 1) â(1 - ²)-±±+2 ´². 3 ¾. Solution for Þ= -1 2 °:ä(¯, Ü)=Ï á ¯-2°+ 2 Ü ±+2 2â+2 Ü°+ 2 ®1 0 Ñá ¯+2°+ 2 Ü ±+2 2(2 ²- 1) â(1 - ²)-±±+2 ´².¼!½ Reference : M. M. Smirnov (1975). 13. ( Ù+Ø)2Ô2 ÕÔ Ö2= å2ÔÔ Ø æ( Ù+Ø)2Ô ÕÔ Ø ç. General solution:ä(¯, Ü)=Ï(¯+ è Ü)+Ñ(¯- è Ü)Þ+¯, whereÏ( é) andÑ( µ) are arbitrary functions. Page 319 4.4. Equations Containing the First Time Derivative 4.4.1. Equations of the Form ê2 ëê ì2+ í ê ëê ì= î2ê2 ëê ï2+ ð ê ëê ï+ ñ ë+ ò(ï,ì) 1. Ô2 ÕÔ Ö2+ ó Ô ÕÔ Ö= å2Ô2 ÕÔ Ø2+ ô(Ø,Ö). For õ( ö, Ü)º 0, this equation governs free transverse vibration of a string, and also longitudinal vibration of a rod in a resisting medium with a velocity-proportional resistance coef®cient.1÷. The substitution ä( ö, Ü)=exp ø-1 2 ù ÜIú»û( ö, Ü) leads to the equationü2ûüÜ2= è2 ü2ûüö2+1 4 ù2û+exp ø1 2 ù ÜIú õ( ö, Ü), which is considered in Subsection 4.1.3.2÷. Fundamental solution: ý ý ( ö, Ü)=1 2 è þ øÿè Ü- | ö| úexp ø-1 2ù ÜIú 0 ø1 2ù Ü2- ö2 è2ú, whereþ( ) is the Heaviside unit step function and 0( ) is the modi®ed Bessel function. Reference : V . S. Vladimirov, V . P. Mikhailov, A. A. Vasharin, et al. (1974). 3 ÷. Domain: - Û< ö< Û. Cauchy problem. Initial conditions are prescribed:ä= ( ö) at Ü= 0,ü Ýä=Ñ( ö) at Ü= 0. Solution:ä( ö, Ü)=1 2exp ø-1 2ù Ü ú ( ö+ è Ü)+ ( ö- è Ü) + ù Ü 4 èexp ø-1 2ù Ü ú + - 1 ø1 2 ù 2-( - )2 2 2-( - )2 2 ( ) +1 2 exp -1 2 + - 0 1 2 2-( - )2 2 ( )+1 2 ( ) +1 2 0 + ( - )- ( - )exp -1 2 (- !) 0 1 2 (- !)2-( - )2 2 "( , !) !, where0( #) and1( #) are the modi®ed Bessel functions of the ®rst kind. 4 $. Domain: 0 £ £ %. First boundary value problem. The following conditions are prescribed:&= 0( ) at= 0 (initial condition),' &= 1( ) at= 0 (initial condition),&= 1() at = 0 (boundary condition),&= 2() at = %(boundary condition). Solution:&( ,)= 0 ( 0 "( , !) )( , ,- !) ! + '' ( 0 0( ) )( , ,) + ( 0 1( )+ 0( ) *)( , ,) + 2 0 1( !) + '' )( , ,- !) , - =0 !- 2 0 2( !) + '' )( , ,- !) , - =( !, Page 320 where( , , )=2 exp - 2 =1sin sin sin( ) , = 2 2 2 2- 2 4. Example. Consider the homogeneous equation ( º 0). The initial shape of the string is a triangle with base 0 £ £ and height at = , that is, ( )= for0 £ £ ,( - )- for £ £ . The initial velocities of the string points are zero, ( )= 0. Solution: ( , )=2 22( - )exp -1 2 "! # $ % =11&2sin ' & (sin ' & ( ) % ( ), where) % ( )= cos( * %)+ 2 * % sin( * %) for <2 & +, 1 + 2for =2 & +, cosh( * %)+ 2 * % sinh( * %) for >2 & +, * % = , - - - - + 2 &2 22- 2 4 - - - - ..0/ References : M. M. Smirnov (1975), B. M. Budak, A. N. Tikhonov, and A. A. Samarskii (1980). 5 1. For the second and third boundary value problems on the interval 0 £ £ , see equation 4.4.1.2 (Items 5 1and6 1with 2= 0). 2. 32 43 52+ 6 3 43 5= 7232 43 82+ 9 4+ :(8,5). Telegraph equation (with>0, 2<0, and ;( , )º 0). 1 1. The substitution <( , )=exp =-1 2 ?>A@( , ) leads to the equationB2@B2=2 B2@B2+( 2+1 42) @+exp=1 2 > ;( , ), which is considered in Subsection 4.1.3.21. Fundamental solutions:C C( , )=1 2 D = - | |>exp=-1 2 >AE0 =GF H 2- 2 I2>for 2+1 42=F2>0,C C( , )=1 2D = - | | >exp =-1 2 ?> J0 =F H 2- 2 I2>for 2+1 42= -F2<0, whereD( K) is the Heaviside unit step function, J0( K) and J1( K) are the Bessel functions, andE0( K) andE1( K) are the modi®ed Bessel functions. 3 1. Domain: - L< < L. Cauchy problem. Initial conditions are prescribed:<= M( ) at = 0,B N<= O( ) at = 0. Page 321 Solution for 2+1 42=F2>0:<( , )=1 2exp =-1 2 ?> PQM( + )+ M( - ) R + F 2exp=-1 2 > S+ TVUS- TVU W1 XGY Z [2-( \- ])2 ^ _2>Z[2-( \- ])2 ^ _2 `( ]) a ] +1 2 _expX-1 2 b [ > cS+ TVUS- TVUW0 XGY Z[2-( \- ])2 ^ _2> dfe( ])+1 2 b`( ]) g a ] +1 2 _ c U 0 cS+ T( U- h)S- T( U- h)expd-1 2 b([- i) gW0 XGY Z([- i)2-( \- ])2 ^ _2> j( ], i) a ] a i. Solution for k+1 4b2= -Y2<0:l( \,[)=1 2expX-1 2b [ > d`( \+ _[)+`( \- _[) g - Ym[ 2 _expX-1 2 b [ > c S+ TVUS- TVU n1 XGY Z [2-( \- ])2 ^ _2>Z[2-( \- ])2 ^ _2 `( ]) a ] +1 2 _expX-1 2 b [ > cS+ TVUS- TVUn0 XGY Z[2-( \- ])2 ^ _2> doe( ])+1 2 b`( ]) g a ] +1 2 _ c U 0 c S+ T( U- h)S- T( U- h)exp d-1 2b([- i) gn0 XY Z([- i)2-( \- ])2 ^ _2>j( ], i) a ] a i. 4 p. Domain: 0 £ \£ q. First boundary value problem. The following conditions are prescribed:l=`0( \) at[= 0 (initial condition),BU l=`1( \) at[= 0 (initial condition),l=e1([) at \= 0 (boundary condition),l=e2([) at \= q(boundary condition). Solution:l( \,[)=c U 0 c r 0 j( ], i) s( \, ],[- i) a ] a i + BB[ cr 0 `0( ]) s( \, ],[) a ]+cr 0 d`1( ])+b`0( ]) gAs( \, ],[) a ] + _2c U 0 e1( i) t BB] s( \, ],[- i) u v =0 a i- _2c U 0 e2( i) t BB] s( \, ],[- i) u v =r a i. Let _2 w2- kmq2-1 4b2q2>0. Thens( \, ],[)=2qexp x- b [ 2 y z { =1sin x w |\q ysin x w |]q ysinX[~} >} , = _2 w2 |2q2- k- b2 4. Let _2 w2 |2- kmq2-1 4 b2q2£ 0for |= 1, , and _2 w2 |2- kmq2-1 4 b2q2>0for |= + 1, + 2, Thens( \, ],[)=2qexp x- b [ 2 y { =1sin x w |\q ysin x w |]q ysinhX[} >} +2qexp x- b [ 2 y z { =+1sin x w |\q ysin x w |]q ysinX[~} >} , = k+ b2 4- _2 w2 |2q2, = _2 w2 |2q2- k- b2 4. Page 322 5 p. Domain: 0 £ \£ q. Second boundary value problem. The following conditions are prescribed:l=`0( \) at[= 0 (initial condition),BU l=`1( \) at[= 0 (initial condition),BS l=e1([) at \= 0 (boundary condition),BS l=e2([) at \= q(boundary condition). Solution:l( \,[)=c U 0 c r 0 j( ], i) s( \, ],[- i) a ] a i + BB[ c r 0 `0( ]) s( \, ],[) a ]+c r 0 d`1( ])+b`0( ]) gAs( \, ],[) a ] - _2c U 0 e1( i) s( \,0,[- i) a i+ _2c U 0 e2( i) s( \, q,[- i) a i. For = k+1 4b2<0,s( \, ],[)=expX-1 2b [ > tsinX[}| |>q}| |+2qz { =1cos( \) cos( ])sinX[ Z _22 - >Z _22 - u, = w |q. For = k+1 4 b2>0,s( \, ],[)=expX-1 2b [ > tsinhX[} >q} +2q z { =1cos( \) cos( ])sinX[ Z _22 - >Z _22 - u, = w |q. If the inequality _22 - <0holds for several ®rst values |= 1, , , then the expressionsZ _22 - should be replaced byZ| _22 - |and the sines by the hyperbolic sines in the corre- sponding terms of the series.6p. Domain: 0 £ \£ q. Third boundary value problem. The following conditions are prescribed:l=`0( \) at[= 0 (initial condition),BU l=`1( \) at[= 0 (initial condition),BS l- 1 l=e1([) at \= 0 (boundary condition),BS l+ 2 l=e2([) at \= q(boundary condition). The solution l( \,[) is determined by the formula in Item 5 pwiths( \, ],[)=expX-1 2 b [ >z { =1 ( \) ( ]) sinX[ Z _22 - > Z _22 - , = k+1 4 b2, ( \)=cos( \)+ 1 sin( \), = 2 2 2 2 + 2 12 + 2 2+ 1 2 2 + q 2 x1 + 2 12 y. Here, the are positive roots of the transcendental equationtan( q)= 1+ 22- 1 2. If the inequality _22 - <0holds for several ®rst values |= 1, , , then the expres- sionsZ _22 - should be replaced byZ| _22 - |and the sines by the hyperbolic sines in the corresponding terms of the series. Page 323 3. 2 2+ = 22 2+ + + (,). 1 p. The substitution l( \,[)=expX-1 2 _-2kV\-1 2b [?A( \,[) leads to the equation2[2= _2 2\2+XGY+1 4b2-1 4 _-2k2 +expX1 2 _-2kV\+1 2b [ j( \,[), which is discussed in Subsection 4.1.3.2p. Fundamental solutions: ( \,[)=1 2 _ X _[- | \|exp x- kV\ 2 _2- b [ 2 yW0 x [2- \2_2yifY+ b2 4- k2 4 _2= 2>0, ( \,[)=1 2 _ X _[- | \|exp x- kV\ 2 _2- b [ 2 yn0 x [2- \2_2yifY+ b2 4- k2 4 _2= - 2<0, where( ) is the Heaviside unit step function,n0( ) andn1( ) are the Bessel functions, andW0( ) andW1( ) are the modi®ed Bessel functions. 3 p. Domain: - < \< . Cauchy problem. Initial conditions are prescribed:l=`( \) at[= 0, l=e( \) at[= 0. Solution for +1 4b2-1 4 _-2k2= 2>0:l( \,[)=1 2exp x- b [ 2 y t`( \+ _[) exp x k[ 2 _y+`( \- _[) exp x- k[ 2 _y u + [ 2 _exp x- kV\ 2 _2- b [ 2 y c + - exp x kV] 2 _2y ¡1 ¢ £ ¤2-( ¥- ¦)2 § ¨2£¤2-( ¥- ¦)2 § ¨2 ©( ¦) ª ¦ +1 2 ¨exp x- « ¥ 2 ¨2- ¬ ¤ 2 y + - exp ® « ¦ 2 ¨2 ¯ °0 ±G² ³ ´2-( µ- ¶)2 · ¸2 ¹ ºf»( ¶)+1 2 ¼ ½( ¶) ¾ ¿ ¶ +1 2 ¸ 0 + ( - À)- ( - À)exp Á Â( ¶- µ) 2 ¸2- ¼(´- Ã) 2 Ä °0 ±² ³(´- Ã)2-( µ- ¶)2 · ¸2 ¹ Å( ¶, Ã) ¿ ¶ ¿ Ã. Solution for +1 4 ¼2-1 4 ¸-2Â2= -²2<0:Æ( µ,´)=1 2exp ®- ¼ ´ 2 ¯ Á½( µ+ ¸´) exp ® Â ´ 2 ¸ ¯+½( µ- ¸´) exp ®- Â ´ 2 ¸ ¯Ä - ² ´ 2 ¸exp ®- Â µ 2 ¸2- ¼ ´ 2 ¯ + - exp ® Â ¶ 2 ¸2 ¯ Ç1 ±G² ³ ´2-( µ- ¶)2 · ¸2 ¹³ ´2-( µ- ¶)2 · ¸2 ½( ¶) ¿ ¶ +1 2 ¸exp ®- Â µ 2 ¸2- ¼ ´ 2 ¯ + - exp ® Â ¶ 2 ¸2 ¯Ç0 ±G²³ ´2-( µ- ¶)2 · ¸2 ¹ ºo»( ¶)+1 2 ¼ ½( ¶) ¾ ¿ ¶ +1 2 ¸ 0 + ( - À)- ( - À)exp Á Â( ¶- µ) 2 ¸2- ¼(´- Ã) 2 ÄÇ0 ±G² ³(´- Ã)2-( µ- ¶)2 · ¸2 ¹ Å( ¶, Ã) ¿ ¶ ¿ Ã.È0É Reference : A. N. Tikhonov and A. A. Samarskii (1990). Page 324 4 Ê. Domain: 0 £ µ£ Ë. First boundary value problem. The following conditions are prescribed:Æ=½0( µ) at´= 0 (initial condition),Ì Æ=½1( µ) at´= 0 (initial condition),Æ= »1(´) at µ= 0 (boundary condition),Æ= »2(´) at µ= Ë(boundary condition). Solution:Æ( µ,´)= 0 Í0 Å( ¶, Ã) Î( µ, ¶,´- Ã) ¿ ¶ ¿ Ã + ÌÌ´ Í0 ½0( ¶) Î( µ, ¶,´) ¿ ¶+ Í0 º½1( ¶)+¼ ½0( ¶) ¾AÎ( µ, ¶,´) ¿ ¶ + ¸2 0 »1( Ã) Á ÌÌ¶ Î( µ, ¶,´- Ã)Ä Ï=0 ¿ Ã- ¸2 0 »2( Ã) Á ÌÌ¶ Î( µ, ¶,´- Ã)Ä Ï=Í ¿ Ã. Let ¸2 Ð2+1 4 ¸-2Â2Ë2- AË2-1 4¼2Ë2>0. ThenÎ( µ, ¶,´)=2Ëexp Á Â( ¶- µ) 2 ¸2- ¼ ´ 2 Ä Ñ ÒÓ =1sin ® Ð ÔµË ¯sin ® Ð Ô¶Ë ¯sin±´~Õ Ö Ó¹Õ Ö Ó ,Ö Ó = ¸2 Ð2 Ô2Ë2+ Â2 4 ¸2- - ¼2 4. Let¸2Ð2Ô2+1 4 ¸-2Â2Ë2- AË2-1 4¼2Ë2£ 0 for Ô= 1, ×××, ;¸2Ð2Ô2+1 4 ¸-2Â2Ë2- AË2-1 4 ¼2Ë2>0for Ô= + 1, + 2, ××× ThenÎ( µ, ¶,´)=2Ëexp Á Â( ¶- µ) 2 ¸2- ¼ ´ 2 Ä ÒÓ =1sin ® Ð ÔµË ¯sin ® Ð Ô¶Ë ¯sinh±´Õ Ø Ó¹ÕØ Ó +2Ëexp Á Â( ¶- µ) 2 ¸2- ¼ ´ 2 Ä Ñ Ò Ó =+1sin ® Ð ÔµË ¯sin ® Ð Ô¶Ë ¯sin±´Õ Ö Ó¹Õ Ö Ó , whereØ Ó = + ¼2 4- ¸2 Ð2 Ô2Ë2- Â2 4 ¸2andÖ Ó = ¸2 Ð2 Ô2Ë2+ Â2 4 ¸2- - ¼2 4.È0É Reference : A. G. Butkovskiy (1979). 5 Ê. Domain: 0 £ µ£ Ë. Second boundary value problem. The following conditions are prescribed:Æ=½0( µ) at´= 0 (initial condition),Ì Æ=½1( µ) at´= 0 (initial condition),Ì Æ= »1(´) at µ= 0 (boundary condition),Ì Æ= »2(´) at µ= Ë(boundary condition). Solution:Æ( µ,´)= 0 Í0 Å( ¶, Ã) Î( µ, ¶,´- Ã) ¿ ¶ ¿ Ã + ÌÌ´ Í0 ½0( ¶) Î( µ, ¶,´) ¿ ¶+ Í0 º½1( ¶)+¼ ½0( ¶) ¾AÎ( µ, ¶,´) ¿ ¶ - ¸2 0 »1( Ã) Î( µ,0,´- Ã) ¿ Ã+ ¸2 0 »2( Ã) Î( µ, Ë,´- Ã) ¿ Ã. Page 325 For Ù= +1 4¼2<0,Î( µ, ¶,´)= Úexp ® Â ¶¸2- ¼ ´ 2 ¯sin±´Õ| Ù| ¹Õ| Ù|+2Ëexp Á Â( ¶- µ) 2 ¸2- ¼ ´ 2 Ä Ñ ÒÓ =1 Û Ó ( µ)Û Ó ( ¶) 1 + Ü2 Ósin±´Õ Ö Ó¹Õ Ö Ó , whereÚ= Â¸2±GÝ ÞÍQßVà2- 1 ¹,Ö Ó = ¸2 Ð2 Ô2Ë2+ Â2 4 ¸2- á- ¼2 4,Û Ó ( µ)=cos ® Ð ÔµË ¯+ Ü Ó sin ® Ð ÔµË ¯, Ü Ó = Â Ë 2 ¸2 Ð Ô. For Ù= á+1 4 ¼2>0,Î( µ, ¶,´)= Úexp ® Â ¶¸2- ¼ ´ 2 ¯sinh±´Õ Ù ¹Õ Ù+2Ëexp Á Â( ¶- µ) 2 ¸2- ¼ ´ 2 Ä Ñ ÒÓ =1 Û Ó ( µ)Û Ó ( ¶) 1 + Ü2 Ósin±´~Õ Ö Ó¹Õ Ö Ó , where the coef®cient Ú,Ö Ó , Ü Ó and the functionsÛ Ó ( µ) remain as before. If the inequalityÖ Ó <0 holds for several ®rst values Ô= 1, ×××, , then the expressionsÕ Ö Ó must be replaced byÕ|Ö Ó |and the sines by the hyperbolic sines in the corresponding terms of the series.6Ê. Domain: 0 £ µ£ Ë. Third boundary value problem. The following conditions are prescribed:Æ=½0( µ) at´= 0 (initial condition),Ì âÆ=½1( µ) at´= 0 (initial condition),Ì ãÆ- ä1 Æ= »1(´) at µ= 0 (boundary condition),Ì ãÆ+ ä2 Æ= »2(´) at µ= Ë(boundary condition). The solution Æ( µ,´) is determined by the formula in Item 5 ÊwithÎ( µ, ¶,´)=exp Á Â( ¶- µ) 2 ¸2- ¼ ´ 2 Ä Ñ ÒÓ =1 Û Ó ( µ)Û Ó ( ¶) sin±G´Õ Ö Ó¹å ÓÕ Ö Ó . Here,Û Ó ( µ)=cos( Ü Óµ)+2 ¸2ä1+Â2 ¸2Ü Ó sin( Ü Óµ),Ö Ó = ¸2Ü2 Ó + Â2 4 ¸2- á- ¼2 4,å Ó =2 ¸2ä2-Â4 ¸2Ü2 Ó4 ¸4Ü2 Ó +(2 ¸2ä1+Â)2 4 ¸4Ü2 Ó +(2 ¸2ä2-Â)2+2 ¸2ä1+Â4 ¸2Ü2 Ó + Ë 2+ Ë(2 ¸2ä1+Â)2 8 ¸4Ü2 Ó , where the Ü Ó are positive roots of the transcendental equation tan( Ü Ë)Ü=4 ¸4( ä1+ ä2) 4 ¸4Ü2-(2 ¸2ä1+Â)(2 ¸2ä2-Â). 4.4.2. Equations of the Formæ2 çæ è2+ é æçæ è= ê( ë) æ2 çæë2+ ì( ë) æçæë+ í( ë) ç+ î( ë, è) 1. ï2 ðï ñ2+ ò ï ðï ñ= ó2 ôï2 ðï õ2+1õ ï ðï õ ö. This equation describes vibration of a circular membrane in a resisting medium with velocity-proportional resistance coef®cient. Page 326 1 ÷. Domain: 0 £ ø£ ù. First boundary value problem. The following conditions are prescribed:ú= û( ø) at ü= 0 (initial condition),ý þú= ÿ( ø) at ü= 0 (initial condition),ú= 0 at = ù(boundary condition). Solution:ú( ø, ü)=exp -1 2 üÑ =1 cos( ü)+ sin( ü) 0 øùö, = 22ù2- 2 4. Here,=2ù2 2 1( ) 0 û( ø) 0 øùö ø ø, = 2 +2 ù2 2 1( ) 0 ÿ( ø) 0 øùö ø ø, where the are positive zeros of the Bessel function, 0()= 0. 2 ÷. For the solution of the second and third boundary value problems, see equation 4.4.2.2 (Items 3 ÷ and4 ÷with = 0). Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 2. 2 2+ = 2 2 2+1 ± + (,). 1 !. The substitution "( #, ü)=exp -1 2 ü %$( #, &) leads to the equationý2$ý&2=2 ý2$ý#2+1# ý$ý#- -1 42 $+exp 1 2 & '( #, &), which is discussed in Subsection 4.2.5.2!. Domain: 0 £ #£ (. First boundary value problem. The following conditions are prescribed:"= )0( #) at &= 0 (initial condition),* +"= )1( #) at &= 0 (initial condition),"= ,( &) at #= ((boundary condition). Solution:"( #, &)= **& 0 )0( -) .( #, -, &) -+ 0 )1( -)+ / )0( -) 0.( #, -, &) - -2 + 0 ,( 1) 2 **- .( #, -, &- 1) 3 4 = 1+ + 0 0 '( -, 1) .( #, -, &- 1) - 1. Here,.( #, -, &)=exp 5-1 2 / &6 7 =12 -(2 2 1( ) 0 #( 0 -(sin 58&:9 69 , = 22(2+ - /2 4, where the are positive zeros of the Bessel function, 0()= 0. The numerical values of the ®rst ten are speci®ed in Paragraph 1.2.1-3. Page 327 3 !. Domain: 0 £ #£ (. Second boundary value problem. The following conditions are prescribed:"= )0( #) at &= 0 (initial condition),* +"= )1( #) at &= 0 (initial condition),* ;"= ,( &) at #= ((boundary condition). Solution:"( #, &)= **& 0 )0( -) .( #, -, &) -+ 0 )1( -)+ / )0( -) .( #, -, &) - +2 + 0 ,( 1) .( #, (, &- 1) 1+ + 0 0 '( -, 1) .( #, -, &- 1) - 1. Here,.( #, -, &)=exp 5-1 2 / &6 22 -sin 5<&:9 0 6(29 0+2(2 7 =1 - 2 0( ) 0 #( 0 -(sin 58&:9 69 3, where 0= -1 4 /2; =22(-2+ -1 4 /2; the are positive zeros of the ®rst-order Bessel function, 1()= 0. The numerical values of the ®rst ten roots are speci®ed in Paragraph 1.2.1-4. 4 !. Domain: 0 £ #£ (. Third boundary value problem. The following conditions are prescribed:"= )0( #) at &= 0 (initial condition),* +"= )1( #) at &= 0 (initial condition),* ;"+ =%"= ,( &) at #= ((boundary condition). The solution "( #, &) is given by the formula in Item 3 !with.( #, -, &)=2(2exp 5-1 2 / & 67 =1 2- ( =2(2+2) 2 0( ) 0 #( 0 -(sin 58&:9 69 . Here, =22(-2+ -1 4 /2and the are positive roots of the transcendental equation 1()- =0( 0()= 0. The numerical values of the ®rst six roots can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984).3.2 2+ = 2 2 2+2 ± + (,). 1 !. The substitution "( #, &)=exp 5-1 2 / &6 $( #, &) leads to the equation*2$*&2=2 *2$*#2+2# *$*#- 5>-1 4 /26 $+exp 51 2 / &6 '( #, &), which is discussed in Subsection 4.2.6. Page 328 2 !. Domain: 0 £ #£ (. First boundary value problem. The following conditions are prescribed:"= )0( #) at &= 0 (initial condition),* +"= )1( #) at &= 0 (initial condition),"= ,( &) at #= ((boundary condition). Solution:"( #, &)= **& 0 )0( -) .( #, -, &) - 0 )1( -)+ / )0( -) 0.( #, -, &) - -2 + 0 ,( 1) 2 **- .( #, -, &- 1) 3 4 = 1+ + 0 0 '( -, 1) .( #, -, &- 1) - 1, where.( #, -, &)=2 -( #exp 5-1 2 / &6 7 =1sin ? @ #(sin ? @ -(sin 5<&:9 69 , = 2@2?2(2+ - /2 4. 3 !. Domain: 0 £ #£ (. Second boundary value problem. The following conditions are prescribed:"= )0( #) at &= 0 (initial condition),* +"= )1( #) at &= 0 (initial condition),* ;"= ,( &) at #= ((boundary condition). Solution:"( #, &)= **& 0 )0( -) .( #, -, &) - 0 )1( -)+ / )0( -) 0.( #, -, &) - +2 + 0 ,( 1) .( #, (, &- 1) 1+ + 0 0 '( -, 1) .( #, -, &- 1) - 1, where.( #, -, &)=exp 5-1 2 / &6 23 -2sin 5& 9 0 6(39 0+2 -( # 7 =1 2+ 129 sin #(sin -(sin 58&0A 6%3. Here, 0= -1 4 /2; =22(-2+ -1 4 /2; and the are positive roots of the transcendental equation tan-= 0. The numerical values of the ®rst ®ve roots are speci®ed in Paragraph 1.2.1-5. 4 !. Domain: 0 £ #£ (. Third boundary value problem. The following conditions are prescribed:"= )0( #) at &= 0 (initial condition),* +"= )1( #) at &= 0 (initial condition),* ;"+ =%"= ,( &) at #= ((boundary condition). The solution "( #, &) is given by the formula in Item 3 !with.( #, -, &)=2 -( #exp 5-1 2 / &6 7 =1 2+( =0(- 1)22+ =0(( =0(- 1)sin #(sin -(sin 58& 9 69 . Here, =22(-2+ -1 4 /2and the are positive roots of the transcendental equationcot+=0(- 1 = 0 . The numerical values of the ®rst six roots can be found in Carslaw and Jaeger (1984). Page 329 4. 2 2+ = 2 B B B± + (B,). 1 !. The substitution "( #, &)=exp 5-1 2 / &6 $( #, &) leads to an equation of the form 4.3.1.2:*2$*&2=2 ** C C *$* C- 5>-1 4 /26 $+exp 51 2 / &6 '( C, &). 2 !. Domain: 0 £ C£ D. First boundary value problem. The following conditions are prescribed:"= )0( C) at &= 0 (initial condition),* +"= )1( C) at &= 0 (initial condition),"= ,( &) at C= D(boundary condition),"¹ E at C= 0 (boundedness condition). Solution:"( C, &)= **& F0 )0( -) .( C, -, &) -+ F0 )1( -)+ / )0( -) 0.( C, -, &) - -2D + 0 ,( 1) 2 **- .( C, -, &- 1) 3 4 =F 1+ + 0 F0 '( -, 1) .( C, -, &- 1) - 1, where.( C, -, &)=1Dexp 5-1 2 / &67 GIH =11J2 1( K H ) J 0 L K H MCD N J 0 L K H M OD Nsin P8&:Q R H SQ R H . Here, R H =1 4 T2K2 HD-1+ U-1 4 /2; the K H are positive zeros of the Bessel function, J 0( K)= 0. 3 V. Domain: 0 £ C£ D. Second boundary value problem. The following conditions are prescribed:W= X0( C) at &= 0 (initial condition),Y ZW= X1( C) at &= 0 (initial condition),Y [W= \( &) at C= D(boundary condition),W¹ E at C= 0 (boundedness condition). Solution:W( C, &)= YY& ] ^0 X0( O ) .( C, O , &) _ O +] ^0 ` X1( O )+ / X0( O ) a _ O +T2D] Z 0 \( 1) .( C, D, &- 1) _ 1+] Z 0 ] ^0 b( O , 1) .( C, O , &- 1) _ O_ 1, where.( c, O , &)=exp P-1 2 / & S d sin P8&Q R0 SDQ R0+1D e fIg =11h2 0( i g ) h 0 L i gMCD N h 0 L i g M OD Nsin P8&:Q R gSQ R g j . Here, R0= U-1 4 /2; R g =1 4 T2i2 gD-1+ U-1 4 /2; the i g are positive zeros of the ®rst-order Bessel function, h 1( i)= 0. The numerical values of the ®rst ten roots i g are speci®ed in Paragraph 1.2.1-5. Page 330 4 V. Domain: 0 £ C£ D. Third boundary value problem. The following conditions are prescribed:W= X0( C) at &= 0 (initial condition),Y ZW= X1( C) at &= 0 (initial condition),Y [W+ / W= \( &) at C= D(boundary condition). The solution W( C, &) is given by the formula in Item 3 Vwith.( c, O , &)=1Dexp P-1 2 / & Se f g =1 i2 g (4 /2D+ i2 g ) h2 0( i g ) h 0 L i gMCD N h 0 L i g M OD Nsin P8&:Q R gSQ R g . Here, R g =1 4 T2i2 gD-1+ U-1 4 /2, and the i g are positive roots of the transcendental equationi h 1( i)- 2 / QD h 0( i)= 0. The numerical values of the ®rst six roots i g can be found in Abramowitz and Stegun (1964) and Carslaw and Jaeger (1984).5.k2 lk m2+ n k lk m= ( o p q+ r) k2 lk p2+1 2 o s p q±1k lk p+ t l. The substitution u=] _ CQT C v+ Uleads to a constant coef®cient equation of the form 4.4.1.2:Y ZwZW+ / Y ZW= Y xyxW+ z W. 4.4.3. Other Equations 1. k2 lk m2+ n± 1m k lk m= k2 lk p2. Darboux equation. Domain: - E< C< E. Cauchy problem. Initial conditions are prescribed:W= X( C) at {= 0,Y ZW= 0 at {= 0. Solution:W( C, {)= | P }2 SQ ~| P }2-1 2 S]1 -1 X( C+ { O )(1 - O 2) }-3 2_ O ( />1). Reference : R. Courant and D. Hilbert (1989). 2. k2 lk m2+2 om k lk m= k2 lk p2± r2l. Domain: - E< C< E. Cauchy problem. Initial conditions are prescribed:W= X( C) at {= 0,{2 Y ZW= \( C) at {= 0. Solution for 0<2T<1:W( C, {)=|(2T)|2(T) ]1 0 X P C+ {(2 O - 1) S Å h-1 P2 Uy{0 O (1 - O ) SO-1(1 - O ) -1_ O +|(2 - 2T) (1 - 2T)|2(1 -T) {1-2 ]1 0 \ P C+ {(2 O - 1) S Å h - P2 Uy{0 O (1 - O ) SO - (1 - O )- _ O , where Å h ( u)= 2 |(1 + ) u- h ( u),|( )=]e0 - -1_ . Reference : M. M. Smirnov (1975). Page 331 3. k2 lk m2+2 om k lk m=m qk2 lk p2. Domain: - E< C< E. Cauchy problem. Initial conditions are prescribed:W= X( C) at {= 0,{2 Y ZW= \( C) at {= 0. Solution for 0 £ 2T<1and >0:W( C, {)=|(2 )|2( ) ]1 0 X C+2 2 + {2+ v 2(2 O - 1)N O -1(1 - O ) -1_ O +|(2 - 2 ) (1 - 2T)|2(1 - ) {1-2 ]1 0 \ C+2 2 + {2+ v 2(2 O - 1)N O - (1 - O )- _ O , where= + 4T 2( + 2),|( u)=]e0 - x -1_ . Reference : M. M. Smirnov (1975). 4.m2k2 lk m2+ nm k lk m= o2k2 lk p2+ r k lk p+ t l. The substitution {= ( ¹ 0) leads to a constant coef®cient equation of the form 4.4.1.3:Y2 WY12+( - 1) YWY1=T2 Y2 WY C2+ U YWY C+ z W. 5.m2k2 lk m2+ nm k lk m= o2p2k2 lk p2+ r p k lk p+ t l. The transformation{= , C= ( ¹ 0, ¹ 0) leads to a constant coef®cient equation of the form 4.4.1.3:Y2 WY12+( - 1) YWY1=T2 Y2 WY O 2+( U-T2) YWY O+ z W. 6.m q k2 lk m2+ om q±1k lk m= k2 lk p2, 0 < s< 2. Domain: - < < . Cauchy problem. Initial conditions are prescribed:W= X( ) at {= 0,{ Y ZW= \( ) at {= 0. 1 V. Solution for1 2 <T<1:W( , {)=|(2 )|2( ) ]1 0 X +2 2 - {2- v 2(2 O - 1)N O -1(1 - O ) -1_ O +|(2 - 2 ) (1 -T)|2(1 - ) {1- ]1 0 \ +2 2 - {2- v 2(2 O - 1)N O - (1 - O )- _ O , Page 332 where=2T- 2(2 - ),|( u)=]e0 - x -1_ . 2 V. Solution forT=1 2 :W( , {)= X( )+ X( u) 2+1 2 ] x \( O ) _ O ,= -2 2 - {2- v 2, u= +2 2 - {2- v 2. Reference : M. M. Smirnov (1975). 7. (m q+ n) k2 lk m2+1 2 sm q±1k lk m= o k2 lk p2+ r k lk p+ t l. The substitution 1=] _ {Q{ v+ leads to the equation Y W=T Y [ [W+ U Y [W+ z W, which is discussed in Subsection 4.1.5. 4.5. Equations Containing Arbitrary Functions 4.5.1. Equations of the Form ( ) 2 2= ( ) ± ( ) + ( ,) It is assumed that the functions , ¡, ¡ ¢£, and ¤are continuous and the inequalities >0, ¡>0hold for 1£ £ 2. 4.5.1-1. General relations to solve linear nonhomogeneous boundary value problems. The solution of the equation in question under the general initial conditions¥= ¦0( ) at {= 0,§ ¨¥= ¦1( ) at {= 0(1) and the arbitrary linear nonhomogeneous boundary conditions© 1 §£ ¥+ ª1 ¥= «1( {) at = 1,© 2 §£ ¥+ ª2 ¥= «2( {) at = 2(2) can be represented as the sum¥( , {)= ¬ ¨ 0 ¬ £2£1 ( ®, ¯) °( ±, ®, ²- ¯) ³ ® ³ ¯ + §§² ¬ £2£1 ´( ®) ¦0( ®) °( ±, ®, ²) ³ ®+ ¬ £2£1 ´( ®) ¦1( ®) °( ±, ®, ²) ³ ® + ¡( ±1) ¬ ¨ 0 «1( ¯) µ1( ±, ²- ¯) ³ ¯+ ¡( ±2) ¬ ¨ 0 «2( ¯) µ2( ±, ²- ¯) ³ ¯. ( 3) Here, the modi®ed Green's function is determined by°( ±, ®, ²)= ¶ ·I¸ =1 ¹ ¸ ( ±)¹ ¸ ( ®) sin º8²:» ¼ ¸ ½¾¹ ¸¾2»¼ ¸ , ¾¹ ¸¾2= ¬ ¿2¿1 ´( ±)¹2 ¸ ( ±) ³ ±, ( 4) Page 333 where the ¼ ¸ and¹ ¸ ( ±) are the eigenvalues and corresponding eigenfunctions of the Sturm±Liouville problem for the second-order linear ordinary differential equation [ À( ±)¹ Á ¿]Á¿+[ ¼´( ±)- Â( ±)]¹= 0,© 1¹Á ¿+ ª1¹= 0 at ±= ±1,© 2¹ Á ¿+ ª2¹= 0 at ±= ±2.(5) The functions µ1( ±, ²) and µ2( ±, ²) that occur in the integrands of the last two terms in solution (3) are expressed in terms of the Green's function of (4). The corresponding formulas will be speci®edbelow in studying speci®c boundary value problems. General properties of the Sturm±Liouville problem (5): 1Ã. There are ®nitely many eigenvalues ¼1< ¼2< ¼3< ÄÅÄÅÄ, with ¼ ¸ Æ Ç as È Æ Ç ; hence the number of negative eigenvalues is ®nite.2Ã. Any two eigenfunctions¹ ¸ ( ±) and¹ É( ±) for È¹ Êare orthogonal to each other with weight´( ±) on the interval ±1£ ±£ ±2; speci®cally,¬ ¿2¿1 ´( ±)¹ ¸ ( ±)¹ É( ±) ³ ±= 0 at È¹ Ê. 3 Ã. If the conditionsÂ( ±)³ 0, Ë1 Ì1£ 0, Ë2 Ì2³ 0 (6) are satis®ed, then there are no negative eigenvalues. If Âº 0andÌ1=Ì2= 0, the least eigenvalue is¼1= 0and the corresponding eigenfunction is Í1=const. In the other cases where conditions (6) are satis®ed, all eigenvalues are positive.Î ÏyÐ Ñ Ò:Ó ÔMore detailed information about the properties of the Sturm±Liouville problem (5) can be found in Subsection 1.8.9. Asymptotic and approximate formulas for eigenvalues andeigenfunctions are also presented there. 4.5.1-2. First boundary value problem (case Ë1= Ë2= 0,Ì1=Ì2= 1). The solution of the ®rst boundary value problem for the equation in question with the initialconditions (1) and the boundary conditionsÕ = Ö1( ²) at ±= ±1, Õ = Ö2( ²) at ±= ±2 is given by relations (3) and (4) in whichµ1( ±, ²)= ×× ® °( ±, ®, ²) Ø Ø Ø =¿1, µ2( ±, ²)= - ×× ® °( ±, ®, ²) Ø Ø Ø =¿2. 4.5.1-3. Second boundary value problem (case Ë1= Ë2= 1,Ì1=Ì2= 0). The solution of the second boundary value problem for the equation in question with the initialconditions (1) and the boundary conditions×¿ Õ = Ö1( ²) at ±= ±1,×¿ Õ = Ö2( ²) at ±= ±2 is given by relations (3) and (4) withµ1( ±, ²)= - °( ±, ±1, ²), µ2( ±, ²)= °( ±, ±2, ²). Page 334 4.5.1-4. Third boundary value problem (case Ë1= Ë2= 1,Ì1¹ 0,Ì2¹ 0). The solution of the third boundary value problem for the equation in question with the initialconditions (1) and the boundary conditions (2) withË1= Ë2= 1is given by relations (3) and (4) in whichµ1( ±, ²)= - °( ±, ±1, ²), µ2( ±, ²)= °( ±, ±2, ²). 4.5.1-5. Mixed boundary value problem (case Ë1=Ì2= 0, Ë2=Ì1= 1). The solution of the mixed boundary value problem for the equation in question with the initialconditions (1) and the boundary conditionsÕ = Ö1( ²) at ±= ±1,×¿ Õ = Ö2( ²) at ±= ±2 is given by relations (3) and (4) withµ1( ±, ²)= ×× ® °( ±, ®, ²) Ø Ø Ø =¿1, µ2( ±, ²)= °( ±, ±2, ²). 4.5.1-6. Mixed boundary value problem (case Ë1=Ì2= 1, Ë2=Ì1= 0). The solution of the mixed boundary value problem with the initial conditions (1) and the boundaryconditions×¿ Õ = Ö1( ²) at ±= ±1, Õ = Ö2( ²) at ±= ±2 is given by relations (3) and (4) withµ1( ±, ²)= - °( ±, ±1, ²), µ2( ±, ²)= - ×× ® °( ±, ®, ²) Ø Ø Ø =¿2.Ù Ú References for Subsection 4.5.1: V . M. Babich, M. B. Kapilevich, S. G. Mikhlin et al. (1964), V . A. Marchenko (1986), V . S. Vladimirov (1988), A. D. Polyanin (2000a). 4.5.2. Equations of the FormÛ2 ÜÛ Ý2+ Þ( Ý) ÛÜÛ Ý= ß( Ý) à ÛÛ á âã( á) ÛÜÛ á ä± å( á) Ü æ+ ç( á, Ý) It is assumed that the functions À, ÀÁ¿, and Âare continuous and À>0for ±1£ ±£ ±2. 4.5.2-1. General relations to solve linear nonhomogeneous boundary value problems. The solution of the equation in question under the general initial conditionsÕ = è0( ±) at ²= 0,× é Õ = è1( ±) at ²= 0(1) and the arbitrary linear nonhomogeneous boundary conditions´1×¿ Õ + ê1 Õ = Ö1( ²) at ±= ±1,´2×¿ Õ + ê2 Õ = Ö2( ²) at ±= ±2(2) Page 335 can be represented as the sumÕ ( ±, ²)= ¬ é 0 ¬ ¿2¿1 ( ®, ¯) ë( ±, ®, ², ¯) ³ ® ³ ¯ - ¬ ¿2¿1 è0( ®) ì ×× ¯ ë( ±, ®, ², ¯) í î =0 ³ ®+ ï ð2ð1 ñ è1( ò)+ ó(0) è0( ò) ô0ë( õ, ò, ö,0) ÷ ò + ø( õ1) ï ù 0 ú1( û) ü( û) ý1( õ, ö, û) ÷ û+ ø( õ2) ï ù 0 ú2( û) ü( û) ý2( õ, ö, û) ÷ û. ( 3) Here, the modi®ed Green's function is determined byþ( õ, ò, ö, û)= ÿ =1¹ ( õ)¹ ( ò)¹ 2 ( ö, û), ¹ 2= ï ð2ð1¹2( õ) ÷ õ, ( 4) where the and¹ ( õ) are the eigenvalues and corresponding eigenfunctions of the Sturm±Liouville problem for the following second-order linear ordinary differential equation with homogeneousboundary conditions: [ø( õ)¹ ð] ð +[ - ( õ)]¹= 0, 1¹ ð+ 1¹= 0 at õ= õ1, 2¹ ð+ 2¹= 0 at õ= õ2.(5) The functions = ( ö, û) are determined by solving the Cauchy problem for the linear ordinary differential equation + ó( ö) + ü( ö) = 0, ù = = 0, ù = = 1.(6) The prime denotes the derivative with respect to ö, and ûis a free parameter occurring in the initial conditions. The functions ý1( õ, ö) and ý2( õ, ö) that occur in the integrands of the last two terms in solution (3) are expressed in terms of the Green's function of (4). The corresponding formulas will be speci®ed below when studying speci®c boundary value problems. The properties of the Sturm±Liouville problem (5) are detailed in Subsection 1.8.9. Asymptotic and approximate formulas for eigenvalues and eigenfunctions are also presented there. 4.5.2-2. First, second, third, and mixed boundary value problems. 1 .First boundary value problem . The solution of the equation in question with the initial condi- tions (1) and boundary conditions (2) for 1= 2= 0and 1= 2= 1is given by relations (3) and (4), whereý1( õ, ö, û)= ò þ( õ, ò, ö, û) =ð1, ý2( õ, ö, û)= - ò þ( õ, ò, ö, û) =ð2. 2 .Second boundary value problem . The solution of the equation with the initial conditions (1) and boundary conditions (2) for 1= 2= 1and 1= 2= 0is given by relations (3) and (4) withý1( õ, ö, û)= - þ( õ, õ1, ö, û), ý2( õ, ö, û)= þ( õ, õ2, ö, û). 3 .Third boundary value problem . The solution of the equation with the initial conditions (1) and boundary conditions (2) for 1= 2= 1and 1 2¹ 0is given by relations (3) and (4) in whichý1( õ, ö, û)= - þ( õ, õ1, ö, û), ý2( õ, ö, û)= þ( õ, õ2, ö, û). 4 .Mixed boundary value problem . The solution of the equation with the initial conditions (1) and boundary conditions (2) for 1= 2= 0and 2= 1= 1is given by relations (3) and (4) withý1( õ, ö, û)= ò þ( õ, ò, ö, û) =ð1, ý2( õ, ö, û)= þ( õ, õ2, ö, û). Page 336 5 .Mixed boundary value problem . The solution of the equation with the initial conditions (1) and boundary conditions (2) for 1= 2= 1and 2= 1= 0is given by relations (3) and (4) withý1( õ, ö, û)= - þ( õ, õ1, ö, û), ý2( õ, ö, û)= - ò þ( õ, ò, ö, û) =ð2. References : V . M. Babich, M. B. Kapilevich, S. G. Mikhlin et al. (1964), A. V . Bitsadze and D. F. Kalinichenko (1985), A. D. Polyanin (2000a). 4.5.3. Other Equations 1. 2 2= ( ) 2 2. This is a special case of the equation of Subsection 4.5.1 with ( õ)= 1 ( õ), ø( õ)= 1, and == 0. 1 . Particular solutions:= 1 õ ö+ 2 ö+ 3 õ+ 4,= 1 ö2+ 2 õ ö+ 3 ö+ 4 õ+ 2 1 ï ð õ- ò( ò) ÷ ò+ 5,= 1 ö3+ 2 õ ö+ 3 ö+ 4 õ+ 6 1 ö ï ð õ- ò( ò) ÷ ò+ 5,=( 1 õ+ 2) ö2+ 3 õ ö+ 4 ö+ 5 õ+ 2 ï ð( õ- ò)( 1 ò+ 2)( ò) ÷ ò+ 6, where 1, 2, 3, 4, 5, and 6are arbitrary constants, and óis an arbitrary real number. 2 . Separable particular solution:=( 1 ù+ 2 -ù) ( õ), where 1, 2, and are arbitrary constants, and the function = ( õ) is determined by the ordinary differential equation ( õ) ð ð- 2 = 0. 3 . Separable particular solution:=[ 1sin( ö)+ 2cos( ö)] !( õ), where 1, 2, and are arbitrary constants, and the function != !( õ) is determined by the ordinary differential equation ( õ) ! ð ð+ 2!= 0. 4 . Particular solutions with even powers of ö:= #" =0 $ " ( õ) ö2 " , where the functions$ " =$ " ( õ) are de®ned by the recurrence relations$ ( õ)= % õ+ & ,$ " -1( õ)= % "õ+ & " + 2 (2 - 1) ï ð( õ- ò)$ " ( ò)( ò) ÷ ò, where % " , & " are arbitrary constants ( = ', ()()(,1). 5 . Particular solutions with odd powers of ö:= " =0 * " ( õ) ö2 " +1, where the functions * " = * " ( õ) are de®ned by the recurrence relations*( õ)= % õ+ & ,* " -1( õ)= % "õ+ & " + 2 (2 + 1) ï ð( õ- ò) * " ( ò)( ò) ÷ ò, where % " , & " are arbitrary constants ( = ', ()()(,1). Page 337 2. 2 2= +( ) ,. This is a special case of the equation of Subsection 4.5.1 with ( õ)= 1, ø( õ)= ( õ), and == 0. 1 . Particular solutions:= 1 ö2+ 2 ö+ 2 ï 1 õ+ 3( õ) ÷ õ+ 4,= 1 ö3+ 2 ö+ 6 ö ï 1 õ+ 3( õ) ÷ õ+ 4,=[ 1( õ)+ 2] ö+ 3( õ)+ 4,( õ)= ï ÷ õ( õ),=[ 1( õ)+ 2] ö2+ 3( õ)+ 4+ 2 ï -1( õ) ï[ 1( õ)+ 2] ÷ õ . ÷ õ, where 1, 2, 3, 4, and 5are arbitrary constants. 2 . Separable particular solution:=( 1 ù+ 2 -ù) ( õ), where 1, 2, and are arbitrary constants, and the function = ( õ) is determined by the ordinary differential equation [ ( õ) ð] ð - 2 = 0. 3 . Separable particular solution:=[ 1sin( ö)+ 2cos( ö)] !( õ), where 1, 2, and are arbitrary constants, and the function != !( õ) is determined by the ordinary differential equation [ ( õ) ! ð] ð + 2!= 0. 4 . Particular solutions with even powers of ö:= #" =0 / " ( õ) ö2 " , where the functions/ " =/ " ( õ) are de®ned by the recurrence relations/ ( õ)= % ( õ)+ & ,( õ)= ï ÷ õ( õ),/ " -1( õ)= % "( õ)+ & " + 2 (2 - 1) ï1( õ) - ï/ " ( õ) ÷ õ . ÷ õ, where % " , & " are arbitrary constants ( = ', ()()(,1). 5 . Particular solutions with odd powers of ö:= #" =0 0 " ( õ) ö2 " +1, where the functions0 " =0 " ( õ) are de®ned by the recurrence relations0 ( õ)= % ( õ)+ & ,( õ)= ï ÷ õ( õ),0 " -1( õ)= % "( õ)+ & " + 2 (2 + 1) ï1( õ) - ï0 " ( õ) ÷ õ . ÷ õ, where % " , & " are arbitrary constants ( = ', ()()(,1). Page 338 3. 2 2= ( ) 2 2+ 1( ) + 2( ,), 0 < ( ) < 3. This equation can be rewritten in the form of the equation from Subsection 4.5.1 with ( õ)º 0:( õ) 2 ö2= õ 4 ø( õ) õ 5+ ( õ)( õ, ö), where( õ)=1( õ)exp4 ïú( õ)( õ) ÷ õ5, ø( õ)=exp4 ïú( õ)( õ) ÷ õ5. 4. 2 2= ( ) 2 2+ 1( ) + 6( ) + 2( ,). This equation can be rewritten in the form of the equation from Subsection 4.5.1:( õ) 2 ö2= õ 4 ø( õ) õ 5- ( õ) + ( õ)( õ, ö), where( õ)=1( õ)exp4 ïú( õ)( õ) ÷ õ5, ø( õ)=exp4 ïú( õ)( õ) ÷ õ5, ( õ)= - 7( õ)( õ)exp4 ïú( õ)( õ) ÷ õ5. 5. 2 2= ( ) 2 2+ 1( ) +ñ 61( ) + 62() ô . 1 . There are separable solutions in the product form ( õ, ö)=$( õ) *( ö), where the functions$=$( õ) and *= *( ö) satisfy the ordinary differential equations ( is an arbitrary constant):( õ)$ ð ð +ú( õ)$ ð +ñ +71( õ) ô$= 0, * ùwù+ñ -72( ö) ô *= 0. 2 . For the solution of various boundary value problems for the original equation, see Subsec- tions 0.4.1 and 0.4.2. 6. 2 2= ( ) 2 2+1 2 8( ) + 9 . The substitution := ï ÷ õ;( õ)leads to the constant coef®cient equation ùwù = <=< + ü that is discussed in Subsection 4.1.3. 7. 2 2= 22 2+ ( 8 >+ 2 1) + ( 1 8 >+ 12) , = ( ), 1= 1( ). The transformation( õ, ö)= ?( ò, ö) exp @- A BC D E F, G= A D EC( E) leads to the wave equation H IJIK?= H LML)?that is discussed in Subsection 4.1.1. 8. 2 N O2+ P N O= Q( R) 2 N R2+1 2 Q S( R) N R+ T N. The substitution U= A D EVC( E)leads to a constant coef®cient equation of the form 4.4.1.2:H IJIKW+ X H IYW= H Z=Z)W+ [=W. Page 339 9. Q(O) 2 N O2+1 2 Q S(O) N O= P 2 N R2+ T N R+ \ N. The substitution ]= A D ^VC( ^)leads to the equation H _ _ W= X Hð ð W+ [`Hð W+ a`Wthat is discussed in Subsection 4.1.5. 10. Q(O) 2 N O2+1 2 QS(O) N O= b( R) 2 N R2+1 2 bS( R) N R+ \ N. The transformation ]= A D ^VC( ^), U= A D EVB( E)leads to the constant coef®cient equationH _ _ W= H Z=Z W+ a`Wthat is discussed in Subsection 4.1.3. Page 340 Chapter 5 Hyperbolic Equations with TwoSpace Variab les 5.1. WaveEquation 2 2= 2 2 5.1.1. Problems inCartesian Coor dinates Thewaveequation with twospace variables intherectangular Cartesian system ofcoordinates has theform 2 2= 2 2 2+ 2 2 . 5.1.1-1. Particular solutions andsome relations. 1 .Particular solutions:( , , )= exp 1 + 2 2 1+ 2 2 ,( , , )= sin( 1 + 1)sin( 2 + 2)sin 2 1+ 2 2 ,( , , )= sin( 1 + 1)sin( 2 + 2)cos 2 1+ 2 2 ,( , , )= sinh( 1 + 1)sinh( 2 + 2)sinh 2 1+ 2 2 ,( , , )= sinh( 1 + 1)sinh( 2 + 2)cosh 2 1+ 2 2 ,( , , )= ( sin + cos + )+ ( sin + cos - ), where , 1, 2, 1, 2,and arearbitrary constants, and ( )and ( )arearbitrary functions. 2 .Particular solutions thatareexpressed interms ofsolutions tosimpler equations:( , , )= cos( )+ sin( ) ( , ), where !"!= 2 # #- 22, (1)( , , )= cosh( )+ sinh( ) ( , ), where !"!= 2 # #+ 22, (2)( , , )= cos( )+ sin( ) $( , ), where # #+ %%=-( & )2, (3)( , , )= cosh( )+ sinh( ) ( , ), where # #+ %%=( & )2, (4)( , , )=exp 2 ' ( , (), (= ) 2,where *= ' # #. (5) Forparticular solutions ofequations (1)and(2)forthefunction ( , ),seetheKlein±Gordon equation 4.1.3. Forparticular solutions ofequations (3)and(4)forthefunction ( , ),see Subsection 7.3.2. Forparticular solutions oftheheat equation (5)forthefunction ( , (),see Subsection 1.1.1. Page341 3 . Fundamental solution: + + ( , , )= ,( - -) 2 . / 2 2- -2,,( )= 01for ³ 0, 0for <0, where -= 2+ 2. 4 . In®nite series solutions that contain arbitrary functions of the space variables:( , , )= 1( , )+ 2 354 =1( )2 4 (2 6)! 7 41( , ),7º 2 2+ 2 2,( , , )= 98 ( , )+ 2 354 =1( )2 4 (2 6+ 1)! 7 48 ( , ), where 1( , ) and 8 ( , ) are any in®nitely differentiable functions. The ®rst solution satis®es the initial conditions ( , ,0)= 1( , ), !( , ,0)= 0and the second solution to the initial conditions ( , ,0)= 0, !( , ,0)= 8 ( , ). The sums are ®nite if 1( , ) and 8 ( , ) are bivariate polynomials.:<; Reference : A. V . Bitsadze and D. F. Kalinichenko (1985). 5 . A wide class of solutions to the wave equation with two space variables are described by the formulas( , , )=Re =( >) and ( , , )=Im =( >). ( 6) Here, =( >) is an arbitrary analytic function of the complex argument >related to the variables ( , , ) by the implicit relation -( - 0) >+( - 0) /1 - >2= ?( >), ( 7) where ?( >) is any analytic function and 0, 0are arbitrary constants. Solutions of the forms (6), (7) ®nd wide application in the theory of diffraction. If the argument >obtained by solving (7) with a prescribed ?( >) is real in some domain @, then one should set Re =( >)= =( >) in relation (6) everywhere in @.:<; Reference : V . I. Smirnov (1974, V ol. 3, Pt. 2). 6 . Suppose = ( , , ) is a solution of the wave equation. Then the functions1= ( A + 1, A + 2, A + 3),2= - B 1 -( B & )2, , - B -2 1 -( B & )2 ,3= | -2- 2 2| -2- 2 2, -2- 2 2, -2- 2 2 ,4= / C + 1( 2 2- -2)C, + 2( 2 2- -2)C, + 3( 2 2- -2)C ,-2= 2+ 2,C= 1 - 2 ( 1 + 2 - 3 )+( 2 1+ 2 2- 2 3)( -2- 2 2), where , 1, 2, 3, 1, 2, 3, B, and A are arbitrary constants, are also solutions of the equation. The signs at A in the expression of 1can be taken independently of one another. The function 2 results from the invariance of the wave equation under the Lorentz transformation. More detailed information about particular solutions and transformations of the wave equation with two space variables can be found in the references cited below.:<; References : E. Kalnins and W. Miller, Jr. (1975, 1976), W. Miller, Jr. (1977). Page 342 2 EDF22 H 5.1.1-2. Domain: - J< < J,- J< < J. Cauchy problem. Initial conditions are prescribed:= 1( , ) at = 0, != 8 ( , ) at = 0. Solution (Poisson's formula):( , , )=1 2 . K KL MON 1( P, Q) R P R Q2 2-( P- )2-( Q- )2+1 2 . K KL MON 8 ( P, Q) R P R Q2 2-( P- )2-( Q- )2, where the integration is performed over the interior of the circle of radius with center at ( , ).:<; References : N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov (1970), A. N. Tikhonov and A. A. Samarskii (1990). 5.1.1-3. Domain: 0 £ £ S1,0 £ £ S2. First boundary value problem. A rectangle is considered. The following conditions are prescribed:= 10( , ) at = 0 (initial condition), != 11( , ) at = 0 (initial condition),= 8 1( , ) at = 0 (boundary condition),= 8 2( , ) at = S1(boundary condition),= 8 3( , ) at = 0 (boundary condition),= 8 4( , ) at = S2(boundary condition). Solution:( , , )= K T1 0 K T2 0 10( P, Q) ?( , , P, Q, ) R Q R P+ K T1 0 K T2 0 11( P, Q) ?( , , P, Q, ) R Q R P + 2K ! 0 KT2 0 8 1( Q, () U P ?( , , P, Q, - () V W =0 R Q R ( - X2K ! 0 K T2 0 8 2( Q, () U YY P ?( Z, [, P, Q, \- () V W =T1 R Q R ( + X2K ! 0 K T1 0 8 3( P, () UYY Q ?( Z, [, P, Q, \- () V ] =0 R P R ( - X2K ! 0 KT1 0 8 4( P, () UYY Q ?( Z, [, P, Q, \- () V ] =T2 R P R (, where?( Z, [, P, Q, \)=4X S1 S2 2 3 4 =1 2 3^=11A 4^sin( _ 4Z) sin( ` ^[) sin( _ 4P) sin( ` ^Q) sin( X A 4^\),_ 4 = 6 .S1, ` ^= a .S2, A 4^= b _2 c+ `2^. The problem of vibration of a rectangular membrane with sides S1and S2rigidly ®xed in its contour is characterized by homogeneous boundary conditions, d eº 0( f= 1,2,3,4).g<h References : M. M. Smirnov (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 343 5.1.1-4. Domain: 0 £ Z£ i1,0 £ [£ i2. Second boundary value problem. A rectangle is considered. The following conditions are prescribed:j= k0( Z, [) at \= 0 (initial condition),Y l j= k1( Z, [) at \= 0 (initial condition),Y m j= d1( [, \) at Z= 0 (boundary condition),Y m j= d2( [, \) at Z= i1(boundary condition),Y n j= d3( Z, \) at [= 0 (boundary condition),Y n j= d4( Z, \) at [= i2(boundary condition). Solution:j( Z, [, \)=YY \ o p1 0 o p2 0 k0( q, r) s( t, u, q, r, v) w r w q+o p1 0 o p2 0 k1( q, r) s( t, u, q, r, v) w r w q - X2o l 0 o p2 0 d1( r, x) s( t, u,0, r, v- x) w r w x + X2o l 0 o p2 0 d2( r, x) s( t, u, i1, r, v- x) w r w x - X2o l 0 o p1 0 d3( q, x) s( t, u, q,0, v- x) w q w x + X2o l 0 o p1 0 d4( q, x) s( t, u, q, i2, v- x) w q w x, wheres( t, u, q, r, v)= vi1 i2+2X i1 i2 y zc=0 y z{=0 | c{}c{cos( ~ ct) cos( {u) cos( ~ cq) cos( {r) sin( X }c{v),~ c= i1, {= i2, }c{= b ~2 c+ 2{,| c{= 0for== 0, 1for = 0(¹), 2for ¹ 0,g<h References : A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 5.1.1-5. Domain: 0 £ t£ i1,0 £ u£ i2. Third boundary value problem. A rectangle is considered. The following conditions are prescribed:j= k0( t, u) at v= 0 (initial condition),Y l j= k1( t, u) at v= 0 (initial condition),Y m j- 1 j= d1( u, v) at t= 0 (boundary condition),Y m j+ 2 j= d2( u, v) at t= i1(boundary condition),Y n j- 3 j= d3( t, v) at u= 0 (boundary condition),Y n j+ 4 j= d4( t, v) at u= i2(boundary condition). The solution j( t, u, v) is determined by the formula in Paragraph 5.1.1-4 wheres( t, u, q, r, v)=4Xy zc=1 y z{=11c{ 2 c+ 2{sin( ct+ c) sin( {u+ {) ´sin( cq+ c) sin( {r+ {) sin X vb 2 c+ 2{ , c=arctan ci1, {=arctan {i2, c{= "i1+( 1 2+ 2c)( 1+ 2) ( 2 1+ 2 c)( 2 2+ 2 c) i2+( 3 4+ 2{)( 3+ 4) ( 2 3+ 2{)( 2 4+ 2{) , Page 344 2 22 where the cand {are positive roots of the transcendental equations2- 1 2=( 1+ 2) cot( i1 ), 2- 3 4=( 3+ 4) cot( i2 ).g<h References : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 5.1.1-6. Domain: 0 £ t£ i1,0 £ u£ i2. Mixed boundary value problems. 1 . A rectangle is considered. The following conditions are prescribed:j= k0( t, u) at v= 0 (initial condition),Y l j= k1( t, u) at v= 0 (initial condition),j= d1( u, v) at t= 0 (boundary condition),j= d2( u, v) at t= i1(boundary condition),Y n j= d3( t, v) at u= 0 (boundary condition),Y n j= d4( t, v) at u= i2(boundary condition). Solution:j( t, u, v)=YY v o p1 0 o p2 0 k0( q, r) s( t, u, q, r, v) w r w q+o p1 0 o p2 0 k1( q, r) s( t, u, q, r, v) w r w q + X2o l 0 o p2 0 d1( r, x) YY q s( t, u, q, r, v- x) W =0 w r w x - X2o l 0 o p2 0 d2( r, x) YY q s( t, u, q, r, v- x) W =p1 w r w x - X2o l 0 o p1 0 d3( q, x) s( t, u, q,0, v- x) w q w x + X2o l 0 o p1 0 d4( q, x) s( t, u, q, i2, v- x) w q w x, wheres( t, u, q, r, v)=2X i1 i2 y zc=1 y z{=0| {}c{sin( ~ ct) cos( {u) sin( ~ cq) cos( {r) sin( X }c{v),~ c= i1, {= i2, }c{= b ~2 c+ 2{,| {= 1for= 0, 2for¹ 0. 2 . A rectangle is considered. The following conditions are prescribed:j= k0( t, u) at v= 0 (initial condition),Y l j= k1( t, u) at v= 0 (initial condition),j= d1( u, v) at t= 0 (boundary condition),Y m j= d2( u, v) at t= i1(boundary condition),j= d3( t, v) at u= 0 (boundary condition),Y n j= d4( t, v) at u= i2(boundary condition). Page 345 Solution:j( t, u, v)=YY v o p1 0 o p2 0 k0( q, r) s( t, u, q, r, v) w r w q+o p1 0 o p2 0 k1( q, r) s( t, u, q, r, v) w r w q + X2o l 0 o p2 0 d1( r, x) YY q s( t, u, q, r, v- x) W =0 w r w x + X2o l 0 o p2 0 d2( r, x) s( t, u, i1, r, v- x) w r w x + X2o l 0 o p1 0 d3( q, x) YY r s( t, u, q, r, v- x) =0 w q w x + X2o l 0 o p1 0 d4( q, x) s( t, u, q, i2, v- x) w q w x, wheres( t, u, q, r, v)=4X i1 i2 y zc=0 y z{=01}c{sin( ~ ct) sin( {u) sin( ~ cq) sin( {r) sin( X }c{v),~ c=(2+ 1) 2 i1, {=(2+ 1) 2 i2, }c{= b ~2 c+ 2{. 5.1.2. Problems in Polar Coordinates The wave equation with two space variables in the polar coordinate system has the formY2 jY v2= X2 Y2 jY 2+1 Y jY +12 Y2 jY 2 ,= t2+ u2. One-dimensional solutions = (, v) that are independent of the angular coordinateare considered in Subsection 4.2.1. 5.1.2-1. Domain: 0 ££ ,0 ££ 2. First boundary value problem. A circle is considered. The following conditions are prescribed:= 0(,) at v= 0 (initial condition),Y ¡ = 1(,) at v= 0 (initial condition),= d(, v) at= (boundary condition). Solution:(,, v)=YY v o2 ¢ 0 o £0 0( q, r) s(,, q, r, v) q w q w r+o2 ¢ 0 o £0 1( q, r) s(,, q, r, v) q w q w r - ¤2o ¡ 0 o2 ¢ 0 d( r, x) ¥¥ q s(,, q, r, v- x) ¦=£ w r w x. Here,s(,, q, r, v)=1 ¤ 2y z¨§ =0 y z{=1| § §{[ © ª § ( §{)]2 © § ( §{) © § ( §{q) cos[(- r)] sin( §{¤ v),|0= 1,| § = 2 (= 1,2, «««), where the © § ( q) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the §{are positive roots of the transcendental equation © § ( )= 0. The problem of vibration of a circular membrane of radius rigidly ®xed in its contour is characterized by the homogeneous boundary condition, d(, v)º 0.¬< References : N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov (1970), A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 346 2 22 5.1.2-2. Domain: 0 ££ ,0 ££ 2. Second boundary value problem. A circle is considered. The following conditions are prescribed:= 0(,) at v= 0 (initial condition),¥¡ = 1(,) at v= 0 (initial condition),¥ ® = d(, v) at= (boundary condition). Solution:(,, v)= ¥¥ v o2 ¢ 0 o £0 0( q, r) s(,, q, r, v) q w q w r+o2 ¢ 0 o £0 1( q, r) s(,, q, r, v) q w q w r + ¤2o ¡ 0 o2 ¢ 0 d( r, x) s(,, , r, v- x) w r w x. Here,s(,, q, r, v)= v 2+1 ¤y z5§ =0 y z{=1| § §{© § ( §{) © § ( §{q) ( 2 §{2-2)[ © § ( §{)]2cos[(- r)] sin( §{¤ v),|0= 1,| § = 2 (= 1,2, «««), where the © § ( q) are the Bessel functions and the §{are positive roots of the transcendental equation© ª § ( )= 0.¬< References : A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 5.1.2-3. Domain: 0 ££ ,0 ££ 2. Third boundary value problem. A circle is considered. The following conditions are prescribed:= 0(,) at v= 0 (initial condition),¥¡ = 1(,) at v= 0 (initial condition),¥ ® + = d(, v) at= (boundary condition). The solution (,, v) is determined by the formula in Paragraph 5.1.2-2 wheres(,, q, r, v)=1 ¤y z5§ =0 y z{=1| § §{© § ( §{) © § ( §{q) ( 2 §{2+ 22-2)[ © § ( §{)]2cos[(- r)] sin( §{¤ v),|0= 1,| § = 2 (= 1,2, «««). Here, the © § ( q) are the Bessel functions and the §{are positive roots of the transcendental equation© ª § ( )+ © § ( )= 0. 5.1.2-4. Domain: 1££ 2,0 ££ 2. First boundary value problem. An annular domain is considered. The following conditions are prescribed:= 0(,) at v= 0 (initial condition),¥¡ = 1(,) at v= 0 (initial condition),= d1(, v) at= 1(boundary condition),= d2(, v) at= 2(boundary condition). Page 347 Solution:(,, v)= ¥¥ v o2 ¢ 0 o £2£1 0( q, r) s(,, q, r, v) q w q w r+o2 ¢ 0 o £2£1 1( q, r) s(,, q, r, v) q w q w r + ¤21o ¡ 0 o2 ¢ 0 d1( r, x) ¥¥ q s(,, q, r, v- x)¦=£1 w r w x - ¤22o ¡ 0 o2 ¢ 0 d2( r, x) ¥¥ q s(,, q, r, v- x) ¦=£2 w r w x. Here,s(,, q, r, v)=2 ¤y z5§ =0 y z{=1| § ¯ §{ ° § ( §{) ° § ( §{q) cos[(- r)] sin( §{¤ v),| § = ±1 ²2for= 0, 1 for¹ 0, ¯ §{= §{©2 § ( §{2)©2 § ( §{1)- ©2 § ( §{2),° § ( §{)= © § ( §{1) ³ § ( §{)- ³ § ( §{1) © § ( §{), where the © § () and ³ § () are the Bessel functions, and the §{are positive roots of the transcen- dental equation© § ( 1) ³ § ( 2)- ³ § ( 1) © § ( 2)= 0. 5.1.2-5. Domain: 1££ 2,0 ££ 2. Second boundary value problem. An annular domain is considered. The following conditions are prescribed:= 0(,) at ´= 0 (initial condition),¥¡ = 1(,) at ´= 0 (initial condition),¥ ® = d1(, ´) at= 1(boundary condition),¥ ® = d2(, ´) at= 2(boundary condition). Solution:(,, ´)= ¥¥ ´ µ2 ¢ 0 µ £2£1 0( ¶, ·) ¸( ¹,, ¶, ·, ´) ¶ º ¶ º ·+µ2 ¢ 0 µ £2£1 1( ¶, ·) ¸( ¹,, ¶, ·, ´) ¶ º ¶ º · - ¤2 » 1µ ¼0 µ2 ½ 0 d1( ·, ¾) ¸( ¹,, » 1, ·, ´- ¾) º · º ¾ + ¿2 » 2µ ¼0 µ2 ½ 0 d2( ·, ¾) ¸( ¹,, » 2, ·, ´- ¾) º · º ¾. Here,¸( ¹,, ¶, ·, ´)= ´( »2 2- »2 1)+1 ¿ À Á5Â =0 À ÁÃ=1 Ä Â Å ÂÃ Æ Â ( Å ÂÃ¹) Æ Â ( Å ÂÃ¶) cos[(- ·)] sin( Å ÂÃ¿ ´) ( Å 2 ÂÃ »2 2-2) Æ2 Â ( Å ÂÃ » 2)-( Å 2 ÂÃ »2 1-2) Æ2 Â ( Å ÂÃ » 1),Æ Â ( Å ÂÃ¹)= Ç È Â ( Å ÂÃ » 1) ³ Â ( Å ÂÃ¹)- ³ È Â ( Å ÂÃ » 1) Ç Â ( Å ÂÃ¹), whereÄ0= 1andÄ Â = 2for= 1,2, ÉÉÉ; the Ç Â ( ¹) and ³ Â ( ¹) are the Bessel functions; and the Å ÂÃare positive roots of the transcendental equationÇ È Â ( Å» 1) ³ È Â ( Å» 2)- ³ È Â ( Å» 1) Ç È Â ( Å» 2)= 0. Page 348 2 ËÊÌ22 Î 5.1.2-6. Domain: » 1£ ¹£ » 2,0 ££ 2. Third boundary value problem. An annular domain is considered. The following conditions are prescribed:Ð= Ñ0( ¹,) at Ò= 0 (initial condition),Ó¼ Ð= Ñ1( ¹,) at Ò= 0 (initial condition),Ó ÔÐ- Õ1 Ð= d1(, Ò) at ¹= » 1(boundary condition),Ó ÔÐ+ Õ2 Ð= d2(, Ò) at ¹= » 2(boundary condition). The solution Ð( ¹,, Ò) is determined by the formula in Paragraph 5.1.2-5 where¸( ¹,, ¶, ·, Ò)=1 ¿À Á5Â =0 À ÁÃ=1 Ä Â Å ÂÃ Æ Â ( Å ÂÃ¹) Æ Â ( Å ÂÃ¶) cos[(- ·)] sin( Å ÂÃ¿ Ò) ( Õ2 2 »2 2+ Å 2 ÂÃ »2 2-2) Æ2 Â ( Å ÂÃ » 2)-( Õ2 1 »2 1+ Å 2 ÂÃ »2 1-2) Æ2 Â ( Å ÂÃ » 1),Æ Â ( Å ÂÃ¹)= Ö Å ÂÃÇ È Â ( Å ÂÃ » 1)- Õ1 Ç Â ( Å ÂÃ » 1) ×Ø Â ( Å ÂÃ¹) -Ö Å ÂÃØ È Â ( Å ÂÃ » 1)- Õ1 Ø Â ( Å ÂÃ » 1)× Ç Â ( Å ÂÃ¹). Here,Ä0= 1andÄ Â = 2for Ù= 1,2, ÉÉÉ; the Ç Â ( ¹) and Ø Â ( ¹) are the Bessel functions; and the Å ÂÃare positive roots of the transcendental equationÖ ÅÇ È Â ( Å» 1)- Õ1 Ç Â ( Å» 1)× Ö ÅØ È Â ( Å» 2)+ Õ2 Ø Â ( Å» 2)× =Ö ÅØ È Â ( Å» 1)- Õ1 Ø Â ( Å» 1)× Ö ÅÇ È Â ( Å» 2)+ Õ2 Ç Â ( Å» 2)×. 5.1.2-7. Domain: 0 £ ¹£ »,0 ££0. First boundary value problem. A circular sector is considered. The following conditions are prescribed:Ð= Ñ0( ¹,) at Ò= 0 (initial condition),Ó¼ Ð= Ñ1( ¹,) at Ò= 0 (initial condition),Ð= d1(, Ò) at ¹= »(boundary condition),Ð= d2( ¹, Ò) at= 0 (boundary condition),Ð= d3( ¹, Ò) at=0(boundary condition). Solution:Ð( ¹,, Ò)= ÓÓÒ µ Ú0 0 µ Û0 Ñ0( ¶, ·) ¸( ¹,, ¶, ·, Ò) ¶ º ¶ º ·+µ Ú0 0 µ Û0 Ñ1( ¶, ·) ¸( ¹,, ¶, ·, Ò) ¶ º ¶ º · - ¿2 »µ ¼0 µ Ú0 0 d1( ·, ¾) Ü ÓÓ¶ ¸( ¹,, ¶, ·, Ò- ¾) Ý Þ =Û º · º ¾ + ¿2µ¼0 µÛ0 d2( ¶, ¾)1¶ Ü ÓÓ· ¸( ¹,, ¶, ·, Ò- ¾) Ý ß =0 º ¶ º ¾ - ¿2µ¼0 µÛ0 d3( ¶, ¾)1¶ Ü ÓÓ· ¸( ¹,, ¶, ·, Ò- ¾) Ý ß =Ú0 º ¶ º ¾. Here,¸( ¹,, ¶, ·, Ò)=4¿ »20 À Á Â =1 À ÁÃ=1 Ç Â½ àÚ0( Å ÂÃ¹) Ç Â½ àÚ0( Å ÂÃ¶) Å ÂÃ[ ÇÈ Â½ àÚ0( Å ÂÃ »)]2sin á Ù â0 ãsin á Ù â ·0 ãsin( Å ÂÃ¿ Ò), where the Ç Â½ àÚ0( ¹) are the Bessel functions and the Å ÂÃare positive roots of the transcendental equation Ç Â½ àÚ0( Å»)= 0. Page 349 5.1.2-8. Domain: 0 £ ¹£ »,0 ££0. Second boundary value problem. A circular sector is considered. The following conditions are prescribed:Ð= Ñ0( ¹,) at Ò= 0 (initial condition),Ó¼ Ð= Ñ1( ¹,) at Ò= 0 (initial condition),Ó ÔÐ= d1(, Ò) at ¹= »(boundary condition),¹-1 ÓÚ Ð= d2( ¹, Ò) at= 0 (boundary condition),¹-1 ÓÚ Ð= d3( ¹, Ò) at=0(boundary condition). Solution:Ð( ¹,, Ò)= ÓÓÒ µ Ú0 0 µ Û0 Ñ0( ¶, ·) ¸( ¹,, ¶, ·, Ò) ¶ º ¶ º ·+µ Ú0 0 µ Û0 Ñ1( ¶, ·) ¸( ¹,, ¶, ·, Ò) ¶ º ¶ º · + ¿2 »µ¼0 µÚ0 0 d1( ·, ¾) ¸( ¹,, », ·, Ò- ¾) º · º ¾ - ¿2µ¼0 µÛ0 d2( ¶, ¾) ¸( ¹,, ¶,0, Ò- ¾) º ¶ º ¾ + ¿2µ¼0 µÛ0 d3( ¶, ¾) ¸( ¹,, ¶,0, Ò- ¾) º ¶ º ¾. Here,¸( ¹,, ¶, ·, Ò)=2 Ò»20+40¿ À Á Â =0 À ÁÃ=1 Å ÂÃÇ Â½ àÚ0( Å ÂÃ¹) Ç Â½ àÚ0( Å ÂÃ¶) ( »220 Å 2 ÂÃ- Ù2â2) ÖäÇ Â½ àÚ0( Å ÂÃ ») ×2 ´cos á Ù â0 ãcos á Ù â ·0 ãsin( Å ÂÃ¿ Ò), where the Ç Â½ àÚ0( ¹) are the Bessel functions and the Å ÂÃare positive roots of the transcendental equation ÇÈ Â½ àÚ0( Å»)= 0. 5.1.2-9. Domain: 0 £ ¹£ »,0 ££0. Mixed boundary value problem. A circular sector is considered. The following conditions are prescribed:Ð= Ñ0( ¹,) at Ò= 0 (initial condition),Ó¼ Ð= Ñ1( ¹,) at Ò= 0 (initial condition),Ó ÔÐ+ Õ Ð= d(, Ò) at ¹= »(boundary condition),ÓÚ Ð= 0 at= 0 (boundary condition),ÓÚ Ð= 0 at=0(boundary condition). Solution:Ð( ¹,, Ò)= ÓÓÒ µ Ú0 0 µ Û0 Ñ0( ¶, ·) ¸( ¹,, ¶, ·, Ò) ¶ º ¶ º ·+µ Ú0 0 µ Û0 Ñ1( ¶, ·) ¸( ¹,, ¶, ·, Ò) ¶ º ¶ º · + ¿2 »µ ¼0 µ Ú0 0 d( ·, ¾) ¸( ¹,, », ·, Ò- ¾) º · º ¾. Here,¸( ¹,, ¶, ·, Ò)=À Á Â =0 À ÁÃ=1Ä ÂÃÇ eäå( Å ÂÃ¹) Ç eäå( Å ÂÃ¶) cos( f Â) cos( f Â·) sin( Å ÂÃ¿ Ò),f Â = Ù â0,Ä ÂÃ=4 Å ÂÃ¿0( Å 2 ÂÃ »2+ Õ2 »2- f2 Â )Ö Ç eäå( Å ÂÃ »)×2, Page 350 2 ËÊÌ22 Î where the Ç eäå( ¹) are the Bessel functions and the Å ÂÃare positive roots of the transcendental equation ÅÇ Èeäå( Å»)+ Õ Ç eäå( Å»)= 0. 5.1.3. Axisymmetric Problems In the axisymmetric case the wave equation in the cylindrical system of coordinates has the formÓ2 ÐÓÒ2= ¿2á Ó2 ÐÓ¹2+1¹ ÓÐÓ¹+ Ó2 ÐÓ æ2ã, ¹= ç è2+ é2. One-dimensional problems with axial symmetry that have solutions Ð= Ð( ¹, Ò) are considered in Subsection 4.2.1. In the solution of the problems considered below, the modi®ed Green's function ê( ¹, æ, ¶, ·, Ò)= 2 â ¶ ¸( ¹, æ, ¶, ·, Ò) is used for convenience. 5.1.3-1. Domain: 0 £ ¹£ »,0 £ æ£ ë. First boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:Ð= Ñ0( ¹, æ) at Ò= 0 (initial condition),Ó¼ Ð= Ñ1( ¹, æ) at Ò= 0 (initial condition),Ð= d1( æ, Ò) at ¹= »(boundary condition),Ð= d2( ¹, Ò) at æ= 0 (boundary condition),Ð= d3( ¹, Ò) at æ= ë(boundary condition). Solution:Ð( ¹, æ, Ò)= ÓÓÒ µ ì0 µ Û0 Ñ0( ¶, ·) ê( ¹, æ, ¶, ·, Ò) º ¶ º ·+µ ì0 µ Û0 Ñ1( ¶, ·) ê( ¹, æ, ¶, ·, Ò) º ¶ º · - ¿2µ¼0 µì0 í1( ·, ¾) Ü ÓÓ¶ ê( ¹, æ, ¶, ·, Ò- ¾) Ý Þ =Û º · º ¾ + ¿2 î¼0 îÛ0í2( ï, ¾) Ü ÓÓ ðê( ñ, æ, ï, ð, Ò- ¾) Ý ß =0 ò ïò ¾ - ¿2 î¼0 îÛ0 í3( ï, ¾) Ü ÓÓ ðê( ñ, æ, ï, ð, Ò- ¾) Ý ß =ì ò ïò ¾. Here,ê( ñ, æ, ï, ð, Ò)=4 ïó2ë ô Á Â =1 ô ÁÃ=11Ç2 1( Å Â ) Ç0 á Å Âñóã Ç0 á Å Âïóãsin á õ â æëãsin á õ â ðëãsin ö÷ Òùø ú ÂÃ û÷ øú ÂÃ,ú ÂÃ= Å 2 Âó2+ â2õ2ë2, where the Å Â are positive zeros of the Bessel function, Ç0( Å )= 0. 5.1.3-2. Domain: 0 £ ñ£ ó,0 £ æ£ ë. Second boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:Ð= Ñ0( ñ, æ) at Ò= 0 (initial condition),Ó üÐ= Ñ1( ñ, æ) at Ò= 0 (initial condition),Ó ÔÐ=í1( æ, Ò) at ñ= ó(boundary condition),Ó ýÐ=í2( ñ, Ò) at æ= 0 (boundary condition),Ó ýÐ=í3( ñ, Ò) at æ= ë(boundary condition). Page 351 Solution:Ð( ñ, æ, Ò)= ÓÓÒ îì0 îÛ0 Ñ0( ï, ð) ê( ñ, æ, ï, ð, Ò)ò ïò ð+ îì0 îÛ0 Ñ1( ï, ð) ê( ñ, æ, ï, ð, Ò)ò ïò ð + ÷2 î ü 0 îì0í1( ð, þ) ê( ñ, æ, ó, ð, Ò- þ)ò ðò þ - ÷2 î ü 0 îÛ0í2( ï, þ) ê( ñ, æ, ï,0, Ò- þ)ò ïò þ + ÷2 î ü 0 îÛ0 í3( ï, þ) ê( ñ, æ, ï, ë, Ò- þ)ò ïò þ. Here,ê( ñ, æ, ï, ð, Ò)=2 Òÿïó2ë+2 ïó2ë ô =0 ô =0 2 0( ) 0 ñó 0 ïó ´cos õ cos õ ð sin ö÷ øú û÷ ø ú ,ú = 2ó2+ 2õ2 2, = 0forõ= 0, = 0, 1forõ= 0, >0, 2forõ>0, where the are zeros of the ®rst-order Bessel function, 1( )= 0( 0= 0). 5.1.3-3. Domain: 0 £ ñ£ ó,0 £ £ . Third boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , ) at = 0 (initial condition), ü= 1( , ) at = 0 (initial condition), + 1 = 1( , ) at = (boundary condition), ý- 2 = 2( , ) at = 0 (boundary condition), ý+ 3 = 3( , ) at = (boundary condition). The solution ( , , ) is determined by the formula in Paragraph 5.1.3-2 where( , , , , )=2 2 =1 =1 2 ( 2 1 2+ 2) 2 0( ) 0 0 ( ) ( ) 2sin ! " ! ,! = 22+ #2, ( )=cos( # )+ 2# sin( # ), 2= 3 2 #2 #2+ 2 2#2+ 2 3+ 2 2 #2+ 2 1 + 2 2#2 . Here, the and # are positive roots of the transcendental equations 1( )- 1 0( )= 0,tan( # )#= 2+ 3#2- 2 3. Page 352 2 %$'&22 ) 5.1.3-4. Domain: 0 £ £ ,0 £ £ . Mixed boundary value problems. 1 +. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , ) at = 0 (initial condition), ,= 1( , ) at = 0 (initial condition),= 1( , ) at = (boundary condition), -= 2( , ) at = 0 (boundary condition), -= 3( , ) at = (boundary condition). Solution:( , , )= .ì0 . /0 0( , ) ( , , , , ) 0 0 +.ì0 . /0 1( , ) ( , , , , ) 0 0 - 2. , 0 . ì0 1( , 1) 2 ( , , , , - 1) 3 4 =/ 0 0 1 - 2. , 0 . /0 2( , 1) ( , , ,0, - 1) 0 0 1 + 2. , 0 . /0 3( , 1) ( , , , , - 1) 0 0 1. Here,( , , , , )=2 2 5=1 5=0 62 1( 7 ) 6 0 7 6 0 7 cos 8 cos 8 sin ! " ! ,! = 722+ 282 2, = 91for8= 0, 2for8>0, where the 7 are positive zeros of the Bessel function, 6 0( 7)= 0. 2 +. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , ) at = 0 (initial condition), ,= 1( , ) at = 0 (initial condition), = 1( , ) at = (boundary condition),= 2( , ) at = 0 (boundary condition),= 3( , ) at = (boundary condition). Solution:( , , )= . ì0 . /0 0( , ) ( , , , , ) 0 0 +. ì0 . /0 1( , ) ( , , , , ) 0 0 + 2. , 0 .ì0 1( , 1) ( , , , , - 1) 0 0 1 + 2. , 0 . /0 2( , 1) 2 ( , , , , - 1) 3 : =0 0 0 1 - 2. , 0 . /0 3( , 1) 2 ( , , , , - 1) 3 : =ì 0 0 1. Page 353 Here,( , , , , )=4 2 5=0 5=1162 0( 7 ) 6 0 7 6 0 7 sin 8 sin 8 sin ! " ! ,! = 722+ 282 2, where the 7 are zeros of the ®rst-order Bessel function, 6 1( 7)= 0( 70= 0). 5.1.3-5. Domain: 1£ £ 2,0 £ £ . First boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , ) at = 0 (initial condition), ,= 1( , ) at = 0 (initial condition),= 1( , ) at = 1(boundary condition),= 2( , ) at = 2(boundary condition),= 3( , ) at = 0 (boundary condition),= 4( , ) at = (boundary condition). Solution:( , , )= .ì0 . /2/1 0( , ) ( , , , , ) 0 0 +.ì0 . /2/1 1( , ) ( , , , , ) 0 0 + 2. , 0 . ì0 1( , 1) 2 ( , , , , - 1) 3 4 =/1 0 0 1 - 2. , 0 .ì0 2( , 1) 2 ( , , , , - 1) 3 4 =/2 0 0 1 + 2. , 0 . /2/1 3( , 1) 2 ( , , , , - 1) 3 : =0 0 0 1 - 2. , 0 . /2/1 4( , 1) 2 ( , , , , - 1) 3 : =ì 0 0 1. Here,( , , , , )= 22 1 5=1 5=1 72 62 0( ;<7 )62 0( 7 )- 62 0( ;<7 ) = ( )= ( ) sin 8 sin 8 sin ! " ! ,= ( )= >0( 7 ) 6 0 7 1 - 6 0( 7 ) >0 7 1 , ;= 21, ! = 722 1+ 282 2, where 6 0( 7) and >0( 7) are the Bessel functions, and the 7 are positive roots of the transcendental equation6 0( 7) >0( ;<7)- 6 0( ;<7) >0( 7)= 0. 5.1.3-6. Domain: 1£ £ 2,0 £ £ . Second boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , ) at = 0 (initial condition), ,= 1( , ) at = 0 (initial condition), = 1( , ) at = 1(boundary condition), = 2( , ) at = 2(boundary condition), -= 3( , ) at = 0 (boundary condition), -= 4( , ) at = (boundary condition). Page 354 2 %$'&22 ) Solution:( , , )= . C0 . /2/1 0( , ) ( , , , , ) 0 0 +. C0 . /2/1 1( , ) ( , , , , ) 0 0 - 2. , 0 .C0 1( , 1) ( , , 1, , - 1) 0 0 1+ 2. , 0 .C0 2( , 1) ( , , 2, , - 1) 0 0 1 - 2. , 0 . /2/1 3( , 1) ( , , ,0, - 1) 0 0 1+ 2. , 0 . /2/1 4( , 1) ( , , , , - 1) 0 0 1. Here,( , , , , )=2 D ( 2 2- 2 1) +4 ( 2 2- 2 1) 5=118cos 8 cos 8 sin 8 + 2 2 2 1 5=1 5=0 72 62 1( ;<7 )62 1( 7 )- 62 1( ;<7 )= ( )= ( ) cos 8 cos 8 sin ! " ! , where= ( )= >1( 7 ) 6 0 7 1 - 6 1( 7 ) >0 7 1 , ;= 21,= 91for8= 0, 2for8>1, ! = 722 1+ 282 2;6 E( 7) and > E( 7) are the Bessel functions ( = 0,1); and the 7 are positive roots of the transcendental equation6 1( 7) >1( ;<7)- 6 1( ;<7) >1( 7)= 0. 5.2. Nonhomogeneous Wave EquationF2 GF H2= I2 J 2 G+ K( L, M, H) 5.2.1. Problems in Cartesian Coordinates 5.2.1-1. Domain: - N< O< N,- N< P< N. Cauchy problem. Initial conditions are prescribed:= Q( O, P) at R= 0,S ,= T( O, P) at R= 0. Solution:( O, P, R)=1 2 U V SSR W WX£ Y , Q( Z, [) \ Z \ []V2R2- ^2+1 2 U V W WX£ Y , T( Z, [) \ Z \ []V2R2- ^2 +1 2 U V W , 0 _ W WX£ Y( , - `) a( Z, [, b) \ Z \ []V2( R- b)2- ^2 c \ b, ^2=( Z- O)2+( [- P)2.dfe Reference : N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov (1970). Page 355 5.2.1-2. Domain: 0 £ O£ g1,0 £ P£ g2. First boundary value problem. A rectangle is considered. The following conditions are prescribed:h= Q0( O, P) at R= 0 (initial condition),S ,h= Q1( O, P) at R= 0 (initial condition),h= T1( P, R) at O= 0 (boundary condition),h= T2( P, R) at O= g1(boundary condition),h= T3( O, R) at P= 0 (boundary condition),h= T4( O, R) at P= g2(boundary condition). The solution h( O, P, R) is given by the formula in Paragraph 5.1.1-3 with the additional termW , 0 W C1 0 W C2 0a( Z, [, b) i( O, P, Z, [, R- b) \ [ \ Z \ b, which allows for the equation's nonhomogeneity; this term is the solution of the nonhomogeneous equation with homogeneous initial and boundary conditions.dfe Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 5.2.1-3. Domain: 0 £ O£ g1,0 £ P£ g2. Second boundary value problem. A rectangle is considered. The following conditions are prescribed:h= Q1( O, P) at R= 0 (initial condition),S ,h= Q2( O, P) at R= 0 (initial condition),S jh= T1( P, R) at O= 0 (boundary condition),S jh= T2( P, R) at O= g1(boundary condition),S kh= T3( O, R) at P= 0 (boundary condition),S kh= T4( O, R) at P= g2(boundary condition). The solution h( O, P, R) is given by the formula in Paragraph 5.1.1-4 with the additional term speci®ed in Paragraph 5.2.1-2 (the Green's function is taken from Paragraph 5.1.1-4).dfe References : A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 5.2.1-4. Domain: 0 £ O£ g1,0 £ P£ g2. Third boundary value problem. A rectangle is considered. The following conditions are prescribed:h= Q1( O, P) at R= 0 (initial condition),S ,h= Q2( O, P) at R= 0 (initial condition),S jh- l1 h= T1( P, R) at O= 0 (boundary condition),S jh+ l2 h= T2( P, R) at O= g1(boundary condition),S kh- l3 h= T3( O, R) at P= 0 (boundary condition),S kh+ l4 h= T4( O, R) at P= g2(boundary condition). The solution h( O, P, R) is the sum of the solution to the homogeneous equation with non- homogeneous initial and boundary conditions (see Paragraph 5.1.1-5) and the solution to thenonhomogeneous equation with homogeneous initial and boundary conditions. This solution isgiven by the formula in Paragraph 5.2.1-2 in which one should substitute the Green's function of Paragraph 5.1.1-5).dfe References : A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 356 2 nm'o22 q 5.2.1-5. Domain: 0 £ O£ g1,0 £ P£ g2. Mixed boundary value problems. 1 s. A rectangle is considered. The following conditions are prescribed:h= Q1( O, P) at R= 0 (initial condition),S th= Q2( O, P) at R= 0 (initial condition),h= T1( P, R) at O= 0 (boundary condition),h= T2( P, R) at O= g1(boundary condition),S kh= T3( O, R) at P= 0 (boundary condition),S kh= T4( O, R) at P= g2(boundary condition). The solution h( O, P, R) is given by the formula in Paragraph 5.1.1-6, Item 1 s, with the additional term speci®ed in Paragraph 5.2.1-2. 2 s. A rectangle is considered. The following conditions are prescribed:h= Q1( O, P) at R= 0 (initial condition),S th= Q2( O, P) at R= 0 (initial condition),h= T1( P, R) at O= 0 (boundary condition),S jh= T2( P, R) at O= g1(boundary condition),h= T3( O, R) at P= 0 (boundary condition),S kh= T4( O, R) at P= g2(boundary condition). The solution h( O, P, R) is given by the formula in Paragraph 5.1.1-6, Item 2 s, with the additional term speci®ed in Paragraph 5.2.1-2. 5.2.2. Problems in Polar Coordinates A nonhomogeneous wave equation in the polar coordinate system has the formS2 hSR2= V2 u v2 hv w2+1w v hv w+1w2 v2 hv x2 y+a(w,x, z),w= ] { 2+ |2. One-dimensional boundary value problems independent of the angular coordinatexare consid- ered in Subsection 4.2.2. 5.2.2-1. Domain: 0 £w£ },0 £x£ 2 ~. First boundary value problem. A circle is considered. The following conditions are prescribed:h= 0(w,x) at z= 0 (initial condition),v th= 1(w,x) at z= 0 (initial condition),h= (x, z) atw= }(boundary condition). The solution h(w,x, z) is given by the formula in Paragraph 5.1.2-1 with the additional termW t 0 W2 0 W 0a( Z, [, b) i(w,x, Z, [, z- b) Z \ Z \ [ \ b, ( 1) which allows for the equation's nonhomogeneity; this term is the solution of the nonhomogeneous equation with homogeneous initial and boundary conditions.dfe References : N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov (1970), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 357 5.2.2-2. Domain: 0 £w£ },0 £x£ 2 ~. Second boundary value problem. A circle is considered. The following conditions are prescribed:h= 0(w,x) at z= 0 (initial condition),v th= 1(w,x) at z= 0 (initial condition),v h= (x, z) atw= }(boundary condition). The solution h(w,x, z) is given by the formula in Paragraph 5.1.2-2 with the additional term (1).dfe References : A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 5.2.2-3. Domain: 0 £w£ },0 £x£ 2 ~. Third boundary value problem. A circle is considered. The following conditions are prescribed:h= 0(w,x) at z= 0 (initial condition),v th= 1(w,x) at z= 0 (initial condition),v h+ l h= (x, z) atw= }(boundary condition). The solution h(w,x, z) is the sum of the solution to the homogeneous equation with nonho- mogeneous initial and boundary conditions (see Paragraph 5.1.2-3) and the solution to the nonho-mogeneous equation with homogeneous initial and boundary conditions [this solution is given by formula (1) in which one should substitute the Green's function in Paragraph 5.1.2-3].dfe References : A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 5.2.2-4. Domain: }1£w£ }2,0 £x£ 2 ~. First boundary value problem. An annular domain is considered. The following conditions are prescribed:h= 0(w,x) at z= 0 (initial condition),v th= 1(w,x) at z= 0 (initial condition),h= 1(x, z) atw= }1(boundary condition),h= 2(x, z) atw= }2(boundary condition). The solution h(w,x, z) is given by the formula in Paragraph 5.1.2-4 with the additional termW t 0 W2 0 W 21a( Z, [, b) i(w,x, Z, [, z- b) Z \ Z \ [ \ b, ( 2) which allows for the equation's nonhomogeneity; this term is the solution of the nonhomogeneous equation with homogeneous initial and boundary conditions. 5.2.2-5. Domain: }1£w£ }2,0 £x£ 2 ~. Second boundary value problem. An annular domain is considered. The following conditions are prescribed:h= 0(w,x) at z= 0 (initial condition),v th= 1(w,x) at z= 0 (initial condition),v h= 1(x, z) atw= }1(boundary condition),v h= 2(x, z) atw= }2(boundary condition). The solution h(w,x, z) is given by the formula in Paragraph 5.1.2-5 with the additional term (2). Page 358 2 nm'o22 q 5.2.2-6. Domain: }1£w£ }2,0 £x£ 2 ~. Third boundary value problem. An annular domain is considered. The following conditions are prescribed:h= 0(w,x) at z= 0 (initial condition),v th= 1(w,x) at z= 0 (initial condition),v h- l1 h= 1(x, z) atw= }1(boundary condition),v h+ l2 h= 2(x, z) atw= }2(boundary condition). The solution h(w,x, z) is the sum of the solution to the homogeneous equation with nonho- mogeneous initial and boundary conditions (see Paragraph 5.1.2-6) and the solution to the nonho-mogeneous equation with homogeneous initial and boundary conditions [this solution is given by formula (2) in which one should substitute the Green's function in Paragraph 5.1.2-6]. 5.2.2-7. Domain: 0 £w£ },0 £x£x0. First boundary value problem. A circular sector is considered. The following conditions are prescribed:h= 0(w,x) at z= 0 (initial condition),v th= 1(w,x) at z= 0 (initial condition),h= 1(x, z) atw= } (boundary condition),h= 2(w, z) atx= 0 (boundary condition),h= 3(w, z) atx=x0(boundary condition). The solution h(w,x, z) is given by the formula in Paragraph 5.1.2-7 with the additional termW t 0 W 0 0 W 0a( Z, [, b) i(w,x, Z, [, z- b) Z \ Z \ [ \ b, ( 3) which allows for the equation's nonhomogeneity. 5.2.2-8. Domain: 0 £w£ },0 £x£x0. Second boundary value problem. A circular sector is considered. The following conditions are prescribed:h= 0(w,x) at z= 0 (initial condition),v th= 1(w,x) at z= 0 (initial condition),v h= 1(x, z) atw= } (boundary condition),w-1v h= 2(w, z) atx= 0 (boundary condition),w-1v h= 3(w, z) atx=x0(boundary condition). The solution h(w,x, z) is given by the formula in Paragraph 5.1.2-8 with the additional term (3). 5.2.2-9. Domain: 0 £w£ },0 £x£x0. Mixed boundary value problem. A circular sector is considered. The following conditions are prescribed:h= 0(w,x) at z= 0 (initial condition),v th= 1(w,x) at z= 0 (initial condition),v h+ l h= (x, z) atw= } (boundary condition),v h= 0 atx= 0 (boundary condition),v h= 0 atx=x0(boundary condition). The solution h(w,x, z) is given by the formula in Paragraph 5.1.2-9 with the additional term (3). Page 359 5.2.3. Axisymmetric Problems In the axisymmetric case, a nonhomogeneous wave equation in the cylindrical system of coordinateshas the formv2 hv z2= 2 uv2 hv w2+1w v hv w+ v2 hv 2 y+a(w,, z),w= ] { 2+ |2. 5.2.3-1. Domain: 0 £w£ },0 ££ g. First boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:h= 0(w,) at z= 0 (initial condition),v th= 1(w,) at z= 0 (initial condition),h= 1(, z) atw= }(boundary condition),h= 2(w, z) at= 0 (boundary condition),h= 3(w, z) at= g(boundary condition). The solution h(w,, z) is given by the formula in Paragraph 5.1.3-1 with the additional termW t 0 WC0 W 0a( Z, [, b) (w,, Z, [, z- b) \ Z \ [ \ b, ( 1) which allows for the equation's nonhomogeneity; this term is the solution of the nonhomogeneous equation with homogeneous initial and boundary conditions. 5.2.3-2. Domain: 0 £w£ },0 ££ g. Second boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:h= 0(w,) at z= 0 (initial condition),v th= 1(w,) at z= 0 (initial condition),v h= 1(, z) atw= }(boundary condition),v h= 2(w, z) at= 0 (boundary condition),v h= 3(w, z) at= g(boundary condition). The solution h(w,, z) is given by the formula in Paragraph 5.1.3-2 with the additional term (1). 5.2.3-3. Domain: 0 £w£ },0 ££ g. Third boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:h= 0(w,) at z= 0 (initial condition),v th= 1(w,) at z= 0 (initial condition),v h+ l1 h= 1(, z) atw= }(boundary condition),v h- l2 h= 2(w, z) at= 0 (boundary condition),v h+ l3 h= 3(w, z) at= g(boundary condition). The solution h(w,, z) is the sum of the solution to the homogeneous equation with nonho- mogeneous initial and boundary conditions (see Paragraph 5.1.3-3) and the solution to the nonho-mogeneous equation with homogeneous initial and boundary conditions [this solution is given by formula (1) in which one should substitute the Green's function in Paragraph 5.1.3-3]. Page 360 2 22 5.2.3-4. Domain: 0 £ £ ,0 £ £ . Mixed boundary value problems. 1 . A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , ) at = 0 (initial condition), = 1( , ) at = 0 (initial condition),= 1( , ) at = (boundary condition), = 2( , ) at = 0 (boundary condition), = 3( , ) at = (boundary condition). The solution ( , , ) is given by the formula in Paragraph 5.1.3-4, Item 1, with the additional term (1).2. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , ) at = 0 (initial condition), = 1( , ) at = 0 (initial condition), = 1( , ) at = (boundary condition),= 2( , ) at = 0 (boundary condition),= 3( , ) at = (boundary condition). The solution ( , , ) is given by the formula in Paragraph 5.1.3-4, Item 2 , with the additional term (1). 5.2.3-5. Domain: 1£ £ 2,0 £ £ . First boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , ) at = 0 (initial condition), = 1( , ) at = 0 (initial condition),= 1( , ) at = 1(boundary condition),= 2( , ) at = 2(boundary condition),= 3( , ) at = 0 (boundary condition),= 4( , ) at = (boundary condition). The solution ( , , ) is given by the formula in Paragraph 5.1.3-5 with the additional term 0 0 21 ( , , ) ( , , , , - ) , ( 2) which allows for the equation's nonhomogeneity; this term is the solution of the nonhomogeneous equation with homogeneous initial and boundary conditions. 5.2.3-6. Domain: 1£ £ 2,0 £ £ . Second boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , ) at = 0 (initial condition), = 1( , ) at = 0 (initial condition), = 1( , ) at = 1(boundary condition), = 2( , ) at = 2(boundary condition), = 3( , ) at = 0 (boundary condition), = 4( , ) at = (boundary condition). The solution ( , , ) is given by the formula in Paragraph 5.1.3-6 with the additional term (2). Page 361 5.3. Equations of the Form !2 "! #2= $2 % 2 "± & "+ '( (, ),#) 5.3.1. Problems in Cartesian Coordinates The two-dimensional nonhomogeneous Klein±Gordon equation with two space variables in the rectangular Cartesian coordinate system is written as2 2= *2 + 2 ,2+ 2 -2 .- / +( ,, -, ). 5.3.1-1. Fundamental solutions. 1. Case /= - 02<0:1 1( ,, -, )= 2( * - ) 2 3 *2cosh 450 6 2- 2 7*2 86 2- 2 7*2, = 6 ,2+ -2, where2( ) is the Heaviside unit step function. 2 . Case /= 02>0:1 1( ,, -, )=2( * - ) 2 3 *2cos 4906 2- 2 7*2 86 2- 2 7*2, = 6 ,2+ -2.:<; References : V . S. Vladimirov, V . P. Mikhailov, A. A. Vasharin, et al. (1974), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). 5.3.1-2. Domain: - =< ,< =,- =< -< =. Cauchy problem. Initial conditions are prescribed:= ( ,, -) at = 0, = ( ,, -) at = 0. 1 . Solution for /= - *2 >2<0:( ,, -, )=1 2 3 * ?£ @ ( , )cosh 4 >6*22- A2 86*22- A2 +1 2 3 * ?£ @ ( , )cosh 4 >6*22- A2 86*22- A2 +1 2 3 * 0 ?£ @( - B) ( , , )cosh 4 >6*2( - )2- A2 86*2( - )2- A2 , A= 6( ,- )2+( -- )2. 2 . Solution for /= *2 >2>0:( ,, -, )=1 2 3 * ?£ @ ( , )cos 4 >6*22- A2 86*22- A2 +1 2 3 * ?£ @ ( , )cos 4 >6*22- A2 86*22- A2 +1 2 3 * 0 ?£ @( - B) ( , , )cos 4 >6*2( - )2- A2 86*2( - )2- A2 , A= 6( ,- )2+( -- )2.:<; Reference : B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 362 2 2 C 5.3.1-3. Domain: 0 £ ,£ 1,0 £ -£ 2. First boundary value problem. A rectangle is considered. The following conditions are prescribed:= 0( ,, -) at = 0 (initial condition), = 1( ,, -) at = 0 (initial condition),= 1( -, ) at ,= 0 (boundary condition),= 2( -, ) at ,= 1(boundary condition),= 3( ,, ) at -= 0 (boundary condition),= 4( ,, ) at -= 2(boundary condition). Solution:( ,, -, )= 1 0 2 0 0( , ) D( ,, -, , , ) + 1 0 2 0 1( , ) D( ,, -, , , ) + *2 0 2 0 1( , ) E D( ,, -, , , - ) F G =0 - *2 0 2 0 2( , ) E D( ,, -, , , - ) F G = 1 + *2 0 1 0 3( , ) E D( ,, -, , , - ) F H =0 - *2 0 1 0 4( , ) E D( ,, -, , , - ) F H = 2 + 0 1 0 2 0( , , ) D( ,, -, , , - ) , whereD( ,, -, , , )=41 2 I JLK =1 I JM=11N KMsin( O K,) sin( P M -) sin( O K) sin( P M) sin( N KM),O K = Q 31, P M= R 32, N KM= S *2O2 K + *2P2M+ /. 5.3.1-4. Domain: 0 £ ,£ T1,0 £ -£ T2. Second boundary value problem. A rectangle is considered. The following conditions are prescribed:U= V0( ,, -) at W= 0 (initial condition),X YU= V1( ,, -) at W= 0 (initial condition),X ZU= [1( -, W) at ,= 0 (boundary condition),X ZU= [2( -, W) at ,= T1(boundary condition),X \U= [3( ,, W) at -= 0 (boundary condition),X \U= [4( ,, W) at -= T2(boundary condition). Page 363 Solution:U( ,, -, W)= XXW ] ^1 0 ] ^2 0 V0( _, `) a( ,, -, _, `, W) b ` b _+] ^1 0 ] ^2 0 V1( _, `) a( ,, -, _, `, W) b ` b _ - c2] Y 0 ] ^2 0 [1( `, d) a( ,, -,0, `, W- d) b ` b d + c2] Y 0 ] ^2 0 [2( `, d) a( ,, -, T1, `, W- d) b ` b d - c2] Y 0 ] ^1 0 [3( _, d) a( ,, -, _,0, W- d) b _ b d + c2] Y 0 ] ^1 0 [4( _, d) a( ,, -, _, T2, W- d) b _ b d +] Y 0 ] ^1 0 ] ^2 0 e( _, `, d) a( ,, -, _, `, W- d) b ` b _ b d, wherea( ,, -, _, `, W)=sin fgWih j kl 1 l 2 h j+2l 1 l 2 m nLo =0 m nM=0 p oMN oMcos( O o,) cos( P M -) cos( O o_) cos( P M`) sin( N oM q),O o = Q r l 1, P M= s r l 2, N oM= t c2O2 o + c2P2M+ j,p oM= u0forQ=s= 0, 1forQs= 0(Q¹s), 2forQs¹ 0. 5.3.1-5. Domain: 0 £ v£ l 1,0 £ -£ l 2. Third boundary value problem. A rectangle is considered. The following conditions are prescribed:w= x0( v, -) at q= 0 (initial condition),X Yw= x1( v, -) at q= 0 (initial condition),X Zw- y1 w= [1( -, q) at v= 0 (boundary condition),X Zw+ y2 w= [2( -, q) at v= l 1(boundary condition),X \w- y3 w= [3( v, q) at -= 0 (boundary condition),X \w+ y4 w= [4( v, q) at -= l 2(boundary condition). The solution w( v, -, q) is determined by the formula in Paragraph 5.3.1-3 wherea( v, -, _, `, q)= 4m n o =1 m nM=11z oMt c2 {2 o + c2 |2M+ jsin( { ov+ } o ) sin( |M -+ ~ M) ´sin( { o_+ } o ) sin( |M`+ ~ M) sinf qt c2 {2 o + c2 |2M+ jk,} o =arctan { ol 1, ~ M=arctan |Ml 2, z oM= l 1+( y1 y2+ {2 o )( y1+ y2) ( y2 1+ {2 o )( y2 2+ {2 o ) l 2+( y3 y4+ |2M)( y3+ y4) ( y2 3+ |2M)( y2 4+ |2M) . Here, the { o and |Mare positive roots of the transcendental equations{2- y1 y2=( y1+ y2) {cot( l 1 {),|2- y3 y4=( y3+ y4) |cot( l 2 |).< References : A. G. Butkovskiy (1979), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). Page 364 2 2 5.3.1-6. Domain: 0 £ v£ l 1,0 £ -£ l 2. Mixed boundary value problems. 1 . A rectangle is considered. The following conditions are prescribed:w= x0( v, -) at q= 0 (initial condition),X Yw= x1( v, -) at q= 0 (initial condition),w= [1( -, q) at v= 0 (boundary condition),w= [2( -, q) at v= l 1(boundary condition),X \w= [3( v, q) at -= 0 (boundary condition),X \w= [4( v, q) at -= l 2(boundary condition). Solution:w( v, -, q)= XXq] ^1 0 ] ^2 0 x0( _, `) a( v, -, _, `, q) b ` b _ +] ^1 0 ] ^2 0 x1( _, `) a( v, -, _, `, q) b ` b _ + c2] Y 0 ] ^2 0 [1( `, d) XX_ a( v, -, _, `, q- d) =0 b ` b d - c2] Y 0 ] ^2 0 [2( `, d) XX_ a( v, -, _, `, q- d) =^1 b ` b d - c2] Y 0 ] ^1 0 [3( _, d) a( v, -, _,0, q- d) b _ b d + c2] Y 0 ] ^1 0 [4( _, d) a( v, -, _, l 2, q- d) b _ b d +] Y 0 ] ^1 0 ] ^2 0e( _, `, d) a( v, -, _, `, q- d) b ` b _ b d, wherea( v, -, _, `, q)=2l 1 l 2 m n o =1 m nM=0 p MN oMsin( O ov) cos( P M -) sin( O o_) cos( P M`) sin( N oM q),O o = Q r l 1, P M= s r l 2, N oM= t c2O2 o + c2P2M+ j,p M= 1fors= 0, 2fors¹ 0. 2 . A rectangle is considered. The following conditions are prescribed:w= x0( v, -) at q= 0 (initial condition),X Yw= x1( v, -) at q= 0 (initial condition),w= [1( -, q) at v= 0 (boundary condition),X Zw= [2( -, q) at v= l 1(boundary condition),w= [3( v, q) at -= 0 (boundary condition),X \w= [4( v, q) at -= l 2(boundary condition). Page 365 Solution:w( v, -, q)= XXq] ^1 0 ] ^2 0 x0( _, `) a( v, -, _, `, q) b ` b _+] ^1 0 ] ^2 0 x1( _, `) a( v, -, _, `, q) b ` b _ + c2] Y 0 ] ^2 0 [1( `, d) XX_ a( v, -, _, `, q- d)=0 b ` b d + c2] Y 0 ] ^2 0 [2( `, d) a( v, -, l 1, `, q- d) b ` b d + c2] Y 0 ] ^1 0 [3( _, d) XX` a( v, -, _, `, q- d) =0 b _ b d + c2] Y 0 ] ^1 0 [4( _, d) a( v, -, _, l 2, q- d) b _ b d +] Y 0 ] ^1 0 ] ^2 0 e( _, `, d) a( v, -, _, `, q- d) b ` b _ b d, wherea( v, -, _, `, q)=4l 1 l 2 m nLo =0 m n=01 osin( ov) sin( -) sin( o_) sin( `) sin( o q), o = r(2Q+ 1) 2 l 1, = r(2s+ 1) 2 l 2, o= c22 o + c22+ j. 5.3.2. Problems in Polar Coordinates A nonhomogeneous Klein±Gordon equation with two space variables in the polar coordinate system has the formX2 wXq2= c2 2 w 2+1 w +12 2 w 2 - w+ (,, ),= v2+ -2. One-dimensional solutions w= w(, ) independent of the angular coordinateare considered in Subsection 4.2.5. 5.3.2-1. Domain: 0 ££ ,0 ££ 2 ¡. First boundary value problem. A circle is considered. The following conditions are prescribed:¢= £0(,) at = 0 (initial condition), Y¢= £1(,) at = 0 (initial condition),¢= [(, ) at= (boundary condition). Solution:¢(,, )= ¤2 ¥ 0 ¤ ¦0 £0( §, ¨) ©(,, §, ¨, ) § ª § ª ¨+¤2 ¥ 0 ¤ ¦0 £1( §, ¨) ©(,, §, ¨, ) § ª § ª ¨ - «2 ¤ Y 0 ¤2 ¥ 0 [( ¨, ¬) § ©(,, §, ¨, - ¬) ® ¯ =¦ ª ¨ ª ¬ +¤ Y 0 ¤2 ¥ 0 ¤ ¦0 ( §, ¨, ¬) ©(,, §, ¨, - ¬) § ª § ª ¨ ª ¬. Page 366 °²2 µ Here,*©(,, §, ¨, )=1¡ 2 » ¼L½ =0 » ¼¾=1 ¿ ½ [ À Á ½ ( Â ½¾ )]2 À ½ ( Â ½¾ Ã) À ½ ( Â ½¾§) cos[ Ä( Å- ¨)]sin ÆgÇÉÈ «2Â2 ½¾+ Ê ËÈ«2Â2 ½¾+ Ê,¿0= 1,¿ ½ = 2 ( Ä= 1,2, ÌÌÌ), where the À ½ ( §) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the Â ½¾are positive roots of the transcendental equation À ½ ( Â )= 0. 5.3.2-2. Domain: 0 £ Ã£ ,0 £ Å£ 2 ¡. Second boundary value problem. A circle is considered. The following conditions are prescribed:¢= £0( Ã, Å) at Ç= 0 (initial condition),Í Y¢= £1( Ã, Å) at Ç= 0 (initial condition),Í Î¢= [( Å, Ç) at Ã= (boundary condition). Solution:¢( Ã, Å, Ç)= ÍÍÇ ¤2 ¥ 0 ¤¦0 £0( §, ¨) ©( Ã, Å, §, ¨, Ç) § ª § ª ¨+¤2 ¥ 0 ¤¦0 £1( §, ¨) ©( Ã, Å, §, ¨, Ç) § ª § ª ¨ + «2 ¤ Y 0 ¤2 ¥ 0 [( ¨, ¬) ©( Ã, Å, , ¨, Ç- ¬) ª ¨ ª ¬ +¤ Y 0 ¤2 ¥ 0 ¤¦0 Ï( §, ¨, ¬) ©( Ã, Å, §, ¨, Ç- ¬) § ª § ª ¨ ª ¬. Here,©( Ã, Å, §, ¨, Ç)=sin ÆgÇiÐ Ê Ë¡ 2Ð Ê +1¡ » ¼L½ =0 » ¼¾=1 ¿ ½Â2 ½¾À ½ ( Â ½¾ Ã) À ½ ( Â ½¾§) ( Â2 ½¾ 2- Ä2)[ À ½ ( Â ½¾ )]2cos[ Ä( Å- ¨)]sin ÆÇ È«2Â2 ½¾+ Ê ËÈ«2Â2 ½¾+ Ê, where¿0= 1and¿ ½ = 2for Ä= 1,2, ÌÌÌ; the À ½ ( §) are the Bessel functions; and the Â ¾are positive roots of the transcendental equation À Á ½ ( Â )= 0. 5.3.2-3. Domain: 0 £ Ã£ ,0 £ Å£ 2 ¡. Third boundary value problem. A circle is considered. The following conditions are prescribed:¢= £0( Ã, Å) at Ç= 0 (initial condition),Í Y¢= £1( Ã, Å) at Ç= 0 (initial condition),Í Î¢+ Ñ ¢= [( Å, Ç) at Ã= Ò(boundary condition). The solution Ó( Ã, Å, Ç) is determined by the formula in Paragraph 5.3.2-2 where©( Ã, Å, §, ¨, Ç)=1Ô» ¼L½ =0 » ¼¾=1 ¿ ½Â2 ½¾À ½ ( Â ½¾ Ã) À ½ ( Â ½¾§) ( Â2 ½¾Ò2+ Ñ2Ò2- Ä2)[ À ½ ( Â ½¾Ò)]2cos[ Ä( Å- ¨)]sin ÆgÇ È«2Â2 ½¾+ Ê ËÈ«2Â2 ½¾+ Ê,¿0= 1,¿ ½ = 2 ( Ä= 1,2, ÌÌÌ). Here, the À ½ ( §) are the Bessel functions and the Â ¾are positive roots of the transcendental equationÂ À Á ½ ( Â Ò)+ Ñ À ½ ( Â Ò)= 0. * In the expressions of the Green's functions speci®ed in Subsection 5.3.2, the ratios sin ÕÖºØ× ³2 Ù2 Ú Û+ ¶ ÜiÝ × ³2 Ù2 Ú Û+ ¶ must be replaced by sinh ÕÞºØ×| ³2 Ù2 Ú Û+ ¶| ÜiÝ ×| ³2 Ù2 Ú Û+ ¶|if ³2Ù2Ú Û+ ¶<0. Page 367 5.3.2-4. Domain: Ò1£ Ã£ Ò2,0 £ Å£ 2 Ô. First boundary value problem. An annular domain is considered. The following conditions are prescribed:Ó= ß0( Ã, Å) at Ç= 0 (initial condition),Í YÓ= ß1( Ã, Å) at Ç= 0 (initial condition),Ó= [1( Å, Ç) at Ã= Ò1(boundary condition),Ó= [2( Å, Ç) at Ã= Ò2(boundary condition). Solution:Ó( Ã, Å, Ç)= ÍÍÇ ¤2 ¥ 0 ¤¦2¦1 ß0( §, ¨) ©( Ã, Å, §, ¨, Ç) § ª § ª ¨+¤2 ¥ 0 ¤¦2¦1 ß1( §, ¨) ©( Ã, Å, §, ¨, Ç) § ª § ª ¨ + «2Ò1¤ Y 0 ¤2 ¥ 0 [1( ¨, ¬) ÍÍ§ ©( Ã, Å, §, ¨, Ç- ¬) ® ¯ =¦1 ª ¨ ª ¬ - «2Ò2¤ Y 0 ¤2 ¥ 0 [2( ¨, ¬) ÍÍ§ ©( Ã, Å, §, ¨, Ç- ¬) ® ¯ =¦2 ª ¨ ª ¬ +¤ Y 0 ¤2 ¥ 0 ¤ ¦2¦1 Ï( §, ¨, ¬) ©( Ã, Å, §, ¨, Ç- ¬) § ª § ª ¨ ª ¬. Here,©( Ã, Å, §, ¨, Ç)= Ô 2 » ¼½ =0 » ¼¾=1¿ ½ à ½¾ á ½ ( Â ½¾ Ã) á ½ ( Â ½¾§) cos[ Ä( Å- ¨)]sin ÆgÇ È«2Â2 ½¾+ Ê ËÈ«2Â2 ½¾+ Ê,¿ ½ = â1 ã2for Ä= 0, 1 for Ä¹ 0, à ½¾= Â2 ½¾À2 ½ ( Â ½¾Ò2)À2 ½ ( Â ½¾Ò1)- À2 ½ ( Â ½¾Ò2),á ½ ( Â ½¾ Ã)= À ½ ( Â ½¾Ò1) ä ½ ( Â ½¾ Ã)- ä ½ ( Â ½¾Ò1) À ½ ( Â ½¾ Ã), where the À ½ ( Ã) and ä ½ ( Ã) are the Bessel functions, and the Â ½¾are positive roots of the transcen- dental equationÀ ½ ( Â Ò1) ä ½ ( Â Ò2)- ä ½ ( Â Ò1) À ½ ( Â Ò2)= 0. 5.3.2-5. Domain: Ò1£ Ã£ Ò2,0 £ Å£ 2 Ô. Second boundary value problem. An annular domain is considered. The following conditions are prescribed:Ó= ß0( Ã, Å) at Ç= 0 (initial condition),Í YÓ= ß1( Ã, Å) at Ç= 0 (initial condition),Í ÎÓ= [1( Å, Ç) at Ã= Ò1(boundary condition),Í ÎÓ= [2( Å, Ç) at Ã= Ò2(boundary condition). Solution:Ó( Ã, Å, Ç)= ÍÍÇ ¤2 ¥ 0 ¤¦2¦1 ß0( §, ¨) ©( Ã, Å, §, ¨, Ç) § ª § ª ¨+¤2 ¥ 0 ¤¦2¦1 ß1( §, ¨) ©( Ã, Å, §, ¨, Ç) § ª § ª ¨ - «2Ò1¤ Y 0 ¤2 ¥ 0 [1( ¨, ¬) ©( Ã, Å, Ò1, ¨, Ç- ¬) ª ¨ ª ¬ + «2Ò2¤ Y 0 ¤2 ¥ 0 [2( ¨, ¬) ©( Ã, Å, Ò2, ¨, Ç- ¬) ª ¨ ª ¬ +¤ Y 0 ¤2 ¥ 0 ¤¦2¦1 Ï( §, ¨, ¬) ©( Ã, Å, §, ¨, Ç- ¬) § ª § ª ¨ ª ¬. Page 368 °² µ Here,©( Ã, Å, §, ¨, Ç)=sin ÆgÇiÐ Ê ËÔ( Ò2 2- Ò2 1)Ð Ê +1Ô» ¼½ =0 » ¼¾=1 ¿ ½Â2 ½¾ á ½ ( Â ½¾ Ã) á ½ ( Â ½¾§) cos[ Ä( Å- ¨)] sin Æ5Ç È«2Â2 ½¾+ Ê Ëå ( Â2 ½¾Ò2 2- Ä2) á2 ½ ( Â ½¾Ò2)-( Â2 ½¾Ò2 1- Ä2) á2 ½ ( Â ½¾Ò1) æ È«2Â2 ½¾+ Ê, whereá ½ ( Â ½¾ Ã)= À Á ½ ( Â ½¾Ò1) ä ½ ( Â ½¾ Ã)- ä Á ½ ( Â ½¾Ò1) À ½ ( Â ½¾ Ã),¿ ½ = â1for Ä= 0, 2for Ä>0, the À ½ ( Ã) and ä ½ ( Ã) are the Bessel functions, and the Â ½¾are positive roots of the transcendental equationÀ Á ½ ( Â Ò1) ä Á ½ ( Â Ò2)- ä Á ½ ( Â Ò1) À Á ½ ( Â Ò2)= 0. 5.3.2-6. Domain: Ò1£ Ã£ Ò2,0 £ Å£ 2 Ô. Third boundary value problem. An annular domain is considered. The following conditions are prescribed:Ó= ß0( Ã, Å) at Ç= 0 (initial condition),Í YÓ= ß1( Ã, Å) at Ç= 0 (initial condition),Í ÎÓ- Ñ1 Ó= [1( Å, Ç) at Ã= Ò1(boundary condition),Í ÎÓ+ Ñ2 Ó= [2( Å, Ç) at Ã= Ò2(boundary condition). The solution Ó( Ã, Å, Ç) is determined by the formula in Paragraph 5.3.2-5 where©( Ã, Å, §, ¨, Ç)=1Ô» ¼L½ =0 » ¼¾=1 ¿ ½Â2 ½¾ á ½¾( Ã) á ½¾( §) cos[ Ä( Å- ¨)] sin( ç ½¾Ç)ç ½¾ å ( Ñ2 2 Ò2 2+ Â2 ½¾Ò2 2- Ä2) á2 ½¾( Ò2)-( Ñ2 1 Ò2 1+ Â2 ½¾Ò2 1- Ä2) á2 ½¾( Ò1)æ,á ½¾( Ã)= åÂ ½¾À Á ½ ( Â ½¾Ò1)- Ñ1 À ½ ( Â ½¾Ò1)æ ä ½ ( Â ½¾ Ã) - åÂ ½¾ä Á ½ ( Â ½¾Ò1)- Ñ1 ä ½ ( Â ½¾Ò1)æ À ½ ( Â ½¾ Ã). Here,¿0= 1and¿ ½ = 2for Ä= 1,2, ÌÌÌ; ç ½¾= È«2Â2 ½¾+ Ê; the À ½ ( Ã) and ä ½ ( Ã) are the Bessel functions; and the Â ½¾are positive roots of the transcendental equationåÂ À Á ½ ( Â Ò1)- Ñ1 À ½ ( Â Ò1)æ åÂ ä Á ½ ( Â Ò2)+ Ñ2 ä ½ ( Â Ò2)æ = åÂ ä Á ½ ( Â Ò1)- Ñ1 ä ½ ( Â Ò1)æ åÂ À Á ½ ( Â Ò2)+ Ñ2 À ½ ( Â Ò2)æ. 5.3.2-7. Domain: 0 £ Ã£ Ò,0 £ Å£ Å0. First boundary value problem. A circular sector is considered. The following conditions are prescribed:Ó= ß0( Ã, Å) at Ç= 0 (initial condition),Í YÓ= ß1( Ã, Å) at Ç= 0 (initial condition),Ó= [1( Å, Ç) at Ã= Ò (boundary condition),Ó= [2( Ã, Ç) at Å= 0 (boundary condition),Ó= [3( Ã, Ç) at Å= Å0(boundary condition). Page 369 Solution:Ó( Ã, Å, Ç)= ÍÍÇ ¤ è0 0 ¤¦0 ß0( §, ¨) ©( Ã, Å, §, ¨, Ç) § ª § ª ¨+¤ è0 0 ¤¦0 ß1( §, ¨) ©( Ã, Å, §, ¨, Ç) § ª § ª ¨ - «2Ò¤ Y 0 ¤ è0 0 [1( ¨, ¬) ÍÍ§ ©( Ã, Å, §, ¨, Ç- ¬) ® ¯ =¦ ª ¨ ª ¬ + «2¤ Y 0 ¤ ¦0 [2( §, ¬)1§ ÍÍ¨ ©( Ã, Å, §, ¨, Ç- ¬) ® é =0 ª § ª ¬ - «2¤ Y 0 ¤¦0 [3( §, ¬)1§ ÍÍ¨ ©( Ã, Å, §, ¨, Ç- ¬) ® é =è0 ª § ª ¬ +¤ Y 0 ¤ è0 0 ¤¦0 Ï( §, ¨, ¬) ©( Ã, Å, §, ¨, Ç- ¬) § ª § ª ¨ ª ¬. Here,©( Ã, Å, §, ¨, Ç)=4Ò2Å0 » ¼L½ =1 » ¼¾=1 À ½¥ êè0( Â ½¾ Ã) À ½¥ êè0( Â ½¾§) [ À Á ½¥ êè0( Â ½¾Ò)]2sin ë Ä ÔÅÅ0 ìsin ë Ä Ô¨Å0 ìsin Æ9ç ½¾ÇíËç ½¾, where the À ½¥ êè0( Ã) are the Bessel functions and the Â ½¾are positive roots of the transcendental equation À ½¥ êè0( Â Ò)= 0, and ç ½¾= È «2Â2 ½¾+ Ê. 5.3.2-8. Domain: 0 £ Ã£ Ò,0 £ Å£ Å0. Second boundary value problem. A circular sector is considered. The following conditions are prescribed:Ó= ß0( Ã, Å) at Ç= 0 (initial condition),Í YÓ= ß1( Ã, Å) at Ç= 0 (initial condition),Í ÎÓ= [1( Å, Ç) at Ã= Ò (boundary condition),Ã-1 Íè Ó= [2( Ã, Ç) at Å= 0 (boundary condition),Ã-1 Íè Ó= [3( Ã, Ç) at Å= Å0(boundary condition). Solution:Ó( Ã, Å, Ç)= ÍÍÇ î ï0 0 î ð0 ñ0( ò, ó) ô( õ, Å, ò, ó, ö) ò ÷ ò ÷ ó+î ï0 0 î ð0 ñ1( ò, ó) ô( õ, Å, ò, ó, ö) ò ÷ ò ÷ ó + ø2 ùî Y 0 î ï0 0 [1( ó, ú) ô( õ, Å, ù, ó, ö- ú) ÷ ó ÷ ú - ø2î Y 0 îð0 [2( ò, ú) ô( õ, Å, ò,0, ö- ú) ÷ ò ÷ ú + ø2î Y 0 îð0 [3( ò, ú) ô( õ, Å, ò, Å0, ö- ú) ÷ ò ÷ ú +î Y 0 î ï0 0 î ð0 û( ò, ó, ú) ô( õ, Å, ò, ó, ö- ú) ò ÷ ò ÷ ó ÷ ú. Here,ô( õ, Å, ò, ó, ö)=2sin ügöiý þ ÿù2Å0 ý þ+ 4 Å0» =0 » =1 2 ï0( õ) ï0( ò) ( ù2Å20 2 - 2 Ô2) å ï0( ù)æ2 ´cos ÔÅÅ0 cos Ôó 0 sin ügö ø2 2 + þ ÿø2 2 + þ, where the ï0( õ) are the Bessel functions and the are positive roots of the transcendental equation ï0( ù)= 0. Page 370 5.3.2-9. Domain: 0 £ õ£ ù,0 £ £ 0. Mixed boundary value problem. A circular sector is considered. The following conditions are prescribed:Ó=ñ0( õ, ) at ö= 0 (initial condition), YÓ=ñ1( õ, ) at ö= 0 (initial condition), Ó+ Ó= [( , ö) at õ= ù(boundary condition),ï Ó= 0 at = 0 (boundary condition),ï Ó= 0 at = 0(boundary condition). Solution:Ó( õ, , ö)= ö î ï0 0 î ð0 ñ0( ò, ó) ô( õ, , ò, ó, ö) ò ÷ ò ÷ ó +î ï0 0 î ð0 ñ1( ò, ó) ô( õ, , ò, ó, ö) ò ÷ ò ÷ ó + ø2 ùî Y 0 î ï0 0 [( ó, ú) ô( õ, , ù, ó, ö- ú) ÷ ó ÷ ú +î Y 0 îï0 0 îð0 û( ò, ó, ú) ô( õ, , ò, ó, ö- ú) ò ÷ ò ÷ ó ÷ ú. Here,ô( õ, , ò, ó, ö)=» =0 » =1 ( õ) ( ò) cos( ) cos( ó) sin üö ø2 2 + þ ÿ, = ! 0, =4 2 0( 2 ù2+ 2 ù2- 2) å ( ù)æ2ø2 2 + þ, where the ( õ) are the Bessel functions and the are positive roots of the transcendental equation ( ù)+ ( ù)= 0. 5.3.3. Axisymmetric Problems In the axisymmetric case, a nonhomogeneous Klein±Gordon equation in the cylindrical system of coordinates has the form2 "ö2= ø2 2 "õ2+1õ "õ+ 2 " #2 - þ "+û( õ, #, ö), õ= $2+ %2. In the solutions of the problems considered below, the modi®ed Green's function &( õ, #, ò, ó, ö)= 2 ! ò ô( õ, #, ò, ó, ö) is used for convenience. 5.3.3-1. Domain: 0 £ õ£ ù,0 £ #£ '. First boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:"=ñ0( õ, #) at ö= 0 (initial condition), Y"=ñ1( õ, #) at ö= 0 (initial condition),"= [1( #, ö) at õ= ù(boundary condition),"= [2( õ, ö) at #= 0 (boundary condition),"= [3( õ, ö) at #= '(boundary condition). Page 371 Solution:"( õ, #, ö)= ö î (0 îð0 ñ0( ò, ó) &( õ, #, ò, ó, ö) ÷ ò ÷ ó +î (0 î ð0 ñ1( ò, ó) &( õ, #, ò, ó, ö) ÷ ò ÷ ó - ø2î Y 0 î (0 [1( ó, ú) ) ò &( õ, #, ò, ó, ö- ú) * + =ð ÷ ó ÷ ú + ø2î Y 0 î ð0 [2( ò, ú) ) ó &( õ, #, ò, ó, ö- ú) * , =0 ÷ ò ÷ ú - ø2î Y 0 îð0 [3( ò, ú) ) ó &( õ, #, ò, ó, ö- ú) * , =( ÷ ò ÷ ú +î Y 0 î (0 î ð0û( ò, ó, ú) &( õ, #, ò, ó, ö- ú) ÷ ò ÷ ó ÷ ú. Here,&( õ, #, ò, ó, ö)=4 òù2' » =1 » =112 1( ) 0 õù 0 òù sin - ! #' sin - ! ó' sin ügöý ç ÿý ç ,ç = ø2 2ù2+ ø2!2-2'2+ þ, where the are positive zeros of the Bessel function, 0( )= 0. 5.3.3-2. Domain: 0 £ õ£ ù,0 £ #£ '. Second boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:"=ñ0( õ, #) at ö= 0 (initial condition), Y"=ñ1( õ, #) at ö= 0 (initial condition), "= [1( #, ö) at õ= ù(boundary condition), ."= [2( õ, ö) at #= 0 (boundary condition), ."= [3( õ, ö) at #= '(boundary condition). Solution:"( õ, #, ö)= ö î (0 î ð0 ñ0( ò, ó) &( õ, #, ò, ó, ö) ÷ ò ÷ ó +î (0 îð0 ñ1( ò, ó) &( õ, #, ò, ó, ö) ÷ ò ÷ ó + ø2î Y 0 î (0 [1( ó, ú) &( õ, #, ù, ó, ö- ú) ÷ ó ÷ ú - ø2î Y 0 î ð0 [2( ò, ú) &( õ, #, ò,0, ö- ú) ÷ ò ÷ ú + ø2î Y 0 îð0 [3( ò, ú) &( õ, #, ò, ', ö- ú) ÷ ò( ÷ ú +î Y 0 î (0 î ð0û( ò, ó, ú) &( õ, #, ò, ó, ö- ú) ÷ ò ÷ ó ÷ ú. Page 372 Here,&( õ, #, ò, ó, ö)=2 òsin ügöý þ ÿù2'9ý þ +2 òù2' » =0 » =0 2 0( ) 0 õù 0 òù cos - ! #' cos - ! ó' sin ügöý ç ÿý ç ,ç = ø2 2ù2+ ø2!2-2'2+ þ, = /0for-= 0, = 0, 1for-= 0, >0, 2for->0, where the are zeros of the ®rst-order Bessel function, 1( )= 0( 0= 0). 5.3.3-3. Domain: 0 £ õ£ ù,0 £ #£ '. Third boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:"=ñ0( õ, #) at ö= 0 (initial condition), Y"=ñ1( õ, #) at ö= 0 (initial condition), "+ 1 "= [1( #, ö) at õ= ù(boundary condition), ."- 2 "= [2( õ, ö) at #= 0 (boundary condition), ."+ 3 "= [3( õ, ö) at #= '(boundary condition). The solution "( õ, #, ö) is determined by the formula in Paragraph 5.3.3-2 where&( õ, #, ò, ó, ö)=2 òù2» =1 » =1 2 ( 2 1 ù2+ 2) 2 0( ) 0 õù 0 òù ( #) ( ó)0 02sin ügöý ç ÿý ç ,ç = ø2 2ù2+ ø2 12+ þ, ( #)=cos( 1 #)+ 21sin( 1 #),0 02= 3 2 12 12+ 2 212+ 2 3+ 2 2 12+ ' 2 1 + 2 212 . Here, the and 1are positive roots of the transcendental equations1( )- 1 ù0( )= 0,tan( 1')1 = 2+ 312- 2 3. 5.3.3-4. Domain: 0 £ õ£ ù,0 £ #£ '. Mixed boundary value problems. 1 2. A circular cylinder of ®nite length is considered. The following conditions are prescribed:"= 30( õ, #) at ö= 0 (initial condition), Y"= 31( õ, #) at ö= 0 (initial condition),"= [1( #, ö) at õ= ù(boundary condition), ."= [2( õ, ö) at #= 0 (boundary condition), ."= [3( õ, ö) at #= '(boundary condition). Page 373 Solution:"( õ, #, ö)= ö 4 (0 4 50 30( 6, 7) &( 8, #, 6, 7, 9) : 6 : 7 +4 (0 450 31( 6, 7) &( 8, #, 6, 7, 9) : 6 : 7 - ;24 Y 0 4 (0 [1( 7, <) ) 6 &( 8, #, 6, 7, 9- <) * + =5 : 7 : < - ;24 Y 0 450 [2( 6, <) &( 8, #, 6,0, 9- <) : 6 : < + ;24 Y 0 4 50 [3( 6, <) &( 8, #, 6, ', 9- <) : 6 : < +4 Y 0 4 (0 450 =( 6, 7, <) &( 8, #, 6, 7, 9- <) : 6 : 7 : <. Here,&( 8, #, 6, 7, 9)=2 6>2' » ?A@ =1 » ?B=0 BC2 1( D @ ) C 0 E D @8> F C 0 E D @6> FcosE G H I JFcosE G H 7JFsin KL9NM O @B PMO @B,O @B= ;2D2 @>2+ ;2H2G2J2+ Q, B= R1forG= 0, 2forG>0, where the D @ are positive zeros of the Bessel function, C 0( D)= 0. 2 S. A circular cylinder of ®nite length is considered. The following conditions are prescribed:T= U0( 8,I) at 9= 0 (initial condition),V YT= U1( 8,I) at 9= 0 (initial condition),V WT= [1(I, 9) at 8= >(boundary condition),T= [2( 8, 9) atI= 0 (boundary condition),T= [3( 8, 9) atI= J(boundary condition). Solution:T( 8,I, 9)= VV9 4 X0 4 50 U0( 6, 7) Y( 8,I, 6, 7, 9) : 6 : 7+4 X0 4 50 U1( 6, 7) Y( 8,I, 6, 7, 9) : 6 : 7 + ;24 Y 0 4X0 [1( 7, <) Y( 8,I, >, 7, 9- <) : 7 : < + ;24 Y 0 4 50 [2( 6, <) Z VV7 Y( 8,I, 6, 7, 9- <) [ \ =0 : 6 : < - ;2 ] Y 0 ]50 [3( 6, ^) Z VV7 Y( 8,I, 6, 7, _- ^) [\ =X ` 6` ^ + ] Y 0 ]X0 ]50 a( 6, 7, ^) Y( 8,I, 6, 7, _- ^)` 6` 7` ^. Here,Y( 8,I, 6, 7, _)=4 6b2 J c ?@ =0 c ?B=11C2 0( D @ ) C 0 E D @8b F C 0 E D @6b FsinE G H I JFsinE G H 7JFsin KL_ MO @B PM O @B,O @B= d2D2 @b2+ d2H2G2J2+ Q, where the D @ are zeros of the ®rst-order Bessel function, C 1( D)= 0( D0= 0). Page 374 eg j 5.3.3-5. Domain: b 1£ 8£ b 2,0 £I£ J. First boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:T= U0( 8,I) at _= 0 (initial condition),V YT= U1( 8,I) at _= 0 (initial condition),T= [1(I, _) at 8= b 1(boundary condition),T= [2(I, _) at 8= b 2(boundary condition),T= [3( 8, _) atI= 0 (boundary condition),T= [4( 8, _) atI= J(boundary condition). Solution:T( 8,I, _)= VV_ ]X0 ]5251 U0( 6, 7) Y( 8,I, 6, 7, _)` 6` 7 + ]X0 ]5251 U1( 6, 7) Y( 8,I, 6, 7, _)` 6` 7 +d2 ] Y 0 ]X0 [1( 7, ^) Z VV6 Y( 8,I, 6, 7, _- ^) [ p =51` 7` ^ -d2 ] Y 0 ]X0 [2( 7, ^) Z VV6 Y( 8,I, 6, 7, _- ^) [ p =52` 7` ^ +d2 ] Y 0 ]5251 [3( 6, ^) Z VV7 Y( 8,I, 6, 7, _- ^) [ \ =0` 6` ^ -d2 ] Y 0 ]5251 [4( 6, ^) Z VV7 Y( 8,I, 6, 7, _- ^) [ \ =X ` 6` ^ + ] Y 0 ]X0 ]5251 a( 6, 7, ^) Y( 8,I, 6, 7, _- ^)` 6` 7` ^. Here,Y( 8,I, 6, 7, _)=H26b2 1 J c ?@ =1 c ?B=1 D2 @C2 0( qrD @ )C2 0( D @ )- C2 0( qrD @ ) s @ ( 8)s @ ( 6) sinE G H I JFsinE G H 7JFsin KL_ MO @B PM O @B,s @ ( 8)= t0( D @ ) C 0 E D @8b 1 F- C 0( D @ ) t0 E D @8b 1 F, q= b 2b 1, O @B= d2D2 @b2 1+ d2H2G2J2+ Q, where C 0( D) and t0( D) are the Bessel functions, and the D @ are positive roots of the transcendental equationC 0( D) t0( qrD)- C 0( qrD) t0( D)= 0. 5.3.3-6. Domain: b 1£ 8£ b 2,0 £I£ J. Second boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:T= U0( 8,I) at _= 0 (initial condition),V YT= U1( 8,I) at _= 0 (initial condition),V WT= [1(I, _) at 8= b 1(boundary condition),V WT= [2(I, _) at 8= b 2(boundary condition),V uT= [3( 8, _) atI= 0 (boundary condition),V uT= [4( 8, _) atI= J(boundary condition). Page 375 Solution:T( 8,I, _)= VV_ ]X0 ]5251 U0( 6, 7) Y( 8,I, 6, 7, _)` 6` 7 + ]X0 ]5251 U1( 6, 7) Y( 8,I, 6, 7, _)` 6` 7 -d2 ] Y 0 ]X0 [1( 7, ^) Y( 8,I, b 1, 7, _- ^)` 7` ^ +d2 ] Y 0 ]X0 [2( 7, ^) Y( 8,I, b 2, 7, _- ^)` 7` ^ -d2 ] Y 0 ]5251 [3( 6, ^) Y( 8,I, 6,0, _- ^)` 6` ^ +d2 ] Y 0 ]5251 [4( 6, ^) Y( 8,I, 6, J, _- ^)` 6` ^ + ] Y 0 ]X0 ]5251 a( 6, 7, ^) Y( 8,I, 6, 7, _- ^)` 6` 7` ^. Here,Y( 8,I, 6, 7, _)=2 6sin KL_ MQ P ( b2 2- b2 1) JMQ+4 6 ( b2 2- b2 1) J c vB=1cosE G H I JFcosE G H 7JFsin KL_M w B PM w B +H26 2 b2 1 J c vAx =1 c vB=0 y B z2 xC2 1( q z x )C2 1( z x )- C2 1( q z x )s x ( 8)s x ( 6) cos {G H I JFcos {G H 7JFsin KL_NM O xB PMO xB, wheres x ( 8)= t1( z x ) C 0 { z x8b 1 F- C 1( z x ) t0 { z x8b 1 F, q= b 2b 1,y B= R1forG= 0, 2forG>1, w B= d2H2G2J2+ Q, O xB= d2z2 xb2 1+ d2H2G2J2+ Q;C |( z) and t |( z) are the Bessel functions ( }= 0,1); and the z x are positive roots of the transcendental equationC 1( z) t1( q z)- C 1( q z) t1( z)= 0. 5.4. Telegraph Equation~2 ~ 2+ ~~ = 2 2 ± + ( , , ) 5.4.1. Problems in Cartesian Coordinates A two-dimensional nonhomogeneous telegraph equation in the rectangular Cartesian coordinatesystem is written as 2 2+ } = 2{ 2 2+ 2 2 - + ( , , ). 5.4.1-1. Reduction to the two-dimensional Klein±Gordon equation. The substitution ( , , )=exp -1 2 } ( , , ) leads to the equation 2 2= 2{ 2 2+ 2 2 - -1 4 }2 +exp 1 2 } ( , , ), which is discussed in Subsection 5.3.1. Page 376 5.4.1-2. Fundamental solutions. 1 ¢. Case -1 4 £2= ¤2>0:¥ ¥( , , )= ¦( - 8) exp -1 2£ cos §¤ ¨ 2- 82 ©2 2 ª 2¨ 2- 82 ©2, where 8= ¨ 2+ 2and ¦( «) is the Heaviside unit step function. 2 ¢. Case -1 4£2= - ¤2<0:¥ ¥( , , )= ¦( - 8) exp -1 2£ cosh ¤ ¨ 2- 82 ©2 2 ª 2¨ 2- 82 ©2.¬® Reference : V . S. Vladimirov, V . P. Mikhailov, A. A. Vasharin, et al. (1974). 5.4.1-3. Domain: - ¯< < ¯,- ¯< < ¯. Cauchy problem. Initial conditions are prescribed:= °( , ) at = 0, ±= ²( , ) at = 0. Solution:( , , )=exp -1 2 £ ³ ³´£ µ ±°( 6, 7) ¶( , , 6, 7, ) · 6 · 7 +exp -1 2£ ³ ³´£ µ ± ¸²( 6, 7)+1 2£ °( 6, 7) ¹¶( , , 6, 7, ) · 6 · 7 + ³ ± 0 · º ³ ³´£ µ( ± - »)exp ¸-1 2£( - º) ¹ ( 6, 7, º) ¶( , , 6, 7, - º) · 6 · 7. Here,¶( , , 6, 7, )= ¼½ ½ ½ ½ ¾½ ½ ½ ½¿ cos §¤ ¨ 2- À2 ©2 2 ª 2¨ 2- À2 ©2for -1 4 £2= ¤2>0, cosh ¤ ¨ 2- À2 ©2 2 ª 2¨ 2- À2 ©2for -1 4 £2= - ¤2<0, where À= ¨( - Á)2+( - Â)2. 5.4.1-4. Domain: 0 £ £ Ã1,0 £ £ Ã2. First boundary value problem. A rectangle is considered. The following conditions are prescribed:= °0( , ) at = 0 (initial condition), ±= °1( , ) at = 0 (initial condition),= ²1( , ) at = 0 (boundary condition),= ²2( , ) at = Ã1(boundary condition),= ²3( , ) at = 0 (boundary condition),= ²4( , ) at = Ã2(boundary condition). Page 377 Solution:( , , )= ³ Ä1 0 ³ Ä2 0 °0( Á, Â) Å( , , Á, Â, ) · Â · Á + ³Ä1 0 ³Ä2 0 ¸°1( Á, Â)+£ °0( Á, Â)¹ Å( , , Á, Â, ) · Â · Á + 2 ³ ± 0 ³Ä2 0 ²1( Â, º) Æ Á Å( , , Á, Â, - º) Ç È =0 · Â · º - 2 ³ ± 0 ³ Ä2 0 ²2( Â, º) Æ ÉÉ Á Å( , , Á, Â, - º) Ç È =Ä1 · Â · º + 2 ³ ± 0 ³Ä1 0 ²3( Á, º) Æ ÉÉ Â Å( , , Á, Â, - º) Ç Ê =0 · Á · º - 2 ³ ± 0 ³ Ä1 0 ²4( Á, º) Æ ÉÉ Â Å( , , Á, Â, - º) ÇÊ =Ä2 · Á · º + ³ ± 0 ³ Ä1 0 ³ Ä2 0 ( Á, Â, º) Å( , , Á, Â, - º) · Â · Á · º, whereÅ( , , Á, Â, )=4Ã1 Ã2exp -1 2£ ËÌAÍ =1 ËÌÎ=11Ï ÍÎsin( Ð Í Ñ ) sin( Ò Î Ó) sin( Ð ÍÁ) sin( Ò ÎÂ) sin( Ï ÍÎ Ô),Ð Í = Õ ªÃ1, Ò Î= Ö ªÃ2, Ï ÍÎ= × Ø2Ð2 Í + Ø2Ò2Î+ Ù-1 4 Ú2. 5.4.1-5. Domain: 0 £ Ñ £ Ã1,0 £ Ó£ Ã2. Second boundary value problem. A rectangle is considered. The following conditions are prescribed:Û= Ü0( Ñ , Ó) at Ô= 0 (initial condition),É Ý Û= Ü1( Ñ , Ó) at Ô= 0 (initial condition),É Þ Û= ß1( Ó, Ô) at Ñ = 0 (boundary condition),É Þ Û= ß2( Ó, Ô) at Ñ = Ã1(boundary condition),É à Û= ß3( Ñ , Ô) at Ó= 0 (boundary condition),É à Û= ß4( Ñ , Ô) at Ó= Ã2(boundary condition). Solution:Û( Ñ , Ó, Ô)= ÉÉ Ô á â1 0 á â2 0 Ü0( ã, ä) å( æ, ç, ã, ä, è) é ä é ã +á â1 0 á â2 0 ê Ü1( ã, ä)+Ú Ü0( ã, ä) ëå( æ, ç, ã, ä, è) é ä é ã - Ø2á Ý 0 á â2 0 ß1( ä, ì) å( æ, ç,0, ä, è- ì) é ä é ì+ Ø2á Ý 0 á â2 0 ß2( ä, ì) å( æ, ç, í1, ä, è- ì) é ä é ì - Ø2á Ý 0 á â1 0 ß3( ã, ì) å( æ, ç, ã,0, è- ì) é ã é ì+ Ø2á Ý 0 á â1 0 ß4( ã, ì) å( æ, ç, ã, í2, è- ì) é ã é ì +á Ý 0 á â1 0 á â2 0 î( ã, ä, ì) å( æ, ç, ã, ä, è- ì) é ä é ã é ì. Page 378 ïñ ðïñ õ Here,å( æ, ç, ã, ä, è)=exp ú-1 2 Ú èû üsin ú Ï 00 èûí1 í2 Ï 00 +2í1 í2 ý þAÿ =0 ý þÎ=0 ÿÎÏ ÿÎcos( Ð ÿæ) cos( Îç) cos( Ð ÿã) cos( Îä) sin( Ï ÿÎè) , whereÐ ÿ = í1, Î= í2, Ï ÿÎ= 2Ð2 ÿ + 22Î+ -1 4 2, ÿÎ= 0for== 0, 1for = 0(¹), 2for ¹ 0. 5.4.1-6. Domain: 0 £ æ£ í1,0 £ ç£ í2. Third boundary value problem. A rectangle is considered. The following conditions are prescribed:Û= Ü0( æ, ç) at è= 0 (initial condition),Ý Û= Ü1( æ, ç) at è= 0 (initial condition), Û- 1 Û= 1( ç, è) at æ= 0 (boundary condition), Û+ 2 Û= 2( ç, è) at æ= í1(boundary condition), Û- 3 Û= 3( æ, è) at ç= 0 (boundary condition), Û+ 4 Û= 4( æ, è) at ç= í2(boundary condition). The solution Û( æ, ç, è) is determined by the formula in Paragraph 5.4.1-5 whereå( æ, ç, ã, ä, è)= 4exp ú-1 2 è ûý þAÿ =1 ý þÎ=11 ÿÎ 2 2 ÿ + 2 2Î+ -1 4 2sin( ÿæ+ ÿ ) sin( Îç+ Î) ´sin( ÿã+ ÿ ) sin( Îä+ Î) sin è 2 2 ÿ + 2 2+ -1 4 2 . Here, ÿ =arctan ÿí1, =arctan í2, ÿ= ü í1+( 1 2+ 2 ÿ )( 1+ 2) ( 2 1+ 2 ÿ )( 2 2+ 2 ÿ ) ü í2+( 3 4+ 2)( 3+ 4) ( 2 3+ 2)( 2 4+ 2) ; the ÿ and are positive roots of the transcendental equations2- 1 2=( 1+ 2) cot( í1 ), 2- 3 4=( 3+ 4) cot( í2 ). 5.4.1-7. Domain: 0 £ æ£ í1,0 £ ç£ í2. Mixed boundary value problems. 1 ¢. A rectangle is considered. The following conditions are prescribed:Û= Ü0( æ, ç) at è= 0 (initial condition),Ý Û= Ü1( æ, ç) at è= 0 (initial condition),Û= 1( ç, è) at æ= 0 (boundary condition),Û= 2( ç, è) at æ= í1(boundary condition), Û= 3( æ, è) at ç= 0 (boundary condition), Û= 4( æ, è) at ç= í2(boundary condition). Page 379 Solution:Û( æ, ç, è)= è á â1 0 á â2 0 Ü0( ã, ä) å( æ, ç, ã, ä, è) é ä é ã +á â1 0 á â2 0 ê Ü1( ã, ä)+ Ü0( ã, ä) ëå( æ, ç, ã, ä, è) é ä é ã + 2á Ý 0 á â2 0 1( ä, ì) ü ã å( æ, ç, ã, ä, è- ì) =0 é ä é ì - 2á Ý 0 á â2 0 2( ä, ì) ü ã å( æ, ç, ã, ä, è- ì) =â1 é ä é ì - 2á Ý 0 á â1 0 3( ã, ì) å( æ, ç, ã,0, è- ì) é ã é ì + 2á Ý 0 á â1 0 4( ã, ì) å( æ, ç, ã, í2, è- ì) é ã é ì +á Ý 0 á â1 0 á â2 0 î( ã, ä, ì) å( æ, ç, ã, ä, è- ì) é ä é ã é ì, whereå( æ, ç, ã, ä, è)=2í1 í2exp ú-1 2 èûý þÿ =1 ý þ=0 ÿsin( ÿæ) cos( ç) sin( ÿã) cos( ä) sin( ÿè), ÿ = í1, = í2, ÿ= 22 ÿ + 22+ -1 4 2, = 1for= 0, 2for¹ 0. 2 ¢. A rectangle is considered. The following conditions are prescribed:Û= Ü0( æ, ç) at è= 0 (initial condition),Ý Û= Ü1( æ, ç) at è= 0 (initial condition),Û= 1( ç, è) at æ= 0 (boundary condition), Û= 2( ç, è) at æ= í1(boundary condition),Û= 3( æ, è) at ç= 0 (boundary condition), Û= 4( æ, è) at ç= í2(boundary condition). Solution:Û( æ, ç, è)= è 1 0 2 0 Ü0( , ) !( ", #, , , $) % % + 1 0 2 0 & Ü1( , )+ Ü0( , ) '(!( ", #, , , $) % % + 2 Ý 0 2 0 1( , )) * !( ", #, , , $- )) =0 % % ) + 2 +0 2 0 2( , )) !( ", #, ,1, , $- )) % % ) + 2+0 1 0 3( , )) * !( ", #, , , $- )) - =0 % % ) + 2+0 1 0 4( , )) !( ", #, , ,2, $- )) % % ) + +0 1 0 2 0 .( , , )) !( ", #, , , $- )) % % % ), Page 380 where ( , , , , )=4 1 2exp -1 2 =0 =01 sin( ) sin( ) sin( ) sin( ) sin( ), = (2 + 1) 2 1, = (2 !+ 1) 2 2, = " #22+ #22+ $-1 42. 5.4.2. Problems in Polar Coordinates A two-dimensional nonhomogeneous telegraph equation in the polar coordinate system has the form%2 &%2+ %&%= #2 ' %2 &% (2+1( %&% (+1(2 %2 &% )2 *- $ &+ +( (, ), ), (= , 2+ 2. For one-dimensional solutions &= &( (, ), see equation 4.4.2.2. 5.4.2-1. Domain: 0 £ (£ -,0 £ )£ 2. First boundary value problem. A circle is considered. The following conditions are prescribed:&= .0( (, )) at = 0 (initial condition),% /&= .1( (, )) at = 0 (initial condition),&= 0( ), ) at (= -(boundary condition). Solution:&( (, ), )= %% 12 2 0 1 30 .0( , ) ( (, ), , , ) 4 4 +12 2 0 1 30 5 .1( , )+ .0( , ) 6 ( (, ), , , ) 4 4 - #2-1 / 0 12 2 0 0( , 7) 8 %% ( (, ), , , - 7) 9 : =3 4 4 7 +1 / 0 12 2 0 1 30 +( , , 7) ( (, ), , , - 7) 4 4 4 7. Here, ( (, ), , , )=1 -2exp -1 2 =0 =1 ; < ( = () < ( = ) [ < >( = -)]2cos[ ( )- )]sin ?A@ @ , = #2=2 + $-1 42,;0= 1,; = 2 ( = 1,2, BBB), where the < ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the = are positive roots of the transcendental equation < ( = -)= 0. 5.4.2-2. Domain: 0 £ (£ -,0 £ )£ 2. Second boundary value problem. A circle is considered. The following conditions are prescribed:&= .0( (, )) at = 0 (initial condition),% /&= .1( (, )) at = 0 (initial condition),% C&= 0( ), ) at (= -(boundary condition). Page 381 Solution:&( (, ), )= %% 12 2 0 1 30 .0( , ) ( (, ), , , ) 4 4 +12 2 0 1 30 5 .1( , )+ .0( , ) 6 ( (, ), , , ) 4 4 + #2-1 / 0 12 2 0 0( , 7) ( (, ), -, , - 7) 4 4 7 +1 / 0 12 2 0 1 30 +( , , 7) ( (, ), , , - 7) 4 4 4 7. Here, ( (, ), , , )=exp -1 2 Dsin ?, $-2 E4 -2, $-2 E4 +1 =0 =1 ; =2 < ( = () < ( = ) ( =2 -2- 2)[ < ( = -)]2cos[ ( )- )]sin @ @ F, = #2=2 + $-1 4 2,;0= 1,; = 2 ( = 1,2, BBB), where the < ( ) are the Bessel functions and the = are positive roots of the transcendental equation< >( = -)= 0. 5.4.2-3. Domain: 0 £ (£ -,0 £ )£ 2. Third boundary value problem. A circle is considered. The following conditions are prescribed:&= .0( (, )) at = 0 (initial condition),% /&= .1( (, )) at = 0 (initial condition),% C&+ G &= 0( ), ) at (= -(boundary condition). The solution &( (, ), ) is determined by the formula in Paragraph 5.4.2-2 where ( (, ), , , )=1exp -1 2 =0 =1 ; =2 < ( = () < ( = ) cos[ ( )- )] sin ?A@ ( =2 -2+ G2-2- 2)[ < ( = -)]2@ , = #2=2 + $-1 42,;0= 1,; = 2 ( = 1,2, BBB). Here, the < ( ) are the Bessel functions and the = are positive roots of the transcendental equation= < >( = -)+ G < ( = -)= 0. 5.4.2-4. Domain: -1£ (£ -2,0 £ )£ 2. First boundary value problem. An annular domain is considered. The following conditions are prescribed:&= .0( (, )) at = 0 (initial condition),% /&= .1( (, )) at = 0 (initial condition),&= 01( ), ) at (= -1(boundary condition),&= 02( ), ) at (= -2(boundary condition). Page 382 Solution:&( (, ), )= %% 12 2 0 1 3231 .0( , ) ( (, ), , , ) 4 4 +12 2 0 1 3231 5 .1( , )+ .0( , ) 6 ( (, ), , , ) 4 4 + #2-11 / 0 12 2 0 01( , 7) 8 %% ( (, ), , , - 7) 9 : =31 4 4 7 - #2-21 / 0 12 2 0 02( , 7) 8 %% ( (, ), , , - 7) 9 : =32 4 4 7 +1 / 0 12 2 0 1 3231 +( , , 7) ( (, ), , , - 7) 4 4 4 7. Here, ( (, ), , , )= 2exp -1 2 =0 =1; H I ( = () I ( = ) cos[ ( )- )]sin ? @ @ ,; = J1 E2for = 0, 1 for ¹ 0, H = =2 <2( = -2)<2( = -1)- <2( = -2),I ( = ()= < ( = -1) K ( = ()- K ( = -1) < ( = (), = #2=2 + $-1 4 2, where the < ( () and K ( () are the Bessel functions, and the = are positive roots of the transcen- dental equation< ( = -1) K ( = -2)- K ( = -1) < ( = -2)= 0. 5.4.2-5. Domain: -1£ (£ -2,0 £ )£ 2. Second boundary value problem. An annular domain is considered. The following conditions are prescribed:&= .0( (, )) at = 0 (initial condition),% /&= .1( (, )) at = 0 (initial condition),% C&= 01( ), ) at (= -1(boundary condition),% C&= 02( ), ) at (= -2(boundary condition). Solution:&( (, ), )= %%12 2 0 1 3231 .0( , ) ( (, ), , , ) 4 4 +12 2 0 1 3231 5 .1( , )+ .0( , ) 6 ( (, ), , , ) 4 4 - #2-11 / 0 12 2 0 01( , 7) ( (, ), -1, , - 7) 4 4 7 + #2-21 / 0 12 2 0 02( , 7) ( (, ), -2, , - 7) 4 4 7 +1 / 0 12 2 0 1 3231 +( , , 7) ( (, ), , , - 7) 4 4 4 7. Page 383 Here, ( (, ), , , )=exp -1 2 Dsin ?, $-2 E4 ( -2 2- -2 1), $-2 E4 +1 =0 =1 ; =2 I ( = () I ( = ) cos[ ( )- )] sin ?, #2=2 + $-2 E4 5( =2 -2 2- 2) I2( = -2)-( =2 -2 1- 2) I2( = -1) 6, #2=2 + $-2 E4 F, whereI ( = ()= < >( = -1) K ( = ()- K >( = -1) < ( = (),; = J1for = 0, 2for >0, the < ( () and K ( () are the Bessel functions, and the = are positive roots of the transcendental equation< >( = -1) K >( = -2)- K >( = -1) < >( = -2)= 0. 5.4.2-6. Domain: -1£ (£ -2,0 £ )£ 2. Third boundary value problem. An annular domain is considered. The following conditions are prescribed:&= .0( (, )) at = 0 (initial condition),% /&= .1( (, )) at = 0 (initial condition),% C&- G1 &= 01( ), ) at (= -1(boundary condition),% C&+ G2 &= 02( ), ) at (= -2(boundary condition). The solution &( (, ), ) is determined by the formula in Paragraph 5.4.2-5 where ( (, ), , , )=1exp -1 2 =0 =1 ; =2 H I ( = () I ( = ) cos[ ( )- )] sin( ). Here,; = J1for = 0, 2for >0, = "#2=2 + $-1 42,H =( G2 2 -2 2+ =2 -2 2- 2) I2( = -2)-( G2 1 -2 1+ =2 -2 1- 2) I2( = -1),I ( = ()=5 = < >( = -1)- G1 < ( = -1) 6LK ( = () -5 = K >( = -1)- G1 K ( = -1) 6 < ( = (), where the < ( () and K ( () are the Bessel functions, and the = are positive roots of the transcen- dental equation5 = < >( = -1)- G1 < ( = -1) 65 = K >( = -2)+ G2 K ( = -2) 6 =5 = K >( = -1)- G1 K ( = -1) 65 = < >( = -2)+ G2 < ( = -2) 6. 5.4.2-7. Domain: 0 £ (£ -,0 £ )£ ) 0. First boundary value problem. A circular sector is considered. The following conditions are prescribed:&= .0( (, )) at = 0 (initial condition),% /&= .1( (, )) at = 0 (initial condition),&= 01( ), ) at (= - (boundary condition),&= 02( (, ) at )= 0 (boundary condition),&= 03( (, ) at )= ) 0(boundary condition). Page 384 Solution:&( (, ), )= %% 1 M0 0 1 30 .0( , ) ( (, ), , , ) 4 4 +1 M0 0 1 30 5 .1( , )+ .0( , ) 6 ( (, ), , , ) 4 4 - #2-1 / 0 1M0 0 01( , 7) 8 %% ( (, ), , , - 7) 9 : =3 4 4 7 + #21 / 0 1 30 02( , 7)1 8 %% ( (, ), , , - 7) 9 N =0 4 4 7 - #21 / 0 1 30 03( , 7)1 8 %% ( (, ), , , - 7) 9 N =M0 4 4 7 +1 / 0 1 M0 0 1 30 +( , , 7) ( (, ), , , - 7) 4 4 4 7. Here, ( (, ), , , )=4-2 ) 0exp -1 2 =1 =1 <2 OM0( = () <2 OM0( = ) [ < >2 OM0( = -)]2 ´sin ' )) 0 *sin ' ) 0 *sin ?, #2=2 + $-2 E4 , #2=2 + $-2 E4, where the <2 OM0( () are the Bessel functions and the = are positive roots of the transcendental equation <2 OM0( = -)= 0. 5.4.2-8. Domain: 0 £ (£ -,0 £ )£ ) 0. Second boundary value problem. A circular sector is considered. The following conditions are prescribed:&= .0( (, )) at = 0 (initial condition),% /&= .1( (, )) at = 0 (initial condition),% C&= 01( ), ) at (= - (boundary condition),(-1 %M &= 02( (, ) at )= 0 (boundary condition),(-1 %M &= 03( (, ) at )= ) 0(boundary condition). Solution:&( (, ), )= %% 1 M0 0 1 30 .0( , ) ( (, ), , , ) 4 4 +1 M0 0 1 30 5 .1( , )+ .0( , ) 6 ( (, ), , , ) 4 4 + #2-1 / 0 1M0 0 01( , 7) ( (, ), -, , - 7) 4 4 7 - #21 / 0 1 30 02( , 7) ( (, ), ,0, - 7) 4 4 7 + #21 / 0 1 30 03( , 7) ( (, ), , ) 0, - 7) 4 4 7 +1 / 0 1 M0 0 1 30 +( , , 7) ( (, ), , , - 7) 4 4 4 7. Page 385 Here, ( (, ), , , )=exp -1 2 D2sin ?, $-2 E4 -2 ) 0 , $-2 E4+ 4 ) 0 =0 =1 =2 <2 OM0( = () <2 OM0( = ) ( -2 )20=2 - 22) <22 OM0( = -) ´cos ' )) 0 *cos ' ) 0 *sin ?, #2=2 + $-2 E4 , #2=2 + $-2 E4 F, where the <2 OM0( () are the Bessel functions and the = are positive roots of the transcendental equation < >2 OM0( = -)= 0. 5.4.2-9. Domain: 0 £ (£ -,0 £ )£ ) 0. Mixed boundary value problem. A circular sector is considered. The following conditions are prescribed:&= .0( (, )) at = 0 (initial condition),% /&= .1( (, )) at = 0 (initial condition),% C&+ P &= 0( ), Q) at (= - (boundary condition),%M &= 0 at )= 0 (boundary condition),%M &= 0 at )= ) 0(boundary condition). Solution:&( (, ), Q)= %%Q 1M0 0 1 30 .0( R, S) T( (, ), R, S, Q) R 4 R 4 S +1M0 0 1 30 5 .1( R, S)+ U .0( R, S) 6LT( (, ), R, S, Q) R 4 R 4 S + #2-1 / 0 1M0 0 0( S, 7) T( (, ), -, S, Q- 7) 4 S 4 7 +1 / 0 1 M0 0 1 30 +( R, S, 7) T( (, ), R, S, Q- 7) R 4 R 4 S 4 7. Here,T( (, ), R, S, Q)=exp V-1 2 U QW X Y[Z =0 X Y\=1; Z\ < ]_^( = Z\ `) < ]_^( = Z\R) cos( G Z a ) cos( G ZS) sin Vcb Z\QW,G Z = d e a 0, f Z\=4 g2 Z\ a 0( g2 Z\ h2+ P2h2- G2 Z ) i_j ]_^( g Z\h) k2b Z\, b Z\= l #2g2 Z\+ $-1 4 U2, where the j ]_^( `) are the Bessel functions and the g Z\are positive roots of the transcendental equationg j m ]_^( g h)+ P j ]_^( g h)= 0. 5.4.3. Axisymmetric Problems In the axisymmetric case, a nonhomogeneoustelegraph equation in the cylindrical coordinate systemhas the form%2 n%Q2+ U %n%Q= #2 o %2 n%`2+1` %n%`+ %2 n% p2 q- $ n+ r( `, p, Q), `= s t2+ u2. Page 386 vx wvx= z - }| In the solutions of the problems considered below, the modi®ed Green's function ( `, p, , , Q)= 2e ( `, p, , , Q) is used for convenience. 5.4.3-1. Domain: 0 £ `£ h,0 £ p£ . First boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:n= 0( `, p) at Q= 0 (initial condition),% n= 1( `, p) at Q= 0 (initial condition),n= 1( p, Q) at `= h(boundary condition),n= 2( `, Q) at p= 0 (boundary condition),n= 3( `, Q) at p= (boundary condition). Solution:n( `, p, Q)= %%Q 0 0 0( , ) ( , p, , , Q) + 0 0 1( , )+ 0( , ) ( , p, , , Q) - 2 0 0 1( , ) %% ( , p, , , Q- ) = + 2 0 0 2( , ) %% ( , , , , Q- ) =0 - 2 0 0 3( , ) %% ( , , , , Q- ) = + 0 0 0 r( , , ) ( , , , , Q- ) . Here,( , , , , Q)=4 - ? 22 [ =1 =11¡2 1( ¢ ) ¡ 0 £ ¢ ¤ ¡ 0 £ ¢ ¥ ¤sin£ ¦ e §¤sin£ ¦ e ¨ §¤sin( © ª)© ,© = « 2¢2 2+ 2e2¦2§2+ $- 2 4, where the ¢ are positive zeros of the Bessel function, ¡ 0( ¢)= 0. 5.4.3-2. Domain: 0 £ £ ,0 £ £ §. Second boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:¬= 0( , ) at ª= 0 (initial condition),% ®¬= 1( , ) at ª= 0 (initial condition),% ¯¬= °1( , ª) at = (boundary condition),% ±¬= °2( , ª) at = 0 (boundary condition),% ±¬= °3( , ª) at = §(boundary condition). Page 387 Solution:¬( , , ª)= %%ª 0 0 0( ¥ ,¨) ²( , , ¥ ,¨, ª) ¥¨ + ³ ´ 0 ³0 µ 1( ¥ ,¨)+ ¶ 0( ¥ ,¨) ·L²( , , ¥ ,¨, ª) ¸ ¥¸¨ + ¹2³ ® 0 ³´ 0 °1(¨, º) ²( , , ,¨, ª- º) ¸¨ ¸ º - ¹2³ ® 0 ³0 °2( ¥ , º) ²( , , ¥ ,0, ª- º) ¸ ¥¸ º + ¹2³ ® 0 ³0 °3( ¥ , º) ²( , , ¥ , §, ª- º) ¸ ¥¸ º + ³ ® 0 ³´ 0 ³0 »( ¥ ,¨, º) ²( , , ¥ ,¨, ª- º) ¸ ¥¸¨ ¸ º. Here,²( , , ¥ ,¨, ª)= 2 ¥ exp ¼-1 2 ¶ ª½ ¾sin ¼ ªA¿ À ½2 §¿ À+12 § Á Â[Ã =0 Á ÂÄ=0 Å ÃÄÆ2 0( Ç Ã ) Æ 0 È Ç Ã É Æ 0 È Ç Ã Ê É ´cosÈ Ë Ì ÍÉcosÈ Ë Ì Î ÍÉsin ¼?Ï ¿ Ð ÃÄ ½¿ Ð ÃÄ Ñ, whereÀ= Ò- ¶2 4, Ð ÃÄ= ¹2Ç2 Ã2+ ¹2Ì2Ë2Í2+ Ò- ¶2 4,Å ÃÄ= Ó0forË= 0, Ô= 0, 1forË= 0, Ô>0, 2forË>0, and the Ç Ã are zeros of the ®rst-order Bessel function, Æ 1( Ç)= 0( Ç0= 0). 5.4.3-3. Domain: 0 £ Õ£ ,0 £ £ Í. Third boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:Ö= ×0( Õ, Ø) at Ï= 0 (initial condition),Ù ÚÖ= ×1( Õ, Ø) at Ï= 0 (initial condition),Ù ÛÖ+ Ü1 Ö= Ý1( Ø, Ï) at Õ= Þ(boundary condition),Ù ßÖ- Ü2 Ö= Ý2( Õ, Ï) at Ø= 0 (boundary condition),Ù ßÖ+ Ü3 Ö= Ý3( Õ, Ï) at Ø= Í(boundary condition). The solution Ö( Õ, Ø, Ï) is determined by the formula in Paragraph 5.4.3-2 whereà( Õ, Ø, Ê ,Î, Ï)=2 ÊÞ2exp ¼-1 2 á Ï ½Á Â Ã =1 Á ÂÄ=1Å ÃÆ 0 È Ç ÃÕÞ É Æ 0 È Ç Ã ÊÞ É â Ä( Ø)â Ä(Î)ãâ Ä ã2sin ¼?Ï ¿ Ð ÃÄ ½¿ Ð ÃÄ. Here,Å Ã = Ç2 Ã ( Ü2 1 Þ2+ Ç2 Ã ) Æ2 0( Ç Ã ), Ð ÃÄ= ä2Ç2 ÃÞ2+ä2 å2Ä+ Ò- á2 4,â Ä( Ø)=cos( åÄØ)+ Ü2åÄsin( åÄØ), ãâ Ä ã2= Ü3 2 å2Ä å2Ä+ Ü2 2å2Ä+ Ü2 3+ Ü2 2 å2Ä+ Í 2 È1 + Ü2 2å2Ä É; the Ç Ã and åÄare positive roots of the transcendental equationsÇ Æ 1( Ç)- Ü1 Þ Æ 0( Ç)= 0,tan( å Í)å = Ü2+ Ü3å2- Ü2 Ü3. Page 388 æè çæè= ê - íì 5.4.3-4. Domain: 0 £ Õ£ Þ,0 £ Ø£ Í. Mixed boundary value problems. 1 ò. A circular cylinder of ®nite length is considered. The following conditions are prescribed:Ö= ×0( Õ, Ø) at Ï= 0 (initial condition),Ù ÚÖ= ×1( Õ, Ø) at Ï= 0 (initial condition),Ö= Ý1( Ø, Ï) at Õ= Þ(boundary condition),Ù ßÖ= Ý2( Õ, Ï) at Ø= 0 (boundary condition),Ù ßÖ= Ý3( Õ, Ï) at Ø= Í(boundary condition). Solution:Ö( Õ, Ø, Ï)= ÙÙÏ ó ô0 ó õ0 ×0( Ê ,Î) à( Õ, Ø, Ê ,Î, Ï) ö ÊöÎ +óô0 óõ0 ÷ ×1( Ê ,Î)+á ×0( Ê ,Î) ø à( Õ, Ø, Ê ,Î, Ï) ö ÊöÎ -ä2ó Ú 0 ó ô0 Ý1(Î, ù) ú ÙÙ Êà( Õ, Ø, Ê ,Î, Ï- ù) û ü =õ öÎ ö ù -ä2ó Ú 0 óõ0 Ý2( Ê , ù) à( Õ, Ø, Ê ,0, Ï- ù) ö Êö ù+ä2ó Ú 0 óõ0 Ý3( Ê , ù) à( Õ, Ø, Ê , Í, Ï- ù) ö Êö ù +ó Ú 0 ó ô0 ó õ0 ý( Ê ,Î, ù) à( Õ, Ø, Ê ,Î, Ï- ù) ö ÊöÎ ö ù. Here,à( Õ, Ø, Ê ,Î, Ï)=2 Ê þ - ÿ Ú 2Þ2 Í =1 =0 Å Æ2 1( Ç ) Æ 0 Ç ÕÞ É Æ 0 Ç ÊÞ Écos Ë Ì ØÍÉcos Ë Ì Î ÍÉsin( Ð Ï)Ð ,Ð = ä2Ç22+ ä2Ì2Ë2 2+ - 2 4, = 1forË= 0, 2forË>0, where the are zeros of the Bessel function, 0( )= 0. 2 . A circular cylinder of ®nite length is considered. The following conditions are prescribed:= ×0( , ) at = 0 (initial condition), = ×1( , ) at = 0 (initial condition), = 1( , ) at = (boundary condition),= 2( , ) at = 0 (boundary condition),= 3( , ) at = (boundary condition). Solution:( , , )= 0 0 ×0( , ) ( , , , , ) + 0 0 ×1( , )+ ×0( , ) !"( , , , , ) +ä2 0 0 1( , #) ( , , , , - #) # +ä2 0 0 2( , #) $ ( , , , , - #) % & =0 # -ä2 0 0 3( , #) $ ( , , , , - #) % & = # + 0 0 0 '( , , #) ( , , , , - #) #. Page 389 Here,( , , , , )=4 (- ÿ 22 )+* =0 ),=112 0( * ) 0 - * . 0 - * .sin- / Ì .sin- / Ì .sin( 0 *,)0 *,,0 *,= 1 222 *2+ 22Ì2/2 2+ 3- 42 4, where the * are zeros of the ®rst-order Bessel function, 1( )= 0( 0= 0). 5.4.3-5. Domain: 1£ £ 2,0 £ £ . First boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 50( , ) at = 0 (initial condition), = 51( , ) at = 0 (initial condition),= 1( , ) at = 1(boundary condition),= 2( , ) at = 2(boundary condition),= 3( , ) at = 0 (boundary condition),= 4( , ) at = (boundary condition). Solution:( , , )= 0 21 50( , ) ( , , , , ) + 0 21 51( , )+4 50( , ) !"( , , , , ) +22 0 0 1( , #) $ ( , , , , - #) % 6 =1 # -22 0 0 2( , #) $ ( , , , , - #) %6 =2 # +22 0 21 3( , #) $ ( , , , , - #) % & =0 # -22 0 21 4( , #) $ ( , , , , - #) % & = # + 0 0 21 '( , , #) ( , , , , - #) #. Here,( , , , , )=Ì22 1 (- ÿ 2 ) * =1 ),=1 2 *2 0( 78 * ) 9 * ( ) 9 * ( )2 0( * )- 2 0( 78 * )sin- / Ì .sin- / Ì .sin :<; 0 *, =; 0 *,,9 * ( )= >0( ? * ) @0 - ? * A 1 .- @0( ? * ) >0 - ? * A 1 ., 7= 2 1, 0 *,= 22?2 *2 1+ 22Ì2/2 2+ 3- 42 4, where @0( ?) and >0( ?) are the Bessel functions, and the ? * are positive roots of the transcendental equation@0( ?) >0( 78?)- @0( 78?) >0( ?)= 0. Page 390 5.5. Other Equations with Two Space Variables 1. B2 CB D2+ E B CB D= F2- B2 CB G2+ B2 CB H2 .+ I1 B CB G+ I2 B CB H+ J C. The transformation K ( L, M, N)= O( L, M, #) exp--1 2 4 N- 31 L+ 32 M 222 ., #=2 N leads to the equation from Subsection 5.1.3:P2OP Q2= P2OPL2+ P2OPM2+ åO, å= R22+ 42 422-1 424( 321+ 322). 2.DTS B2 CB D2+ U2 DVS±1B CB D= B2 CB G2+ B2 CB H2. Domain: - W< L< W,- W< M< W. Cauchy problem. Initial conditions are prescribed: K = 5( L, M) at N= 0,N , X2 P Y K = Z( L, M) at N= 0. Solution for 1 £/<2: K ( L, M, N)=1 2 [ N , X2 PPN \ \] ^ 5( _, `) a _ a `b42cN2- c - d2+1 2 [ \ \] ^ Z( _, `) a _ a `b42cN2- c - d2,4 c=2 2 - e, d= b ( L- _)2+( M- `)2, where f Y= { d2£42cN2- c }is the circle with center at ( L, M) and radius4 cN1 XVgVh.ikj Reference : M. M. Smirnov (1975). Page 391 Chapter 6 Hyperbolic Equations with Three orMore Space Variab les 6.1. WaveEquation l2 ml n2= o2 p 3 m 6.1.1. Problems inCartesian Coor dinates Thewaveequation with three space variables intherectangular Cartesian coordinate system hasthe formP2 KPN2= q2 r s2 ts u2+ s2 ts v2+ s2 ts w2 x. This equation isoffundamental importance insound propagation theory ,thepropagation ofelec- tromagnetic ®elds theory ,andanumber ofother areas ofphysics andmechanics. 6.1.1-1. Particular solutions andtheir properties. 1 y.Particular solutions:t(u,v,w, z)= {exp | }1u+ }2v+ }3w ~ q z b}2 1+ }2 2+ }2 3 ,t(u,v,w, z)= {sin( }1u+ f1)sin( }2v+ f2)sin( }3w+ f3)sin |"q z b}2 1+ }2 2+ }2 3 ,t(u,v,w, z)= {sin( }1u+ f1)sin( }2v+ f2)sin( }3w+ f3)cos |q z b}2 1+ }2 2+ }2 3 ,t(u,v,w, z)= {sinh( }1u+ f1)sinh( }2v+ f2)sinh( }3w+ f3)sinh |"q z b}2 1+ }2 2+ }2 3 ,t(u,v,w, z)= {sinh( }1u+ f1)sinh( }2v+ f2)sinh( }3w+ f3)cosh |"q z b}2 1+ }2 2+ }2 3 , where {, f1, f2, f3, }1, }2,and }3arearbitrary constants. 2 y.Fundamental solution: (u,v,w, z)=1 2 q ( q2z2- 2), = u2+v2+w2, where( )istheDirac delta function.k Refer ence:V.S.Vladimiro v(1988). 3 y.In®nite series solutions containing arbitrary functions ofspace variables:t(u,v,w, z)= (u,v,w)+ + =1( z)2 (2 )! (u,v,w),º s2s u2+ s2s v2+ s2s w2,t(u,v,w, z)= z(u,v,w)+ z + =1( z)2 (2 +1)! (u,v,w), Page393 where (u,v,w) and (u,v,w) are any in®nitely differentiable functions. The ®rst solution satis®es the initial conditions t(u,v,w,0)= (u,v,w),s t(u,v,w,0)= 0, and the second solution initial conditions t(u,v,w,0)= 0,s t(u,v,w,0)= (u,v,w). The sums are ®nite if (u,v,w) and (u,v,w) are polynomials inu,v,w.k Reference : A. V . Bitsadze and D. F. Kalinichenko (1985). 4 y. Suppose t= t(u,v,w, z) is a solution of the wave equation. Then the functionst1= { t(~ u+ 1,~ v+ 2,~ w+ 3,~ z+ 4),t2= { t r u- z1 -( )2,v,w, z- -2u1 -( )2 x,t3= {2- 2z2 t r u2- 2z2, v2- 2z2, w2- 2z2, z2- 2z2 x, where {, , , andare arbitrary constants, are also solutions of the equation. The signs atin the expression of t1can be taken independently of one another. The function t2is a consequence of the invariance of the wave equation under the Lorentz transformation.k References : G. N. Polozhii (1964), W. Miller, Jr. (1977), A. V . Bitsadze and D. F. Kalinichenko (1985). 6.1.1-2. Domain: - <u< ,- <v< ,- <w< . Cauchy problem. Initial conditions are prescribed:t= (u,v,w) at z= 0,s t= (u,v,w) at z= 0. Solution (Kirchhoff's formula):t(u,v,w, z)=1 4 ss z ( , , ) +1 4 ( , , ) ,= ( -u)2+( -v)2+( -w)2, where the integration is performed over the surface of the sphere of radius zwith center at (u,v,w).k References : N. S. Koshlyakov, E. B. Glizer, and M. M. Smirnov (1970), A. N. Tikhonov and A. A. Samarskii (1990). 6.1.1-3. Domain: 0 £u£ 1,0 £v£ 2,0 £w£ 3. First boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:t= 0(u,v,w) at z= 0 (initial condition),s t= 1(u,v,w) at z= 0 (initial condition),t= 1(v,w, z) atu= 0 (boundary condition),t= 2(v,w, z) atu= 1(boundary condition),t= 3(u,w, z) atv= 0 (boundary condition),t= 4(u,w, z) atv= 2(boundary condition),t= 5(u,v, z) atw= 0 (boundary condition),t= 6(u,v, z) atw= 3(boundary condition). Page 394 ¡"£ Solution:t(u,v,w, z)= ss z §3 0 §2 0 §1 0 0( , , ) ¨(u,v,w, , , , z) + §3 0 §2 0 §1 0 1( , , ) ¨(u,v,w, , , , z) + 2 0 §3 0 §2 0 1( , , ©) ª «« ¨( ¬, , ®, , , , ¯- ©) ° ± =0 © - 2 0 §3 0 §2 0 2( , , ©) ª «« ¨( ¬, , ®, , , , ¯- ©) ° ± =§1 © + 2 0 §3 0 §1 0 3( , , ©) ª «« ¨( ¬, , ®, , , , ¯- ©) ° ² =0 © - 2 0 §3 0 §1 0 4( , , ©) ª «« ¨( ¬, , ®, , , , ¯- ©) °² =§2 © + 2 0 §2 0 §1 0 5( , , ©) ª «« ¨( ¬, , ®, , , , ¯- ©) ° ³ =0 © - 2 0 §2 0 §1 0 6( , , ©) ª «« ¨( ¬, , ®, , , , ¯- ©) ° ³ =§3 ©. Here,¨( ¬, , ®, , , , ¯)=8 1 2 3 =1 ´=1 ¶µ =11 ´ µ sin( · ¬) sin( ¸ ´) sin( ¹ µ®) ´sin( · ) sin( ¸ ´) sin( ¹ µ) sin( ´ µ¯), where· = º 1, ¸ ´= » º 2, ¹ µ = ¼ º 3, ´ µ = ½ ·2 + ¸2´+ ¹2 µ . 6.1.1-4. Domain: 0 £ ¬£ 1,0 £ £ 2,0 £ ®£ 3. Second boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:¾= 0( ¬, , ®) at ¯= 0 (initial condition),« ¾= 1( ¬, , ®) at ¯= 0 (initial condition),« ¿ ¾= 1( , ®, ¯) at ¬= 0 (boundary condition),« ¿ ¾= 2( , ®, ¯) at ¬= 1(boundary condition),« À ¾= 3( ¬, ®, ¯) at = 0 (boundary condition),« À ¾= 4( ¬, ®, ¯) at = 2(boundary condition),« Á ¾= 5( ¬, , ¯) at ®= 0 (boundary condition),« Á ¾= 6( ¬, , ¯) at ®= 3(boundary condition). Page 395 Solution:¾( ¬, , ®, ¯)= «« ¯ §3 0 §2 0 §1 0 0( , , ) ¨( ¬, , ®, , , , ¯) + §3 0 §2 0 §1 0 1( , , ) ¨( ¬, , ®, , , , ¯) - 2 0 §3 0 §2 0 1( , , ©) ¨( ¬, , ®,0, , , ¯- ©) © + 2 0 §3 0 §2 0 2( , , ©) ¨( ¬, , ®, 1, , , ¯- ©) © - 2 0 §3 0 §1 0 3( , , ©) ¨( ¬, , ®, ,0, , ¯- ©) © + 2 0 §3 0 §1 0 4( , , ©) ¨( ¬, , ®, , 2, , ¯- ©) © - 2 0 §2 0 §1 0 5( , , ©) ¨( ¬, , ®, , ,0, ¯- ©) © + 2 0 §2 0 §1 0 6( , , ©) ¨( ¬, , ®, , , 3, ¯- ©) ©, where¨( ¬, , ®, , , , ¯)= ¯ 1 2 3+1 1 2 3 + =0 ´=0 µ =0 Â Â ´Â µ ´ µ cos( · ¬) cos( ¸ ´) cos( ¹ µ®) ´cos( · ) cos( ¸ ´) cos( ¹ µ) sin( ´ µ¯),· = º 1, ¸ ´=» º 2, ¹ µ =¼ º 3, ´ µ = ½·2 + ¸2´+ ¹2 µ ,Â = Ã1for = 0, 2for >0. The summation here is performed over the indices satisfying the condition +»+¼>0; the term corresponding to =»=¼= 0is singled out. 6.1.1-5. Domain: 0 £ ¬£ 1,0 £ £ 2,0 £ ®£ 3. Third boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:¾= Ä0( ¬, , ®) at ¯= 0 (initial condition),« ¾= Ä1( ¬, , ®) at ¯= 0 (initial condition),« ¿ ¾- Å1 ¾= 1( , ®, ¯) at ¬= 0 (boundary condition),« ¿ ¾+ Å2 ¾= 2( , ®, ¯) at ¬= 1(boundary condition),« À ¾- Å3 ¾= 3( ¬, ®, ¯) at = 0 (boundary condition),« À ¾+ Å4 ¾= 4( ¬, ®, ¯) at = 2(boundary condition),« Á ¾- Å5 ¾= 5( ¬, , ¯) at ®= 0 (boundary condition),« Á ¾+ Å6 ¾= 6( ¬, , ¯) at ®= 3(boundary condition). The solution ¾( ¬, , ®, ¯) is determined by the formula in Paragraph 6.1.1-4 where¨( ¬, , Æ, , ¯)=8 Ç + =1 Ç ´=1 Ç ¶µ =11È ´ µ½ ·2 + ¸2´+ ¹2 µsin( · ¬+ É ) sin( ¸ ´+ Ê ´) sin( ¹ µ®+ Ë µ ) ´sin( · Æ+ É ) sin( ¸ ´+ Ê ´) sin( ¹ µ+ Ë µ ) sin Ì" ¯ ½·2 + ¸2´+ ¹2 µ Í Page 396 ¡"£ withÉ =arctan · 1, Ê ´=arctan ¸ ´ 2, Ë µ =arctan ¹ µ 3,È ´ µ = ªÎ 1+( Å1 Å2+ ·2 )( Å1+ Å2) ( Å2 1+ ·2 )( Å2 2+ ·2 ) ° ªÎ 2+( Å3 Å4+ ¸2´)( Å3+ Å4) ( Å2 3+ ¸2´)( Å2 4+ ¸2´) ° ªÎ 3+( Å5 Å6+ ¹2 µ )( Å5+ Å6) ( Å2 5+ ¹2 µ )( Å2 6+ ¹2 µ ) °. Here, the · , ¸ ´, and ¹ µ are positive roots of the transcendental equations·2- Å1 Å2=( Å1+ Å2) ·cot( 1 ·), ¸2- Å3 Å4=( Å3+ Å4) ¸cot( 2 ¸), ¹2- Å5 Å6=( Å5+ Å6) ¹cot( 3 ¹). 6.1.1-6. Domain: 0 £ ¬£ 1,0 £ £ 2,0 £ ®£ 3. Mixed boundary value problems. 1 Ï. A rectangular parallelepiped is considered. The following conditions are prescribed:¾= Ä0( ¬, , ®) at ¯= 0 (initial condition),« ¾= Ä1( ¬, , ®) at ¯= 0 (initial condition),¾= 1( , ®, ¯) at ¬= 0 (boundary condition),¾= 2( , ®, ¯) at ¬= 1(boundary condition),« À ¾= 3( ¬, ®, ¯) at = 0 (boundary condition),« À ¾= 4( ¬, ®, ¯) at = 2(boundary condition),« Á ¾= 5( ¬, , ¯) at ®= 0 (boundary condition),« Á ¾= 6( ¬, , ¯) at ®= 3(boundary condition). Solution:¾( ¬, , ®, ¯)= «« ¯ §3 0 §2 0 §1 0 Ä0( Æ, , ) ¨( ¬, , ®, Æ, , , ¯) Æ + §3 0 §2 0 §1 0 Ä1( Æ, , ) ¨( ¬, , ®, Æ, , , ¯) Æ + 2 0 §3 0 §2 0 1( , , ©) ª «« Æ ¨( ¬, , ®, Æ, , , ¯- ©) ° ± =0 © - 2 0 §3 0 §2 0 2( , , ©) ª «« Æ ¨( ¬, , ®, Æ, , , ¯- ©) ° ± =§1 © - 2 0 §3 0 §1 0 3( Æ, , ©) ¨( ¬, , ®, Æ,0, , ¯- ©) Æ © + 2 0 §3 0 §1 0 4( Æ, , ©) ¨( ¬, , ®, Æ, 2, , ¯- ©) Æ © - 2 0 §2 0 §1 0 5( Æ, , ©) ¨( ¬, , ®, Æ, ,0, ¯- ©) Æ © + 2 0 §2 0 §1 0 6( Æ, , ©) ¨( ¬, , ®, Æ, , 3, ¯- ©) Æ ©. Here,¨( ¬, , ®, Æ, , , ¯)=2 1 2 3 Ç Ð+Ñ =1 Ç Ð´=0 Ç Ð µ =0 Â ´Â µÒ Ñ´ µ sin( · Ñ¬) cos( ¸ ´) cos( ¹ µ®) ´sin( · ÑÆ) cos( ¸ ´ Ó) cos( ¹ µ Ô ) sin( Ò Ñ´ µ¯), Page 397 where· Ñ = ºÕ 1, ¸ ´=» ºÕ 2, ¹ µ =¼ ºÕ 3,Ò Ñ´ µ = ½·2 Ñ + ¸2´+ ¹2 µ ,Â ´= Ã1for»= 0, 2for»>0. 2 Ï. A rectangular parallelepiped is considered. The following conditions are prescribed:¾= Ä0( ¬, , ®) at ¯= 0 (initial condition),« Ö ¾= Ä1( ¬, , ®) at ¯= 0 (initial condition),¾= ×1( , ®, ¯) at ¬= 0 (boundary condition),« ¿ ¾= ×2( , ®, ¯) at ¬= Õ 1(boundary condition),¾= ×3( ¬, ®, ¯) at = 0 (boundary condition),« À ¾= ×4( ¬, ®, ¯) at = Õ 2(boundary condition),¾= ×5( ¬, , ¯) at ®= 0 (boundary condition),« Á ¾= ×6( ¬, , ¯) at ®= Õ 3(boundary condition). Solution:¾( ¬, , ®, ¯)= «« ¯ Ø Ù3 0 Ø Ù2 0 Ø Ù1 0 Ä0( Æ, Ó, Ô ) Ú( ¬, , ®, Æ, Ó, Ô , Û) Ü Æ Ü ÓÜ Ô +Ø Ù3 0 Ø Ù2 0 Ø Ù1 0 Ä1( Æ, Ó, Ô ) Ú( ¬, , ®, Æ, Ó, Ô , Û) Ü Æ Ü ÓÜ Ô + 2Ø Ö 0 Ø Ù3 0 Ø Ù2 0 ×1( Ó, Ô , Ý) Þ ßß à Ú( ¬, , ®,à, Ó, Ô , Û- Ý) á â =0 Ü ÓÜ ÔÜ Ý + 2Ø Ö 0 Ø Ù3 0 Ø Ù2 0 ×2( Ó, Ô , Ý) Ú( ¬, , ®, Õ 1, Ó, Ô , Û- Ý) Ü ÓÜ ÔÜ Ý + 2Ø Ö 0 Ø Ù3 0 Ø Ù1 0 ×3(à, Ô , Ý) Þ ßß Ó Ú( ¬, , ®,à, Ó, Ô , Û- Ý) á ã =0 Üà Ü ÔÜ Ý + 2Ø Ö 0 Ø Ù3 0 Ø Ù1 0 ×4(à, Ô , Ý) Ú( ¬, , ®,à, Õ 2, Ô , Û- Ý) Üà Ü ÔÜ Ý + 2Ø Ö 0 Ø Ù2 0 Ø Ù1 0 ×5(à, Ó, Ý) Þ ßß ÔÚ( ¬, , ®,à, Ó, Ô , Û- Ý) á ä =0 Üà Ü ÓÜ Ý + å2Ø Ö 0 Ø Ù2 0 Ø Ù1 0 ×6(à, Ó, Ý) Ú( ¬, , ®,à, Ó, Õ 3, Û- Ý) Üà Ü ÓÜ Ý. Here,Ú( ¬, , ®,à, Ó, Ô , Û)=8å Õ 1 Õ 2 Õ 3 æ Ð+Ñ =1 æ Ðç=1 æ Ð¶è =11Ò Ñç è sin( é Ñ¬) sin( ê ç) sin( ¹ è®) ´sin( é Ñà) sin( ê ç Ó) sin( ¹ è Ô ) sin( å Ò Ñç èÛ), whereé Ñ = º(2 ë+ 1) 2 Õ 1, ê ç= º(2»+ 1) 2 Õ 2, ¹ è = º(2¼+ 1) 2 Õ 3, Ò Ñç è = ì é2 í+ ê2ç+ ¹2 è . Page 398 î"ð 6.1.2. Problems in Cylindrical Coordinates The three-dimensional wave equation in the cylindrical coordinate system is written asß2 ôß Û2= å2Þ1 õßß õ ö õß ôß õ ÷+1 õ 2 ß2 ôß ø2+ ß2 ôß ù2 á, õ = ú û2+ ü2. One-dimensional problems with axial symmetry that have solutions ô= ô( õ , Û) are considered in Subsection 4.2.1. Two-dimensional problems whose solutions have the form ô= ô( õ ,ø, Û) orô= ô( õ ,ù, Û) are discussed in Subsections 5.1.2 and 5.1.3. 6.1.2-1. Domain: 0 £ õ £ ý,0 £ø£ 2 þ,0 £ù£ ÿ. First boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:ô= 0( õ ,ø,ù) at Û= 0 (initial condition),ß ô= 1( õ ,ø,ù) at Û= 0 (initial condition),ô= 1(ø,ù, Û) at õ = ý(boundary condition),ô= 2( õ ,ø, Û) atù= 0 (boundary condition),ô= 3( õ ,ø, Û) atù= ÿ(boundary condition). Solution:ô( õ ,ø,ù, Û)= ßß Û Ø Ù0 Ø2 0 Ø 0 à 0(à, Ó, ) Ú( õ ,ø,ù,à, Ó, , Û) Üà Ü ÓÜ + 0 2 0 0 à 1(à, Ó, ) ( õ ,ø,ù,à, Ó, , ) Üà Ü ÓÜ - å2ý 0 0 2 0 1( Ó, , Ý) ( õ ,ø,ù,, Ó, , - Ý) = Ü ÓÜ Ü Ý + 2 0 2 0 0 2(, Ó, Ý) ( õ ,ø,ù,, Ó, , - Ý) =0 Ü Ü ÓÜ Ý - 2 0 2 0 0 3(, Ó, Ý) ( õ ,ø,ù,, Ó, , - Ý) = Ü Ü ÓÜ Ý. Here,( õ ,ø,ù,, Ó, , )=2þ ý2ÿæ í=0 æ ç=1 æ =1 í [ í( íçý)]2 íç í( íç õ ) í( íç) ´cos[ (ø- )] sin ö þùÿ ÷sin ö þ ÿ ÷sin ú íç , íç = 2íç+ 2þ2ÿ2, í= 1for = 0, 2for >0, where the !() are the Bessel functions (the prime denotes the derivative with respect to the argument) and the ! çare positive roots of the transcendental equation !( ý)= 0. 6.1.2-2. Domain: 0 £ õ £ ý,0 £ "£ 2 þ,0 £ #£ ÿ. Second boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:$= 0( õ , ", #) at = 0 (initial condition), $= 1( õ , ", #) at = 0 (initial condition), % $= 1( ", #, ) at õ = &(boundary condition), ' $= (2( õ , ", ) at #= 0 (boundary condition), ' $= (3( õ , ", ) at #= )(boundary condition). Page 399 Solution:$( õ , ", #, )= 0 2 * 0 + 0 ,0(, , -) ( õ , ", #,, , -, ) Ü Ü Ü - + 0 2 * 0 + 0 ,1(, , -) ( õ , ", #,, , -, ) Ü Ü Ü - + 2& . 0 0 2 * 0 (1( , -, Ý) ( õ , ", #, &, , -, - Ý) Ü Ü - Ü Ý - 2 . 0 2 * 0 + 0 (2(, , Ý) ( õ , ", #,, ,0, - Ý) Ü Ü Ü Ý + 2 . 0 2 * 0 + 0 (3(, , Ý) ( õ , ", #,, , ), - Ý) Ü Ü Ü Ý. Here,( õ , ", #,, , -, )= /&2)+2/2 &2æ 0 =11cos 1 / 2) 3cos 1 /) 3sin 1 / 4) 3 +1/)æ !=0 æ ç=1 æ =0 ! 2! ç !( ! ç õ ) !( ! ç) ( 2! ç&2- 2)[ !( ! ç&)]2cos[ ( "- )] cos 1 / 2) 3cos 1 /) 3sin( ! ç 4)! ç ,! ç = 5 62! ç+ 72 /2)2, != 1for 8= 0, 2for 8>0, where the 9 !( :) are the Bessel functions and the 6 ! çare positive roots of the transcendental equation9 ;!( 6 &)= 0. 6.1.2-3. Domain: 0 £ <£ &,0 £ "£ 2 /,0 £ #£ ). Third boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:$=,0( <, ", #) at 4= 0 (initial condition),=. $=,1( <, ", #) at 4= 0 (initial condition),=% $+71 $= (( ", #, 4) at <= &(boundary condition),=' $-72 $= (2( <, ", 4) at #= 0 (boundary condition),=' $+73 $= (3( <, ", 4) at #= )(boundary condition). The solution $( <, ", #, 4) is determined by the formula in Paragraph 6.1.2-2 where>( <, ", #, :, ?, -, 4)=1/æ @BA =0 æ @ç=1 æ @DC =1 A62 Aç9 A ( 6 Aç<) 9 A ( 6 Aç:) cos[ 8( "- ?)] E C ( #) E C ( -) sin( F Aç C4) ( 62 Aç G2+72 1 G2- 82)[ 9 A ( 6 AçG)]2 HE CH2F Aç C ,F Aç C = I J 62 Aç+ K2 C , E C ( #)=cos( K C#)+ 72K C sin( K C#), HE CH2= 73 2 K2 CK2 C +72 2K2 C +72 3+ 72 2 K2 C + L2 M1+ 72 2K2 C N . Here,0= 1and A = 2for 8= 1,2, OPOPO; the 9 A ( :) are the Bessel functions; and the 6 Açand K C are positive roots of the transcendental equations6 9 ; A ( 6 G)+71 9 A ( 6 G)= 0,tan( KL)K= 72+73K2-7273. Page 400 6.1.2-4. Domain: 0 £ £ ,0 £ £ 2 ,0 £ £ . Mixed boundary value problems. 1 . A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition),= 1( , , ) at = (boundary condition), = 2( , , ) at = 0 (boundary condition), = 3( , , ) at = (boundary condition). Solution:( , , , )= 0 2 0 0 0(, , ) ( , , ,, , , ) + 0 2 0 0 1(, , ) ( , , ,, , , ) - 2 0 0 2 0 1( , , ) ( , , ,, , , - ) ! " = - 2 0 2 0 0 2(, , ) ( , , ,, ,0, - ) + 2 0 2 0 0 3(, , ) ( , , ,, , , - ) . Here,( , , ,, , , )=1 2 # $&% =0 # $'=1 # $)( =0 * %* ( [ + , % ( - %')]2 . / %' (+ % ( - %') + % ( - %') ´cos[ 0( - )] cos 1 2 3cos 1 2 3sin 45 76 / %' ( 8 ,/ %' ( = -2 %'+ 22 22,* % = 91for 0= 0, 2for 0>0, where the + % () are the Bessel functions (the prime denotes the derivative with respect to the argument) and the - %'are positive roots of the transcendental equation + % ( - )= 0. 2 . A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition), := 1( , , ) at = (boundary condition),= 2( , , ) at = 0 (boundary condition),= 3( , , ) at = (boundary condition). Solution:( , , , )= 0 2 0 0 0(, , ) ( , , ,, , , ) + 0 2 0 0 1(, , ) ( , , ,, , , ) + 2 0 0 2 0 1( , , ) ( , , , , , , - ) + 2 0 2 0 0 2(, , ) ( , , ,, , , - )! ; =0 - 2 0 2 0 0 3(, , ) ( , , ,, , , - )! ; = . Page 401 Here,( , , ,, , , )=2 2 2# $)( =112sin 1 2 3sin 1 2 3sin 1 2 3 +2 # $% =0 # $'=1 # $)( =1 * %-2 %' ( -2 %'2- 02)[ + % ( - %')]2. / %' (+ % ( - %') + % ( - %') ´cos[ 0( - )] sin 1 2 3sin 1 2 3sin 45 6 / %' ( 8 ,/ %' ( = -2 %'+ 22 22,* % = 91for 0= 0, 2for 0>0, where the + % () are the Bessel functions and the - %'are positive roots of the transcendental equation+ , % ( - )= 0. 6.1.2-5. Domain: 1£ £ 2,0 £ £ 2 ,0 £ £ . First boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition),= 1( , , ) at = 1(boundary condition),= 2( , , ) at = 2(boundary condition),= 3( , , ) at = 0 (boundary condition),= 4( , , ) at = (boundary condition). Solution:( , , , )= 0 2 0 21 0(, , ) ( , , ,, , , ) + 0 2 0 21 1(, , ) ( , , ,, , , ) + 21 0 0 2 0 1( , , ) ( , , ,, , , - )! " = 1 - 22 0 0 2 0 2( , , ) ( , , ,, , , - )! " = 2 + 2 0 2 0 21 3(, , ) ( , , ,, , , - )! ; =0 - 2 0 2 0 21 4(, , ) ( , , ,, , , - )! ; = . Here,( , , ,, , , )= 2 # $<% =0 # $'=1 # $ ( =1 * %-2 %'+2 % ( - %'2)+2 % ( - %'1)- +2 % ( - %'2) = %'( )= %'() ´cos[ 0( - )] sin 1 2 3sin 1 2 3sin 45 . / %' ( 8 . / %' ( ,* % = 91for 0= 0, 2for 0¹ 0, / %' ( = -2 %'+ 22 22,= %'( )= + % ( - %'1) > % ( - %')- > % ( - %'1) + % ( - %'), Page 402 2= 2 where the + % ( ) and > % ( ) are the Bessel functions, and the - %'are positive roots of the transcen- dental equation+ % ( - 1) > % ( - 2)- > % ( - 1) + % ( - 2)= 0. 6.1.2-6. Domain: 1£ £ 2,0 £ £ 2 ,0 £ £ . Second boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition), := 1( , , ) at = 1(boundary condition), := 2( , , ) at = 2(boundary condition), = 3( , , ) at = 0 (boundary condition), = 4( , , ) at = (boundary condition). Solution:( , , , )= 0 2 0 21 0(, , ) ( , , ,, , , ) + 0 2 0 21 1(, , ) ( , , ,, , , ) - 21 0 0 2 0 1( , , ) ( , , , 1, , , - ) + 22 0 0 2 0 2( , , ) ( , , , 2, , , - ) - 2 0 2 0 21 3(, , ) ( , , ,, ,0, - ) + 2 0 2 0 21 4(, , ) ( , , ,, , , - ) . Here,( , , ,, , , )= ( 2 2- 2 1) +2 2( 2 2- 2 1) # $)( =112cos 1 2 3cos 1 2 3sin 1 2 3 +1 # $% =0 # $'=1 # $)( =0 * %* (-2 %'= %'( )= %'() ( -2 %'2 2- 02)=2 %'( 2)-( -2 %'2 1- 02)=2 %'( 1) ´cos[ 0( - )] cos 1 2 3cos 1 2 3sin 4 . / %' ( 8 . / %' ( , where* % = 91for 0= 0, 2for 0¹ 0, / %' ( = -2 %'+ 22 22,= %'( )= + , % ( - %'1) > % ( - %')- > , % ( - %'1) + % ( - %'); the + % ( ) and > % ( ) are the Bessel functions, and the - %'are positive roots of the transcendental equation+ , % ( - 1) > , % ( - 2)- > , % ( - 1) + , % ( - 2)= 0. Page 403 6.1.2-7. Domain: 1£ £ 2,0 £ £ 2 ,0 £ £ . Third boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition), :-21 = 1( , , ) at = 1(boundary condition), :+22 = 2( , , ) at = 2(boundary condition), -23 = 3( , , ) at = 0 (boundary condition), +24 = 4( , , ) at = (boundary condition). The solution ( , , , ) is determined by the formula in Paragraph 6.1.2-6 where( , , ,, , , )=1 # $&% =0 # $'=1 # $@? =1* %-2 %'AB ?A26 -2 %'+ /2 ? ´ = %'( )= %'() cos[ 0( - )] B ? ( ) B ? ( ) sin 45 6 -2 %'+ /2 ? 8 (22 2 2 2+ -2 %'2 2- 02)=2 %'( 2)-(22 1 2 1+ -2 %'2 1- 02)=2 %'( 1). Here,* % = 91for 0= 0, 2for 0¹ 0, = %'( )= CD- %'+ , % ( - %'1)-21 + % ( - %'1) E7> % ( - %') - CD- %'> , % ( - %'1)-21 > % ( - %'1) E + % ( - %'),B ? ( )=cos( / ?)+ 23/ ? sin( / ?), AB ?A2= 24 2 /2 ?/2 ? +22 3/2 ? +22 4+ 23 2 /2 ? + 2 11 + 22 3/2 ?3, where the + % ( ) and > % ( ) are the Bessel functions; the - %'are positive roots of the transcendental equationC - + , % ( - 1)-21 + % ( - 1)E C - > , % ( - 2)+22 > % ( - 2)E =C - > , % ( - 1)-21 > % ( - 1)E C - + , % ( - 2)+22 + % ( - 2)E; and the / ? are positive roots of the transcendental equationtan( /)/= 23+24/2-2324. 6.1.2-8. Domain: 1£ £ 2,0 £ £ 2 ,0 £ £ . Mixed boundary value problems. 1 . A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition),= 1( , , ) at = 1(boundary condition),= 2( , , ) at = 2(boundary condition), = 3( , , ) at = 0 (boundary condition), = 4( , , ) at = (boundary condition). Page 404 2= Solution:( , , , )= 0 2 0 21 0(, , ) ( , , ,, , , ) + 0 2 0 21 1(, , ) ( , , ,, , , ) + 21 0 0 2 0 1( , , ) ( , , ,, , , - )! " = 1 - 22 0 0 2 0 2( , , ) ( , , ,, , , - )! " = 2 - 2 0 2 0 21 3(, , ) ( , , ,, ,0, - ) + 2 0 2 0 21 4(, , ) ( , , ,, , , - ) . Here,( , , ,, , , )= 4 # $% =0 # $'=1 # $)( =0 * %* (-2 %'+2 % ( - %'2)+2 % ( - %'1)- +2 % ( - %'2) = %'( )= %'() ´cos[ 0( - )] cos 1 2 3cos 1 2 3sin 4 . / %' ( 8 . / %' ( ,* % = 91for 0= 0, 2for 0¹ 0, / %' ( = -2 %'+ 22 22,= %'( )= + % ( - %'1) > % ( - %')- > % ( - %'1) + % ( - %'), where the + % ( ) and > % ( ) are the Bessel functions, and the - %'are positive roots of the transcen- dental equation+ % ( - 1) > % ( - 2)- > % ( - 1) + % ( - 2)= 0. 2 . A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition), := 1( , , ) at = 1(boundary condition), := 2( , , ) at = 2(boundary condition),= 3( , , ) at = 0 (boundary condition),= 4( , , ) at = (boundary condition). Solution:( , , , )= 0 2 0 21 0(, , ) ( , , ,, , , ) + 0 2 0 21 1(, , ) ( , , ,, , , ) - 21 0 0 2 0 1( , , ) ( , , , 1, , , - ) + 22 0 0 2 0 2( , , ) ( , , , 2, , , - ) + 2 0 2 0 21 3(, , ) ( , , ,, , , - )! ; =0 - 2 0 2 0 21 4(, , ) ( , , ,, , , - )! ; = . Page 405 Here,( , , ,, , , )=2 2( 2 2- 2 1) # $)( =112sin 1 2 3sin 1 2 3sin 1 2 3 +2 # $&% =0 # $'=1 # $ ( =1 * %-2 %'= %'( )= %'() ( -2 %'2 2- 02)=2 %'( 2)-( -2 %'2 1- 02)=2 %'( 1) ´cos[ 0( - )] sin 1 2 3sin 1 2 3sin 45 . / %' ( 8 . / %' ( , where* % = 91for 0= 0, 2for 0¹ 0, / %' ( = -2 %'+ 22 22,= %'( )= + , % ( - %'1) > % ( - %')- > , % ( - %'1) + % ( - %'); the + % ( ) and > % ( ) are the Bessel functions, and the - %'are positive roots of the transcendental equation+ , % ( - 1) > , % ( - 2)- > , % ( - 1) + , % ( - 2)= 0. 6.1.2-9. Domain: 0 £ £ ,0 £ £ 0,0 £ £ . First boundary value problem. A cylindrical sector of ®nite thickness is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition),= 1( , , ) at = (boundary condition),= 2( , , ) at = 0 (boundary condition),= 3( , , ) at = 0(boundary condition),= 4( , , ) at = 0 (boundary condition),= 5( , , ) at = (boundary condition). Solution:( , , , )= 0 F0 0 0 0(, , ) ( , , ,, , , ) + 0 F0 0 0 1(, , ) ( , , ,, , , ) - 2 0 0 F0 0 1( , , ) ( , , ,, , , - )! " = + 2 0 0 0 2(, , )1 ( , , ,, , , - )! G =0 - 2 0 0 0 3(, , )1 ( , , ,, , , - )! G = F0 + 2 0 F0 0 0 4(, , ) ( , , ,, , , - )! ; =0 - 2 0 F0 0 0 5(, , ) ( , , ,, , , - )! ; = . Page 406 Here,( , , ,, , , )=82H 0 # $&% =1 # $'=1 # $)( =1 + % I F0( - %') + % I F0( - %') [ + , % I F0( - %')]2sin 1 0 0 3sin 1 0 0 3 ´sin 1 2 3sin 1 2 3sin 45 6 -2 %'+22 2 J2 86 -2 %'+22 2 J2, where the + % I F0( ) are the Bessel functions and the - %'are positive roots of the transcendental equation + % I F0( - )= 0. 6.1.2-10. Domain: 0 £ £ ,0 £ £ 0,0 £ £ . Mixed boundary value problem. A cylindrical sector of ®nite thickness is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition),= 1( , , ) at = (boundary condition),= 2( , , ) at = 0 (boundary condition),= 3( , , ) at = 0(boundary condition), = 4( , , ) at = 0 (boundary condition), = 5( , , ) at = (boundary condition). Solution:( , , , )= 0 F0 0 0 0(, , ) ( , , ,, , , ) + 0 F0 0 0 1(, , ) ( , , ,, , , ) - 2 0 0 F0 0 1( , , ) ( , , ,, , , - )! " = + 2 0 0 0 2(, , )1 ( , , ,, , , - )! G =0 - 2 0 0 0 3(, , )1 ( , , ,, , , - )! G = F0 - 2 0 F0 0 0 4(, , ) ( , , ,, ,0, - ) + 2 0 F0 0 0 5(, , ) ( , , ,, , , - ) . Here,( , , ,, , , )=42H 0 # $&% =1 # $'=1 # $ ( =0 * (+ % I F0( - %') + % I F0( - %') [ + , % I F0( - %')]2sin 1 0 0 3sin 1 0 0 3 ´cos 1 2 3cos 1 2 3sin 45 6 -2 %'+22 2 J2 86 -2 %'+22 2 J2, where*0= 1and* ( = 2for2³ 1; the + % I F0( ) are the Bessel functions; and the - %'are positive roots of the transcendental equation + % I F0( - )= 0. Page 407 6.1.3. Problems in Spherical Coordinates The three-dimensional wave equation in the spherical coordinate system is represented as2 2= 2 12 1 2 3+12sin K K 1sin K K 3+12sin2K 2 2 !, = 6 L2+ M2+ 2. One-dimensional problems with central symmetry that have solutions = ( , ) are considered in Subsection 4.2.3. 6.1.3-1. Domain: 0 £ £ ,0 £ K£ ,0 £ £ 2 . First boundary value problem. A spherical domain is considered. The following conditions are prescribed:= 0( , K, ) at = 0 (initial condition), = 1( , K, ) at = 0 (initial condition),= ( K, , ) at = (boundary condition). Solution:( , K, , )= 2 0 0 0 0(, , ) ( , K, ,, , , )2sin + 2 0 0 0 1(, , ) ( , K, ,, , , )2sin - 22 0 2 0 0 ( , , ) ( , K, ,, , , - )! " = sin , where( , K, ,, , , )=1 2 2. # $&% =0 # $'=1 %$)( =0* ( N%' (+ % +1 I2( / %') + % +1 I2( / %') ´ O (% (cos K) O (% (cos ) cos[2( - )] sin( / %' ),* ( = 91for2= 0, 2for2¹ 0, N%' ( =(2 0+ 1)( 0-2)! ( 0+2)! CP+ , % +1 I2( / %') E2/ %'. Here, the + % +1 I2( ) are the Bessel functions, the O (% ( -) are the associated Legendre functions expressed in terms of the Legendre polynomials O % ( -) asO (% ( -)=(1 - -2) (I2 ( - (O % ( -), O % ( -)=10!2 % % - % ( -2- 1) % , and the / %'are positive roots of the transcendental equation + % +1 I2( /)= 0. 6.1.3-2. Domain: 0 £ £ ,0 £ K£ ,0 £ £ 2 . Second boundary value problem. A spherical domain is considered. The following conditions are prescribed:= 0( , K, ) at = 0 (initial condition), = 1( , K, ) at = 0 (initial condition), := ( K, , ) at = (boundary condition). Page 408 Solution:( , K, , )= 2 0 0 0 0(, , ) ( , K, ,, , , )2sin + 2 0 0 0 1(, , ) ( , K, ,, , , )2sin + 22 0 2 0 0 ( , , ) ( , K, , , , , - ) sin , where( , K, ,, , , )=3 4 3+1 2 . # $&% =0 # $'=1 %$Q( =0* ( N%' (+ % +1 I2( / %') + % +1 I2( / %') ´ O (% (cos K) O (% (cos ) cos[2( - )] sin( / %' ),* ( = 91for2= 0, 2for2¹ 0, N%' ( = / %'(2 0+ 1)( 0-2)! ( 0+2)!C 2 /2 %'- 0( 0+ 1)E C + % +1 I2( / %')E2. Here, the + % +1 I2( ) are the Bessel functions, the O (% ( -) are the associated Legendre functions (see Paragraph 6.1.3-1), and the / %'are positive roots of the transcendental equation 2 / + , % +1 I2( /)- + % +1 I2( /)= 0.RTS Reference : M. M. Smirnov (1975). 6.1.3-3. Domain: 0 £ £ ,0 £ K£ ,0 £ £ 2 . Third boundary value problem. A spherical domain is considered. The following conditions are prescribed:= 0( , K, ) at = 0 (initial condition), = 1( , K, ) at = 0 (initial condition), :+2 = ( K, , ) at = (boundary condition). The solution ( , K, , ) is determined by the formula in Paragraph 6.1.3-2 where( , K, ,, , , )=1 2 . # $% =0 # $'=1 %$? =0* ? N%' ?+ % +1 I2( / %') + % +1 I2( / %') ´ O ?% (cos K) O ?% (cos ) cos[ U( - )] sin( / %' ),* ? = 91for U= 0, 2for U¹ 0, N%' ? = / %'(2 0+ 1)( 0- U)! ( 0+ U)!C 2 /2 %'+(2 + 0)(2 - 0- 1)E C + % +1 I2( / %')E2. Here, the + % +1 I2( ) are the Bessel functions, the O ?% ( -) are the associated Legendre functions (see Paragraph 6.1.3-1), and the / %'are positive roots of the transcendental equation/ + , % +1 I2( /)+ 42 -1 2 8+ % +1 I2( /)= 0. 6.1.3-4. Domain: 1£ £ 2,0 £ K£ ,0 £ £ 2 . First boundary value problem. A spherical layer is considered. The following conditions are prescribed:= 0( , K, ) at = 0 (initial condition), = 1( , K, ) at = 0 (initial condition),= 1( K, , ) at = 1(boundary condition),= 2( K, , ) at = 2(boundary condition). Page 409 Solution:( , K, , )= 2 0 0 21 0(, , ) ( , K, ,, , , )2sin + 2 0 0 21 1(, , ) ( , K, ,, , , )2sin + 22 1 0 2 0 0 1( , , ) ( , K, ,, , , - )! " = 1sin - 22 2 0 2 0 0 2( , , ) ( , K, ,, , , - )! " = 2sin , where( , K, ,, , , )= 8 . # $&% =0 # $'=1 %$)( =0* ( N%' (= % +1 I2( / %')= % +1 I2( / %') ´ O (% (cos K) O (% (cos ) cos[2( - )] sin( / %' ). Here,= % +1 I2( / %')= + % +1 I2( / %'1) > % +1 I2( / %')- > % +1 I2( / %'1) + % +1 I2( / %'),* ( = 91for2= 0, 2for2¹ 0, N%' ( = / %'(2 0+ 1)( 0-2)! +2 % +1 I2( / %'2) ( 0+2)!C +2 % +1 I2( / %'1)- +2 % +1 I2( / %'2)E, where the + % +1 I2( ) are the Bessel functions, the O (% ( -) are the associated Legendre functions expressed in terms of the Legendre polynomials O % ( -) asO (% ( -)=(1 - -2) (I2 ( - (O % ( -), O % ( -)=10!2 % % - % ( -2- 1) % , and the / %'are positive roots of the transcendental equation= % +1 I2( /2)= 0. 6.1.3-5. Domain: 1£ £ 2,0 £ K£ ,0 £ £ 2 . Second boundary value problem. A spherical layer is considered. The following conditions are prescribed:= 0( , K, ) at = 0 (initial condition), = 1( , K, ) at = 0 (initial condition), := 1( K, , ) at = 1(boundary condition), := 2( K, , ) at = 2(boundary condition). Solution:( , K, , )= 2 0 0 21 0(, , ) ( , K, ,, , , )2sin + 2 0 0 21 1(, , ) ( , K, ,, , , )2sin - 22 1 0 2 0 0 1( , , ) ( , K, , 1, , , - ) sin + 22 2 0 2 0 0 2( , , ) ( , K, , 2, , , - ) sin , Page 410 where( , K, ,, , , )=3 4 ( 3 2- 3 1)+1 4 . # $&% =0 # $'=1 %$)( =0 * (N%' (= % +1 I2( / %')= % +1 I2( / %') ´ O (% (cos K) O (% (cos ) cos[2( - )] sin( / %' ),* ( = 91for2= 0, 2for2¹ 0, N%' ( = / %'( 0+2)! (2 0+ 1)( 0-2)! 21 =2 % +1 I2( / %') ,= % +1 I2( / %')= / %'+ , % +1 I2( / %'1)-1 2 1 + % +1 I2( / %'1)! > % +1 I2( / %') - / %'> , % +1 I2( / %'1)-1 2 1 > % +1 I2( / %'1) ! + % +1 I2( / %'). Here, the + % +1 I2( ) and > % +1 I2( ) are the Bessel functions, the O (% ( -) are the associated Legendre functions (see Paragraph 6.1.3-4), and the / %'are positive roots of the transcendental equation/= , % +1 I2( /2)-1 2 2 = % +1 I2( /2)= 0. 6.1.3-6. Domain: 1£ £ 2,0 £ K£ ,0 £ £ 2 . Third boundary value problem. A spherical layer is considered. The following conditions are prescribed:= 0( , K, ) at = 0 (initial condition), = 1( , K, ) at = 0 (initial condition), :-21 = 1( K, , ) at = 1(boundary condition), :+22 = 2( K, , ) at = 2(boundary condition). The solution ( , K, , ) is determined by the formula in Paragraph 6.1.3-5 where( , K, ,, , , )=1 4 . # $&% =0 # $'=1 %$@? =0 * ?N%' ?= % +1 I2( / %')= % +1 I2( / %') ´ O ?% (cos K) O ?% (cos ) cos[ U( - )] sin( / %' ). Here,* ? = 91for U= 0, 2for U¹ 0, N%' ? = / %'( 0+ U)! (2 0+ 1)( 0- U)! 21 =2 % +1 I2( / %') ,= % +1 I2( /)= /+ , % +1 I2( /1)- 121+1 2 1 3 + % +1 I2( /1)! > % +1 I2( /) - /> , % +1 I2( /1)- 121+1 2 1 3 > % +1 I2( /1)! + % +1 I2( /), where the + % +1 I2( ) and > % +1 I2( ) are the Bessel functions, the O ?% ( -) are the associated Legendre functions (see Paragraph 6.1.3-4), and the / %'are positive roots of the transcendental equation/= , % +1 I2( /2)+ 122-1 2 2 3 = % +1 I2( /2)= 0. Page 411 6.2. Nonhomogeneous Wave EquationV2 WV X2= Y2 Z 3 W+ [( \, ], ^, X) 6.2.1. Problems in Cartesian Coordinates 6.2.1-1. Domain: - _< L< _,- _< M< _,- _< `< _. Cauchy problem. Initial conditions are prescribed: a = b( L, M, `) at c= 0,d e a = f( L, M, `) at c= 0. Solution:a ( L, M, `, c)=1 4 g h ddc i i : = j e b( k, l, m)n o p+1 4 g h i i : = j e f( k, l, m)n o p +1 4 g h2i i i : £ j e1n q r k, l, m, c- nh s o ko lo m, n= t( k- u)2+( l- v)2+( m- `)2, where the integration is performed over the surface of the sphere ( n= h c) and the volume of the sphere ( n£ h c) with center at ( u, v, `).wTS Reference : N. S. Koshlyakov, E. B. Glizer, and M. M. Smirnov (1970). 6.2.1-2. Domain: 0 £ u£ x1,0 £ v£ x2,0 £ `£ x3. Different boundary value problems. 1 y. The solution of the ®rst boundary value problem for a parallelepiped is given by the formula from Paragraph 6.1.1-3 with the additional termi e 0 i z1 0 i z2 0 i z3 0 q( k, l, m, {) |( u, v, `, k, l, m, c- {)o mo lo ko {, which allows for the equation's nonhomogeneity; this term is the solution of the nonhomogeneous equation with homogeneous initial and boundary conditions.2y. The solution of the second boundary value problem for a parallelepiped is given by the formula from Paragraph 6.1.1-4 with the additional term speci®ed in Paragraph 6.2.1-2, Item 1 y; the Green's function is taken from Paragraph 6.1.1-4. 3 y. The solution of the third boundary value problem for a parallelepiped is the sum of the so- lution of the homogeneous equation with nonhomogeneous initial and boundary conditions (see Paragraph 6.1.1-5) and the solution of the nonhomogeneous equation with homogeneous initial and boundary conditions. The latter solution is given by the formula from Paragraph 6.2.1-2, Item 1 y, in which one should substitute the Green's function from Paragraph 6.1.1-5. 4 y. The solutions of mixed boundary value problems for a parallelepiped are given by the formulas from Paragraph 6.1.1-6 to which one should add the term speci®ed in Paragraph 6.2.1-2, Item 1 y. 6.2.2. Problems in Cylindrical Coordinates A three-dimensional nonhomogeneous wave equation in the cylindrical coordinate system is written asd2 adc2= h2 }1n ddn ~ n d adn +1n2 d2 ad 2+ d2 ad`2 +q( n, , `, c). Page 412 6.2.2-1. Domain: 0 £ n£ ,0 £ £ 2 ,0 £ £ x. Different boundary value problems. 1 y. The solution of the ®rst boundary value problem for a circular cylinder of ®nite length is given by the formula from Paragraph 6.1.2-1 with the additional termi 0 i z0 i2 0 i 0 q( k, l, m, {) |( n, , , k, l, m, - {) ko ko lo mo {, ( 1) which allows for the equation's nonhomogeneity. 2 y. The solution of the second boundary value problem for a circular cylinder of ®nite length is given by the formula from Paragraph 6.1.2-2 with the additional term (1). 3 y. The solution of the third boundary value problem for a circular cylinder of ®nite length is the sum of the solution speci®ed in Paragraph 6.1.2-3 and expression (1). 4 y. The solutions of mixed boundary value problems for a circular cylinder of ®nite length are given by the formulas from Paragraph 6.1.2-4 with additional terms of the form (1). 6.2.2-2. Domain: 1£ n£ 2,0 £ £ 2 ,0 £ £ x. Different boundary value problems. 1 y. The solution of the ®rst boundary value problem for a hollow cylinder of ®nite dimensions is given by the formula from Paragraph 6.1.2-5 with the additional termi 0 i z0 i2 0 i 21 q( k, l, m, {) |( n, , , k, l, m, - {) ko ko lo mo {, ( 2) which allows for the equation's nonhomogeneity. 2 y. The solution of the second boundary value problem for a hollow cylinder of ®nite dimensions is given by the formula from Paragraph 6.1.2-6 with the additional term (2). 3 y. The solution of the third boundary value problem for a hollow cylinder of ®nite dimensions is the sum of the solution speci®ed in Paragraph 6.1.2-7 and expression (2). 4 y. The solutions of mixed boundary value problems for a hollow cylinder of ®nite dimensions are given by the formulas from Paragraph 6.1.2-8 with additional terms of the form (2). 6.2.2-3. Domain: 0 £ n£ ,0 £ £ 0,0 £ £ x. Different boundary value problems. 1 y. The solution of the ®rst boundary value problem for a cylindrical sector of ®nite thickness is given by the formula from Paragraph 6.1.2-9 with the additional termi 0 i z0 i 0 0 i 0 q( k, l, m, {) |( n, , , k, l, m, - {) ko ko lo mo {, ( 3) which allows for the equation's nonhomogeneity. 2 y. The solution of a mixed boundary value problem for a cylindrical sector of ®nite thickness is given by the formula from Paragraph 6.1.2-10 with the additional term (2). 6.2.3. Problems in Spherical Coordinates A three-dimensional nonhomogeneous wave equation in the spherical coordinate system is repre- sented as2 2= 2 }1n2 n ~ n2 n +1n2sin ~sin +1n2sin2 2 2 +q( n, , , ). Page 413 6.2.3-1. Domain: 0 £ n£ ,0 £ £ ,0 £ £ 2 . Boundary value problem. 1 y. The solution of the ®rst boundary value problem for a sphere is given by the formula from Paragraph 6.1.3-1 with the additional termi0 i2 0 i 0 i0 q( k, l, m, {) |( n, , , k, l, m, - {) k2sin lo ko lo mo {, ( 1) which allows for the equation's nonhomogeneity. 2 y. The solution of the second boundary value problem for a sphere is given by the formula from Paragraph 6.1.3-2 with the additional term (1). 3 y. The solution of the third boundary value problem for a sphere is the sum of the solution speci®ed in Paragraph 6.1.3-3 and expression (1). 6.2.3-2. Domain: 1£ n£ 2,0 £ £ ,0 £ £ 2 . Boundary value problems. 1 y. The solution of the ®rst boundary value problem for a spherical layer is given by the formula from Paragraph 6.1.3-4 with the additional termi0 i2 0 i 0 i21 q( k, l, m, {) |( n, , , k, l, m, - {) k2sin lo ko lo mo {, ( 2) which allows for the equation's nonhomogeneity. 2 y. The solution of the second boundary value problem for a spherical layer is given by the formula from Paragraph 6.1.3-5 with the additional term (2). 3 y. The solution of the third boundary value problem for a spherical layer is the sum of the solution speci®ed in Paragraph 6.1.3-6 and expression (2). 6.3. Equations of the Form V2 V 2= 2 3 ± + ( , , ¡, ) 6.3.1. Problems in Cartesian Coordinates Athree-dimensional nonhomogeneous Klein±Gordon equation in the rectangular Cartesian system of coordinates has the form2 2= 2~ 2 u2+ 2 v2+ 2 2 - ¢ + £( u, v, , ). 6.3.1-1. Fundamental solutions. 1 y. For ¢= - ¤2<0,¥ ¥( u, v, , )=1 4 2 } ¦( - § ¨ )§- ¤ ©1 ª ¤ « 2- §2¨ 2 ¬« 2- §2¨ 2 ( - § ¨ ), where §= « ®2+ ¯2+ 2,¦( °) is the Dirac delta function,( °) is the Heaviside unit step function, and©1( ) is the modi®ed Bessel function. Page 414 2 ². For ¢= ¤2>0,¥ ¥( ®, ¯, , )=1 4 2 ³ ¦( - § ¨ )§- ¤ ´1 ª ¤ « 2- §2¨ 2 ¬« 2- §2¨ 2 ( - § ¨ ), where´1( ) is the Bessel function.µT¶ Reference : V . S. Vladimirov, V . P. Mikhailov, A. A. Vasharin, et al. (1974). 6.3.1-2. Domain: - ·< ®< ·,- ·< ¯< ·,- ·< < ·. Cauchy problem. Initial conditions are prescribed:= ¸( ®, ¯, ) at = 0, = ¹( ®, ¯, ) at = 0. Let = 1and £( ®, ¯, , )º 0. 1 ². Solution for ¢= - ¤2<0:( ®, ¯, , )= ³1 º »0 §2©0 ª ¤ ¼ 2- §2 ¬7½ ¾ ¿¸( ®, ¯, À) Á Â § Ã +1 º»0 §2©0 ª ¤ ¼ 2- §2 ¬ ½ ¾¿¹( ®, ¯, À)Á Â §. Here,©0( À) is the modi®ed Bessel function and ½ ¾¿TÄ( ®, ¯, À)Áis the average of Ä( ®, ¯, À) over the spherical surface with center at ( ®, ¯, À) and radius §:½ ¾ ¿Ä( ®, ¯, À) Á=1 4 Å º2 0 º 0 Ä( ®+ §sin Æcos Ç, ¯+ §sin Æsin Ç, À+ §cos Æ) sin Æ Â Æ Â Ç. 2 ². Solution for È= É2>0:Ê( ®, ¯, À, )= ³1 º»0 Ë2´0 ª É ¼ 2-Ë2 ¬ ½ ¾¿¸( ®, ¯, À)Á ÂË Ã +1 º»0 Ë2´0 ª É¼ 2-Ë2 ¬ ½ ¾¿¹( ®, ¯, À)Á ÂË, where´0( À) is the Bessel function.µT¶ Reference : V . I. Smirnov (1974, V ol. 2). 6.3.1-3. Domain: 0 £ ®£ Ì1,0 £ ¯£ Ì2,0 £ À£ Ì3. First boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:Ê= ¸0( ®, ¯, À) at = 0 (initial condition),» Ê= ¸1( ®, ¯, À) at = 0 (initial condition),Ê= ¹1( ¯, À, ) at ®= 0 (boundary condition),Ê= ¹2( ¯, À, ) at ®= Ì1(boundary condition),Ê= ¹3( ®, À, ) at ¯= 0 (boundary condition),Ê= ¹4( ®, À, ) at ¯= Ì2(boundary condition),Ê= ¹5( ®, ¯, ) at À= 0 (boundary condition),Ê= ¹6( ®, ¯, ) at À= Ì3(boundary condition). Page 415 Solution:Ê( ®, ¯, À, )= º Í3 0 º Í2 0 º Í1 0 ¸0( °, Î, Ï) Ð( ®, ¯, À, °, Î, Ï, ) Â ° Â Î Â Ï +ºÍ3 0 ºÍ2 0 ºÍ1 0 ¸1( °, Î, Ï) Ð( ®, ¯, À, °, Î, Ï, ) Â ° Â Î Â Ï + Ñ2º »0 º Í3 0 º Í2 0 ¹1( Î, Ï, Ò)³ ° Ð( ®, ¯, À, °, Î, Ï, - Ò) Ã Ó =0 Â Î Â Ï Â Ò - Ñ2º»0 ºÍ3 0 ºÍ2 0 ¹2( Î, Ï, Ò)³ ° Ð( ®, ¯, À, °, Î, Ï, - Ò) Ã Ó =Í1 Â Î Â Ï Â Ò + Ñ2º »0 º Í3 0 º Í1 0 ¹3( °, Ï, Ò)³ Î Ð( ®, ¯, À, °, Î, Ï, - Ò) Ã Ô =0 Â ° Â Ï Â Ò - Ñ2º »0 º Í3 0 º Í1 0 ¹4( °, Ï, Ò)³ Î Ð( ®, ¯, À, °, Î, Ï, - Ò) Ã Ô =Í2 Â ° Â Ï Â Ò + Ñ2º»0 ºÍ2 0 ºÍ1 0 ¹5( °, Î, Ò)³ Ï Ð( ®, ¯, À, °, Î, Ï, - Ò) Ã Õ =0 Â ° Â Î Â Ò - Ñ2º »0 º Í2 0 º Í1 0 ¹6( °, Î, Ò)³ Ï Ð( ®, ¯, À, °, Î, Ï, - Ò) Ã Õ =Í3 Â ° Â Î Â Ò +º»0 ºÍ3 0 ºÍ2 0 ºÍ1 0 Ö( °, Î, Ï, Ò) Ð( ®, ¯, À, °, Î, Ï, - Ò) Â ° Â Î Â Ï Â Ò. Here,Ð( ®, ¯, À, °, Î, Ï, )=8Ì1 Ì2 Ì3 × Ø&Ù =1 × ØÚ=1 × Ø)Û =11¼ Ü ÙÚ Û sin( Ý Ù Þ ) sin( ß Ú à) sin( á ÛÀ) ´sin( Ý Ù â ) sin( ß ÚÎ) sin( á ÛÏ) sin ãH7äÜ ÙÚ Û¬, whereÝ Ù = å ÅÌ1, ß Ú= æ ÅÌ2, á Û = ç ÅÌ3,Ü ÙÚ Û = Ñ2( Ý2 Ù + ß2Ú+ á2 Û )+ È. 6.3.1-4. Domain: 0 £ Þ £ Ì1,0 £ à£ Ì2,0 £ À£ Ì3. Second boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:Ê= è0( Þ , à, À) at é= 0 (initial condition),ê» Ê= è1( Þ , à, À) at é= 0 (initial condition),ê ëÊ= ì1( à, À, é) at Þ = 0 (boundary condition),ê ëÊ= ì2( à, À, é) at Þ = Ì1(boundary condition),ê íÊ= ì3( Þ , À, é) at à= 0 (boundary condition),ê íÊ= ì4( Þ , À, é) at à= Ì2(boundary condition),ê îÊ= ì5( Þ , à, é) at À= 0 (boundary condition),ê îÊ= ì6( Þ , à, é) at À= Ì3(boundary condition). Page 416 ïñ ô Solution:Ê( Þ , à, À, é)= êêé û ü3 0 û ü2 0 û ü1 0 è0( â , ý, þ) ÿ( Þ , à, , â , ý, þ, é) âý þ +û ü3 0 û ü2 0 û ü1 0 è1( â , ý, þ) ÿ( Þ , à, , â , ý, þ, é) âý þ - 2û 0 û ü3 0 û ü2 0 ì1( ý, þ, ) ÿ( Þ , à, ,0, ý, þ, é- ) ý þ + 2û 0 û ü3 0 û ü2 0 ì2( ý, þ, ) ÿ( Þ , à, , 1, ý, þ, é- ) ý þ - 2û 0 û ü3 0 û ü1 0 ì3( â , þ, ) ÿ( Þ , à, , â ,0, þ, é- ) âþ + 2û 0 û ü3 0 û ü1 0 ì4( â , þ, ) ÿ( Þ , à, , â , 2, þ, é- ) âþ - 2û 0 û ü2 0 û ü1 0 ì5( â , ý, ) ÿ( Þ , à, , â , ý,0, é- ) âý + 2û 0 û ü2 0 û ü1 0 ì6( â , ý, ) ÿ( Þ , à, , â , ý, 3, é- ) âý +û 0 û ü3 0 û ü2 0 û ü1 0Ö( â , ý, þ, ) ÿ( Þ , à, , â , ý, þ, é- ) âý þ , whereÿ( Þ , à, , â , ý, þ, é)=sin ãHé ¬1 2 3 +11 2 3× Ù =0 × Ú=0 × Û =0 Ù Ú Û ÙÚ Û cos( Ù ) cos( Ú ) cos( Û) ´cos( Ù ) cos( Úý) cos( Ûþ) sin ÙÚ Û¬, Ù = 1, Ú= 2, Û = ç 3, ÙÚ Û = 2( 2 Ù + 2Ú+ 2 Û )+ , Ù = 1for= 0, 2for>0. The summation is performed over the indices satisfying the condition++ç>0; the term corresponding to==ç= 0is singled out. 6.3.1-5. Domain: 0 £ £ 1,0 £ £ 2,0 £ £ 3. Third boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition), - 1 = 1( , , ) at = 0 (boundary condition), + 2 = 2( , , ) at = 1(boundary condition), - 3 = 3( , , ) at = 0 (boundary condition), + 4 = 4( , , ) at = 2(boundary condition), !- 5 = 5( , , ) at = 0 (boundary condition), !+ 6 = 6( , , ) at = 3(boundary condition). The solution ( , , , ) is determined by the formula in Paragraph 6.3.1-4 whereÿ( , , , ý, )= 8× &Ù =1 × Ú=1 × )Û =11" ÙÚ Û ÙÚ Û sin( Ù + # Ù ) sin( Ú + $ Ú) sin( Û+ % Û ) ´sin( Ù + # Ù ) sin( Úý+ $ Ú) sin( Ûþ+ % Û ) sin ÙÚ Û¬, Page 417 # Ù =arctan Ù1, $ Ú=arctan Ú2, % Û =arctan Û3, ÙÚ Û = 2( 2 Ù + 2Ú+ 2 Û )+ ," ÙÚ Û = &'1+( 1 2+ 2 Ù )( 1+ 2) ( 2 1+ 2 Ù )( 2 2+ 2 Ù ) ( &'2+( 3 4+ 2Ú)( 3+ 4) ( 2 3+ 2Ú)( 2 4+ 2Ú) ( &'3+( 5 6+ 2 Û )( 5+ 6) ( 2 5+ 2 Û )( 2 6+ 2 Û ) (. Here, the Ù , Ú, and Û are positive roots of the transcendental equations2- 1 2=( 1+ 2) cot( 1 ), 2- 3 4=( 3+ 4) cot( 2 ), 2- 5 6=( 5+ 6) cot( 3 ). 6.3.1-6. Domain: 0 £ £ 1,0 £ £ 2,0 £ £ 3. Mixed boundary value problems. 1 ). A rectangular parallelepiped is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition),= 1( , , ) at = 0 (boundary condition),= 2( , , ) at = 1(boundary condition), = 3( , , ) at = 0 (boundary condition), = 4( , , ) at = 2(boundary condition), != 5( , , ) at = 0 (boundary condition), != 6( , , ) at = 3(boundary condition). Solution:( , , , )= û ü3 0 û ü2 0 û ü1 0 0( , ý, þ) ÿ( , , , , ý, þ, ) ý þ +û ü3 0 û ü2 0 û ü1 0 1( , ý, þ) ÿ( , , , , ý, þ, ) ý þ + 2û 0 û ü3 0 û ü2 0 1( ý, þ, ) & ÿ( , , , , ý, þ, - )( *=0 ý þ - 2û 0 û ü3 0 û ü2 0 2( ý, þ, ) & ÿ( , , , , ý, þ, - )( *=ü1 ý þ - 2û 0 û ü3 0 û ü1 0 3( , þ, ) ÿ( , , , ,0, þ, - ) þ + 2û 0 û ü3 0 û ü1 0 4( , þ, ) ÿ( , , , , 2, þ, - ) þ - 2û 0 û ü2 0 û ü1 0 5( , ý, ) ÿ( , , , , ý,0, - ) ý + 2û 0 û ü2 0 û ü1 0 6( , ý, ) ÿ( , , , , ý, 3, - ) ý +û 0 û ü3 0 û ü2 0 û ü1 0 Ö( , ý, þ, ) ÿ( , , , , ý, þ, - ) ý þ . Here,ÿ( , , , , ý, þ, )=21 2 3× Ù =1 × Ú=0 × Û =0 Ú Û ÙÚ Û sin( Ù ) cos( Ú ) cos( Û) ´sin( Ù ) cos( Úý) cos( Ûþ) sin ÙÚ Û¬, Page 418 +.- 1 where Ú= 81for= 0, 2for>0, Û = 81forç= 0, 2forç>0, Ù = 1, Ú= 2, Û = ç 3, ÙÚ Û = 2( 2 Ù + 2Ú+ 2 Û )+ . 2 ). A rectangular parallelepiped is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition),= 1( , , ) at = 0 (boundary condition), = 2( , , ) at = 1(boundary condition),= 3( , , ) at = 0 (boundary condition), = 4( , , ) at = 2(boundary condition),= 5( , , ) at = 0 (boundary condition), != 6( , , ) at = 3(boundary condition). Solution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ere,=( , , >, , ;, <, )=8C1 C2 C3 G Ù =1 G Ú=1 G Û =11 ÙÚ Û sin( Ù ) sin( Ú ) sin( Û>) ´sin( Ù ) sin( Ú;) sin( Û<) sin ÙÚ Û H , where Ù = (2+ 1) 2 C1, Ú= (2+ 1) 2 C2, Û = (2 I+ 1) 2 C3, ÙÚ Û = @2( 2 Ù + 2Ú+ 2 Û )+ . Page 419 6.3.2. Problems in Cylindrical Coordinates Anonhomogeneous Klein±Gordon equation in the cylindrical coordinate system is written as2 2= @2&1 J J K J J L+1 J 2 2 M2+ 2 >2(- +F( J , M, >, ), J = 2+ 2. One-dimensional problems with axial symmetry that have solutions = ( J , ) are treated in Subsection 4.2.5. Two-dimensional problems whose solutions have the form = ( J , M, ) or= ( J , >, ) are considered in Subsections 5.3.2 and 5.3.3. 6.3.2-1. Domain: 0 £ J £ N,0 £ M£ 2,0 £ >£ C. First boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( J , M, >) at = 0 (initial condition),A = 1( J , M, >) at = 0 (initial condition),= 1( M, >, ) at J = N(boundary condition),= 2( J , M, ) at >= 0 (boundary condition),= 3( J , M, ) at >= C(boundary condition). Solution:( J , M, >, )= 9 :0 92 O 0 9 P0 0( , ;, <) =( J , M, >, , ;, <, ) ? ? ; ? < +9 :0 92 O 0 9P0 1( , ;, <) =( J , M, >, , ;, <, ) ? ? ; ? < - @2N9 A0 9 :0 92 O 0 1( ;, <, B) & =( J , M, >, , ;, <, - B)( *=P ? ; ? < ? B + @29 A0 92 O 0 9 P0 2( , ;, B) & < =( J , M, >, , ;, <, - B)( E=0 ? ? ; ? B - @29 A0 92 O 0 9 P0 3( , ;, B) & < =( J , M, >, , ;, <, - B)( E=: ? ? ; ? B +9 A0 9 :0 92 O 0 9P0 F( , ;, <, B) =( J , M, >, , ;, <, - B) ? ? ; ? < ? B. Here,=( J , M, >, , ;, <, )=2 N2CG Q Ù =0 G QÚ=1 G Q Û =1 R Ù [ S T Ù ( U ÙÚN)]2 V ÙÚ ÛS Ù ( U ÙÚ J ) S Ù ( U ÙÚ W) ´cos[ X( M- ;)] sin K I Y >C Lsin K I Y <C Lsin Z[\ V ÙÚ Û H , whereV ÙÚ Û = @2U2 ÙÚ+ @2I2Y2C2+ ],R Ù = ^1for X= 0, 2for X>0, the S Ù ( W) are the Bessel functions (the prime denotes the derivative with respect to the argument), and the U ÙÚare positive roots of the transcendental equation S Ù ( U _)= 0. Page 420 6.3.2-2. Domain: 0 £ £ ,0 £ £ 2 ,0 £ £ . Second boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition), = 1( , , ) at = (boundary condition), = 2( , , ) at = 0 (boundary condition), = 3( , , ) at = (boundary condition). Solution:( , , , )= 0 2 0 0 0(, , !) "( , , ,, , !, ) # # # ! + 0 2 0 0 1(, , !) "( , , ,, , !, ) # # # ! + $2 0 0 2 0 1( , !, %) "( , , , , , !, - %) # # ! # % - $2 0 2 0 0 2(, , %) "( , , ,, ,0, - %) # # # % + $2 0 2 0 0 3(, , %) "( , , ,, , , - %) # # # % + 0 0 2 0 0 &(, , !, %) "( , , ,, , !, - %) # # # ! # %. Here,"( , , ,, , !, )=sin '(*) + , 2) ++2 2 - .0/ =11) 1 / cos 2 3 4 5cos 2 3 5sin' 761 /, +1 - .98 =0 - .:=1 - . / =0 ; 8; /=< 2 8: > 8 ( <8: ) > 8 ( <8:) ( < 2 8:2- ?2)[ > 8 ( <8:)]2cos[ ?( - )] cos 2 3 4 5cos 2 3 5sin( @ 8: /)@ 8: / ,1 / = $23222+ +, @ 8: / = A $2 < 2 8:+ $23222+ +,; 8 = B1for ?= 0, 2for ?>0, where the > 8 () are the Bessel functions and the <8:are positive roots of the transcendental equation> C 8 ( <)= 0. 6.3.2-3. Domain: 0 £ £ ,0 £ £ 2 ,0 £ £ . Third boundary value problem. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition), +31 = ( , , ) at = (boundary condition), -32 = 2( , , ) at = 0 (boundary condition), +33 = 3( , , ) at = (boundary condition). The solution ( , , , ) is determined by the formula in Paragraph 6.3.2-2 where"( , , ,, , !, )=1 - .98 =0 - .:=1 - .ED =1; 8 < 2 8:> 8 ( <8: ) > 8 ( <8:) cos[ ?( - )] F D ( ) F D ( !) sin( @ 8: D) ( < 2 8:2+32 1 2- ?2)[ > 8 ( <8:)]2 GF DG2@ 8: D . Page 421 Here,; 8 = B1for ?= 0, 2for ?>0, @ 8: D = H $2 < 2 8:+ $212 D + +,F D ( )=cos(1 D)+ 321 D sin(1 D), GF DG2= 33 212 D12 D +32 212 D +32 3+ 32 212 D + 2 21 + 32 212 D5, the > 8 () are the Bessel functions, and the <8:and1 D are positive roots of the transcendental equations <> C 8 ( <)+31 > 8 ( <)= 0,tan(1 )1= 32+3312-3233. 6.3.2-4. Domain: 0 £ £ ,0 £ £ 2 ,0 £ £ . Mixed boundary value problems. 1 I. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition),= 1( , , ) at = (boundary condition), = 2( , , ) at = 0 (boundary condition), = 3( , , ) at = (boundary condition). Solution:( , , , )= 0 2 0 0 0(, , !) "( , , ,, , !, ) # # # ! + 0 2 0 0 1(, , !) "( , , ,, , !, ) # # # ! - $2 0 0 2 0 1( , !, %) J "( , , ,, , !, - %) K L = # # ! # % - $2 0 2 0 0 2(, , %) "( , , ,, ,0, - %) # # # % + $2 0 2 0 0 3(, , %) "( , , ,, , , - %) # # # % + 0 0 2 0 0 &(, , !, %) "( , , ,, , !, - %) # # # ! # %. Here,"( , , ,, , !, )=1 2 - .98 =0 - .:=1 - .0/ =0 ; 8; / [ >C 8 ( <8:)]2) @ 8: /> 8 ( <8: ) > 8 ( <8:) ´cos[ ?( - )] cos 2 3 5cos 2 3 ! 5sin' 6@ 8: /,,@ 8: / = $2 < 2 8:+ $23222+ +,; 8 = B1for ?= 0, 2for ?>0, where the > 8 () are the Bessel functions (the prime denotes the derivative with respect to the argument) and the <8:are positive roots of the transcendental equation > 8 ( <)= 0. Page 422 2 I. A circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition), = 1( , , ) at = (boundary condition),= 2( , , ) at = 0 (boundary condition),= 3( , , ) at = (boundary condition). Solution:( , , , )= 0 2 0 0 0(, , !) "( , , ,, , !, ) # # # ! + 0 2 0 0 1(, , !) "( , , ,, , !, ) # # # ! + $2 0 0 2 0 1( , !, %) "( , , , , , !, - %) # # ! # % + $2 0 2 0 0 2(, , %) J ! "( , , ,, , !, - %) K M =0 # # # % - $2 0 2 0 0 3(, , %) J ! "( , , ,, , !, - %) K M = # # # % + 0 0 2 0 0 &(, , !, %) "( , , ,, , !, - %) # # # ! # %. Here,"( , , ,, , !, )=2 2 - .0/ =11)1 / sin 2 3 5sin 2 3 ! 5sin '(761 /, +2 - .98 =0 - .:=1 - .N/ =1 ; 8 < 2 8: ( < 2 8:2- ?2)[ > 8 ( <8:)]2) @ 8: /> 8 ( <8: ) > 8 ( <8:) ´cos[ ?( - )] sin 2 3 5sin 2 3 ! 5sin' 6@ 8: /,,1 / = $23222+ +, @ 8: / = $2 < 2 8:+ $23222+ +,; 8 = B1for ?= 0, 2for ?>0, where the > 8 () are the Bessel functions and the <8:are positive roots of the transcendental equation>C 8 ( <)= 0. 6.3.2-5. Domain: 1£ £ 2,0 £ £ 2 ,0 £ £ . First boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition),= 1( , , ) at = 1(boundary condition),= 2( , , ) at = 2(boundary condition),= 3( , , ) at = 0 (boundary condition),= 4( , , ) at = (boundary condition). Page 423 Solution:( , , , )= 0 2 0 21 0(, , !) "( , , ,, , !, ) # # # ! + 0 2 0 21 1(, , !) "( , , ,, , !, ) # # # ! + $21 0 0 2 0 1( , !, %) J "( , , ,, , !, - %) K L = 1 # # ! # % - $22 0 0 2 0 2( , !, %) J "( , , ,, , !, - %) K L = 2 # # ! # % + $2 0 2 0 21 3(, , %) J ! "( , , ,, , !, - %) K M =0 # # # % - $2 0 2 0 21 4(, , %) J ! "( , , ,, , !, - %) K M = # # # % + 0 0 2 0 21 &(, , !, %) "( , , ,, , !, - %) # # # ! # %. Here,"( , , ,, , !, )= 2 - .98 =0 - .:=1 - .0/ =1 ; 8 < 2 8:>2 8 ( <8:2)>2 8 ( <8:1)- >2 8 ( <8:2) O 8:( )O 8:() ´cos[ ?( - )] sin 2 3 5sin 2 3 ! 5sin' ) @ 8: /,) @ 8: / , where; 8 = B1for ?= 0, 2for ?¹ 0, @ 8: / = $2 < 2 8:+ $23222+ +,O 8:( )= > 8 ( <8:1) P 8 ( <8: )- P 8 ( <8:1) > 8 ( <8: ); the > 8 ( ) and P 8 ( ) are the Bessel functions, and the <8:are positive roots of the transcendental equation> 8 ( <1) P 8 ( <2)- P 8 ( <1) > 8 ( <2)= 0. 6.3.2-6. Domain: 1£ £ 2,0 £ £ 2 ,0 £ £ . Second boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition), = 1( , , ) at = 1(boundary condition), = 2( , , ) at = 2(boundary condition), = 3( , , ) at = 0 (boundary condition), = 4( , , ) at = (boundary condition). Page 424 Solution:( , , , )= 0 2 0 21 0(, , !) "( , , ,, , !, ) # # # ! + 0 2 0 21 1(, , !) "( , , ,, , !, ) # # # ! - $21 0 0 2 0 1( , !, %) "( , , , 1, , !, - %) # # ! # % + $22 0 0 2 0 2( , !, %) "( , , , 2, , !, - %) # # ! # % - $2 0 2 0 21 3(, , %) "( , , ,, ,0, - %) # # # % + $2 0 2 0 21 4(, , %) "( , , ,, , , - %) # # # % + 0 0 2 0 21 &(, , !, %) "( , , ,, , !, - %) # # # ! # %. Here,"( , , ,, , !, )=sin' ) +,( 2 2- 2 1) Q) ++2( 2 2- 2 1) - .0/ =1cos 2 3 5cos 2 3 ! 5sin' ) 1 /,)1 / +1 - .8 =0 - .:=1 - .0/ =0 ; 8; / < 2 8:O 8:( )O 8:() ( < 2 8:2 2- ?2)O2 8:( 2)-( < 2 8:2 1- ?2)O2 8:( 1) ´cos[ ?( - )] cos 2 3 5cos 2 3 ! 5sin '() @ 8: /,) @ 8: / , where; 8 = B1for ?= 0, 2for ?¹ 0, 1 / = $23222+ +, @ 8: / = $2 < 2 8:+ $23222+ +,O 8:( )= > C 8 ( <8:1) P 8 ( <8: )- P C 8 ( <8:1) > 8 ( <8: ); the > 8 ( ) and P 8 ( ) are the Bessel functions, and the <8:are positive roots of the transcendental equation> C 8 ( <1) P C 8 ( <2)- P C 8 ( <1) > C 8 ( <2)= 0. 6.3.2-7. Domain: 1£ £ 2,0 £ £ 2 ,0 £ £ . Third boundary value problem. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , , ) at = 0 (initial condition), = 1( , , ) at = 0 (initial condition), -31 = 1( , , ) at = 1(boundary condition), +32 = 2( , , ) at = 2(boundary condition), -33 = 3( , , ) at = 0 (boundary condition), +34 = 4( , , ) at = (boundary condition). Page 425 The solution ( , , , ) is determined by the formula in Paragraph 6.3.2-6 where"( , , ,, , !, )=1 - .98 =0 - .:=1 - .RD =1; 8 < 2 8:GF DG26 $2 < 2 8:+ $2@2 D + + ´ O 8:( )O 8:() cos[ ?( - )] F D ( ) F D ( !) sin' 6 $2 < 2 8:+ $2@2 D + +, (32 2 2 2+ < 2 8:2 2- ?2)O2 8:( 2)-(32 1 2 1+ < 2 8:2 1- ?2)O2 8:( 1). Here,O 8:( )= S <8:> C 8 ( <8:1)-31 > 8 ( <8:1) T7P 8 ( <8: U) - S <8:P C 8 ( <8: V1)-31 P 8 ( <8: V1) T > 8 ( <8: U),; 8 = B1for ?= 0, 2for ?¹ 0, F D ( W)=cos( @ DW)+ 33@ D sin( @ DW), GF DG2= 34 2 @2 D@2 D +32 3@2 D +32 4+ 33 2 @2 D + X2 21 + 32 3@2 D5, where the > 8 ( U) and P 8 ( U) are the Bessel functions, and the <8:are positive roots of the transcen- dental equationS <> C 8 ( <V1)-31 > 8 ( <V1) T S <P C 8 ( <V2)+32 P 8 ( <V2) T = S <P C 8 ( <V1)-31 P 8 ( <V1) T S <> C 8 ( <V2)+32 > 8 ( <V2) T, and the @ D are positive roots of the transcendental equationtan( @X)@= 33+34@2-3334. 6.3.2-8. Domain: V1£ U£ V2,0 £ Y£ 2 Z,0 £ W£X. Mixed boundary value problems. 1 I. A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:[= \0( U, Y, W) at ]= 0 (initial condition),^ _[= \1( U, Y, W) at ]= 0 (initial condition),[= `1( Y, W, ]) at U= V1(boundary condition),[= `2( Y, W, ]) at U= V2(boundary condition),^ a[= `3( U, Y, ]) at W= 0 (boundary condition),^ a[= `4( U, Y, ]) at W=X(boundary condition). Solution:[( U, Y, W, ])= ^^] b c0 b2 d 0 b e2e1 \0( f, g, h) i( U, Y, W, f, g, h, ]) f j f j g j h +b c0 b2 d 0 b e2e1 \1( f, g, h) i( U, Y, W, f, g, h, ]) f j f j g j h + k2V1b _ 0 b c0 b2 d 0 `1( g, h, l) J ^^f i( U, Y, W, f, g, h, ]- l) K L =e1 j g j h j l - k2V2b _ 0 b c0 b2 d 0 `2( g, h, l) J ^^f i( U, Y, W, f, g, h, ]- l) K L =e2 j g j h j l - k2b _ 0 b2 d 0 b e2e1 `3( f, g, l) i( U, Y, W, f, g,0, ]- l) f j f j g j l + k2b _ 0 b2 d 0 b e2e1 `4( f, g, l) i( U, Y, W, f, g,X, ]- l) f j f j g j l +b _ 0 b c0 b2 d 0 b e2e1 m( f, g, h, l) i( U, Y, W, f, g, h, ]- l) f j f j g j h j l. Page 426 np s Here,i( U, Y, W, f, g, h, ])= Z 4X z {9| =0z {}=1z {0~ =0 | ~ 2 |} 2 | ( |} 2)2 | ( |} 1)- 2 | ( |} 2) |}( ) |}( f) ´cos[ ( - g)] cos 5cos h5sin () |} ~ ) |} ~ , where | = 1for = 0, 2for ¹ 0, |} ~ = k2 2 |}+ k2222+ , |}( )= | ( |} 1) | ( |})- | ( |} 1) | ( |}); the | ( ) and | ( ) are the Bessel functions, and the |}are positive roots of the transcendental equation | ( 1) | ( 2)- | ( 1) | ( 2)= 0. 2 . A hollow circular cylinder of ®nite length is considered. The following conditions are prescribed:= 0( , ,) at = 0 (initial condition), = 1( , ,) at = 0 (initial condition), = 1( ,, ) at = 1(boundary condition), = 2( ,, ) at = 2(boundary condition),= 3( , , ) at= 0 (boundary condition),= 4( , , ) at= (boundary condition). Solution:( , ,, )= b c0 b2 d 0 b e2e1 0( f, g, h) i( , ,, f, g, h, ) f j f j g j h +b c0 b2 d 0 b e2e1 1( f, g, h) i( , ,, f, g, h, ) f j f j g j h - k21b 0 b c0 b2 d 0 1( g, h, l) i( , ,, 1, g, h, - l) j g j h j l + k22b 0 b c0 b2 d 0 2( g, h, l) i( , ,, 2, g, h, - l) j g j h j l + k2b 0 b2 d 0 b e2e1 3( f, g, l) h i( , ,, f, g, h, - l) =0 f j f j g j l - k2b 0 b2 d 0 b e2e1 4( f, g, l) h i( , ,, f, g, h, - l) =c f j f j g j l +b 0 b c0 b2 d 0 b e2e1 m( f, g, h, l) i( , ,, f, g, h, - l) f j f j g j h j l. Here,i( , ,, f, g, h, )=2( 2 2- 2 1) z {0~ =1sin sin h sin () ~ ) ~ +2 z {9| =0z {}=1z {~ =1 | 2 |} |}( ) |}( f) ( 2 |} 2 2- 2)2 |}( 2)-( 2 |} 2 1- 2)2 |}( 1) ´cos[ ( - g)] sin sin h sin () |} ~ ) |} ~ , Page 427 where | = 1for = 0, 2for ¹ 0, ~ = k2222+ , |} ~ = k2 2 |}+ k2222+ , |}( )= | ( |} 1) | ( |})- | ( |} 1) | ( |}); the | ( ) and | ( ) are the Bessel functions, and the |}are positive roots of the transcendental equation | ( 1) | ( 2)- | ( 1) | ( 2)= 0. 6.3.2-9. Domain: 0 £ £ ,0 £ £ 0,0 ££ . First boundary value problem. A cylindrical sector of ®nite thickness is considered. The following conditions are prescribed:= 0( , ,) at = 0 (initial condition), = 1( , ,) at = 0 (initial condition),= 1( ,, ) at = (boundary condition),= 2( ,, ) at = 0 (boundary condition),= 3( ,, ) at = 0(boundary condition),= 4( , , ) at= 0 (boundary condition),= 5( , , ) at= (boundary condition). Solution:( , ,, )= c0 ¡0 0 e0 0( f, g, ¢) £( , ,, f, g, ¢, ) f ¤ f ¤ g ¤ ¢ + c0 ¡0 0 e0 1( f, g, ¢) £( , ,, f, g, ¢, ) f ¤ f ¤ g ¤ ¢ - ¥2 0 c0 ¡0 0 1( g, ¢, ¦) f £( , ,, f, g, ¢, - ¦) § =e ¤ g ¤ ¢ ¤ ¦ + ¥2 0 c0 e0 2( f, ¢, ¦)1f g £( , ,, f, g, ¢, - ¦) ¨ =0 ¤ f ¤ ¢ ¤ ¦ - ¥2 0 c0 e0 3( f, ¢, ¦)1f g £( , ,, f, g, ¢, - ¦) ¨ =¡0 ¤ f ¤ ¢ ¤ ¦ + ¥2 0 ¡0 0 e0 4( f, g, ¦) ¢ £( , ,, f, g, ¢, - ¦) =0 f ¤ f ¤ g ¤ ¦ - ¥2 0 ¡0 0 e0 5( f, g, ¦) ¢ £( , ,, f, g, ¢, - ¦) =c f ¤ f ¤ g ¤ ¦ + 0 c0 ¡0 0 e0 ©( f, g, ¢, ¦) £( , ,, f, g, ¢, - ¦) f ¤ f ¤ g ¤ ¢ ¤ ¦. Here,£( , ,, f, g, ¢, )=82 0 ª «9¬ =1 ª «=1 ª «0® =1 ¯ ¬d °¡0( ± ¬ ²)¯ ¬d °¡0( ± ¬f) [¯ ¬d °¡0( ± ¬ ³)]2sin ´ µ ¶ ··0 ¸sin ´ µ ¶ g·0 ¸ ´sin ´ ¹ ¶ º »¸sin ´ ¹ ¶ ¢»¸sin ¼(½7¾ ¥2±2 ¬+ ¥2¹2¶2 »-2+ ¿ À¾ ¥2±2 ¬+ ¥2¹2¶2 »-2+ ¿, where the¯ ¬d °¡0( ²) are the Bessel functions and the ± ¬are positive roots of the transcendental equation¯ ¬d °¡0( ± ³)= 0. Page 428 ÁÃ Æ 6.3.2-10. Domain: 0 £ ²£ ³,0 £·£·0,0 £º£ ». Mixed boundary value problem. A cylindrical sector of ®nite thickness is considered. The following conditions are prescribed:= 0( ²,·,º) at ½= 0 (initial condition),Ì Í= 1( ²,·,º) at ½= 0 (initial condition),= Î1(·,º, ½) at ²= ³ (boundary condition),= Î2( ²,º, ½) at·= 0 (boundary condition),= Î3( ²,º, ½) at·=·0(boundary condition),Ì Ï= Î4( ²,·, ½) atº= 0 (boundary condition),Ì Ï= Î5( ²,·, ½) atº= »(boundary condition). Solution:( ²,·,º, ½)= ÌÌ½ Ð Ñ0 Ð Ò0 0 Ð e0 0( f, g, Ó) Ô( ²,·,º, f, g, Ó, ½) f Õ f Õ g Õ Ó +Ð Ñ0 Ð Ò0 0 Ð e0 1( f, g, Ó) Ô( ²,·,º, f, g, Ó, ½) f Õ f Õ g Õ Ó - Ö2³Ð Í 0 Ð Ñ0 Ð Ò0 0 Î1( g, Ó, ×) Ø ÌÌf Ô( Ù,·,º, f, g, Ó, ½- ×) Ú Û =e Õ g Õ Ó Õ × + Ö2Ð Í 0 Ð Ñ0 Ð e0 Î2( f, Ó, ×)1f Ø ÌÌg Ô( Ù,·,º, f, g, Ó, ½- ×) Ú Ü =0 Õ f Õ Ó Õ × - Ö2Ð Í 0 Ð Ñ0 Ð e0 Î3( f, Ó, ×)1f Ø ÌÌg Ô( Ù,·,º, f, g, Ó, ½- ×) Ú Ü =Ò0 Õ f Õ Ó Õ × - Ö2Ð Í 0 Ð Ò0 0 Ð e0 Î4( f, g, ×) Ô( Ù,·,º, f, g,0, ½- ×) f Õ f Õ g Õ × + Ö2Ð Í 0 Ð Ò0 0 Ð e0 Î5( f, g, ×) Ô( Ù,·,º, f, g, », ½- ×) f Õ f Õ g Õ × +Ð Í 0 Ð Ñ0 Ð Ò0 0 Ð e0 Ý( f, g, Ó, ×) Ô( Ù,·,º, f, g, Ó, ½- ×) f Õ f Õ g Õ Ó Õ ×. Here,Ô( Ù,·,º, f, g, Ó, ½)=4³2 »·0 Þ ß9à =1 Þ ßá=1 Þ ß0â =0 ã â¯ àd äÒ0( å àáÙ)¯ àd äÒ0( å àáf) [¯ æ àd äÒ0( å àá ç)]2sin ´ µ ¶ ··0 ¸sin ´ µ ¶ g·0 ¸ ´cos ´ ¹ ¶ º »¸cos ´ ¹ ¶ Ó»¸sin ¼(½7¾ Ö2å2 àá+ Ö2¹2¶2 »-2+ ¿ À¾ Ö2å2 àá+ Ö2¹2¶2 »-2+ ¿, whereã0= 1andã â = 2for¹³ 1; the¯ àd äÒ0( Ù) are the Bessel functions; and the å àáare positive roots of the transcendental equation¯ àd äÒ0( å ç)= 0. 6.3.3. Problems in Spherical Coordinates Anonhomogeneous Klein±Gordon equation in the spherical coordinate system is written asÌ2 èÌ½2= Ö2Ø1Ù2 ÌÌÙ ´Ù2 ÌèÌÙ¸+1Ù2sin é ÌÌé ´sin é ÌèÌé¸+1Ù2sin2é Ì2 èÌ·2 Ú- ¿ è+Ý( Ù, é,·, ½). One-dimensional problems with central symmetry that have solutions of the form è= è( Ù, ½) are treated in Subsection 4.2.6. Page 429 430 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 6.3.3-1. Domain: 0£ Ù£ ç,0£ é£¶,0£·£2¶.First boundary value problem. Aspherical domain isconsidered. Thefollowing conditions areprescribed:è= ê0( Ù, é,·)at ½=0 (initial condition ),Ì Íè= ê1( Ù, é,·)at ½=0 (initial condition ),è= Î( é,·, ½)at Ù= ç(boundary condition ). Solution:è( Ù, é,·, ½)= ÌÌ½ Ð2 d 0 Ð d 0 Ð e0 ê0( f, g, Ó) Ô( Ù, é,·, f, g, Ó, ½) f2sin g Õ f Õ g Õ Ó +Ð2 d 0 Ð d 0 Ð e0 ê1( f, g, Ó) Ô( Ù, é,·, f, g, Ó, ½) f2sin g Õ f Õ g Õ Ó - Ö2ç2Ð Í 0 Ð2 d 0 Ð d 0 Î( g, Ó, ×) Ø ÌÌf Ô( Ù, é,·, f, g, Ó, ½- ×) Ú Û =esin g Õ g Õ Ó Õ × +Ð Í 0 Ð2 d 0 Ð d 0 Ð e0Ý( f, g, Ó, ×) Ô( Ù, é,·, f, g, Ó, ½- ×) f2sin g Õ f Õ g Õ Ó Õ ×. Here,Ô( Ù, é,·, f, g, Ó, ½)=1 2¶ ç2 ëÙ fÞ ß à =0 Þ ßá=1 àß0â =0ã â ì àá â¯ à +1 ä2( í àáÙ)¯ à +1 ä2( í àáf) ´ î âà (cos é) î âà (cos g)cos[¹(·- Ó)]sin ¼(½7ï Ö2í2 àá+ ð ñ, whereã â = ò1for ó=0, 2for ó¹0, ì àá â =(2 ô+1)( ô- ó)! ( ô+ ó)![ õæ à +1 ä2( í àá ç)]2 ö ÷2í2 àá+ ð; the õ à +1 ä2( ø)aretheBessel functions, the î âà ( å)aretheassociated Legendre functions expressed interms oftheLegendre polynomials î à ( å)asî âà ( å)=(1- å2) âä2 ù úù åú î û( å), î û( å)=1ô!2 û ù ûù å û( å2-1) û, andthe í û üarepositi veroots ofthetranscendental equation õû+1 ý2( í þ)=0. 6.3.3-2. Domain: 0£ ø£ þ,0£ ÿ£ ,0£ £2 .Second boundary value problem. Aspherical domain isconsidered. Thefollowing conditions areprescribed:= 0( ø, ÿ, )at =0 (initial condition ),Ì Í= 1( ø, ÿ, )at =0 (initial condition ),Ì = ( ÿ, , )at ø= þ(boundary condition ). Solution:( ø, ÿ, , )= ÌÌ 2 0 0 0 0( , g, ) ( ø, ÿ, , , g, , ) 2sin gù ù gù + 2 0 0 0 1( , g, ) ( ø, ÿ, , , g, , ) 2sin gù ù gù + ÷2þ2 Í 0 2 0 0 ( g, , ) ( ø, ÿ, , þ, g, , - )sin gù gù ù + Í 0 2 0 0 0 ( , g, , ) ( ø, ÿ, , , g, , - ) 2sin gù ù gù ù . Page430 6.3. EQUATIONS OFTHEFORM 2 2= 2 3 - + ( , , , Ë) 431 Here,( ø, ÿ, , , g, , )=3sin ð ñ 4 þ3 ð+1 2 ø û=0 ü=1 û ú=0 ! ú " û üúö ÷2 #2û ü+ ð õû+1 ý2( #û ü ø) õû+1 ý2( #û ü ) ´ $ú û(cos ÿ) $ú û(cos g)cos[ ó( - )]sin 7ï ÷2 #2û ü+ ð ñ, where! ú= ò1for ó=0, 2for ó¹0," û üú= #2û ü(2 ô+1)( ô- ó)! ( ô+ ó)! % þ2 #2û ü- ô( ô+1) & % õû+1 ý2( #û ü þ) &2; the õû+1 ý2( ø)aretheBessel functions, the $ú û( ')aretheassociated Legendre functio ns(seeParagraph 6.3.3-1 ),andthe #û üarepositi veroots ofthetranscendental equation 2 #þ õ (û+1 ý2( #þ)- õû+1 ý2( #þ)=0. 6.3.3-3. Domain: 0£ ø£ þ,0£ ÿ£ ,0£ £2 .Third boundary value problem. Aspherical domain isconsidered. Thefollowing conditions areprescribed:= 0( ø, ÿ, )at =0 (initial condition ),) *= 1( ø, ÿ, )at =0 (initial condition ),) + ó = ( ÿ, , )at ø= þ(boundary condition ). Thesolution ( ø, ÿ, , )isdetermined bytheformula inParagraph 6.3.3-2 where( ø, ÿ, , , g, , )=1 2 ø û=0 ü=1 û ,+ =0! +" û ü +ö ÷2 #2û ü+ ð õû+1 ý2( #û ü ø) õû+1 ý2( #û ü ) ´ $ +û(cos ÿ) $ +û(cos g)cos[ -( - )]sin /. ÷2 #2û ü+ 0 1. Here,! + = 21for -=0, 2for -¹0," û ü + = #2û ü(2 3+1)( 3- -)! ( 3+ -)!% þ2 #2û ü+( 4 þ+ 3)( 4 þ- 3-1)& %65û+1 ý2( #û ü þ)&2; the5û+1 ý2( ø)aretheBessel functions, the $ +û( ')aretheassociated Legendre functions (see Paragraph 6.3.3-1 ),andthe #û üarepositi veroots ofthetranscendental equation#þ5 (û+1 ý2( #þ)+ 74 þ-1 2 15û+1 ý2( #þ)=0. 6.3.3-4. Domain: þ1£ ø£ þ2,0£ ÿ£ ,0£ £2 .First boundary value problem. Aspherical layer isconsidered. Thefollowing conditions areprescribed:= 0( ø, ÿ, )at =0 (initial condition ),) *= 1( ø, ÿ, )at =0 (initial condition ),= 1( ÿ, , )at ø= þ1(boundary condition ),= 2( ÿ, , )at ø= þ2(boundary condition ). Page431 432 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Solution:( ø, ÿ, , )= )) 2 0 0 2 1 0( , g, ) ( ø, ÿ, , , g, , ) 2sin gù ù gù + 2 0 0 2 1 1( , g, ) ( ø, ÿ, , , g, , ) 2sin gù ù gù + ÷2þ2 1 * 0 2 0 0 1( g, , ) 8 )) ( ø, ÿ, , , g, , - ) 9 : = 1sin gù gù ù - ÷2þ2 2 * 0 2 0 0 2( g, , ) 8 )) ( ø, ÿ, , , g, , - ) 9 : = 2sin gù gù ù + * 0 2 0 0 2 1 ( , g, , ) ( ø, ÿ, , , g, , - ) 2sin gù ù gù ù , where( ø, ÿ, , , g, , )= 8 ø û=0 ü=1 û ú=0 ! ú " û üú ; < 2 #2 = >+ 0 ? =+1 @2( # = > A)? =+1 @2( # = > ) ´ $ú =(cos B) $ú =(cos g)cos[ 4( C- )]sin /. < 2 #2 = >+ 0 1. Here,? =+1 @2( # = > A)=5 =+1 @2( # = > D1) E =+1 @2( # = > A)- E =+1 @2( # = > D1)5 =+1 @2( # = > A),! ú= 21for 4=0, 2for 4¹0," = >ú= # = >(2 3+1)( 3- 4)!52=+1 @2( # = > D2) ( 3+ 4)![52=+1 @2( # = > D1)-52=+1 @2( # = > D2)], where the5 =+1 @2( A)aretheBessel functions, the $ú =( ')aretheassociated Legendre functions expressed interms oftheLegendre polynomials $ =( ')as$ú =( ')=(1- '2)ú @2 F úF 'ú $ =( '), $ =( ')=13!2 = F =F ' =( '2-1) = , andthe # = >arepositi veroots ofthetranscendental equation? =+1 @2( # D2)=0. 6.3.3-5. Domain: D1£ A£ D2,0£ B£ G,0£ C£2 G.Second boundary value problem. Aspherical layer isconsidered. Thefollowing conditions areprescribed:H= I0( A, B, C)at J=0 (initial condition ),) *H= I1( A, B, C)at J=0 (initial condition ),) KH= L1( B, C, J)at A= D1(boundary condition ),) KH= L2( B, C, J)at A= D2(boundary condition ). Solution:H( A, B, C, J)= ))J M2 N 0 M N 0 M O2O1 I0( P, Q, R) S( A, B, C, P, Q, R, J) P2sin QF PF QF R +M2 N 0 M N 0 MO2O1 I1( P, Q, R) S( A, B, C, P, Q, R, J) P2sin QF PF QF R - T2D2 1M U0 M2 N 0 M N 0 L1( Q, R, V) S( W, X, C, D1, Q, R, J- V)sin QF QF RF V + T2D2 2M U0 M2 N 0 M N 0 L2( Q, R, V) S( W, X, C, D2, Q, R, J- V)sin QF QF RF V +M U0 M2 N 0 M N 0 MO2O1 Y( P, Q, R, V) S( W, X, Z, P, Q, R, J- V) P2sin QF PF QF RF V, Page432 6.3. EQUATIONS OFTHEFORM [2 \[]2= ^2 _3 `- a`+ b( c, d, , e) 433 whereS( W, X, Z, P, Q, R, J)=3sin fJg h i 4 j( k3 2- k3 1)g h+1 4 jg W P l mon =0 l mp=1 nmrq =0 s qt np q u n +1 v2( w npW) u n +1 v2( w npP) ´ x qn (cos X) x qn (cos Q)cos[ y( Z- R)]sin fz/{ T2w2 np+ h i{ T2w2 np+ h. Here,s q = |1for y=0, 2for y¹0, t np q =( }+ y)! (2 }+1)( }- y)! M O2O1 W u 2 n +1 v2( w npW) ~ W,u n +1 v2( w npW)= w np n +1 v2( w npk1)-1 2 k1 n +1 v2( w npk1) n +1 v2( w npW) - w np n +1 v2( w npk1)-1 2 k1 n +1 v2( w npk1) n +1 v2( w npW), where the n +1 v2( W)and n +1 v2( W)aretheBessel functions, the x qn ( )aretheassociated Legendre functions (seeParagraph 6.3.3-4), andthe w nparepositi veroots ofthetranscendental equationw u n +1 v2( w k2)-1 2 k2 u n +1 v2( w k2)=0. 6.3.3-6. Domain: k1£ W£ k2,0£ X£ j,0£ Z£2 j.Third boundary value problem. Aspherical layer isconsidered. Thefollowing conditions areprescribed:= 0( W, X, Z)at z=0 (initial condition ),U = 1( W, X, Z)at z=0 (initial condition ), - y1 = 1( X, Z, z)at W= k1(boundary condition ), + y2 = 2( X, Z, z)at W= k2(boundary condition ). Thesolution ( W, X, Z, z)isdetermined bytheformula inParagraph 6.3.3-5 whereS( W, X, Z, P, Q, R, z)=1 4 jg W P l mn =0 l mp=1 nm, =0 s t np u n +1 v2( w npW) u n +1 v2( w npP) ´ x n (cos X) x n (cos Q)cos[ ( Z- R)]sin fz/{ T2w2 np+ h i{ T2w2 np+ h. Here,s = |1for =0, 2for ¹0, t np =( }+ )! (2 }+1)( }- )! MO2O1 W u 2 n +1 v2( w npW) ~ W,u n +1 v2( w W)= w n +1 v2( w k1)- y1+1 2 k1 n +1 v2( w k1) n +1 v2( w W) - w n +1 v2( w k1)- y1+1 2 k1 n +1 v2( w k1) n +1 v2( w W), where the n +1 v2( W)and n +1 v2( W)aretheBessel functions, the x n ( )aretheassociated Legendre functions (seeParagraph 6.3.3-4), andthe w nparepositi veroots ofthetranscendental equationw u n +1 v2( w k2)+ y2-1 2 k2 u n +1 v2( w k2)=0. Page433 434 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 6.4. Telegraph Equation2 2+ = 2 3 ± + ( , , , ) 6.4.1. Problems inCartesian Coor dinates Athree-dimensional nonhomo geneous telegraphequation intherectangular Cartesian system of coordinates hastheform2 2+ = 2 2 2+ 2 2+ 2 2- ¡ + ¢( , , , ). 6.4.1-1. Reduction tothethree-dimensional Klein±Gordon equation. Thesubstitution ( , , , )=exp £-1 2 ¥¤/¦( , , , )leads totheequation2 ¦ 2= 2 2 ¦ 2+ 2 ¦ 2+ 2 ¦ 2- £7¡-1 4 2 ¤/¦+exp £1 2 ¥¤¢( , , , ), which isdiscussed inSubsection 6.3.1. 6.4.1-2. Domain: 0£ £ §1,0£ £ §2,0£ £ §3.First boundary value problem. Arectangular parallelepiped isconsidered. Thefollowing conditions areprescribed:¨= ©0( , , )at =0 (initial condition ),ª «¨= ©1( , , )at =0 (initial condition ),¨= ¬1( , , )at =0(boundary condition ),¨= ¬2( , , )at = §1(boundary condition ),¨= ¬3( , , )at =0(boundary condition ),¨= ¬4( , , )at = §2(boundary condition ),¨= ¬5( , , )at =0(boundary condition ),¨= ¬6( , , )at = §3(boundary condition ). Solution:¨( , , , )= ªª ®3 0 ®2 0 ®1 0 ©0( ¯, °, ±) ²( , , , ¯, °, ±, ) ³ ¯ ³ ° ³ ± + ®3 0 ®2 0 ®1 0 ´ ©1( ¯, °, ±)+ ©0( ¯, °, ±) µ/²( , , , ¯, °, ±, ) ³ ¯ ³ ° ³ ± + 2 « 0 ®3 0 ®2 0 ¬1( °, ±, ¶) · ªª¯ ²( , , , ¯, °, ±, - ¶) ¸ ¹ =0 ³ ° ³ ± ³ ¶ - 2 « 0 ®3 0 ®2 0 ¬2( °, ±, ¶) · ªª¯ ²( , , , ¯, °, ±, - ¶) ¸ ¹ =®1 ³ ° ³ ± ³ ¶ + 2 « 0 ®3 0 ®1 0 ¬3( ¯, ±, ¶) · ªª° ²( , , , ¯, °, ±, - ¶) ¸ º =0 ³ ¯ ³ ± ³ ¶ - 2 « 0 ®3 0 ®1 0 ¬4( ¯, ±, ¶) · ªª° ²( , , , ¯, °, ±, - ¶) ¸ º =®2 ³ ¯ ³ ± ³ ¶ Page434 6.4. TELEGRAPH EQUATION »2 ¼»½2+ ¾ » ¼»½= ¿2 À3 Á- ÂÁ+ Ã( Ä, Å, Æ, Ç) 435 + 2 « 0 ®2 0 ®1 0 ¬5( ¯, °, ¶) · ªª± ²( , , , ¯, °, ±, - ¶) ¸ È =0 ³ ¯ ³ ° ³ ¶ - 2 « 0 ®2 0 ®1 0 ¬6( ¯, °, ¶) · ªª± ²( , , , ¯, °, ±, - ¶) ¸ È =®3 ³ ¯ ³ ° ³ ¶ + « 0 ®3 0 ®2 0 ®1 0 ¢( ¯, °, ±, ¶) ²( , , , ¯, °, ±, - ¶) ³ ¯ ³ ° ³ ± ³ ¶. Here,²( , , , ¯, °, ±, )=8§1 §2 §3exp £-1 2 ¤ É ÊË =1 É ÊÌ=1 É Ê,Í =11Î Ï ËÌ Í sin( Ð Ë)sin( Ñ Ì )sin( Ò Í ) ´sin( Ð Ë¯)sin( Ñ Ì°)sin( Ò Í±)sin £ /Ó Ï ËÌ Í Ô , whereÐ Ë = Õ Ö × 1, Ñ Ì= Ø Ö × 2, Ò Í = ÙÚÖ × 3, Ï ËÌ Í = Û2( Ð2 Ë + Ñ2Ì+ Ò2 Í )+ Ü-1 4 Ý2. 6.4.1-3. Domain: 0£ Þ£ × 1,0£ ß£ × 2,0£ à£ × 3.Second boundary value problem. Arectangular parallelepiped isconsidered. Thefollowing conditions areprescribed:á= â0( Þ, ß, à)at ã=0 (initial condition ),ä åá= â1( Þ, ß, à)at ã=0 (initial condition ),ä æá= ç1( ß, à, ã)at Þ=0(boundary condition ),ä æá= ç2( ß, à, ã)at Þ= × 1(boundary condition ),ä èá= ç3( Þ, à, ã)at ß=0(boundary condition ),ä èá= ç4( Þ, à, ã)at ß= × 2(boundary condition ),ä éá= ç5( Þ, ß, ã)at à=0(boundary condition ),ä éá= ç6( Þ, ß, ã)at à= × 3(boundary condition ). Solution:á( Þ, ß, à, ã)= ääã ê ë3 0 ê ë2 0 ê ë1 0 â0( ì, í, î) ï( Þ, ß, à, ì, í, î, ã) ð ì ð í ð î +ê ë3 0 ê ë2 0 ê ë1 0 ñ â1( ì, í, î)+Ý â0( ì, í, î) òÚï( Þ, ß, à, ì, í, î, ã) ð ì ð í ð î - Û2ê å 0 ê ë3 0 ê ë2 0 ç1( í, î, ó) ï( Þ, ß, à,0, í, î, ã- ó) ð í ð î ð ó + Û2ê å 0 ê ë3 0 ê ë2 0 ç2( í, î, ó) ï( Þ, ß, à, × 1, í, î, ã- ó) ð í ð î ð ó - Û2ê å 0 ê ë3 0 ê ë1 0 ç3( ì, î, ó) ï( Þ, ß, à, ì,0, î, ã- ó) ð ì ð î ð ó + Û2ê å 0 ê ë3 0 ê ë1 0 ç4( ì, î, ó) ï( Þ, ß, à, ì, × 2, î, ã- ó) ð ì ð î ð ó - Û2ê å 0 ê ë2 0 ê ë1 0 ç5( ì, í, ó) ï( Þ, ß, à, ì, í,0, ã- ó) ð ì ð í ð ó + Û2ê å 0 ê ë2 0 ê ë1 0 ç6( ì, í, ó) ï( Þ, ß, à, ì, í, × 3, ã- ó) ð ì ð í ð ó +ê å 0 ê ë3 0 ê ë2 0 ê ë1 0 ô( ì, í, î, ó) ï( Þ, ß, à, ì, í, î, ã- ó) ð ì ð í ð î ð ó, Page435 436 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES whereï( Þ, ß, à, ì, í, î, ã)= õ- ö å÷ 2× 1 × 2 × 3 øsin ùãú û Ôú û+ É üý =0 É üþ=0 É ü,ÿ =0 ý þ ÿú ýþ ÿ cos( Ð ýÞ)cos( Ñ þ )cos( Ò ÿ) ´cos( Ð ýì)cos( Ñ þí)cos( Ò ÿî)sin ù ýþ ÿ , ý = 1for =0, 2for >0, Ð ý = 1, Ñ þ= 2, Ò ÿ = 3,û= Ü-1 4 Ý2, ýþ ÿ = Û2( Ð2 ý + Ñ2þ+ Ò2 ÿ )+ Ü-1 4 Ý2. The summation isperformed overtheindices satisfying thecondition ++>0;theterm corresponding to ===0issingled out. 6.4.1-4. Domain: 0£ Þ£ 1,0£ £ 2,0£ £ 3.Third boundary value problem. Arectangular parallelepiped isconsidered. Thefollowing conditions areprescribed:= 0( Þ, , )at =0 (initial condition ), = 1( Þ, , )at =0 (initial condition ), -1 = 1( , , )at Þ=0(boundary condition ), +2 = 2( , , )at Þ= 1(boundary condition ), -3 = 3( Þ, , )at =0(boundary condition ), +4 = 4( Þ, , )at = 2(boundary condition ), -5 = 5( Þ, , )at =0(boundary condition ), +6 = 6( Þ, , )at = 3(boundary condition ). Thesolution ( Þ, , , )isdetermined bytheformula inParagraph 6.4.1-3 whereï( Þ, , ì, í, )=8exp ù-1 2 Ý üý =1 ü =1 ü =11 ý ý sin( Ð ýÞ+ ý )sin( Ñ + )sin( Ò + ) ´sin( Ð ýì+ ý )sin( Ñ í+ )sin( Ò î+ )sin ù! ý . Here, ý =arctan Ð ý 1, =arctan Ñ 2, =arctan Ò 3, ý = Û2( Ð2 ý + Ñ2 + Ò2 )+ Ü-1 4 Ý2, ý =ø 1+(12+ Ð2 ý )(1+2) (2 1+ Ð2 ý )(2 2+ Ð2 ý ) ø 2+(34+ Ñ2 )(3+4) (2 3+ Ñ2 )(2 4+ Ñ2 ) ø 3+(56+ Ò2 )(5+6) (2 5+ Ò2 )(2 6+ Ò2 ) , where the Ð ý , Ñ ,and Ò arepositi veroots ofthetranscendental equationsÐ2-12=(1+2) Ðcot( 1 Ð),Ñ2-34=(3+4) Ñcot( 2 Ñ),Ò2-56=(5+6) Òcot( 3 Ò). Page436 6.4. TELEGRAPH EQUATION "2 #"%$2+ & " #"%$= '2 (3 )- *)+ +( ,, -, ., /) 437 6.4.1-5. Domain: 0£ Þ£ 1,0£ £ 2,0£ £ 3.Mixedboundary value problems. 1 0.Arectangular parallelepiped isconsidered. Thefollowing conditions areprescribed:= 0( Þ, , )at =0 (initial condition ), = 1( Þ, , )at =0 (initial condition ),= 1( , , )at Þ=0(boundary condition ),= 2( , , )at Þ= 1(boundary condition ), = 3( Þ, , )at =0(boundary condition ), = 4( Þ, , )at = 2(boundary condition ), = 5( Þ, , )at =0(boundary condition ), = 6( Þ, , )at = 3(boundary condition ). Solution:( Þ, , , )= ê ë3 0 ê ë2 0 ê ë1 0 0( ì, í, î) ï( Þ, , , ì, í, î, ) ð ì ð í ð î +ê ë3 0 ê ë2 0 ê ë1 0 ñ 1( ì, í, î)+Ý 0( ì, í, î)ò ï( Þ, , , ì, í, î, ) ð ì ð í ð î + Û2ê 0 ê ë3 0 ê ë2 0 1( í, î, ó)ø ì ï( Þ, , , ì, í, î, - ó) 1 =0 ð í ð î ð ó - Û2ê 0 ê ë3 0 ê ë2 0 2( í, î, ó)ø ì ï( Þ, , , ì, í, î, - ó) 1 =ë1 ð í ð î ð ó - Û2ê 0 ê ë3 0 ê ë1 0 3( ì, î, ó) ï( Þ, , , ì,0, î, - ó) ð ì ð î ð ó + Û2ê 0 ê ë3 0 ê ë1 0 4( ì, î, ó) ï( Þ, , , ì, 2, î, - ó) ð ì ð î ð ó - Û2ê 0 ê ë2 0 ê ë1 0 5( ì, í, ó) ï( Þ, , , ì, í,0, - ó) ð ì ð í ð ó + Û2ê 0 ê ë2 0 ê ë1 0 6( ì, í, ó) ï( Þ, , , ì, í, 3, - ó) ð ì ð í ð ó +ê 0 ê ë3 0 ê ë2 0 ê ë1 0 ô( ì, í, î, ó) ï( Þ, , , ì, í, î, - ó) ð ì ð í ð î ð ó, whereï( Þ, , , ì, í, î, )=2 1 2 3exp ù-1 2 Ý üý =1 üþ=0 ü,ÿ =0 þ ÿú ýþ ÿ sin( 2 ýÞ)cos( 3 þ )cos( 4 ÿ) ´sin( 2 ýì)cos( 3 þí)cos( 4 ÿî)sin ù ýþ ÿ , þ= 1for=0, 2for>0, 2 ý = 1, 3 þ= 2, 4 ÿ = 3, ýþ ÿ = Û2( 22 ý + 32þ+ 42 ÿ )+ Ü-1 4 Ý2. Page437 438 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 2 0.Arectangular parallelepiped isconsidered. Thefollowing conditions areprescribed:= 0( Þ, , )at =0 (initial condition ), = 1( Þ, , )at =0 (initial condition ),= 1( , , )at Þ=0(boundary condition ), = 2( , , )at Þ= 1(boundary condition ),= 3( Þ, , )at =0(boundary condition ), = 4( Þ, , )at = 2(boundary condition ),= 5( Þ, , )at =0(boundary condition ), = 6( Þ, , )at = 3(boundary condition ). Solution:( Þ, , , )= 5 63 0 5 62 0 5 61 0 0( 7, 8, 9) :( ;, , , 7, 8, 9, ) < 7 < 8 < 9 +563 0 562 0 561 0 = 1( 7, 8, 9)+ > 0( 7, 8, 9) ? :( ;, , , 7, 8, 9, ) < 7 < 8 < 9 + @25 0 5 63 0 5 62 0 1( 8, 9, A) B 7 :( ;, , , 7, 8, 9, - A) 1 =0 < 8 < 9 < A + @25 0 563 0 562 0 2( 8, 9, A) :( ;, , , 1, 8, 9, - A) < 8 < 9 < A + @25 0 5 63 0 5 61 0 3( 7, 9, A) B 8 :( ;, , , 7, 8, 9, - A) C =0 < 7 < 9 < A + @25 0 5 63 0 5 61 0 4( 7, 9, A) :( ;, , , 7, 2, 9, - A) < 7 < 9 < A + @25 0 562 0 561 0 5( 7, 8, A) B 9 :( ;, , , 7, 8, 9, - A) D =0 < 7 < 8 < A + @25 0 5 62 0 5 61 0 6( 7, 8, A) :( ;, , , 7, 8, 3, - A) < 7 < 8 < A +5 0 563 0 562 0 561 0 ô( 7, 8, 9, A) :( ;, , , 7, 8, 9, - A) < 7 < 8 < 9 < A, where:( ;, , , 7, 8, 9, )=8 1 2 3exp E-1 2 > FG =1 H FI=1 H FKJ =11L GI J sin( 2 G;)sin( M I )sin( N J) ´sin( 2 G7)sin( M I8)sin( N J9)sin EOQP GI J R ,2 G = S(2 T+1) 2 U1, M I= S(2 V+1) 2 U2, N J = S(2 W+1) 2 U3, GI J = @2( 22 G + M2I+ N2 J )+ X-1 4 >2. 6.4.2. Problems inCylindrical Coor dinates Athree-dimensional nonhomogeneous telegraph equation inthecylindrical coordinate system is written asY2 ZYO2+ > YZYO= @2B1 [ YY [ \ [ YZY [ ]+1 [ 2 Y2 ZY ^2+ Y2 ZY2 _- X Z+ `( [ , ^, , O), [ = P a2+ 2. One-dimensional problems with axial symmetry thathavesolutions Z= Z( [ , O)aretreated in Subsection 4.4.2. Two-dimensional problems whose solutions havetheform Z= Z( [ , ^, O)orZ= Z( [ , , O)areconsidered inSubsections 5.4.2 and5.4.3. Page438 6.4. TELEGRAPH EQUATION b2 cb%d2+ e b cb%d= f2 g3 h- ih+ j( k, l, ., /) 439 6.4.2-1. Domain: 0£ [ £ m,0£ ^£2S,0£ £ U.First boundary value problem. Acircular cylinder of®nite length isconsidered. Thefollowing conditions areprescribed:Z= n0( [ , ^, )at O=0(initial condition ),Y oZ= n1( [ , ^, )at O=0(initial condition ),Z= p1( ^, , O)at [ = m(boundary condition ),Z= p2( [ , ^, O)at =0(boundary condition ),Z= p3( [ , ^, O)at = U(boundary condition ). Solution:Z( [ , ^, , O)= YYO q r0 q2 s 0 q t0 u n0(u, v, w) x( [ , ^, ,u, v, w, O) yu y v y w +qr0 q2 s 0 qt0 u z n1(u, v, w)+ { n0(u, v, w) |Qx( [ , ^, ,u, v, w, O) yu y v y w - }2mq o 0 qr0 q2 s 0 p1( v, w, ~) YYu x( [ , ^, ,u, v, w, - ~)_ =t y v y w y ~ + }2q o 0 q2 s 0 qt0 u p2(u, v, ~) YYw x( [ , ^, ,u, v, w, - ~)_ =0 yu y v y ~ - }2q o 0 q2 s 0 qt0 u p3(u, v, ~) YYw x( [ , ^, ,u, v, w, - ~)_ =r yu y v y ~ +q o 0 q r0 q2 s 0 q t0 u `(u, v, w, ~) x( [ , ^, ,u, v, w, - ~) yu y v y w y ~. Here,x( [ , ^, ,u, v, w, )=2 - o 2S m2U =0 =1 K =1 [ ( m)]2 ( ) ( u)cos[ ( ^- v)] ´sin \ ]sin \ w ]sin , where = }22 + }2222+ -1 4 {2, = 1for =0, 2for >0, the (u)aretheBessel functions (theprime denotes thederivativewith respect totheargument), andthe arepositi veroots ofthetranscendental equation ( m)=0. 6.4.2-2. Domain: 0£ £ m,0£ ^£2 ,0£ £ .Second boundary value problem. Acircular cylinder of®nite length isconsidered. Thefollowing conditions areprescribed:= n0( , ^, )at =0 (initial condition ), o= n1( , ^, )at =0 (initial condition ), = p1( ^, , )at = m(boundary condition ), = p2( , ^, )at =0(boundary condition ), = p3( , ^, )at = (boundary condition ). Page439 440 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Solution:( , ^, , )= qr0 q2 s 0 qt0 u n0(u, v, w) x( , ^, ,u, v, w, ) yu y v y w +q r0 q2 s 0 q t0 u z n1(u, v, w)+ { n0(u, v, w) |x( , ^, ,u, v, w, ) yu y v y w + }2mq o 0 qr0 q2 s 0 p1( v, w, ~) x( , ^, , m, v, w, - ~) y v y w y ~ - }2q o 0 q2 s 0 qt0 u p2(u, v, ~) x( , ^, ,u, v,0, - ~) yu y v y ~ + }2q o 0 q2 s 0 qt0 u p3(u, v, ~) x( , ^, ,u, v, , - ~) yu y v y ~ +q o 0 qr0 q2 s 0 qt0 u `(u, v, w, ~) x( , ^, ,u, v, w, - ~) yu y v y w y ~. Here,x( , ^, ,u, v, w, )=exp -1 2 { sin m2 +2 m2 K =11 ¡ cos ¢ £ ¤cos ¢ ¥ ¤sin Q¦¡ +1 =0 =1 =0 2 ( ) ( ¥) ( 2 §2- 2)[ ( §)]2cos[ ( ¨- ©)]cos ¢ £ ¤cos ¢ ¥ ¤sin( ) ª , = - «2 4,¡ = ¬2222+ - «2 4, = ¬22 + ¬2222+ - «2 4, = 1for =0, 2for >0, where the ( ¥)aretheBessel functions andthe arepositi veroots ofthetranscendental equation ( §)=0. 6.4.2-3. Domain: 0£ £ §,0£ ¨£2 ,0£ £ .Third boundary value problem. Acircular cylinder of®nite length isconsidered. Thefollowing conditions areprescribed:= ®0( , ¨, )at ¯=0(initial condition ), °= ®1( , ¨, )at ¯=0(initial condition ), + 1 = ±( ¨, , ¯)at = §(boundary condition ), - 2 = ±2( , ¨, ¯)at =0(boundary condition ), + 3 = ±3( , ¨, ¯)at = (boundary condition ). Thesolution ( , ¨, , ¯)isdetermined bytheformula inParagraph 6.4.2-2 where²( , ¨, , ¥, ©, ³, ¯)=1exp -1 2« ¯ ´µ¶ =0 ´µ·=1 ´µ¸ =1 ¹ ¶ º 2 ¶· » ¶ ( º ¶· ¼) » ¶ ( º ¶·¥) ( º 2 ¶·§2+ ½2 1 §2- ¾2)[ » ¶ ( º ¶·§)]2 ´cos[ ¾( ¨- ©)] ¿ ¸ ( À)¿ ¸ ( ³)Á¿ ¸Á2sin Â¯¦ Ã ¶· ¸ Ä¦ Ã ¶· ¸ . Here, the » ¶ ( ¥)aretheBessel functions,¹ ¶ = Å1for ¾=0, 2for ¾>0, Ã ¶· ¸ =¬2 º 2 ¶·+¬2 Æ2 ¸ + Ç-1 4«2,¿ ¸ ( À)=cos( Æ ¸À)+ ½2Æ ¸ sin( Æ ¸À), Á¿ ¸Á2= ½3 2 Æ2 ¸Æ2 ¸ + ½2 2Æ2 ¸ + ½2 3+ ½2 2 Æ2 ¸ + È2 É1+ ½2 2Æ2 ¸ Ê ; Page440 6.4. TELEGRAPH EQUATION 2 2+ = 2 3 - + ( , , , ) 441 the and arepositi veroots ofthetranscendental equations ( )+ 1 ( )=0,tan( )= 2+ 32- 2 3. 6.4.2-4. Domain: 0£ £ ,0£ £2 ,0£ £ .Mixedboundary value problems. 1 .Acircular cylinder of®nite length isconsidered. Thefollowing conditions areprescribed:= 0( , , )at =0 (initial condition ), != 1( , , )at =0 (initial condition ),= "1( , , )at = (boundary condition ), #= "2( , , )at =0(boundary condition ), #= "3( , , )at = (boundary condition ). Solution:( , , , )= $ % 0 $2 & 0 $ ' 0 ( 0((, ), *) +( , , ,(, ), *, ) ,( , ) , * + $ % 0 $2 & 0 $ ' 0( - 1((, ), *)+ . 0((, ), *) /0+( , , ,(, ), *, ) ,( , ) , * - 12 $ ! 0 $ % 0 $2 & 0 "1( ), *, 2) 3 ( +( , , ,(, ), *, - 2) 4 5 = ' , ) , * , 2 - 12 $ ! 0 $2 & 0 $ ' 0 ( "2((, ), 2) +( , , ,(, ),0, - 2) ,( , ) , 2 + 12 $ ! 0 $2 & 0 $ ' 0 ( "3((, ), 2) +( , , ,(, ), , - 2) ,( , ) , 2 + $ ! 0 $ % 0 $2 & 0 $ ' 0 ( 6((, ), *, 2) +( , , ,(, ), *, - 2) ,( , ) , * , 2. Here,+( , , ,(, ), *, )=1 2exp 7-1 2 . 98 : ;=0 : ;=1 : ;=< =0 > > < [ ( )]2 ? @ < ( ) ( () ´cos[ A( - ))]cos B C Dcos B C * Dsin 7E0F @ <8,@ < = 122 + 12222+ G-1 4 .2,> = H1for A=0, 2for A>0, where the (()aretheBessel functions (the prime denotes thederivativewith respect tothe argument) andthe arepositi veroots ofthetranscendental equation ( )=0. 2 .Acircular cylinder of®nite length isconsidered. Thefollowing conditions areprescribed:= 0( , , )at =0(initial condition ), != 1( , , )at =0(initial condition ), I= "1( , , )at = (boundary condition ),= "2( , , )at =0(boundary condition ),= "3( , , )at = (boundary condition ). Page441 442 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Solution:( , , , )= $ % 0 $2 & 0 $ ' 0( 0((, ), *) +( , , ,(, ), *, ) ,( , ) , * + $ % 0 $2 & 0 $ ' 0 (- 1((, ), *)+ . 0((, ), *)/ +( , , ,(, ), *, ) ,( , ) , * + 12 $ ! 0 $ % 0 $2 & 0 "1( ), *, 2) +( , , , , ), *, - 2) , ) , * , 2 + 12 $ ! 0 $2 & 0 $ ' 0 ( "2((, ), 2) 3 * +( , , ,(, ), *, - 2) 4 J =0 ,( , ) , 2 - 12 $ ! 0 $2 & 0 $ ' 0 ( "3((, ), 2) 3 * +( , , ,(, ), *, - 2) 4 J = % ,( , ) , 2 + $ ! 0 $ % 0 $2 & 0 $ ' 0( 6((, ), *, 2) +( , , ,(, ), *, - 2) ,( , ) , * , 2. Here,+( , , ,(, ), *, )=2 2exp 7-1 2 . 8: ; < =11? < sin B C Dsin B C * Dsin 7F <8 +2 exp 7-1 2 . 98: ;=0 : ;=1 : ; < =1 > 2 ( 2 2- A2)[ ( )]2 ( ) ( () ´cos[ A( - ))]sin B C Dsin B C * Dsin 7E ? @ <8?@ < , < = 12222+ G-1 4 .2,> =H1for A=0, 2for A>0,@ < = 122 + 12222+ G-1 4 .2, where the (()aretheBessel functions andthe arepositi veroots ofthetranscendental equation( )=0. 6.4.2-5. Domain: 1£ £ 2,0£ £2 ,0£ £ .First boundary value problem. Ahollo wcircular cylinder of®nite length isconsidered. Thefollowing conditions areprescribed:= 0( , , )at =0 (initial condition ), != 1( , , )at =0 (initial condition ),= "1( , , )at = 1(boundary condition ),= "2( , , )at = 2(boundary condition ),= "3( , , )at =0 (boundary condition ),= "4( , , )at = (boundary condition ). Page442 6.4. TELEGRAPH EQUATION 2 2+ = 2 3 - + ( , , , ) 443 Solution:( , , , )= $ % 0 $2 & 0 $ '2'1 0((, ), *) +( , , ,(, ), *, )( ,( , ) , * + $ % 0 $2 & 0 $ '2'1 - 1((, ), *)+ . 0((, ), *) /0+( , , ,(, ), *, )( ,( , ) , * + 121 $ ! 0 $ % 0 $2 & 0 "1( ), *, 2) 3 ( +( , , ,(, ), *, - 2) 4 5 = '1 , ) , * , 2 - 122 $ ! 0 $ % 0 $2 & 0 "2( ), *, 2) 3 ( +( , , ,(, ), *, - 2) 4 5 = '2 , ) , * , 2 + 12 $ ! 0 $2 & 0 $ '2'1 "3((, ), 2) 3 * +( , , ,(, ), *, - 2) 4 J =0 ( ,( , ) , 2 - 12 $ ! 0 $2 & 0 $ '2'1 "4((, ), 2) 3 * +( , , ,(, ), *, - 2) 4 J = %( ,( , ) , 2 + $ ! 0 $ % 0 $2 & 0 $ '2'1 6((, ), *, 2) +( , , ,(, ), *, - 2)( ,( , ) , * , 2. Here,+( , , ,(, ), *, )= 2 exp 7-1 2 . 98 : ;=0 : ;=1 : ; < =1> 2 2( 2)2( 1)- 2( 2) K ( )K (() ´cos[ A( - ))]sin B C Dsin B C * Dsin 7 ?@ <8? @ < ,> =H1for A=0, 2for A¹0, @ < = 122 + 12222+ G-1 4 .2,K ( )= ( 1) L ( )- L ( 1) ( ), where the ( )and L ( )aretheBessel functions, andthe arepositi veroots ofthetranscen- dental equation ( 1) L ( 2)- L ( 1) ( 2)=0. 6.4.2-6. Domain: 1£ £ 2,0£ £2 ,0£ £ .Second boundary value problem. Ahollo wcircular cylinder of®nite length isconsidered. Thefollowing conditions areprescribed:= 0( , , )at =0 (initial condition ), != 1( , , )at =0 (initial condition ), I= "1( , , )at = 1(boundary condition ), I= "2( , , )at = 2(boundary condition ), #= "3( , , )at =0 (boundary condition ), #= "4( , , )at = (boundary condition ). Page443 444 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Solution:( , , , )= $ % 0 $2 & 0 $ '2'1 0((, ), *) +( , , ,(, ), *, )( ,( , ) , * + $ % 0 $2 & 0 $ '2'1 - 1((, ), *)+ . 0((, ), *) /0+( , , ,(, ), *, )( ,( , ) , * - 121 $ ! 0 $ % 0 $2 & 0 "1( ), *, 2) +( , , , 1, ), *, - 2) , ) , * , 2 + 122 $ ! 0 $ % 0 $2 & 0 "2( ), *, 2) +( , , , 2, ), *, - 2) , ) , * , 2 - 12 $ ! 0 $2 & 0 $ '2'1 "3((, ), 2) +( , , ,(, ),0, - 2)( ,( , ) , 2 + 12 $ ! 0 $2 & 0 $ '2'1 "4((, ), 2) +( , , ,(, ), , - 2)( ,( , ) , 2 + $ ! 0 $ % 0 $2 & 0 $ '2'1 6((, ), *, 2) +( , , ,(, ), *, - 2)( ,( , ) , * , 2. Here,+( , , ,(, ), *, )= M- N !EO 2( 2 2- 2 1) 3sin 7E ? P8?P+2 : ; < =1cos B C Dcos B C * Dsin 7E ? <8? <4 + M- N !EO 2 : ;=0 : ;=1 : ; < =0> > <2 K ( )K (() ( 2 2 2- A2)K2 ( 2)-( 2 2 1- A2)K2 ( 1) ´cos[ A( - ))]cos B C Dcos B C * Dsin 7Q ?@ <8? @ < , where> =H1for A=0, 2for A¹0, P= G-1 4 .2, < = 12222+ G-1 4 .2, @ < = 122 + 12222+ G-1 4 .2,K ( )= ( 1) L ( )- L ( 1) ( ); the ( )and L ( )aretheBessel functions, andthe arepositi veroots ofthetranscendental equation ( 1) L ( 2)- L ( 1) ( 2)=0. 6.4.2-7. Domain: 1£ £ 2,0£ £2 ,0£ £ .Third boundary value problem. Ahollo wcircular cylinder of®nite length isconsidered. Thefollowing conditions areprescribed:= 0( , , )at =0 (initial condition ), != 1( , , )at =0 (initial condition ), I- 1 = "1( , , )at = 1(boundary condition ), I+ 2 = "2( , , )at = 2(boundary condition ), #- 3 = "3( , , )at =0 (boundary condition ), #+ 4 = "4( , , )at = (boundary condition ). Page444 6.4. TELEGRAPH EQUATION 2 2+ = 2 3 - + ( , , , ) 445 Thesolution ( , , , )isdetermined bytheformula inParagraph 6.4.2-6 where+( , , ,(, ), *, )=1exp 7-1 2 . 98: ;=0 : ;=1 : ;=1 > 2 RS R2F122 + 12@2+ G- .2 T4 ´ K ( )K (()cos[ A( - ))] S( ) S( *)sin 7E F122 + 12@2+ G- .2 T4 8 ( 2 2 2 2+ 2 2 2- A2)K2 ( 2)-( 2 1 2 1+ 2 2 1- A2)K2 ( 1). Here,K ( )=- ( 1)- 1 ( 1)/ L ( ) -- L ( 1)- 1 L ( 1)/ ( ),> =H1for A=0, 2for A¹0, S( )=cos( @ )+ 3@sin( @ ), RS R2= 4 2 @2 @2+ 2 3@2+ 2 4+ 3 2 @2+ 2 B1+ 2 3@2 D, where the ( )and L ( )aretheBessel functions, the arepositi veroots ofthetranscendental equation- ( 1)- 1 ( 1)/- L ( 2)+ 2 L ( 2)/ =- L ( 1)- 1 L ( 1)/- ( 2)+ 2 ( 2)/, andthe @arepositi veroots ofthetranscendental equationtan( @)@= 3+ 4@2- 3 4. 6.4.2-8. Domain: 1£ £ 2,0£ £2 ,0£ £ .Mixedboundary value problems. 1 .Ahollo wcircular cylinder of®nite lengthisconsidered .Thefollowing conditions areprescrib ed:= 0( , , )at =0 (initial condition ), != 1( , , )at =0 (initial condition ),= "1( , , )at = 1(boundary condition ),= "2( , , )at = 2(boundary condition ), #= "3( , , )at =0 (boundary condition ), #= "4( , , )at = (boundary condition ). Solution:( , , , )= $ % 0 $2 & 0 $ '2'1 0((, ), *) +( , , ,(, ), *, )( ,( , ) , * + $ % 0 $2 & 0 $ '2'1 - 1((, ), *)+ . 0((, ), *) /0+( , , ,(, ), *, )( ,( , ) , * + 121 $ ! 0 $ % 0 $2 & 0 "1( ), *, 2) 3 ( +( , , ,(, ), *, - 2) 4 5 = '1 , ) , * , 2 - 122 $ ! 0 $ % 0 $2 & 0 "2( ), *, 2) 3 ( +( , , ,(, ), *, - 2) 4 5 = '2 , ) , * , 2 - 12 $ ! 0 $2 & 0 $ '2'1 "3((, ), 2) +( , , ,(, ),0, - 2)( ,( , ) , 2 + 12 $ ! 0 $2 & 0 $ '2'1 "4((, ), 2) +( , , ,(, ), , - 2)( ,( , ) , 2 + $ ! 0 $ % 0 $2 & 0 $ '2'1 6((, ), *, 2) +( , , ,(, ), *, - 2)( ,( , ) , * , 2. Page445 446 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Here,+( , , ,(, ), *, )= 4 exp 7-1 2 . 98 : ;=0 : ;=1 : ; < =0> > <2 2( 2)2( 1)- 2( 2) K ( )K (() ´cos[ A( - ))]cos B C Dcos B C * Dsin 7 ?@ <8? @ < ,> =H1for A=0, 2for A¹0, @ < = 122 + 12222+ G-1 4 .2,K ( )= ( 1) L ( )- L ( 1) ( ), where the ( )and L ( )aretheBessel functions, andthe arepositi veroots ofthetranscen- dental equation ( 1) L ( 2)- L ( 1) ( 2)=0. 2 .Ahollo wcircular cylinder of®nite lengthisconsidered .Thefollowing conditions areprescrib ed:= 0( , , )at =0 (initial condition ), != 1( , , )at =0 (initial condition ), I= "1( , , )at = 1(boundary condition ), I= "2( , , )at = 2(boundary condition ),= "3( , , )at =0 (boundary condition ),= "4( , , )at = (boundary condition ). Solution:( , , , )= $ % 0 $2 & 0 $ '2'1 0((, ), *) +( , , ,(, ), *, )( ,( , ) , * + $ % 0 $2 & 0 $ '2'1 - 1((, ), *)+ . 0((, ), *)/ +( , , ,(, ), *, )( ,( , ) , * - 121 $ ! 0 $ % 0 $2 & 0 "1( ), *, 2) +( , , , 1, ), *, - 2) , ) , * , 2 + 122 $ ! 0 $ % 0 $2 & 0 "2( ), *, 2) +( , , , 2, ), *, - 2) , ) , * , 2 + 12 $ ! 0 $2 & 0 $ '2'1 "3((, ), 2) 3 * +( , , ,(, ), *, - 2) 4 J =0 ( ,( , ) , 2 - 12 $ ! 0 $2 & 0 $ '2'1 "4((, ), 2) 3 * +( , , ,(, ), *, - 2) 4J = %( ,( , ) , 2 + $ ! 0 $ % 0 $2 & 0 $ '2'1 6((, ), *, 2) +( , , ,(, ), *, - 2)( ,( , ) , * , 2. Here,+( , , ,(, ), *, )=2M- N !EO 2( 2 2- 2 1) : ; < =1sin B C Dsin B C * Dsin 7 ? <8? < +2M- N !EO 2 : ;=0 : ;=1 : ; < =1> 2 K ( )K (() ( 2 2 2- A2)K2 ( 2)-( 2 2 1- A2)K2 ( 1) ´cos[ A( - ))]sin B C Dsin B C * Dsin 7E ?@ <8? @ < , Page446 6.4. TELEGRAPH EQUATION 2 2+ = 2 3 - + ( , , , ) 447 where> =H1for A=0, 2for A¹0, < = 12222+ G-1 4 .2, @ < = 122 + 12222+ G-1 4 .2,K ( )= ( 1) L ( )- L ( 1) ( ); the ( )and L ( )aretheBessel functions, andthe arepositi veroots ofthetranscendental equation ( 1) L ( 2)- L ( 1) ( 2)=0. 6.4.2-9. Domain: 0£ £ ,0£ £ 0,0£ £ .First boundary value problem. Acylindrical sector of®nite thickness isconsidered. Thefollowing conditions areprescribed:= 0( , , )at =0 (initial condition ), != 1( , , )at =0 (initial condition ),= "1( , , )at = (boundary condition ),= "2( , , )at =0 (boundary condition ),= "3( , , )at = 0(boundary condition ),= "4( , , )at =0 (boundary condition ),= "5( , , )at = (boundary condition ). Solution:( , , , )= $ % 0 $ U0 0 $ ' 0 0((, ), *) +( , , ,(, ), *, )( ,( , ) , * + $ % 0 $ U0 0 $ ' 0- 1((, ), *)+ . 0((, ), *) /0+( , , ,(, ), *, )( ,( , ) , * - 12 $ ! 0 $ % 0 $ U0 0 "1( ), *, 2) 3 ( +( , , ,(, ), *, - 2) 4 5 = ' , ) , * , 2 + 12 $ ! 0 $ % 0 $ ' 0 "2((, *, 2)1( 3 ) +( , , ,(, ), *, - 2) 4 V =0 ,( , * , 2 - 12 $ ! 0 $ % 0 $ ' 0 "3((, *, 2)1( 3 ) +( , , ,(, ), *, - 2) 4 V = U0 ,( , * , 2 + 12 $ ! 0 $ U0 0 $ ' 0 "4((, ), 2) 3 * +( , , ,(, ), *, - 2) 4J =0 ( ,( , ) , 2 - 12 $ ! 0 $ U0 0 $ ' 0 "5((, ), 2) 3 * +( , , ,(, ), *, - 2) 4 J = %( ,( , ) , 2 + $ ! 0 $ % 0 $ U0 0 $ ' 06((, ), *, 2) +( , , ,(, ), *, - 2)( ,( , ) , * , 2. Here,+( , , ,(, ), *, )=8M- N !EO 22E0 : ;=1 : ;=1 : ; < =1 & OU0( ) & OU0( () [ & OU0( )]2sin B A 0 Dsin B A )0 D ´sin B C Dsin B C * Dsin 7EF 122 + 1222-2+ G- .2 T4 8F 122 + 1222-2+ G- .2 T4, where the & OU0( )aretheBessel functions andthe arepositi veroots ofthetranscendental equation & OU0( )=0. Page447 448 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 6.4.2-10. Domain: 0£ £ ,0£ £ 0,0£ £ .Mixedboundary value problem. Acylindrical sector of®nite thickness isconsidered. Thefollowing conditions areprescribed:= 0( , , )at =0 (initial condition ), != 1( , , )at =0 (initial condition ),= "1( , , )at = (boundary condition ),= "2( , , )at =0 (boundary condition ),= "3( , , )at = 0(boundary condition ), #= "4( , , )at =0 (boundary condition ), #= "5( , , )at = (boundary condition ). Solution:( , , , )= $ % 0 $ U0 0 $ ' 0 0((, ), *) +( , , ,(, ), *, )( ,( , ) , * + $ % 0 $ U0 0 $ ' 0- 1((, ), *)+ . 0((, ), *)/ +( , , ,(, ), *, )( ,( , ) , * - 12 $ ! 0 $ % 0 $ U0 0 "1( ), *, 2) 3 ( +( , , ,(, ), *, - 2) 4 5 = ' , ) , * , 2 + 12 $ ! 0 $ % 0 $ ' 0 "2((, *, 2)1( 3 ) +( , , ,(, ), *, - 2) 4 V =0 ,( , * , 2 - 12 $ ! 0 $ % 0 $ ' 0 "3((, *, 2)1( 3 ) +( , , ,(, ), *, - 2) 4 V = U0 ,( , * , 2 - 12 $ ! 0 $ U0 0 $ ' 0 "4((, ), 2) +( , , ,(, ),0, - 2)( ,( , ) , 2 + 12 $ ! 0 $ U0 0 $ ' 0 "5((, ), 2) +( , , ,(, ), , - 2)( ,( , ) , 2 + $ ! 0 $ % 0 $ U0 0 $ ' 0 6((, ), *, 2) +( , , ,(, ), *, - 2)( ,( , ) , * , 2. Here,+( , , ,(, ), *, )=4M- N !EO 22E0 : ;=1 : ;=1 : ;W< =0 > <& OU0( ) & OU0( () [ & OU0( )]2sin B A 0 Dsin B A )0 D ´cos B C Dcos B C * Dsin 7EF 122 + 1222-2+ G- .2 T4 8F 122 + 1222-2+ G- .2 T4, where>0=1and> < =2for ³1;the & OU0( )aretheBessel functions; andthe arepositi ve roots ofthetranscendental equation & OU0( )=0. 6.4.3. Problems inSpherical Coor dinates Athree-dimensional nonhomogeneous telegraph equation inthespherical coordinate system is written as 2 2+ . = 12312 B 2 D+12sin X X Bsin X X D+12sin2X 2 2 4- G +6( , X, , ). Page448 6.4. TELEGRAPH EQUATION 2 2+ = 2 3 - + ( , , , ) 449 6.4.3-1. Domain: 0£ £ ,0£ X£ ,0£ £2 .First boundary value problem. Aspherical domain isconsidered. Thefollowing conditions areprescribed:= 0( , X, )at =0 (initial condition ), != 1( , X, )at =0 (initial condition ),= "( X, , )at = (boundary condition ). Solution:( , X, , )= $2 & 0 $& 0 $ ' 0 0((, ), *) +( , X, ,(, ), *, )(2sin ) ,( , ) , * + $2 & 0 $& 0 $ ' 0- 1((, ), *)+ . 0((, ), *)/ +( , X, ,(, ), *, )(2sin ) ,( , ) , * - 122 $ ! 0 $2 & 0 $& 0 "( ), *, 2) 3 ( +( , X, ,(, ), *, - 2) 4 5 = 'sin ) , ) , * , 2 + $ ! 0 $2 & 0 $& 0 $ ' 06((, ), *, 2) +( , X, ,(, ), *, - 2)(2sin ) ,( , ) , * , 2, where+( , X, ,(, ), *, )=1 2 2?(exp 7-1 2 . 98 : ;=0 : ;=1 ;W< =0> <ZY <+1 O 2( @ ) +1 O 2( @ () ´ [ <(cos X) [ <(cos ))cos[ ( - *)]sin 7EF 12 @2 + G- .2 T4 8F 12 @2 + G- .2 T4,> < =H1for =0, 2for ¹0, Y < =(2 A+1)( A- )! ( A+ )!- +1 O 2( @ )/2. Here, the +1 O 2( )aretheBessel functions, the [ <( )aretheassociated Legendre functions expressed interms oftheLegendre polynomials [ ( )as[ <( )=(1- 2) <O 2 , <, <[ ( ), [ ( )=1A!2 , , ( 2-1) , andthe @ arepositi veroots ofthetranscendental equation +1 O 2( @)=0. 6.4.3-2. Domain: 0£ £ ,0£ X£ ,0£ £2 .Second boundary value problem. Aspherical domain isconsidered. Thefollowing conditions areprescribed:= 0( , X, )at =0 (initial condition ), != 1( , X, )at =0 (initial condition ), I= "( X, , )at = (boundary condition ). Solution:( , X, , )= $2 & 0 $& 0 $ ' 0 0((, ), *) +( , X, ,(, ), *, )(2sin ) ,( , ) , * + $2 & 0 $& 0 $ ' 0- 1((, ), *)+ . 0((, ), *)/ +( , X, ,(, ), *, )(2sin ) ,( , ) , * + 122 $ ! 0 $2 & 0 $& 0 "( ), *, 2) +( , X, , , ), *, - 2)sin ) , ) , * , 2 + $ ! 0 $2 & 0 $& 0 $ ' 06((, ), *, 2) +( , X, ,(, ), *, - 2)(2sin ) ,( , ) , * , 2, Page449 450 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES where+( , X, ,(, ), *, )=3M- N !EO 2 4 3sin 7E ?P8?P+ M- N !EO 2 2 ?( : ;=0 : ;=1 ; < =0> <\Y <+1 O 2( @ ) ´ +1 O 2( @ () [ <(cos X) [ <(cos ))cos[ ( - *)]sin 7EF 12 @2 + P8F 12 @2 + P,> < =H1for =0, 2for ¹0, Y < = @2 (2 A+1)( A- )! ( A+ )!- 2 @2 - A( A+1)/- +1 O 2( @ )/2, P= G-1 4 .2. Here, the +1 O 2( )aretheBessel functions, the [ <( )aretheassociated Legendre functions (see Paragraph 6.4.3-1), andthe @ arepositi veroots ofthetranscendental equation 2 @ +1 O 2( @)- +1 O 2( @)=0. 6.4.3-3. Domain: 0£ £ ,0£ X£ ,0£ £2 .Third boundary value problem. Aspherical domain isconsidered. Thefollowing conditions areprescribed:= 0( , X, )at =0 (initial condition ), != 1( , X, )at =0 (initial condition ), I+ = "( X, , )at = (boundary condition ). Thesolution ( , X, , )isdetermined bytheformula inParagraph 6.4.3-2 where+( , X, ,(, ), *, )= M- N !EO 2 2 ?( : ;=0 : ;=1 ;%=0> % Y %+1 O 2( @ ) +1 O 2( @ () ´ [ %(cos X) [ %(cos ))cos[ ( - *)]sin 7EF 12 @2 + G- .2 T4 8F 12 @2 + G- .2 T4,> %=H1for =0, 2for ¹0, Y %= @2 (2 A+1)( A- )! ( A+ )!- 2 @2 +( 0+ A)( 0- A-1)/- +1 O 2( @ )/2. Here, the +1 O 2( )aretheBessel functions, the [ %( )aretheassociated Legendre functions (see Paragraph 6.4.3-1), andthe @ arepositi veroots ofthetranscendental equation@ +1 O 2( @)+ 70-1 2 8+1 O 2( @)=0. 6.4.3-4. Domain: 1£ £ 2,0£ X£ ,0£ £2 .First boundary value problem. Aspherical layer isconsidered. Thefollowing conditions areprescribed:= 0( , X, )at =0 (initial condition ), != 1( , X, )at =0 (initial condition ),= "1( X, , )at = 1(boundary condition ),= "2( X, , )at = 2(boundary condition ). Page450 6.4. TELEGRAPH EQUATION 2 2+ = 2 3 - + ( , , , ) 451 Solution:( , X, , )= $2 & 0 $& 0 $ '2'1 0((, ), *) +( , X, ,(, ), *, )(2sin ) ,( , ) , * + $2 & 0 $& 0 $ '2'1 - 1((, ), *)+ . 0((, ), *)/ +( , X, ,(, ), *, )(2sin ) ,( , ) , * + 122 1 $ ! 0 $2 & 0 $& 0 "1( ), *, 2) 3 ( +( , X, ,(, ), *, - 2) 4 5 = '1sin ) , ) , * , 2 - 122 2 $ ! 0 $2 & 0 $& 0 "2( ), *, 2) 3 ( +( , X, ,(, ), *, - 2) 4 5 = '2sin ) , ) , * , 2 + $ ! 0 $2 & 0 $& 0 $ '2'1 6((, ), *, 2) +( , X, ,(, ), *, - 2)(2sin ) ,( , ) , * , 2, where+( , X, ,(, ), *, )= M- N !EO 2 8 ?( : ;=0 : ;=1 ; < =0> <ZY <K +1 O 2( @ )K +1 O 2( @ () ´ [ <(cos X) [ <(cos ))cos[ ( - *)]sin 7EF 12 @2 + G- .2 T4 8F 12 @2 + G- .2 T4,K +1 O 2( @ )= +1 O 2( @ 1) L+1 O 2( @ )- L+1 O 2( @ 1) +1 O 2( @ ),> < =H1for =0, 2for ¹0, Y < = @ (2 A+1)( A- )! 2+1 O 2( @ 2) ( A+ )!- 2+1 O 2( @ 1)- 2+1 O 2( @ 2)/. Here, the +1 O 2( )aretheBessel functions, the [ <( )aretheassociated Legendre functions expressed interms oftheLegendre polynomials [ ( )as[ <( )=(1- 2) <O 2 , <, <[ ( ), [ ( )=1A!2 , , ( 2-1) , andthe @ arepositi veroots ofthetranscendental equationK +1 O 2( @2)=0. 6.4.3-5. Domain: 1£ £ 2,0£ X£ ,0£ £2 .Second boundary value problem. Aspherical layer isconsidered. Thefollowing conditions areprescribed:= 0( , X, )at =0 (initial condition ), != 1( , X, )at =0 (initial condition ), I= "1( X, , )at = 1(boundary condition ), I= "2( X, , )at = 2(boundary condition ). Solution:( , X, , )= $2 & 0 $& 0 $ '2'1 0((, ), *) +( , X, ,(, ), *, )(2sin ) ,( , ) , * + $2 & 0 $& 0 $ '2'1 - 1((, ), *)+ . 0((, ), *)/ +( , X, ,(, ), *, )(2sin ) ,( , ) , * - 122 1 $ ! 0 $2 & 0 $& 0 "1( ), *, 2) +( , X, , 1, ), *, - 2)sin ) , ) , * , 2 + 122 2 $ ! 0 $2 & 0 $& 0 "2( ), *, 2) +( , X, , 2, ), *, - 2)sin ) , ) , * , 2 + $ ! 0 $2 & 0 $& 0 $ '2'1 6((, ), *, 2) +( , X, ,(, ), *, - 2)(2sin ) ,( , ) , * , 2, Page451 452 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES where+( , X, ,(, ), *, )=3M- N !EO 2sin 7E ?P8 4 ( 3 2- 3 1) ?P+ M- N !EO 2 4 ?( : ;=0 : ;=1 ; < =0> <Y <K +1 O 2( @ )K +1 O 2( @ () ´ [ <(cos X) [ <(cos ))cos[ ( - *)]sin 7EF 12 @2 + P8F 12 @2 + P. Here,> < =H1for =0, 2for ¹0, Y < =( A+ )! (2 A+1)( A- )! $ '2'1 K2+1 O 2( @ ) , , P= G-1 4 .2,K +1 O 2( @ )= 3 @ +1 O 2( @ 1)-1 2 1 +1 O 2( @ 1) 4 L+1 O 2( @ ) - 3 @ L +1 O 2( @ 1)-1 2 1 L+1 O 2( @ 1) 4 +1 O 2( @ ), where the +1 O 2( )and L+1 O 2( )aretheBessel functions, the [ <( )aretheassociated Legendre functions (seeParagraph 6.4.3-4), andthe @ arepositi veroots ofthetranscendental equation@K +1 O 2( @2)-1 2 2 K +1 O 2( @2)=0. 6.4.3-6. Domain: 1£ £ 2,0£ X£ ,0£ £2 .Third boundary value problem. Aspherical layer isconsidered. Thefollowing conditions areprescribed:= 0( , X, )at =0 (initial condition ), != 1( , X, )at =0 (initial condition ), I- 1 = "1( X, , )at = 1(boundary condition ), I+ 2 = "2( X, , )at = 2(boundary condition ). Thesolution ( , X, , )isdetermined bytheformula inParagraph 6.4.3-5 where+( , X, ,(, ), *, )= M- N !EO 2 4 ?( : ;=0 : ;=1 ;%=0 > % Y %K +1 O 2( @ )K +1 O 2( @ () ´ [ %(cos X) [ %(cos ))cos[ ( - *)]sin 7EF 12 @2 + P8F 12 @2 + P. Here,> %=H1for =0, 2for ¹0, Y %=( A+ )! (2 A+1)( A- )! $ '2'1 K2+1 O 2( @ ) , , P= G-1 4 .2,K +1 O 2( @)= 3 @ +1 O 2( @1)- B 1+1 2 1 D +1 O 2( @1) 4 L+1 O 2( @) - 3 @L +1 O 2( @1)- B 1+1 2 1 D L+1 O 2( @1) 4 +1 O 2( @), where the +1 O 2( )and L+1 O 2( )aretheBessel functions, the [ %( )aretheassociated Legendre functions (seeParagraph 6.4.3-4), andthe @ arepositi veroots ofthetranscendental equation@K +1 O 2( @2)+ B 2-1 2 2 D K +1 O 2( @2)=0. Page452 6.5. OTHER EQUATIONS WITH THREE SPACEVARIABLES 453 6.5. Other Equations with Three Space Variab les 6.5.1. Equations Containing Arbitrar yParameter s 1. ]2 ^] _2= ]] ` B a` b ] ^] ` D+ ]] c B dc e ] ^] c D+ ]] f B gf h ] ^] f D. This equation admits separable solutions. Inaddition, for A¹2, i¹2,and .¹2,there areparticular solutions oftheform j = j ((, k),(2=4 3 l2- m1(2- A)2+ n2- oG(2- i)2+ p2- NP(2- .)2 4, where j ((, k)isdetermined bytheone-dimensional nonstationary equationq2 jqk2= q2 jq r2+> r q jq r,>=2 B1 2- A+1 2- i+1 2- . D-1. 2. ]2 ^] _2+ s ] ^] _= a2B ]2 ^] `2+ ]2 ^] c2+ ]2 ^] f2D+ d1 ] ^] `+ d2 ] ^] c+ d3 ] ^] f+ g ^. Thetransformation j (l,n,p, k)= t(l,n,p, u)exp B-1 2 v k- G1l+ G2n+ G3p2 w2D, u= w k leads totheequation inSubsection 6.3.1:q2tqu2= q2tql2+ q2tqn2+ q2tqp2+ x t, x= Pw2+ v2 4 w2-1 4 w4 y G2 1+ G2 2+ G2 3 z. 6.5.2. Equation oftheForm{( |, }, ~) 2 2=div[ ( |, }, ~)Ñ ]± ( |, }, ~) + ( |, }, ~,) Such equations areencountered when studying vibration of®nite volumes. Theequation iswritten using thenotation div w(r)Ñ j = qql w(r) q jql + qqn w(r) q jqn + qqp w(r) q jqp , r={l,n,p}. Theproblems fortheequation inquestion areconsidered belowfortheinterior ofabounded domain with smooth surface .Inwhat follows,itisassumed that (r)>0, w(r)>0,and (r)³0. 6.5.2-1. First boundary value problem. Thesolution oftheequation inquestion with theinitial conditions j = 0(r)at k=0,q j = 1(r)at k=0(1) andthenonhomogeneous boundary conditions ofthe®rstkind j = (r, k)for r (2) Page453 454 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES canbewritten asthesum j (r, k)= qqk 0( ) ( ) (r, , ) + 1( ) ( ) (r, , ) - 0 ( , ) ( ) (r, , - ) + 0 ( , ) (r, , - ) .(3) Here, themodi®ed Green' sfunction isexpressed as(r, , )= ¢¡ =11£ ¤ ¡ ¥C¦ ¡ ¥ 2 ¦ ¡ (r) ¦ ¡ ( )sin( § ¤ ¡),¥C¦ ¡ ¥ 2= ¨(r) ¦ 2 ¡ (r) ©, ={ r 1, r 2, r 3},(4) where the ¤ ¡ and ¦ ¡ (r)aretheeigen values andcorresponding eigenfunctions oftheSturm±Liouville problem forthefollowing second-order elliptic equation with homogeneous boundary conditions of the®rstkind: div ª(r)Ñ ¦ « - ¬(r) ¦ + ¤¨(r) ¦ =0, (5) ¦ =0for r ®. (6) Theintegration insolution (3)isperformed with respect to ¯1, ¯2,and ¯3; °° ± ²isthederivativealong theoutw ardnormal tothesurface ®with respect to ¯1, ¯2,and ¯3. General properties oftheSturm±Liouville problem (5)±(6): 1 ³.There are®nitely manyeigen values. Alleigen values arereal andcanbeordered sothat¤ 1£ ¤ 2£ ¤ 3£ ´´´and ¤ ¡ µ ¶ as · µ ¶ ;therefore thenumber ofnegativeeigen values is®nite. 2 ³.If¨(r)>0, (r)>0,and ¬(r)³0,then alleigen values arepositi ve, ¤ ¡ >0. 3 ³.Aneigenfunction isdetermined uptoaconstant multiplier .Twoeigenfunctions ¦ ¡ (r)and ¦ ¸ (r) corresponding todifferent eigen values ¤ ¡ and ¤ ¸ areorthogonal with weight¨(r)inthedomain ©, thatis, ¨(r) ¦ ¡ (r) ¦ ¸ (r) ©=0for ·¹ ¹. 4 ³.Anarbitrary function º(r)twice continuously differentiable andsatisfying theboundary con- dition oftheSturm±Liouville problem ( º=0forr ®)canbeexpanded intoanabsolutely and uniformly convergent series intheeigenfunctions:º(r)= ¡ =1 º ¡ ¦ ¡ (r), º ¡ =1 ¥C¦ ¡ ¥ 2 º(r)¨(r) ¦ ¡ (r) ©, where ¥C¦ ¡ ¥ 2isde®ned in(4).» ¼¾½ ¿ ÀÂÁ ÃInathree-dimensional problem, ®nitely manylinearly independent eigenfunctions ¦ (1) ¡ , ÄÄÄ, ¦ ( ¸ ) ¡ generally correspond toeach eigen value ¤ ¡ .These functions canalwaysbereplaced bytheir linear combinations Å ¦ ( Å) ¡ = ÆÅ,1 ¦ (1) ¡ + ´´´+ ÆÅ, Å-1 ¦ ( Å-1) ¡ + ¦ ( Å) ¡ , Ç=1, ÄÄÄ, ¹, sothat Å ¦ (1) ¡ , ÄÄÄ,Å ¦ ( ¸ ) ¡ arenoworthogonal pairwise. Forthisreason, without lossofgenerality ,all eigenfunctions canbeassumed orthogonal. Page454 6.6. EQUATIONS WITH ÈSPACEVARIABLES 455 6.5.2-2. Second boundary value problem. Thesolution oftheequation with theinitial conditions (1)andnonhomogeneous boundary conditions ofthesecond kind, É = Ê(r, )for r ®, canberepresented asthesumÉ(r, )= Ë0( )¨( ) (r, , ) © + Ë1( )¨( ) (r, , ) © + Ì0 Ê( , ) ( ) (r, , - ) ® + Ì0 ( , ) (r, , - ) © . (7) Here, themodi®ed Green' sfunction isgivenbyrelation (4),the ¤ ¡ and ¦ ¡ (r)aretheeigen values andcorresponding eigenfunctions oftheSturm±Liouville problem forthesecond-order elliptic equation (5)with homogeneous boundary conditions ofthesecond kind, ¦ =0for r ®. (8) For ¬(r)>0,thegeneral properties oftheeigen value problem (5),(8)arethesame asthose of the®rstboundary value problem (all ¤ ¡ arepositi ve). 6.5.2-3. Third boundary value problem. Thesolution oftheequation with theinitial conditions (1)andnonhomogeneous boundary conditions ofthethird kind, É + Ç(r)É= Ê(r, )for r ®, isdetermined byrelations (7)and(4),where the ¤ ¡ and ¦ ¡ (r)aretheeigen values andeigenfunc- tions oftheSturm±Liouville problem forthesecond-order elliptic equation (5)with homogeneous boundary conditions ofthethird kind, ¦ + Ç(r) ¦ =0for r ®. (9) If ¬(r)³0and Ç(r)>0,thegeneral properties oftheeigen value problem (5),(9)arethesame asthose ofthe®rstboundary value problem (seeParagraph 6.5.2-1). Suppose Ç(r)= Ç=const .Denote theGreen' sfunctions ofthesecond andthird boundary value problems by Í2(r, , )and Í3(r, , , Ç),respecti vely.If ¬(r)>0,thelimit relation Í2(r, , )= limÅ0Î0 Í3(r, , , Ç)holds.ÏÐ Refer ences forSubsection 6.5.2: V.S.Vladimiro v(1988), A.D.Polyanin (2000a). 6.6. Equations with ÑSpace Variab les Throughout thissection thefollowing notation isused:Ò ¡É= ¡ÓÅ=12É Ô2Å,x={Ô1, ÄÄÄ,Ô ¡ },y={ Õ1, ÄÄÄ, Õ ¡ },|x|= ÖÔ2 1+ ´´´+Ô2 ¡ . Page455 456 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 6.6.1. WaveEquation ×2 Ø× Ù2= Ú2 Û Ü Ø 6.6.1-1. Fundamental solution:Ý Ý(x, Þ)= ßàààààáàààààâ(-1) ¡ -2 2 2 ã ä ¡ +1 2 å æ ·-1 2 ç è( ã Þ-|x|) ( ã2Þ2-|x|2) é-1 2if ê³2iseven; 1 2 ä ãæ1 2 ä ã2Þ ëë Þ ç é-3 2 ì ( ã2Þ2-|x|2) if ê³3isodd; whereè( í)istheHeaviside unitstepfunction and ì ( í)istheDirac delta function.îï Refer ence:V.S.Vladimiro v(1988). 6.6.1-2. Properties ofsolutions. Suppose ð( ñ1, òòò, ñé, Þ)isasolution ofthewaveequation. Then thefunctionsð1= ó ð( ô õ ñ1+ ö1, òòò, ô õ ñé+ öé, ô õ Þ+ öé+1),ð2= ó ðæ ñ1- ÷ Þø 1-( ÷ ù ú)2, ñ2, òòò, ñé, û- ÷ ú-2ñ1ø 1-( ÷ ù ú)2ç,ð3= ó üþý2- ú2û2ü-é-1 2ð ÿ ñ1ý2- ú2û2, òòò, ñéý2- ú2û2, ûý2- ú2û2ç, ý=|x|, arealso solutions ofthisequation everywhere theyarede®ned; ó, ö1, òòò, öé+1, ÷,and õare arbitrary constants. The signs at õintheexpression of ð1canbetakenindependently ofone another . 6.6.1-3. Domain: - < ñ < ; =1, òòò, ê.Cauchy problem. Initial conditions areprescribed:ð= (x)atû=0,ë ð= (x)atû=0. Solution:ð(x,û)=1úé-1( ê-2)! ë é-1ë û é-1 0 ú2û2- ý2 é-3 2ý [ (x)] ý +1úé-1( ê-2)! ë é-2ë û é-2 0 ú2û2- ý2 é-3 2ý [ (x)] ý. Here, [ (x)]istheaverage of overthesurfaceofthesphere ofradius ýwith center atx: [ (x)]º1 é ýé-1 |x-y|= (y) , é=2 é 2( ê ù2), where é ýé-1isthearea ofthesurfaceofan ê-dimensional sphere ofradius ý, isthearea element ofthissurface,and|x-y|2=( ñ1- 1)2+ +( ñé- é)2. Forodd ê,thesolution canbealternati velyrepresented asð(x,û)=1 1´3 òòò( ê-2) ëë û ÿ1û ëë û ç é-3 2 û é-2 [ (x)] +1 1´3 òòò( ê-2) ÿ1û ëë û ç é-3 2 û é-2 [ (x)] . Page456 6.6. EQUATIONS WITH SPACEVARIABLES 457 Foreven ê,thesolution canbealternati velyrepresented asð(x,û)=1 2´4 òòò( ê-2) úé-1ëë û ÿ1û ëë û ç é-2 2 0 [ (x)] ýé-1 ýú2û2- ý2 +1 2´4 òòò( ê-2) úé-1 ÿ1û ëë û ç é-2 2 0 [ (x)] ýé-1 ýú2û2- ý2.îï Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), R.Courant andD.Hilbert (1989), D.Zwillinger (1998). 6.6.1-4. Domain: 0£ ñ £ ; =1, òòò, ê.Boundary value problems. Forsolutions ofthe®rst, second, third, andmixedboundary value problems with nonhomoge- neous conditions ofgeneral form, seeParagraphs 6.6.2-2, 6.6.2-3, 6.6.2-4, and6.6.2-5 for º0, respecti vely. 6.6.2. Nonhomog eneous WaveEquation2 2= 2 + !( "1, # # #, " , ) 6.6.2-1. Domain: - < ñ < ; =1, òòò, ê.Cauchy problem. Initial conditions areprescribed:ð= (x)atû=0,ë ð= (x)atû=0. Solution:ð(x,û)=1úé-1( ê-2)! ë é-1ë û é-1 0 ú2û2- ý2 é-3 2ý [ (x)] ý +1úé-1( ê-2)! ë é-2ë û é-2 0 ú2û2- ý2 é-3 2ý [ (x)] ý +1úé-1( ê-2)! ë é-2ë û é-2 0 $ &% 0 ú2$2- ý2 é-3 2ý [ (x,û- $)] ý. Here, [ (x)]istheaverage of overthespherical surfaceofradius ýwith center atx: [ (x)]º1 é ýé-1 |x-y|= (y) , é=2 é 2( ê ù2), where é ýé-1isthearea ofthesurfaceofan ê-dimensional sphere ofradius ýand isthearea element ofthissurface.îï Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), R.Courant andD.Hilbert (1989). 6.6.2-2. Domain: '={0£ ñ £ ; =1, òòò, ê}.First boundary value problem. Thefollowing conditions areprescribed:ð= 0(x) atû=0 (initial condition ),ë ð= 1(x) atû=0 (initial condition ),ð= (x,û)at ñ =0(boundary conditions ),ð= ( (x,û)at ñ = (boundary conditions ). Page457 458 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Solution:ð(x,û)= 0 )(y, $) *(x,y,û- $) y $ +ëë û )0(y) *(x,y,û) y+ )1(y) *(x,y,û) y + ú2 é +=1 0 , ( -) . (y, $) // *(x,y,û- $) 0-=0 ( ) $ - ú2 1 +=1 0 , ( -) . ( (y, $) // *(x,y,û- $) 0-= 2- ( ) $, where x={ 31, 444, 31},y={ 1, 444, 1}, y= 1 2 4445 1, ( )= 1 4445 -1 +1 4446 1,( )={0£ 7£ 87for 9=1, 444, -1, +1, 444, :}. Green' sfunction:*(x,y, ;)=211 2 44451 <+ = 1=1 <+ = 2=1 444 <+ =?> =1sin( @ = 1 31)sin( @ = 2 32) 444sin( @ =?>31) ´sin( @ = 1 1)sin( @ = 2 2) 444sin( @ =?>1)sinBA ;DC @2 = 1+ + @2 =?> A C @2 = 1+ + @2 =?> , where@ = 1= E1 1, @ = 2= E2 2, 444, @ =?> = E1 1. 6.6.2-3. Domain: '={0£ 3 F£ F; G=1, 444, :}.Second boundary value problem. Thefollowing conditions areprescribed:H= I0(x) at ;=0 (initial condition ),/ J H= I1(x) at ;=0 (initial condition ),/ K - H= L F(x, ;)at 3 F=0(boundary conditions ),/ K - H= ( F(x, ;)at 3 F= F(boundary conditions ). Solution:H(x, ;)= M J 0 M N O(y, $) P(x,y, ;- $) Qy Q $ + M NI0(y) P(x,y, ;) Qy+ M NI1(y) P(x,y, ;) Qy -A2 1 +F=1 M J 0 M , ( -) R L F(y, $) P(x,y, ;- $) ST-=0 Q U( F)TQ $ +A2 1 +F=1 M J 0 M , ( -)RWV F(y, $) P(x,y, ;- $)S T-= 2- Q U( F)TQ $. Here,P(x,y, ;)= ;X 1 X 2 444 X1+1X 1 X 2 444 X1 <+ = 1=0 <+ = 2=0 444 <+ =?> =0 Y = 1Y = 2 444Y =?>A C @2 = 1+ + @2 =?> sin ZA ;DC @2 = 1+ + @2 =?> [ ´cos( @ = 1 31)cos( @ = 2 32) 444cos( @ =?>31)cos( @ = 1 1)cos( @ = 2 2) 444cos( @ =?>1), Page458 6.6. EQUATIONS WITH SPACEVARIABLES 459 where@ = 1= E1 \X 1, @ = 2= E2 \X 2, 444, @ =?> = E1 \X1;Y =?] = ^1for _`7=0, 2for _`7¹0, a=1,2, 444, b. Thesummation isperformed overtheindices satisfying thecondition _1+ + _`c>0;theterm corresponding to _1= = _`c=0issingled out. 6.6.2-4. Domain: '={0£ 3 d£ Xd; e=1, 444, b}.Third boundary value problem. Thefollowing conditions areprescribed:H= f0(x) at g=0 (initial condition ),h iH= f1(x) at g=0 (initial condition ),h j kH- lDd H= m d(x, g)at 3 d=0(boundary conditions ),h j kH+ nd H=V d(x, g)at 3 d= Xd(boundary conditions ). Thesolution H(x, g)isdetermined bytheformula inParagraph 6.6.2-3 whereP(x,y, g)=2 c o p q 1=1 o p q 2=1 444 o p q?r =1sin ZBsgDt u2 q 1+ u2 q 2+ + u2 q?r [s v q 1 v q 2 444 v q?rt u2 q 1+ u2 q 2+ + u2 q?r ´sin( u q 1 31+ w q 1)sin( u q 2 32+ w q 2) 444sin( u q?r3 c+ w q?r ) ´sin( u q 1 x1+ w q 1)sin( u q 2 x2+ w q 2) 444sin( u q?rx c+ w q?r ). Here,w q?] =arctan u q?]X7, v q?] = X7+( l&7 nD7+ u2 q?] )( ly7+ nD7) ( l27+ u2 q?] )( n27+ u2 q?] ),a=1,2, 444, b; the u q?] arepositi veroots ofthetranscendental equations 1l&7+ nD7 z u- l&7 nD7u {=cot( |}7 u),a=1,2, 444, b. 6.6.2-5. Domain: '={0£ 3 d£ |d; e=1, 444, b}.Mixedboundary value problem. Thefollowing conditions areprescribed:H= f0(x) at g=0 (initial condition ),h iH= f1(x) at g=0 (initial condition ),H= m d(x, g)at 3 d=0(boundary conditions ),h j kH= ~ d(x, g)at 3 d= |d(boundary conditions ). Solution:H(x, g)= i 0 (y, $) (x,y, g- $) y $ + hhg f0(y) (x,y, g) y+ f1(y) (x,y, g) y + s2 cpd=1 i 0 ( k ) m d(y, $) hhx d (x,y, g- $) k =0 U( d) $ + s2 cpd=1 i 0 ( k ) ~ d(y, $) (x,y, g- $) k = k U( d) $, Page459 460 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES where(x,y, g)=2 c|1 |2 4445|8c o p q 1=1 o p q 2=1 444 o p q?r =1sin( u q 1 31)sin( u q 2 32) 444sin( u q?r3 c) ´sin( u q 1 x1)sin( u q 2 x2) 444sin( u q?rx c)sin sgDt u2 q 1+ + u2 q?r st u2 q 1+ + u2 q?r ,u q 1= (2 _1+1) 2 |1, u q 2= (2 _2+1) 2 |2, 444, u q?r = (2 _`c+1) 2 |8c. 6.6.3. Equations oftheForm 2 2= 2 ± + ( 1, , ,) 6.6.3-1. Domain: - < 3 < ; =1, 444, .Cauchy problem. Initial conditions areprescribed: = (x)at =0, = (x)at =0, where x={ 31, 444, 3 }. 1 ¡.Let ¢=- £2<0and º0.Thesolution issought bythedescent method intheform (x, )=1 exp( £&3 +1) ¤(x, 3 +1, ), (1) where¤isthesolution oftheCauchy problem fortheauxiliary ( +1)-dimensional waveequation2¤ 2= ¥ +1¤(2) with theinitial conditions¤=exp( £&3 +1) (x)at =0, ¤=exp( £&3 +1) (x)at =0.(3) Forthesolution ofproblem (2),(3),seeParagraph 6.6.1-3. 2¡.Let ¢= £2>0and º0.Inthiscase thefunction exp( £&3 +1)in(1)and(3)must bereplaced bycos( £&3 +1).¦¨§ Refer ence:R.Courant andD.Hilbert (1989). 6.6.3-2. Domain: ©={0£ 3 £ |; =1, 444, }.First boundary value problem. Thefollowing conditions areprescribed: = 0(x) at =0 (initial condition ), = 1(x) at =0 (initial condition ), = (x, )at 3 =0(boundary conditions ), = ~ (x, )at 3 = |(boundary conditions ). Solution: (x, )= 0 (y, ª) (x,y, - ª) y ª + 0(y) (x,y, ) y+ 1(y) (x,y, ) y + «2 ¬=1 0 ( ) (y, ª) ® (x,y, - ª) =0 ¯( ) ª - «2 ¬=1 0 ( ) ~ (y, ª) ® (x,y, - ª) = ¯( ) ª, Page460 6.6. EQUATIONS WITH SPACEVARIABLES 461 where x={ 1, , },y={ 1, , }, y= 1 2 , ( ) = 1 -1 +1 ,( )={0£ £ for =1, , -1, +1, , }. Green' sfunction:(x,y, )=2 1 2 1=1 2=1 =1sin( 1 1)sin( 2 2) sin( ) ´sin( 1 1)sin( 2 2) sin( )sin 2( 2 1+ + 2)+ ! 2( 2 1+ + 2)+ , where 1= "1 # 1, 2= "2 # 2, , = " # . 6.6.3-3. Domain: $={0£ £ ; =1, , }.Second boundary value problem. Thefollowing conditions areprescribed:%= &0(x) at =0 (initial condition ),' (%= &1(x) at =0 (initial condition ),' ) *%= + (x, )at =0(boundary conditions ),' ) *%= , (x, )at = (boundary conditions ). Solution:%(x, )= - ( 0 - . /(y, 0) (x,y, - 0) y 0 + - . &0(y) (x,y, ) y+ - . &1(y) (x,y, ) y - 2 =1 - ( 0 - 1 ( * ) 2 + (y, 0) (x,y, - 0) 3 * =0 ( ) 0 + 2 =1 - ( 0 - 1 ( * ) 2 , (y, 0) (x,y, - 0) 3 * = 4 * ( ) 0. Here,(x,y, )=1 1 2 1=0 2=0 =0 5 15 2 5 cos( 1 1)cos( 2 2) cos( ) ´cos( 1 1)cos( 2 2) cos( )sin 2( 2 1+ + 2)+ ! 2( 2 1+ + 2)+ , where 1= "1 # 1, 2= "2 # 2, , = " # ;5 6= 71for" =0, 2for" ¹0, =1,2, , . Page461 462 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 6.6.3-4. Domain: $={0£ £ ; =1, , }.Third boundary value problem. Thefollowing conditions areprescribed:%= &0(x) at =0 (initial condition ),' (%= &1(x) at =0 (initial condition ),' ) *%- %= + (x, )at =0(boundary conditions ),' ) *%+ 8 %= , (x, )at = (boundary conditions ). Thesolution %(x, )isdetermined bytheformula inParagraph 6.6.3-3 where(x,y, )=2 1=1 2=1 =1sin 2( 2 1+ 2 2+ + 2)+ !9 1 9 2 9 2( 2 1+ 2 2+ + 2)+ ´sin( 1 1+ : 1)sin( 2 2+ : 2) sin( + : ) ´sin( 1 1+ : 1)sin( 2 2+ : 2) sin( + : ). Here,: 6=arctan 6 , 96= +( ; 8+ 26)( <+ 8) ( 2+ 26)( 82+ 26), =1,2, , ; the 6arepositi veroots ofthetranscendental equations 1 ;+ 8 = - ; 8 >=cot( ? ), =1,2, , . 6.6.3-5. Domain: $={0£ £ ; =1, , }.Mixedboundary value problem. Thefollowing conditions areprescribed:%= &0(x) at =0 (initial condition ),' (%= &1(x) at =0 (initial condition ),%= + (x, )at =0(boundary conditions ),' ) *%= , (x, )at = (boundary conditions ). Solution:%(x, )= - ( 0 - . /(y, 0) (x,y, - 0) y 0 + '' - . &0(y) (x,y, ) y+ - . &1(y) (x,y, ) y + 2 =1 - ( 0 - 1 ( * ) @ + (y, 0) '' (x,y, - 0) A * =0 ( ) 0 + 2 =1 - ( 0 - 1 ( * ) @ , (y, 0) (x,y, - 0) A * = 4 * ( ) 0. Here,(x,y, )=2 1 2 1=1 2=1 =1sin( 1 1)sin( 2 2) sin( ) ´sin( 1 1)sin( 2 2) sin( )sin 2( 2 1+ + 2)+ ! 2( 2 1+ + 2)+ , where 1= #(2"1+1) 2 1, 2= #(2"2+1) 2 2, , = #(2" +1) 2 . Page462 6.6. EQUATIONS WITH SPACEVARIABLES 463 6.6.4. Equations Containing theFirstTime Deriv ative 1. B2 CB D2+ E B CB D= F2 G HC± I C+ J( K1, L L L, K H,D). Nonhomo geneous telegraphequation with space variables. 1 M.Thesubstitution %=exp-1 2 N !Oleads totheequation'2O'2= 2 PO- -1 4N2!O+exp1 2N ! /( 1, , , ), which isconsidered inSubsection 6.6.3. 2 M.Domain: $={0£ £ ; =1, , }.First boundary value problem. Thefollowing conditions areprescribed:%= &0(x) at =0 (initial condition ),' (%= &1(x) at =0 (initial condition ),%= + (x, )at =0(boundary conditions ),%= , (x, )at = (boundary conditions ). Solution:%(x, )= - ( 0 - . /(y, 0) (x,y, - 0) y 0 + '' - . &0(y) (x,y, ) y+ - .2 &1(y)+N &0(y) 3 (x,y, ) y + 2 =1 - ( 0 - 1 ( * ) @ + (y, 0) '' (x,y, - 0) A * =0 ( ) 0 - 2 =1 - ( 0 - 1 ( * ) @ , (y, 0) '' (x,y, - 0) A * = 4 * ( ) 0, where x={ 1, , },y={ 1, , }, y= 1 2 , ( ) = 1 -1 +1 ,( )={0£ £ for =1, , -1, +1, , }. Green' sfunction:(x,y, )=2 Q- R (S 2 1 2 1=1 2=1 =1sin( 1 1)sin( 2 2) sin( ) ´sin( 1 1)sin( 2 2) sin( )sin 2( 2 1+ + 2)+ -N2 T4! 2( 2 1+ + 2)+ -N2 T4, where 1= "1 # 1, 2= "2 # 2, , = " # . 3 M.Domain: $={0£ £ ; =1, , }.Second boundary value problem. Thefollowing conditions areprescribed:%= &0(x) at =0 (initial condition ),' (%= &1(x) at =0 (initial condition ),' ) *%= + (x, )at =0(boundary conditions ),' ) *%= , (x, )at = (boundary conditions ). Page463 464 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Solution:%(x, )= - ( 0 - . /(y, 0) (x,y, - 0) y 0 + - .&0(y) (x,y, ) y+ - .2 &1(y)+N &0(y) 3 (x,y, ) y - 2 =1 - ( 0 - 1 ( * ) 2 + (y, 0) (x,y, - 0) 3 * =0 ( ) 0 + 2 =1 - ( 0 - 1 ( * ) 2 , (y, 0) (x,y, - 0) 3 * = 4 * ( ) 0. Here,(x,y, )= Q- R (S 2 1 2 1=0 2=0 =05 15 2 5 cos( 1 1)cos( 2 2) cos( ) ´cos( 1 1)cos( 2 2) cos( )sin 2( 2 1+ + 2)+ -N2 T4! 2( 2 1+ + 2)+ -N2 T4, where 1= "1 # 1, 2= "2 # 2, , = " # ;5 6= 71for" =0, 2for" ¹0, =1,2, , . 4 M.Domain: $={0£ £ ; =1, , }.Third boundary value problem. Thefollowing conditions areprescribed:%= &0(x) at =0 (initial condition ),' (%= &1(x) at =0 (initial condition ),' ) *%- %= + (x, )at =0(boundary conditions ),' ) *%+ 8 %= , (x, )at = (boundary conditions ). Thesolution %(x, )isgivenbytheformula inItem 3 Mwith(x,y, )=2 Q- R (S 2 1=1 2=1 =1sin 2( 2 1+ 2 2+ + 2)+ -N2 T4!9 1 9 2 9 2( 2 1+ 2 2+ + 2)+ -N2 T4 ´sin( 1 1+ : 1)sin( 2 2+ : 2) sin( + : ) ´sin( 1 1+ : 1)sin( 2 2+ : 2) sin( + : ). Here,: 6=arctan 6 , 96= +( ; 8+ 26)( <+ 8) ( 2+ 26)( 82+ 26), =1,2, , ; the 6arepositi veroots ofthetranscendental equation 1 ;+ 8 = - ; 8 >=cot( ? ), =1,2, , . Page464 6.6. EQUATIONS WITH SPACEVARIABLES 465 5 M.Domain: $={0£ £ ; =1, , }.Mixedboundary value problem. Thefollowing conditions areprescribed:%= &0(x) at =0 (initial condition ),' (%= &1(x) at =0 (initial condition ),%= + (x, )at =0(boundary conditions ),' ) *%= , (x, )at = (boundary conditions ). Solution:%(x, )= - ( 0 - . /(y, 0) (x,y, - 0) y 0 + '' - .&0(y) (x,y, ) y+ - .2 &1(y)+N &0(y) 3 (x,y, ) y + 2 =1 - ( 0 - 1 ( * ) U + (y, 0) '' (x,y, - 0) V * =0 ( ) 0 + 2 =1 - ( 0 - 1 ( * )U , (y, 0) (x,y, - 0)V * = 4 * ( ) 0, where(x,y, )=2 Q- R (S 2 1 2 1=1 2=1 =1sin( 1 1)sin( 2 2) sin( ) ´sin( 1 1)sin( 2 2) sin( )sin 2( 2 1+ + 2)+ -N2 T4! 2( 2 1+ + 2)+ -N2 T4, 1= #(2"1+1) 2 1, 2= #(2"2+1) 2 2, , = #(2" +1) 2 . 2. B2 CB D2+ E B CB D= F2 G HC+ HXW =1 I WB CB K W + Y C. Thetransformation%( 1, , , )=O( 1, , , 0)exp=-1 2 N -1 2 2 =1 >, 0= leads totheequation'2O'02= PO+ O, = 82+ N2 4 2-1 4 4 =1 2 , which isconsidered inSubsection 6.6.3.Z\[ Refer ence:R.Courant andD.Hilbert (1989). 3. B2 CB D2+ ]±1D B CB D= G HC. Darboux equation. Cauchy problem. Initial conditions areprescribed:%= &(x)at =0,' (%=0 at =0. Page465 466 HYPERBOLIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Solution:%(x, )=1^ _ _ -1 - |x-y|= ( &(y) ` a b, ^ _=2# _ S 2c( d T2), where ^ _ e _ -1isthearea ofthesurfaceofan d-dimensional sphere ofradius e,and ` a bisthearea element ofthissurface(i.e., thesolution %istheaverage ofthefunction &overthesphere for radius ewith center atx).Z\[ Refer ence:R.Courant andD.Hilbert (1989). Page466 Chapter 7 Elliptic Equations with TwoSpace Variab les 7.1. Laplace Equation f2 g=0 The Laplace equation isoften encountered inheat and mass transfer theory ,¯uid mechanics, elasticity ,electrostatics, andother areas ofmechanics andphysics. Forexample, inheat andmass transfer theory ,thisequation describes steady-state temperature distrib ution intheabsence ofheat sources andsinks inthedomain under study . Aregular solution oftheLaplace equation iscalled aharmonic function. The®rstboundary valueproblem fortheLaplace equation isoften referred toastheDirichlet problem, andthesecond boundary value problem astheNeumann problem. Extremum principle :Givenadomain h,aharmonic function iin hthatisnotidentically constant in hcannot attain itsmaximum orminimum value atanyinterior point of h. 7.1.1. Problems inCartesian Coor dinate System TheLaplace equation with twospace variables intherectangular Cartesian system ofcoordinates is written as j 2ij k 2+ j 2ij l 2=0. 7.1.1-1. Particular solutions andamethod fortheir construction. 1 M.Particular solutions:i( k , l )= m k + n l + o,i( k , l )= m( k2- l2)+ n k l ,i( k , l )= m( k3-3 k l2)+ n(3 k2 l - l3),i( k , l )= m k + n lk 2+ l 2+ o,i( k , l )=exp( p q k )( mcos q l + nsin q l ),i( k , l )=( mcos q k + nsin q k )exp( p q l ),i( k , l )=( msinh q k + ncosh q k )( ocos q l + hsin q l ),i( k , l )=( mcos q k + nsin q k )( osinh q l + hcosh q l ),i( k , l )= mln r( k - k 0)2+( l - l 0)2 s+ n, where m, n, o, h, k 0, l 0,and qarearbitrary constants. 2 M.Fundamental solution: t t ( k , l )=1 2 uln1 v, v = w k 2+ l 2. Page467 468 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 3 x.If i( k , l )isasolution oftheLaplace equation, then thefunctionsi1= m i( p y k + o1, p y l + o2),i2= m i( k cos z+ l sin z,- k sin z+ l cos z),i3= m i { kk 2+ l 2, lk 2+ l 2 |, arealso solutions everywhere theyarede®ned; m, o1, o2, z,and yarearbitrary constants. The signs at yin i1aretakenindependently ofeach other . 4 x.Afairly general method forconstructing particular solutions involvesthefollowing. Let }( ~)=( k , l )+ ( k , l )beanyanalytic function ofthecomple xvariable ~= k + l ( and arereal functions oftherealvariables k and l ; 2=-1).Then therealandimaginary parts of }both satisfy thetwo-dimensional Laplace equation, 2 =0, 2 =0. Recall thattheCauchy±Riemann conditionsjj k= jj l, jj l=- jj k arenecessary andsuf®cient conditions forthefunction }tobeanalytic. Thus, byspecifying analytic functions }( ~)andtaking their realandimaginary parts, oneobtains various solutions of thetwo-dimensional Laplace equation.\[ Refer ences :M.A.Lavrent'e vandB.V.Shabat (1973), A.G.Sveshnik ovandA.N.Tikhono v(1974), A.V.Bitsadze andD.F.Kalinichenk o(1985). 7.1.1-2. Speci®c features ofstating boundary value problems fortheLaplace equation. 1 x.Forouter boundary value problems ontheplane, itis(usually) required tosettheadditional condition thatthesolution oftheLaplace equation must bebounded atin®nity . 2 x.Thesolution ofthesecond boundary value problem isdetermined uptoanarbitrary additi ve term. 3 x.Letthesecond boundary value problem inaclosed bounded domain hwith piece wise smooth boundary becharacterized bytheboundary condition*jij = }(r)for r , where isthederivativealong the(outw ard)normal to .Thenecessary andsuf®cient condition ofsolvability oftheproblem hastheform }(r) =0. Thesame solvability condition occurs fortheouter second boundary value problem ifthedomain isin®nite buthasa®nite boundary .\[ Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). *More rigorously , must satisfy theLyapuno vcondition [seeBabich, Kapile vich, Mikhlin, etal.(1964) andTikhono v andSamarskii (1990)]. Page468 7.1. LAPLA CEEQUATION 2 =0 469 7.1.1-3. Domain: - < < ,0£ < .First boundary value problem. Ahalf-plane isconsidered. Aboundary condition isprescribed:= }( )at =0. Solution:( , )=1 - }( ) ( - )2+ 2=1 ¡ 2 - ¡ 2 }( + tan ¢) ¢.\[ Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), H.S.Carsla wandJ.C.Jaeger(1984). 7.1.1-4. Domain: - < < ,0£ < .Second boundary value problem. Ahalf-plane isconsidered. Aboundary condition isprescribed:£ ¤= }( )at =0. Solution:( , )=1 - }( )lnw( - )2+ 2 + ¥, where ¥isanarbitrary constant.\[ Refer ence:V.S.Vladimiro v(1988). 7.1.1-5. Domain: 0£ < ,0£ < .First boundary value problem. Aquadrant oftheplane isconsidered. Boundary conditions areprescribed:= }1( )at =0, = }2( )at =0. Solution:( , )=4 0 }1( ¦) ¦ ¦ [ 2+( - ¦)2][ 2+( + ¦)2]+4 0 }2( ) [( - )2+ 2][( + )2+ 2].\[ Refer ence:V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974). 7.1.1-6. Domain: - < < ,0£ £ §.First boundary value problem. Anin®nite strip isconsidered. Boundary conditions areprescribed:= }1( )at =0, = }2( )at = §. Solution:( , )=1 2 §sin { § | - }1( ) cosh[ ( - ) ¨ §]-cos( ¨ §) +1 2 §sin { § | - }2( ) cosh[ ( - ) ¨ §]+cos( ¨ §).\[ Refer ence:H.S.Carsla wandJ.C.Jaeger(1984). 7.1.1-7. Domain: - < < ,0£ £ §.Second boundary value problem. Anin®nite strip isconsidered. Boundary conditions areprescribed:£ ¤= }1( )at =0, £ ¤= }2( )at = §. Solution:( , )=1 2 - }1( )ln ©cosh[ ( - ) ¨ §]-cos( ¨ §) ª -1 2 - }2( )ln©cosh[ ( - ) ¨ §]+cos( ¨ §)ª + ¥, where ¥isanarbitrary constant. Page469 470 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.1.1-8. Domain: 0£ < ,0£ £ §.First boundary value problem. Asemiin®nite strip isconsidered. Boundary conditions areprescribed:= }1( )at =0, = }2( )at =0, = }3( )at = §. Solution:( , )=2§ «X¬ =1exp {- § |sin { § | ® 0 }1( ¦)sin { ¦§ | ¦ +1 2 §sin { § | 0 ¯1 cosh[ ( - ) ¨ §]-cos( ¨ §)-1 cosh[ ( + ) ¨ §]-cos( ¨ §) ° ±2( ) +1 2 §sin ² § ³ 0 ¯1 cosh[ ( - ) ¨ §]+cos( ¨ §)-1 cosh[ ( + ) ¨ §]+cos( ¨ §) ° ±3( ) . Example. Consider the®rstboundary value problem fortheLaplace equation inasemiin®nite strip with ´1( µ)=1and´2( ¶)= ´3( ¶)=0. Using thegeneral formula andcarrying outtransformations, weobtain thesolution( ¶, µ)=2·arctan ¸sin( ·µ ¹º) sinh( ·¶ ¹º) ».¼\[ Refer ence:H.S.Carsla wandJ.C.Jaeger(1984). 7.1.1-9. Domain: 0£ £ §,0£ £ ½.First boundary value problem. Arectangle isconsidered. Boundary conditions areprescribed:=±1( )at =0, =±2( )at = §,=±3( )at =0, =±4( )at = ½. Solution:( , )= «X¬ =1 ¾ ¬ sinh ¿ ½( §- ) Àsin ² ½ ³+ «X¬ =1 Á ¬ sinh ² ½ ³sin ² ½ ³ + «X¬ =1 ¥ ¬ sin ² § ³sinh ¿ §( ½- ) À+ «X¬ =1 Â ¬ sin ² § ³sinh ² § ³, where thecoef®cients¾ ¬ ,Á ¬ , ¥ ¬ ,andÂ ¬ areexpressed as¾ ¬ =2Ã ¬ Ä Å 0 ±1( Æ)sin ² Ç Æ½ ³ È Æ,Á É=2ÃÉ Ä Å 0 ±2( Æ)sin ² Ç Æ½ ³ È Æ,ÊÉ=2ËÉ Ä ® 0 ±3( Æ)sin ² Ç ÆÌ³ È Æ,Â É=2ËÉ Ä ® 0 ±4( Æ)sin ² Ç ÆÌ³ È Æ,ÃÉ= ½sinh ² Ç Ì½ ³, ËÉ= Ìsinh ² Ç ½Ì³.¼\[ Refer ences :M.M.Smirno v(1975), H.S.Carsla wandJ.C.Jaeger(1984). Page470 7.1. LAPLA CEEQUATION Í2 Î=0 471 7.1.1-10. Domain: 0£ Ï£ Ì,0£ Ð£ ½.Second boundary value problem. Arectangle isconsidered. Boundary conditions areprescribed:£ Ñ Ò=±1( Ð)at Ï=0, £ Ñ Ò=±2( Ð)at Ï= Ì,£ ¤ Ò=±3( Ï)at Ð=0, £ ¤ Ò=±4( Ï)at Ð= ½. Solution:Ò( Ï, Ð)=-¾0 4 Ì( Ï- Ì)2+Á0 4 Ì Ï2- Ê 0 4 ½( Ï- ½)2+Â0 4 ½ Ð2+ Ó - ½ Ô ÕÉ=1 Ö É ×Écosh Ø Ç Ù( Ì- Ï) Úcos Û Ç ÙÐ Ü+ ÙÔ ÕÉ=1 Ý É ×Écosh Û Ç ÙÏ Ücos Û Ç ÙÐ Ü - ÌÔ ÕÉ=1 ÊÉ ËÉcos Û ÇÌ Ï Ücosh Ø ÇÌ( Ù- Ð) Ú+ ÌÔ ÕÉ=1 Â É ËÉcos Û ÇÌ Ï Ücosh Û ÇÌ Ð Ü, where Óisanarbitrary constant, andthecoef®cientsÖ É,Ý É, ÊÉ,Â É, ×É,and ËÉareexpressed asÖ É=2Ù Ä Å 0 Þ1( Æ)cos Û Ç ÆÙÜÈ Æ,Ý É=2Ù Ä Å 0 Þ2( Æ)cos Û Ç ÆÙÜÈ Æ,ÊÉ=2Ì Ä ® 0 Þ3( Æ)cos Û Ç ÆÌ ÜÈ Æ,Â É=2Ì Ä ® 0 Þ4( Æ)cos Û Ç ÆÌ ÜÈ Æ,×É= Çsinh Û Ç ÌÙÜ, ËÉ= Çsinh Û Ç ÙÌ Ü. The solvability condition fortheproblem inquestion hastheform (see Paragraph 7.1.1-2, Item 3 ß)ÄÅ 0 Þ1( Ð)È Ð+ ÄÅ 0 Þ2( Ð)È Ð- Ä ® 0 Þ3( Ï)È Ï- Ä ® 0 Þ4( Ï)È Ï=0. 7.1.1-11. Domain: 0£ Ï£ Ì,0£ Ð£ Ù.Third boundary value problem. Arectangle isconsidered. Boundary conditions areprescribed:£ Ñ Ò- à1 Ò=Þ1( Ð)at Ï=0, £ Ñ Ò+ à2 Ò=Þ2( Ð)at Ï= Ì,£ ¤ Ò- à3 Ò=Þ3( Ï)at Ð=0, £ ¤ Ò+ à4 Ò=Þ4( Ï)at Ð= Ù. Forthesolution, seeParagraph 7.2.2-14 with áº0. 7.1.1-12. Domain: 0£ Ï£ Ì,0£ Ð£ Ù.Mixedboundary value problems. 1 ß.Arectangle isconsidered. Boundary conditions areprescribed:£ Ñ Ò=Þ( Ð)at Ï=0, £ Ñ Ò= â( Ð)at Ï= Ì,Ò= ã( Ï)at Ð=0, Ò= ä( Ï)at Ð= Ù. Solution:Ò( Ï, Ð)=- ÙÇ Ô ÕÉ=1 ÞÉ ×Écosh Ø Ç Ù( Ì- Ï) Úsin Û Ç ÐÙÜ+ ÙÇ Ô ÕÉ=1 âÉ ×Écosh Û Ç ÏÙÜsin Û Ç ÐÙÜ + Ô ÕÉ=1 ãÉ ËÉcos Û Ç ÏÌ Üsinh Ø Ç Ì( Ù- Ð) Ú+ Ô ÕÉ=1 äÉ ËÉcos Û Ç ÏÌ Üsinh Û Ç ÐÌ Ü + Ù- ÐÌ Ù Ä ® 0 ã( Ï)È Ï+ ÐÌ Ù Ä ® 0 ä( Ï)È Ï, Page471 472 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES whereÞÉ=2Ù ÄÅ 0 Þ( Æ)sin Û Ç ÆÙÜÈ Æ, âÉ=2Ù ÄÅ 0 â( Æ)sin Û Ç ÆÙÜÈ Æ,ãÉ=2Ì Ä ® 0 ã( Æ)cos Û Ç ÆÌ ÜÈ Æ, äÉ=2Ì Ä ® 0 ä( Æ)cos Û Ç ÆÌ ÜÈ Æ,×É=sinh Û Ç ÌÙÜ, ËÉ=sinh Û Ç ÙÌ Ü,å\[ Refer ence:M.M.Smirno v(1975). 2 ß.Arectangle isconsidered. Boundary conditions areprescribed:Ò=Þ( Ð)at Ï=0, £ Ñ Ò= â( Ð)at Ï= Ì,Ò= ã( Ï)at Ð=0, £ ¤ Ò= ä( Ï)at Ð= Ù, whereÞ(0)= ã(0). Solution:Ò( Ï, Ð)= Ô ÕÉ=0 ÞÉcosh ×Écosh Û ×É Ì- ÏÌ Üsin Û ×É ÐÌ Ü+ ÌÔ ÕÉ=0 âÉ ×Écosh ×Ésinh Û ×É ÏÌ Üsin Û ×É ÐÌ Ü + Ô ÕÉ=0 ãÉcosh ËÉsin Û ËÉ ÏÙÜcosh Û ËÉ Ù- ÐÙÜ+ ÙÔ ÕÉ=0 äÉ ËÉcosh ËÉsin Û ËÉ ÏÙÜsinh Û ËÉ ÐÙÜ, whereÞÉ=2Ù æ ç 0 Þ( è)sin Ø é(2 ê+1)Ùè Ú ë è, â ì=2Ù æ ç 0 â( è)sin Ø é(2 ê+1)Ùè Ú ë è,ã ì=2 íæ î 0 ã( è)sin Ø é(2 ê+1) íè Ú ë è, äì=2 íæ î 0 ä( è)sin Ø é(2 ê+1) íè Ú ë è,×ì= é(2 ê+1) í 2 Ù , ï ì= é(2 ê+1) Ù 2 í.å\[ Refer ence:M.M.Smirno v(1975). 7.1.2. Problems inPolar Coor dinate System Thetwo-dimensional Laplace equation inthepolar coordinate system iswritten as 1 ð ££ ðÛ ð £ Ò£ ðÜ+1 ð 2 £2 Ò£ ñ2=0, ð = ò Ï2+ Ð2. 7.1.2-1. Particular solutions:Ò( ð )=Öln ð +Ý,Ò( ð , ñ)= ÛÖ ð ó +Ý ðóÜ( ôcos õ ñ+Âsin õ ñ), where õ=1,2, ööö;Ö,Ý, ô,andÂarearbitrary constants. Page472 7.1. LAPLA CEEQUATION ÷2 ø=0 473 7.1.2-2. Domain: 0£ ð £ ùor ù£ ð < ú.First boundary value problem. TheconditionÒ=Þ( ñ)at ð = ù issetattheboundary ofthecircle;Þ( ñ)isagivenfunction. 1 ß.Solution oftheinner problem ( ð £ ù):Ò( ð , ñ)=1 2é æ2 û 0Þ( ü) ù2- ð 2ð 2-2 ù ð cos( ñ- ü)+ ù2 ë ü. This formula isconventionally referred toasthePoisson integral. Solution oftheouter problem inseries form:Ò( ð , ñ)= í 0 2+ ý Õì=1 Û ðù Ü ì ( íìcos ê ñ+ Ùìsin ê ñ),íì=1é æ2 û 0 Þ( ü)cos( ê ü) ë ü, ê=0,1,2, ööö,Ùì=1é æ2 û 0 Þ( ü)sin( ê ü) ë ü, ê=1,2,3, ööö 2 ß.Bounded solution oftheouter problem ( ð ³ ù):Ò( ð , ñ)=1 2é æ2 û 0 Þ( ü) ð 2- ù2ð 2-2 ù ð cos( ñ- ü)+ ù2 ë ü. Bounded solution oftheouter problem inseries form:Ò( ð , ñ)= í 0 2+ ý Õì=1 Û ù ðÜ ì ( íìcos ê ñ+ Ùìsin ê ñ), where thecoef®cients í 0, íì,and Ùìarede®ned bythesame relations asintheinner problem. Inhydrodynamics andother applications, outer problems aresometimes encountered inwhich onehastoconsider unbounded solutions for ð þú. Example. The potential ¯owofanideal (inviscid) incompressible ¯uid about acircular cylinder ofradius ÿwith a constant incident velocity atin®nity ischaracterized bythefollowing boundary conditions forthestream function:ø=0at = ÿ,ø sin as . Solution:ø( , )= - ÿ2 sin .å\[ Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), A.N.Tikhono vandA.A.Samarskii (1990). 7.1.2-3. Domain: 0£ ð £ ùor ù£ ð < ú.Second boundary value problem. Thecondition£ Ò=Þ( ñ)at ð = ù issetattheboundary ofthecircle. ThefunctionÞ( ñ)must satisfy thesolvability conditionæ2 û 0Þ( ñ) ë ñ=0. Page473 474 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 1 ß.Solution oftheinner problem ( ð £ ù):Ò( ð , ñ)= ù 2é æ2 û 0 Þ( ü)ln ð 2-2 ù ð cos( ñ- ü)+ ù2ù2 ë ü+ ô, where ôisanarbitrary constant; thisformula isknownastheDini integral. Series solution oftheinner problem:Ò( ð , ñ)= ý ì=1 ùê ðù ì ( íìcos ê ñ+ <ìsin ê ñ)+ ô,íì=1é æ2 û 0 ( ü)cos( ê ü) ë ü, ì=1é æ2 û 0 ( ü)sin( ê ü) ë ü, where ôisanarbitrary constant. 2 .Solution oftheouter problem ( ð ³ ù):( ð , ñ)=- ù 2é æ2 û 0 ( ü)ln ð 2-2 ù ð cos( ñ- ü)+ ù2ð 2 ë ü+ ô, where ôisanarbitrary constant. Series solution oftheouter problem:( ð , ñ)=- ý ì=1 ùê ù ð ì ( íìcos ê ñ+ <ìsin ê ñ)+ ô, where thecoef®cient íìand <ìarede®ned bythesame relations asintheinner problem, and ôis anarbitrary constant.\[ Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). 7.1.2-4. Domain: 0£ ð £ ùor ù£ ð < ú.Third boundary value problem. Thecondition + =( ñ)at ð = ù. issetatthecircle boundary;( ñ)isagivenfunction. 1 .Solution oftheinner problem ( ð £ ù):( ð , ñ)= í 0 2 + ý ì=1 ù ù+ ê ðù ì ( íìcos ê ñ+ <ìsin ê ñ),íì=1é 2 0 ( )cos( ) , =0,1,2, ,=12 0 ( )sin( ) , =1,2,3, 2 .Solution oftheouter problem ( ð ³ ):( ð , ñ)= 0 2 + =1 - ð ( cos ñ+ sin ñ), where thecoef®cient0, ,and arede®ned bythesame relations asintheinner problem.\[ Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). Page474 7.1. LAPLA CEEQUATION 2 !=0 475 7.1.2-5. Domain: 1£ ð £ 2.First boundary value problem. Anannular domain isconsidered. Boundary conditions areprescribed:=1( ñ)at ð = 1, =2( ñ)at ð = 2. Solution:( ð , ñ)= "0+ #0ln ð + =1 ð( " cos ñ+ # sin ñ)+ =11 ð( $ cos ñ+ % sin ñ), where thecoef®cient "0, #0, " , # , $ ,and % areexpressed as"0=1 2 (1) 0ln 2-(2) 0ln 1 ln 2-ln 1," = 2(2)- 1(1)2 2- 2 1,$ =( 1 2) 2(1)- 1(2)2 2- 2 1, #0=1 2 (2) 0-(1) 0 ln 2-ln 1,# = 2 (2)- 1 (1)2 2- 2 1,% =( 1 2) 2 (1)- 1 (2)2 2- 2 1. Here, the( &)and ( &)( '=1,2)arethecoef®cients oftheFourier series expansions ofthefunctions1( ñ)and2( ñ):( &)=12 0 &( )cos( ) , =0,1,2, ,( &)=12 0 &( )sin( ) , =1,2,3, )( Refer ence:M.M.Smirno v(1975). 7.1.2-6. Domain: 1£ ð £ 2.Second boundary value problem. Anannular domain isconsidered. Boundary conditions areprescribed: *=1( ñ)at ð = 1, *=2( ñ)at ð = 2. Solution:( ð , ñ)= #ln ð + =1 ð( " cos ñ+ # sin ñ)+ =11 ð( $ cos ñ+ % sin ñ)+ +. Here, thecoef®cients #, " , # , $ ,and % areexpressed as#=1 2 1(1) 0, " = +1 2(2)- +1 1(1)( 2 2- 2 1), # = +1 2 (2)- +1 1 (1)( 2 2- 2 1),$ =( 1 2) +1 -1 1(2)- -1 2(1)( 2 2- 2 1), % =( 1 2) +1 -1 1 (2)- -1 2 (1)( 2 2- 2 1), where theconstants( &)and ( &)( '=1,2)arede®ned bythesame relations asinthe®rstboundary value problem; +isanarbitrary constant., -/. 0 132 4Note thatthecondition(1) 0 1=(2) 0 2must hold; thisrelation isaconsequence of thesolvability condition fortheproblem, * = 51 1 6- * = 52 2 6=0. Page475 476 ELLIPTIC EQUATIONS WITH TWOSPACE VARIABLES TABLE 21 Two-dimensional Laplace operator in some curvilinear orthogonal systems of coordinates Coordinates Transformation ( 7>0) Laplace operator, 82 Parabolic coordinates9, : ;= 7 9:, <=1 2 7( :2- 92) - =< 9< =,0 £ :< =172( 92+ :2) > ?2 ? 92+?2 ? :2 @ Elliptic coordinatesA, B ;= 7cosh Acos B, <= 7sinh Asin B 0 £ A< =,0 £ B<2 172(sinh2 A+sin2B) > ?2 ? A2+?2 ? B2@ Bipolar coordinatesC, D;= 7sinh D cosh D-cos C, <= 7sin C cosh D-cos C 0 £ C<2 ,- =< D< =172(cosh D-cos C)2> ?2 ? C2+?2 ? D2@ 7.1.2-7. Domain: 1£ E£ 2. Mixed boundary value problem. An annular domain is considered. Boundary conditions are prescribed:? *= F1( G) at E= 1, = F2( G) at E= 2. Solution:( E, G)=1 2(2) 0+1 2(1) 0 1ln E2+ H=1 E ( " cos G+ # sin G)+ H=11E ( $ cos G+ % sin G). Here, the coef®cients " , # , $ , and % are expressed as" = 2(2)+ +1 1(1)( 2 2+ 2 1), # = 2 I(2)+ +1 1 I(1)( 2 2+ 2 1),$ = +1 1 2 -1 1(2)- 2(1)( 2 2+ 2 1), % = +1 1 2 -1 1 I(2)- 2 I(1)( 2 2+ 2 1), where the constants( &)andI( &)( '= 1,2) are de®ned by the same formulas as in the ®rst boundary value problem.J)( Reference : M. M. Smirnov (1975). 7.1.3. Other Coordinate Systems. Conformal Mappings Method 7.1.3-1. Parabolic, elliptic, and bipolar coordinate systems. In a number of applications, it is convenient to solve the Laplace equation in other orthogonal system sofcoordinates .Someofthosecommonl yencountere daredisplaye dinTable21.Inallthe coordinate systems presented, the Laplace equation 82 = 0is reduced to the equation considered in Paragraph 7.1.1-1 in detail (particular solutions and solutions to boundary value problems are given there). Theorthogona ltransformation spresente dinTable21canbewritte ninthelanguag eofcompl ex variables as follows:;+ 'K<= -1 2 'L7( 9+ 'K:)2(parabolic coordinates),;+ 'K<= 7cosh( A+ 'KB) (elliptic coordinates),;+ 'K<= 'L7cot M1 2( C+ 'KD) N (bipolar coordinates). Page 476 7.1. LAPLA CEEQUATION O2 P=0 477 Therealparts, aswell astheimaginary parts, inboth sides ofthese relations must beequated to each other ( '2=-1). Example. Plane hydrodynamic problems ofpotential ¯owsofideal (inviscid) incompressible ¯uid arereduced tothe Laplace equation forthestream function. Inparticular ,themotion ofanelliptic cylinder with semiax es Qand Ratavelocity inthedirection parallel tothemajor semiaxis ( Q> R)inideal ¯uid isdescribed bythestream functionP( S, T)=- R U Q+ RQ- R V1 W2 XZY sin T, [2= Q2- R2, where Sand Taretheelliptic coordinates.J)\ Refer ences :G.Lamb (1945), J.Happel andH.Brenner (1965), G.KornandT.Korn(1968). 7.1.3-2. Domain ofarbitrary shape. Method ofconformal mappings. 1 .Let ]= ]( ^)beananalytic function thatde®nes aconformal mapping from thecomple xplane^= _+ `Kaintoacomple xplane ]= b+ `Kc,where b= b( _, a)and c= c( _, a)arenewindependent variables. Withreference tothefactthattherealandimaginary parts ofananalytic function satisfy theCauchy±Riemann conditions, wehave? d b=? e cand? e b=-? d c,andhence?2 f? _2+?2 f? a2= gh] i( ^) g2> ?2 f? b2+?2 f? c2@. Therefore, theLaplace equation inthe _ a-plane transforms under aconformal mapping into the Laplace equation inthe b c-plane. 2 j.Anysimply connected domain kinthe _ a-plane with apiece wise smooth boundary canbe mapped, with appropriate conformal mappings, onto theupper half-plane orintoaunitcircle intheb c-plane. Consequently ,a®rstandasecond boundary value problem fortheLaplace equation in k canbereduced, respecti vely,toa®rstandasecond boundary valueproblem fortheupper half-space oracircle; such problems areconsidered inSubsections 7.1.1 and7.1.2. Subsection 7.2.4 presents conformal mappings ofsome domains onto theupper half-plane or aunit circle. Moreo ver,examples ofsolving speci®c boundary value problems forthePoisson equation bytheconformal mappings method aregiventhere; theGreen' sfunctions forasemicircle andaquadrant ofacircle areobtained. Alargenumber ofconformal mappings ofvarious domains canbefound, forexample, inthe references cited below.J)\ Refer ences :V.I.Lavrik andV.N.Savenkov(1970), M.A.Lavrent'e vandB.V.Shabat (1973), V.I.Ivanovand M.K.Trubetsk ov(1994). 7.1.3-3. Reduction ofthetwo-dimensional Neumann problem totheDirichlet problem. Lettheposition ofanypoint ( _ l, a l)located ontheboundary mofadomain kbespeci®ed by aparameter n,sothat _ l= _ l( n)and a l= a l( n).Then afunction oftwovariables, F( _, a),is determined on mbytheparameter naswell, F( _, a) gpo= F( _ l( n), a l( n))= F l( n). Thesolution ofthetwo-dimensional Neumann problem fortheLaplace equation q2 f=0in k with theboundary condition ofthesecond kind? f? r= F l( n)for r s m canbeexpressed interms ofthesolution ofthetwo-dimensional Dirichlet problem fortheLaplace equation q2 b=0in kwith theboundary condition ofthe®rstkindb= t l( n)for r s m, where t l( n)= u v l( n) w n,asfollows:f( _, a)= udd0 x bx a( y, a0) w y- uee0 x bx _( _, y) w y+ z. Here, ( _0, a0)arethecoordinates ofanypoint in k,and zisanarbitrary constant.J)\ Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). Page477 478 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.2. Poisson Equation {2 |=± }(x) 7.2.1. Preliminar yRemarks. Solution Structure JustastheLaplace equation, thePoisson equation isoften encountered inheat andmass transfer theory ,¯uid mechanics, elasticity ,electrostatics, andother areas ofmechanics andphysics. For example, itdescribes steady-state temperature distrib ution inthepresence ofheat sources orsinks inthedomain under study . TheLaplace equation isaspecial case ofthePoisson equation with ~º0. Inwhat follows,weconsider a®nite domain with asuf®ciently smooth boundary .Letr s and s ,where r={ _, a}, ={ , },|r- |2=( _- )2+( a- )2. 7.2.1-1. First boundary value problem. Thesolution ofthe®rstboundary value problem forthePoisson equationq2 f=- ~(r) (1) inthedomain with thenonhomogeneous boundary conditionf= v(r)for r s canberepresented asf(r)= u ~( ) (r, ) w - u v( )x x r w . (2) Here, (r, )istheGreen' sfunction ofthe®rst boundary value problem, isthederivative oftheGreen' sfunction with respect to , along theoutw ardnormal Ntotheboundary .The integration isperformed with respect to , ,with w = w w . The Green' sfunction = (r, )ofthe®rst boundary value problem isdetermined bythe following conditions. 1 j.Thefunction satis®es theLaplace equation in _, ainthedomain everywhere except forthe point ( , ),atwhich hasasingularity oftheform1 2 ln1 |r- |. 2 j.Withrespect to _, a,thefunction satis®es thehomogeneous boundary condition ofthe®rst kind atthedomain boundary ,i.e.,thecondition | =0. TheGreen' sfunction canberepresented intheform(r, )=1 2 ln1 |r- |+ b, (3) where theauxiliary function b= b(r, )isdetermined bysolving the®rstboundary value problem fortheLaplace equation q2 b=0with theboundary condition b g =-1 2 ln1 |r- |;inthisproblem,istreated asatwo-dimensional freeparameter . TheGreen' sfunction issymmetric with respect toitsarguments: (r, )= ( ,r). / 3 When using thepolar coordinate system, oneshould set r={ , }, ={ , },|r- |2= 2+ 2-2 cos( - ), w = w w inrelations (2)and(3). Page478 7.2. POISSON EQUATION 2 =- (x) 479 7.2.1-2. Second boundary value problem. Thesecond boundary value problem forthePoisson equation (1)ischaracterized bytheboundary conditionx fx r= v(r)for r s . Thenecessary solvability condition forthisproblem isu ~(r) w + u v(r) w =0. (4) Thesolution ofthesecond boundary value problem, provided thatcondition (4)issatis®ed, can berepresented asf(r)= u ~( ) (r, ) w + u v( ) (r, ) w + z, (5) where zisanarbitrary constant. TheGreen' sfunction = (r, )ofthesecond boundary value problem isdetermined bythe following conditions: 1 j.Thefunction satis®es theLaplace equation in _, ainthedomain everywhere except forthe point ( , ),atwhich hasasingularity oftheform1 2 ln1 |r- |. 2 j.Withrespect to _, a,thefunction satis®es thehomogeneous boundary condition ofthesecond kind atthedomain boundary:x x r =10, where 0isthelength oftheboundary of . TheGreen' sfunction isunique uptoanadditi veconstant. / 3 TheGreen' sfunction cannot bedetermined bycondition 1 jandthehomogeneous boundary condition =0.The point isthattheproblem isunsolv able for inthiscase, because, onrepresenting intheform (3),for weobtain aproblem with anonhomogeneous boundary condition ofthesecond kind forwhich thesolvability condition (4)nowisnotsatis®ed. 7.2.1-3. Third boundary value problem. Thesolution ofthethird boundary value problem forthePoisson equation (1)inthedomain with thenonhomogeneous boundary conditionx fx r+ = v(r)for r s isgivenbyformula (5)with z=0,where = (r, )istheGreen' sfunction ofthethird boundary value problem andisdetermined bythefollowing conditions: 1 ¡.Thefunction satis®es theLaplace equation in ¢, £inthedomain everywhere except forthe point ( , ),atwhich hasasingularity oftheform1 2 ln1 |r- |. 2¡.Withrespect to ¢, £,thefunction satis®es thehomogeneous boundary condition ofthethird kind atthedomain boundary ,i.e.,thecondition ¤ + ¥ =0. TheGreen' sfunction canberepresented intheform (3);theauxiliary function isidenti®ed by solving thecorresponding third boundary value problem fortheLaplace equation ¦2 =0. TheGreen' sfunction issymmetric with respect toitsarguments: (r, )= ( ,r).§)¨ Refer ences forSubsection 7.2.1: V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), N.S.Koshlyak ov, E.B.Gliner ,andM.M.Smirno v(1970). Page479 480 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.2.2. Problems inCartesian Coor dinate System Thetwo-dimensional Poisson equation intherectangular Cartesian coordinate system hastheformx2 x ¢2+x2 x £2+ ~( ¢, £)=0. 7.2.2-1. Particular solutions ofthePoisson equation with aspecial right-hand side. 1 ¡.If ~( ¢, £)= © ª«=1 © ª¬=1 «®¬exp( ¯ «¢+ ° ¬£),theequation hassolutions oftheform ( ¢, £)=- © ±«=1 © ±¬=1 «®¬¯2«+ °2¬exp( ¯ «¢+ ° ¬£). 2¡.If ~( ¢, £)=© ª«=1 © ª¬=1 «®¬sin( ¯ «¢+ ² «)sin( ° ¬£+ ³ ¬),theequation admits solutions oftheform ( ¢, £)= © ±«=1 © ±¬=1 «®¬¯2«+ °2¬sin( ¯ «¢+ ² «)sin( ° ¬£+ ³ ¬). 7.2.2-2. Domain: - ´< ¢< ´,- ´< £< ´. Solution: ( ¢, £)=1 2 µ ¶-¶ µ ¶-¶ ~( , )ln1· ( ¢- )2+( £- )2 ¸ ¸ . 7.2.2-3. Domain: - ´< ¢< ´,0£ £< ´.First boundary value problem. Ahalf-plane isconsidered. Aboundary condition isprescribed: = ¹( ¢)at £=0. Solution: ( ¢, £)=1µ ¶-¶ £ ¹( º)¸ º ( ¢- º)2+ £2+1 2 µ ¶0 µ ¶-¶ »( º, ¼)ln · ( ¢- º)2+( £+ ¼)2· ( ¢- º)2+( £- ¼)2 ¸ º¸ ¼.§)¨ Refer ence:A.G.Butk ovskiy (1979). 7.2.2-4. Domain: - ´< ¢< ´,0£ £< ´.Second boundary value problem. Ahalf-plane isconsidered. Aboundary condition isprescribed:½ ¾ = ¹( ¢)at £=0. Solution: ( ¢, £)=1¿µ ¶-¶ ¹( º)ln · ( ¢- º)2+ £2¸ º +1 2 ¿µ¶0 µ¶-¶ »( º, ¼) Àln1· ( ¢- º)2+( £- ¼)2+ln1· ( ¢- º)2+( £+ ¼)2 Á ¸ º¸ ¼+ Â, where Âisanarbitrary constant.§)¨ Refer ence:V.S.Vladimiro v(1988). Page480 7.2. POISSON EQUATION 2 =- (x) 481 7.2.2-5. Domain: - < < ,0£ £ .First boundary value problem. Anin®nite strip isconsidered. Boundary conditions areprescribed:= 1( )at =0, = 2( )at = . Solution:( , )=1 2 sin - 1( ) cosh[ ( - ) ]-cos( ) +1 2 sin - 2( ) cosh[ ( - ) ]+cos( ) +1 4 0 - ( , )lncosh[ ( - ) ]-cos[ ( + ) ] cosh[ ( - ) ]-cos[ ( - ) ] . Refer ence:H.S.Carsla wandJ.C.Jaeger(1984). 7.2.2-6. Domain: - < < ,0£ £ .Second boundary value problem. Anin®nite strip isconsidered. Boundary conditions areprescribed: = 1( )at =0, = 2( )at = . Solution:( , )=- - 1( ) ( , , ,0) + - 2( ) ( , , , ) + 0 - ( , ) ( , , , ) + . Here,( , , , )=1 4 ln1 cosh[ ( - ) ]-cos[ ( - ) ]+1 4 ln1 cosh[ ( - ) ]-cos[ ( + ) ], where isanarbitrary constant. 7.2.2-7. Domain: - < < ,0£ £ .Third boundary value problem. Anin®nite strip isconsidered. Boundary conditions areprescribed: - 1 = 1( )at =0, + 2 = 2( )at = . Thesolution ( , )isdetermined bytheformula inParagraph 7.2.2-6 where( , , , )=1 2 =1 ( ) ( ) 2 ! exp "- ! | - | #, ( )= ! cos( ! )+ 1sin( ! ), 2=1 2( !2 + 2 1) $%+( 1+ 2)( !2 + 1 2) ( !2 + 2 1)( !2 + 2 2) &. Here, the ! arepositi veroots ofthetranscendental equation tan( !)=( 1+ 2) !!2- 1 2. Page481 482 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.2.2-8. Domain: - < < ,0£ £ .Mixedboundary value problem. Anin®nite strip isconsidered. Boundary conditions areprescribed:= 1( )at =0, = 2( )at = . Solution:( , )= - 1( ) $ ( , , , )& '=0 + - 2( ) ( , , , ) + 0 - ( , ) ( , , , ) , where( , , , )=1 =01! exp "- ! | - | #sin( ! )sin( ! ), ! = (2 (+1) 2 . 7.2.2-9. Domain: 0£ < ,0£ £ .First boundary value problem. Asemiin®nite strip isconsidered. Boundary conditions areprescribed:= 1( )at =0, = 2( )at =0, = 3( )at = . Solution:( , )= 0 1( ) $ ( , , , )& )=0 + 0 2( ) $ ( , , , )& '=0 - 0 3( ) $ ( , , , )& '= + 0 0 ( , ) ( , , , ) , where( , , , )=1 4 lncosh[ ( - ) ]-cos[ ( + ) ] cosh[ ( - ) ]-cos[ ( - ) ]-1 4 lncosh[ ( + ) ]-cos[ ( + ) ] cosh[ ( + ) ]-cos[ ( - ) ]. Alternati vely,theGreen' sfunction canberepresented intheseries form( , , , )=1 =11* + exp "- * | - | #-exp "- * | + | #-,sin( * )sin( * ), * = (. Refer ences :N.N.Lebede v,I.P.Skal' skaya, andYa.S.U¯yand (1955), A.G.Butk ovskiy (1979). 7.2.2-10. Domain: 0£ < ,0£ £ .Third boundary value problem. Asemiin®nite strip isconsidered. Boundary conditions areprescribed: .- 1 = 1( )at =0, - 2 = 2( )at =0, + 3 = 3( )at = . Solution:( , )= 0 0 ( , ) ( , , , ) - 0 1( ) ( , ,0, ) - 0 2( ) ( , , ,0) + 0 3( ) ( , , , ) , Page482 7.2. POISSON EQUATION 2 =- (x) 483 where( , , , )= =1 ( ) ( ) 2 ! ( ! + 1) / ( , ), ( )= ! cos( ! )+ 2sin( ! ), 2=1 2( !2 + 2 2) $%+( 2+ 3)( !2 + 2 3) ( !2 + 2 2)( !2 + 2 3)&,/ ( , )= 0exp(- ! ) +! cosh( ! )+ 1sinh( ! ) ,for > , exp(- ! ) +! cosh( ! )+ 1sinh( ! ) ,for > . Here, the ! arepositi veroots ofthetranscendental equation tan( !)=( 2+ 3) !!2- 2 3. 7.2.2-11. Domain: 0£ < ,0£ £ .Mixedboundary value problems. 1 1.Asemiin®nite strip isconsidered. Boundary conditions areprescribed:= 1( )at =0, = 2( )at =0, = 3( )at = . Solution:( , )= 0 1( ) $ ( , , , )& )=0 - 0 2( ) ( , , ,0) + 0 3( ) ( , , , ) + 0 0 ( , ) ( , , , ) , where( , , , )=1 2 =0 2 * + exp "- * | - | #-exp "- * | + | #-,cos( * )cos( * ),* = (,2= 31for (=0, 2for (¹0. 2 1.Asemiin®nite strip isconsidered. Boundary conditions areprescribed: .= 1( )at =0, = 2( )at =0, = 3( )at = . Solution:( , )=- 0 1( ) ( , ,0, ) + 0 2( ) $ ( , , , )& '=0 - 0 3( ) $ ( , , , )& '= + 0 0 ( , ) ( , , , ) , where( , , , )=1 =11* + exp "- * | - | #+exp "- * | + | #-,sin( * )sin( * ), * = (. 7.2.2-12. Domain: 0£ < ,0£ < .First boundary value problem. Aquadrant oftheplane isconsidered. Boundary conditions areprescribed:= 1( )at =0, = 2( )at =0. Solution:( , )=4 0 1( ) [ 2+( - )2][ 2+( + )2]+4 0 2( ) [( - )2+ 2][( + )2+ 2] +1 2 0 0 ( , )ln 4( - )2+( + )24( + )2+( - )24( - )2+( - )24( + )2+( + )2 . Refer ences :V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974), A.G.Butk ovskiy (1979). Page483 484 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.2.2-13. Domain: 0£ £ ,0£ £ 5.First boundary value problem. Arectangle isconsidered. Boundary conditions areprescribed:= 1( )at =0, = 2( )at = ,= 3( )at =0, = 4( )at = 5. Solution:( , )= 0 60 ( , ) ( , , , ) +60 1( ) $ ( , , , )& )=0 -60 2( ) $ ( , , , )& )= + 0 3( ) $ ( , , , )& '=0 - 0 4( ) $ ( , , , )& '=6 . Twoforms ofrepresentation oftheGreen' sfunction:( , , , )=2 =1sin( 7 )sin( 7 )7 sinh( 7 5) / ( , )=25 8=1sin( *8)sin( *8)*8sinh( *8) 9 8( , ), where7 = (,/ ( , )= 0sinh( 7 )sinh[ 7 ( 5- )] for 5³ > ³0, sinh( 7 )sinh[ 7 ( 5- )] for 5³ > ³0,*8= :5,9 8( , )= 0sinh( *8)sinh[ *8( - )]for ³ > ³0, sinh( *8)sinh[ *8( - )]for ³ > ³0. TheGreen' sfunction canbewritten inform ofadouble series:( , , , )=4 5 =1 8=1sin( 7 )sin( *8)sin( 7 )sin( *8)72 + *28, 7 = (, *8= :5. Refer ence:A.G.Butk ovskiy (1979). 7.2.2-14. Domain: 0£ £ ,0£ £ 5.Third boundary value problem. Arectangle isconsidered. Boundary conditions areprescribed: .- 1 = 1( )at =0, .+ 2 = 2( )at = , - 3 = 3( )at =0, + 4 = 4( )at = 5. Solution:( , )= 0 60 ( , ) ( , , , ) - 60 1( ) ( , ,0, ) + 60 2( ) ( , , , ) - 0 3( ) ( , , ,0) + 0 4( ) ( , , , 5) . Here,( , , , )= =1 8=1 ( ) ( ) ; 8( ) ; 8( ) 2 ; 8 2( !2 + <28), ( )=cos( ! )+ 1! sin( ! ), 2= 2 2 !2 !2 + 2 1!2 + 2 2+ 1 2 !2 + 2 1+ 2 1!2 ,; 8( )=cos( < 8)+ 3< 8sin( < 8), ; 8 2= 4 2 <28 <28+ 2 3<28+ 2 4+ 3 2 <28+ 5 2 1+ 2 3<28, Page484 7.2. POISSON EQUATION 2 =- (x) 485 where the ! and < 8arepositi veroots ofthetranscendental equations tan( !)!= 1+ 2!2- 1 2,tan( < 5)<= 3+ 4<2- 3 4. 7.2.2-15. Domain: 0£ £ ,0£ £ 5.Mixedboundary value problem. Arectangle isconsidered. Boundary conditions areprescribed:= 1( )at =0, .= 2( )at = ,= 3( )at =0, = 4( )at = 5. Solution:( , )= 0 60 ( , ) ( , , , ) + 60 1( ) $ ( , , , )& )=0 + 60 2( ) ( , , , ) + 0 3( ) $ ( , , , )& '=0 + 0 4( ) ( , , , 5) . Twoforms ofrepresentation oftheGreen' sfunction:( , , , )=2 =0sin( 7 )sin( 7 )7 cosh( 7 5) / ( , )=25 8=0sin( *8)sin( *8)*8cosh( *8) 9 8( , ), where7 = (2 (+1),/ ( , )= 0sinh( 7 )cosh[ 7 ( 5- )] for 5³ > ³0, sinh( 7 )cosh[ 7 ( 5- )] for 5³ > ³0,*8= (2 :+1)5,9 8( , )= 0sinh( *8)cosh[ *8( - )]for ³ > ³0, sinh( *8)cosh[ *8( - )]for ³ > ³0. TheGreen' sfunction canbewritten inform ofadouble series:( , , , )=4 5 =0 8=0sin( 7 )sin( *8)sin( 7 )sin( *8)72 + *28,7 = (2 (+1) 2 , *8= (2 :+1) 2 5. 7.2.3. Problems inPolar Coor dinate System Thetwo-dimensional Poisson equation inthepolar coordinate system iswritten as 1 = = = =+1 = 2 2 2+ ( = ,)=0, = =4 2+ 2. 7.2.3-1. Domain: 0£ = £ >,0££2 .First boundary value problem. Acircle isconsidered. Aboundary condition isprescribed:= ()at = = >. Page485 486 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES Solution:( = ,)=1 2 2 ? 0 ( ) >2- = 2= 2-2 > = cos(- )+ >2 +2 ? 0 @0 ( , ) ( = ,, , ) , where( = ,, , )=1 2 ln1 |r-r0|-1 2 ln >= 0|( > = 0)2r0-r|, r={ , }, = = cos, = = sin, r0={ 0, 0}, 0= cos , 0= sin . Themagnitude ofavector difference iscalculated as| r- 5r0|2= 2 = 2-2 5 =cos(- )+ 522 ( and 5areanyscalars). Thus, weobtain( = ,, , )=1 4 ln = 22-2 >2 =cos(- )+ >4>2[ = 2-2 =cos(- )+ 2]. Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), A.G.Butk ovskiy (1979). 7.2.3-2. Domain: 0£ = £ >,0££2 .Third boundary value problem. Acircle isconsidered. Aboundary condition isprescribed: A+ = ()at = = >. Solution:( = ,)= >2 ? 0 ( ) ( = ,, >, ) +2 ? 0 @0 ( , ) ( = ,, , ) , where( = ,, , )=1 =0 8=1 B C ( ! 8 = ) C ( ! 8) ( !2 8>2+ 2>2- (2)[ C ( ! 8>)]2cos[ ((- )],B0=1,B =2( (=1,2, DEDED). Here, the C ( )aretheBessel functions andthe ! 8arepositi veroots ofthetranscendental equation! C F ( !>)+ C ( !>)=0. 7.2.3-3. Domain: >£ = < ,0££2 .First boundary value problem. Theexterior ofacircle isconsidered. Aboundary condition isprescribed:= ()at = = >. Solution:( = ,)=1 2 2 ? 0 ( ) = 2- >2= 2-2 > = cos(- )+ >2 +2 ? 0 @ ( , ) ( = ,, , ) , where theGreen' sfunction ( = ,, , )isde®ned bytheformula presented inParagraph 7.2.3-1. Refer ence:A.G.Butk ovskiy (1979). Page486 7.2. POISSON EQUATION G2 =- (x) 487 7.2.3-4. Domain: >1£ = £ >2,0££2 .First boundary value problem. Anannular domain isconsidered. Boundary conditions areprescribed:= 1()at = = >1, = 2()at = = >2. Solution:( = ,)= >12 ? 0 1( ) $ ( = ,, , )& )=@1 - >22 ? 0 2( ) $ ( = ,, , )& )=@2 +2 ? 0 @2@1 ( , ) ( = ,, , ) . Here,( = ,, , )=1 2 =0 ln1 = -ln >1 = H , where= 2 = = 2+ I2 -2 =I cos(- ),( = H )2= = 2+( I H )2-2 =I H cos(- ),I = 0( >1 >2)2 J for (=2 , ( >2 >1)2 J+2for (=2 +1, I H = >2 1I . Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 7.2.3-5. Domain: 0£ = £ >,0££ .First boundary value problem. Asemicircle isconsidered. Boundary conditions areprescribed:= 1()at = = >, = 2( = )at=0, = 3( = )at= . Solution:( = ,)=- > ? 0 1( ) $ ( = ,, , )& )=@ + @0 2( )1 $ ( = ,, , )& '=0 - @0 3( )1 $ ( = ,, , )& '= ? + ? 0 @0 ( , ) ( = ,, , ) , where( = ,, , )=1 4 ln = 22-2 >2 =cos(- )+ >4>2[ = 2-2 =cos(- )+ 2]-1 4 ln = 22-2 >2 =cos(+ )+ >4>2[ = 2-2 =cos(+ )+ 2]. SeealsoExample 2inParagraph 7.2.4-2. Refer ences :V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 7.2.3-6. Domain: 0£ = £ >,0££ 2.First boundary value problem. Aquadrant ofacircle isconsidered. Boundary conditions areprescribed:= 1()at = = >, = 2( = )at=0, = 3( = )at= 2. Solution:( = ,)=- > ? K2 0 1( ) $ ( = ,, , )& )=@ + @0 2( )1 $ ( = ,, , )& '=0 - @0 3( )1 $ ( = ,, , )& '= ? K2 + ? K2 0 @0 ( , ) ( = ,, , ) , Page487 488 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES where( = ,, , )= 1( = ,, , )- 1( = ,, ,2 - )- 1( = ,, , - )+ 1( = ,, , + ),1( = ,, , )=1 4 ln = 22-2 >2 =cos(- )+ >4>2[ = 2-2 =cos(- )+ 2]. SeealsoExample 3inParagraph 7.2.4-2. Refer ences :V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 7.2.3-7. Domain: 0£ = £ >,0££ L.First boundary value problem. Acircular sector isconsidered. Boundary conditions areprescribed:= 1()at = = >, = 2( = )at=0, = 3( = )at= L. Solution:( = ,)=- > M0 1( ) $ ( = ,, , )& )=@ + @0 2( )1 $ ( = ,, , )& '=0 - @0 3( )1 $ ( = ,, , )& '=M + M0 @0 ( , ) ( = ,, , ) . 1 1.For L= (,where (isapositi veinteger,theGreen' sfunction isexpressed as( = ,, , )= -1J=0 +1( = ,, ,2 L+ )- 1( = ,, ,2 L- ) ,,1( = ,, , )=1 4 ln = 22-2 >2 =cos(- )+ >4>2[ = 2-2 =cos(- )+ 2]. Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 2 1.Forarbitrary L,theGreen' sfunction isgivenby( = ,, , )=1 2 ln NNPO ? KM-Å Q? KM NN NN >2 ? KM-(Å QO) ? KM NNNN O ? KM- Q? KM NN NN >2 ? KM-( QO) ? KM NN, whereO= = R SUT , Q= R S',Å Q= R - S',and V2=-1. 7.2.3-8. Domain: 0£ = < ,0££ L.First boundary value problem. Awedge domain isconsidered. Boundary conditions areprescribed:= 1( = )at=0, = 2( = )at= L. Solution:( = ,)= 0 1( )1 $ ( = ,, , )& '=0 - 0 2( )1 $ ( = ,, , )& '=M +M0 0 ( , ) ( = ,, , ) , where( = ,, , )=1 4 ln = 2 ? KM-2( =) ? KMcos[ (+ ) L]+ 2 ? KM = 2 ? KM-2( =) ? KMcos[ (- ) L]+ 2 ? KM. Alternati vely,theGreen' sfunction canberepresented inthecomple xform( = ,, , )=1 2 ln NNPO ? KM-Å Q? KM NNNNWO ? KM- Q? KM NN,O= = R SUT , Q= R S',Å Q= R - S', V2=-1. Page488 7.2. P OISSON EQUATION X2 Y= - Z(x) 489 7.2.4. Arbitrary Shape Domain. Conformal Mappings Method 7.2.4-1. Description of the method. Tables of conformal mappings. Any simply connected domain [in the \ ]-plane with a piecewise smooth boundary can be mapped in a mutually unique way, with an appropriate conformal mapping, onto the upper half-plane orinto a unit circle in a^ _-plane. Under a conformal mapping, a Poisson equation in the \ ]-plane transforms into a Poisson equation in the ^ _-plane; what is changed is the function , as well as the function `in the boundary condition. Consequently, a ®rst and a second boundary value problem for the plane domain [can be reduced, respectively, to a ®rst and a second boundary value problem for the upper half-plane or a unit circle. The latter problems are considered above (see Subsections7.2.2 and 7.2.3). A large number of conformal mappings (mappings de®ned by analytic functions) of various domains onto the upper half-plane or a unit circle can be found, for example, in Lavrik and Savenkov(1970), Lavrent'ev and Shabat (1973), and Ivanov and Trubetskov (1994). Table22present sconforma lmapping sofsomedomain s[inthecompl explaneOonto the upper half-plane Im a³ 0in the complex plane a. In the relations involving square roots, it is assumed that b Q= b| Q| + cos "1 2 #+ Vsin "1 2 #-,, where=arg Q(i.e., the ®rst branch of b Qis taken). Table23present sconforma lmapping sofsomedomain s [inthecompl explaneOonto the unit circle | a| £ 1 in the complex plane a. 7.2.4-2. General formula for the Green's function. Example boundary value problems. Let a function a= a(O) de®ne a conformal mapping of a domain [in the complex planeOonto the upper half-plane in the complex plane a. Then the Green's function of the ®rst boundary value problem in [for the Poisson (Laplace) equation is expressed as( \, ], , )=1 2 clnN N N N a (O)-Å a( Q)a(O)- a( Q) N N N N ,O= \+ Vd], Q= + Vd, ( 1) where a(O)= ^( \, ])+ Vd_( \, ]) and Å a(O)= ^( \, ])- Vd_( \, ]). The solution of the ®rst boundary value problem for the Poisson equation is determined by the above Green's function in accordance with formula (2) speci®ed in Paragraph 7.2.1-1. Example 1. Consider the ®rst boundary value problem for the Poisson equation in the strip - e< f< e,0 £ g£ h. Thefunctio nthatmapsthisstripontotheuppe rhalf-plan ehastheform i( j)=exp( k jElm h)(seethesecon drowofTable 22 ). Substitutin gthisexpressio ninto relation (1) and performing elementary transformations, we obtain the Green's functionn( f, g, o, p)=1 4 klncosh[ k( f- o) lmh]-cos[ k( g+ p) lmh] cosh[ k( f- o) lmh]-cos[ k( g- p) lmh]. Example 2. Consider the ®rst boundary value problem for the Poisson equation in a semicircle of radius hsuch thatq= { f2+ g2£ h2, g³ 0}. The domain qis conformally mapped onto the upper half-plane by the function i( j)= -( jElmh+ h lmj) (seethesixthrowinTable22).Substitutin gthisexpressio ninto(1),wearriveattheGreen 'sfunctionn( f, g, o, p)=1 2 kln r j-Å srtr h2- jÅ srr j- srur h2- j sr, j= f+ vUg, s= o+ vUp. Example 3. Consider the ®rst boundary value problem for the Poisson equation in a quadrant of a circle of radius h, so that q= { f2+ g2£ h2, f³ 0, g³ 0}. The conformal mapping of the domain qonto the upper half-plane is performed withthefunctio n i( j)=-( jElm h)2-( h lm j)2(seetheseventhrowofTable22).Substitutin gthisexpressio ninto(1)yieldsn( f, g, o, p)=1 2 kln r j2-Å s2rtr h4- j2Å s2rr j2- s2rtr h4- j2 s2r, j= f+ vUg, s= o+ vUp. 3 1. Let a function a= a(O) de®ne a conformal mapping of a domain [in the complex planeOonto the unit circle wxa w£ 1in the complex plane a. Then the Green's function of the ®rst boundary value problem in [for the Laplace equation is given byy( \, ], z, {)=1 2 clnN N N N 1 -Å a( Q) a(O)a(O)- a( Q) N N N N ,O= \+ Vd], Q= z+ Vd{. ( 2)|} References for Subsection 7.2.4: N. N. Lebedev, I. P. Skal'skaya, and Ya. S. U¯yand (1955), A. G. Sveshnikov and A. N. Tikhonov (1974). Page 489 490 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES TABLE 22 Conformal mapping ofsome domains [intheO-plane onto theupper half-plane Im a³0inthe a-plane. Notation:O= \+ Vd]and a= ^+ Vd_ No Domain [intheO-plane Transformation 1First quadrant: 0£ \< ~,0£ ]< ~ a= 2O2+ 5,, 5arerealnumbers 2In®nite strip ofwidth : - ~< \< ~,0£ ]£ a=exp( cO ) 3Semiin®nite strip ofwidth : 0£ \< ~,0£ ]£ a=cosh( cO ) 4Plane with thecut intherealaxis a=bO 5Interior ofanin®nite sector with angle L: 0£argO£ L,0£|O|< ~(0< L£2 c) a=O ? KM 6Upper halfofacircle ofradius :\2+ ]2£ 2, ]³0 a=-O- O 7Quadrant ofacircle ofradius :\2+ ]2£ 2, \³0, ]³0 a=-O22- 2O2 8Sector ofacircle ofradius with angle L:\2+ ]2£ 2,0£argO£ L a=- O ? KM- O ? KM 9Upper half-plane with acircular domain orradius remo ved: ]³0, \2+ ]2³ 2 a=O+ O 10Exterior ofaparabola:]2-2 7 \³0 a= O-1 2 - -1 2 11Interior ofaparabola:2-2 £0 = cosh c 1 2O -1 4 7.3. Helmholtz Equation 2 + =± (x) Manyproblems related tosteady- state oscillation s(mec hanical,acoustical, thermal, electromagnetic, etc.) leadtothetwo-dimensional Helm holtzequatio n.For <0,thisequation describes mass transfer processes with volume chemical reactions ofthe®rstorder .Moreo ver,anyelliptic equation with constant coef®cients canbereduced totheHelmholtz equation. 7.3.1. General Remarks, Results, and Form ulas 7.3.1-1. Some de®nitions. The Helmholtz equation iscalled homogeneous if =0andnonhomogeneous if ¹0.A homogeneous boundary valueproblem isaboundary valueproblem forthehomogeneous Helmholtz equation with homogeneous boundary conditions; aparticular solution ofahomogeneous boundary value problem is =0. The values oftheparameter forwhich there arenontri vial solutions (solutions other Page490 7.3. HELMHOL TZEQUATION 2 + =- (x) 491 TABLE 23 Conformal mapping ofsome domains intheO-plane onto theunitcircle ||£1.Notation:O=+ ,= + d,O0=0+ 0,andÅO0=0- 0 No Domain inO-plane Transformation 1Upper half-plane: - << ,0£ < = R SUO-O0O-ÅO0,isarealnumber 2Acircle ofunitradius:2+ 2£1 = R SUO-O0 1-ÅO0O,isarealnumber 3Exterior ofacircle ofradius :2+ 2³ 2= O 4In®nite strip ofwidth : - << ,0£ £ =exp( cO )-exp( cO0 ) exp( cO )-exp( cÅO0 ) 5Semicircle ofradius :2+ 2£ 2,³0= O2+2 O- 2O2-2 O- 2 6Sector ofaunitcircle with angle : |O|£1,0£argO£ =(1+O t)2- (1-O t)2 (1+O t)2+ (1-O t)2 7Exterior ofanellipse with semiax es and : ( )2+( )2³1O=1 2 ¡( - )+ + ¢ than identical zero) ofthehomogeneous boundary value problem arecalled eigen values andthe corresponding solutions, = ,arecalled eigenfunctions oftheboundary value problem. Inwhat follows,the®rst, second, andthird boundary value problems forthetwo-dimensional Helmholtz equation ina®nite two-dimensional domain £with boundary ¤areconsidered. Forthe third boundary value problem with theboundary condition¥¥ ¦+ § =0for r ¨ ¤, itisassumed that §>0.Here, © ª© «isthederivativealong theoutw ardnormal tothecontour ¤,and r={, }. 7.3.1-2. Properties ofeigen values andeigenfunctions. 1 ¬.There arein®nitely manyeigen values { };thesetofeigen values forms adiscrete spectrum forthegivenboundary value problem. 2 ¬.Alleigen values arepositi ve,except fortheeigen value 0=0existing inthesecond boundary valueproblem (thecorresponding eigenfunction is 0=const). Wenumber theeigen values inorder ofincreasing magnitudes, 1< 2< 3< EE. 3 ¬.Theeigen values tend toin®nity asthenumber ®increases. Thefollowing asymptotic estimate holds: lim ¯ ° ® = £2 4 c, where £2isthearea ofthetwo-dimensional domain under study . Page491 492 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 4 ¬.Theeigenfunctions = (, )arede®ned uptoaconstant multiplier .Anytwoeigenfunctions corresponding todifferent eigen values, ¹ ±,areorthogonal:² ³ ± ´ £=0. 5 ¬.Anytwice continuously differentiable function µ= µ(r)thatsatis®es theboundary conditions of aboundary valueproblem canbeexpanded intoauniformly convergent series intheeigenfunctions oftheboundary value problem:µ= °¶=1 µ , where µ =1· ·2 ²³µ ´ £, · ·2= ²³2 ´ £. If µissquare summable, then theseries convergesinmean. 6 ¬.Theeigen values ofthe®rstboundary value problem donotincrease ifthedomain isextended.¸ ¹mº » ¼¾½ ¿ ÀInatwo-dimensional problem, generally correspond toeach eigen value ®nitely manylinearly independent eigenfunctions (1), (2), ÁEÁEÁ, ( Â).These functions canalwaysbe replaced bytheir linear combinations Å ( Ã)= ÄÃ,1 (1)+ EE+ ÄÃ, Ã-1 ( Ã-1)+ ( Ã), Å=1,2, ÁEÁEÁ, Æ, sothattheneweigenfunctions Å (1),Å (2), ÁEÁEÁ,Å ( Â)nowarepairwise orthogonal. Therefore, without lossofgenerality ,weassume thatalltheeigenfunctions areorthogonal.ÇÈ Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). 7.3.1-3. Nonhomogeneous Helmholtz equation with homogeneous boundary conditions. Three cases arepossible. 1 ¬.Iftheequation parameter isnotequal toanyoneoftheeigen values, then there exists theseries solution= °¶=1 É Ê ËÊ- Ë ÌÊ,whereÉ Ê=1·ÌÊ ·2 ²³ ÍÌÊ ´ £, ·ÌÊ ·2= ²³Ì2Ê ´ £. 2 ¬.If Ëisequal tosome eigen value, Ë= Ë±,then thesolution ofthenonhomogeneous problem exists only ifthefunction Í isorthogonal to Ì±,i.e.,² ³ÍÌ± ´ £=0. Inthiscase thesystem isexpressed asÌ= ±-1¶Ê=1 É Ê ËÊ- Ë± ÌÊ+ °¶Ê= ±+1 É Ê ËÊ- Ë± ÌÊ+ Î Ì±,É Ê=1·ÌÊ ·2 ²³ ÍÌÊ ´ £, where ·ÌÊ ·2= ² ³Ì2Ê ´ £,and Îisanarbitrary constant. 3 ¬.If Ë= Ë±and ²³ ÍÌ± ´ £¹0,then theboundary value problem forthenonhomogeneous equation does nothavesolutions.¸ ¹mº » ¼¾½ Ï ÀIf ÆÊmutually orthogonal eigenfunctions Ì( Ã)Ê( Å=1,2, ÁEÁEÁ, ÆÊ)correspond to each eigen value ËÊ,then, for Ë¹ ËÊ,thesolution iswritten asÌ= °¶Ê=1 ÂEÐ¶Ã=1 É( Ã)Ê ËÊ- Ë Ì( Ã)Ê,whereÉ( Ã)Ê=1·Ì( Ã)Ê ·2 ²³ ÍÌ( Ã)Ê ´ Ñ, ·Ì( Ã)Ê ·2= ²³ ÒÌ( Ã)Ê Ó2´ Ñ.ÇÈ Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). Page492 7.3. HELMHOL TZEQUATION Ô2 Õ+ ÖÕ=- ×(x) 493 7.3.1-4. Solution ofnonhomogeneous boundary value problem ofgeneral form. 1 Ø.Thesolution ofthe®rstboundary value problem fortheHelmholtz equation with theboundary conditionÌ= µ(r)for r Ù Ú canberepresented intheformÌ(r)= Û Ü Í ( Ý) y(r, Ý) Þ Ñ ß- Û à á( Ý) ââ ã ß y(r, Ý) Þ Ú ß. (1) Here, r={ ä, å}and Ý={ æ, ç}(r Ù Ñ, Ý Ù Ñ); èè é êdenotes thederivativealong theoutw ardnormal tothecontour Úwith respect tothevariables æand ç.TheGreen' sfunction isgivenbytheseriesy(r, Ý)= ë ¶Ê=1 ÌÊ(r) ÌÊ( Ý)·ÌÊ ·2( ËÊ- Ë), Ë¹ ËÊ, (2) where the ÌÊand ËÊaretheeigenfunctions andeigen values ofthehomogeneous ®rst boundary value problem. 2 Ø.Thesolution ofthesecond boundary value problem with theboundary conditionâ Ìâ ã= á(r)for r Ù Ú canbewritten asÌ(r)= Û Ü Í ( Ý) y(r, Ý) Þ Ñ ß+ Û à á( Ý) y(r, Ý) Þ Ú ß. (3) Here, theGreen' sfunction isgivenbytheseriesy(r, Ý)=-1Ñ2 Ë+ë ¶Ê=1 ÌÊ(r) ÌÊ( Ý)·ÌÊ ·2( ËÊ- Ë), Ë¹ ËÊ, (4) where Ñ2isthearea ofthetwo-dimensional domain under consideration, andthe ËÊand ÌÊarethe positi veeigen values andthecorresponding eigenfunctions ofthehomogeneous second boundary value problem. Forclarity ,theterm corresponding tothezero eigen value Ë 0=0( Ì0=const) is singled outin(4). 3 Ø.Thesolution ofthethird boundary valueproblem fortheHelmholtz equation with theboundary conditionâ Ìâ ã+ ì Ì= á(r)for r Ù Ú isgivenbyformula (3),where theGreen' sfunction isde®ned byseries (2),which involvesthe eigenfunctions ÌÊandeigen values ËÊofthehomogeneous third boundary value problem. 7.3.1-5. Boundary conditions atin®nity inthecase ofanin®nite domain. Inwhat follows,thefunction Í isassumed tobe®nite orsuf®ciently rapidly decaying as í î ï. 1 Ø.For Ë<0,inthecase ofanin®nite domain, thevanishing condition ofthesolution atin®nity is set,Ìî0as í î ï. 2 Ø.For Ë>0,ifthedomain isunbounded, theradiation conditions (Sommerfeld conditions) at in®nity areused. Intwo-dimensional problems, these conditions arewritten as limðmñë ò í Ì=const , limðmñë ò í ó â Ìâ í+ ôò ËÌ õ=0, where ô2=-1. Toidentify asingle solution, theprinciple oflimit absorption andtheprinciple oflimit amplitude arealsoused.ö÷ Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). Page493 494 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.3.2. Problems inCartesian Coor dinate System Atwo-dimensional nonhomogeneous Helmholtz equation intherectangular Cartesian system of coordinates hastheformâ2Ìâ ä2+ â2Ìâ å2+ ËÌ=- Í ( ä, å). 7.3.2-1. Particular solutions andsome relations. 1 Ø.Particular solutions ofthehomogeneous equation ( Í º0):Ì=(É ä+ ø)( Îcos ù å+ úsin ù å), û= ù2,ü=( ý ä+ ø)( Îcosh ù å+ úsinh ù å), û=- ù2,ü=( ýcos ù ä+ øsin ù ä)( Î å+ ú), û= ù2,ü=( ýcosh ù ä+ øsinh ù ä)( Î å+ ú), û=- ù2,ü=( ýcos ù1 ä+ øsin ù1 ä)( Îcos ù2 å+ úsin ù2 å), û= ù2 1+ ù2 2,ü=( ýcos ù1 ä+ øsin ù1 ä)( Îcosh ù2 å+ úsinh ù2 å), û= ù2 1- ù2 2,ü=( ýcosh ù1 ä+ øsinh ù1 ä)( Îcos ù2 å+ úsin ù2 å), û=- ù2 1+ ù2 2,ü=( ýcosh ù1 ä+ øsinh ù1 ä)( Îcosh ù2 å+ úsinh ù2 å), û=- ù2 1- ù2 2, where ý, ø, Î,and úarearbitrary constants. 2 Ø.Fundamental solutions: þ þ ( ä, å)=1 2 ÿ 0( -í) if û=- 2<0,þ þ ( ä, å)= ô 4 (1) 0( ì í)if û= ì2>0,þ þ ( ä, å)=- ô 4 (2) 0( ì í)if û= ì2>0, where í= ä2+ å2,0( )isthemodi®ed Bessel function ofthesecond kind, (1) 0( )and (2) 0( ) aretheHank elfunctions ofthe®rstandsecond kind oforder 0, ä0and å0arearbitrary constants, and ô2=-1.Theleading term oftheasymptotic expansion ofthefundamental solutions, as í î0, isgivenby1 2 ln1ð. 3 Ø.Suppose ü= ü( ä, å)isasolution ofthehomogeneous Helmholtz equation. Then thefunctionsü 1= ü( ä+ 1, å+ 2),ü 2= ü(- ä+ 1, å+ 2),ü 3= ü( äcos + åsin + 1,- äsin + åcos + 2), where 1, 2,and arearbitrary constants, arealsosolutions oftheequation.ö÷ Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). 7.3.2-2. Domain: - ï< ä< ï,- ï< å< ï. 1 .Solution for û=- 2<0:ü( ä, å)=1 2 ÿ Ûë -ë Ûë -ë ( æ, ç)0( ) Þ æ Þ ç, = ( ä- æ)2+( å- ç)2. 2 .Solution for û= ì2>0:ü( ä, å)=- ô 4 - - ( , ) (2) 0( ) , = ( - )2+( - )2. Theradiation conditions (Sommerfeld conditions) atin®nity were used toobtain thissolution (see Paragraph 7.3.1-5, Item 2 ).ö÷ Refer ences :B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980), A.N.Tikhono vandA.A.Samarskii (1990). Page494 7.3. HELMHOL TZEQUATION Ô2 Õ+ Õ=- (x) 495 7.3.2-3. Domain: - ï< < ï,0£ < ï.First boundary value problem. Ahalf-plane isconsidered. Aboundary condition isprescribed:ü= ( )at =0. Solution:ü( , )= - ( ) ( , , , ) =0 + 0 - ( , )( , , , ) . 1 .TheGreen' sfunction for û=- 2<0:( , , , )=1 2 ÿ 0( 1)-0( 2) ,1= ( - )2+( - )2, 2= ( - )2+( + )2. 2 .TheGreen' sfunction for û= 2>0:( , , , )=- 4 (2) 0( 1)- (2) 0( 2) . Theradiation conditions atin®nity were used toobtain thisrelation (seeParagraph 7.3.1-5, Item 2 ).ö÷ Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 7.3.2-4. Domain: - ï< < ï,0£ < ï.Second boundary value problem. Ahalf-plane isconsidered. Aboundary condition isprescribed: ü= ( )at =0. Solution:ü( , )=- - ( )( , , ,0) + 0 - ( , )( , , , ) . 1 .TheGreen' sfunction for û=- 2<0:( , , , )=1 2 ÿ 0( 1)+0( 2) ,1= ( - )2+( - )2, 2= ( - )2+( + )2. 2 .TheGreen' sfunction for û= 2>0:( , , , )=- 4 (2) 0( 1)+ (2) 0( 2) . Theradiation conditions atin®nity were used toobtain thisrelation (seeParagraph 7.3.1-5, Item 2 ).ö÷ Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). Page495 496 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.3.2-5. Domain: 0£ < ï,0£ < ï.First boundary value problem. Aquadrant oftheplane isconsidered. Boundary conditions areprescribed:ü= 1( )at =0, ü= 2( )at =0. Solution:ü( , )= 0 1( ) ( , , , ) ! =0 + 0 2( ) ( , , , ) =0 + 0 0 ( , )( , , , ) . 1 .TheGreen' sfunction for û=- 2<0:( , , , )=1 2 ÿ "0( 1)-0( 2)-0( 3)+0( 4) ,1= ( - )2+( - )2, 2= ( - )2+( + )2,3= ( + )2+( - )2, 4= ( + )2+( + )2. 2 .TheGreen' sfunction for û= 2>0:( , , , )=- 4 (2) 0( 1)- (2) 0( 2)- (2) 0( 3)+ (2) 0( 4) . 7.3.2-6. Domain: 0£ < ï,0£ < ï.Second boundary value problem. Aquadrant oftheplane isconsidered. Boundary conditions areprescribed: # ü= 1( )at =0, ü= 2( )at =0. Solution:ü( , )=- 0 1( )( , ,0, ) - 0 2( )( , , ,0) + 0 0 ( , )( , , , ) . 1 .TheGreen' sfunction for û=- 2<0:( , , , )=1 2 ÿ "0( 1)+0( 2)+0( 3)+0( 4) ,1= ( - )2+( - )2, 2= ( - )2+( + )2,3= ( + )2+( - )2, 4= ( + )2+( + )2. 2 .TheGreen' sfunction for û= 2>0:( , , , )=- 4 (2) 0( 1)+ (2) 0( 2)+ (2) 0( 3)+ (2) 0( 4) . Page496 7.3. HELMHOL TZEQUATION Ô2 Õ+ Õ=- (x) 497 7.3.2-7. Domain: - ï< < ï,0£ £ $.First boundary value problem. Anin®nite strip isconsidered. Boundary conditions areprescribed:ü= 1( )at =0, ü= 2( )at = $. Solution:ü( , )= - 1( ) ( , , , ) =0 - - 2( ) ( , , , ) = % + % 0 - ( , )( , , , ) . Green' sfunction:( , , , )=1$ &(' =11) ' exp *- ) ' | - | +sin( , ')sin( , '), , ' = - .$, ) ' = / ,2 ' - 0. Alternati vely,theGreen' sfunction for 0=- 12<0canberepresented as( , , , )=1 2- & ' =- "20( 1 31 ' )-20( 1 32 ' ) ,3 ' 1= 4( - )2+( - -2. $)2, 3 ' 2= 4( - )2+( + +2. $)2. 7.3.2-8. Domain: - ï< < ï,0£ £ $.Second boundary value problem. Anin®nite strip isconsidered. Boundary conditions areprescribed: 5= 1( )at =0, 5= 2( )at = $. Solution:5( , )=- - 1( )( , , ,0) + - 2( )( , , , $) + % 0 - 6( , )( , , , ) . Green' sfunction:( , , , )=1 2 $ &(' =0 7 ') ' exp *- ) ' | - | +cos( , ')cos( , '),, ' = - .$, ) ' =/ ,2 ' - 0,7= 81for.=0, 2for.¹0. Alternati vely,theGreen' sfunction for 0=- 12<0canberepresented as( , , , )=1 2- & ' =- "20( 1 31 ' )+20( 1 32 ' ) ,3 ' 1= 4( - )2+( - ' 1)2, ' 1=2. $+ ,3 ' 2= 4( - )2+( - ' 2)2, ' 2=2. $- .9: Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). Page497 498 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.3.2-9. Domain: - ;< << ;,0£ =£ $.Third boundary value problem. Anin®nite strip isconsidered. Boundary conditions areprescribed: 5- >15= ?1( <)at ==0, @ 5+ >25= ?2( <)at == $. Thesolution5( <, =)isdetermined bytheformula inParagraph 7.3.2-8 whereA( <, =, B, C)=1 2 D & ' =1 E ' ( =)E ' ( C)FE 'F2 ) ' exp *- ) ' | <- B| +, ) ' = / ù2 ' - 0,E ' ( =)= ù ' cos( ù '=)+ >1sin( ù '=), FE 'F2=1 2( ù2 ' + >2 1) GIH+( >1+ >2)( ù2 ' + >1 >2) ( ù2 ' + >2 1)( ù2 ' + >2 2) J. Here, the ù ' arepositi veroots ofthetranscendental equation tan( ù H)=( >1+ >2) ùù2- >1 >2. 7.3.2-10. Domain: - ;< << ;,0£ =£ H.Mixedboundary value problem. Anin®nite strip isconsidered. Boundary conditions areprescribed:5= ?1( <)at ==0, @ K5= ?2( <)at == H. Solution:5( <, =)= LD-D ?1( B) G @@ C A( <, =, B, C)J M=0 N B+ LD-D ?2( B) A( <, =, B, H)N B + L O 0 LD-D 6( B, C) A( <, =, B, C)N BN C, whereA( <, =, B, C)=1HD &(' =01) ' exp *- ) ' | <- B| +sin( , '=)sin( , 'C), , ' = -(2.+1) 2 H, ) ' = / ,2 ' - 0. 7.3.2-11. Domain: 0£ << ;,0£ =£ H.First boundary value problem. Asemiin®nite strip isconsidered. Boundary conditions areprescribed:5= ?1( =)at <=0,5= ?2( <)at ==0,5= ?3( <)at == H. Solution:5( <, =)= LO 0 LD0 6( B, C) A( <, =, B, C)N BN C+ LO 0 ?1( C) G @@ B A( <, =, B, C)J P=0 N C + LD0 ?2( B) G @@ C A( <, =, B, C)J M=0 N B- LD0 ?3( B) G @@ C A( <, =, B, C)J M=O N B, whereA( <, =, B, C)=1H D & ' =11) ' Q exp *- ) ' | <- B| +-exp *- ) ' | <+ B| +SRsin( , '=)sin( , 'C),, ' = - .H, ) ' = / ,2 ' - 0. Page498 7.3. HELMHOL TZEQUATION T2 U+ VU=- W(x) 499 7.3.2-12. Domain: 0£ << ;,0£ =£ H.Second boundary value problem. Asemiin®nite strip isconsidered. Boundary conditions areprescribed:@ X5= ?1( =)at <=0, @ K5= ?2( <)at ==0, @ K5= ?3( <)at == H. Solution:5( <, =)= LO 0 LD0 6( B, C) A( <, =, B, C)N BN C- LO 0 ?1( C) A( <, =,0, C)N C - LD0 ?2( B) A( <, =, B,0)N B+ LD0 ?3( B) A( <, =, B, H)N B, whereA( <, =, B, C)=1 2 H D Y(Z =0 7 Z[ ZQ exp \- [ Z | <- B| ]+exp \- [ Z | <+ B| ]SRcos( ^ Z=)cos( ^ ZC),^ Z = _ `H, [ Z = / ^2 Z - 0,7= 81for`=0, 2for`¹0. 7.3.2-13. Domain: 0£ << ;,0£ =£ H.Third boundary value problem. Asemiin®nite strip isconsidered. Boundary conditions areprescribed:@ X5- >15= ?1( =)at <=0, @ K5- >25= ?2( <)at ==0, @ K5+ >35= ?3( <)at == H. Thesolution5( <, =)isdetermined bytheformula inParagraph 7.3.2-12 whereA( <, =, B, C)=D Y Z =1 E Z ( =)E Z ( C)FE ZF2 [ Z ( [ Z + >1) a Z ( <, B), [ Z = b ù2 Z - c,E Z ( =)= ù Z cos( ù Z=)+ >2sin( ù Z=), FE ZF2=1 2( ù2 Z + >2 2) GIH+( >2+ >3)( ù2 Z + >2 >3) ( ù2 Z + >2 2)( ù2 Z + >2 3) J,a Z ( <, B)= dexp(- [ Z e ) Q[ Z cosh( [ ZB)+ >1sinh( [ ZB) Rfor e > B, exp(- [ ZB) Q[ Z cosh( [ Z e )+ >1sinh( [ Z e ) Rfor B> e . Here, the ù Z arepositi veroots ofthetranscendental equation tan( ù H)=( >2+ >3) ùù2- >2 >3. 7.3.2-14. Domain: 0£ e < f,0£ =£ H.Mixedboundary value problems. 1 g.Asemiin®nite strip isconsidered. Boundary conditions areprescribed:h= i1( =)at e =0, j K h= i2( e )at ==0, j K h= i3( e )at == k. Solution:h( e , =)= L O 0 i1( l) m jj n o( e , =, n, l) p q =0 r l- s t 0 i2( n)o( e , =, n,0)r n + s t 0 i3( n)o( e , =, n, k)r n+ s u 0 s t 0 v( n, l)o( e , =, n, l)r nr l, whereo( e , =, n, l)=1 2 k t w(x =0 y x[ x z exp \- [ x | e - n| ]-exp \- [ x | e + n| ]S{cos( ^ x=)cos( ^ xl),^ x = _ `k, [ x = b ^2 x - c,y= |1for`=0, 2for`¹0. Page499 500 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 2g.Asemiin®nite strip isconsidered. Boundary conditions areprescribed:j } h= i1( =)at e =0, h= i2( e )at ==0, h= i3( e )at == k. Solution:h( e , =)=- su 0 i1( l)o( e , =,0, l)r l+ st 0 i2( n)m jj l o( e , =, n, l)p ~ =0 r n - st 0 i3( n)m jj l o( e , =, n, l)p ~ =u r n+ su 0 st 0 v( n, l)o( e , =, n, l)r nr l, whereo( e , =, n, l)=1k t wx =11[ x z exp \- [ x | e - n| ]+exp \- [ x | e + n| ]S{sin( ^ x=)sin( ^ xl),^ x = _ `k, [ x = b ^2 x - c. 7.3.2-15. Domain: 0£ e £ k,0£ =£ .First boundary value problem. Arectangle isconsidered. Boundary conditions areprescribed:h= i1( =)at e =0, h= i2( =)at e = k,h= i3( e )at ==0, h= i4( e )at == . 1g.Eigen values oftheone-dimensional problem (itisconvenient tolabel them with adouble subscript):c x =_2 `2k2+ 22 ;`=1,2, ;=1,2, Eigenfunctions andthenorm squared:h x =sin ` _ ek sin _ = , h x 2= k 4. Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 2g.Solution for ¹ x :h( e , )= s u 0 s 0 v( n, l)o( e , , n, l)r lr n + s 0 i1( l)m jj n o( e , , n, l)p q =0 r l- s 0 i2( l)m jj n o( e , , n, l)p q =u r l + su 0 i3( n)m jj l o( e , , n, l)p ~ =0 r n- su 0 i4( n)m jj l o( e , , n, l)p ~ = r n. Twoforms ofrepresentation oftheGreen' sfunction:o( e , , n, l)=2k t w(x =1sin( x e )sin( xn)[ x sinh( [ x) x ( , l)=2 t w =1sin( ^ )sin( ^ l) sinh( k) ( , n), where x = _ `, [ x = 2 x - , x ( , )= sinh( [ x)sinh[ [ x ( - )] for ³ > ³0, sinh( [ x)sinh[ [ x ( - )] for ³ > ³0,^ = _ , = ^2 - , ( , )= sinh( )sinh[ ( - )]for ³ > ³0, sinh( )sinh[ ( - )]for ³ > ³0. Alternati vely,theGreen' sfunction canbewritten asthedouble series( , , , )=4 t wx =1 t w =1sin( x)sin( ^ )sin( x)sin( ^ )2 x + ^2 - , x = _ `, ^ = _ . Page500 7.3. HELMHOL TZEQUATION 2 + =- (x) 501 7.3.2-16. Domain: 0£ £ ,0£ £ .Second boundary value problem. Arectangle isconsidered. Boundary conditions areprescribed: = 1( )at =0, = 2( )at = , = 3( )at =0, = 4( )at = . 1 .Eigen values ofthehomogeneous problem: = 2 22+ 22 ;=0,1,2, ;=0,1,2, Eigenfunctions andthenorm squared: =cos cos , 2= 4(1+ 0)(1+ 0), 0= 1for=0, 0for¹0. Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 2 .Solution for ¹ : ( , )= 0 0 !( ", #) $( , , ", #) % # % " - 0 1( #) $( , ,0, #) % #+ 0 2( #) $( , , , #) % # - 0 3( ") $( , , ",0) % "+ 0 4( ") $( , , ", ) % ". Twoforms ofrepresentation oftheGreen' sfunction:$( , , ", #)=1 & ' =0 ( cos( ) )cos( ) ")*sinh( *) + ( , #)=1 & ' =0 ( cos( , )cos( , #)- sinh( - ) . ( , "), where) = ,+ ( , #)= /cosh( *#)cosh[ *( - )] for > #, cosh( *)cosh[ *( - #)] for #> ,, = ,. ( , ")= /cosh( - ")cosh[ - ( - )]for > ", cosh( - )cosh[ - ( - ")]for "> ,*= 0 )2 - , - = 0 ,2 - ,( = 1for=0, 2for¹0. TheGreen' sfunction canalsobewritten asthedouble series$( , , ", #)=1 & ' =0 & ' =0( ( cos( ) )cos( , )cos( ) ")cos( , #))2 + ,2 - , ) = , , = .1InParagraphs 7.3.2-17 through 7.3.2-20, only theeigenvalues andeigenfunctions ofhomo ge- neous boundary value problems forthehomo geneous Helmholtz equation (with!º0)aregiven. Thesolutions ofthecorresponding nonhomo geneous problems canbeconstructed using formulas presented inParagraphs 7.3.1-3 and7.3.1-4. Page501 502 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.3.2-17. Domain: 0£ £ ,0£ £ .Third boundary value problem. Arectangle isconsidered. Boundary conditions areprescribed: - 21 =0at =0, + 22 =0at = , - 23 =0at =0, + 24 =0at = . Eigen values: = -2 + 32 , where the - and 3 arepositi veroots ofthetranscendental equations tan( -)=( 21+ 22) --2- 21 22, tan( 3 )=( 23+ 24) 332- 23 24. Eigenfunctions: =( - cos - + 21sin - )( 3 cos 3 + 23sin 3 ). Thesquare ofthenorm ofaneigenfunction: 2=1 4( -2 + 22 1)( 32 + 22 3) 45+( 21+ 22)( -2 + 21 22) ( -2 + 22 1)( -2 + 22 2) 6 45+( 23+ 24)( 32 + 23 24) ( 32 + 22 3)( 32 + 22 4) 6. Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 7.3.2-18. Domain: 0£ £ ,0£ £ .Mixedboundary value problems. 1 .Arectangle isconsidered. Boundary conditions areprescribed: =0at =0, =0at = , =0at =0, =0at = . Eigen values: = 2 22+ 22 ;=1,2,3, ;=0,1,2, Eigenfunctions andthenorm squared: =sin cos , 2= 4(1+ 0), 0=1for=0, 0for¹0. 2 .Arectangle isconsidered. Boundary conditions areprescribed: =0at =0, =0at = , =0at =0, =0at = . Eigen values: = 2 4 4(2+1)22+(2+1)226;=0,1,2, ;=0,1,2, Eigenfunctions andthenorm squared: =sin 4 (2+1) 2 6sin 4 (2+1) 2 6, 2= 4. Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). Page502 7.3. HELMHOL TZEQUATION 2 + =- (x) 503 7.3.2-19. First boundary value problem foratriangular domain. Thesides ofthetriangle arede®ned bytheequations=0, =0, = - . Theunkno wnquantity iszero forthese sides. Eigen values: = 22 7(+)2+2 8;=1,2, ;=1,2, Eigenfunctions: =sin 4 (+) 6sin -(-1) sin sin 4 (+) 6. Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). 7.3.2-20. Second boundary value problem foratriangular domain. Thesides ofthetriangle arede®ned bytheequations=0, =0, = - . Thenormal derivativeoftheunkno wnquantity forthese sides iszero. Eigen values: = 227(+)2+2 8;=0,1, ;=0,1, Eigenfunctions: =cos 4 (+) 6cos -(-1) cos cos 4 (+) 6. 7.3.3. Problems inPolar Coor dinate System Atwo-dimensional nonhomogeneous Helmholtz equation inthepolar coordinate system iswritten as 1 9 9 9 9+1 9 2 2 :2+ =-!( 9 , :, ;), 9 = < 2+ 2. 7.3.3-1. Particular solutions ofthehomogeneous equation (!º0): =[ = >0( - 9 )+ ? @0( - 9 )]( A :+ B), = -2, =[ = C0( - 9 )+ ? D0( - 9 )]( A :+ B), =- -2, =[ = > ( - 9 )+ ? @ ( - 9 )]( Acos :+ Bsin :), = -2, =[ = C ( - 9 )+ ? D ( - 9 )]( Acos :+ Bsin :), =- -2, where=1,2, ; =, ?, A, Barearbitrary constants; the > ( -)and @ ( -)aretheBessel functions; andthe C ( -)and D ( -)arethemodi®ed Bessel functions.1InParagraphs 7.3.3-2 through 7.3.3-11, only theeigenvalues andeigenfunctions ofhomo ge- neous boundary value problems forthehomo geneous Helmholtz equation (with!º0)aregiven. Thesolutions ofthecorresponding nonhomo geneous problems canbeconstructed using formulas presented inParagraphs 7.3.1-3 and7.3.1-4. Page503 504 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.3.3-2. Domain: 0£ 9 £ E.First boundary value problem. Acircle isconsidered. Aboundary condition isprescribed: =0at 9 = E. Eigen values: = -2 E2;=0,1,2, ;=1,2,3, Here, the - arepositi vezeros oftheBessel functions, > ( -)=0. Eigenfunctions: (1) = > F 9< Gcos :, (2) = > F 9< Gsin :. Eigenfunctions possessing theaxial symmetry property: (1) 0 = >0 F 9 H I 0 J G. Thesquare ofthenorm ofaneigenfunction isgivenbyLK( M)NJ 2=1 2 O E2(1+ N0)[ > P N( -NJ)]2, 2=1,2; RQTS= /1for U= V, 0for U¹ V. Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 7.3.3-3. Domain: 0£ 9 £ E.Second boundary value problem. Acircle isconsidered. Aboundary condition isprescribed:W XK=0at 9 = E. Eigen values:INJ= -2NJE2, where the -NJareroots ofthetranscendental equation >P N( -)=0. Eigenfunctions:K(1)NJ= > N( 9< INJ)cos Y :, K(2)NJ= > N( 9< INJ)sin Y :. Here, Y=0,1,2, ZZZ;for Y¹0,theparameter [assumes thevalues [=1,2,3, ZZZ;for Y=0, aroot - 00=0(thecorresponding eigenfunction is K00=1). Eigenfunctions possessing theaxial symmetry property: K(1) 0 J= >0 F 9 H I 0 J G. Thesquare ofthenorm ofaneigenfunction isgivenby\K( M)NJ \2= O2E2(1+ ] N0) 2 -2NJ( -2NJ- Y2)[ > N( -NJ)]2, \K00 \2=O E2, where 2=1,2; ] QTS= /1for U= V, 0for U¹ V.^_ Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). Page504 7.3. HELMHOL TZEQUATION `2 a+ ba=- c(x) 505 7.3.3-4. Domain: 0£ 9 £ E.Third boundary value problem. Acircle isconsidered. Aboundary condition isprescribed:W XK+ 2 K=0at 9 = E. Eigen values:INJ= -2NJE2; Y=0,1,2, ZZZ; [=1,2,3, ZZZ Here, the -NJisthe [throotofthetranscendental equation ->P N( -)+ 2 E > N( -)=0. Eigenfunctions:K(1)NJ= > N F 9< INJ Gcos Y :, K(2)NJ= > N F 9< INJ Gsin Y :. Thesquare ofthenorm ofaneigenfunction isgivenby\K(1)NJ \2= \K(2)NJ \2= O E2(1+ ] N0) 2 -2NJ( 22E2+ -2NJ- Y2)[ > N( -NJ)]2, ]QTS= /1for U= V, 0for U¹ V.^_ Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 7.3.3-5. Domain: E1£ 9 £ E2.First boundary value problem. Anannular domain isconsidered. Boundary conditions areprescribed:K=0at 9 = E1, K=0at 9 = E2. Eigen values: INJ= -2NJ; Y=0,1,2, ZZZ; [=1,2,3, ZZZ Here, the -NJarepositi veroots ofthetranscendental equation> N( -E1) @ N( -E2)- > N( -E2) @ N( -E1)=0. Eigenfunctions:K(1)NJ=[ > N( -NJ 9 ) @ N( -NJ E1)- > N( -NJ E1) @ N( -NJ 9 )]cos Y :,K(2)NJ=[ > N( -NJ 9 ) @ N( -NJ E1)- > N( -NJ E1) @ N( -NJ 9 )]sin Y :. Thesquare ofthenorm ofaneigenfunction isgivenby\K(1)NJ \2= \K(2)NJ \2=2(1+ ] N0)O -2NJ >2N( -NJ E1)- >2N( -NJ E2)>2N( -NJ E2), ]QTS= /1for U= V, 0for U¹ V.^_ Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 7.3.3-6. Domain: E1£ 9 £ E2.Second boundary value problem. Anannular domain isconsidered. Boundary conditions areprescribed:W XK=0at 9 = E1, W XK=0at 9 = E2. Eigen values: INJ= -2NJ; Y=0,1,2, ZZZ; [=0,1,2, ZZZ Page505 506 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES Here, the -NJareroots ofthetranscendental equation>P N( -E1) @P N( -E2)- >P N( -E2) @P N( -E1)=0. If Y=0,there isaroot - 00=0andthecorresponding eigenfunction is K(1) 00=1. Eigenfunctions:K(1)NJ=[ > N( -NJ 9 ) @P N( -NJ E1)- >P N( -NJ E1) @ N( -NJ 9 )]cos Y :,K(2)NJ=[ > N( -NJ 9 ) @ P N( -NJ E1)- > P N( -NJ E1) @ N( -NJ 9 )]sin Y :. Thesquare ofthenorm ofaneigenfunction isgivenby\K(1)NJ \2= \K(2)NJ \2=2(1+ ] N0)O -2NJ / d1- Y2E2 2 e2NJ f g h P N(e NJ i1)h P N(e NJ i2) j2 - d1- Y2i2 1 e2NJ f k,lK(1) 00 l2= m(i2 2-i2 1); npoTq= r1for s= t, 0for s¹ t.uv Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 7.3.3-7. Domain:i1£ w£i2.Third boundary value problem. Anannular domain isconsidered. Boundary conditions areprescribed:x yK- z K=0at w=i1, x yK+ z K=0at w=i2. Eigen values: { |J=e2 |J; }=0,1,2, ~~~; =1,2,3, ~~~; where thee |Jarepositi veroots ofthetranscendental equation 1(e i1) 2(e i2)- 2(e i2) 1(e i1)=0. Here, weusethenotation 1(e i)=h | (e i)- zeh | (e i), 1(e i)= | ( i)- z | ( i), 2( i)=h | ( i)+ zh | ( i), 2( i)= | ( i)+ z | ( i). Eigenfunctions:K(1) |J=[ 1( |J i1)h | ( |J w)- 1( |J i1) | ( |J w)]cos } :,K(2) |J=[ 1( |J i1)h | ( |J w)- 1( |J i1) | ( |J w)]sin } :. Thesquare ofthenorm ofaneigenfunction isgivenby( =1,2)lK( ) |J l2=1 2 m |i2 2 r |J(i2) 2+ 1- }2i2 2 2 |J f 2 |J(i2)k -1 2 m |i2 1 r |J(i1) 2+ 1- }2i2 1 2 |J f 2 |J(i1)k, |J( w)= 1( |J i1)h | ( |J w)- 1( |J i1) | ( |J w), oTq= r2for s= t, 1for s¹ t.uv Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). Page506 7.3. HELMHOL TZEQUATION 2 + =- (x) 507 7.3.3-8. Domain: 0£ w£i,0£ £ .First boundary value problem. Acircular sector isconsidered. Boundary conditions areprescribed:K=0at w=i, K=0at =0, K=0at = . Eigen values:{ |J= 2 |Ji2; }=1,2,3, ~~~; =1,2,3, ~~~ Here, the |Jarepositi vezeros oftheBessel functions,h | ( )=0. Eigenfunctions:K |J=h | |J wifsin } m f. Thesquare ofthenorm ofaneigenfunction isgivenbylK |J l2= i2 4 h | ( |J) 2 .uv Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 7.3.3-9. Domain: 0£ w£i,0£ £ .Second boundary value problem. Acircular sector isconsidered. Boundary conditions areprescribed:x yK=0at w=i, x K=0at =0, x K=0at = . Eigen values:{ |J= 2 |Ji2; }=0,1,2, ~~~; =0,1,2, ~~~ Here, the |Jareroots ofthetranscendental equationh | ( )=0. Eigenfunctions:K |J=h | |J wi cos } m , K00=1. Thesquare ofthenorm ofaneigenfunction isgivenbylK |J l2= i2 4(1+ n | 0) 1- }22 |Jf h | ( |J) 2 , lK00 l2= i2 2.uv Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 7.3.3-10. Domain: 0£ w£i,0£ £ .Third boundary value problem. Acircular sector isconsidered. Boundary conditions areprescribed:x yK+ z1 K=0at w=i, x K- z2 K=0at =0, x K+ z3 K=0at = . Eigen values:{ |J= 2 |Ji2; }=1,2,3, ~~~; =1,2,3, ~~~ Here, the |Jarepositi veroots ofthetranscendental equation h ( )+ z1 ih ( )=0;the | arepositi veroots ofthetranscendental equation tan( )=( z2+ z3) 2- z2 z3. Eigenfunctions:K |J=h |J wi | cos( |)+ z2sin( |)2 | + z2 2. Thesquare ofthenorm ofaneigenfunction isgivenbylK |J l2= 2 4 +( z2+ z3)( 2 | + z2 z3) ( 2 | + z2 2)( 2 | + z2 3) ¡ 1+ z2 12- 2 |2 | ¢ £2( | ¢ ).uv Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). Page507 508 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.3.3-11. Domain:1£ w£2,0£ £ .First boundary value problem. Boundary conditions areprescribed: ¤ =0at w=1, ¤ =0at =0, ¤ =0at w=2, ¤ =0at = . Eigen values: { | ¢ = 2 | ¢ , where the | ¢ arepositi veroots ofthetranscendental equation£ ( 1) ( 2)-£ ( 2) ( 1)=0, | = } m. Eigenfunctions:¤| ¢ = £ ( | ¢w) ( | ¢1)-£ ( | ¢1) ( | ¢w) sin( |). Thesquare ofthenorm ofaneigenfunction isgivenbyl ¤| ¢l2= m22 | ¢ £ ( | ¢1) 2- £ ( | ¢2) 2£ ( | ¢2) 2.uv Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 7.3.4. Other Orthogonal Coor dinate Systems. Elliptic Domain InParagraphs 7.3.4-1 and7.3.4-2, twoother orthogonal systems ofcoordinates aredescribed in which thehomogeneous Helmholtz equation admits separation ofvariables. 7.3.4-1. Parabolic coordinate system. Intheparabolic coordinates thatareintroduced bytherelations¥=1 2( ¦2- §2), ¨= ¦ § (0£ ¦< ©,- ©< §< ©), theHelmholtz equation hastheformx2 ¤x¦2+ x2 ¤x§2+ { ( ¦2+ §2) ¤ =0. Setting ¤ = ª( ¦) «( §),wearriveatthefollowing linear ordinary differential equations for ª= ª( ¦) and «= «( §):ª¬+( {¦2+ z) ª=0, «¬+( {§2- z) «=0, where zistheseparation constant. Thegeneral solutions ofthese equations aregivenbyª( ¦)= 1 ®-1 ¯2( ° ¦)+ 2 ®-1 ¯2(- ° ¦), «( §)= 1 - ®-1 ¯2( ° §)+ 2 - ®-1 ¯2(- ° §),=1 2 z(- { )-1 ¯2, °=(-4 { )1 ¯4. Here, 1, 1, 2,and 2arearbitrary constants, and ( ±)istheparabolic cylinder function, ( ±)=21 ¯2exp ²-1 4 ±2 ³ ´ ²1 2 ³´ ²1 2- 2 ³ µ ²- 2,1 2;1 2 ±2 ³+2-1 ¯2´ ²-1 2 ³´ ²- 2 ³ ±µ ²1 2- 2,3 2;1 2 ±2 ³¡. For = }=0,1,2, ~~~,wehave | ( ±)=2- |¯2exp²-1 4 ±2 ³¶ |²2-1 ¯2± ³,where ¶ | ( ±)=(-1) | exp² ±2 ³ · |± | exp²- ±2 ³.uv Refer ences :M.Abramo witz andI.Stegun(1964), W.Miller ,Jr.(1977). Page508 7.3. HELMHOL TZEQUATION 2 + =- (x) 509 7.3.4-2. Elliptic coordinate system. Intheelliptic coordinates thatareintroduced bytherelations¥= ¸cosh ¹cos º, ¨= ¸sinh ¹sin º (0£ ¹< ©,0£ º<2 », ¸>0), theHelmholtz equation isexpressed asx2 ¤x¦2+ x2 ¤x§2+ ¸2 { (cosh2¹-cos2º) ¤ =0. Setting ¤ = ( ¹) ¼( º),wearriveatthefollowing linear ordinary differential equations for= ( ¹)and ¼= ¼( º): ½¬½+²1 2 ¸2 ¾cosh 2 ¹- ¿ ³À=0, ¼ ½¬½-²1 2 ¸2 ¾cos2 º- ¿ ³¼=0, where ¿istheseparation constant. Thesolutions ofthese equations periodic in ºaregivenbyÀ( ¹)= ÁCe Â( ¹, Ã), Se Â( ¹, Ã), ¼( º)= Áce Â( º, Ã), se Â( º, Ã), Ã=1 4 ¸2 ¾, where Ce Â( ¹, Ã)andSe Â( ¹, Ã)arethemodi®ed Mathieu functions, andce Â( º, Ã)andse Â( º, Ã)are theMathieu functions; toeach value of Ãthere isacorresponding ¿= ¿ Â( Ã).ÄÅ Refer ences :M.Abramo witz andI.Stegun(1964), W.Miller ,Jr.(1977). 7.3.4-3. Domain: ( ¥ Æ¸)2+( ¨ Æ Ç)2£1.First boundary value problem. Theunkno wnquantity iszero attheboundary oftheelliptic domain:¤ =0if( ¥ Æ¸)2+( ¨ Æ Ç)2=1 ( ¸³ Ç). The®rstthree eigen values andeigenfunctions aregivenbytheapproximate relations¾1= È2 10 2 É1¸2+1Ç2 Ê, Ë1( Ì)= Í0(È10 Ì),¾(c) 2= È2 11 4 É3Î2+1Ç2 Ê, Ë(c) 2( Ì, Ï)= Í1(È11 Ì)cos Ï,¾(s) 2= È2 11 4 É1Î2+3Ç2 Ê, Ë(s) 2( Ì, Ï)= Í1(È11 Ì)sin Ï, whereÈ10=2.4048 andÈ11=3.8317 arethe®rst roots oftheBessel functions Í0and Í1,i.e.,Í0(È10)=0and Í1(È11)=0; Ì= Ð( Ñ Æ Î)2+( Ò Æ Ç)2. Theaboverelations were obtained using thegeneralized (nonorthogonal) polar coordinates Ì, Ï de®ned byÑ= ÎÌcos Ï, Ò= ÇÌsin Ï (0£ Ì£1,0£ Ï£2 Ó) andthevariational method. For Ô= Ð1-( ÇÆ Î)2£0.9,theaboveformulas provide anaccurac yof1%for ¾1and2%for ¾(c) 2 and ¾(s) 2.For Ô£0.5,theerrors incalculating ¾1and ¾(c) 2donotexceed 0.01%, andthemaximum error indetermining ¾(s) 2is0.12%. Inthelimit case Ô=0thatcorresponds toacircular domain, the aboveformulas areexact.ÄÅ Refer ence:L.D.Akulenk oandS.V.Nestero v(2000). Page509 510 ELLIPTIC EQUATIONS WITH TWOSPACE VARIABLES TABLE 24 Transformations reducing equation 7.4.1.3 to the Helmholtz equation Õ2 ÖÕ×2+ Õ2 ÖÕ Ø2= ÇË No Exponent ¿ Transformation Factor Ç 1¿= 1 Ù=1 2( Ñ2- Ò2), Ú= Ñ Ò Ç= Î 2¿= 2 Ù=1 3 Ñ3- Ñ Ò2, Ú= Ñ2Ò-1 3 Ò3 Ç= Î 3¿= -1 Ù=1 2ln( Ñ2+ Ò2), Ú=arctan ÒÑ Ç= Î 4¿= -2 Ù= - ÑÑ2+ Ò2, Ú= ÒÑ2+ Ò2 Ç= Î 5 ¿= -1 2 Ñ=1 2(Ù2- Ú2), Ò=Ù Ú Ç= 2 Î 6 ¿= Û3, Û4, ÜÜÜÙ=( Ñ+ ÝÞÒ) ß+1+( Ñ- ÝÞÒ) ß+1 2( ¿+1), Ú=( Ñ+ ÝÞÒ) ß+1-( Ñ- ÝÞÒ) ß+1 2( ¿+1) Ý Ç= Î 7 ¿is any ( ¿¹ -1 ) Ù= à ß+1cos[( ¿+ 1) Ï]¿+ 1, Ú= à ß+1sin[( ¿+ 1) Ï]¿+ 1Ñ=àcos Ï, Ò=àsin Ï Ç= Î 7.4. Other Equations 7.4.1. Stationary Schr Èodinger Equation á2 â= ã( ä, å)â 1. æ2 çæ è2+ æ2 çæ é2= ê(è2+é2) ç. The transformation ë =1 2( Ñ2- Ò2), ì= Ñ Ò leads to the Helmholtz equation í 2Ëí ë 2+ í 2Ëíì2- ÎË= 0, which is discussed in Subsection 7.3.2. 2. æ2 çæ è2+ æ2 çæ é2= ê(è2+é2)2ç. The transformationë =1 3 Ñ3- Ñ Ò2, ì= Ñ2Ò-1 3 Ò3 leads to the Helmholtz equation í 2Ëí ë 2+ í 2Ëíì2- ÎË= 0, which is discussed in Subsection 7.3.2. 3. æ2çæ è2+ æ2çæ é2= ê(è2+é2) î ç. Thisisaspecia lcaseofequatio n7.4.1. 7for ï( ð)= Îð ß.Table24present stransformation sthat reduce this equation to the Helmholtz equation that is discussed in Subsection 7.3.2; the sixth row involves the imaginary unit, Ý2= -1 . Page 510 7.4. OTHER EQUATIONS 511 4. æ2 çæ è2+ æ2 çæ é2= ê ñ ò ó ç. Thetransformationð( Ñ, Ò)=exp ô1 2 õ Ñ öcos ô1 2 õ Ò ö, ÷( Ñ, Ò)=exp ô1 2 õ Ñ ösin ô1 2 õ Ò ö leads totheHelmholtz equationí 2Ëíð2+ í 2Ëí÷2=4 Îõ-2Ë, which isdiscussed inSubsection 7.3.2. 5. æ2 çæ è2+ æ2 çæ é2= ø ñ ù ó+ úüûç. ThetransformationÙ= ÎÑ+ ýþÒ, Ú= ýþÑ- ÎÒ leads toanequation oftheform 7.4.1.4:í 2ËíÙ2+ í 2ËíÚ2= ÿÎ2+ ý2 ×Ë. 6. æ2 çæ è2+ æ2 çæ é2= ( êè+ é) ç. This isaspecial case ofequation 7.4.1.9 for ( ð)=0.Particular solutions:Ë( Ñ, Ò)= 1cos[ÿ( ýþÑ- ÎÒ)]+ 2sin[ÿ( ýþÑ- ÎÒ)] Ï( ÎÑ+ ýþÒ), where 1, 2,andÿarearbitrary constants, andthefunction Ï= Ï(Ù)isdetermined bytheordinary differential equationÏ ××- 1Î2+ ý2 ï(Ù)+ÿ2 Ï=0. 7. æ2 çæ è2+ æ2 çæ é2= (è2+é2) ç. 1 .This equation admits separation ofvariables inthepolarcoordinatesà, Ï( Ñ=àcos Ï, Ò=àsin Ï). Particular solution:Ë( Ñ, Ò)= 1cos(ÿ Ï)+ 2sin(ÿ Ï) (à), where 1, 2,andÿarearbitrary constants, andthefunction = (à)isdetermined bytheordinary differential equationà(à ) -ÿ2+à2ï(à2) =0. 2 .Thetransformationë =1 2( Ñ2- Ò2), ì= Ñ Ò leads toasimilar equationí 2Ëí ë 2+ í 2Ëíì2= ( ë 2+ ì2) Ë, ( ð)= ï(2 ð) 2 ð. Inthespecial case ï( ð)=2 Î,wehave ( ð)= Î ð.For ï( ð)= ýþð3,weobtain anequation ofthe form 7.4.1.1 with ( ð)=4 ýþð. Page511 512 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 8. æ2 çæ è2+ æ2 çæ é2=[ (è)+ (é)] ç. Aparticular separable solution: ( Ñ, Ò)= Ï( Ñ) ( Ò), where thefunctions Ï( Ñ)and ( Ò)aredetermined bythesecond-order ordinary differential equationsÏ -[ ï( Ñ)- ] Ï=0, -[ ( Ò)+ ] =0, where isanarbitrary constant. 9. æ2 çæ è2+ æ2 çæ é2=[ ( êè+ é)+ ( è± êé)] ç. ThetransformationÙ= ÎÑ+ ýþÒ, Ú= ýþÑ- ÎÒ leads toanequation oftheform 7.4.1.8:í 2 íÙ2+ í 2 íÚ2= ï(Ù)Î2+ ý2+ ( Ú)Î2+ ý2 . 10. æ2 çæ è2+ æ2 çæ é2=(è2+é2)[ (è2±é2)+ (è é)] ç. Thetransformationë =1 2( Ñ2- Ò2), ì= Ñ Ò leads toanequation oftheform 7.4.1.8:í 2 í ë 2+ í 2 íì2=[ ï(2 ë )+ ( ì)] . 7.4.2. Convective Heat and Mass Transf erEquations 1. æ2 çæ è2+ æ2 çæ é2= æ çæ è. This isaconvectiveheat andmass transfer equation. Itdescribes astationary temperature (concen- tration) ®eld inacontinuous medium moving with aconstant velocity along the Ñ-axis. Inparticular , itmodels convective-molecular heat transfer from aheated ¯atplate ina¯owofathermal-transfer ideal ¯uid moving along theplate. This occurs, forexample, ifaliquid-metal coolant ¯owspast a ¯atplate orifaplate isinaseepage ¯owthrough agranular medium. Inthesequel, itisassumed thattheequation iswritten indimensionless variables Ñ, Òrelated to thecharacteristic length (fora¯atplate oflength 2 ,thecharacteristic length istakentobe ). 1 .The substitution ( Ñ, Ò)=exp ô1 2 Ñ ö ( Ñ, Ò)brings theoriginal equation totheHelmholtz equationí 2íÑ2+ í 2íÒ2=1 4 2. Particular solutions ofthisequation inCartesian andpolar coordinates canbefound inSubsections 7.3.2 and7.3.3. Page512 7.4. OTHER EQUATIONS 513 2 .Intheelliptic coordinatesÑ=cosh ìcos Ú, Ò=sinh ìsin Ú awide class ofparticular solutions (vanishing as ì )canbeindicated; thisclass ofsolutions of theoriginal equation isrepresented inseries form as =exp ô1 2 Ñ ö !=0 " !ce !( #,- $)Fek !( ì,- $), $=-1 16 2, where the" !arearbitrary constants, thece !( #,- $)aretheMathieu functions, andtheFek !( ì,- $) arethemodi®ed Mathieu functions [e.g., seeMcLachlan (1947) andBateman andErdÂelyi(1955)]. 3 .Consider the®rstboundary value problem intheupper half-plane (- < %< ,0£ &< ). Weassume thatthesurfaceofaplate of®nite length ismaintained ataconstant temperature 0and themedium hasatemperature =const farawayfrom theplate: = 0 for &=0,| %|<1,í =0 for &=0,| %|>1, for %2+ &2 . Thesolution ofthisproblem intheelliptic coordinates ì, #(seeItem 2 )hastheform ( #, ì)= +( 0- )exp ô1 2cos #cosh ì ö !=0 ' !ce !( #,- $)Fek !( (,- $) Fer !(0,- $), where'2 )=2ce2 )(0,- $) ce2 )(0, $)"(2 )) 0,'2 )+1=-1 2ce2 )+1(0,- $) ce2 )+1(0, $) *(2 )+1) 1, $=-1 16 2. Here, the"(2 )) 0and*(2 )+1) 1 arethecoef®cients intheseries expansions oftheMathieu functions; these canbefound inMcLachlan (1947). 4 .Consider thesecond boundary valueproblem intheupper half-plane (- < %< ,0£ &< ). Weassume thatathermal ¯uxisprescribed onthesurfaceofaplate of®nite length andthemedium hasaconstant temperature farawayfrom theplate:+ = ,( %)for &=0,| %|<1,+ =0 for &=0,| %|>1, as %2+ &2 . Thesolution ofthisproblem intheCartesian coordinates hastheform ( %, &)= -1- .1 -1 ,( /)exp 01 2 1( %- /) 230 41 2 1 5( 6- /)2+ 72 8 9/, where 30( :)isthemodi®ed Bessel function ofthesecond kind.;=< Refer ences :P.V.Cherpak ov(1975), A.A.Borzykh andG.P.Cherepano v(1978). 2. >2 ?> @2+ >2 ?> A2= B > ?> @+ C > ?> A+ D ?. This equation describes astationary temperature ®eld inamedium moving with aconstant velocity , provided there isvolume release heat (orabsorption) proportional totemperature. Thesubstitution E ( 6, 7)=exp 01 2(1 6+ F 7) 2 G( 6, 7) brings theoriginal equation totheHelmholtz equationH2GH62+ H2GH72= IKJ+1 412+1 4 F2 LG, which isdiscussed inSubsections 7.3.1 through 7.3.3. Page513 514 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 3. >2 ?> @2+ >2 ?> A2=Pe(1±A2) > ?> @. TheGraetz±Nusselt equation .Itgoverns steady-state heat exchange inalaminar ¯uid ¯owwith a parabolic velocity pro®le inaplane channel. Theequation iswritten interms ofthedimensionless Cartesian coordinates 6, 7related tothechannel half-width M;Pe= G M N OisthePeclet number andGisthe¯uid velocity atthechannel axis( 7=0).Thewallsofthechannel correspond to 7= P1. 1 Q.Particular solutions: E ( 7)= R+ S 7,E ( 6, 7)=12 R 6+ RPe(6 72- 74)+ S,E ( 6, 7)= T UWV =1 R V exp X- Y2 V Pe 6 Z [ V ( 7). Here, R, S, R V ,andY V arearbitrary constants, andthefunctions [ V arede®ned by[ V ( 7)=expI-1 2Y V72 L \I1 V ,1 2;Y V72 L,1 V =1 4-1 4Y V -1 4Y3 V Pe-2, (1) where \(1, F; /)=1+ ] ^_=1 `(`+1) ababa(`+ _-1)c( c+1) ababa( c+ _-1) dfe _!isthedegenerate hyper geometric function. 2 Q.Letthewallsofthechannel bemaintained ataconstant temperature, E =0for 6<0and E = E 0 for 6>0.Due tothesymmetry oftheproblem about the 6-axis, itsuf®ces toconsider only halfof thedomain, 0£ 7£1.Theboundary conditions arewritten as7=0, H EH7=0; 7=1, E = g0 for 6<0, E 0for 6>0;6 h- i, Eh0; 6 h i, Eh E 0. Thesolution oftheoriginal equation under these boundary conditions issought intheformE ( 6, 7)= E 0 ] UWV =1 S V exp X j2 V Pe 6 Z k V ( 7) for 6<0,E ( 6, 7)= E 0 l1- ] UWV =1 R V exp X- Y2 V Pe 6 Z[ V ( 7) mfor 6>0. Theseries coef®cients must satisfy thematching conditions attheboundary:E ( 6, 7) nnpo q0,o<0- E ( 6, 7) nnpo q0,o>0=0,Ho E ( 6, 7) nnpo q0,o<0- Ho E ( 6, 7) nnpo q0,o>0=0. For 6>0,thefunction [ V ( 7)isde®ned byrelation (1),where theeigen valuesY V areroots of thetranscendental equation\Isr V ,1 2;Y VL=0,wherer V =1 4-1 4Y V -1 4Y3 V Pe-2. ForPe h i,itisconvenient tousethefollowing approximate relation toidentify theY V :Y V =4( t-1)+1.68 ( t=1,2,3, uvuvu). (2) The error ofthisformula does notexceed 0.2%. The corresponding numerical values ofthe coef®cients R V arerather well approximated bytherelationsR1=1.2, R V =2.27(-1) V -1Y-7 w6 V for t=2,3,4, uvuvu, whose maximum error islessthan 0.1%, provided thattheY V arecalculated by(2). ForPe h0,thefollowing asymptotic relations hold:Y V = x y z{t-1 2 |Pe, } V =4(-1) V -1y2(2 t-1)2, [ V ( ~)=cos by zst-1 2 | ~ ( t=1,2,3, uvuvu). Noresults for <0aregivenhere, because theyareofsecondary importance inapplications. Page514 7.4. OTHER EQUATIONS 515 3 .Letaconstant thermal ¯uxbeprescribed atthewallsfor >0andlet,for <0,thewallsbe insulated from heat andthetemperature vanishes as h- i.Then theboundary conditions have theform~=0, H H~=0; ~=1, H H~= g0for <0,for >0; h- i, h0. Inthedomain ofthermal stabilization, theasymptotic behavior ofthesolution (as h i)isas follows:( , ~)= 3 2 Pe+3 4 ~2-1 8 ~4+9 4Pe2-39 280 .= Refer ences :L.Graetz (1883), W.Nusselt (1910), C.A.Deavours (1974), A.D.Polyanin, A.M.Kutepo v,A.V.Vyazmin, andD.A.Kazenin (2001). 4. 2 2+1 + 2 2=Pe(1±2) . This equation governssteady-state heatexchange inalaminar ¯uid ¯owwith parabolic (Poiseuille' s) velocity pro®le inacircular tube. Theequation iswritten interms ofthedimensionless cylindrical coordinates , ~related tothetube radius ;Pe= isthePeclet number and isthe¯uid velocity atthetube axis(at =0).Thewallsofthetube correspond to =1. 1 .Particular solutions:( )= }+ ln ,( , :)=16 } :+ }Pe(4 2- 4)+ ,( , :)= W =1 } exp - 2 Pe ( ). Here, }, , } ,and arearbitrary constants, andthefunctions arede®ned by ( )=exp z-1 2 2| \zs ,1; 2|, =1 2-1 4 -1 43 Pe-2, (1) where \( , ; )isthedegenerate hyper geometric function (seeequation 7.4.2.3, Item 1 ). 2 .Letthetube wallbemaintained ataconstant temperature such that =0for<0and = 0 for>0.Theboundary conditions arewritten as=0, H H=0; =1, = 0 for<0, 0for>0; - , 0; , 0. Thesolution oftheoriginal equation under these boundary conditions issought intheform( ,)= 0 ] =1 exp 2 Pe ( ) for<0,( ,)= 0 ¡1- ] =1 } exp - 2 Pe ( ) ¢for>0. Theseries coef®cients must satisfy thematching conditions attheboundary ,( ,) ££¤¦¥ 0,¤<0- ( ,) ££¤¦¥ 0,¤>0=0,§¤ ( ,) ££p¤¦¥ 0,¤<0- §¤ ( ,) ££¤¦¥ 0,¤>0=0. For>0,thefunctions ( )arede®ned byrelations (1),where theeigen values areroots of thetranscendental equation\zs ,1; |=0,where =1 2-1 4 -1 43 Pe-2. Page515 516 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES ForPe ,itisconvenient tousethefollowing approximate relation toidentify the : =4( ¨-1)+2.7 ( ¨=1,2,3, ©v©v©). (2) The error ofthisformula does notexceed 0.3%. The corresponding numerical values ofthe coef®cients } arerather well approximated bytherelations} =2.85(-1) -1-2 ª3 for ¨=1,2,3, ©v©v©, whose maximum error is0.5%, Noresults for<0aregivenhere, since theyareofsecondary importance inapplications. 3 .Letaconstant thermal ¯uxbeprescribed atthewallfor>0andlet,for<0,thetube surface beinsulated from heat andthetemperature vanishes as - .Then theboundary conditions havetheform=0, § §=0; =1, § §= 0for<0,for>0; - , 0. Inthedomain ofthermal stabilization, theasymptotic behavior ofthesolution (as )isas follows:( ,)= 4Pe+ 2-1 4 4+8 Pe2-7 24 .= Refer ences :C.A.Deavours (1974), A.D.Polyanin, A.M.Kutepo v,A.V.Vyazmin, andD.A.Kazenin (2001). 5. 2 «2+ 2 ¬2= (¬) «. This equation describes steady-state heat exchange inalaminar ¯uid ¯owwith anarbitrary velocity pro®le=( ~)inaplane channel. 1 .Particular solutions:( , ~)= } + } ® ¯¯0( °- )( ) ± + °+ ², (1)³( ´, °)= + =1 } exp(- ´) µ ( °). (2) Here, }, , ², °0, } ,and arearbitrary constants, andthefunctions µ = µ ( °)aredetermined bythesecond-order linear ordinary differential equation±2µ ± °2+ ¶· ( °)+ 2 ¸µ =0. 2 ¹.Solution (1)describes thetemperature distrib ution farawayfrom theinlet section ofthetube, in thedomain ofthermal stabilization, provided thataconstant thermal ¯uxisprescribed atthechannel walls. 6. º » ¼2 ½¼ «2+ ¼2 ½¼ ¬2 ¾= ¿1(«,¬) ¼ ½¼ «+ ¿2(«,¬) ¼ ½¼ ¬. This isanequation ofsteady-state convectiveheat andmass transfer intheCartesian coordinate system. Here, À1= À1( ´, °)and À2= À2( ´, °)arethecomponents ofthe¯uid velocity thatare assumed tobeknownfrom thesolution ofthehydrodynamic problem. Page516 7.4. OTHER EQUATIONS 517 1 ¹.Inplane problems ofconvectiveheat exchange inliquid metals modeled byanideal ¯uid, as well asindescribing seepage (®ltration) streams emplo ying themodel ofpotential ¯ows,the¯uid velocity components À1( ´, °)and À2( ´, °)canbeexpressed interms ofthepotential Á= Á( ´, °)and stream function Â= Â( ´, °)asfollows:À1= §Á§´=- §Â§°, À2= §Á§°= §Â§´. (1) Thefunction Áisdetermined bysolving theLaplace equation Ã Á=0.Inspeci®c problems, the potential Áandstream function Âmay beidenti®ed byinvoking thecomple xvariable theory [e.g., seeLavrent'e vandShabat (1973) andSedo v(1980)]. Bypassing intheconvectiveheat exchange equation from ´, °tothenewvariables Á, Â (Boussinesq transformation) andtaking intoaccount (1),wearriveasimpler equation with constant coef®cients oftheform 7.4.2.1:§2 ³§Á2+ §2 ³§Â2=1 Ä §³§Á. (2) TheBoussinesq transformation brings anyplane contour inapotential ¯owtoacutinthe Á-axis, simultaneously with thereduction oftheoriginal equation totheform (2). Consequently ,theheat transfer problem ofapotential ¯owabout thiscontour isreduced totheheat exchange problem ofa longitudinal ¯owofanideal ¯uid pasta¯atplate (seeequation 7.4.2.1, Items 3 ¹and4 ¹). 2 ¹.Asymptotic analyses ofplane problems onheat/mass exchange ofbodies ofvarious shape with laminar translational andshear ¯owsofaviscous (and ideal) incompressible ¯uid forlargeand small Peclet numbers were carried outinthereferences cited below.Inthethermal boundary layer approximation, thesolution oftheheatexchange problemfora¯atplate inalongitudinaltranslationa l ¯owofaviscous incompressible ¯uid atlargeReynolds numbers ispresented in1.9.1.4, Item 3 ¹.Å=Æ Refer ences :V.G.Levich (1962), P.V.Cherpak ov(1975), A.A.Borzykh andG.P.Cherepano v(1978), Yu.P.Gupalo, A.D.Polyanin, andYu.S.Ryazantse v(1985), A.D.Polyanin, A.M.Kutepo v,A.V.Vyazmin, andD.A.Kazenin (2001). 7.1Ç2 ¼¼ Ç » Ç2¼ ½¼ Ç¾+1Ç2sin È ¼¼ È »sin È ¼ ½¼ È ¾=cos È ¼ ½¼ Ç±sin ÈÇ ¼ ½¼ È. This isaspecial case ofequation 7.4.1.8 with Ä =1, À É=cos Ê,and À Ë=-sin Ê.This equation is obtained from theequation Ì Í Í ³+ Ì¯¯ ³+ Ì ÎfÎ ³= Ì Í ³bythepassage tothespherical coordinate system intheaxisymmetric case. Thegeneral solution satisfying thedecay condition ( ³ Ï0as Ð Ï Ñ)isexpressed as³( Ð, Ê)= Ò ÓÐ Ô1 Õ2 exp Ò Ðcos Ê 2 Ô Ö ×WØ =0 Ù Ø ÚØ +1 2 Ò Ð 2 Ô Û Ø (cos Ê), where theÙ Ø arearbitrary constants. The Legendre polynomialsÛ Ø ( Ü)andthemodi®ed Bessel functions ÚØ +1 2( Ý)aregivenbyÛ Ø ( Ü)=1Þ!2 Ø ß ØßÜ Ø ( Ü2-1) Ø , ÚØ +1 2 Ò Ð 2 Ô= Ò ÓÐ Ô1 Õ2 exp Ò- Ð 2 Ô Ø×à=0( Þ+ á)! ( Þ- á)! á! Ð à.Å=Æ Refer ence:P.L.Rimmer (1968). 8. â ã1Ç2 ¼¼ Ç » Ç2¼ ½¼ Ç¾+1Ç2sin È ¼¼ È »sin È ¼ ½¼ È ¾ ä= ¿ å ¼ ½¼ Ç+ ¿ æÇ ¼ ½¼ È. This equation isoften encountered inaxisymmetric problems ofconvectiveheat andmass exchange ofsolid particles, drops, andbubbles with a¯owofaviscous incompressible ¯uid. The ¯uid velocity components À É= À É( Ð, Ê)and À Ë= À Ë( Ð, Ê)canbeexpressed interms ofthestream functionÂ= Â( Ð, Ê)asÀ É=1Ð2sin Ê Ì ÂÌ Ê, À Ë=-1Ðsin Ê Ì ÂÌ Ð. (1) Page517 518 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES Asymptotic analyses forawide class ofaxisymmetric problems onheat/mass exchange of solid particles, drops, andbubbles ofvarious shape with alaminar translational orstraining ¯ow ofaviscous incompressible ¯uid atlargeandsmall Peclet numbers Pe= ç è é Ä areperformed inthebooks cited below.The Peclet number iswritten interms ofthecharacteristic velocity ç (e.g., theunperturbed ¯uid velocity farawayfrom theparticle inthecase oftranslation ¯ow), thecharacteristic size oftheparticle è(e.g., theradius foraspherical particle), andthethermal conducti vityordiffusion coef®cient Ä . Thefollowing boundary conditions areusually speci®ed:ê= ê 0at Ð= è, ê Ï êÖas Ð Ï Ñ, (2) where èistheparticle radius, ê 0thetemperature attheparticle surface,and êÖthetemperature farawayfrom theparticle ( ê 0and êÖareconstant). Convectivemass transfer problems arecharacterized bylargePeclet numbers. Tosolvesuch problems, thediffusion boundary layer approximation isoften used; inthiscase, theleft-hand sideof theequation takesintoaccount only thediffusion mass transfer inthenormal direction totheparticle surface(thetangential mass transfer isneglected). Theconvectiveterms ontheright-hand sideare partially preserv edÐthe¯uid velocity components areapproximated bytheir leading terms ofthe asymptotic expansion near thephase surface. Presented belowaresome important results obtained bysolving theoriginal equation under theboundary conditions (2)inthediffusion boundary layer approximation. Example 1.Forthetranslational Stokes¯owofaviscous incompressible ¯uid about aspherical bubble, thestream function isexpressed as ë ( ì, í)=1 2 î ì( ì- ï)sin2í. Here,îistheunperturbed ¯uid velocity intheincident ¯ow, ïthebubble radius (thevalue í= ðcorresponds tothefront critical point atthebubble surface). Inthiscase, thesolution oftheconvectiveheat/mass transfer equation with theboundary conditions (2)forPe=î ï ñfò ó1inthediffusion boundary layer approximation isgivenbyô( ì, í)= ô 0+( ô õ- ô 0)erf ö, ö= ÷3 8Pe ø ìï-1 ù1-cos íú 2-cos í, where erf öistheerror function. Example 2.Forthetranslational Stokes¯owofaviscous incompressible ¯uid about asolid spherical particle, the stream function isexpressed asë ( ì, í)=1 4 î( ì- ï)2ø2+ ïì ùsin2í. Here, thenotation isthesame asinthecase ofabubble above. Forasolid particle, thesolution oftheconvectiveheat/mass transfer equation with theboundary conditions (2)for Pe=î ï ñfò ó1inthediffusion boundary layer approximation isgivenbyô( ì, í)= ô 0+( ô õ- ô 0) ûpü ý1 3 þ ÿ-1ý1 3, öþ, ö=Pe( ì- ï)3sin3í 3 ï3ýð- í+1 2sin2 íþ, where ü( )isthegamma function and ( , ö)= 0 - -1 istheincomplete gamma function. Refer ences :V.G.Levich (1962), Yu.P.Gupalo, A.D.Polyanin, andYu.S.Ryazantse v(1985), A.D.Polyanin, A.M.Kutepo v,A.V.Vyazmin, andD.A.Kazenin (2001). 7.4.3. Equations ofHeat and Mass Transf erinAnisotr opic Media 1. â + =0. This isatwo-dimensional equation oftheheat and mass transfer theory inainhomogeneous anisotropic medium. Here, 1( )= and 2( )= aretheprincipal thermal diffusivities. Page518 7.4. OTHER EQUATIONS 519 1 .Particular solutions ( , !, "arearbitrary constants):#( , )= 1-+ ! 1-+ ",#( , )= $ 2-(2- %)- 2-(2- &) '+ !,#( , )= 1-1-+ !. 2 .For %¹2and &¹2,there areparticular solutions oftheform#= #( (), (= )(2- &)22-+ (2- %)22- *1 +2. Thefunction #= #( ()isdetermined bytheordinary differential equation# ,-,./.+ ( # ,.=0, =4- % & (2- %)(2- &). (1) Thegeneral solution ofequation (1)isgivenby#( ()= 0 "1 (1- 1+ "2for ¹1,"1ln (+ "2for =1, where "1and "2arearbitrary constants. 3 .There aremultiplicati velyseparable particular solutions intheform#( , )= 2( ) 3( ), (2) where 2( )and 3( )aredetermined bythefollowing second-order linear ordinary differential equations ( 1isanarbitrary constant): ( 2 ,4) ,4=- 1 2, (3) ( 3 ,5) ,5= 1 3. (4) Thesolution ofequation (3)isgivenby2( )= 67 879 1-2 :"1 ; < =?> 2-2 @+ "2 A < =?> 2-2 @ Bfor 1>0,1-2 :"1 C?< => 2-2 @+ "2 D < =?> 2-2 @ Bfor 1<0,E=|1- %| 2- %,>=2 2- % F| 1|, where "1and "2arearbitrary constants,; <( G)andA <( G)aretheBessel functions, andC <( G)andD <( G)arethemodi®ed Bessel functions. Thesolution ofequation (4)isexpressed as3( )= 67 879 1-2 :"1 ; H =I 2-2 @+ "2 A H =?I 2-2 @ Bfor 1<0,1-2 :"1 CH =?I 2-2 @+ "2 D H =?I 2-2 @ Bfor 1>0,J=|1- &| 2- &,I=2 2- & F| 1|, where "1and "2arearbitrary constants. Thesum ofsolutions oftheform (2)corresponding todifferent values oftheparameter 1is also asolution oftheoriginal equation; thesolutions ofsome boundary value problems may be obtained byseparation ofvariables. 4 .Seeequation 7.4.3.3, Item 4 ,for K=0. Page519 520 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 2. LL M N O M P L QL M + LL N L QL = R. This isatwo-dimensional equation oftheheat andmass transfer theory with constant volume release ofheat inaninhomogeneous anisotropic medium. Here, 1( )= and 2( )= are theprincipal thermal diffusivities. 1 .For %¹2and &¹2,there areparticular solutions oftheform#= #( (), (=) (2- &)22-+ (2- %)22- *1 +2. (1) Thefunction #= #( ()isdetermined bytheordinary differential equation# ,-,./.+ ( # ,.= !, (2) where =4- % & (2- %)(2- &), !=4 K (2- %)2(2- &)2. (3) Thegeneral solution ofequation (2)isgivenby#( ()= 67 879 "1 (1- 1+ "2+ S2( 1+1) (2for ¹ T1,"1ln (+ "2+1 4 ! (2for =1,"1 (2+ "2+1 2 ! (2ln (for =-1, where "1and "2arearbitrary constants. 2 .Thesubstitution#( , )= U( , )+ K(2- %) 2- leads toahomogeneous equation oftheform 7.4.3.1:VVN VUV W+ VV XN Y X Z VUV XW=0. 3. LL M N O M P L QL M W+ LL [ N [ L QL [ W= RQ. This isatwo-dimensional equation oftheheat andmass transfer theory with alinear source inan inhomogeneous anisotropic medium. 1 \.For ]¹2and &¹2,there areparticular solutions oftheform^= ^( _), _= `Y(2- &)2 a2- b+ (2- ])2 X2- Z c1 d2. Thefunction ^= ^( _)isdetermined bytheordinary differential equation^ e-ef/f+ g_ ^ ef= h ^, (1) whereg=4- ] & (2- ])(2- &), h=4 KY(2- ])2(2- &)2. Thegeneral solution ofequation (1)isgivenby^( _)= _1- i 2 jlk1 m n o _ p| h| q+ k2 r n o _ p| h| q sfor h<0,^( _)= _1- i 2 jlk1 t?n o _ u h q+ k2 v n o _ u h q s for h>0, where w=1 2|1-g|; k1and k2arearbitrary constants;m n( x)andr n( x)aretheBessel functions; andt?n( x)andv n( x)arethemodi®ed Bessel functions. Page520 7.4. OTHER EQUATIONS 521 2 .There aremultiplicati velyseparable particular solutions oftheform( , )= ( ) ( ), where ( )and ( )aredetermined bythefollowing second-order linear ordinary differential equations ( 1isanarbitrary constant): ( ) = 1 , ( ) =( - 1) . (2) Thesolutions ofequations (2)areexpressed interms oftheBessel functions (ormodi®ed Bessel functions); seeequation 7.4.3.1, Item 3 . 3 .There areadditi velyseparable particular solutions oftheform( , )= ( )+ ( ), where ( )and ( )aredetermined bythefollowing second-order linear ordinary differential equations ( 2isanarbitrary constant): ( ) - = 2, ( ) - =- 2. (3) Thesolutions ofequations (3)areexpressed interms oftheBessel functions (ormodi®ed Bessel functions). 4 .Thetransformation (speci®ed byA.I.Zhuro v,2001)2-2= cos , 2- 2= sin , where 2= (2- )2and 2= (2- )2,leads totheequation2 2+4- (2- )(2- )1 +12 2 2-22( - - )cos2 +( - ) (2- )(2- )sin2 =4 , which admits separable solutions oftheform ( , )= 1( ) 2( ). 4. ( + ) ! "+ # $ ( #+s) % ! # "= &. Thetransformation '= + (, )= + *leads toanequation oftheform 7.4.3.2:' + ' ' ,+ ) + ) ) ,= . 5. ( + ) ! "+ # $ ( #+s) % ! # "= & !. Thetransformation '= + (, )= + *leads toanequation oftheform 7.4.3.3:' + ' ' ,+ ) + ) ) ,= . 6. + - . / ! ,+ #+ $ - 0 1 ! #,=0. This isatwo-dimensional equation oftheheat andmass transfer theory inaninhomogeneous anisotropic medium. Here, 1( )= 243 and 2( )= 52 6 aretheprincipal thermal diffusivities. Page521 522 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 1 .Particular solutions ( , , 7arearbitrary constants):( , )= 2-3 + 2-6 + 7,( , )= 82( 8 +1) 2-3 - 92( 9 +1) 2-6 + ,( , )= 2-3 -6 + . 2 .There aremultiplicati velyseparable particular solutions oftheform( , )= ( ) ( ), (1) where ( )and ( )aredetermined bythefollowing second-order linear ordinary differential equations ( 1isanarbitrary constant): ( 2 3 ) =- 1 , (2) ( 52 6 ) = 1 . (3) Thesolution ofequation (2)isgivenby( )= : 2-3 ;2 <71 =1 > ( 2-3 ;2 ?+ 72 @1 > ( 2-3 ;2 ?5Afor 1>0,2-3 ;2 <71 B1 > ( 2-3 ;2 ?+ 72 C1 > ( 2-3 ;2 ?5Afor 1<0, where (=-(2 D 8) E| 1| D ; 71and 72arearbitrary constants;=1( F)and@1( F)aretheBessel functions; andB1( F)andC1( F)arethemodi®ed Bessel functions. Thesolution ofequation (3)isgivenby( )= : 2-6 ;2 <71 =1 > *G2-6 ;2 ?+ 72 @1 > *G2-6 ;2 ?5Afor 1<0,2-6 ;2 <71 B1 > *G2-6 ;2 ?+ 72 C1 > *G2-6 ;2 ?5Afor 1>0, where *=-(2 D 9) E| 1| D ; 71and 72arearbitrary constants. Thesum ofsolutions oftheform (1)corresponding todifferent values oftheparameter 1is alsoasolution oftheoriginal equation. 3 .Seeequation 7.4.3.8, Item 3 ,for =0. 7. + - . / ! ,+ #+ $ - 0 1 ! #,= &. This isatwo-dimensional equation oftheheatandmass transfer theory with constant volume release ofheat inaninhomogeneous anisotropic medium. Here, 1( )= 2 3 and 2( )= 52 6 arethe principal thermal diffusivities. Thesubstitution( , )= H( , )- 82( 8 +1) 2-3 leads toahomogeneous equation oftheform 7.4.3.6: + 2 3 H ,+ + 52 6 H ,=0. 8. + -. / ! ,+ #+ $ -0 1 ! #,= & !. This isatwo-dimensional equation oftheheat andmass transfer theory with alinear source inan inhomogeneous anisotropic medium. Page522 7.4. OTHER EQUATIONS 523 1 .For 8 9¹0,there areparticular solutions oftheform= ( I), I=> 922-3 + 822-6 )1 ;2. Thefunction = ( I)isdetermined bytheordinary differential equation J KLK-1I K= , =4 8292. Forthesolution ofthisequation, see7.4.3.3 (Item 1 for =-1). 2 .Theoriginal equation admits multiplicati vely(and additi vely)separable solutions. Seeequation 7.4.3.12 with ( )= 2 3 and ( )= 52 6 . 3 .Thetransformation (speci®ed byA.I.Zhuro v,2001)2-3 ;2= cos , 2-6 ;2= sin , where 2= 82and 2= 92,leads totheequation2 2-1 +12 2 2-22cot2 =4 , which admits separable solutions oftheform ( , )= 1( ) 2( ). 9. + ! ,+ #+ $ -. 1 ! ,= & !. 1 .For ¹2and 8¹0,there areparticular solutions oftheform= ( ), 2= 2-(2- )2+ 2-3 82. Thefunction = ( )isdetermined bytheordinary differential equation2 2+ 2- 1 =4 . Forthesolution ofthisequation, see7.4.3.3 (Item 1 ). 2 .Theoriginal equation admits multiplicati vely(and additi vely)separable solutions. Seeequation 7.4.3.12 with ( )= and ( )= 52M3 . 3 .Thetransformation (speci®ed byA.I.Zhuro v,2001)1-1 2= cos , 2-1 23 = sin , where 2= (2- )2and 2= 82,leads totheequation2 2+ 2- 1 +12 2 2-22(1- )cos2 +1 (2- )sin2 =4 , which admits separable solutions oftheform ( , )= 1( ) 2( ). 10. ON ( ) ! "+ 2 ! #2=0. 1 .Particular solutions:= 71 2+ 72 -2 P 71 + 73( ) Q + 74,= 71 3+ 72 -6 P 71 + 73( ) Q + 74,=[ 71 R( )+ 72] + 73 R( )+ 74,R( )= PQ ( ),=[ 71 R( )+ 72] 2+ 73 R( )+ 74-2 P S1( ) P[ 71 R( )+ 72]Q TQ U, where V1, V2, V3, V4,and V5arearbitrary constants. Page523 524 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 2 W.Separable particular solution: X =( V1 Y Z [+ V2 Y-Z [) \(U), where V1, V2,and ]arearbitrary constants, andthefunction \= \(U)isdetermined bytheordinary differential equation [ ^(U) \ _ `] _`+ ]2\=0. 3 W.Separable particular solution: X =[ V1sin( ] a)+ V2cos( ] a)] b(U), where V1, V2,and ]arearbitrary constants, andthefunction b= b(U)isdetermined bytheordinary differential equation [ ^(U) b _ `] _`- ]2b=0. 4 W.Particular solutions with evenpowers of a:X = c dfe =0 g e (U) a2 e , where thefunctionsg e =g e (U)arede®ned bytherecurrence relationsgc(U)= hc i(U)+ jc,i(U)= k lU^(U),g e -1(U)= h ei(U)+ j e -2 m(2 m-1) k1^(U) n kg e (U)lU TlU, where h e and j e arearbitrary constants ( m= o, ppp,1). 5 W.Particular solutions with oddpowers of a:X = c d e =0 ) e (U) a2 e +1, where thefunctions ) e = ) e (U)arede®ned bytherecurrence relations)c(U)= hc i(U)+ jc,i(U)= k lU^(U),) e -1(U)= h ei(U)+ j e -2 m(2 m+1) k1^(U) n k ) e (U)lU TlU, where h e and j e arearbitrary constants ( m= o, ppp,1). 11. qq r sOt(r) q uq r v+ qq w syx(w) q uq w v=0. This isatwo-dimensional sourceless equation oftheheat andmass transfer theory inaninho- mogeneous anisotropic medium. The functions ^= ^(U)and z= z( a)aretheprincipal thermal diffusivities. 1 W.Particular solutions:X (U, a)= h1 k lU^(U)+ j1 k l az( a)+ V1,X (U, a)= h2 kU lU^(U)- h2 k al az( a)+ j2,X (U, a)= h3 k lU^(U) k l az( a)+ j3, where the h e , j e ,and V1arearbitrary constants. Alinear combination ofthese solutions isalsoa solution oftheoriginal equation. Page524 7.4. OTHER EQUATIONS 525 2 W.There aremultiplicati velyseparable particular solutions oftheformX (U, a)= {(U) |( a), (1) where {(U)and |( a)aredetermined bythefollowing second-order linear ordinary differential equations ( hisanarbitrary constant): ( ^ { _`) _`= h {, ^= ^(U), ( z | _[) _[=- h |, z= z( a).(2) Thesum ofsolutions oftheform (1)corresponding todifferent values oftheparameter hin(2) isalso asolution oftheoriginal equation (thesolutions ofsome boundary value problems may be obtained byseparation ofvariables). 12. qq r sOt(r) q uq r v+ qq w syx(w) q uq w v= }u. This isatwo-dimensional equation oftheheat andmass transfer theory with alinear source inan inhomogeneous anisotropic medium. Thefunctions ^= ^(U)and z= z( a)aretheprincipal thermal diffusivities. 1 W.There aremultiplicati velyseparable particular solutions oftheformX (U, a)= {(U) |( a), (1) where {(U)and |( a)aredetermined bythefollowing second-order linear ordinary differential equations ( hisanarbitrary constant): ( ^ { _`) _`= h {, ^= ^(U), ( z | _[) _[=( ~- h) |, z= z( a).(2) Thesum ofsolutions oftheform (1)corresponding todifferent values oftheparameter hin(2) isalso asolution oftheoriginal equation; thesolutions ofsome boundary value problems may be obtained byseparation ofvariables. 2 W.There areadditi velyseparable particular solutions oftheformX (U, a)=i(U)+ ( a), wherei(U)and ( a)aredetermined bythefollowing second-order linear ordinary differential equations ( Visanarbitrary constant): ( ^i _`) _`- ~i= V, ^= ^(U), ( z _[) _[- ~ =- V, z= z( a). Inthespecial case ~=0,thesolutions ofthese equations canberepresented asi(U)= V kU lU^(U)+ h1 k lU^(U)+ j1,( a)=- V k al az( a)+ h2 k l az( a)+ j2, where h1, h2, j1,and j2arearbitrary constants. Page525 526 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 7.4.4. Other Equations Arising inApplications 1.w q2uq r2+ q2uq w2=0. Tricomi equation. Itisused todescribe near-sonic ¯owsofgas. 1 W.Particular solutions: X = hU a+ jU+ V a+ , X = h(3U2- a3)+ j(U3-U a3)+ V(6 aU2- a4), where h, j, V,and arearbitrary constants. 2 W.Particular solutions with evenpowers ofU:X = c d e =0 { e ( a)U2 e , where thefunctions { e = { e ( a)arede®ned bytherecurrence relations{c( a)= hc a+ jc, { e -1( a)= h ea+ j e -2 m(2 m-1) k [ 0( a- ) { e ( )l , where h e and j e arearbitrary constants ( m= o, ppp,1). 3 W.Particular solutions with oddpowers ofU:X = c de =0 | e ( a)U2 e +1, where thefunctions | e = | e ( a)arede®ned bytherecurrence relations|c( a)= hc a+ jc, | e -1( a)= h ea+ j e -2 m(2 m+1) k [ 0( a- ) | e ( )l , where h e and j e arearbitrary constants ( m= o, ppp,1). 4 W.Separable particular solutions: X (U, a)= yhsinh(3 ]U)+ jcosh( 3 ]U) a yV 1 3(2 ] a3 2)+ 1 3(2 ] a3 2) , X (U, a)= yhsin(3 ]U)+ jcos(3 ]U) a yV 1 3(2 ] a3 2)+ 1 3(2 ] a3 2) , where h, j, V, ,and ]arearbitrary constants, 1 3( F)and 1 3( F)aretheBessel functions, and1 3( F)and 1 3( F)arethemodi®ed Bessel functions. 5 W.For a>0,seealso equation 7.4.4.2 with o=1.For a<0,thechange ofvariable a=- leads toanequation oftheform 4.3.3.11 with o=1.y Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). 2.w q2uq r2+ q2uq w2=0. 1 W.Particular solutions: X = hU a+ jU+ V a+ ,X = hU2-2 h ( o+1)( o+2) ac+2,X = hU3-6 h ( o+1)( o+2) U ac+2,X = h aU2-2 h ( o+2)( o+3) ac+3, where h, j, V,and arearbitrary constants. Page526 7.4. OTHER EQUATIONS 527 2 W.Particular solutions with evenpowers ofU:X = dfe =0 { e ( a)U2 e , where thefunctions { e = { e ( a)arede®ned bytherecurrence relations{( a)= h a+ j, { e -1( a)= h ea+ j e -2 m(2 m-1) k [( a- ) c { e ( )l , where h e and j e arearbitrary constants ( m= , ppp,1), isanynumber . 3 W.Particular solutions with oddpowers ofU:X = de =0 | e ( a)U2 e +1, where thefunctions | e = | e ( a)arede®ned bytherecurrence relations|( a)= h a+ j, | e -1( a)= h ea+ j e -2 m(2 m+1) k [( a- ) c | e ( )l , where h e and j e arearbitrary constants ( m= , ppp,1), isanynumber . 4 W.Separable particular solutions:X (U, a)= hsinh( ] U)+ jcosh( ] U) a V 1 2 ( ] a )+ 1 2 ( ] a ) , =1 2( o+2), X (U, a)= hsin( ] U)+ jcos( ] U) a V 1 2 ( ] a )+ 1 2 ( ] a ) , where h, j, V, ,and ]arearbitrary constants, ( F)and ( F)aretheBessel functions, and ( F) and ( F)arethemodi®ed Bessel functions. 5 W.Fundamental solutions (for a>0):X 1( , a, 0, a0)= m1( 2 1)- ( ~, ~,2 ~;1- ), ~= o 2( o+2), = 2 22 1, 2( , a, 0, a0)= m2( 2 1)- (1- )1-2 (1- ~,1- ~,2-2 ~;1- ). Here, ( , , ; )isthehyper geometric function and2 1=( - 0)2+4 ( o+2)2 ac+2 2+ ac+2 2 0 ¡, m1=1 4 ¢ 4o+2 ¡2 £2( ~)£(2 ~),2 2=( - 0)2+4 ( o+2)2 ¤ ¥+2 2-¤ ¥+2 2 0 ¡, m2=1 4 ¢ 4o+2 ¡2 £2(1- ~)£(2-2 ~), where £( ~)isthegamma function; 0and¤0arearbitrary constants. Thefundamental solutions satisfy theconditions¦ § 1 ¨¨ § =0=0, 2 ¨¨ § =0=0 ( and 0areany,¤0>0). Thesolutions ofsome boundary value problems canbefound inthe®rstbook cited below.y Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), A.D.Polyanin (2001a). Page527 528 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES 3. q2uq ©2+ ª© q uq ©+ q2uq «2=0. Elliptic analo gueoftheEuler ±Poisson±Darboux equation. 1 ¬.For =1,seeSubsections 8.1.2 and8.2.3 with = ( , F).For ¹1,thetransformation=(1- ) F,¤= 1- ®leads toanequation oftheform 7.4.4.1:¤2 ® 1- ® ¦2 ¦2+ ¦2 ¦¤2=0. 2 ¬.Suppose ®= ®( , F)isasolution oftheequation inquestion fora®xedvalue ofthe parameter .Then thefunctions ¯ ®de®ned bytherelations¯ ®= ¦®¦F,¯ ®= ¦®¦+ F ¦®¦F,¯ ®=2 F ¦®¦+( F2- 2) ¦®¦F+ F ® arealsosolutions ofthisequation. 3 ¬.Suppose ®= ®( , F)isasolution oftheequation inquestion fora®xedvalue ofthe parameter .Using this ®,onecanconstruct solutions oftheequation with other values ofthe parameter bytheformulas 2- ®= ®-1 ®,®-2= ¦®¦+( -1) ®,®-2= F ¦®¦- 2 ¦®¦F+( -1) F ®,®-2= ( 2- F2) ¦®¦+2 2F ¦®¦F+ °2-( -1) F2 ®,®+2=1 ¦®¦,®+2= F ¦®¦- ¦®¦F,®+2= 2- F2 ¦®¦+2 F ¦®¦F+ ®.y Refer ence:A.V.Akseno v(2001). 4. q2uq r2+t(r) q2uq w2=0. 1 ¬.Particular solutions:= ±1 ¤+ ±2¤+ ±3 + ±4,= ±1¤2+ ±2 ¤+ ±3¤+ ±4 -2 ±1 ² ³( ´- µ) ¶( µ) · µ+ ±5,¸= ±1¤3+ ±2 ´¤+ ±3¤+ ±4 ´-6 ±1¤ ² ³( ´- µ) ¶( µ) · µ+ ±5,¸=( ±1 ´+ ±2)¤2+ ±3 ´¤+ ±4¤+ ±5 ´-2²³( ´- µ)( ±1 µ+ ±2) ¶( µ) · µ+ ±6, where ±1, ±2, ±3, ±4, ±5,and ±6arearbitrary constants, isanynumber . Page528 7.4. OTHER EQUATIONS 529 2 ¬.Separable particular solution:¸=( ±1 ¹ º § + ±2 ¹-º § ) »( ´), where ±1, ±2,and ¼arearbitrary constants, andthefunction »= »( ´)isdetermined bytheordinary differential equation » ½J½³ ³+ ¼2¶( ´) »=0. 3 ¬.Separable particular solution:¸=[ ±1sin( ¼¤)+ ±2cos( ¼¤)] ¾( ´), where ±1, ±2,and ¼arearbitrary constants, andthefunction ¾= ¾( ´)isdetermined bytheordinary differential equation ¾ ½J½³ ³- ¼2¶( ´) ¾=0. 4 ¬.Particular solutions with evenpowers of¤:¸= ¥ ¿fÀ =0 Á À ( ´) Â2 À , where thefunctionsÁ À =Á À ( ´)arede®ned bytherecurrence relationsÁ Ã( ´)= ÄÃ ´+ ÅÃ,Á À -1( ´)= Ä À´+ Å À -2 Æ(2 Æ-1)² ³( ´- µ) ¶( µ)Á À ( µ) · µ, where Ä À and Å À arearbitrary constants ( Æ= Ç, ppp,1), isanynumber . 5 È.Particular solutions with oddpowers of Â:¸= Ã ¿À =0 É À ( ´) Â2 À +1, where thefunctionsÉ À =É À ( ´)arede®ned bytherecurrence relationsÉ Ã( ´)= ÄÃ ´+ ÅÃ,É À -1( ´)= Ä À´+ Å À -2 Æ(2 Æ+1)² ³( ´- µ) ¶( µ)É À ( µ) · µ, where Ä À and Å À arearbitrary constants ( Æ= Ç, ppp,1),and isanynumber . 5. qq r ÊOË1(r) q Ìq r Í+ qq Î ÊOË2(Î) q Ìq Î Í+ Ï ÐÒÑ1(r)+ Ñ2(Î) ÓÌ=0. This equation isencountered inthetheory ofvibration ofinhomogeneous membranes. Itsseparable solutions aresought intheform ¸( ´, Â)=Á( ´)É( Â). Thearticle cited belowpresents analgorithm foraccelerated convergence ofsolutions toeigen- value boundary value problems forthisequation.ÔyÕ Refer ence:L.D.Akulenk oandS.V.Nestero v(1999). 7.4.5. Equations oftheFormÖ( ×) Ø2 ÙØ ×2+ Ø2 ÙØ Ú2+ Û( ×) Ø ÙØ ×+ Ü( ×) Ù=± Ý( ×,Ú) 7.4.5-1. Statements ofboundary value problems. Relations fortheGreen' sfunction. Consider two-dimensional boundary value problems fortheequationÞ( ´) ß2 ¸ß ´2+ ß2 ¸ß Â2+ à( ´) ß ¸ß ´+ á( ´) ¸=- â( ´, Â) (1) Page529 530 ELLIPTIC EQUATIONS WITH TWOSPACE VARIABLES with general boundary conditions in ´,ã 1ß³ ¸- Æ1 ¸= ¶1( Â) at ´= ´1,ã 2ß³ ¸+ Æ2 ¸= ¶2( Â) at ´= ´2,(2) and different boundary conditions in Â. We assume that the coef®cients of equation (1) and the boundary conditions (2) meet the requirementÞ( ´), à( ´), á( ´) are continuous functions ( ´1£ ´£ ´2); Þ>0,| ã 1| + | Æ1|>0,| ã 2| + | Æ2|>0. In the general case, the Green's function can be represented asä( ´, Â, å, æ)= ç( å) è ¿Ã=1 é Ã( ´)é Ã( å)êé Ã ê2 ëÃ( Â, æ; ¼Ã). ( 3) Here,ç( ´)=1Þ( ´)expÊOì à( í)Þ( í) î íÍ, êé Ã ê2=ì ï2ï1 ç( í)é2Ã( í)î í, ( 4) and the ðÃandé Ã( í) are the eigenvalues and eigenfunctions of the homogeneous boundary value problem for the ordinary differential equationÞ( í)é ½J½ ï ï + à( í)é ½ ï +[ ð+ á( í)]é= 0, ( 5)ã 1é ½ ï - Æ1é= 0 at í= í1, ( 6)ã 2é ½ ï + Æ2é= 0 at í= í2. ( 7) The functionsëÃforvariou sboundar ycondition sin Âarespeci®e dinTable25. Equation (5) can be rewritten in self-adjoint form as [ ñ( í)é ½ ï ] ½ ï +[ ð ç( í)- ò( í)]é= 0, ( 8) where the functions ñ( í) and ò( í) are given byñ( í)=expÊOì à( í)Þ( í) î íÍ, ò( í)= - á( í)Þ( í)expÊOì à( í)Þ( í) î íÍ, and ç( í) is de®ned in (4). The eigenvalue problem (8), (6), (7) possesses the following properties: 1 È. All eigenvalues ð1, ð2, pppare real and ðÃ ó ôas Çó ô. 2 È. The system of eigenfunctions {é1( í),é2( í), ppp}is orthogonal on the interval í1£ í£ í2with weight ç( í), that is,ì ï2ï1 ç( í)é Ã( í)é õ( í)î í= 0 for Ç¹ ö. 3 È. If the conditionsò( í)³ 0, ã 1 Æ1³ 0, ã 2 Æ2³ 0 (9) are satis®ed, there are no negative eigenvalues. If òº 0and Æ1= Æ2= 0, then the least eigenvalue isð0= 0and the corresponding eigenfunction isé0=const; in this case, the summation in (3) must start with Ç= 0. In the other cases, if conditions (9) are satis®ed, all eigenvalues are positive; for example, the ®rst inequality in (9) holds if á( í)£ 0. Page 530 7.4. O THER EQUATIONS 531 TABLE 25 The functionsëÃin (3) for various boundary conditions.* Notation: ÷Ã= ø ðÃ Domain Boundary conditions FunctionëÃ( Â, æ; ðÃ) -ô< Â<ô| ù|<ôfor Âó ú ô1 2 û ü ý- û ü| þ- ÿ| 0 £ <ô ù= 0for = 01û ü ý- û üGþsinh( ÷ æ) for > æ,ý- û üGÿsinh( ÷ ) for æ> 0 £ <ô ß þ ù= 0for = 01û ü ý- û üGþcosh( ÷ æ) for > æ,ý- û üGÿcosh( ÷ ) for æ> 0 £ <ô ß þ ù- 3 ù= 0for = 01û ü( û ü+ 3) ý- û üGþ[ ÷ cosh( ÷ æ)+ 3sinh( ÷ æ)] for > æ,ý- û üGÿ[ ÷ cosh( ÷ )+ 3sinh( ÷ )] for æ> 0 £ £ ù= 0 at = 0,ù= 0 at = 1û üsinh( û ü ) sinh( ÷ æ) sinh[ ÷ ( - )] for > æ, sinh( ÷ ) sinh[ ÷ ( - æ)] for æ> 0 £ £ ß þ ù= 0 at = 0,ß þ ù= 0 at = 1û üsinh( û ü) cosh( ÷ æ) cosh[ ÷ ( - )] for > æ, cosh( ÷ ) cosh[ ÷ ( - æ)] for æ> 0 £ £ ù= 0 at = 0,ß þ ù= 0 at = 1û ücosh( û ü ) sinh( ÷ æ) cosh[ ÷ ( - )] for > æ, sinh( ÷ ) cosh[ ÷ ( - æ)] for æ> Subsection 1.8.9 presents some relations for estimating the eigenvalues ð and eigenfunc- tionsé ( í). The Green's function of the two-dimensional third boundary value problem (1)±(2) augmented by the boundary conditionsß ùß - 3 ù= 0 at = 0, ß ùß + 4 ù= 0 at = is given by relation (3) withë ( , æ; ð )= ÷ cosh( ÷ )+ 3sinh( ÷ ) ÷ cosh[ ÷ ( - )]+ 4sinh[ ÷ ( - )] ÷ ÷ ( 3+ 4) cosh( ÷ )+( ÷2+ 3 4) sinh( ÷ ) for > ,÷ cosh( ÷ )+ 3sinh( ÷ ) ÷ cosh[ ÷ ( - )]+ 4sinh[ ÷ ( - )] ÷ ÷ ( 3+ 4) cosh( ÷ )+( ÷2+ 3 4) sinh( ÷ ) for < . 7.4.5-2. Representation of solutions to boundary value problems using the Green's function. 1 . The solution of the ®rst boundary value problem for equation (1) with the boundary conditionsù= 1( ) at í= í1, ù= 2( ) at í= í2,ù= 3( í) at = 0, ù= 4( í) at = *Forunbounde ddomains ,theconditio nofboundednes softhesolutio nas isset;inTable25,thisconditio nis omitted. Page 531 532 ELLIPTIC EQUATIONS WITH TWOSPACEVARIABLES isexpressed interms oftheGreen' sfunction asù( í, )= ( í1)ì 0 1( ) ( , ,, ) = 1 - ( 2) ! 0 2( ) ( , ",, ) = 2 + ! 21 3() ( , ",, ) # =0 - ! 21 4() ( , ",, ) # = + ! 21 ! 0 $(, )( , ",, ) . 2.Thesolution ofthesecond boundary value problem forequation (1)with boundary conditions %= 1( ")at = 1, %= 2( ")at = 2, & %= 3( )at "=0, & %= 4( )at "= isexpressed interms oftheGreen' sfunction as%( , ")=- ( 1) ! 0 1( )( , ", 1, ) + ( 2) ! 0 2( )( , ", 2, ) - ! 21 3()( , ",,0) + ! 21 4()( , ",, ) + ! 21 ! 0$(, )( , ",, ) . 3.The solution ofthethird boundary value problem forequation (1)interms oftheGreen' s function isrepresented inthesame wayasthesolution ofthesecond boundary value problem (the Green' sfunction isnowdifferent). Page532 Chapter 8 Elliptic Equations with Three orMore Space Variab les 8.1. Laplace Equation '3 (=0 Thethree-dimensional Laplace equation isoften encountered inheat andmass transfer theory ,¯uid mechanics, elasticity ,electrostatics, andother areas ofmechanics andphysics. Forexample, inheat andmass transfer theory ,thisequation describes stationary temperature distrib ution intheabsence ofheat sources andsinks inthedomain under study . Aregular solution oftheLaplace equation iscalled aharmonic function. The®rstboundary valueproblem fortheLaplace equation isoften referred toastheDirichlet problem, andthesecond boundary value problem, astheNeumann problem. Extremum principle :Givenadomain ),aharmonic function%in )thatisnotidentically constant in )cannot attain itsmaximum orminimum value atanyinterior point of ). 8.1.1. Problems inCartesian Coor dinates Thethree-dimensional Laplace equation intherectangular Cartesian system ofcoordinates iswrit- tenas2% 2+ 2% "2+ 2% *2=0. 8.1.1-1. Particular solutions andsome relations. 1.Particular solutions:%( , ",*)= + + , "+ -*+ ),%( , ",*)= + 2+ , "2-( ++ ,)*2+ - "+ ) *+ . "*,%( , ",*)=cos( /1 + /2 ")exp( 0 /*),%( , ",*)=sin( /1 + /2 ")exp( 0 /*),%( , ",*)=exp( /1 + /2 ")cos( /*+ +),%( , ",*)=exp( 0 / )cos( /1 "+ +)cos( /2*+ ,),%( , ",*)=cosh( /1 )cosh( /2 ")cos( /*+ ,),%( , ",*)=cosh( /1 )sinh( /2 ")cos( /*+ ,),%( , ",*)=cosh( / )cos( /1 "+ +)cos( /2*+ ,),%( , ",*)=sinh( /1 )sinh( /2 ")sin( /*+ ,),%( , ",*)=sinh( / )sin( /1 "+ +)sin( /2*+ ,), where +, ,, -, ), ., /1,and /2arearbitrary constants, and /= 1 /2 1+ /2 2. 2 2.Fundamental solution: 3 3 ( , ",*)=1 4 41 2+ "2+*2. Page533 534 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 3 2.Suppose%=%( , ",*)isasolution oftheLaplace equation. Then thefunctions%1= +%( 0 5 + -1, 0 5 "+ -2, 0 5 + -3),%2= + 6% 7 6 2, " 6 2, * 6 2 8, 6 =1 2+ "2+*2,%3= +9 :% 7 - ; 6 2: , "- < 6 2:, *- = 6 2:8, : =1-2( ; + <>"+ =*)+( ;2+ <2+ =2) 6 2, where +, - ?, ;, <, =,and 5arearbitrary constants, arealso solutions ofthisequation. Thesigns at 5intheexpression of%1canbetakenindependently ofoneanother .@BA Refer ences :W.Miller ,Jr.(1977), R.Courant andD.Hilbert (1989). 8.1.1-2. Domain: - C< D< C,- C< "< C,0£*< C.First boundary value problem. Ahalf-space isconsidered. Aboundary condition isprescribed:%= E( D, ")at*=0. Solution:%( D, ",*)=1 2 4 ! F -F ! F -F * E( G, H) G HI ( D- G)2+( "- H)2+*2 J3 K2.@BA Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). 8.1.1-3. Domain: - C< D< C,- C< "< C,0£*< C.Second boundary value problem. Ahalf-space isconsidered. Aboundary condition isprescribed:L M%= E( D, ")at*=0. Solution:%( D, ",*)=-1 2 4 ! F -F ! F -F E( G, H) N G N H1( D- G)2+( "- H)2+*2+ -, where -isanarbitrary constant.@BA Refer ence:V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974). 8.1.1-4. Domain: 0£ D£ ;,0£ "£ <,0£*£ =.First boundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:%= E1( ",*)at D=0,%= E2( ",*)at D= ;,%= E3( D,*)at "=0,%= E4( D,*)at "= <,%= E5( D, ")at*=0,%= E6( D, ")at*= =. Solution:%( D, ",*)= F O?=1 F OP=1 E2? Psinh( 51? PD)+ E1? Psinh[ 51? P( ;- D)] sinh( 51? P;)sin Q 4 R S< Tsin Q 4 U*= T + F O?=1 F OP=1 E4? Psinh( 52? PS)+ E3? Psinh[ 52? P( <- S)] sinh( 52? P<)sin Q 4 R D; Tsin Q 4 U*= T + F O?=1 F OP=1 E6? Psinh( 53? P*)+ E5? Psinh[ 53? P( =-*)] sinh( 53? P=)sin Q 4 R D; Tsin Q 4 U S< T, Page534 8.1. LAPLA CEEQUATION V3 W=0 535 where theconstant coef®cients aregivenby51? P= 4 X R2<2+ U2=2, 52? P= 4 X R2;2+ U2=2, 53? P= 4 X R2;2+ U2<2,E Y? P= Z[[[[[[[\[[[[[[[]4<^= _ `0 _ a0 b Y( c, d)sin e 4 f c< gsin e 4 h d= g N c N d for i=1,2; 4j= _ k0 _ a0 b Y( l, d)sin e 4 f ljgsin e 4 h d= g N l N dfor i=3,4; 4j< _ k0 _ `0 b Y( l, c)sin e 4 f ljgsin e 4 h c< g N l N cfor i=5,6. Example. The planes m=0and m= nhaveconstant temperaturesW1andW2,respecti vely.The other planes are maintained atzero temperature ( o3= o4= o5= o6=0). Solution:W=16p2 q r s =1 q rt=1 W2sinh( u stm)+W1sinh[ u st( n- m)] (2 v+1)(2 w+1)sinh( u stn)sin( x s y )sin( z t {),x s = p(2 v+1)| , z t= p(2 w+1)}, u st= ~ x2 s + z2t.B Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980), H.S.Carsla wandJ.C.Jaeger(1984).Forthesolution ofother boundary value problems forthethree-dimensional Laplace equation intheCartesian coordinate system, seeSubsection 8.2.2 for º0. 8.1.2. Problems inCylindrical Coor dinates Thethree-dimensional Laplace equation inthecylindrical coordinate system iswritten as 3 º1 LL LL +1 2 L2L 2+ L2Ld2=0, = l2+ c2. 8.1.2-1. Particular solutions:( , , d)= + ( cos h + sin h )( + d),( , , d)= ( )( cos h +sin h )( cosh d+ sinh d),( , , d)= ( )( cos h +sin h )( cosh d+ sinh d),( , , d)= ( )( cos h +sin h )( cos d+ sin d),( , , d)= ( )( cos h +sin h )( cos d+ sin d), where h=0,1,2, ; ,, , , , ,and arearbitrary constants; the ( )and ( )are theBessel functions; andthe ( )and ( )arethemodi®ed Bessel functions. 8.1.2-2. Domain: 0£ £ j,0£ £2 4,- < d< .First boundary value problem. Anin®nite circular cylinder isconsidered. Aboundary condition isprescribed:=b( , d)at = j. Solution:( , , d)=-1j ?=0 =1 ?( ? ) ?( ? j) - B ?( ¡)cos ¢ £+ ?( ¡)sin ¢ £ ¤exp ¥- ¦ ? §| ¡- ¨| © ª ¡, Page535 536 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES where the « ?( ¬)aretheBessel functions and ¦ § ?arepositi veroots ofthetranscendental equation« ?( ¦)=0.The functions ?( ¨)and ?( ¨)arethecoef®cients oftheFourier series expansion of ®( £, ¨), ?( ¨)= ¯ ?4 2 ° 0 ®( £, ¨)cos( ¢ £) ª £, ?( ¨)=14 2 ° 0 ®( £, ¨)sin( ¢ £) ª £, where¯0=1 ±2and¯ ?=1for ¢=1,2, ²²² Ifthesurfacetemperature isindependent of £,i.e., ®( ¨, £)= ®( ¨),then thesolution takesthe form ³ ( ¬, £, ¨)=1 ´§=1 «0( ¦ § ¬)«1( ¦ § ) 0 ®( ¨+ µ)+ ®( ¨- µ)¤exp¥- ¦ § µ© ª µ, where the ¦ §arepositi veroots ofthetranscendental equation «0( ¦)=0.¶B· Refer ence:H.S.Carsla wandJ.C.Jaeger(1984). 8.1.2-3. Domain: 0£ ¬£ ,0£ ££2 4,0£ ¨£ ¸.First boundary value problem. Acircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:³ = ®( £, ¨)at ¬= , ³ = ¹1( ¬, £)at ¨=0, ³ = ¹2( ¬, £)at ¨= ¸. Solution:³ ( ¬, £, ¨)= ´?=0 ´§=1 º ? » ¼ ½ ¬¸ ¾º ? » ¼ ½ ¸ ¾ ¥ ? §cos ¢ £+ ? §sin ¢ £©sin » ¼ ½ ¨¸ ¾ + ´?=0 ´§=1 « ? » ¿ § ? ¬ ¾ ¥ÁÀ(1)? §cos ¢ £+ Â(1)? §sin ¢ £©sinh » ¿ § ?( ¸- ¨) ¾ sinh » ¿ § ? ¸ ¾ + ´?=0 ´§=1 « ? » ¿ § ? ¬ ¾ ¥ÁÀ(2)? §cos ¢ £+ Â(2)? §sin ¢ £©sinh » ¿ § ? ¨ ¾ sinh » ¿ § ? ¸ ¾, where the « ?( ¬)aretheBessel functions, theº ?( ¬)arethemodi®ed Bessel functions, and¿ § ?is the½throotoftheequation « ?(¿)=0.Thecoef®cients ? §, ? §,À( Ã)? §,and Â( Ã)? §arede®ned by ? §= ¯ ?¼ ¸ 2 ° 0 Ä0 ®( £, ¨)cos( ¢ £)sin » ¼ ½ ¨¸ ¾ ª £ ª ¨, ? §=2¼ ¸ 2 ° 0 Ä0 ®( £, ¨)sin( ¢ £)sin » ¼ ½ ¨¸ ¾ ª £ ª ¨,À( Ã)? §= ¯ ?¼ 2[ « Å?(¿ § ?)]22 ° 0 Æ0 ¹Ã( ¬, £)cos( ¢ £) « ? » ¿ § ? ¬ ¾ ¬ ª ¬ ª £,Â( Ã)? §=2¼ 2[ « Å?(¿ § ?)]22 ° 0 Æ0 ¹Ã( ¬, £)sin( ¢ £) « ? » ¿ § ? ¬ ¾ ¬ ª ¬ ª £,¯ ?= Ç1for ¢=0, 2for ¢¹0, È=1,2.¶B· Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980).ÉForthesolution ofother boundary value problems forthethree-dimensional Laplace equation inthecylindrical coordinate system, seeSubsection 8.2.3 for Êº0. Page536 8.1. LAPLA CEEQUATION Ë3 ³ =0 537 8.1.3. Problems inSpherical Coor dinates Thethree-dimensional Laplace equation inthespherical coordinate system iswritten as 1¬2 ÌÌ ¬ Í ¬2Ì ³Ì ¬ Î+1¬2sin Ï ÌÌ Ï Ísin ÏÌ ³Ì Ï Î+1¬2sin2Ï Ì2 ³Ì £2=0, ¬= Ð Ñ2+ Ò2+ ¨2. 8.1.3-1. Particular solutions:³ ( ¬)= + Ó¬,³ ( ¬, Ï)=Í ¬ Ô+ Ó¬Ô+1Î Õ Ô(cos Ï),³ ( ¬, Ï, £)=Í ¬ Ô+ Ó¬Ô+1Î Õ §Ô(cos Ï)(Àcos½ £+ Âsin½ £), where ¢=0,1,2, ²²²;½=0,1,2, ²²², ¢; ,Ó,À, Âarearbitrary constants; theÕ Ô( ¡)arethe Legendre polynomials; andtheÕ §Ô( ¡)aretheassociated Legendre functions thatareexpressed asÕ Ô( Ñ)=1¢!2Ô ªÔª ÑÔ( Ñ2-1) Ô,Õ §Ô( Ñ)=(1- Ñ2) § Ö2 ª §ª Ñ §Õ Ô( Ñ). 8.1.3-2. Domain: 0£ ¬£ ×or ×£ ¬< Ø.First boundary value problem. Aboundary condition atthesphere surfaceisprescribed:³ = ®( Ï, £)at ¬= ×. 1 Ù.Solution oftheinner problem (for ¬£ ×):³ ( ¬, Ï, £)= × 4¼ 2 ° 0 ° 0 ®( Ï0, £0) ×2- ¬2 ( ¬2-2 × ¬cos Ú+ ×2)3 Ö2sin Ï0 ª Ï0 ª £0, cos Ú=cos Ïcos Ï0+sin Ïsin Ï0cos( £- £0). This formula isconventionally called thePoisson integralforasphere. Series solution:³ ( ¬, Ï, £)= ´Ô=0 Í ¬× Î Ô ÛÔ( Ï, £), ÛÔ( Ï, £)= Ô ´§=0( Ô §cos½ £+Ó Ô §sin½ £)Õ §Ô(cos Ï), where 00=1 4¼ 2 ° 0 ° 0 ®( Ï, £)sin Ï ª Ï ª £, Ô §=(2 ¢+1)( ¢-½)! 2¼( ¢+½)! 2 ° 0 ° 0 ®( Ï, £)Õ §Ô(cos Ï)cos½ £sin Ï ª Ï ª £,Ó Ô §=(2 ¢+1)( ¢-½)! 2¼( ¢+½)! 2 ° 0 ° 0 ®( Ï, £)Õ §Ô(cos Ï)sin½ £sin Ï ª Ï ª £. 2 Ù.Solution oftheouter problem (for ¬³ ×):³ ( ¬, Ï, £)= × 4¼ 2 ° 0 ° 0 ®( Ï0, £0) ¬2- ×2 ( ¬2-2 × ¬cos Ú+ ×2)3 Ö2sin Ï0 ª Ï0 ª £0, where cos Úisexpressed inthesame wayasintheinner problem. Page537 538 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Series solution:³ ( ¬, Ï, £)= ´Ô=0 Í ×¬ Î Ô+1ÛÔ( Ï, £),ÛÔ( Ï, £)= Ô ´§=0( Ô §cos½ £+Ó Ô §sin½ £)Õ §Ô(cos Ï), where thecoef®cients Ô §andÓ Ô §arede®ned bythesame relations asintheinner problem.¶B· Refer ences :G.N.Polozhii (1964), V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980), A.N.Tikhono vandA.A.Samarskii (1990). 8.1.3-3. Domain: 0£ ¬£ ×or ×£ ¬< Ø.Second boundary value problem. Aboundary condition atthesphere surfaceisprescribed:Ì ³Ì ¬= ®( Ï, £)at ¬= ×. Thefunction ®( Ï, £)must satisfy thesolvability condition2 ° 0 ° 0 ®( Ï, £)sin Ï ª Ï ª £=0. 1 Ù.Solution oftheinner problem (for ¬£ ×):³ ( ¬, Ï, £)= × 4¼ 2 ° 0 ° 0 ®( Ï0, £0) Ü1×ln( ×+ ¬1- ¬cos Ú)-2¬1 Ýsin Ï0 ª Ï0 ª £0,¬1= Ð¬2-2 × ¬cos Ú+ ×2,cos Ú=cos Ïcos Ï0+sin Ïsin Ï0cos( £- £0). Series solution:³ ( ¬, Ï, £)= ´Ô=1 Ô ´§=0 ×¢ Í ¬× Î Ô( Ô §cos½ £+Ó Ô §sin½ £)Õ §Ô(cos Ï)+À, where thecoef®cients Ô §andÓ Ô §areexpressed inthesame wayasintheinner ®rstboundary value problem (seeParagraph 8.1.3-2), andÀisanarbitrary constant. 2 Ù.Solution oftheouter problem (for ¬³ ×):³ ( ¬, Ï, £)=- × 4¼ 2 ° 0 ° 0 ®( Ï0, £0) Ü1×ln ×+ ¬1- ¬cos Ú¬(1-cos Ú)-2¬1 Ýsin Ï0 ª Ï0 ª £0,¬1= Ð ¬2-2 × ¬cos Ú+ ×2,cos Ú=cos Ïcos Ï0+sin Ïsin Ï0cos( £- £0). Series solution:³ ( ¬, Ï, £)=- ´Ô=0 Ô ´§=0 ×¢+1 Í ×¬ Î Ô+1 ( Ô §cos½ £+Ó Ô §sin½ £)Õ §Ô(cos Ï)+À, where thecoef®cients Ô §andÓ Ô §areexpressed inthesame wayasintheinner ®rstboundary value problem, andÀisanarbitrary constant. Page538 8.2. P OISSON EQUATION V3 W+ Þ(x)= 0 539 3 Ù. Outer boundary value problems where unbounded solutions as ¬ ß Øare sought are also encountered in applications. Example. A potential translational ¯ow of an ideal incompressible ¯uid about a sphere of radius àis governed by the Laplace equation with the boundary conditions:áâW= 0 at ã= à, |W- ä ãcos å| æ0as ã æ ç, where èis the potential, äthe unperturbed ¯ow velocity at in®nity; the ¯uid velocity is expressed in terms of the potential asv= Ñ é. Solution:è= ä ã ê1 + à3 2 ã3 ëcos å. This solution is a special case of the second formula from Paragraph for 8.1.3-1 for ì= 1.¶B· References : G. N. Polozhii (1964), V . M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), L. G. Loitsyanskii (1996).ÉFor the solution of other boundary value problems for the three-dimensional Laplace equation in the spherical coordinate system, see Subsection 8.2.4 for Êº 0. 8.1.4. Other Orthogonal Curvilinear Systems of Coordinates The three-dimensional Laplace equation admits separation of variables in the eleven orthogonalcoordinat esystem sthatarelistedinTable26. For the general ellipsoidal and conical coordinate systems, the functions®, ¹, and íare deter- mined by Lam Âe equations that involve the Jacobian elliptic function sn ¨=sn( ¨, î). The solutions of these equations under some conditions can be represented in the form of ®nite series called Lam Âe polynomials. For details about the Lam Âe equation and its solutions, see Whittaker and Watson (1963), Bateman and Erd Âelyi (1955), Arscott (1964), and Miller, Jr. (1977). There are also coordinate systems that allow the so-called ï-separation of variables of the three-dimensional Laplace equation. Such solutions in the new coordinate system,¿, ð, ñ, can be represented in the form ò= ó ï( ô, ð, ñ) õ( ô) ö( ð) í( ñ). Coordinates that allow the ï-separation of variable sarelistedinTable27. Only the bicylindrical and toroidal coordinate systems are fairly widely used in applications. In three subsequent coordinate systems, the functions õ= õ( ô) and ö= ö( ñ) are determined by identical equations. With the change of variables ô=sn2( ÷, î), ñ=sn2( ø, î), where î= ù-1 Ö2, these equations are reduced to Lam Âe equations ( ÷and øare the new independent variables).úBû References for Subsection 8.1.4: M. B Ãocher (1894), F. M. Morse and H. Feshbach (1953, V ols. 1±2), N. N. Lebedev, I. P. Skal'skaya, and Ya. S. U¯yand (1955), P. Moon and D. Spencer (1961), A. Makarov, J. Smorodinsky, K. Valiev, and P. Winternitz (1967), W. Miller, Jr. (1977). 8.2. Poisson Equation ü3 ý+ þ(x) = 0 8.2.1. Preliminary Remarks. Solution Structure Like the three-dimensional Laplace equation, the three-dimensional Poisson equation is often en- countered in heat and mass transfer theory, ¯uid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. In particular, the Poisson equation describes stationary temperature distribution in the presence of thermal sources or sinks in the domain under consideration. The Laplace equation is a special case of the Poisson equation with ÿº 0. Throughout this section, we consider a three-dimensional bounded domain with a suf®ciently smooth boundary . We assume that r and , where r= { , , }and = { , , }. Page 539 540 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES TABLE 26 Orthogonal coordinates Å ,Å ,Å thatallowseparable solutions oftheformò= õ(Å ) ö(Å ) (Å )forthethree-dimensional Laplace equation 3 =0 Coordinates Transformations Particular solutions (orequations for õ, ö, ) Cartesian, , = ,= ,= =cos( 1 + 1)cos( 2 + 2)cosh( 3 + 3), where 2 1+ 2 2= 2 3; seealsoParagraph 8.1.1-1 Cylindrical , , = cos ,= sin ,= =[ Ô( )+ Ô( )]cos( + )exp( ),Ô( )and Ô( )aretheBessel functions; seealsoParagraph 8.1.2-1 Parabolic cylindrical, , =1 2( 2- 2),= ,= = -1 2( ) - -1 2( )cos( + ), where = ó2 , ( )istheparabolic cylinder function Elliptic cylindrical , , = ùcosh cos ,= ùsinh sin ,= = CeÔ( ,- )ceÔ( ,- )cos( + ), SeÔ( ,- )seÔ( ,- )cos( + ), CeÔandSeÔarethemodi®ed Mathieu functions, ceÔandseÔaretheMathieu functions, =1 4 ù2 2 Spherical , , = sin cos ,= sin sin ,= cos =( Ô+ -Ô-1) ! "Ô(cos )cos( + ),! "Ô( )aretheassociated Legendre functions, seealsoParagraph 8.1.3-1 Prolate spheroidal , , = ùsinh sin cos ,= ùsinh sin sin ,= ùcosh cos = ! "Ô(cosh ) ! "Ô(cos )cos( + ),! "Ô( )aretheassociated Legendre functions Oblate spheroidal , , = ùcosh sin cos ,= ùcosh sin sin ,= ùsinh cos = ! "Ô(- #sinh ) ! "Ô(cos )cos( + ),! "Ô( )aretheassociated Legendre functions Parabolic, , = ù cos ,= ù sin ,=1 2 ù( 2- 2) = $&%"( ' ) %"( ' )cos( + ),"( )aretheBessel functions,$"( )arethemodi®ed Bessel functions Paraboloidal , , =2 (cosh cos sinh ,=2 (sinh sin cosh ,=1 2 ((cosh 2 +cos2 -cosh 2 ) = CeÔ( ,- ))ceÔ( ,- ))CeÔ( + #+* ,2,- )), SeÔ( ,- ))seÔ( ,- ))SeÔ( + #+* ,2,- )),)=1 2 ( ';ceÔandseÔaretheMathieu functions, CeÔandSeÔarethemodi®ed Mathieu functions General ellipsoidal - , ., / = 0( - 1)( 2- 1)( 3- 1)1( 1-1),= 0( -1)( 2-1)( 3-1) 1- 1,= 4 2531 6 7879:9+( '2+ '1sn2) 6=0(Lam Âeequation),; 787<5<+( '2+ '1sn2) ;=0(Lam Âeequation), 787=&=+( '2+ '1sn2 ) ;=0(Lam Âeequation),- =sn2( , ), .=sn2( , ), /=sn2( , ), = (-1 2 Conical/, - , . = / 0( 1 -1)( 152-1) 1- 1,= / 0 1( -1)( 2-1)1-1,= / > ( -. 6( /)= /Ô+ /-Ô-1, =0,1, ?@?@?,; 7879:9+[ '- ( +1) 2sn2] ;=0(Lam Âeequation), 787<5<+[ '- ( +1) 2sn2] =0(Lam Âeequation), where - =sn2( , ), .=sn2( , ), = (1 2,;and areexpressed interms oftheLam Âepolynomials Page540 8.2. POISSON EQUATION 3 + (x)=0 541 TABLE 27 Coordinates Å ,Å ,Å thatallow -separated solutions oftheform= (Å ,Å ,Å ) (Å ) (Å ) (Å )forthethree-dimensional Laplace equation 3 =0 Newcoordinates, function Transformations ofcoordinatesFunctions , , (equations for , , ) Bicylindrical coordinates , , ,=cosh -cos = -1sin cos ,= -1sin sin ,= -1sinh ; 0£ £ , isany, 0£ <2 ( )= 1 (cos )+ 2 (cos ), ( )= 1cosh ( +1 2) + 2sinh ( +1 2) ,( )= 1cos( )+ 2sin( ),=0,1,2, ; =0,1,2, Toroidal coordinates , , ,=cosh -cos = -1sinh cos ,= -1sinh sin ,= -1sin ; ³0,- £ £ ,0£ <2 ( )= 1 -1 2(cosh )+ 2 -1 2(cosh ), ( )= 1cos( )+ 2sin( ),( )= 1cos( )+ 2sin( ),=0,1,2, ; =0,1,2, Coordinates !, ", ,= #( $- %)( %- &)%( %-1) - #( $-1)(1- &)%-1 = -1cos ,= -1sin ,= -1 '- ! " ( );!> )>1, "<0,0£ <2 *( !)[ *( !) +] ++ (1 4- 2) !- , =0, *( ")[ *( ") +] ++ (1 4- 2) "- , =0,( )= 1cos( )+ 2sin( ),*( -)=4 -( --1)( -- )) Coordinates !, ", ,= ' $ &% + #( $-1)( &-1)%-1 = -1cos ,= -1sin ,= -1#( $- %)( %- &)%( %-1); 1< "< )< !,0£ <2 *( !)[ *( !) +] ++ (1 4- 2) !- , =0, *( ")[ *( ") +] ++ (1 4- 2) "- , =0,( )= 1cos( )+ 2sin( ),*( -)=4 -( --1)( -- )) Coordinates !, ", ,=2Re# .( $- %)( &- %)%( %- /),)=Å 0= + 12, , arerealnumbers = -1cos ,= -1sin ,= -1'- ! " (( ) 0);!>0, "<0,0£ <2 *( !)[ *( !) +] ++ (1 4- 2) !- , =0, *( ")[ *( ") +] ++ (1 4- 2) "- , =0,( )= 1cos( )+ 2sin( ),*( -)=4 -( -- ))( -- 0) Coordinates !, 3, ",=1+ ' $ 45&%5/ = -1#( $- %)( 4- %)( &- %) ( /- %)( %-1) %,= -1#( $- /)( 4- /)( &- /) ( %- /)( /-1) /,= -1#( $-1)( 4-1)( &-1) ( %-1)( /-1); 0< "<1< 3< 0< !< ) *( !)[ *( !) +] +-(3 !2+ ,1 !+ ,2) =0,*( 3)[ *( 3) +] +-(3 32+ ,1 3+ ,2) =0, *( ")[ *( ") +] +-(3 "2+ ,1 "+ ,2) =0,*( -)=16 -( --1)( -- ))( -- 0) Coordinates !, 3, ",=2Re #( $- %)( 4- %)( &- %). %( %-1)( %- /),)=Å 0= + 12, , arerealnumbers = -1#( $-1)( 4-1)( &-1) ( %-1)( /-1),= -1'- $ 45&%5/,= -1;"<0< !<1< 3 *( !)[ *( !) +] +-(3 !2+ ,1 !+ ,2) =0,*( 3)[ *( 3) +] +-(3 32+ ,1 3+ ,2) =0, *( ")[ *( ") +] +-(3 "2+ ,1 "+ ,2) =0,*( -)=16 -( --1)( -- ))( -- 0) 8.2.1-1. First boundary value problem. Thesolution ofthe®rstboundary value problem forthePoisson equation3 + 6(r)=0 (1) inadomain 7with thenonhomogeneous boundary condition= (r)for r 8 9 Page541 542 ELLIPTIC EQUATIONS WITH THREE OR MORESPACE VARIABLES TABLE 28 The volume elements and distances occurring in relations (2) and (5) in some coordinate systems. In all cases, := { ;, <, =} Coordinate systemV olume element, > 7&Gradient, Ñ&? (|i @| = |i A| = |i B| = 1)Distance,>= |r- :| Cartesian r={ , , } > ; > < > = i @ C DC @+i A C DC A+i B C DC B >= '( - ;)2+( - <)2+( - =)2 Cylindrical r={ E, , } ; > ; > < > = i @ C DC @+i A1@ C DC A+i B C DC B >= 'E2+ ;2-2 E ;cos( - <)+( - =)2 Spherical r={ E, F, } ;2sin < > ; > < > =i @ C DC @+i A1@ C DC A+i B1@sin A C DC B >= 'E2+ ;2- 2 E ;cos G, where cos G=cos Fcos <+sin Fsin <cos( - =) can be represented in the form(r)= H I 6( :) J(r, :) > 7&- H K ( :) L JL M & > 9&. ( 2) Here, J(r, :) is the Green's function of the ®rst boundary value problem, C NC O Pis the derivative of the Green's function with respect to ;, <, =along the outward normal Nthe boundary 9of the domain 7. Integration is everywhere with respect to ;, <, =. Thevolum eelement sinsolutio n(2)forbasiccoordinat esystem sarepresente dinTable28.In addition, the expressions of the gradients are given, which enable one to ®nd the derivative along the normal in accordance with the formulaC NC O P=(N× Ñ& J). The Green's function J= J(r, :) of the ®rst boundary value problem is determined by the following conditions:1Q. The function Jsatis®es the Laplace equation with respect to , , in the domain 7everywhere except for the point ( ;, <, =), at which it can have a singularity of the form1 4 R1 |r- S|. 2 Q. The function J, with respect to , , , satis®es the homogeneous boundary condition of the ®rst kind at the boundary, i.e., the condition J T K= 0. The Green's function can be represented asJ(r, :)=1 4 1 |r- :|+?, ( 3) where the auxiliary function?=?(r, :) is determined by solving the ®rst boundary value problem for the Laplace equation 3 ?= 0with the boundary condition? T K= -1 4 R1 |r- S|; the vector quantity : in this problem is treated as a three-dimensional free parameter. The Green's function possesses the symmetry property with respect to their arguments: J(r, :)=J( :,r). The construction of Green's functions is discussed in Paragraphs 8.3.1-4 and 8.3.1-6 through 8.3.1-8 for ,= 0.U VW X Y[Z \ ]For outer ®rst boundary value problems for the Laplace equation, the following condition is usually set at in®nity: | ^|< (|r|(|r| _ `, a=const). 8.2.1-2. Second boundary value problem. The second boundary value problem for the Poisson equation (1) is characterized by the boundaryconditionL ^L M= b(r) for r 8 9. Page 542 8.2. POISSON EQUATION c3 d+ e(x)=0 543 Necessary condition solvability oftheinner problem:HI 6(r) >r+ HK b(r) > 9=0. (4) Thesolution ofthesecond boundary value problem canbewritten as^(r)= H I 6( :) J(r, :) > 7&+ H K b( :) J(r, :) > 9&+ f, (5) where fisanarbitrary constant, provided thatthesolvability condition ismet. TheGreen' sfunction J= J(r, :)ofthesecond boundary value problem isdetermined bythe following conditions: 1 Q.Thefunction Jsatis®es theLaplace equation with respect to g, h, iinthedomain 7everywhere except forthepoint ( ;, <, =)atwhich ithasasingularity oftheform1 4 R1 |r- S|. 2 Q.Thefunction J,with respect to g, h, i,satis®es thehomogeneous condition ofthesecond kind attheboundary ,i.e.,theconditionL JL M jjjj K=190, where 90isthearea ofthesurface 9. TheGreen' sfunction isunique uptoanadditi veconstant.U VW X Y[Z k ]TheGreen' sfunction cannot beidenti®ed with condition 1 Qandthehomogeneous boundary conditionC NC O jj K=0;thisproblem for Jhasnosolution, because, onrepresenting Jin theform (3),for?weobtain aproblem with anonhomogeneous boundary condition ofthesecond kind, forwhich thesolvability condition (2)isnotmet.U VW X Y[Z l ]Condition (4)isnotextended totheouter second boundary value problem (for in®nite domain). 8.2.1-3. Third boundary value problem. Thesolution ofthethird boundary valueproblem forthePoisson equation (1)inabounded domain 7 with thenonhomogeneous boundary conditionL ^L M+ m ^= b(r)for r 8 9 isgivenbyrelation (5)with f=0,where J= J(r, :)istheGreen' sfunction ofthethird boundary value problem; theGreen' sfunction isdetermined bythefollowing conditions: 1 Q.Thefunction Jsatis®es theLaplace equation with respect to g, h, iin 7everywhere except for thepoint ( ;, <, =)atwhich ithasasingularity oftheform1 4 R1 |r- S|. 2 Q.Thefunction J,with respect to g, h, i,satis®es thehomogeneous boundary condition ofthe third kind attheboundary ,i.e.,thecondition nC NC O+ m J o K=0. The Green' sfunction canberepresented intheform (3),where theauxiliary function?is determined bysolving thecorresponding third boundary value problem fortheLaplace equationp 3 q=0. Theconstruction ofGreen' sfunctions isdiscussed inParagraphs 8.3.1-4 and8.3.1-6 through 8.3.1-8 for r=0.sut Refer ences forSubsection 8.2.1: V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), N.S.Koshlyak ov, E.B.Gliner ,andM.M.Smirno v(1970). Page543 544 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.2.2. Problems inCartesian Coor dinates Thethree-dimensional Poisson equation intherectangular Cartesian system ofcoordinates hasthe formL2^L g2+ L2^L h2+ L2^L i2+ v( g, h, i)=0. 8.2.2-1. Domain: - `< g< `,- `< h< `,- `< i< `. Solution:^( g, h, i)=1 4 w x y-y x y-y x y-y v( z, {, |) } z } { } |~ ( g- z)2+( h- {)2+( i- |)2.sut Refer ence:R.Courant andD.Hilbert (1989). 8.2.2-2. Domain: - `< g< `,- `< h< `,0£ i< `.First boundary value problem. Ahalf-space isconsidered. Aboundary condition isprescribed:= b( g, h)at i=0. Solution:( g, h, i)=1 2 w xy-y xy-y i b( z, {) } z } {n( g- z)2+( h- {)2+ i2o3 2 +1 4 w xy0 xy-y xy-y 1 --1 + v( z, {, |) } z } { } |, where -= ~ ( - z)2+( - {)2+( - |)2, += ~ ( - z)2+( - {)2+( + |)2.sut Refer ences :A.G.Butk ovskiy (1979), B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 8.2.2-3. Domain: - < < ,- < < ,0£ < .Third boundary value problem. Ahalf-space isconsidered. Aboundary condition isprescribed: - = ( , )at =0. Solution:( , , )=-x y-y x y-y ( z, {) ( , , , z, {,0) } z } { +xy0 xy-y xy-y v( z, {, |) ( , , , z, {, |) } z } { } |, where( , , , z, {, |)=1 4 w 1~ ( - z)2+( - {)2+( - |)2+1~ ( - z)2+( - {)2+( + |)2 -2 xy0exp(- ) } ~ ( - z)2+( - {)2+( + |+ )2 . Page544 8.2. POISSON EQUATION c3 d+ (x)=0 545 8.2.2-4. Domain: - < < ,0£ < ,0£ < .First boundary value problem. Adihedral angle isconsidered. Boundary conditions areprescribed:= 1( , )at =0, = 2( , )at =0. Solution:( , , )=x y0 x y-y 1( z, |) { ( , , , z, {, |) =0 } z } | +x y0 x y-y 2( z, {) | ( , , , z, {, |) =0 } z } { +x y0 x y0 x y-y v( z, {, |) ( , , , z, {, |) } z } { } |, where( , , , z, {, |)=1 4 w 1~ ( - z)2+( - {)2+( - |)2-1~ ( - z)2+( - {)2+( + |)2 -1~ ( - z)2+( + {)2+( - |)2+1~ ( - z)2+( + {)2+( + |)2.sut Refer ences :V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974), A.G.Butk ovskiy (1979). 8.2.2-5. Domain: 0£ < ,0£ < ,0£ < .First boundary value problem. Anoctant isconsidered. Boundary conditions areprescribed:= 1( , )at =0, = 2( , )at =0, = 3( , )at =0. Solution:( , , )=x y0 x y0 1( {, |) z ( , , , z, {, |) =0 } { } | +x y0 x y0 2( z, |) { ( , , , z, {, |) =0 } z } | +x y0 x y0 3( z, {) | ( , , , z, {, |) =0 } z } { +x y0 x y0 x y0 v( z, {, |) ( , , , z, {, |) } z } { } |, where( , , , z, {, |)=1 4 w 1~ ( - z)2+( - {)2+( - |)2-1~ ( - z)2+( - {)2+( + |)2 -1~ ( - z)2+( + {)2+( - |)2+1~ ( - z)2+( + {)2+( + |)2 -1~ ( + z)2+( - {)2+( - |)2+1~ ( + z)2+( - {)2+( + |)2 +1~ ( + z)2+( + {)2+( - |)2-1~ ( + z)2+( + {)2+( + |)2 .sut Refer ences :V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974), A.G.Butk ovskiy (1979). Page545 546 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.2.2-6. Domain: - < < ,- < < ,0£ £ .First boundary value problem. Anin®nite layer isconsidered. Boundary conditions areprescribed:= 1( , )at =0, = 2( , )at = . Solution:( , , )=xy-y xy-y 1( z, {) | ( , , , z, {, |) =0 } z } { -xy-y xy-y 2( z, {) | ( , , , z, {, |) = } z } { +x 0 xy-y xy-y v( z, {, |) ( , , , z, {, |) } z } { } |. Green' sfunction:( , , , z, {, |)=1 4 w y =-y 11-12 , where1= ~ ( - z)2+( - {)2+( - |-2 )2,2= ~ ( - z)2+( - {)2+( + |-2 )2.u Refer ences :A.G.Butk ovskiy (1979), B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 8.2.2-7. Domain: - < < ,- < < ,0£ £ .Mixedboundary value problem. Anin®nite layer isconsidered. Boundary conditions areprescribed:= 1( , )at =0, = 2( , )at = . Solution:( , , )=x y-y x y-y 1( z, {) | ( , , , z, {, |) =0 } z } { +x y-y x y-y 2( z, {) ( , , , z, {, ) } z } { +x 0 xy-y xy-y ( z, {, |) ( , , , z, {, |) } z } { } |. Green' sfunction:( , , , z, {, |)=1 4 y =-y 11-12 , where1= ~ ( - z)2+( - {)2+[ -(-1) |-2 ]2,2= ~ ( - z)2+( - {)2+[ +(-1) |-2 ]2.u Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). Page546 8.2. POISSON EQUATION c3 d+ (x)=0 547 8.2.2-8. Domain: 0£ < ,- < < ,0£ £ .First boundary value problem. Asemiin®nite layer isconsidered. Boundary conditions areprescribed:= 1( , )at =0, = 2( , )at =0, = 3( , )at = . Solution:( , , )=x 0 xy-y 1( {, |) z ( , , , z, {, |) =0 } { } | +xy-y xy0 2( z, {) | ( , , , z, {, |) =0 } z } { -x y-y x y0 3( z, {) | ( , , , z, {, |) = } z } { +x 0 x y-y x y0 ( z, {, |) ( , , , z, {, |) } z } { } |. Green' sfunction:( , , , z, {, |)=1 4 y =-y 11-12-13+14 , where1= ~ ( - z)2+( - {)2+( - |-2 )2,2= ~ ( - z)2+( - {)2+( + |-2 )2,3= ~ ( + z)2+( - {)2+( - |-2 )2,4= ~ ( + z)2+( - {)2+( + |-2 )2.u Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 8.2.2-9. Domain: 0£ £ ,0£ £ ,- < < .First boundary value problem. Anin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed:= 1( , )at =0, = 2( , )at = ,= 3( , )at =0, = 4( , )at = . Solution:( , , )=x 0 x y-y 1( {, |) z ( , , , z, {, |) =0 } | } { -x0 xy-y 2( {, |) z ( , , , z, {, |) = } | } { +x 0 xy-y 3( z, |) { ( , , , z, {, |) =0 } | } z -x 0 xy-y 4( z, |) { ( , , , z, {, |) = } | } z +x 0 x0 xy-y ( z, {, |) ( , , , z, {, |) } | } { } z. Green' sfunction:( , , , z, {, |)=2 y =1 y =11¡ sin( ¢ )sin( £ )sin( ¢ z)sin( £ {)exp(- ¡ | - ||),¢ = , £ = ¤ , ¡ = ¥ ¢2+ £2 . Page547 548 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Alternati vely,theGreen' sfunction canberepresented as( , , , z, {, |)=1 4 y =-y y =-y 1(1) -1(2) -1(3) +1(4) , where(1) = ~ ( - z-2 )2+( - {-2¤ )2+( - |)2,(2) = ~ ( + z-2 )2+( - {-2¤ )2+( - |)2,(3) = ~ ( - z-2 )2+( + {-2¤ )2+( - |)2,(4) = ~ ( + z-2 )2+( + {-2¤ )2+( - |)2. 8.2.2-10. Domain: 0£ £ ,0£ £ ,- < < .Third boundary value problem. Anin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed: ¦- 1 = 1( , )at =0, ¦+ 2 = 2( , )at = , §- 3 = 3( , )at =0, §+ 4 = 4( , )at = . Solution:( , , )=- ¨0 ¨y-y 1( {, |) ( , , ,0, {, |) } | } {+ ¨0 ¨y-y 2( {, |) ( , , , , {, |) } | } { - ¨ 0 ¨y-y 3( z, |) ( , , , z,0, |) } | } z+ ¨ 0 ¨y-y 4( z, |) ( , , , z, , |) } | } z + ¨ 0 ¨0 ¨y-y ( z, {, |) ( , , , z, {, |) } | } { } z. Green' sfunction:( , , , z, {, |)=1 2 y =1 y =1 © ( , )© ( z, {)ª© ª2 ¡ exp(- ¡ | - ||), where « ( , )=( ¬ cos ¬ + 1sin ¬ )( cos + 3sin ), ¡ = ¥ ¬2+ 2 ,ª « ª2=1 4( ¬2+ 2 1)( 2 + 2 3) +( 1+ 2)( ¬2+ 1 2) ( ¬2+ 2 1)( ¬2+ 2 2) +( 3+ 4)( 2 + 3 4) ( 2 + 2 3)( 2 + 2 4) . Here, the ¬ and arepositi veroots ofthetranscendental equations tan( ¬ )=( 1+ 2) ¬¬2- 1 2, tan( )=( 3+ 4) 2- 3 4. 8.2.2-11. Domain: 0£ £ ,0£ £ ,- < < .Mixedboundary value problems. 1 ®.Anin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed: « = 1( , )at =0, ¦ « = 2( , )at = , « = 3( , )at =0, § « = 4( , )at = . Page548 8.2. POISSON EQUATION ¯3 °+ (x)=0 549 Solution: « ( , , )= ¨0 ¨y-y 1( {, |) z ( , , , z, {, |) =0 } | } { + ¨0 ¨y-y 2( {, |) ( , , , , {, |) } | } { + ¨ 0 ¨y-y 3( z, |) { ( , , , z, {, |) =0 } | } z + ¨ 0 ¨y-y 4( z, |) ( , , , z, , |) } | } z + ¨ 0 ¨0 ¨y-y ( z, {, |) ( , , , z, {, |) } | } { } z. Green' sfunction:( , , , z, {, |)=2 y =0 y =01¡ sin( ¢ )sin( £ )sin( ¢ z)sin( £ {)exp(- ¡ | - ||), where¢ =(2 +1) 2 , £ =(2¤+1) 2 , ¡ = ¥ ¢2+ £2 . 2 ®.Anin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed: « = 1( , )at =0, « = 2( , )at = , § « = 3( , )at =0, § « = 4( , )at = . Solution: « ( , , )= ¨0 ¨y-y 1( {, |) z ( , , , z, {, |) =0 } | } { - ¨0 ¨y-y 2( {, |) z ( , , , z, {, |) = } | } { - ¨ 0 ¨y-y 3( z, |) ( , , , z,0, |) } | } z + ¨ 0 ¨y-y 4( z, |) ( , , , z, , |) } | } z + ¨ 0 ¨0 ¨y-y ( z, {, |) ( , , , z, {, |) } | } { } z. Green' sfunction:( , , , z, {, |)=1 y =1 y =0 ± ¡ sin( ¢ )cos( £ )sin( ¢ z)cos( £ {)exp(- ¡ | - ||), where¢ = , £ = ¤ , ¡ = ¥ ¢2+ £2 ,± = ²1for¤=0, 2for¤¹0.³Paragraphs 8.2.2-12 through 8.2.2-17 present only Green'sfunctions; thecomplete solution is constructed with theformulas given inParagraphs 8.2.1-1 through 8.2.1-3. 8.2.2-12. Domain: 0£ £ ,0£ £ ,- < < .First boundary value problem. Anin®nite cylindrical domain oftriangular cross-section isconsidered. Boundary conditions are prescribed: « = 1( , )at =0, « = 2( , )at =0, « = 3( , )at = . Page549 550 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Green' sfunction:( , , , z, {, |)= ´( , , , z, {, |)- ´( , , , {, z, |), where´( , , , z, {, |)=22y =1 y =11¡ sin( ¢ )sin( ¢ )sin( ¢ z)sin( ¢ {)exp(- ¡ | - ||),¢ = , ¢ = ¤ , ¡ =¥ ¢2+ ¢2 . Analternati verepresentation oftheGreen' sfunction canbeobtained bysetting´( , , , z, {, |)=1 4 y =-y y =-y µ1(1) -1(2) -1(3) +1(4) ¶, where thefunctions ·( ¸)¹ º( »=1,2,3,4)arespeci®ed inParagraph 8.2.2-9 for ¼= .½u¾ Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 8.2.2-13. Domain: 0£ ¿£ ¼,0£ À£ ,0£ < .First boundary value problem. Asemiin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed: « = Á1( À, )at ¿=0, « = Á2( À, )at ¿= ¼, « = Á3( ¿, )at À=0, « = Á4( ¿, )at À= , « = Á5( ¿, À)at =0. Green' sfunction:Â( ¿, À, , Ã, Ä, Å)=4¼ Æ Ç¹=1 Æ Çº=11È¹ ºsin( É ¹¿)sin( Ê ºÀ)sin( É ¹Ã)sin( Ê ºÄ) Ë ¹ º( Ì, Å),É ¹= Í Î¼, Ê º= Ï Î, È¹ º= Ð É2¹+ Ê2º,Ë ¹ º( Ì, Å)= Ñexp(- È¹ ºÌ)sinh( È¹ ºÅ)for Ì> Å³0, exp(- È¹ ºÅ)sinh( È¹ ºÌ)for Å> Ì³0. Analternati verepresentation oftheGreen' sfunction:Â( ¿, À, Ì, Ã, Ä, Å)=1 4Î Æ Ç¹=-Æ Æ Çº=-Æ µ1·(1)¹ º-1·(2)¹ º-1·(3)¹ º+1·(4)¹ º-1·(5)¹ º+1·(6)¹ º+1·(7)¹ º-1·(8)¹ º¶, where·(1)¹ º= Ò( ¿- Ã-2Í ¼)2+( À- Ä-2Ï )2+( Ì- Å)2,·(2)¹ º= Ò( ¿+ Ã-2Í ¼)2+( À- Ä-2Ï )2+( Ì- Å)2,·(3)¹ º= Ò( ¿- Ã-2Í ¼)2+( À+ Ä-2Ï )2+( Ì- Å)2,·(4)¹ º= Ò( ¿+ Ã-2Í ¼)2+( À+ Ä-2Ï )2+( Ì- Å)2,·(5)¹ º= Ò( ¿- Ã-2Í ¼)2+( À- Ä-2Ï )2+( Ì+ Å)2,·(6)¹ º= Ò( ¿+ Ã-2Í ¼)2+( À- Ä-2Ï )2+( Ì+ Å)2,·(7)¹ º= Ò( ¿- Ã-2Í ¼)2+( À+ Ä-2Ï )2+( Ì+ Å)2,·(8)¹ º= Ò( ¿+ Ã-2Í ¼)2+( À+ Ä-2Ï )2+( Ì+ Å)2. Page550 8.2. POISSON EQUATION Ó3 Ô+ Õ(x)=0 551 8.2.2-14. Domain: 0£ ¿£ ¼,0£ À£ ,0£ Ì< Ö.Third boundary value problem. Asemiin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed: × Ø « - »1 « = Á1( À, Ì)at ¿=0, × Ø « + »2 « = Á2( À, Ì)at ¿= ¼, × Ù « - »3 « = Á3( ¿, Ì)at À=0, × Ù « + »4 « = Á4( ¿, Ì)at À= , × Ú « - »5 « = Á5( ¿, À)at Ì=0. Green' sfunction:Â( ¿, À, Ì, Ã, Ä, Å)=Æ Ç¹=1 Æ Çº=1 Û ¹ º( ¿, À)Û ¹ º( Ã, Ä)ªÛ ¹ º ª2 Ë ¹ º( Ì, Å), where «¹ º( ¿, À)=( Ü ¹cos Ü ¹¿+ »1sin Ü ¹¿)( Ý ºcos Ý ºÀ+ »3sin Ý ºÀ),ª «¹ º ª2=1 4( Ü2¹+ »2 1)( Ý2º+ »2 3) Þß¼+( »1+ »2)( Ü2¹+ »1 »2) ( Ü2¹+ »2 1)( Ü2¹+ »2 2) à Þß+( »3+ »4)( Ý2º+ »3 »4) ( Ý2º+ »2 3)( Ý2º+ »2 4) à,Ë ¹ º( Ì, Å)= áââ ãââäexp(- È¹ ºÌ) å È¹ ºcosh( È¹ ºÅ)+ »5sinh( È¹ ºÅ) æÈ¹ º( È¹ º+ »5)for Ì> Å, exp(- È¹ ºÅ)å È¹ ºcosh( È¹ ºÌ)+ »5sinh( È¹ ºÌ)æÈ¹ º( È¹ º+ »5)for Å> Ì, È¹ º= Ð Ü2¹+ Ý2º. Here, the Ü ¹and Ý ºarepositi veroots ofthetranscendental equations tan( Ü ¼)=( »1+ »2) ÜÜ2- »1 »2, tan( Ý )=( »3+ »4) ÝÝ2- »3 »4. 8.2.2-15. Domain: 0£ ¿£ ¼,0£ À£ ,0£ Ì< Ö.Mixedboundary value problems. 1 ç.Asemiin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed: « = Á1( À, Ì)at ¿=0, « = Á2( À, Ì)at ¿= ¼, « = Á3( ¿, Ì)at À=0, « = Á4( ¿, Ì)at À= , × Ú « = Á5( ¿, À)at Ì=0. Green' sfunction:Â( ¿, À, Ì, Ã, Ä, Å)=4¼ Æ Ç¹=1 Æ Çº=11È¹ ºsin( É ¹¿)sin( Ê ºÀ)sin( É ¹Ã)sin( Ê ºÄ) Ë ¹ º( Ì, Å),É ¹= Í Î¼, Ê º= Ï Î, È¹ º= Ð É2¹+ Ê2º,Ë ¹ º( Ì, Å)= Ñexp(- È¹ ºÌ)cosh( È¹ ºÅ)for Ì> Å³0, exp(- È¹ ºÅ)cosh( È¹ ºÌ)for Å> Ì³0. 2 ç.Asemiin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed: × Ø « = Á1( À, Ì)at ¿=0, × Ø « = Á2( À, Ì)at ¿= ¼, × Ù « = Á3( ¿, Ì)at À=0, × Ù « = Á4( ¿, Ì)at À= , « = Á5( ¿, À)at Ì=0. Page551 552 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Green' sfunction:Â( ¿, À, Ì, Ã, Ä, Å)=1¼ Æ Ç¹=0 Æ Çº=0 è ¹è ºÈ¹ ºcos( É ¹¿)cos( Ê ºÀ)cos( É ¹Ã)cos( Ê ºÄ) Ë ¹ º( Ì, Å),É ¹= Í Î¼, Ê º= Ï Î, È¹ º= Ð É2¹+ Ê2º,è ¹= Ñ1forÍ=0, 2forÍ¹0,Ë ¹ º( Ì, Å)= Ñexp(- È¹ ºÌ)sinh( È¹ ºÅ)for Ì> Å³0, exp(- È¹ ºÅ)sinh( È¹ ºÌ)for Å> Ì³0. 8.2.2-16. Domain: 0£ ¿£ ¼,0£ À£ ,0£ Ì£ é.First boundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:« = Á1( À, Ì)at ¿=0, « = Á2( À, Ì)at ¿= ¼, « = Á3( ¿, Ì)at À=0, « = Á4( ¿, Ì)at À= , « = Á5( ¿, À)at Ì=0, « = Á6( ¿, À)at Ì= é. 1 ç.Representation oftheGreen' sfunction intheform ofadouble series:Â( ¿, À, Ì, Ã, Ä, Å)=4¼ Æ Ç¹=1 Æ Çº=1sin( É ¹¿)sin( Ê ºÀ)sin( É ¹Ã)sin( Ê ºÄ) ê ¹ º( Ì, Å),ê ¹ º( Ì, Å)= áââ ãââäsinh( È¹ ºÅ)sinh[ È¹ º( é- Ì)]È¹ ºsinh( È¹ ºé)for é³ Ì> Å³0, sinh( È¹ ºÌ)sinh[ È¹ º( é- Å)]È¹ ºsinh( È¹ ºé)for é³ Å> Ì³0,É ¹= Î Í¼, Ê º= Î Ï, È¹ º= Ð É2¹+ Ê2º. This relation canbeused toobtain twoother representations oftheGreen' sfunction bymeans of thefollowing cyclic permutations: ( ¿, Ã, ¼)ë ì ( Ì, Å, é) í î( À, Ä, ) 2 ç.Representation oftheGreen' sfunction intheform ofatriple series:Â( ¿, À, Ì, Ã, Ä, Å)=8¼ ïé Æ Ç¹=1 Æ Çº=1 Æ Ç¸=1sin( É ¹¿)sin( Ê ºÀ)sin( ð¸ Ì)sin( É ¹Ã)sin( Ê ºÄ)sin( ð¸ Å)É2¹+ Ê2º+ ð2¸,É ¹= Î Í¼, Ê º= Î Ï, ð¸= Î »é. 3 ç.Analternati verepresentation oftheGreen' sfunction intheform ofatriple series:Â( ¿, À, Ì, Ã, Ä, Å)=1 4Î Æ Ç¹=-Æ Æ Çº=-Æ Æ Ç¸=-Æ ñ1ò(1)ó ô õ-1ò(2)ó ô õ-1ò(3)ó ô õ+1ò(4)ó ô õ -1ò(5)ó ô õ+1ò(6)ó ô õ+1ò(7)ó ô õ-1ò(8)ó ô õ ö, Page552 8.2. POISSON EQUATION Ó3 Ô+ Õ(x)=0 553 whereò(1)ó ô õ= Ò( ÷- ø-2Í ¼)2+( ù- ú-2Ï )2+( Ì- û-2 ü é)2,ò(2)ó ô õ= Ò( ÷+ ø-2Í ¼)2+( ù- ú-2Ï )2+( Ì- û-2 ü é)2,ò(3)ó ô õ= Ò( ÷- ø-2Í ¼)2+( ù+ ú-2Ï )2+( Ì- û-2 ü é)2,ò(4)ó ô õ= Ò( ÷+ ø-2Í ¼)2+( ù+ ú-2Ï )2+( Ì- û-2 ü é)2,ò(5)ó ô õ= Ò( ÷- ø-2Í ¼)2+( ù- ú-2Ï )2+( Ì+ û-2 ü é)2,ò(6)ó ô õ= Ò( ÷+ ø-2Í ¼)2+( ù- ú-2Ï )2+( Ì+ û-2 ü é)2,ò(7)ó ô õ= Ò( ÷- ø-2Í ¼)2+( ù+ ú-2Ï )2+( Ì+ û-2 ü é)2,ò(8)ó ô õ= Ò( ÷+ ø-2Í ¼)2+( ù+ ú-2Ï )2+( Ì+ û-2 ü é)2. 8.2.2-17. Domain: 0£ ÷£ ¼,0£ ù£ ,0£ Ì£ é.Third boundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:× Ø ý - ü1 ý = þ1( ù, Ì)at ÷=0, × Ø ý + ü2 ý = þ2( ù, Ì)at ÷= ¼, × Ù ý - ü3 ý = þ3( ÷, Ì)at ù=0, × Ù ý + ü4 ý = þ4( ÷, Ì)at ù= , × Úý - ü5 ý = þ5( ÷, ù)at Ì=0, × Úý + ü6 ý = þ6( ÷, ù)at Ì= é. Green' sfunction: ÿ ( ÷, ù, Ì, ø, ú, û)= ó=1 ô=1 =1 ó( ÷) ó( ø) ô( ù) ô( ú) ( ) ( û)ó 2 ô 2 ( û) 2( 2ó+ 2ô+ 2),ó( ÷)=cos( ó÷)+ ü1 ósin( ó÷), ó 2= ü2 2 2ó 2ó+ ü2 12ó+ ü2 2+ ü1 2 2ó+ 2 ñ 1+ ü2 12óö, ô( ù)=cos( ôù)+ ü3 ôsin( ôù), ô 2= ü4 2 2ô 2ô+ ü2 3 2ô+ ü2 4+ ü3 2 2ô+ 2 ñ 1+ ü2 3 2ôö, ( )=cos( )+ ü5 sin( ), 2= ü6 2 2 2+ ü2 5 2+ ü2 6+ ü5 2 2+ 2 ñ 1+ ü2 5 2ö, where the ó, ô,and arepositi veroots ofthetranscendental equations tan( )= ü1+ ü22- ü1 ü2,tan( ) = ü3+ ü4 2- ü3 ü4,tan( ) = ü5+ ü6 2- ü5 ü6. 8.2.2-18. Domain: 0£ ÷£,0£ ù£ ,0£ £.Mixedboundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:ý = þ1( ù, )at ÷=0, × Ø ý = þ2( ù, )at ÷=, ý = þ3( ÷, )at ù=0, × ý = þ4( ÷, )at ù= , ý = þ5( ÷, ù)at =0, × ý = þ6( ÷, ù)at =. Page553 554 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Solution:ý ( ÷, ù, )= 0 0 0 ( ø, ú, û) ÿ ( ÷, ù, , ø, ú, û) û ú ø + 0 0 þ1( ú, û) ××ø ÿ ( ÷, ù, , ø, ú, û) =0 + 0 0 2( , ) ( , , ,, , ) + 0 0 3( , ) ×× ( , , , , , ) =0 + 0 0 4( , ) ( , , , , , ) + 0 0 5( , ) ×× ( , , , , , ) =0 + 0 0 6( , ) ( , , , , ,) . 1 !.Adouble-series representation oftheGreen' sfunction:( , , , , , )=4 " $# =0 " %=0sin( & #)sin( ' %)sin( & #)sin( ' %) ê #%( , ),ê #%( , )= ()) *))+sinh( , #%)cosh[ , #%(- )], #%cosh( , #%)for³ > ³0, sinh( , #%)cosh[ , #%(- )], #%cosh( , #%)for³ > ³0,& # = -(2 .+1) 2, ' %= -(2 /+1) 2 , , #%= 0 &2 # + '2%. This relation canbeused toobtain twoother representations oftheGreen' sfunction bymeans of thefollowing cyclic permutations: ( , ,)1 2 ( , ,) 3 4( , , ) 2!.Atriple series representation oftheGreen' sfunction:( , , , , , )=8 " $# =0 " %=0 " 65 =0sin( & #)sin( ' %)sin( 7 5)sin( & #)sin( ' %)sin( 7 5)&2 # + '2%+ 72 5 ,& # = -(2 .+1) 2, ' %= -(2 /+1) 2 , 7 5 = -(2 8+1) 2. 8.2.3. Problems inCylindrical Coor dinates Thethree-dimensional Poisson equation inthecylindrical coordinate system iswritten as 1 9 :: 9 ; 9: <: 9 =+1 9 2 :2<: 2+ :2<: 2=-( 9 , , ), 9 = > 2+ 2. 8.2.3-1. Domain: 0£ 9 £ ?,0£ @£2-,- A< B< A.First boundary value problem. Anin®nite circular cylinder isconsidered. Aboundary condition isprescribed:<=( @, B)at 9 = ?. Solution:<( 9 , @, B)=- ? C2 D 0 C"-" ( , ) E :: ( 9 , @, B, , , ) = F G G + C F 0 C2 D 0 C"-" H( , , ) ( 9 , @, B, , , ) G G G . Page554 8.2. POISSON EQUATION I3 J+ K(x)=0 555 Green' sfunction:( 9 , @, B, , , L)=1 2- ?2" M$N =0 O MP=1 Q N R N ( S NP 9 ) R N ( S NP T)U R VN ( S NP?) W2S NPcos[ X( @- Y)]exp Z- S NP| B- L| [, whereQ0=1andQ N =2for X¹0;the R N ( T)aretheBessel functions; andthe S NParepositi ve roots ofthetranscendental equation R N ( S ?)=0. 8.2.3-2. Domain: 0£ \£ ?,0£ @£2 ],- A< B< A.Third boundary value problem. Anin®nite circular cylinder isconsidered. Aboundary condition isprescribed:^ _a`+ b `=( @, B)at \= ?. Solution:`( \, @, B)= ? C2 D 0 CO-O ( Y, L) c( \, @, B, ?, Y, L)G LG Y + C F 0 C2 D 0 CO-O H( T, Y, L) c( \, @, B, T, Y, L) TG LG YG T. Green' sfunction:c( \, @, B, T, Y, L)=1 2 ]O M$N =0 O MP=1 Q NS NP R N ( S NP\) R N ( S NP T)cos[ X( @- Y)] ( S2 NP?2+ b2?2- X2) R 2 N ( S NP?)exp Z- S NP| B- L| [, whereQ0=1andQ N =2for X¹0;the R N ( T)aretheBessel functions; andthe S NParepositi ve roots ofthetranscendental equationS R VN ( S ?)+ b R N ( S ?)=0. 8.2.3-3. Domain: 0£ \£ ?,0£ @£2 ],0£ B< A.First boundary value problem. Asemiin®nite circular cylinder isconsidered. Boundary conditions areprescribed:`=1( @, B)at \= ?, `=2( \, @)at B=0. Solution:`( \, @, B)=- ? C2 D 0 CO0 1( Y, L) E ^^T c( \, @, B, T, Y, L) d e = F G LG Y + C2 D 0 C F 0 2( T, Y) E ^^L c( \, @, B, T, Y, L) d f =0 TG TG Y + C F 0 C2 D 0 CO0 H( T, Y, L) c( \, @, B, T, Y, L) TG LG YG T. Green' sfunction:c( \, @, B, T, Y, L)=1 2 ] ?2O MN =0 O MP=1 Q N R N ( S NP\) R N ( S NP T)U R VN ( S NP?) W2S NPcos[ X( @- Y)] g NP( B, L),g NP( B, L)=exp(- S NP| B- L|)-exp(- S NP| B+ L|),Q N = h1for X=0, 2for X¹0, where the S NParepositi veroots ofthetranscendental equation R N ( S ?)=0. Page555 556 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.2.3-4. Domain: 0£ \£ ?,0£ @£2 ],0£ B< A.Third boundary value problem. Asemiin®nite circular cylinder isconsidered. Boundary conditions areprescribed:^ _i`+ b1 `=1( @, B)at \= ?, ^ jk`- b2 `=2( \, @)at B=0. Solution:`( \, @, B)= ? C2 D 0 CO0 1( Y, L) c( \, @, B, ?, Y, L)G LG Y - C2 D 0 C F 0 2( T, Y) c( \, @, B, ?, Y,0) TG TG Y + C F 0 C2 D 0 CO0 H( T, Y, L) c( \, @, B, T, Y, L) TG LG YG T. Green' sfunction:c( \, @, B, T, Y, L)=1]O M$N =0 O MP=1Q NS2 NP R N ( S NP\) R N ( S NP T)cos[ X( @- Y)] ( S2 NP?2+ b2 1 ?2- X2) R 2 N ( S NP?) g NP( B, L),Q N = h1for X=0, 2for X¹0, g NP( B, L)= lmm nmmoexp(- S NPB) US NPcosh( S NPL)+ b2sinh( S NPL) WS NP( S NP+ b2)for B> L, exp(- S NPL) US NPcosh( S NPB)+ b2sinh( S NPB) WS NP( S NP+ b2)for L> B, where the R N ( T)aretheBessel functions andthe S NParepositi veroots ofthetranscendental equationS R VN ( S ?)+ b1 R N ( S ?)=0. 8.2.3-5. Domain: 0£ \£ ?,0£ @£2 ],0£ B< A.Mixedboundary value problem. Asemiin®nite circular cylinder isconsidered. Boundary conditions areprescribed:`=1( @, B)at \= ?, ^ jk`=2( \, @)at B=0. Solution:`( \, @, B)=- ? C2 D 0 CO0 1( Y, L) E ^^T c( \, @, B, T, Y, L) d e = F G LG Y - C2 D 0 C F 0 2( T, Y) c( \, @, B, T, Y,0) TG TG Y + C F 0 C2 D 0 CO0 H( T, Y, L) c( \, @, B, T, Y, L) TG LG YG T. Green' sfunction:c( \, @, B, T, Y, L)=1 2 ] ?2O M$N =0 O MP=1 Q N R N ( S NP\) R N ( S NP T)U R VN ( S NP?) W2S NPcos[ X( @- Y)] g NP( B, L),g NP( B, L)=exp(- S NP| B- L|)+exp(- S NP| B+ L|),Q N = h1for X=0, 2for X¹0, where the R N ( T)aretheBessel functions andthe S NPareroots ofthetranscendental equation R N ( S ?)=0.pParagraphs 8.2.3-6 through 8.3.3-10 present only Green'sfunctions; thecomplete solution is constructed with theformulas given inSubsection 8.2.1. Seealso Paragraphs 8.3.1-4 and8.3.1-8 for q=0. Page556 8.2. POISSON EQUATION I3 J+ K(x)=0 557 8.2.3-6. Domain: 0£ \£ ?,0£ @£2 ],0£ B£ r.First boundary value problem. Acircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:`=1( @, B)at \= ?, `=2( \, @)at B=0, `=3( \, @)at B= r. Adouble series representation oftheGreen' sfunction:c( \, @, B, T, Y, L)=1] ?2O MN =0 O MP=1 Q N R N ( S NP\) R N ( S NP T)U R VN ( S NP?) W2S NPsinh( S NPr)cos[ X( @- Y)] g NP( B, L),g NP( B, L)= hsinh( S NPL)sinh[ S NP( r- B)]for r³ B> L³0, sinh( S NPB)sinh[ S NP( r- L)]for r³ L> B³0,Q N = h1for X=0, 2for X¹0, where the R N ( T)aretheBessel functions (the prime denotes thederivativewith respect tothe argument) andthe S NParepositi veroots ofthetranscendental equation R N ( S ?)=0. Atriple series representation oftheGreen' sfunction:c( \, @, B, T, Y, L)=2 r] ?2O MN =0 O MP=1 O M6s =1 Q N [ R VN ( S NP t)]2 U ( r S NP)2+( ] b)2W R N ( S NP\) R N ( S NP T) ´cos[ X( @- Y)]sin u b ] vr wsin u b ] Lr w. 8.2.3-7. Domain: 0£ \£ t,0£ x£2 ],0£ v£ r.Third boundary value problem. Acircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:^ _i`+ b1 `=1( x, v)at \= t,^ j `- b2 `=2( \, x)at v=0,^ j `+ b3 `=3( \, x)at v= r. Green' sfunction:c( \, x, v, T, Y, L)=1]O M$N =0 O MP=1 O Mzy =1 Q NS2 NP R N ( S NP\) R N ( S NP T)cos[ X( x- Y)] { y ( v) { y ( L) ( S2 NP t2+ b2 1 t2- X2)( S2 NP+ q2 y )[ R N ( S NP t)]2 |{ y|2,{ y ( v)=cos( q yv)+ b2q y sin( q yv), |{ y|2= b3 2 q2 yq2 y + b2 2q2 y + b2 3+ b2 2 q2 y + r 2 u1+ b2 2q2 yw. Here,Q0=1andQ N =2for X¹0;the R N ( T)aretheBessel functions; andthe S NPand q y are positi veroots ofthetranscendental equationsS R VN ( S t)+ b1 R N ( S t)=0,tan( q r)q= b2+ b3q2- b2 b3. 8.2.3-8. Domain: 0£ \£ t,0£ x£2 ],0£ v£ r.Mixedboundary value problem. Acircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:`=1( x, v)at \= t, ^ jk`=2( \, x)at v=0, ^ j `=3( \, x)at v= r. Green' sfunction:c( \, x, v, T, Y, L)= r] t2O MN =0 O MP=1 O M6s =1 Q NQ s [ R VN ( S NP t)]2 U ( r S NP)2+( ] b)2W R N ( S NP\) R N ( S NP T) ´cos[ X( x- Y)]cos u b ] vr wcos u b ] Lr w, whereQ0=1andQ N =2for X¹0;the R N ( T)aretheBessel functions (the prime denotes the derivativewith respect totheargument); andthe S NParepositi veroots ofthetranscendental equation R N ( S t)=0. Page557 558 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.2.3-9. Domain: 0£ \£ t,0£ x£ x0,0£ v£ r.First boundary value problem. Acylindrical sector of®nite thickness isconsidered. Boundary conditions areprescribed:`=1( \, v)at x=0, `=2( \, v)at x= x0, `=3( x, v)at \= t,`=4( \, x)at v=0, `=5( \, x)at v= r. Green' sfunction:c( \, x, v, T, Y, L)=8 rt2x0O MN =1 O MP=1 O M6s =1 RN } ~ 0( S NP\) RN } ~ 0( S NP T) [ R VN } ~ 0( S NP t)]2 U ( r S NP)2+( ] b)2W ´sin u X ] xx0 wsin u X ] Yx0 wsin u b ] vr wsin u b ] Lr w, where the RN } ~ 0( \)aretheBessel functions andthe S NParepositi veroots ofthetranscendental equation RN } ~ 0( S t)=0. 8.2.3-10. Domain: 0£ \£ t,0£ x£ x0,0£ v£ r.Mixedboundary value problem. Acylindrical sector of®nite thickness isconsidered. Boundary conditions areprescribed:`=1( \, v)at x=0, `=2( \, v)at x= x0, `=3( x, v)at \= t,^ jk`=4( \, x)at v=0, ^ j `=5( \, x)at v= r. Green' sfunction:c( \, x, v, T, Y, L)=4 rt2x0O M$N =1 O MP=1 O M s =0 Q sRN } ~ 0( S NP\) RN } ~ 0( S NP T) [ R VN } ~ 0( S NP t)]2 U ( r S NP)2+( ] b)2W ´sin u X ] xx0 wsin u X ] Yx0 wcos u b ] vr wcos u b ] Lr w, whereQ0=1andQ s =2for b¹0;the RN } ~ 0( \)aretheBessel functions; andthe S NParepositi ve roots ofthetranscendental equation RN } ~ 0( S t)=0. 8.2.4. Problems inSpherical Coor dinates Thethree-dimensional Poisson equation inthespherical coordinate system iswritten as 1\2 ^^\ u \2 ^ `^\ w+1\2sin ^^ usin ^ `^ w+1\2sin2 ^2 `^x2=- ( \, , x), \= 2+ 2+ v2.pOnly Green'sfunctions arepresented below; thecomplete solutions canbeconstructed with the formulas given inSubsection 8.2.1. 8.2.4-1. Domain: 0£ \£ t,0£ £ ],0£ x£2 ].First boundary value problem. Aspherical domain isconsidered. Aboundary condition isprescribed:`=( x, )at \= t. Green' sfunction:c( \, , x, T, Y, )=1 4 ] \2-2 \ Tcos + T2-1 4 ] \2T2-2 t2\ Tcos + t4, cos =cos cos Y+sin sin Ycos( x- ). Page558 8.2. POISSON EQUATION 3 + (x)=0 559 Analternati verepresentation oftheGreen' sfunction:c(r,r0)=1 4 ]1 |r-r0|-1 4 ] t\0|( t \0)2r0-r|, \0=|r0|, where r={ , , v}, = \sin cos x, = \sin sin x, v= \cos r0={ 0, 0, v0}, 0= Tsin Ycos , 0= Tsin Ysin , v0= Tcos Y. Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 8.2.4-2. Domain: 0£ \£ t,0£ £ ],0£ x£2 ].Second boundary value problem. Aspherical domain isconsidered. Aboundary condition isprescribed:^ _i`=( x, )at \= t. Green' sfunction:c( \, , x, T, Y, )=1 4 ] h1 |r-r0|+ t |r0||r1|+1tln2 t2t2+|r0||r1|-(r×r0) , where |r-r0|= \2-2 \ Tcos + T2,|r0||r1|= \2T2-2 t2\ Tcos + t4, |r0|= T,(r×r0)= \ Tcos ,cos =cos cos Y+sin sin Ycos( x- ). Forasolution ofthesecond boundary value problem toexistthesolvability condition must be satis®ed (seeParagraph 8.2.1-2). Refer ence:N.S.Koshlyak ov,E.B.Gliner ,andM.M.Smirno v(1970). 8.2.4-3. Domain: 0£ \£ t,0£ £ ],0£ x£2 ].Third boundary value problem. Aspherical domain isconsidered. Aboundary condition isprescribed:^ _i`+ b `=( , x)at \= t. Green' sfunction:c( \, , x, T, Y, )=1 2 ] \ TO =0 O =1 zy =0 y y +1 ~ 2( ) +1 ~ 2( ) ´ (cos ) (cos Y)cos[ ( - )], = ¡1for =0, 2for ¹0, =(2 ¢+1)( ¢- )! ( ¢+ )! £¤22 +( ¥ ¤+ ¢)( ¥ ¤- ¢-1) ¦ £ +1 §2( ¤) ¦2. Here, the +1 §2( )aretheBessel functions, the ( ¨)aretheassociated Legendre functions that areexpressed interms oftheLegendre polynomials ( ¨)as ( ¨)=(1- ¨2) §2 © © ¨ ( ¨), ( ¨)=1¢!2 © © ¨ ( ¨2-1) , andthe arepositi veroots ofthetranscendental equation ¤ ª +1 §2( ¤)+ Z«¥ ¤-1 2 ¬ +1 §2( ¤)=0. Page559 560 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.2.4-4. Domain: ¤£ < ,0£ ®£ ¯,0£ £2 ¯.First boundary value problem. Three-dimensional space with aspherical cavityisconsidered. Aboundary condition isprescribed:°= ±( , ®)at = ¤. TheGreen' sfunction oftheouter ®rstboundary value problem isgivenbythesame relation as thatfortheinner ®rstboundary value problem (seeParagraph 8.2.4-1), except that ³ ¤and ³ ¤. 8.2.4-5. Domain: ¤£ < ,0£ ®£ ¯,0£ £2 ¯.Second boundary value problem. Three-dimensional space with aspherical cavityisconsidered. Aboundary condition isprescribed:² ³°= ±( , ®)at = ¤. Green' sfunction:´( , ®, , , Y, µ)=1 4 ¯ ¡1 |r-r0|+ ¤ |r0||r1|+1¤ln(1-cos ¶)|r||r0|¤2+|r0||r1|-(r×r0) ·, where |r|= ,|r0|= ,|r-r0|= ¸ 2-2 cos ¶+ 2,|r0||r1|= ¸ 22-2 ¤2 cos ¶+ ¤4, (r×r0)= cos ¶,cos ¶=cos ®cos ¹+sin ®sin ¹cos( - µ).º» Refer ence:N.S.Koshlyak ov,E.B.Gliner ,andM.M.Smirno v(1970). 8.2.4-6. Domain: ¤1£ £ ¤2,0£ ®£ ¯,0£ £2 ¯.First boundary value problem. Aspherical layer isconsidered. Boundary conditions areprescribed:°= ±1( ®, )at = ¤1, °= ±2( ®, )at = ¤2. Green' sfunction:´( , ®, , , ¹, µ)= ¯ 8 ¼ O =0 O =1 6½ =0 ½ ½ ¾ +1 §2( ) ¾ +1 §2( ) ´ ½ (cos ®) ½ (cos ¹)cos[ ¥( - µ)], where ¾ +1 §2( )= +1 §2( ¤1) ¿ +1 §2( )- ¿ +1 §2( ¤1) +1 §2( ), ½ = ¡1for ¥=0, 2for ¥¹0, ½ =(2 ¢+1)( ¢- ¥)! 2 +1 §2( ¤2) ( ¢+ ¥)! £ 2 +1 §2( ¤1)- 2 +1 §2( ¤2) ¦; the +1 §2( )aretheBessel functions, the ½ ( ¨)aretheassociated Legendre functio ns(seeParagraph 8.2.4-3 ),andthe arepositi veroots ofthetranscendental equation ¾ +1 §2( ¤2)=0. 8.2.4-7. Domain: 0£ £ ¤,0£ ®£ ¯ À2,0£ £2 ¯.First boundary value problem. Ahemisphere isconsidered. Boundary conditions areprescribed:°= ±1( , ®)at = ¤, °= ±2( , )at ®= ¯ À2. Green' sfunction inthespherical coordinate system:´( , ®, , , ¹, µ)= ´ s( , ®, , , ¹, µ)- ´ s( , ®, , , ¯- ¹, µ), where ´ s( , ®, , , ¹, µ)istheGreen' sfunctions forasphere; seeParagraph 8.2.4-1, where ´must bereplaced by ´ s. Green' sfunction intheCartesian coordinate system:´( Á, Â, Ã, Á0, Â0, Ã0)=1 4 ¯ Ä1 |r-r0|- ¤ |r0||r-r Å0| Æ-1 4 ¯ Ä1 |r-r1|- ¤ |r0||r-r Å1| Æ, r={ Á, Â, Ã},r0={ Á0, Â0, Ã0},r1={ Á0, Â0,- Ã0},r Å ½ =( ¤ À 0)2r ½ , ¥=0,1.º» Refer ences :V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). Page560 8.3. HELMHOL TZEQUATION 3 + =- (x) 561 8.2.4-8. Domain: 0£ £ ,0£ £ 2,0£ £ .First boundary value problem. Aquarter ofasphere isconsidered. Boundary conditions areprescribed: = 1( , )at = , = 2( , )at = 2, = 3( , )at =0, = 4( , )at = . Green' sfunction inthespherical coordinate system:( , , , , , )= s( , , , , , )- s( , , , , - , ) + s( , , , , - ,2 - )- s( , , , , ,2 - ), where s( , , , , , )istheGreen' sfunction forasphere; seeParagraph 8.2.4-1, where must bereplaced by s. Green' sfunction intheCartesian coordinate system:( , , , 0, 0, 0)=1 4 1, =0(-1) + 1 |r-r |- |r0||r-r | , r={ , , },r0={ 0, 0, 0},r ={ 0,(-1) 0,(-1) 0},r =( 0)2r , where 0=|r0|; =0,1; =0,1. Refer ences :V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 8.3. Helmholtz Equation 3 + =± !(x) Avariety ofproblems related tosteady-state oscillations (mechanical, acoustic, thermal, electro- magnetic, etc.) lead tothethree-dimensional Helmholtz equation with ">0.This equation governs mass transfer phenomena with volume chemical reaction ofthe®rstorder for "<0.Anyelliptic equation with constant coef®cients canbereduced totheHelmholtz equation. 8.3.1. General Remarks, Results, and Form ulas 8.3.1-1. Some de®nitions. TheHelmholtz equation iscalled homogeneous if #=0andnonhomogeneous if #¹0.Ahomo- geneous boundary value problem isaboundary value problem forahomogeneous equation with homogeneous boundary conditions; =0isaparticular solution ofahomogeneous boundary value problem. Thevalues " oftheparameter "forwhich there arenontri vialsolutions (i.e., notidentically zero solutions) ofahomogeneous boundary value problem arecalled eigen values. Thecorresponding solutions, = ,arecalled eigenfunctions ofthisboundary value problem. Inwhat follows,weconsider simultaneously the®rst, second, andthird boundary value prob- lems forthethree-dimensional Helmholtz equation ina®nite three-dimensional domain $with a suf®ciently smooth surface %.Itisassumed that >0forthethird boundary value problem with theboundary condition & & '+ =0for r ( %, where ) *) +isthederivativealong theoutw ardnormal tothesurface %,andr={ , , }. Page561 562 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.3.1-2. Properties ofeigen values andeigenfunctions. 1 ,.There arein®nitely manyeigen values { " };theyform adiscrete spectrum oftheboundary value problem. 2 ,.Alleigen values arepositi ve,except foroneeigen value "0=0ofthesecond boundary value problem (thecorresponding eigenfunction is 0=const). Theeigen values areassumed tobeordered sothat "1< "2< "3< -.-.-. 3 ,.Theeigen values tend toin®nity asthenumber increases. Thefollowing asymptotic estimate holds: lim / 0 "3 12= $3 6 2, where $3isthevolume ofthedomain under consideration. 4 ,.Theeigenfunctions arede®ned uptoaconstant multiplier .Anytwoeigenfunctions, and 2, thatcorrespond todifferent eigen values " ¹ " 2areorthogonal, thatis,3 4 2 5$=0. 5 ,.Anytwice continuously differentiable function = (r)thatsatis®es theboundary conditions of aboundary valueproblem canbeexpanded intoauniformly convergent series intheeigenfunctions ofthisboundary value problem, speci®cally ,= 0=1 6 , where6 =17 72 34 5$, 7 72= 34 2 5$. If issquare summable, then theseries isconvergent inmean. 6 ,.Theeigen values ofthe®rstboundary value problem donotincrease ifthedomain isextended.8 9;: < =?> @ AInathree-dimensional problem, toeach eigen value " ®nitely manylinearly inde- pendent eigenfunctions (1), B.B.B, ( C)generally correspond. These functions canalwaysbereplaced bytheir linear combinations Å ( D)= ED,1 (1)+ -.-.-+ ED, D-1 ( D-1)+ ( D), F=1,2, B.B.B, G, such that Å (1), B.B.B,Å ( C)arenowpairwise orthogonal. Therefore, without loss ofgenerality ,we assume thatalleigenfunctions areorthogonal. Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1984). 8.3.1-3. Nonhomogeneous Helmholtz equation with homogeneous boundary conditions. Three cases arepossible. 1 ,.If "isnotequal toanyoneoftheeigen values, then thesolution oftheproblem isgivenby = 0=1 H " - " ,whereH =17 72 34# 5$, 7 72= 34 2 5$. 2 ,.If "coincides with oneoftheeigen values, "= " 2,then thecondition oftheorthogonality of thefunction #totheeigenfunction 2,34# 2 5$=0, isanecessary condition forasolution ofthenonhomogeneous problem toexist. Thesolution is then givenby = 2 -1=1H " - " 2 + 0= 2 +1H " - " 2 + I 2,H =17 72 34# 5$, where Iisanarbitrary constant and 7 72= J 4 2 5$. Page562 8.3. HELMHOL TZEQUATION 3 + =- (x) 563 3 ,.If "= " 2and J 4# 2 5$¹0,then theboundary value problem forthenonhomogeneous equation hasnosolution.8 9;: < =?> K AIftoeach eigen value " there arecorresponding G mutually orthogonal eigen- functions ( D)( F=1, B.B.B, G ),then thesolution iswritten as = 0=1 C.LD=1H( D)" - " ( D),whereH( D)=17 ( D) 72 34# ( D) 5$, 7 ( D) 72= 34 M ( D) N25$, provided that "¹ " . Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1984). 8.3.1-4. Solution ofnonhomogeneous boundary value problems ofgeneral form. 1 ,.Thesolution ofthe®rstboundary value problem fortheHelmholtz equation with theboundary condition = (r)for r ( % canberepresented intheform (r)= 34#( O) (r, O) 5$ P- 3 Q( O) && 'P (r, O) 5% P. (1) Here, r={ , , }, O={ , , }(r ( $, O ( $);)) + Rdenotes thederivativealong theoutw ard normal tothesurface %with respect to , , .TheGreen' sfunction isgivenbytheseries(r, O)= 0=1 (r) ( O)7 72( " - "), "¹ " , (2) where the and " aretheeigenfunctions andeigen values ofthehomogeneous ®rst boundary value problem. 2 ,.Thesolution ofthesecond boundary value problem with theboundary condition& & '= (r)for r ( % canberepresented intheform (r)= 34#( O) (r, O) 5$ P+ 3 Q( O) (r, O) 5% P. (3) Here, theGreen' sfunction isgivenbytheseries(r, O)=-1$3 "+ 0=1 (r) ( O)7 72( " - "), (4) where $3isthevolume ofthethree-dimensional domain under consideration, andthe " and are thepositi veeigen values andcorresponding eigenfunctions ofthehomogeneous second boundary value problem. Forclarity ,theterm corresponding tothezero eigen value "0=0( 0=const) is singled outin(4).Itisassumed that "¹0and "¹ " . 3 ,.Thesolution ofthethird boundary valueproblem fortheHelmholtz equation with theboundary condition & & '+ = (r)for r ( % isgivenbyrelation (3)inwhich theGreen' sfunction isde®ned byseries (2)with theeigenfunc- tions andeigen values " ofthehomogeneous third boundary value problem. Page563 564 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 4 ,.Letnonhomogeneous boundary conditions ofvarious types besetondifferent portions % Sof thesurface %= 2TS=1 % S, US[ ]= S(r)for r ( % S. Then thesolution ofthecorresponding mixedboundary value problem canbewritten as (r)= 34#( O) (r, O) 5$ P+ 2S=1 3 Q V S( O) W S(r, O) 5%( S)P, whereW S(r, O)= X Y[Z- && 'P \(r, O)ifa®rst-kind boundary condition isseton % S,\(r, O) ifasecond- orthird-kind boundary condition isseton % S. TheGreen' sfunction isexpressed byseries (2)thatinvolvestheeigenfunctions ] ^andeigen val- ues " ^ofthehomogeneous mixedboundary value problem. 8.3.1-5. Boundary conditions atin®nity inthecase ofanunbounded domain. Belowitisassumed thatthefunction #is®nite orsuf®ciently rapidly decaying as _ ` a. 1 ,.If "<0andthedomain isunbounded, theadditional condition thatthesolution must vanish at in®nity isset:] `0as _ ` a. 2 ,.If ">0,theradiation conditions (Sommerfeld conditions) areoften used atin®nity .Inthree- dimensional problems, these conditions areexpressed as limb;/ 0 _ ]=const , limb;/ 0 _ c d ]d _+ egf h ] i=0, where e2=-1. The principle oflimit absorption andtheprinciple oflimit amplitude arealso emplo yedto separate asingle solution.j Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). 8.3.1-6. Green' sfunction foranin®nite cylindrical domain ofarbitrary cross-section. Consider thethree-dimensional Helmholtz equationk 3 ]+ h ]=- l(r) (5) inside anin®nite cylindrical domain m={( n, o) p q,- a< r< a}with arbitrary cross-section q. Onthesurfaceofthisdomain, let s={( n, o) p t,- a< r< a},where tistheboundary of q, thehomogeneous boundary condition ofgeneral formud vd '+ wv=0for r p s (6) beset,with uw³0.Byappropriately choosing theconstants uand win(6),onecanobtain boundary conditions ofthe®rst( u=0, w=1),second ( u=1, w=0),andthird ( uw¹0)kind. The Green' sfunction ofthe®rst orthird boundary value problem canberepresented inthe form* x ( n, o, r, y, z, {)=1 2 | }~ =1 ~ ( n, o) ~ ( y, z) ~2f ~ - h -f - | - |, ~2= 2 ~ ( n, o) n o,(7) *InParagraphs 8.3.1-6 through 8.3.1-8, thecross-section isassumed tohave®nite dimensions. Page564 8.3. HELMHOL TZEQUATION 3 + =- (x) 565 where the ~ and ~ aretheeigen values andeigenfunctions ofthecorresponding two-dimensional boundary value problem in q,k 2+=0 for( n, o) p q,u + w=0 for( n, o) p t.(8) Recall thatall ~ arepositi ve. Inthesecond boundary value problem, thezero eigen value0=0appears, andhence the summation in(7)must start with =0.Inthiscase,0=1and 0 2= q2,where q2isthearea ofthecross-section q.j Refer ences :B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980), A.N.Tikhono vandA.A.Samarskii (1990). 8.3.1-7. Green' sfunction forasemiin®nite cylindrical domain. 1 .TheGreen' sfunction ofthethree-dimensional ®rstboundary value problem forequation (5)in asemiin®nite cylindrical domain m={( n, o) p q,0£ r< }with arbitrary cross-section qis givenby x ( n, o, r, y, z, {)=| }~ =1 ~ ( n, o) ~ ( y, z) ~2 ~ ( r, {), (9) where ~ ( r, {)=1 2 ~ exp(- ~ | r- {|)-exp(- ~ | r+ {|) = 1 ~ exp(- ~r)sinh( ~{)for r> {³0, 1 ~ exp(- ~{)sinh( ~r)for {> r³0, ~ = ~ - .(10) Relations (9)and(10) involvetheeigenfunctions ~ andeigen values ~ ofthetwo-dimensional ®rstboundary value problem (8)with u=0and w=1. 2 .TheGreen' sfunction ofthethree-dimensional second boundary value problem forequation (5) inasemiin®nite cylindrical domain m={( n, o) p q,0£ r< }with arbitrary cross-section qis givenby x ( n, o, r, y, z, {)=1q2 0( r, {)+| }~ =1 ~ ( n, o) ~ ( y, z) ~2 ~ ( r, {), (11) where ~ ( r, {)=1 2 ~ exp(- ~ | r- {|)+exp(- ~ | r+ {|) = 1 ~ exp(- ~r)cosh( ~{)for r> {³0, 1 ~ exp(- ~{)cosh( ~r)for {> r³0, ~ = ~ - .(12) Relations (11) and(12) involvetheeigenfunctions ~ andeigen values ~ ofthetwo-dimensional second boundary value problem (8)with u=1and w=0.Note thatin(11) theterm corresponding tothezero eigen value0=0isspecially singled out; q2isthearea ofthecross-section q. Page565 566 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 3 .TheGreen' sfunction ofthethree-dimensional third boundary value problem forequation (5) with theboundary conditionsv r- w1v=0for r=0, v + w2v=0for r p s inasemiin®nite cylindrical domain m={( n, o) p q,0£ r< }with arbitrary cross-section q andlateral surface sisgivenbyrelation (9)with ~ ( r, {)= exp(- ~r) ~ cosh( ~{)+ w1sinh( ~{) ~ ( ~ + w1)for r> {³0, exp(- ~{) ~ cosh( ~r)+ w1sinh( ~r) ~ ( ~ + w1)for {> r³0, ~ = ~ - . (13) Relations (9)and(13) involvetheeigenfunctions ~ andeigen values ~ ofthetwo-dimensional third boundary value problem (8)with u=1and w= w2. 4 .TheGreen' sfunction ofthethree-dimensional mixedboundary value problem forequation (5) with asecond-kind boundary condition attheendfaceanda®rst-kind boundary condition atthe lateral surfaceisgivenbyrelations (9)and(12), where the ~ and ~ aretheeigen values and eigenfunctions ofthetwo-dimensional ®rstboundary value problem (8)with u=0and w=1. TheGreen' sfunctions ofother mixedboundary value problems canbeconstructed likewise. 8.3.1-8. Green' sfunction foracylindrical domain of®nite dimensions. 1 .TheGreen' sfunction ofthethree-dimensional ®rstboundary value problem forequation (5)in acylindrical domain of®nite dimensions m={( n, o) p q,0£ r£ }with arbitrary cross-section q isgivenbyrelation (9)with ~ ( r, {)= sinh( ~{)sinh[ ~ ( - r)] ~ sinh( ~ )for ³ r> {³0, sinh( ~r)sinh[ ~ ( - {)] ~ sinh( ~ )for ³ {> r³0, ~ = ~ - . (14) Relations (9)and(14) involvetheeigenfunctions ~ andeigen values ~ ofthetwo-dimensional ®rstboundary value problem (8)with u=0and w=1. Another representation oftheGreen' sfunction:x ( n, o, r, y, z, {)=2 | }~ =1 | }¡=1 ~ ( n, o) ~ ( y, z)sin( ¢ ¡r)sin( ¢ ¡{) ~2( ~ + ¢2¡- ), ¢ ¡= £ ¤ . Itisaconsequence offormula (2). 2 .TheGreen' sfunction ofthethree-dimensional second boundary value problem forequation (5) inacylindrical domain of®nite dimensions ¥={( ¦, §) ¨ ©,0£ ª£ }with arbitrary cross-section © isgivenbyrelation (11) with ~ ( ª, {)= cosh( ~{)cosh[ ~ ( - ª)] ~ sinh( ~ )for ³ ª> {³0, cosh( ~ª)cosh[ ~ ( - {)] ~ sinh( ~ )for ³ {> ª³0, ~ = ~ - . (15) Relations (11) and(15) involvetheeigenfunctions ~ andeigen values ~ ofthetwo-dimensional second boundary value problem (8)with «=1and ¬=0. Page566 8.3. HELMHOL TZEQUATION 3 + =- (x) 567 Another representation oftheGreen' sfunction:x ( ¦, §, ª, y, z, {)=1 | }~ =0 | }¡=0 ¡ ~ ( ¦, §) ~ ( y, z)cos( ¢ ¡ª)cos( ¢ ¡{) ~2( ~ + ¢2¡- ),¢ ¡= £ ¤ , ¡= ®1for¤=0, 2for¤¹0, 0=0,0=1. Itisaconsequence offormula (4).¯ Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 3 .TheGreen' sfunction ofthethree-dimensional third boundary value problem forequation (5) with theboundary conditions ° ª- ¬1°=0at ª=0, ° ª+ ¬2°=0at ª= , ° + ¬3°=0for r ¨ ± inacylindrical domain of®nite dimensions ¥={( ¦, §) ¨ ©,0£ ª£ }with arbitrary cross-section © andlateral surface ±isgivenbyrelation (9)with ²( ª, ³)= [ ²cosh( ² ³)+ ¬1sinh( ² ³)] ´ ²cosh[ ²( - ª)]+ ¬2sinh[ ²( - ª)] µ²[ ²( ¬1+ ¬2)cosh( ² )+( 2²+ ¬1 ¬2)sinh( ² )]for ª> ³, [ ²cosh( ² ª)+ ¬1sinh( ² ª)] ´ ²cosh[ ²( - ³)]+ ¬2sinh[ ²( - ³)] µ²[ ²( ¬1+ ¬2)cosh( ² )+( 2²+ ¬1 ¬2)sinh( ² )]for ª< ³,(16)²= ¶²- (0£ ª£ ,0£ ³£ ). Relations (9)and(16) involvetheeigenfunctions ·²andeigen values ¶²ofthetwo-dimensional third boundary value problem (8)with «=1and ¬= ¬3. 4 .TheGreen' sfunction ofthethree-dimensional mixedboundary value problem forequation (5) with second-kind boundary conditions attheendfaces anda®rst-kind boundary condition atthe lateral surfaceisgivenbyrelations (9)and(15), where the ¶²and ·²aretheeigen values and eigenfunctions ofthetwo-dimensional ®rstboundary value problem (8)with «=0and ¬=1. TheGreen' sfunction ofthethree-dimensional mixedboundary value problem forequation (5) with theboundary conditions°=0for ª=0, ¸ °=0for ª= ,°=0for r ¨ ± inacylindrical domain of®nite dimensions ¥={( ¦, §) ¨ ©,0£ ª£ }with arbitrary cross-section © andlateral surface ±isgivenbyrelation (9)with ²( ª, ³)= sinh( ² ³)cosh[ ²( - ª)]²cosh( ² )for ³ ª> ³³0, sinh( ² ª)cosh[ ²( - ³)]²cosh( ² )for ³ ³> ª³0, ²= ¶²- . (17) Relations (9)and(17) involvetheeigenfunctions ·²andeigen values ¶²ofthetwo-dimensional ®rstboundary value problem (8)with «=0and ¬=1. TheGreen' sfunctions ofother mixedboundary value problems canbeconstructed likewise. 8.3.2. Problems inCartesian Coor dinates Thethree-dimensional nonhomogeneous Helmholtz equation intherectangular Cartesian system of coordinates hastheform2° ¦2+ 2° §2+ 2° ª2+ °=- ¹( ¦, §, ª). Page567 568 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.3.2-1. Particular solutions ofthehomogeneous equation ( ¹º0):°=( º1cos ¬ ¦+ º2sin ¬ ¦)( »1cos¤ §+ »2sin¤ §)( ¼1 ª+ ¼2), = ¬2+¤2;°=( º1cos ¬ ¦+ º2sin ¬ ¦)( »1cosh¤ §+ »2sinh¤ §)( ¼1 ª+ ¼2), = ¬2-¤2;°=( º1cos ¬ ¦+ º2sin ¬ ¦)( »1cos¤ §+ »2sin¤ §)( ¼1cos ª+ ¼2sin ª), = ¬2+¤2+ 2;°=( º1cosh ¬ ¦+ º2sinh ¬ ¦)( »1cos¤ §+ »2sin¤ §)( ¼1cos ª+ ¼2sin ª), =- ¬2+¤2+ 2;°=( º1cosh ¬ ¦+ º2sinh ¬ ¦)( »1cosh¤ §+ »2sinh¤ §)( ¼1cos ª+ ¼2sin ª), =- ¬2-¤2+ 2;°=( º1cosh ¬ ¦+ º2sinh ¬ ¦)( »1cosh¤ §+ »2sinh¤ §)( ¼1cosh ª+ ¼2sinh ª), =- ¬2-¤2- 2, where º1, º2, »1, »2, ¼1,and ¼2arearbitrary constants. Fundamental solutions: ½ ½ ( ¦, §, ª)=1 4£ ¾exp(- ¬¾), =- ¬2<0,½ ½ ( ¦, §, ª)=1 4£ ¾exp( ¿ ÀÁ¬¾), = ¬2>0, where¾= ¦2+ §2+ ª2, ¬>0, À2=-1. 8.3.2-2. Domain: - < ¦< ,- < §< ,- < ª< . 1 .Solution for =- ¬2<0:°( ¦, §, ª)=1 4£ Â Ã-Ã Â Ã-Ã Â Ã-Ã ¹( Ä, Å, ³)exp Æ- Ç È( É- Ä)2+( Ê- Å)2+( Ë- ³)2È( É- Ä)2+( Ê- Å)2+( Ë- ³)2 Ì ÄÌ ÅÌ ³. 2 Í.Solution for Î= Ç2>0:°( É, Ê, Ë)=1 4 ÏÂÃ-Ã ÂÃ-Ã ÂÃ-Ã ¹( Ä, Å, ³)exp Æ- ÀÁÇ È( É- Ä)2+( Ê- Å)2+( Ë- ³)2È( É- Ä)2+( Ê- Å)2+( Ë- ³)2 Ì ÄÌ ÅÌ ³. This solution wasobtained taking into account theradiation condition atin®nity (see Paragraph 8.3.1-5 ,Item 2 Í).ÐÑ Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). 8.3.2-3. Domain: - Ò< É< Ò,- Ò< Ê< Ò,0£ Ë< Ò.First boundary value problem. Ahalf-space isconsidered. Aboundary condition isprescribed:Ó= Ô( É, Ê)at Ë=0. Solution:Ó( É, Ê, Ë)=Â Ã-Ã Â Ã-Ã Ô( Ä, Å) Õ ÖÖ × Ø( É, Ê, Ë, Ä, Å,×) Ù Ú =0 Ì ÄÌ Å +Â Ã0Â Ã-Ã Â Ã-Ã Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì ÄÌ ÅÌ×. Green' sfunction for Î=- Ç2<0:Ø( É, Ê, Ë, Ä, Å,×)=exp(- Ç Ü1) 4 Ï Ü1-exp(- Ç Ü2) 4 Ï Ü2,Ü1= È( É- Ä)2+( Ê- Å)2+( Ë-×)2, Ü2= È( É- Ä)2+( Ê- Å)2+( Ë+×)2.ÐÑ Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). Page568 8.3. HELMHOL TZEQUATION Ý3 Þ+ ßÞ=- à(x) 569 8.3.2-4. Domain: - Ò< É< Ò,- Ò< Ê< Ò,0£ Ë< Ò.Second boundary value problem. Ahalf-space isconsidered. Aboundary condition isprescribed:Ö á Ó= Ô( É, Ê)at Ë=0. Solution:Ó( É, Ê, Ë)=-Â Ã-Ã Â Ã-Ã Ô( Ä, Å)Ø( É, Ê, Ë, Ä, Å,0)Ì ÄÌ Å +ÂÃ0ÂÃ-Ã ÂÃ-Ã Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì ÄÌ ÅÌ×. Green' sfunction for Î=- Ç2<0:Ø( É, Ê, Ë, Ä, Å,×)=exp(- Ç Ü1) 4 Ï Ü1+exp(- Ç Ü2) 4 Ï Ü2,Ü1= È( É- Ä)2+( Ê- Å)2+( Ë-×)2, Ü2= È( É- Ä)2+( Ê- Å)2+( Ë+×)2.ÐÑ Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 8.3.2-5. Domain: - Ò< É< Ò,0£ Ê< Ò,0£ Ë< Ò.First boundary value problem. Adihedral angle isconsidered. Boundary conditions areprescribed:Ó= Ô1( É, Ë)at Ê=0, Ó= Ô2( É, Ê)at Ë=0. Solution:Ó( É, Ê, Ë)=ÂÃ0ÂÃ-Ã Ô1( Ä,×) ÕÖÖ ÅØ( É, Ê, Ë, Ä, Å,×) Ù â =0 Ì ÄÌ× +ÂÃ0ÂÃ-Ã Ô2( Ä, Å) ÕÖÖ × Ø( É, Ê, Ë, Ä, Å,×) Ù Ú =0 Ì ÄÌ Å +ÂÃ0ÂÃ0ÂÃ-Ã Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì ÄÌ ÅÌ×. Green' sfunction for Î=- Ç2<0:Ø( É, Ê, Ë, Ä, Å,×)=exp(- Ç Ü1) 4 Ï Ü1-exp(- Ç Ü2) 4 Ï Ü2-exp(- Ç Ü3) 4 Ï Ü3+exp(- Ç Ü4) 4 Ï Ü4,Ü1= È( É- Ä)2+( Ê- Å)2+( Ë-×)2, Ü2= È( É- Ä)2+( Ê- Å)2+( Ë+×)2,Ü3= È( É- Ä)2+( Ê+ Å)2+( Ë-×)2, Ü4= È( É- Ä)2+( Ê+ Å)2+( Ë+×)2. 8.3.2-6. Domain: - Ò< É< Ò,0£ Ê< Ò,0£ Ë< Ò.Second boundary value problem. Adihedral angle isconsidered. Boundary conditions areprescribed:Ö ã Ó= Ô1( É, Ë)at Ê=0,Ö á Ó= Ô2( É, Ê)at Ë=0. Solution:Ó( É, Ê, Ë)=-ÂÃ0ÂÃ-Ã Ô1( Ä,×)Ø( É, Ê, Ë, Ä,0,×)Ì ÄÌ× -ÂÃ0ÂÃ-Ã Ô2( Ä, Å)Ø( É, Ê, Ë, Ä, Å,0)Ì ÄÌ Å +ÂÃ0ÂÃ0ÂÃ-Ã Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì ÄÌ ÅÌ×. Green' sfunction for Î=- Ç2<0:Ø( É, Ê, Ë, Ä, Å,×)=exp(- Ç Ü1) 4 Ï Ü1+exp(- Ç Ü2) 4 Ï Ü2+exp(- Ç Ü3) 4 Ï Ü3+exp(- Ç Ü4) 4 Ï Ü4,Ü1= È( É- Ä)2+( Ê- Å)2+( Ë-×)2, Ü2= È( É- Ä)2+( Ê- Å)2+( Ë+×)2,Ü3= È( É- Ä)2+( Ê+ Å)2+( Ë-×)2, Ü4= È( É- Ä)2+( Ê+ Å)2+( Ë+×)2. Page569 570 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.3.2-7. Domain: - Ò< É< Ò,- Ò< Ê< Ò,0£ Ë£ ä.First boundary value problem. Anin®nite layer isconsidered. Boundary conditions areprescribed:Ó= Ô1( É, Ê)at Ë=0, Ó= Ô2( É, Ê)at Ë= ä. Solution:Ó( É, Ê, Ë)=Â Ã-Ã Â Ã-Ã Ô1( Ä, Å) ÕÖÖ ×Ø( É, Ê, Ë, Ä, Å,×) ÙÚ =0 Ì ÄÌ Å -ÂÃ-Ã ÂÃ-Ã Ô2( Ä, Å) ÕÖÖ × Ø( É, Ê, Ë, Ä, Å,×) Ù Ú = å Ì ÄÌ Å +Â å 0ÂÃ-Ã ÂÃ-Ã Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì ÄÌ ÅÌ×. Green' sfunction for Î=- Ç2<0:Ø( É, Ê, Ë, Ä, Å,×)=Ã æç=-Ã Õexp(- Ç Ü ç1) 4 Ï Ü ç1-exp(- Ç Ü2 ç) 4 Ï Ü2 ç Ù,Ü1 ç= È( É- Ä)2+( Ê- Å)2+( Ë-×-2 è ä)2,Ü2 ç= È( É- Ä)2+( Ê- Å)2+( Ë+×-2 è ä)2. 8.3.2-8. Domain: - Ò< É< Ò,- Ò< Ê< Ò,0£ Ë£ ä.Second boundary value problem. Anin®nite layer isconsidered. Boundary conditions areprescribed:Ö á Ó= Ô1( É, Ê)at Ë=0,Ö á Ó= Ô2( É, Ê)at Ë= ä. Solution:Ó( É, Ê, Ë)=-Â Ã-Ã Â Ã-Ã Ô1( Ä, Å)Ø( É, Ê, Ë, Ä, Å,0)Ì ÄÌ Å +ÂÃ-Ã ÂÃ-Ã Ô2( Ä, Å)Ø( É, Ê, Ë, Ä, Å, ä)Ì ÄÌ Å +Â å 0ÂÃ-Ã ÂÃ-Ã Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì ÄÌ ÅÌ×. Green' sfunction for Î=- Ç2<0:Ø( É, Ê, Ë, Ä, Å,×)=Ã æç=-Ã Õexp(- Ç Ü ç1) 4 Ï Ü ç1+exp(- Ç Ü2 ç) 4 Ï Ü2 ç Ù,Ü1 ç= È( É- Ä)2+( Ê- Å)2+( Ë-×-2 è ä)2,Ü2 ç= È( É- Ä)2+( Ê- Å)2+( Ë+×-2 è ä)2.ÐÑ Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 8.3.2-9. Domain: 0£ É£ ä,0£ Ê£ é,- Ò< Ë< Ò.First boundary value problem. Anin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed:Ó= Ô1( Ê, Ë)at É=0, Ó= Ô2( Ê, Ë)at É= ä,Ó= Ô3( É, Ë)at Ê=0, Ó= Ô4( É, Ë)at Ê= é. Page570 8.3. HELMHOL TZEQUATION Ý3 Þ+ ßÞ=- à(x) 571 Solution:Ó( É, Ê, Ë)=Â ê0ÂÃ-Ã Ô1( Å,×) ÕÖÖ ÄØ( É, Ê, Ë, Ä, Å,×) Ù ë =0 Ì× Ì Å -Â ê0ÂÃ-Ã Ô2( Å,×) ÕÖÖ ÄØ( É, Ê, Ë, Ä, Å,×) Ùë = å Ì× Ì Å +Â å 0ÂÃ-Ã Ô3( Ä,×) ÕÖÖ ÅØ( É, Ê, Ë, Ä, Å,×) Ù â =0 Ì× Ì Ä -Â å 0ÂÃ-Ã Ô4( Ä,×) ÕÖÖ ÅØ( É, Ê, Ë, Ä, Å,×) Ù â =ê Ì× Ì Ä +Â å 0Â ê0Â Ã-Ã Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì× Ì ÅÌ Ä. Green' sfunction:Ø( É, Ê, Ë, Ä, Å,×)=2ä éÃ æç=1Ã æì=11íç ìsin( î çÉ)sin( ï ìÊ)sin( î çÄ)sin( ï ìÅ)exp(- íç ì| Ë-×|),î ç= è Ïä, ï ì= ð Ïé, íç ì= ñ î2ç+ ï2ì- Î.ÐÑ Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). 8.3.2-10. Domain: 0£ É£ ä,0£ Ê£ é,- Ò< Ë< Ò.Second boundary value problem. Anin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed:Ö ò Ó= Ô1( Ê, Ë)at É=0,Ö ò Ó= Ô2( Ê, Ë)at É= ä,Ö ã Ó= Ô3( É, Ë)at Ê=0,Ö ã Ó= Ô4( É, Ë)at Ê= é. Solution:Ó( É, Ê, Ë)=-Â ê0ÂÃ-Ã Ô1( Å,×)Ø( É, Ê, Ë,0, Å,×)Ì× Ì Å+Â ê0ÂÃ-Ã Ô2( Å,×)Ø( É, Ê, Ë, ä, Å,×)Ì× Ì Å -Â å 0ÂÃ-Ã Ô3( Ä,×)Ø( É, Ê, Ë, Ä,0,×)Ì× Ì Ä+Â å 0ÂÃ-Ã Ô4( Ä,×)Ø( É, Ê, Ë, Ä, é,×)Ì× Ì Ä +Â å 0Â ê0Â Ã-Ã Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì× Ì ÅÌ Ä. Green' sfunction:Ø( É, Ê, Ë, Ä, Å,×)=1 2 ä éÃ æç=0Ã æì=0 ó çó ìíç ìcos( î çÉ)cos( ï ìÊ)cos( î çÄ)cos( ï ìÅ)exp(- íç ì| Ë-×|),î ç= è Ïä, ï ì= ð Ïé, íç ì= ñ î2ç+ ï2ì- Î,ó ç= ô1for è=0, 2for è¹0.ÐÑ Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). 8.3.2-11. Domain: 0£ É£ ä,0£ Ê£ é,- Ò< Ë< Ò.Third boundary value problem. Anin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed:Ö ò Ó- Ç1 Ó= Ô1( Ê, Ë)at É=0,Ö ò Ó+ Ç2 Ó= Ô2( Ê, Ë)at É= ä,Ö ã Ó- Ç3 Ó= Ô3( É, Ë)at Ê=0,Ö ã Ó+ Ç4 Ó= Ô4( É, Ë)at Ê= é. Page571 572 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Thesolution Ó( É, Ê, Ë)isdetermined bytheformula inParagraph 8.3.2-10 whereØ( É, Ê, Ë, Ä, Å,×)=1 2Ã æç=1Ã æì=1 õ ç ì( É, Ê)õ ç ì( Ä, Å)öõ ç ì ö2 íç ìexp(- íç ì| Ë-×|). Here,Óç ì( É, Ê)=( ÷ çcos ÷ çÉ+ Ç1sin ÷ çÉ)( ø ìcos ø ìÊ+ Ç3sin ø ìÊ), íç ì= ñ ÷2ç+ ø2ì- Î,öÓç ì ö2=1 4( ÷2ç+ Ç2 1)( ø2ì+ Ç2 3) Õùä+( Ç1+ Ç2)( ÷2ç+ Ç1 Ç2) ( ÷2ç+ Ç2 1)( ÷2ç+ Ç2 2) Ù Õúé+( Ç3+ Ç4)( ø2ì+ Ç3 Ç4) ( ø2ì+ Ç2 3)( ø2ì+ Ç2 4) Ù, where the ÷ çand ø ìarepositi veroots ofthetranscendental equations tan( ÷ ä)=( Ç1+ Ç2) ÷÷2- Ç1 Ç2, tan( ø é)=( Ç3+ Ç4) øø2- Ç3 Ç4. 8.3.2-12. Domain: 0£ É£ ä,0£ Ê£ é,- Ò< Ë< Ò.Mixedboundary value problems. 1 Í.Anin®nite cylindrical domain ofarectangularcross-section isconsider ed.Boundary conditions areprescribed:Ó= Ô1( Ê, Ë)at É=0,Ö ò Ó= Ô2( Ê, Ë)at É= ä,Ó= Ô3( É, Ë)at Ê=0,Ö ã Ó= Ô4( É, Ë)at Ê= é. Solution:Ó( É, Ê, Ë)=Â ê0Â Ã-Ã Ô1( Å,×) ÕÖÖ ÄØ( É, Ê, Ë, Ä, Å,×) Ù ë =0 Ì× Ì Å +Â ê0ÂÃ-Ã Ô2( Å,×)Ø( É, Ê, Ë, ä, Å,×)Ì× Ì Å +Â å 0ÂÃ-Ã Ô3( Ä,×) ÕÖÖ ÅØ( É, Ê, Ë, Ä, Å,×) Ù â =0 Ì× Ì Ä +Â å 0ÂÃ-Ã Ô4( Ä,×)Ø( É, Ê, Ë, Ä, é,×)Ì× Ì Ä +Â å 0Â ê0Â Ã-Ã Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì× Ì ÅÌ Ä. Green' sfunction:Ø( É, Ê, Ë, Ä, Å,×)=2ä éÃ æç=0Ã æì=01íç ìsin( î çÉ)sin( ï ìÊ)sin( î çÄ)sin( ï ìÅ)exp(- íç ì| Ë-×|),î ç=(2 è+1) Ï 2 ä, ï ì=(2ð+1) Ï 2 é, íç ì= ñ î2ç+ ï2ì- Î. 2 Í.Anin®nite cylindrical domain ofarectangularcross-section isconsider ed.Boundary conditions areprescribed:Ó= Ô1( Ê, Ë)at É=0, Ó= Ô2( Ê, Ë)at É= ä,Ö ã Ó= Ô3( É, Ë)at Ê=0,Ö ã Ó= Ô4( É, Ë)at Ê= é. Page572 8.3. HELMHOL TZEQUATION Ý3 Þ+ ßÞ=- à(x) 573 Solution:Ó( É, Ê, Ë)=Â ê0ÂÃ-Ã Ô1( Å,×) ÕÖÖ ÄØ( É, Ê, Ë, Ä, Å,×) Ù ë =0 Ì× Ì Å -Â ê0ÂÃ-Ã Ô2( Å,×) ÕÖÖ ÄØ( É, Ê, Ë, Ä, Å,×) Ùë = å Ì× Ì Å -Â å 0Â Ã-Ã Ô3( Ä,×)Ø( É, Ê, Ë, Ä,0,×)Ì× Ì Ä +Â å 0Â Ã-Ã Ô4( Ä,×)Ø( É, Ê, Ë, Ä, é,×)Ì× Ì Ä +Â å 0Â ê0ÂÃ-Ã Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì× Ì ÅÌ Ä. Green' sfunction:Ø( É, Ê, Ë, Ä, Å,×)=1ä éÃ æç=1Ã æì=0ó ìíç ìsin( î çÉ)cos( ï ìÊ)sin( î çÄ)cos( ï ìÅ)exp(- íç ì| Ë-×|),î ç= è Ïä, ï ì= ð Ïé, íç ì= ñ î2ç+ ï2ì- Î,ó ì= ô1forð=0, 2forð¹0. 8.3.2-13. Domain: 0£ É£ ä,0£ Ê£ é,0£ Ë< Ò.First boundary value problem. Asemiin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed:Ó= Ô1( Ê, Ë)at É=0, Ó= Ô2( Ê, Ë)at É= ä,Ó= Ô3( É, Ë)at Ê=0, Ó= Ô4( É, Ë)at Ê= é,Ó= Ô5( É, Ê)at Ë=0. Solution:Ó( É, Ê, Ë)=Â ê0ÂÃ0 Ô1( Å,×) ÕÖÖ ÄØ( É, Ê, Ë, Ä, Å,×) Ùë =0 Ì× Ì Å -Â ê0Â Ã0 Ô2( Å,×) ÕÖÖ ÄØ( É, Ê, Ë, Ä, Å,×) Ù ë = å Ì× Ì Å +Â å 0Â Ã0 Ô3( Ä,×) ÕÖÖ ÅØ( É, Ê, Ë, Ä, Å,×) Ù â =0 Ì× Ì Ä -Â å 0Â Ã0 Ô4( Ä,×) ÕÖÖ ÅØ( É, Ê, Ë, Ä, Å,×) Ù â =ê Ì× Ì Ä +Â å 0Â ê0 Ô5( Ä, Å) ÕÖÖ × Ø( É, Ê, Ë, Ä, Å,×) Ù Ú =0 Ì ÅÌ Ä +Â å 0Â ê0ÂÃ0 Û( Ä, Å,×)Ø( É, Ê, Ë, Ä, Å,×)Ì× Ì ÅÌ Ä. Green' sfunction:Ø( É, Ê, Ë, Ä, Å,×)=4ä éÃ æç=1Ã æì=11íç ìsin( î çÉ)sin( ï ìÊ)sin( î çÄ)sin( ï ìÅ) û ç ì( Ë,×),î ç= è Ïä, ï ì= ð Ïé, íç ì= ñ î2ç+ ï2ì- Î,û ç ì( Ë,×)= üexp(- íç ìË)sinh( íç ì×)for Ë>×³0, exp(- íç ì×)sinh( íç ìË)for×> Ë³0. Page573 574 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.3.2-14. Domain: 0£ É£ ä,0£ Ê£ é,0£ Ë< Ò.Second boundary value problem. Asemiin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed:Ö ò Ó= Ô1( Ê, Ë)at É=0,Ö ò Ó= Ô2( Ê, Ë)at É= ä,Ö ã Ó= Ô3( É, Ë)at Ê=0,Ö ã Ó= Ô4( É, Ë)at Ê= é,Ö á Ó= Ô5( É, Ê)at Ë=0. Solution:Ó( É, Ê, Ë)= ý å 0 ýê0 ýÃ0 Û( Ä, þ,×)Ø( ÿ, , , Ä, þ,×) × þ Ä - ýê0 ýÃ0 Ô1( þ,×)Ø( ÿ, , ,0, þ,×) × þ+ ýê0 ýÃ0 Ô2( þ,×)Ø( ÿ, , , ä, þ,×) × þ - ý å 0 ýÃ0 Ô3( Ä,×)Ø( ÿ, , , Ä,0,×) × Ä+ ý å 0 ýÃ0 Ô4( Ä,×)Ø( ÿ, , , Ä, é,×) × Ä - ý å 0 ýê0 Ô5( Ä, þ)Ø( ÿ, , , Ä, þ,0) þ Ä. Green' sfunction:Ø( ÿ, , , Ä, þ,×)=1ä éÃ æç=0Ã æì=0 ó çó ìíç ìcos( î çÿ)cos( ï ì)cos( î çÄ)cos( ï ìþ) û ç ì( ,×),î ç= è ä, ï ì= ð é, íç ì= ñ î2ç+ ï2ì- ,ó ç= ü1for è=0, 2for è¹0,û ç ì( ,×)= üexp(- íç ì)cosh( íç ì×)for >×³0, exp(- íç ì×)cosh( íç ì)for×> ³0. 8.3.2-15. Domain: 0£ ÿ£ ä,0£ £ é,0£ < Ò.Third boundary value problem. Asemiin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed:Ö ò Ó- 1 Ó= Ô1( , )at ÿ=0,Ö ò Ó+ 2 Ó= Ô2( , )at ÿ= ä,Ö ã Ó- 3 Ó= Ô3( ÿ, )at =0,Ö ã Ó+ 4 Ó= Ô4( ÿ, )at = é,Ö á Ó- 5 Ó= Ô5( ÿ, )at =0. Thesolution Ó( ÿ, , )isdetermined bytheformula inParagraph 8.3.2-14 whereØ( ÿ, , , Ä, þ,×)=Ã æç=1Ã æì=1 õ ç ì( ÿ, )õ ç ì( Ä, þ)öõ ç ì ö2 û ç ì( ,×). Here,Óç ì( ÿ, )=( ÷ çcos ÷ çÿ+ 1sin ÷ çÿ)( ø ìcos ø ì+ 3sin ø ì),öÓç ì ö2=1 4( ÷2ç+ 2 1)( ø2ì+ 2 3) Õ ä+( 1+ 2)( ÷2ç+ 1 2) ( ÷2ç+ 2 1)( ÷2ç+ 2 2) Ù Õúé+( 3+ 4)( ø2ì+ 3 4) ( ø2ì+ 2 3)( ø2ì+ 2 4) Ù,û ç ì( ,×)= exp(- íç ì)[ íç ìcosh( íç ì×)+ 5sinh( íç ì×)]íç ì( íç ì+ 5)for >×, exp(- íç ì×)[ íç ìcosh( íç ì)+ 5sinh( íç ì)]íç ì( íç ì+ 5)for×> , íç ì= ñ ÷2ç+ ø2ì- , where the ÷ çand ø ìarepositi veroots ofthetranscendental equations tan( ÷ ä)=( 1+ 2) ÷÷2- 1 2, tan( ø é)=( 3+ 4) øø2- 3 4. Page574 8.3. HELMHOL TZEQUATION Ý3 Þ+ ßÞ=- à(x) 575 8.3.2-16. Domain: 0£ ÿ£ ä,0£ £ é,0£ < Ò.Mixedboundary value problems. 1 .Asemiin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed:Ó= Ô1( , )at ÿ=0, Ó= Ô2( , )at ÿ= ä,Ó= Ô3( ÿ, )at =0, Ó= Ô4( ÿ, )at = é,Ö á Ó= Ô5( ÿ, )at =0. Solution:Ó( ÿ, , )= ýê0 ýÃ0 Ô1( þ,×) ÕÖÖ ÄØ( ÿ, , , Ä, þ,×) Ù ë =0 × þ - ýê0 ýÃ0 Ô2( þ,×) ÕÖÖ ÄØ( ÿ, , , Ä, þ,×) Ù ë = å × þ + ý å 0 ýÃ0 Ô3( Ä,×) ÕÖÖ þØ( ÿ, , , Ä, þ,×) Ù â =0 × Ä - ý å 0 ýÃ0 Ô4( Ä,×) ÕÖÖ þØ( ÿ, , , Ä, þ,×) Ù â =ê × Ä - ý å 0 ýê0 Ô5( Ä, þ)Ø( ÿ, , , Ä, þ,0) þ Ä + ý å 0 ýê0 ýÃ0 Û( Ä, þ,×)Ø( ÿ, , , Ä, þ,×) × þ Ä. Green' sfunction:Ø( ÿ, , , Ä, þ,×)=4ä éÃ æç=1Ã æì=11íç ìsin( î çÿ)sin( ï ì)sin( î çÄ)sin( ï ìþ) û ç ì( ,×),î ç= è ä, ï ì= ð é, íç ì= ñ î2ç+ ï2ì- ,û ç ì( ,×)= üexp(- íç ì)cosh( íç ì×)for >×³0, exp(- íç ì×)cosh( íç ì)for×> ³0. 2 .Asemiin®nite cylindrical domain ofarectangular cross-section isconsidered. Boundary conditions areprescribed:Ö ò Ó= Ô1( , )at ÿ=0,Ö ò Ó= Ô2( , )at ÿ= ä,Ö ã Ó= Ô3( ÿ, )at =0,Ö ã Ó= Ô4( ÿ, )at = é,Ó= Ô5( ÿ, )at =0. Solution:Ó( ÿ, , )= ý å 0 ýê0 ýÃ0 Û( Ä, þ,×)Ø( ÿ, , , Ä, þ,×) × þ Ä - ýê0 ýÃ0 Ô1( þ,×)Ø( ÿ, , ,0, þ,×) × þ+ ýê0 ýÃ0 Ô2( þ,×)Ø( ÿ, , , ä, þ,×) × þ - ý å 0 ýÃ0 Ô3( Ä,×)Ø( ÿ, , , Ä,0,×) × Ä+ ý å 0 ýÃ0 Ô4( Ä,×)Ø( ÿ, , , Ä, é,×) × Ä + ý å 0 ýê0 Ô5( Ä, þ) ÕÖÖ × Ø( ÿ, , , Ä, þ, Ä) Ù Ú =0 þ Ä. Page575 576 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Green' sfunction:Ø( ÿ, , , Ä, þ,×)=1ä éÃ æç=0Ã æì=0 ó çó ìíç ìcos( î çÿ)cos( ï ì)cos( î çÄ)cos( ï ìþ) û ç ì( ,×). Here,î ç= è ä, ï ì= ð é, íç ì= ñ î2ç+ ï2ì- ,ó ç= ü1for è=0, 2for è¹0,û ç ì( ,×)= üexp(- íç ì)sinh( íç ì×)for >×³0, exp(- íç ì×)sinh( íç ì)for×> ³0.Paragraphs 8.3.2-17 through 8.3.2-23 present only theeigenvalues andeigenfunctions ofho- mogeneous boundary value problems forthehomo geneous Helmholtz equation (with Ûº0).The solutions ofthecorresponding nonhomo geneous boundary value problems (with Û 0)canbe constructed bytherelations speci®ed inParagraphs 8.3.1-4 and8.3.1-8. 8.3.2-17. Domain: 0£ ÿ£ ä,0£ £ é,0£ £ .First boundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:Ó= Ô1( , )at ÿ=0, Ó= Ô2( , )at ÿ= ä,Ó= Ô3( ÿ, )at =0, Ó= Ô4( ÿ, )at = é,Ó= Ô5( ÿ, )at =0, Ó= Ô6( ÿ, )at = . 1 .Eigen values ofthehomogeneous problem: ç ì = 2 è2ä2+ ð2é2+ 2 2 ; , , =1,2,3, Eigenfunctions andthenorm squared: =sin ÿä sin sin , ö ö2= ä 8. 2 .Adouble-series representation oftheGreen' sfunction:( ÿ, , , Ä, þ, )=4ä Ã =1Ã =1sin( ÿ)sin( )sin( )sin( þ) ( , ), ( , )= sinh( )sinh[ ( - )] sinh( )for ³ > ³0, sinh( )sinh[ ( - )] sinh( )for ³ > ³0, = ä, = , = ! 2 + 2 - . This relation canbeused toobtain twoother representations oftheGreen' sfunction with theaidof thecyclic permutations oftriples: ( ÿ, , ä)" # ( , , ) $ %( , þ, ) Atriple series representation oftheGreen' sfunction:( ÿ, , , , þ, )=8ä & =1 & =1 & =1sin( ÿ)sin( )sin( ' )sin( )sin( þ)sin( ' )2 + 2 + '2- , = ä, = , ' = .(*) Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). Page576 8.3. HELMHOL TZEQUATION +3 ,+ -,=- .(x) 577 8.3.2-18. Domain: 0£ ÿ£ ä,0£ /£ ,0£ 0£ 1.Second boundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:2 3= 41( /, 0)at ÿ=0, 2 3= 42( /, 0)at ÿ= ä,2ã = 43( ÿ, 0)at /=0, 2ã = 44( ÿ, 0)at /= ,2 5= 45( ÿ, /)at 0=0, 2 5= 46( ÿ, /)at 0= 1. 1 6.Eigen values ofthehomogeneous problem:7 8= 92 2ä2+ 22+ :212 ; , ,:=0,1,2, Eigenfunctions: 8=cos 9 ÿä cos 9 /cos 9: 01 . Thesquare ofthenorm ofaneigenfunction isde®ned as; 8 ;2= ä 1 8(1+ < 0)(1+ < 0)(1+ < 8 0), < 0= =1for =0, 0for ¹0. 2 6.Adouble series representation oftheGreen' sfunction:( ÿ, /, 0, , >, )=1? & =0 & =0 @ @ cos( ÿ)cos( /)cos( )cos( >) ( 0, ), ( 0, )= ABB CBBDcosh( )cosh[ ( 1- 0)] sinh( 1)for 1³ 0> ³0, cosh( 0)cosh[ ( 1- )] sinh( 1)for 1³ > 0³0, = 9 ?, = 9 , = ! 2 + 2 - 7,@ ==1for =0, 2for ¹0. This relation canbeused toobtain twoother representations oftheGreen' sfunction with theaidof thecyclic permutations: ( ÿ, , ?)" # ( 0, , 1) $ %( /, >, ) Atriple series representation oftheGreen' sfunction:( ÿ, /, 0, , >, )=1? 1& =0 & =0 & 8 =0 @ @ @ 8cos( ÿ)cos( /)cos( ' 80)cos( )cos( >)cos( ' 8)2 + 2 + '28- 7 , = 9 ?, = 9 , ' 8= 9:1.(*) Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 8.3.2-19. Domain: 0£ ÿ£ ?,0£ /£ ,0£ 0£ 1.Third boundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:2 3-:1 = 41( /, 0)at ÿ=0, 2 3+:2 = 42( /, 0)at ÿ= ?,2 E-:3 = 43( ÿ, 0)at /=0, 2 E+:4 = 44( ÿ, 0)at /= ,2 5-:5 = 45( ÿ, /)at 0=0, 2 5+:6 = 46( ÿ, /)at 0= 1. Page577 578 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Eigen values ofthehomogeneous problem:7 F= G2+ H2+ I2F; , , J=1,2,3, Here, the G , H ,and I Farepositi veroots ofthetranscendental equations tan( G ?)=(:1+:2) GG2-:1:2,tan( H )=(:3+:4) HH2-:3:4,tan( I 1)=(:5+:6) II2-:5:6. Eigenfunctions: F=1K L M F( G Ncos G N ÿ+:1sin G N ÿ)( H Ocos H O /+:3sin H O /)( I Fcos I F0+:5sin I F0),KN= ! G2N+:2 1, LO= ! H2O+:2 3, M F= ! I2F+:2 5. Thesquare ofthenorm ofaneigenfunction isde®ned as;QPN O F ;2=1 8 R ?+(:1+:2)( G2N+:1:2) ( G2N+:2 1)( G2N+:2 2) S RUT+(:3+:4)( H2O+:3:4) ( H2O+:2 3)( H2O+:2 4) S R 1+(:5+:6)( I2F+:5:6) ( I2F+:2 5)( I2F+:2 6) S.(*) Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 8.3.2-20. Domain: 0£ ÿ£ ?,0£ /£T,0£ 0£ 1.Mixedboundary value problems. 1 6.Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:P= 41( /, 0)at ÿ=0, P= 42( /, 0)at ÿ= ?,P= 43( ÿ, 0)at /=0, P= 44( ÿ, 0)at /=T,2 5 P= 45( ÿ, /)at 0=0, 2 5 P= 46( ÿ, /)at 0= 1. Eigen values ofthehomogeneous problem:7N O 8= 92 V W2?2+ X2T2+ :212 Y;W,X=1,2,3, ZZZ;:=0,1,2, ZZZ Eigenfunctions:PN O 8=sin V 9W ÿ?Ysin V 9X /T Ycos V 9: 01 Y. Thesquare ofthenorm ofaneigenfunction isde®ned as;QPN O 8 ;2= ?T 1 8(1+ < 8 0), < 8 0==1for:=0, 0for:¹0. 2 6.Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:P= 41( /, 0)at ÿ=0, P= 42( /, 0)at ÿ= ?,2 E P= 43( ÿ, 0)at /=0, 2 E P= 44( ÿ, 0)at /=T,2 5 P= 45( ÿ, /)at 0=0, 2 5 P= 46( ÿ, /)at 0= 1. Eigen values ofthehomogeneous problem:7N O 8= 92 V W2?2+ X2T2+ :212 Y;W=1,2,3, ZZZ;X,:=0,1,2, ZZZ Eigenfunctions:PN O 8=sin V 9W ÿ?Ycos V 9X /T Ycos V 9: 01 Y. Thesquare ofthenorm ofaneigenfunction isde®ned as;QPN O 8 ;2= ?T 1 8(1+ <[O0)(1+ < 8 0), <[O0= =1forX=0, 0forX¹0. Page578 8.3. HELMHOL TZEQUATION +3 ,+ -,=- .(x) 579 8.3.2-21. Domain: 0£ ÿ£ ?,0£ /£ ÿ,0£ 0£ 1.First boundary value problem. Aright prism whose base isanisosceles right-angled triangle isconsidered. Boundary conditions areprescribed:P= 41( /, 0)at ÿ=0, P= 42( ÿ, 0)at /=0, P= 43( ÿ, 0)at /= ÿ,P= 44( ÿ, /)at 0=0, P= 45( ÿ, /)at 0= 1. Eigen values ofthehomogeneous problem:7N O 8= 92?2 \(W+X)2+X2 ]+ 92:212;W,X,:=1,2,3, ZZZ Eigenfunctions:PN O 8= ^sinR 9?(W+X) ÿSsin V 9?X /Y-(-1) Nsin V 9?X ÿYsinR 9?(W+X) /S _sin V 9: 01 Y. 8.3.2-22. Domain: 0£ ÿ£ ?,0£ /£ ÿ,0£ 0£ 1.Second boundary value problem. Aright prism whose base isanisosceles right-angled triangle isconsidered. Boundary conditions areprescribed:2 3 P= 41( /, 0)at ÿ=0, 2 E P= 42( ÿ, 0)at /=0, 2 ` P= 43( ÿ, 0)at /= ÿ,2 5 P= 44( ÿ, /)at 0=0, 2 5 P= 45( ÿ, /)at 0= 1, where 2 ` P=N×Ñ P=1a 2( 2 3 P+ 2 E P). Eigen values ofthehomogeneous problem:7N O 8= 92?2 \(W+X)2+X2 ]+ 92:212;W,X,:=0,1,2, ZZZ Eigenfunctions:PN O 8= ^cosR 9?(W+X) ÿScos V 9?X /Y-(-1) Ncos V 9?X ÿYcosR 9?(W+X) /S _cos V 9: 01 Y. 8.3.2-23. Domain: 0£ ÿ£ ?,0£ /£ ÿ,0£ 0£ 1.Mixedboundary value problems. 1 6.Aright prism whose base isanisosceles right-ang ledtriangleisconsidered .Boundary conditions areprescribed:P= 41( /, 0)at ÿ=0, P= 42( ÿ, 0)at /=0, P= 43( ÿ, 0)at /= ÿ,2 5 P= 44( ÿ, /)at 0=0, 2 5 P= 45( ÿ, /)at 0= 1. Eigen values ofthehomogeneous problem:7N O 8= 92?2 \(W+X)2+X2 ]+ 92:212;W,X=1,2,3, ZZZ;:=0,1,2, ZZZ Eigenfunctions:PN O 8= ^sinR 9?(W+X) ÿSsin V 9?X /Y-(-1) Nsin V 9?X ÿYsinR 9?(W+X) /S _cos V 9: 01 Y. 2 6.Aright prism whose base isanisosceles right-ang ledtriangleisconsidered .Boundary conditions areprescribed:2 3 P= 41( /, 0)at ÿ=0, 2 E P= 42( ÿ, 0)at /=0, 2 ` P= 43( ÿ, 0)at /= ÿ,P= 44( ÿ, /)at 0=0, P= 45( ÿ, /)at 0= 1. Eigen values ofthehomogeneous problem:7N O 8= 92?2 \(W+X)2+X2 ]+ 92:212;W,X=0,1,2, ZZZ;:=1,2,3, ZZZ Eigenfunctions:PN O 8= ^cosR 9?(W+X) ÿScos V 9?X /Y-(-1) Ncos V 9?X ÿYcosR 9?(W+X) /S _sin V 9: 01 Y. Page579 580 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.3.3. Problems inCylindrical Coor dinates Thethree-dimensional nonhomogeneous Helmholtz equation inthecylindrical coordinate system is written as 1 b 22 bV b 2 P2 bY+1 b 2 22 P2 c2+ 22 P202+ 7 P=- d( b , c, 0), b = e ÿ2+ f2. 8.3.3-1. Particular solutions ofthehomogeneous equation ( dº0):P=\ K g 0 h b i j k + l m0 h b i j k]( n1 c+ o1)( n2 p+ o2),q= g rh be j - s2 k ( tcosX c+ lsinX c)( ncos sp+ osin sp), j > s2,q= m rh be j - s2 k ( tcosX c+ lsinX c)( ncos sp+ osin sp), j > s2,q= g rh be j + s2 k ( tcosX c+ lsinX c)( ncosh sp+ osinh sp), j >- s2,q= m rh be j + s2 k ( tcosX c+ lsinX c)( ncosh sp+ osinh sp), j >- s2,q= u rh be s2- j k ( tcosX c+ lsinX c)( ncos sp+ osin sp), j < s2,q= m rh be s2- j k ( tcosX c+ lsinX c)( ncos sp+ osin sp), j < s2, whereX=0,1,2, ZZZ; t, l, n, o, n1, n2, o1, o2,and sarearbitrary constants; the g r( v)andm r( v)aretheBessel functions; andthe u r( v)and w r( v)arethemodi®ed Bessel functions. 8.3.3-2. Domain: 0£ b £ x,0£ c£2 y,- z<p< z.First boundary value problem. Anin®nite circular cylinder isconsidered. Aboundary condition isprescribed:q= {( c,p)at b = x. Solution:q( b , c,p)=- x |2 } 0 |&-& {( ~, ) v ( b , c,p, v, ~, ) = ~ + | 0 |2 } 0 | - ( v, ~, )( , ,p, v, ~, ) v ~ v. Here,( , ,p, v, ~, )=1 2 y x2 =0 r =1 t g ( s r) g ( s rv)g ( s rx) 2 r cos[ ( - ~)]exph- r|p- | k , r= s2 r- j , t = 1for =0, 2for ¹0, where the g ( v)aretheBessel functions andthe s rarepositi veroots ofthetranscendental equationg ( s x)=0.* Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). 8.3.3-3. Domain: 0£ £ x,0£ £2 y,- z<p< z.Second boundary value problem. Anin®nite circular cylinder isconsidered. Aboundary condition isprescribed: q= {( ,p)at = x. Page580 8.3. HELMHOL TZEQUATION 3 + =- (x) 581 Solution:( , , )= 2 0 - ( , ) ( , , , , , ) + 0 2 0 - ( , , ) ( , , , , , ) . Here,( , , , , , )=exp - - | - | 2 2- +1 2 =0 =1 2 ( ) ( )cos[ ( - )] ( 2 2- 2) 2 ( ) ! exp - ! | - | ,! = " 2 - , = #1for =0, 2for ¹0, where the ( )aretheBessel functions andthe arepositi veroots ofthetranscendental equation $ ( )=0.%'& Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). 8.3.3-4. Domain: 0£ £ ,0£ £2 ,- (< < (.Third boundary value problem. Anin®nite circular cylinder isconsidered. Aboundary condition isprescribed:) *+ + =( , )at = . Solution:( , , )= 2 0 - ( , ) ( , , , , , ) + 0 2 0 - ( , , ) ( , , , , , ) . Here,( , , , , , )=1 2 =0 =1 2 ( ) ( )cos[ ( - )] ( 2 2+ +22- 2) 2 ( ) ! exp - ! | - | ,! = " 2 - , = #1for =0, 2for ¹0, where the ( )aretheBessel functions andthe arepositi veroots ofthetranscendental equation $ ( )+ + ( )=0. 8.3.3-5. Domain: 0£ £ ,0£ £2 ,0£ < (.First boundary value problem. Asemiin®nite circular cylinder isconsidered. Boundary conditions areprescribed:=1( , )at = , =2( , )at =0. Page581 582 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES Solution:( , , )=- 2 0 0 1( , ) , )) ( , , , , , ) - . = + 2 0 0 2( , ) , )) ( , , , , , ) - / =0 + 0 2 0 0 ( , , ) ( , , , , , ) . Here,( , , , , , )=1 2 2 =0 =1 ( ) ( )0$ ( ) 12! cos[ ( - )] 2 ( , ),2 ( , )=exp(- ! | - |)-exp(- ! | + |), ! = " 2 - , = #1for =0, 2for ¹0, where the ( )aretheBessel functions andthe arepositi veroots ofthetranscendental equation ( )=0. 8.3.3-6. Domain: 0£ £ ,0£ £2 ,0£ < (.Second boundary value problem. Asemiin®nite circular cylinder isconsidered. Boundary conditions areprescribed:) *=1( , )at = , ) 3=2( , )at =0. Solution:( , , )= 2 0 0 1( , ) ( , , , , , ) - 2 0 0 2( , ) ( , , , , ,0) + 0 2 0 0 ( , , ) ( , , , , , ) . Here,( , , , , , )=exp - - | - | +exp - - | + | 2 2- +1 2 =0 =1 2 ( ) ( )cos[ ( - )] ( 2 2- 2) 2 ( ) ! 2 ( , ),2 ( , )=exp(- ! | - |)+exp(- ! | + |), ! = " 2 - , = #1for =0, 2for ¹0, where the ( )aretheBessel functions andthe arepositi veroots ofthetranscendental equation $ ( )=0. 8.3.3-7. Domain: 0£ £ ,0£ £2 ,0£ < (.Third boundary value problem. Asemiin®nite circular cylinder isconsidered. Boundary conditions areprescribed:) *+ +1 =( , )at = , ) 3- +2 =2( , )at =0. Page582 8.3. HELMHOL TZEQUATION 3 + =- (x) 583 Solution:( , , )= 2 0 0 1( , ) ( , , , , , ) - 2 0 0 2( , ) ( , , , , ,0) + 0 2 0 0 ( , , ) ( , , , , , ) . Here,( , , , , , )=1 =0 =1 2 ( ) ( )cos[ ( - )] ( 2 2+ +2 1 2- 2) 2 ( ) 2 ( , ), = #1for =0, 2for ¹0, 2 ( , )= 455 6557exp(- ! )[ ! cosh( ! )+ +2sinh( ! )]! ( ! + +2)for > , exp(- ! )[ ! cosh( ! )+ +2sinh( ! )]! ( ! + +2)for > , where the ( )aretheBessel functions, ! = 8 2 - ,andthe arepositi veroots ofthe transcendental equation $ ( )+ +1 ( )=0. 8.3.3-8. Domain: 0£ £ ,0£ £2 ,0£ < (.Mixedboundary value problem. Asemiin®nite circular cylinder isconsidered. Boundary conditions areprescribed:=1( , )at = , ) 3=2( , )at =0. Solution:( , , )=- 2 0 0 1( , ) , )) ( , , , , , ) - . = - 2 0 0 2( , ) ( , , , , ,0) + 0 2 0 0 ( , , ) ( , , , , , ) . Here,( , , , , , )=1 2 2 =0 =1 ( ) ( )0$ ( ) 12! cos[ ( - )] 2 ( , ),2 ( , )=exp(- ! | - |)+exp(- ! | + |), ! = " 2 - , = #1for =0, 2for ¹0, where the ( )aretheBessel functions andthe arepositi veroots ofthetranscendental equation ( )=0.9Paragraphs 8.3.3-9 through 8.3.3-16 present only theeigenvalues andeigenfunctions ofhomo ge- neous boundary value problems forthehomo geneous Helmholtz equation (withº0).Thesolutions ofthecorresponding nonhomo geneous boundary value problems ( :0)canbeconstructed bythe relations speci®ed inParagraphs 8.3.1-4 and8.3.1-8. Page583 584 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.3.3-9. Domain: 0£ £ ,0£ £2 ,0£ £ ;.First boundary value problem. Acircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:=0at = , =0at =0, =0at = ;. Eigen values: <= 2+2;2+ 2 2; =0,1, =>=>=; ?, +=1,2, =>=>= Here, the arepositi vezeros oftheBessel functions, ( )=0. Eigenfunctions:(1) <= @ Acos( )sin @ + ; A,(2) <= @ Asin( )sin @ + ; A. Eigenfunctions possessing theaxial symmetry property:(1) 0 <= 0 @ 0 Asin @ + ; A. Thesquare ofthenorm ofaneigenfunction isde®ned asB(1) < B2= B(2) < B2= 2; 4(1+ C 0) 0 $ ( ) 12, C = D1for = ?, 0for ¹ ?.%'& Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 8.3.3-10. Domain: 0£ £ ,0£ £2 ,0£ £ ;.Second boundary value problem. Acircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:) *=0at = , ) 3=0at =0, ) 3=0at = ;. Eigen values:000=0, <= 2+2;2+ 2 2; =0,1, =>=>=; +, ?=0,1, =>=>= Here, the areroots ofthetranscendental equation $ ( )=0. Eigenfunctions:(1) <= @ Acos( )cos @ + ; A, (1) 000=1,(2) <= @ Asin( )cos @ + ; A. Thesquare ofthenorm ofaneigenfunction isde®ned asB(1) < B2= B(2) < B2= 2; 4 2 (1+ C 0)( 2 - 2) 0 ( ) 12, B(1) 000 B2= 2;, where C 0istheKroneck erdelta.%'& Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). Page584 8.3. HELMHOL TZEQUATION 3 + =- (x) 585 8.3.3-11. Domain: 0£ £ ,0£ £2 ,0£ £ ;.Third boundary value problem. Acircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:) *+ +1 =0at = , ) 3- +2 =0at =0, ) 3+ +3 =0at = ;. Eigen values: E= F2E+ 2 2, where the F Eand arepositi veroots ofthetranscendental equations tan( F ;)=( +2+ +3) FF2- +2 +3, $ ( )+ +1 ( )=0. Eigenfunctions:(1) E= @ Acos( ) F Ecos F E+ +2sin F E" F2E+ +2 2,(2) E= @ Asin( ) F Ecos F E+ +2sin F E" F2E+ +2 2. Thesquare ofthenorm ofaneigenfunction isde®ned asB( G) E B2= 2 4 2 (1+ C 0)( 2+2 1+ 2 - 2) 0 ( ) 12,H;+( +2+ +3)( F2E+ +2 +3) ( F2E+ +2 2)( F2E+ +2 3) -, where C 0istheKroneck erdelta. 8.3.3-12. Domain: 1£ £ 2,0£ £2 ,0£ £ ;.First boundary value problem. Ahollo wcircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:=0at = 1, =0at = 2,=0at =0, =0at = ;. Eigen values: <= 2+2;2+ 2 ; =0,1,2, =>=>=; ?, +=1,2,3, =>=>= Here, the arepositi veroots ofthetranscendental equation ( 1) I ( 2)- ( 2) I ( 1)=0. Eigenfunctions:(1) <=[ ( ) I ( 1)- ( 1) I ( )]cos( )sin @ + ; A,(2) <=[ ( ) I ( 1)- ( 1) I ( )]sin( )sin @ + ; A. Thesquare ofthenorm ofaneigenfunction isde®ned asB(1) < B2= B(2) < B2= ; 2 (1+ C 0)[ ( 1) 12-[ ( 2) 12 [ ( 2) 12, CGKJ= #1for L= M, 0for L¹ M.%'& Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). Page585 586 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.3.3-13. Domain: 1£ £ 2,0£ £2 ,0£ £ ;.Second boundary value problem. Ahollo wcircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:) *=0at = 1, ) *=0at = 2,) 3=0at =0, ) 3=0at = ;. Eigen values: <= 2+2;2+ 2 ; , ?, +=0,1,2, =>=>= Here, the areroots ofthetranscendental equation $ ( 1) I $ ( 2)- $ ( 2) I $ ( 1)=0. Eigenfunctions:(1) <=[ ( ) I $ ( 1)- $ ( 1) I ( )]cos( )cos @ + ; A,(2) <=[ ( ) I $ ( 1)- $ ( 1) I ( )]sin( )cos @ + ; A. Tothezero eigen value 000=0there isacorresponding eigenfunction (1) 000=1. Thesquare ofthenorm ofaneigenfunction isde®ned asB(1) < B2= B(2) < B2= ;(1+ C 0)(1+ C <0) 2 # @ 1- 22 2 2 A , $ ( 1)$ ( 2) -2 - @ 1- 22 1 2 A N, where C 0istheKroneck erdelta.%'& Refer ences :V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964), B.M.Budak, A.A.Samarskii, and A.N.Tikhono v(1980). 8.3.3-14. Domain: O1£ P£ O2,0£ Q£2 ,0£ R£ ;.Mixedboundary value problems. 1 S.Ahollo wcircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:T=0at P= O1, T=0at P= O2,) 3T=0at R=0, ) 3T=0at R= ;. Eigen values: <= 2+2;2+ 2 ; , +=0,1,2, =>=>=; ?=1,2,3, =>=>= Here, the areroots ofthetranscendental equation ( O1) I ( O2)- ( O2) I ( O1)=0. Eigenfunctions:T(1) <=[ ( P) I ( O1)- ( O1) I ( P)]cos( Q)cos @ + R; A,T(2) <=[ ( P) I ( O1)- ( O1) I ( P)]sin( Q)cos @ + R; A. Thesquare ofthenorm ofaneigenfunction isde®ned asBT(1) < B2= BT(2) < B2= ; U U < 2 [ ( O1)]2-[ ( O2)]2 [ ( O2)]2, U = D2for =0, 1for ¹0. Page586 8.3. HELMHOL TZEQUATION V3 W+ XW=- Y(x) 587 2S.Ahollo wcircular cylinder of®nite length isconsidered. Boundary conditions areprescribed:) *T=0at P= O1, ) *T=0at P= O2,T=0at R=0, T=0at R= ;. Eigen values: <= 2+2;2+ 2 ; =0,1,2, =>=>=; ?, +=1,2,3, =>=>= Here, the areroots ofthetranscendental equation $ ( O1) I $ ( O2)- $ ( O2) I $ ( O1)=0. Eigenfunctions:T(1) <=[ ( P) I $ ( O1)- $ ( O1) I ( P)]cos( Q)sin @ + R; A,T(2) <=[ ( P) I $ ( O1)- $ ( O1) I ( P)]sin( Q)sin @ + R; A. Thesquare ofthenorm ofaneigenfunction isde®ned asBT(1) < B2= BT(2) < B2= ; U 2 # @ 1- 2O2 2 2 A , $ ( O1)$ ( O2) -2 - @ 1- 2O2 1 2 A N, where U isde®ned inItem 1S. 8.3.3-15. Domain: 0£ P£ O,0£ Q£ Q0,0£ R£ ;.First boundary value problem. Acylindrical sector of®nite thickness isconsidered. Boundary conditions areprescribed:T=0at Q=0, T=0at Q= Q0, T=0at P= O,T=0at R=0, T=0at R= ;. Eigen values: <= 2+2;2+ 2 O2; , ?, +=1,2,3, =>=>= Here, the arepositi veroots ofthetranscendental equation Z []\ 0( )=0. Eigenfunctions:T <= Z []\ 0 @ PO Asin @ QQ0 Asin @+ R; A. Thesquare ofthenorm ofaneigenfunction isde®ned asBT < B2=1 8 ; O2Q0 0 $ Z []\ 0( ) 12. 8.3.3-16. Domain: 0£ P£ O,0£ Q£ Q0,0£ R£ ;.Mixedboundary value problem. Acylindrical sector of®nite thickness isconsidered. Boundary conditions areprescribed:T=0at Q=0, T=0at Q= Q0, T=0at P= O,) 3T=0at R=0, ) 3T=0at R= ;. Eigen values: <= 2+2;2+ 2 O2; , ?=1,2,3, =>=>=; +=0,1,2, =>=>= Here, the arepositi veroots ofthetranscendental equation Z []\ 0( )=0. Eigenfunctions:T <= Z []\ 0 @ PO Asin @ QQ0 Acos @+ R; A. Thesquare ofthenorm ofaneigenfunction isde®ned asBT < B2=1 8 ; O2Q0(1+ C <0) 0 $ Z []\ 0( ) 12, C <0= D1for +=0, 0for +¹0. Page587 588 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.3.4. Problems inSpherical Coor dinates The three-dimensional homogeneous Helmholtz equation inthespherical coordinate system is written as 1P2 ))P @P2 )T)P A+1P2sin ^ ))^ @ sin ^ )T)^ A+1P2sin2^ )2 T)Q2+ _ T=0, P= 8 `2+ a2+ R2. 8.3.4-1. Particular solutions:T=1P(sin P+ bcos P), _= 2,T=1P(sinh P+ bcosh P), _=- 2,T=1cP d e+1 [ 2( f P) g he(cos ^)( icos ? Q+ bsin ? Q), _= f2,T=1cP Ie+1 [ 2( f P) g he(cos ^)( icos ? Q+ bsin ? Q), _= f2,T=1cP j e+1 [ 2( f P) g he(cos ^)( icos ? Q+ bsin ? Q), _=- f2,T=1cP k e+1 [ 2( f P) g he(cos ^)( icos ? Q+ bsin ? Q), _=- f2, where l, ?=0,1,2, =>=>=; iand barearbitrary constants;d m( n)and Im( n)aretheBessel functions;j m( n)andk m( n)arethemodi®ed Bessel functions; andthe ghe( n)aretheassociated Legendre functions thatareexpressed interms oftheLegendre polynomials ge( n)asg he( n)=(1- n2) h [ 2 o ho nh ge( n), ge( n)=1l!2e o eo ne( n2-1)e. 8.3.4-2. Domain: 0£ P£ O.First boundary value problem. 1S.Aspherical domain isconsidered. Ahomogeneous boundary condition isprescribed,T=0at P= O. Eigen values:_e <= f2e <O2; l=0,1,2, =>=>=; p=1,2,3, =>=>= Here, the fe <arepositi vezeros oftheBessel functions,d e+1 [ 2( f)=0.Note thatthed e+1 [ 2( f)can beexpressed interms ofelementary functions, seeBateman andErdÂelyi(1953, Vol.2). Eigenfunctions:T(1)e h <=1cP d e+1 [ 2 @fe < PO A g he(cos ^)cos ? Q, ?=0,1,2, =>=>=;T(2)e h <=1cP d e+1 [ 2 @fe < PO A g he(cos ^)sin ? Q, ?=1,2,3, =>=>= Here, the ghe( n)aretheassociated Legendre functions. Eigenfunctions possessing central symmetry (i.e., independent of ^and Q):T(1) 00 <=d1 [ 2 @f0 < PO A. Page588 8.3. HELMHOL TZEQUATION V3 W+ XW=- Y(x) 589 Eigenfunctions possessing axial symmetry (i.e., independent of Q):T(1)e0 <=d e+1 [ 2 @fe < PO A ge(cos ^). Thesquare ofthenorm ofaneigenfunction:BT(1)e h < B2= q O2(1+ Ch0)( l+ r)! (2 l+1)( l- r)! sHd te+1 [ 2( fe u) v2, Ch0= w1for r=0, 0for r¹0,xT(1)e hu x2= xT(2)e hu x2, r=1,2,3, y>y>y 2S.Aspherical domain isconsidered. Anonhomogeneous boundary condition isprescribed,T= z( ^, Q)at P= O. Solution:T( P, ^, Q)= { |e=0 e |h=-e ze h } e( ~ )} ( ) ( , ),} ( )=1 +1 2( ), wherez=1x x 2 0 0 z( , ) ( , )sin o o , x x=2 U2 +1( + )! ( - )!, ( , )= (cos ) for =0,(cos )sin for =1,2, >>,||(cos )cos for =-1,-2, >>, U= 2for =0, 1for ¹0. Thesolution waswritten outunder theassumption that+1 2( )¹0for =0,1,2, >>' Refer ences :M.M.Smirno v(1975), A.N.Tikhono vandA.A.Samarskii (1990).Paragraphs 8.3.4-3 through 8.3.4-6 present only theeigenvalues andeigenfunctions ofhomo ge- neous boundary value problems forthehomo geneous Helmholtz equation (with º0).Thesolutions ofthecorresponding nonhomo geneous boundary value problems ( 0)canbeconstructed bythe relations speci®ed inParagraph8.3.1-4. 8.3.4-3. Domain: 0£ ~£ .Second boundary value problem. Aspherical domain isconsidered. Aboundary condition isprescribed: =0at ~= . Eigen values:00=0, = 2 2; =0,1,2, >>; p=1,2,3, >> Here, the areroots ofthetranscendental equation 2 ¡+1 2( )-+1 2( )=0. Eigenfunctions:(1) 000=1, (1) =1 ~ +1 2 ¢ ~ £ (cos )cos , =0,1,2, >>;(2) =1 ~ +1 2 ¢ ~ £ (cos )sin , =1,2,3, >> Thesquare ofthenorm ofaneigenfunction:¤(1) 000 ¤2=4 3 3, ¤(1) ¤2= 2 ¥( + )! (2 +1)( - )! ¦1- ( +1) 2 §2+1 2( ),¤(1) ¤2= ¤(2) ¤2, =1,2,3, >>, where ¥= 2for =0, 1for ¹0.' Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). Page589 590 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.3.4-4. Domain: 0£ ~£ .Third boundary value problem. Aspherical domain isconsidered. Aboundary condition isprescribed: + ¨ =0at ~= . Eigen values: = 2 2; =0,1,2, >>; p=1,2,3, >> Here, the arepositi veroots ofthetranscendental equation 2 ¡+1 2( )-(1-2 ¨)+1 2( )=0. Eigenfunctions:(1) =1 ~ +1 2 ¢ ~ £ (cos )cos , =0,1,2, >>;(2) =1 ~ +1 2 ¢ ~ £ (cos )sin , =1,2,3, >> Here, the( n)aretheassociated Legendre functions. Thesquare ofthenorm ofaneigenfunction:¤(1) ¤2= 2¥( + )! (2 +1)( - )! ¦1+( ¨+ )( ¨- -1) 2 §2+1 2( ), ¥= ©2for =0, 1for ¹0,¤(1) ¤2= ¤(2) ¤2, =1,2,3, >>' Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). 8.3.4-5. Domain: £ ~< ª.First boundary value problem. Aspherical cavityisconsidered andthedependent variable isprescribed atitssurface:= «( , )at ¬= , andtheradiation conditions areprescribed atin®nity (seeParagraph 8.3.1-5, Item 2 ). Solution for = p2>0:( ¬, , )= ® ¯=0 ¯°=- « ° ±( p ¬)±( p ) ² °( ³, ´),±( µ)=1¶µ ·(2)+1 ¸2( µ), where·(2)+1 ¸2( µ)istheHank elfunction ofthesecond kind andtheother quantities arede®ned just asinParagraph 8.3.4-2, Item 2 .' Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). 8.3.4-6. Domain: 1£ ¬£ 2.First boundary value problem. Aspherical layer isconsidered. Boundary conditions areprescribed:=0at ¬= 1, =0at ¬= 2. Eigen values: ¹ = 2 ; =0,1,2, >>; º=1,2,3, >> Here, the arepositi veroots ofthetranscendental equation»+1 ¸2( 1)²+1 ¸2( 2)- »+1 ¸2( 2)²+1 ¸2( 1)=0. Page590 8.3. H ELMHOLTZ EQUATION ¼3 ½+ ¾½= - ¿(x) 591 Eigenfunctions:(1) °=1¶¬ À+1 ¸2( ¬) °(cos ³) cos Á ´, Á= 0,1,2, >>;(2) °=1¶¬ À+1 ¸2( ¬) °(cos ³) sin Á ´, Á= 1,2,3, >> Here, the °( Â) are the associated Legendre functions andÀ+1 ¸2( ¬)= »+1 ¸2( 1)²+1 ¸2( ¬)-²+1 ¸2( 1) »+1 ¸2( ¬). The square of the norm of an eigenfunction:¤(1) ° ¤2=4 ¥°( Ã+ Á)!Ä(2 Ã+ 1)( Ã- Á)! »2+1 ¸2( 1)- »2+1 ¸2( 2) 2 »2+1 ¸2( 2), ¥°= ©2for Á= 0, 1for Á¹ 0,¤(1) ° ¤2= ¤(2) ° ¤2, Á= 1,2,3, >> 8.3.5. Other Orthogonal Curvilinear Coordinates The homogenous three-dimensional Helmholtz equation admits separation of variables in the eleven orthogona lsystem sofcoordinate slistedinTable29. For the parabolic cylindrical system of coordinates, the multipliers «and Åare expressed in terms of the parabolic cylinder functions as«( Â)= Æ1 Ç È-1 ¸2( É Â)+ Æ2 Ç È-1 ¸2(- É Â), Å( Ê)= b1 Ç- È-1 ¸2( É Ê)+ b2 Ç- È-1 ¸2(- É Ê), =1 2 Ë( º2- ¹ )-1 ¸2, É= Ì4( º2- ¹ ) Í1 ¸4, where Æ1, b1, Æ2, and b2are arbitrary constants. For the elliptic cylindrical system of coordinates, the functions «and Åare determined by the modi®ed Mathieu equation and Mathieu equation, respectively, so that«( Î)= ©Ce( Î, Ï), Se( Î, Ï), Å( Ð)= ©ce( Ð, Ï), se( Ð, Ï), Ï=1 4 Ñ2( ¹ - º2), where Ce( Î, Ï) and Se( Î, Ï) are the modi®ed Mathieu functions, and ce( Ð, Ï) and se( Ð, Ï) are the Mathieu functions; to each value of the parameter Ïthere are certain corresponding eigenvaluesË=Ë( Ï) [see Abramowitz and Stegun (1964)]. In the prolate and oblate spheroidal systems of coordinates, the equations for «and Åare different forms of the spheroidal wave equation, whose bounded solutions are given by«( Î)=Ps| Ò|(cosh Î,Ñ2 ¹ ), Å( Î)=Ps| Ò|(cos Ð,Ñ2 ¹ ) for prolate spheroid,«( Î)=Ps| Ò|(- Ósinh Î,Ñ2 ¹ ), Å( Î)=Ps| Ò|(cos Ð,-Ñ2 ¹ ) for oblate spheroid,ºis an integer, Ã= 0,1,2, Ô>Ô>Ô,- Ã£ º£ Ã, where Ps Ò( Õ,Ñ) are the spheroidal wave functions; see Bateman and Erd Âelyi (1955, V ol. 3), Arscott (1964), and Meixner and Sch Èafke (1965). The separation of variables for the Helmholtz equation in modi®ed prolate and oblate spheroidal systems of coordinates, as well as the spheroidal wave functions, are discussed in Abramowitz and Stegun (1964). In the parabolic coordinate system, the solutions of the equations for «and Åare expressed in terms of the degenerate hypergeometric functions [see Miller, Jr. (1977)] as follows:«( Â)= Â Òexp ÖØ×1 2 Ù Â2 Ú Û Ü- Ë 4Ù+ º+ 1 2, º+ 1; ÝÙ Â2 Þ,Ù= ¶ - ¹ ,Å( Ê)= Ê Òexp ÖØ×1 2 Ù Ê2 Ú Û ÜË 4Ù+ º+ 1 2, º+ 1; ÝÙ Ê2 Þ. Page 591 592 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES TABLE 29 Orthogonal coordinates Å ß,Å à,Å Õthatallowseparable solutions oftheformá= â(Å ß) Å(Å à) ã(Å Õ)forthethree-dimensional Helmholtz equation ä3 á+ ¹á=0 Coordinates Transformations Particular solutions (orequations for â, Å, ã) Cartesianß, à, Õ ß= ß,à= à,Õ= Õ á=cos( º1 ß+ å1)cos( º2 à+ å2)cos( º3 Õ+ å3), where º2 1+ º2 2+ º2 3= ¹ ; seealsoParagraph 8.3.2-1 Cylindrical æ , ´, Õ ß= æ cos ´,à= æ sin ´,Õ= Õ á=[ Æ » ç(Ë æ )+ b² ç(Ë æ )]cos( Ã ´+ è)cos( º Õ+ å), where º2+Ë2= ¹ ,seealsoParagraph 8.3.3-1 ( » çand² çaretheBessel functions) Parabolic cylindricalÂ, Ê, Õ ß=1 2( Â2- Ê2),à= Â Ê,Õ= Õ á= â( Â) Å( Ê)cos( º Õ+ å),â éé+[( ¹ - º2) Â2+Ë] â=0,Å éé+[( ¹ - º2) Ê2-Ë] Å=0 Elliptic cylindricalÎ, Ð, Õ ß=Ñcosh Îcos Ð,à=Ñsinh Îsin Ð,Õ= Õ á= â( Î) Å( Ð)cos( º Õ+ å),â éé+[1 2 Ñ2( ¹ - º2)cosh 2 Î-Ë] â=0,Å éé-[1 2 Ñ2( ¹ - º2)cos2 Ð-Ë] Å=0 Spherical æ , ³, ´ ß= æ sin ³cos ´,à= æ sin ³sin ´,Õ= æ cos ³ á= æ -1 ¸2 »ç +1 ¸2(Ë æ ) ê ë ç(cos ³)cos( Á ´+ å),á= æ -1 ¸2² ç +1 ¸2(Ë æ ) ê ë ç(cos ³)cos( Á ´+ å), where ¹ =Ë2;seealsoParagraph 8.3.4-1 Prolate spheroidalÎ, Ð, ´ ß=Ñsinh Îsin Ðcos ´,à=Ñsinh Îsin Ðsin ´,Õ=Ñcosh Îcos Ð á= â( Î) Å( Ð)cos( º ´+ å),â éé+ â écoth Î+(-Ë+Ñ2 ¹ sinh2Î- º2 ìsinh2Î) â=0,Å éé+ Å écot Ð+(Ë+Ñ2 ¹ sin2Ð- º2 ìsin2Ð) Å=0 Oblate spheroidalÎ, Ð, ´ ß=Ñcosh Îsin Ðcos ´,à=Ñcosh Îsin Ðsin ´,Õ=Ñsinh Îcos Ð á= â( Î) Å( Ð)cos( º ´+ å),â éé+ â étanh Î+(-Ë+Ñ2 ¹ cosh2Î+ º2 ìcosh2Î) â=0,Å éé+ Å écot Ð+(Ë-Ñ2 ¹ sin2Ð- º2 ìsin2Ð) Å=0 ParabolicÂ, Ê, ´ ß= Â Êcos ´,à= Â Êsin ´,Õ=1 2( Â2- Ê2) á= â( Â) Å( Ê)cos( º ´+ å),Â2â éé+ Â â é+( ¹Â4-Ë Â2- º2) â=0,Ê2Å éé+ Ê Å é+( ¹Ê4+Ë Ê2- º2) Å=0 ParaboloidalÎ, Ð, ´ ß=2Ñcosh Îcos Ðsinh ´,à=2Ñsinh Îsin Ðcosh ´,Õ=1 2 Ñ(cosh 2 Î +cos2 Ð-cosh 2 ´) â éé+(- º-Ñ Ëcosh 2 Î+1 2Ñ2 ¹ cosh 4 Î) â=0,Å éé+( º+Ñ Ëcos2 Ð-1 2 Ñ2 ¹ cos4 Ð) Å=0,ã éé+(- º+Ñ Ëcosh 2 ´-1 2Ñ2 ¹ cosh 4 ´) ã=0 General ellipsoidal í , î, ï ß= ð( ñ- ò)( ó- ò)( ô- ò)ò( ò-1),à= ð( ñ-1)( ó-1)( ô-1) 1- ò,õ= ö ñ óôò4 ÷ ø( í )[ ÷ ø( í ) â é] é+( ù í 2+ ú1 í + ú2) â=0, 4 ÷ ø( î)[ ÷ ø( î) û é] é+( ù î2+ ú1 î+ ú2) û=0, 4 ÷ ø( ï)[ ÷ ø( ï) ã é] é+( ù ï2+ ú1 ï+ ú2) ã=0,ø( ü)= ü( ü-1)( ü- ý) Conicalï, í , î ß= ï ð( òþñ-1)( òó-1) 1- ò,à= ï ð ò( ñ-1)( ó-1)ò-1,õ= ï÷ ý íî á= ï-1 ÿ2 ( ç +1 ÿ2) ï ÷ù þû( ) ( ),û +[ ú- ( +1) 2sn2] û=0, +[ ú- ( +1) 2sn2] =0, where =sn2( , ), =sn2( , ), =÷ ý Page592 8.4. OTHER EQUATIONS WITH THREE SPACEVARIABLES 593 Inthecase oftheparaboloidal coordinate system, theequations for , û,and arereduced to theWhittak er±Hill equation + +1 8 2+cos2 -1 8 2cos4 =0. Denote bygc ( ;,)andgs ( ;,),respecti vely,theevenandodd 2 -periodic solutions of theWhittak er±Hill equation, which isageneralization oftheMathieu equation. The subscript =0,1,2, labels thediscrete eigen values = .Each ofthesolutions gc andgs can berepresented intheform ofanin®nite convergent trigonometric series incos andsin , respecti vely;seeUrvin andArscott (1970). Thefunctions , û,and canbeexpressed interms of theperiodic solutions oftheWhittak er±Hill equation asfollows[Miller ,Jr.(1977)]: ( Î)= gc Î;2 ý ,1 2 ú , gs Î;2 ý ,1 2 ú , û( )= gc ;2 ý ,1 2 ú , gs ;2 ý ,1 2 ú , ( ø)= gc ø+ 2;2 ý ,1 2 ú , gs ø+2;2 ý ,1 2 ú , where = ÷ùand = -1 2 ý2ù. Forthegeneral ellipsoidal coordinates, thefunctions , û,and areexpressed interms ofthe ellipsoidal wavefunctions; fordetails, seeArscott (1964) andMiller ,Jr.(1977). Fortheconical coordinate system, thefunctions ûand aredetermined bytheLam Âeequations thatinvolvetheJacobian elliptic function sn õ=sn( õ, ). Theunambiguity conditions forthetransformation yield =0,1,2, Itisknownthat, forany positi veinteger ,there existexactly 2 +1solutions corresponding to2 +1different eigen values ú. These solutions canberepresented theform of®nite series knownasLam Âepolynomials. Formore details about theLam Âeequation anditssolutions, seeWhittak erandWatson (1963), Arscott (1964), Bateman andErdÂelyi(1955), andMiller ,Jr.(1977). Unlik etheLaplace equation, there arenonontri vialtransformations forthethree-dimensional Helmholtz equation thatallowthe -separation ofvariables.!#" Refer ences forSubsection 8.3.5: F.M.Morse andH.Feshbach (1953, Vols.1±2), P.Moon andD.Spencer (1961), A.Makaro v,J.Smorodinsk y,K.Valiev,andP.Winternitz (1967), W.Miller ,Jr.(1977). 8.4. Other Equations with Three Space Variab les 8.4.1. Equations Containing Arbitrar yFunctions 1. $2 %$ &2+ $2 %$ '2+ $2 %$ (2+ ) *+ + , - %=0, ,2=&2+'2+(2. SchrÈoding er'sequation. Itgoverns themotion ofanelectron intheCoulomb ®eld ofanucleus ( ý>0). Thedesired solutions must satisfy thenormalizing condition. / - / . / - / . / - /| 0( 1, 2, õ)|2 31 32 3 õ=1. Eigen values:ù =- ý2 4 2; =1,2,3, Normalized eigenfunctions (inthespherical coordinate system 4, , ø):0 5 6= )2 -3 ÿ2 7(2 8+1)( 8- 9)!( :- 8-1)! 4 ; < :( :+ 8)!( 9+ 8)! = > ? : @ Aexp=-> ?2 : @ B2A+1-A-1= > ? : @ C( <)A( , D),:=1,2,3, ; 9=0, E1, E2, , E 8; 8=0,1,2 , :-1; Page593 594 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES where; <= F2for 9=0, 1for 9¹0,C( <)A( , D)= G HI JA(cos ) for 9=0,J <A(cos )sin 9 D for 9=1,2, ,J| <|A(cos )cos 9 Dfor 9=-1,-2, ,B KA( 1)=18! 1-KML N OAO 1A P 1A+KQL-N R,J <A( 1)=(1- 12) < S2O <O 1 < JA( 1),JA( 1)=18!2A OAO 1A( 12-1)A. These relations involvethegeneralized Laguerre polynomialsBKA( 1)andtheassociated Legendre functionsJ <( T);theJ ( T)aretheLegendre polynomials.U#V Refer ences :G.KornandT.Korn(1968), A.N.Tikhono vandA.A.Samarskii (1990). 2. $2 %$ &2+ $2 %$ '2+ $2 %$ W2= X' $ %$ &. This equation isencountered inproblems ofconvectiveheat andmass transfer inasimple shear ¯ow. Fundamental solution:Y Y( 1, Z, [, T, \, ])=1 (4 ^)3 S2 _ `0exp F-[ 1- T-1 2> a( Z+ \)]2 4a(1+1 12>2a2)-( Z- \)2+( [- ])2 4a b O a ca3(1+1 12>2a2).U#V Refer ences :E.A.Novikov(1958), D.E.Elrick (1962). 3. d2 ed &2+ d2 ed '2+ d2 ed W2+ X1& d ed &+ X2' d ed '+ X3W d ed W=0. This equation isencountered inproblems ofconvectiveheat andmass transfer inastraining ¯ow. Fundamental solution:Y Y( f, Z, [, T, \, ])=_`0 g( f, T,a;>1)g( Z, \,a;>2)g( [, ],a;>3)O a,g( f, T,a;>)= h2 ^>P L2 ij-1Rk-1 S2 exp h->( fL ij- T)2 2(L2 ij-1) k. 4. d2 ed &2 1+ d2 ed &2 2+ d2 ed &2 3=3lm, n=1 X mn& md ed & n. This equation isencountered inproblems ofconvectiveheat andmass transfer inanarbitrary linear shear ¯ow. Thesolution thatcorresponds toasource ofunitpowerattheorigin ofcoordinates isgivenbyo( f1, f2, f3)=1 (4 ^)3 p2 _`0exp h-3lq, r=1 s qr(a) f qfr 4 t(a) k u a vt(a). Here, t= t(a)isthedeterminant ofthematrix B={ w qr};thes qr=s qr(a)arethecofactors oftheentries w qr= w qr(a);the w qraredetermined bysolving thefollowing system ofordinary differential equations with constant coef®cients:u w qru a= x qr+3ly=1 z q ywr y+3ly=1 z r yw q y,w qr { x qraasa {0(initial conditions ), where x q q=1and x qr=0if |¹ }.~# Refer ence:G.K.Batchelor (1979). Page594 8.4. OTHER EQUATIONS WITH THREE SPACEVARIABLES 595 5. dd h1() d ed k+ dd h2() d ed k+ dd h3() d ed k= e. This isathree-dimensional linear equation ofheat andmass transfer theory with asource inan inhomogeneous anisotropic medium. Here, 1= 1( f), 2= 2( ),and 3= 3( )aretheprincipal thermal diffusivities. 1 .Theequation admits multiplicati velyseparable solutions, o( f, , )= 1( f) 2( ) 3( ). 2 .There arealsoadditi velyseparable solutions, o( f, , )= 1( f)+ 2( )+ 3( ). 3 .If 1=z f q, 2=s y,and 3= r( |¹2, ¹2, }¹2),there areparticular solutions ofthe formo= o( ), 2=4 h 2- qz(2- |)2+ 2- ys(2- )2+ 2- r(2- })2k, where thefunction o( )isdetermined bytheordinary differential equationu2 ou 2+ u ou = o,=2 1 2- |+1 2- +1 2- } -1, whose solutions areexpressed interms oftheBessel functions. 8.4.2. Equations oftheForm div[ ( , , )Ñ ]± ( , , ) =± ( , , ) Equations ofthissort areoften encountered inheat andmass transfer theory .Forbrevity,the equation iswritten using thenotation div[z(r)Ñ o]= z(r) o + z(r) o + z(r) o , r={, , }. Inwhat follows,theproblems fortheequation inquestion will beconsidered inabounded domain with asuf®ciently smooth surface .Itisassumed thatz(r)>0and ¡(r)³0. 8.4.2-1. First boundary value problem. Thefollowing boundary condition ofthe®rstkind isimposed:o= (r)for r ¢ . Solution:o(r)= £ ¤ ¥( ¦) §(r, ¦)u ¨- £ © ( ¦)z( ¦) ª ¨ §(r, ¦)u ¨. (1) Here, theGreen' sfunction isgivenby§(r, ¦)= « ¬q=1 q(r) q( ¦)® q ®2 ¯q, ® q ®2= ° ±2q(r) ² ³, ´={ µ, ¶, ·}, (2) where the ¯qand q(r)aretheeigen values andeigenfunctions oftheSturm±Liouville problem for thefollowing second-order elliptic equation with ahomogeneous boundary condition ofthe®rst kind: div ¸#¹(r)Ñ º- »(r)+ ¯=0, (3)=0for r ¼ ½. (4) The integration in(1)isperformed with respect to µ, ¶, ·; ¾¾ ¿ Àdenotes thederivativealong the outw ardnormal tothesurface ½with respect to µ, ¶, ·. General properties oftheSturm±Liouville problem (3)±(4): Page595 596 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 1 Á.There arecountably manyeigen values. Alleigen values arerealandcanbeordered sothat¯1£ ¯2£ ¯3£ ÂÂÂ,with ¯q Ã Äas Å Ã Ä;therefore thenumber ofnegativeeigen values is®nite. 2 Á.If ¹(r)>0and »(r)³0,alleigen values arepositi ve, ¯q>0. 3 Á.Theeigenfunctions arede®ned uptoaconstant multiplier .Anytwoeigenfunctions, q(r)and Æ(r),corresponding todifferent eigen values, ¯qand ¯Æ,areorthogonal toeach other in ³:°± q(r) Æ(r) ² ³=0for Å¹ . 4 Á.Anarbitrary function Ç(r)thatistwice continuously differentiable andsatis®es theboundary condition oftheSturm±Liouville problem ( Ç=0forr ¼ ½)canbeexpanded intoanabsolutely and uniformly convergent series intheeigenfunctions; speci®cally ,Ç(r)= È ¬q=1 Ç q q(r), Ç q=1® q ®2 ° ± Ç(r) q(r) ² ³, where thenorm squared ® q ®2isde®ned in(2).É ÊË Ì ÍÏÎ ÐInathree-dimensional problem, toeach eigen value ¯q®nitely manylinearly inde- pendent eigenfunctions(1)q, ÑÑÑ,(Æ)qgenerally correspond. These functions canalwaysbereplaced bytheir linear combinations Å( r)q= Òr,1(1)q+ ÂÂÂ+ Òr, r-1( r-1)q+( r)q, Ó=1,2, ÑÑÑ, Ô, such that Å(1)q, ÑÑÑ,Å(Æ)qarenowpairwise orthogonal. Therefore, without lossofgenerality ,wecan assume thatalleigenfunctions areorthogonal. 8.4.2-2. Second boundary value problem. Aboundary condition ofthesecond kind isimposed,Õ ÖÕ ×= Ø(r)for r ¼ ½. Itisassumed that »(r)>0. Solution:Ö(r)= °± Ù( ´) Ú(r, ´) ² ³ Û+ ° Ü Ø( ´) ¹( ´) Ú(r, ´) ² ½ Û. (5) Here, theGreen' sfunction isde®ned byrelation (2),where the ¯qand q(r)aretheeigen values andeigenfunctions oftheSturm±Liouville problem forthesecond-order elliptic equation (3)with thefollowing homogeneous boundary condition ofthesecond kind:Õ Õ ×=0for r ¼ ½. (6) If »(r)>0,thegeneral properties oftheeigen valueproblem (3),(6)arethesame asthose ofthe ®rstboundary value problem (seeParagraph 8.4.2-1). 8.4.2-3. Third boundary value problem. Thefollowing boundary condition ofthethird kind isset:Õ ÖÕ ×+ Ó(r) Ö= Ø(r)for r ¼ ½. The solution ofthethird boundary value problem isgivenbyrelations (5)and(2),where the ¯qand q(r)aretheeigen values andeigenfunctions oftheSturm±Liouville problem forthe Page596 8.5. EQUATIONS WITH ÝSPACEVARIABLES 597 second-order elliptic equation (3)with thefollowing homogeneous boundary condition ofthethird kind:Õ Õ ×+ Ó(r)=0for r ¼ ½. (7) If »(r)³0and Ó(r)>0,thegeneral properties oftheeigen value problem (3),(7)arethesame asthose ofthe®rstboundary value problem (seeParagraph 8.4.2-1). Let Ó(r)= Ó=const .Denote theGreen' sfunctions ofthesecond andthird boundary value problems by Ú2(r, ´)and Ú3(r, ´, Ó),respecti vely.For »(r)>0,thefollowing limit relation holds:Ú2(r, ´)=limrÞ0 Ú3(r, ´, Ó). 8.5. Equations with ßSpace Variab les 8.5.1. Laplace Equation à á â=0 The Å-dimensional Laplace equation intherectangular Cartesian system ofcoordinates ã1, ÑÑÑ, ã q hastheformÕ2 ÖÕã2 1+ Õ2 ÖÕã2 2+ ÂÂÂ+ Õ2 ÖÕã2q=0. For Å=2and Å=3,seeSubsections 7.1.1 and8.1.1. Aregular solution oftheLaplace equation iscalled aharmonic function. Inwhat followsweusethenotation: x={ ã1, ÑÑÑ, ã q}and|x|= ä ã2 1+ ÂÂÂ+ ã2q. 8.5.1-1. Particular solutions. 1 Á.Fundamental solution: å å (x)=-1 ( Å-2) æ q|x| q-2, æ q=2 ç qp2è( Å é2)( Å³3). 2 Á.Solution containing arbitrary functions of Å-1variables:Ö( ã1, ÑÑÑ, ã q)=È ¬r=0(-1) r ê ã2 rq (2 Ó)! ë rØ( ã1, ÑÑÑ, ã q-1)+ ã2 r+1q (2 Ó+1)! ë r ì( ã1, ÑÑÑ, ã q-1) í, where Ø( ã1, ÑÑÑ, ã q-1)and ì( ã1, ÑÑÑ, ã q-1)arearbitrary in®nitely differentiable functions. 3 Á.Let Ö( ã1, ÑÑÑ, ã q)beaharmonic function. Then thefunctionsÖ 1= Ò Ö( î ¯ã1+ ï1, ÑÑÑ, î ð ã q+ ï q),Ö 2= Ò |x| q-2 Ö ñã1 |x|2, ÑÑÑ, ã q |x|2 ò, arealso harmonic functions everywhere theyarede®ned; Ò, ï1, ÑÑÑ, ï q,and ðarearbitrary constants. Thesigns at ðintheexpression of Ö 1canbetakenindependently ofoneanother .ó#ô Refer ences :A.V.Bitsadze andD.F.Kalinichenk o(1985), R.Courant andD.Hilbert (1989). 8.5.1-2. Domain: - õ< ã1< õ, ööö,- õ< ã q-1< õ,0£ ã q< õ. The ®rst boundary value problem foran ÷-dimensional half-space isconsidered. Aboundary condition isprescribed:Ö= Ø( ã1, ööö, ã q-1)at ã q=0. Solution:Ö( ã1, ööö, ã q)= è( ÷ é2)ç qp2 ø ù-ù öööø ù-ù ê q-1úr=1( ûr- ür)2+ ü2q ý- q þ2ü q ÿ( û1, ööö, û q-1) û1 ööö û q-1, where ( )isthegamma function.ó#ô Refer ence:A.V.Bitsadze andD.F.Kalinichenk o(1985). Page597 598 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 8.5.1-3. Domain: |x|£1.First boundary value problem. Asphere ofunitradius inthe ÷-dimensional space isconsidered. Aboundary condition isprescribed:= ÿ(x)for |x|=1. Solution (Poisson integral):(x)= ( ÷ 2) 2 q þ2ø |y|=11-|x|2 |y-x| q ÿ(y) y.ó#ô Refer ence:A.V.Bitsadze andD.F.Kalinichenk o(1985). 8.5.2. Other Equations 1. =± ( 1, , ). This isthePoisson equation in ÷independent variables. For ÷=2and ÷=3,seeSections 7.2 and8.2. 1 .Solution:( ü1, ööö, ü q)= ( ÷ 2) 2( ÷-2) q þ2ø ( û1, ööö, û q) û1 ööö û q ( ü1- û1)2+ +( ü q- û q)2 q-2 2.ó#ô Refer ence:S.G.Krein (1972). 2 .Domain: 0£ ür£ r; =1, ööö, ÷.First boundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:= ÿr( ü1, ööö, ür-1, ür+1, ööö, ü q)at ür=0,= r( ü1, ööö, ür-1, ür+1, ööö, ü q)at ür= r. Green' sfunction:( ü1, ööö, ü q, û1, ööö, û q)=2 q1 ööö q ù r1=1 öööù r =1sin( r1 ü1)sin( r1 û1) ööösin( r ü q)sin( r û q)2r1+ + 2r ,r1= 11, r2= 22, ööö, r = q q. 3 .Domain: 0£ ür£ r; =1, ööö, .Mixedboundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:= ÿr( ü1, ööö, ür-1, ür+1, ööö, ü q)at ür=0, = r( ü1, ööö, ür-1, ür+1, ööö, ü q)at ür= r. Green' sfunction:( ü1, ööö, ü q, û1, ööö, û q)=2 q1 ööö q ù r1=0 öööù r =0sin( r1 ü1)sin( r1 û1) ööösin( r ü q)sin( r û q)2r1+ + 2r ,r1= (2 1+1) 2 1, r2= (2 2+1) 2 2, ööö, r = (2 q+1) 2 q. 2. + ! =0. This istheHelmholtz equation in independent variables. For =2and =3,seeSections 7.3 and8.3. Page598 8.5. EQUATIONS WITH "SPACEVARIABLES 599 1 .Fundamental solution for #= 2>0:$ $(x,y)= q-2 2 4(2 ) q-2 2 %- q-2 2 & q-2 2( %),%=|x-y| foreven ,$ $(x,y)= q-2 2 4(2 ) q-2 2sin '1 2 ( %- q-2 2 )- q-2 2( %) forodd , where ) *( )and & *( )aretheBessel functions. 2 .Domain: 0£ ür£ r; =1, ööö, .First boundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:= ÿr( ü1, ööö, ür-1, ür+1, ööö, ü q)at ür=0,= r( ü1, ööö, ür-1, ür+1, ööö, ü q)at ür= r. Green' sfunction:( ü1, ööö, ü q, û1, ööö, û q)=2 q1 2 ööö q ù r1=1 öööù r =1sin( r1 ü1)sin( r1 û1) ööösin( r ü q)sin( r û q)2r1+ + 2r - #,r1= 11, r2= 22, ööö, r = q q. 3 .Domain: 0£ ür£ r; =1, ööö, .Second boundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed: = ÿr( ü1, ööö, ür-1, ür+1, ööö, ü q)at ür=0, = r( ü1, ööö, ür-1, ür+1, ööö, ü q)at ür= r. Green' sfunction:( ü1, ööö, ü q, û1, ööö, û q)=ù r1=0 öööù r =0 +r1 r2 ,-,-, r cos( r1 ü1)cos( r1 û1) öööcos( r ü q)cos( r û q)2r1+ + 2r - #,+r1 r2 ,-,-, r = . r1. r2 ööö. r 1 2 ööö q, r1= 11, r2= 22, ööö, r = q q,. /= 01for 1=0, 2for 1¹0. 4 .Domain: 0£ ür£ r; =1, ööö, .Mixedboundary value problem. Arectangular parallelepiped isconsidered. Boundary conditions areprescribed:= ÿr( ü1, ööö, ür-1, ür+1, ööö, ü q)at ür=0, = r( ü1, ööö, ür-1, ür+1, ööö, ü q)at ür= r. Green' sfunction:( ü1, ööö, ü q, û1, ööö, û q)=2 q1 2 ööö q ù r1=0 öööù r =0sin( r1 ü1)sin( r1 û1) ööösin( r ü q)sin( r û q)2r1+ + 2r - #,r1= (2 1+1) 2 1, r2= (2 2+1) 2 2, ööö, r = (2 q+1) 2 q.2#ô Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). Page599 600 ELLIPTIC EQUATIONS WITH THREE ORMORESPACEVARIABLES 3. 3, 4=1 5 34 62 6 36 4=0. Itisassumed thatforanyrealnumbers û1, ööö, û qtherelation 777 qú8, 9=1 89 û 8û9 777³ qú8=1 û28holds, whereissome positi veconstant. Fundamental solution:$ $( ü1, ööö, ü q, û1, ööö, û q)= :;;;;;;<;;;;;;= ( 2) 2( -2) q þ2 > + ? q8, 9=1s 89( ü 8- û 8)( ü9- û9) ý- q-2 2 for ³3, 1 2 > +ln?28, 9=1s 89( ü 8- û 8)( ü9- û9) ý-1 þ2 for =2, where +isthedeterminant ofthematrix A={ 89}andthes 89aretheentries oftheinverseofA.2#ô Refer ence:V.M.Babich, M.B.Kapile vich, S.G.Mikhlin, etal.(1964). 4. ±13=1 62 6 23+66 @ A 6 6 B+ ! =0. Domain: 8£ ü 8£s 8( C=1, ööö, -1),0£ ü q£ D. 1 .Case 0< E<1.First boundary valueproblem. Thecondition =0issetontheentire boundary ofthedomain. Eigen values andeigenfunctions:#r1,,-,-,, r -1,/= q-18=1 282 (s 8- 8)2+(2- E)2 F*/4 D2- G, H 1,,-,-,, H-1,/= ü1- G 2I) *@ F*/ J ü I K2- G 2B I-1L8=1sin M 8ON( P 8- Q 8)R8- Q 8, where F*/isthe 1thpositi verootoftheequation ) *( F)=0,M1, ööö,M I-1=1,2, ööö; 1=1,2, ööö; S=1- E 2- E. 2 T.Case 1£ E<2.Boundary conditions: thesolution must bebounded at P I=0,andthecondition=0must hold ontherestoftheboundary ofthedomain. Theeigen values andeigenfunctions ofthisproblem aregivenbytherelations ofItem 1 TwithS=( E-1) U(2- E).2#ô Refer ence:M.M.Smirno v(1975). Page600 Chapter 9 Higher -OrderPartialDifferential Equations 9.1. Third-Or derPartialDifferential Equations 1. + 3 3=0. Linearized Corte veg±de Vries equation . 1 .Particular solutions:( , )= ( 3-6 )+ 2+ + ,( , )= ( 5-60 2)+ ( 4-24 ),( , )= sin( + 3)+ cos( + 3)+ ,( , )= sinh( - 3)+ cosh( - 3)+ ,( , )=exp - 3 exp + exp -1 2 sin 3 2 + , where , , , ,and arearbitrary constants. 2.Domain: - < £0.Boundary value problem. Initial andboundary conditions areprescribed:=0at =0, = ( )at =0, 0as - . Solution:( , )=-3 2 0Ai ( - )1 !3 " ( )- # , where Ai ( $)isthesecond derivativeoftheAiry function. 3 .Domain: 0£ < .Thefunction( , )=3 0Ai ( - )1 !3" ( )- # , satis®es theequation andthe®rsttwoconditions speci®ed inItem 2.%& Refer ence:A.V.Faminskii (1999). 2. = ' 6 3 3. Thetransformation ( ( $, )= -2, $=1 ) , = leads toaconstant coef®cient equation oftheform 9.1.1:* (* =- *3 (*$3. Page601 602 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 3. = +( ) 3 3+ ,( )+ -( ) + .( ) . Thetransformation( , )= ( ( $, )exp / 0 ( )# 21, $= 3( )+ 4 ( ) 3( )# , = ( ) 33( )# , where 3( )=exp / ( )# 21,leads toaconstant coef®cient equation oftheform 9.1.1:* (* = *3 (*$3. 4. =( ' 2+ 5 + 6)3 3 3. This isaspecial case ofequation 9.6.4.4 with =1and 7=3.Thetransformation( , )= ( ( $, )( 2+ + ), $= # 2+ + leads totheconstant coef®cient equation* (*= *3 (*$3+(4 - 2) * (*$. 5. 2 2= ' 6 3 3. Thetransformation $=1 ) , ( = -2leads totheconstant coef®cient equation*2 (*2=- *3 (*$3. 6. 2 2=( ' 2+ 5 + 6)3 3 3. This isaspecial case ofequation 9.6.4.4 with =2and 7=3.Thetransformation( , )= ( ( $, )( 2+ + ), $= # 2+ + leads totheconstant coef®cient equation*2 (*2= *3 (*$3+(4 - 2) * (*$. 9.2. Four th-Or derOne-Dimensional Nonstationar y Equations 9.2.1. Equations oftheForm 8 98 :+ ;28498 <4= =(<,:) 9.2.1-1. Particular solutions ofthehomogeneous equation ( >º0):( )= ? 3+ @ 2+ A + B,( , )= ?( 5-120 2 )+ @( 4-24 2),( , )= ?sin( )+ @cos( )+ Asinh( )+ Bcosh( ) exp(- 42), where ?, @, A, B,and arearbitrary constants. Page602 9.2. FOUR TH-ORDER ONE-DIMENSION ALNONST ATION ARYEQUATIONS 603 9.2.1-2. Domain: 0£ £ C.Solution interms oftheGreen' sfunction. 1.Weconsider problems onaninterv al0£ £ Cwith thegeneral initial condition= ( )at =0 andvarious homogeneous boundary conditions. The solution canberepresented interms ofthe Green' sfunction as( , )= D 0 ( E) F( , E, )# E+ 0 D 0 >( E, ) F( , E, - )# E# . 2.Paragraphs 9.2.1-3 through 9.2.1-10 present theGreen' sfunctions forvarious types ofboundary conditions. TheGreen' sfunctions canbeevaluated from theformulaF( , E, )= G HJI =1 K I ( )K I ( E)LK IL2exp(- 4 I2), (1) where the I andK I ( )aredetermined bysolving theself-adjoint eigen value problem forthe fourth-order ordinary differential equationK - 4K=0 subject toappropriate boundary conditions; theprime denotes differentiation with respect to .The norms ofeigenfunctions canbecalculated bytheformulaLK IL2= D 0K2 I ( )# = C 4K2 I ( C)+ C 4 4 IK I ( C) 2- C 2 4 IK I( C)K I ( C). (2) Relations (1)and(2)arewritten under theassumption that =0isnotaneigen value. 9.2.1-3. The function andits®rst derivativeareprescribed attheboundaries:= * M=0at =0, = * M=0at = C. Green' sfunction:F( , E, )=4C G HJI =1 4 IK I ( C) 2K I ( )K I ( E)exp(- 4 I2), whereK I ( )= sinh( IC)-sin( IC) cosh( I)-cos( I) - cosh( IC)-cos( IC) sinh( I)-sin( I) ; the I arepositi veroots ofthetranscendental equation cosh( C)cos( C)=1.Thenumerical values oftheroots canbecalculated from theformulas giveninParagraph 9.2.3-2. 9.2.1-4. The function anditssecond derivativeareprescribed attheboundaries:= * M M=0at =0, = * M M=0at = C. Green' sfunction:F( , E, )=2C G HI =1sin( I)sin( IE)exp(- 4 I2), I = N 7C. Page603 604 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.2.1-5. The ®rst andthird derivativesareprescribed attheboundaries:* M= * M M M=0at =0, * M= * M M M=0at = C. Green' sfunction:F( , E, )=1C+2C G HJI =1cos( I)cos( IE)exp(- 4 I2), I = N 7C. 9.2.1-6. The second andthird derivativesareprescribed attheboundaries: M M= * M M M=0at =0, M M= * M M M=0at = C. Green' sfunction:F( , E, )=1C+3C3(2 - C)(2 E- C)+4C G HJI =1 K I ( )K I ( E)K2 I ( C)exp(- 4 I2), whereK I ( )= sinh( IC)-sin( IC) cosh( I)+cos( I) - cosh( IC)-cos( IC) sinh( I)+sin( I) ; the I arepositi veroots ofthetranscendental equation cosh( C)cos( C)=1.Thenumerical values oftheroots canbecalculated from theformulas giveninParagraph 9.2.3-2. 9.2.1-7. Mixedconditions areprescribed attheboundaries (case 1):= * M=0at =0, = * M M=0at = C. Green' sfunction:F( , E, )=2C G HJI =1 4 IK I ( )K I ( E) |K I( C)K I ( C)|exp(- 4 I2), whereK I ( )= sinh( IC)-sin( IC) cosh( I)-cos( I) - cosh( IC)-cos( IC) sinh( I)-sin( I) ; the I arepositi veroots ofthetranscendental equation tan( C)-tanh( C)=0. 9.2.1-8. Mixedconditions areprescribed attheboundaries (case 2):= * M=0at =0, * M M= * M M M=0at = C. Green' sfunction:F( , E, )=4C G HJI =1K I ( )K I ( E)K2 I ( C)exp(- 4 I2), whereK I ( )= sinh( IC)+sin( IC) cosh( I)-cos( I) - cosh( IC)+cos( IC) sinh( I)-sin( I) ; the I arepositi veroots ofthetranscendental equation cosh( C)cos( C)=-1. Page604 9.2. FOUR TH-ORDER ONE-DIMENSION ALNONST ATION ARYEQUATIONS 605 9.2.1-9. Mixedconditions areprescribed attheboundaries (case 3):= * M M=0at =0, * M= * M M M=0at = C. Green' sfunction:F( , E, )=2C G HI =0sin( I)sin( IE)exp(- 4 I2), I = N(2 7+1) 2 C. 9.2.1-10. Mixedconditions areprescribed attheboundaries (case 4):= * M M=0at =0, * M M= * M M M=0at = C. Green' sfunction:F( , E, )=4C G HJI =1 K I ( )K I ( E)K2 I ( C)exp(- 4 I2), whereK I ( )=sin( IC)sinh( I)+sinh( IC)sin( I); the I arepositi veroots ofthetranscendental equation tan( C)-tanh( C)=0. 9.2.2. Equations oftheForm 8298 :2+ ;28498 <4=0 This equation isencountered instudying transv ersevibration ofelastic rods. 9.2.2-1. Particular solutions:( , )=( ? 3+ @ 2+ A + B) + ?1 3+ @1 2+ A1 + B1,( , )= O?sin( )+ @cos( )+ Asinh( )+ Bcos( ) sin( 2 ),( , )= O?sin( )+ @cos( )+ Asinh( )+ Bcos( ) cos( 2 ), where ?, @, A, B, ?1, @1, A1, B1,and arearbitrary constants. 9.2.2-2. Domain: - < < .Cauchy problem. Initial conditions areprescribed:= ( )at =0, * = 4( )at =0. Boussinesq solution:( , )=1P 2N G -G -2 E P cos E2+sin E2# E +1 P 2N G -G 4-2 E P cos E2-sin E2# E.%& Refer ence:I.Sneddon (1951). 9.2.2-3. Domain: 0£ < .Free vibration ofasemiin®nite rod. Thefollowing conditions areprescribed:=0 at =0, * =0at =0 (initial conditions ),= ( )at =0, * M M=0at =0 (boundary conditions ). Boussinesq solution:( , )=1PN G M!2 Q - 2 2 E2" sin E2 2+cos E2 2 "# E.%& Refer ence:I.Sneddon (1951). Page605 606 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.2.2-4. Domain: 0£ £ C.Boundary value problems. Forsolutions ofvarious boundary value problems, seeSubsection 9.2.3 for >º0. 9.2.3. Equations oftheForm 8298 :2+ ;28498 <4= =(<,:) This equation isencountered instudying forced (transv erse) vibration ofelastic rods. 9.2.3-1. Domain: 0£ £ C.Solution interms oftheGreen' sfunction. 1.Weconsider boundary value problems onaninterv al0£ £ Cwith thegeneral initial condition= ( )at =0, * = 4 ( )at =0 andvarious homogeneous boundary conditions. The solution canberepresented interms ofthe Green' sfunction as( , )= ** D 0 ( E) F( , E, )# E+ D 0 4 ( E) F( , E, )# E+ 0 D 0 >( E, ) F( , E, - )# E# . 2.Paragraphs 9.2.3-2 through 9.2.3-9 present theGreen' sfunctions forvarious types ofboundary conditions. TheGreen' sfunctions canbeevaluated from theformulaF( , E, )=1 G HI =1 K I ( )K I ( E)2 ILK IL2sin( 2 I ), (1) where the I andK I ( )aredetermined bysolving theself-adjoint eigen value problem forthe fourth-order ordinary differential equationK - 4K=0 subject toappropriate boundary conditions; theprime denotes differentiation with respect to .The norms ofeigenfunctions canbecalculated byKrylo v'sformula [seeKrylo v(1949)]:LK IL2= D 0K2 I ( )# = C 4K2 I ( C)+ C 4 4 IK I ( C) 2- C 2 4 IK I( C)K I ( C). (2) Relations (1)and(2)arewritten under theassumption that =0isnotaneigen value. 9.2.3-2. Both ends oftherodareclamped. Boundary conditions areprescribed:= * M=0at =0, = * M=0at = C. Green' sfunction:F( , E, )=4 C G HJI =1 2 IK I ( C) 2K I ( )K I ( E)sin( 2 I ), whereK I ( )= sinh( IC)-sin( IC) cosh( I)-cos( I) - cosh( IC)-cos( IC) sinh( I)-sin( I) ; the I arepositi veroots ofthetranscendental equation cosh( C)cos( C)=1.Thenumerical values oftheroots canbecalculated from theformulas I = R IC,whereR1=1.875,R2=4.694,R I = N2(2 7-1)for 7³3.%& Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). Page606 9.2. FOUR TH-ORDER ONE-DIMENSION ALNONST ATION ARYEQUATIONS 607 9.2.3-3. Both ends oftherodarehinged. Boundary conditions areprescribed:= * M M=0at =0, = * M M=0at = C. Green' sfunction:F( , E, )=2 CN2 G HJI =1172sin( I)sin( IE)sin( 2 I ), I = N 7C.%& Refer ences :A.N.Krylo v(1949), B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 9.2.3-4. Both ends oftherodarefree. Boundary conditions areprescribed: M M= * M M M=0at =0, M M= * M M M=0at = C. Green' sfunction:F( , E, )= C+3 C3(2 - C)(2 E- C)+4 C G HI =1 K I ( )K I ( E)2 IK2 I ( C)sin( 2 I ), whereK I ( )= sinh( IC)-sin( IC) cosh( I)+cos( I) - cosh( IC)-cos( IC) sinh( I)+sin( I) ; the I arepositi veroots ofthetranscendental equation cosh( C)cos( C)=1.Forthenumerical values oftheroots, seeParagraph 9.2.3-2. The®rsttwoterms intheexpression oftheGreen' sfunction correspond tothezero eigen value0=0,towhich twoorthogonal eigenfunctions (1) 0=1and (2) 0=2 - Ccorrespond with L(1) 0 L2= C and L(2) 0 L2=1 3 C3.%& Refer ence:A.N.Krylo v(1949). 9.2.3-5. One endoftherodisclamped andtheother ishinged. Boundary conditions areprescribed:= * M=0at =0, = * M M=0at = C. Green' sfunction:F( , E, )=2 C G HI =1 2 IK I ( )K I ( E) |K I( C)K I ( C)|sin( 2 I ), whereK I ( )= sinh( IC)-sin( IC) cosh( I)-cos( I) - cosh( IC)-cos( IC) sinh( I)-sin( I) ; the I arepositi veroots ofthetranscendental equation tan( C)-tanh( C)=0. Page607 608 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.2.3-6. One endoftherodisclamped andtheother isfree. Boundary conditions areprescribed:= * M=0at =0, * M M= * M M M=0at = C. Green' sfunction:F( , E, )=4 C G HJI =1 K I ( )K I ( E)2 IK2 I ( C)sin( 2 I ), whereK I ( )= sinh( IC)+sin( IC) cosh( I)-cos( I) - cosh( IC)+cos( IC) sinh( I)-sin( I) ; the I arepositi veroots ofthetranscendental equation cosh( C)cos( C)=-1. 9.2.3-7. One endoftherodishinged andtheother isfree. Boundary conditions areprescribed:= * M M=0at =0, * M M= * M M M=0at = C. Green' sfunction:F( , E, )=4 C G HJI =1 K I ( )K I ( E)2 IK2 I ( C)sin( 2 I ), whereK I ( )=sin( IC)sinh( I)+sinh( IC)sin( I); the I arepositi veroots ofthetranscendental equation tan( C)-tanh( C)=0. 9.2.3-8. The ®rst andthird derivativesareprescribed attheends:* M= * M M M=0at =0, * M= * M M M=0at = C. Green' sfunction:F( , E, )= C+2 C G HJI =112 I cos( I)cos( IE)sin( 2 I ), I = N 7C. 9.2.3-9. Mixedboundary conditions areprescribed attheends:= * M M=0at =0, * M= * M M M=0at = C. Green' sfunction:F( , E, )=2 C G HJI =012 I sin( I)sin( IE)sin( 2 I ), I = N(2 7+1) 2 C. 9.2.4. Equations oftheForm 8298 :2+ ;28498 <4+ S9= =(<,:) 9.2.4-1. Particular solutions ofthehomogeneous equation ( >º0):( , )=( ? 3+ @ 2+ A + B)sin T P ,( , )=( ? 3+ @ 2+ A + B)cos T P ,( , )= ?sin( )+ @cos( )+ Asinh( )+ Bcos( ) sin P24+ ,( , )= ?sin( )+ @cos( )+ Asinh( )+ Bcos( ) cos P24+ , where ?, @, A, B,and arearbitrary constants. Page608 9.2. FOUR TH-ORDER ONE-DIMENSION ALNONST ATION ARYEQUATIONS 609 9.2.4-2. Domain: 0£ £ C.Solution interms oftheGreen' sfunction. 1.Weconsider boundary value problems onaninterv al0£ £ Cwith thegeneral initial condition= ( )at =0, * = 4 ( )at =0 andvarious homogeneous boundary conditions. The solution canberepresented interms ofthe Green' sfunction as( , )= ** D 0 ( E) F( , E, )# E+ D 0 4 ( E) F( , E, )# E+ 0 D 0 >( E, ) F( , E, - )# E# . 2.Paragraphs 9.2.4-3 through 9.2.4-10 present theGreen' sfunctions forvarious types ofboundary conditions. TheGreen' sfunctions canbeevaluated from theformulaF( , E, )= G HJI =1 K I ( )K I ( E)LK IL2sin TVU 24 I + U 24 I + , where the I andK I ( )aredetermined bysolving theself-adjoint eigen valueproblem forthefourth- order ordinary differential equationK - 4K=0subject toappropriate boundary conditions. The norms ofeigenfunctions canbecalculated byformula (2)from Paragraph 9.2.3-1. 9.2.4-3. The function andits®rst derivativeareprescribed attheends:= * M=0at =0, = * M=0at = C. Green' sfunction:F( , E, )=4C G HJI =1 4 IK I ( )K I ( E)K I ( C) 2sin VU 24 I + U 24 I + ,K I ( )=#2K I# 2, whereK I ( )= sinh( IC)-sin( IC) cosh( I)-cos( I) - cosh( IC)-cos( IC) sinh( I)-sin( I) ; the I arepositi veroots ofthetranscendental equation cosh( C)cos( C)=1. 9.2.4-4. The function anditssecond derivativeareprescribed attheends:= * M M=0at =0, = * M M=0at = C. Green' sfunction:F( , E, )=2C G HJI =1sin( I)sin( IE)sin TVU 24 I + U 24 I + , I = N 7C. 9.2.4-5. The ®rst andthird derivativesareprescribed attheends:* M= * M M M=0at =0, * M= * M M M=0at = C. Green' sfunction:F( , E, )=sin T P C P +2C G HI =1cos( I)cos( IE)sin TVU 24 I + U 24 I + , I = N 7C. Page609 610 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.2.4-6. The second andthird derivativesareprescribed attheends: M M= * M M M=0at =0, M M= * M M M=0at = C. Green' sfunction:F( , E, )= /1+3C2(2 - C)(2 E- C) 1sin P C P +4C G HJI =1K I ( )K I ( E)K2 I ( C)sin VU 24 I + U 24 I + , whereK I ( )= sinh( IC)-sin( IC) cosh( I)+cos( I) - cosh( IC)-cos( IC) sinh( I)+sin( I) ; the I arepositi veroots ofthetranscendental equation cosh( C)cos( C)=1.Forthenumerical values oftheroots, seeParagraph 9.2.3-2. 9.2.4-7. Mixedboundary conditions areprescribed attheends (case 1):= * M=0at =0, = * M M=0at = C. Green' sfunction:F( , E, )=2C G HJI =1 4 IK I ( )K I ( E) |K I( C)K I ( C)|sin TVU 24 I + U 24 I + , whereK I ( )= sinh( IC)-sin( IC) cosh( I)-cos( I) - cosh( IC)-cos( IC) sinh( I)-sin( I) ; the I arepositi veroots ofthetranscendental equation tan( C)-tanh( C)=0. 9.2.4-8. Mixedboundary conditions areprescribed attheends (case 2):= * M=0at =0, * M M= * M M M=0at = C. Green' sfunction:F( , E, )=4C G HJI =1 K I ( )K I ( E)K2 I ( C)sin TVU 24 I + U 24 I + , whereK I ( )= sinh( IC)+sin( IC) cosh( I)-cos( I) - cosh( IC)+cos( IC) sinh( I)-sin( I) ; the I arepositi veroots ofthetranscendental equation cosh( C)cos( C)=-1. 9.2.4-9. Mixedboundary conditions areprescribed attheends (case 3):= * M M=0at =0, * M= * M M M=0at = C. Green' sfunction:F( , E, )=2C G HJI =0sin( I)sin( IE)sin TVU 24 I + U 24 I + , I = N(2 7+1) 2 C. Page610 9.2. FOUR TH-ORDER ONE-DIMENSION ALNONST ATION ARYEQUATIONS 611 9.2.4-10. Mixedboundary conditions areprescribed attheends (case 4):= * M M=0at =0, * M M= * M M M=0at = C. Green' sfunction:F( , E, )=4C G HI =1 K I ( )K I ( E)K2 I ( C)sin VU 24 I + U 24 I + , whereK I ( )=sin( IC)sinh( I)+sinh( IC)sin( I); the I arepositi veroots ofthetranscendental equation tan( C)-tanh( C)=0. 9.2.5. Other Equations 9.2.5-1. Equations containing the®rst derivativewith respect to . 1. + '2 4 4+ + = W( , ). Thechange ofvariable ( , )= X- Y ( ( , )leads totheequation* (*+ 2 *4 (*4= X Y >( , ), which isdiscussed inSubsection 9.2.1. 2. Z [Z \= ' ]8Z4[Z ]4. This isaspecial case ofequation 9.6.4.2 with ^=1and 7=4. 3. Z [Z \= +(\) Z4[Z ]4+[ ] ,(\)+ -(\)] Z [Z ]+ .(\)[. This isaspecial case ofequation 9.6.4.1 with 7=4.Thetransformation_( `, a)= ( ( $, )exp /cb d( a) e a2f, $= ` g( a)+ b 4 ( a) g( a) e a, h= b ^( a) g4( a) e a, where g( a)=exp icb j( a) e a2f,leads totheconstant coef®cient equationk lkh= k4 lk$4, which isdiscussed inSubsection 9.2.1. 4. Z [Z \=( m ]2+ 5 ]+ 6)4 n4 on p4. This isaspecial case ofequation 9.6.4.4 with q=1and 7=4. Page611 612 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.2.5-2. Equations containing thesecond derivativewith respect to r. 5. n2 on s2+ t n on s+ m2 n4 on p4= W(p,s). With u( v, r)º0thisequation governs transv ersevibration ofanelastic rodinaresisting medium with velocity-proportional resistance coef®cient. Thechange ofvariable w( v, r)=exp x-1 2 q ry l( v, r)leads totheequationk2 lkr2+ z2 k4 lkv4-1 4 q2 l=exp x1 2 q ry u( v, r), which isdiscussed inSubsection 9.2.4. 6. n2 on s2= mp8 n4 on p4. This isaspecial case ofequation 9.6.4.2 with q=2and 7=4. 7. n2 on s2=( mp2+ 5p+ 6)4 n4 on p4. This isaspecial case ofequation 9.6.4.4 with q=2and 7=4. 8. { nn s± n2n p2 |2o=0. 1 }.General solution (tworepresentations):w( v, r)= r l 1( v, r)+ l 0( v, r),w( v, r)= v l 1( v, r)+ l 0( v, r), where l ~= l ~( v, r)isanarbitrary function satisfying theheat equation k l ~- k l ~=0; q=1,2. 2 }.Fundamental solution: ( v, r)= r 2 exp {- v2 4 r |. 3 }.Domain: - < v< .Cauchy problem. Initial conditions areprescribed:w=0at r=0, k w= j( v)at r=0. Solution:w( v, r)= r 2 b -exp i-( v- E)2 4 r f j( E) e E.& Refer ence:G.E.Shilo v(1965). 9. n4 on s4± n4 on p4=0. 1 }.Fundamental solution: ( v, r)=1 2 rln v2+ r2- varctan vr-1 2( r+ v)ln| r+ v| -1 2( r- v)ln| r- v|+1 8| r+ v|+1 8| r- v| . 2 }.Domain: - < v< .Cauchy problem. Initial conditions areprescribed:w=0at r=0, k w=0at r=0, k w= j( v)at r=0. Solution:w( v, r)= b - ( v- E, r) j( E) e E.& Refer ence:G.E.Shilo v(1965). Page612 9.3. TWO-DIMENSION ALNONST ATION ARYFOUR TH-ORDER EQUATIONS 613 10. n4 on s4±2 n4 on s2n p2+ n4 on p4=0. General solution (three representations):w( v, r)= j1( r- v)+ j2( r+ v)+ r 1( r- v)+ 2( r+ v) ,w( v, r)= j1( r- v)+ j2( r+ v)+ v 1( r- v)+ 2( r+ v) ,w( v, r)= j1( r- v)+ j2( r+ v)+( r+ v) 1( r- v)+( r- v) 2( r+ v), where j1( ), j2( ), 1( ),and 2( )arearbitrary functions.& Refer ence:A.V.Bitsadze andD.F.Kalinichenk o(1985). 9.3. Two-Dimensional Nonstationar yFour th-Or der Equations 9.3.1. Equations oftheForm + 2 4 4+ 4 4 = (,,) 9.3.1-1. Domain: 0£ v£ 1,0£ £ 2.Solution interms oftheGreen' sfunction. Weconsider boundary valueproblems inarectangular domain 0£ v£ 1,0£ £ 2with thegeneral initial conditionw= j( v, )at r=0 andvarious homogeneous boundary conditions. The solution canberepresented interms ofthe Green' sfunction asw( v, , r)= b 1 0 b 2 0 j( , ) ( v, , , , r) e e + b 0 b 1 0 b 2 0 u( , , h) ( v, , , , r- h) e e e h. BelowaretheGreen' sfunctions forvarious types ofboundary conditions. 9.3.1-2. The function andits®rst derivativesareprescribed atthesides ofarectangle:w= k w=0at v=0, w= k w=0at v= 1,w= k w=0at =0, w= k w=0at = 2. Green' sfunction:( v, , , , r)=161 2 J =1 =1 ¡4 ¢ 4 O£ ¤¤ ( 1) ¥ ¤¤ ( 2) 2 £ ( v) ¥ ( ) £ ( ) ¥ ( )exp -(¡4 + ¢ 4 ) z2r,£ ¤¤ ( v)= ¦2£ ¦ v2, ¥ ¤¤ ( )= ¦2¥ ¦ 2. Here,£ ( v)= sinh(¡ 1)-sin(¡ 1) cosh(¡ v)-cos(¡ v) - cosh(¡ 1)-cos(¡ 1) sinh(¡ v)-sin(¡ v) ,¥ ( )= sinh( ¢ 2)-sin( ¢ 2) cosh( ¢ )-cos( ¢ ) - cosh( ¢ 2)-cos( ¢ 2) sinh( ¢ )-sin( ¢ ) , where the¡ and ¢ arepositi veroots ofthetranscendental equations cosh(¡ 1)cos(¡ 1)=1,cosh( ¢2)cos( ¢2)=1 ( ¢ =¡ 1 § 2). Page613 614 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.3.1-3. The function anditssecond derivativesareprescribed atthesides ofarectangle:w= ¨ w=0at v=0, w= ¨ w=0at v= 1,w= ¨ Vw=0at =0, w= ¨ Vw=0at = 2. Green' sfunction:( v, , , , r)=41 2 =1 =1sin(¡ v)sin( ¢ )sin(¡ )sin( ¢ )exp -(¡4 + ¢ 4 ) z2r ,¡ = ©1, ¢ = ª2. 9.3.1-4. The ®rst andthird derivativesareprescribed atthesides ofarectangle:w = ¨ w=0at v=0, w = ¨ w=0at v= 1,w = ¨ VVw=0at =0, w = ¨ VVw=0at = 2. Green' sfunction:( v, , , , «)=11 2 J =0 =0 ¬ ¬ cos(¡ )sin( ¢ )cos(¡ )cos( ¢ )exp -(¡4 + ¢ 4 ) z2«,¡ = ©1, ¢ = ª2,¬ = ®1for©=0, 2for©¹0. 9.3.1-5. The second andthird derivativesareprescribed atthesides ofarectangle:¯ = ¨ ¯=0at =0, ¯ = ¨ ¯=0at = 1,¯ V= ¨ VV¯=0at =0, ¯ V= ¨ VV¯=0at = 2. Green' sfunction:( , , , , «)= 1( , , «) 2( , , «),1( , , «)=11+33 1(2 - 1)(2 - 1)+41 =1 £ ( ) £ ( )£2 ( 1)exp(-¡4 z2«),2( , , «)=12+33 2(2 - 2)(2 - 2)+42 =1 ¥ ( ) ¥ ( )¥2 ( 2)exp(- ¢ 4 z2«). Here,£ ( )= sinh(¡ 1)-sin(¡ 1) cosh(¡ )+cos(¡ ) - cosh(¡ 1)-cos(¡ 1) sinh(¡ )+sin(¡ ) ,¥ ( )= sinh( ¢ 2)-sin( ¢ 2) cosh( ¢ )+cos( ¢ ) - cosh( ¢ 2)-cos( ¢ 2) sinh( ¢ )+sin( ¢ ) , where the¡ and ¢ arepositi veroots ofthetranscendental equations cosh(¡ 1)cos(¡ 1)=1,cosh( ¢2)cos( ¢2)=1. Page614 9.3. TWO-DIMENSION ALNONST ATION ARYFOUR TH-ORDER EQUATIONS 615 9.3.1-6. Mixedboundary conditions areprescribed atthesides ofarectangle:¯= ¨ ¯=0at =0, ¨ ¯= ¨ ¯=0at = 1,¯= ¨ V¯=0at =0, ¨ ¯= ¨ VV¯=0at = 2. Green' sfunction:( , , , , «)=41 2 =0 =0sin(¡ )sin( ¢ )sin(¡ )sin( ¢ )exp -(¡4 + ¢ 4 ) z2«,¡ =(2©+1) 2 1, ¢ =(2ª+1) 2 2. 9.3.2. Two-Dimensional Equations oftheForm 2 2+ 2 ° °=0 This equation governs two-dimensional free transv erse vibration ofathin elastic plate; theun- known ¯isthede¯ection (transv erse displacement) oftheplate' smidplane points relati vetothe original plane position. Here, ± ±= ±2and ±istheLaplace operator thatisde®ned as±= ² ³2³ ´2+³2³ 2 intheCartesian coordinate system,³2³ µ2+1µ ³³ µ+1µ2³2³ ¶2inthepolar coordinate system. 9.3.2-1. Particular solutions:¯( , ·, «)= ¸¹1sin( º1 )+ »1cos( º1 ) ¼ ¸¹2sin( º2 ·)+ »2cos( º2 ·) ¼sin ¸( º2 1+ º2 2) z «¼,¯( , ·, «)= ¸¹1sin( º1 )+ »1cos( º1 ) ¼ ¸¹2sin( º2 ·)+ »2cos( º2 ·) ¼cos ¸( º2 1+ º2 2) z «¼,¯( , ·, «)= ¸¹1sinh( º1 )+ »1cosh( º1 ) ¼ ¸¹2sinh( º2 ·)+ »2cosh( º2 ·) ¼sin ¸( º2 1+ º2 2) z «¼,¯( , ·, «)= ¸¹1sinh( º1 )+ »1cosh( º1 ) ¼ ¸¹2sinh( º2 ·)+ »2cosh( º2 ·) ¼cos ¸( º2 1+ º2 2) z «¼,¯( ½, ¾, «)= ¸¹1 ¿ À( º ½)+ ¹2 Á À( º ½)+ ¹3 ÂVÀ( º ½)+ ¹4 Ã À( º ½) ¼cos( Ä ¾)sin( º2 Å Æ),Ç( ½, ¾, Æ)= ¸¹1 ¿ À( º ½)+ ¹2 Á À( º ½)+ ¹3 ÂVÀ( º ½)+ ¹4 Ã À( º ½) ¼sin( Ä ¾)cos( º2 Å Æ), where ¹1, ¹2, ¹3, ¹4, »1, »2, º, º1, º2arearbitrary constants, the¿ À( È)andÁ À( È)aretheBessel functions ofthe®rstandsecond kind, theÂ À( È)andÃ À( È)arethemodi®ed Bessel functions ofthe ®rstandsecond kind, ½= É Ê2+ ·2,and Ä=0,1,2, ËËË 9.3.2-2. Domain: - Ì< Ê< Ì,- Ì< ·< Ì.Cauchy problem. Initial conditions areprescribed:Ç= Í( Ê, ·)at Æ=0, Î Ï Ç= Ð( Ê, ·)at Æ=0. Poisson solution:Ç( Ê, ·, Æ)=1Ñ Ò Ó -Ó Ò Ó -Ó Í ÔÕÊ+2 È Ö Å Æ, ·+2 × Ö Å Æ Øsin ÔTÈ2+ ×2 Ø ÙÈ Ù× +1Ñ Ò Ï 0 Ù ÚÒ Ó -Ó Ò Ó -Ó Ð ÔTÊ+2 ÈÖ Å Ú, ·+2 ×Ö Å Ú Øsin ÔTÈ2+ ×2 Ø ÙÈ Ù×. Green' sfunction: Û ( Ê, ·, È, ×, Æ)=1 4 ÑÅ Ò Ï 0sin Ü( Ê- È)2+( ·- ×)2 4 Å Ú Ý Ù ÚÚ.Þß Refer ences :A.N.Krylo v(1949), I.Sneddon (1951), B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). Page615 616 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.3.2-3. Domain: 0£ Ê£ à1,0£ ·£ à2.Solution interms oftheGreen' sfunction. Weconsider boundary valueproblems inarectangular domain 0£ Ê£ à1,0£ ·£ à2with thegeneral initial conditionsÇ= Í( Ê, ·)at Æ=0, Î Ï Ç= Ð( Ê, ·)at Æ=0 andvarious homogeneous boundary conditions. The solution canberepresented interms ofthe Green' sfunction asÇ( Ê, ·, Æ)= ÎÎ Æ Ò á1 0 Ò á2 0 Í( È, ×) Û ( Ê, ·, È, ×, Æ) Ù× ÙÈ+Ò á1 0 Ò á2 0 Ð( È, ×) Û ( Ê, ·, È, ×, Æ) Ù× ÙÈ. Paragraphs 9.3.2-4 through 9.3.2-6 present theGreen' sfunctions forthree types ofboundary conditions. 9.3.2-4. Domain: 0£ Ê£ à1,0£ ·£ à2.Allsides oftheplate arehinged. Boundary conditions areprescribed:Ç= Î´ ´ Ç=0at Ê=0, Ç= Î´ ´ Ç=0at Ê= à1,Ç= Î VÇ=0at ·=0, Ç= Î VÇ=0at ·= à2. Green' sfunction:Û ( Ê, ·, È, ×, Æ)=4Åà1 à2 Ó âÀ=1 Ó âã=1sin( äÀ Ê)sin( å ã·)sin( äÀ È)sin( å ã×)sin( æÀ ã Å Æ)æÀ ã,äÀ= ÑÄà1, å ã= Ñ çà2, æÀ ã= ä2À+ å2ã. 9.3.2-5. Domain: 0£ Ê£ à1,0£ ·£ à2.The1stand3rdderivativesareprescribed atthesides:Î´ Ç= Î´ ´ ´ Ç=0at Ê=0, Î´ Ç= Î´ ´ ´ Ç=0at Ê= à1,Î è Ç= Î èVèVè Ç=0at ·=0, Î è Ç= Î èVèVè Ç=0at ·= à2. Green' sfunction:Û ( Ê, ·, È, ×, Æ)=1Åà1 à2 Ó âÀ=0 Ó âã=0 é Àé ãcos( äÀ Ê)cos( å ã·)cos( äÀ È)cos( å ã×)sin( æÀ ã Å Æ)æÀ ã,äÀ= ÑÄà1, å ã= Ñ çà2, æÀ ã= ä2À+ å2ã,é À= ê1for Ä=0, 2for Ä¹0. If Ä= ç=0,theratio sin( æÀ ã Å Æ) ë æÀ ãmust bereplaced by Å Æ. 9.3.2-6. Domain: 0£ Ê£ à1,0£ ·£ à2.Mixedboundary conditions aresetatthesides:Ç= Î´ ´ Ç=0 at Ê=0, Ç= Î´ ´ Ç=0 at Ê= à1,Î è Ç= Î èVèVè Ç=0at ·=0, Î è Ç= Î èVèVè Ç=0at ·= à2. Green' sfunction:Û ( Ê, ·, È, ×, Æ)=2Åà1 à2 Ó âÀ=1 Ó âã=0é ãsin( äÀ Ê)cos( å ã·)sin( äÀ È)cos( å ã×)sin( æÀ ã Å Æ)æÀ ã,äÀ= ÑÄà1, å ã= Ñ çà2, æÀ ã= ä2À+ å2ã,é ã= ê1for ç=0, 2for ç¹0. Page616 9.3. TWO-DIMENSION ALNONST ATION ARYFOUR TH-ORDER EQUATIONS 617 9.3.2-7. Domain: 0£ ½< Ì,0£ ¾£2 Ñ.Cauchy problem. Initial conditions forthesymmetric case inthepolar coordinate system:Ç= Í( ½)at Æ=0, Î Ï Ç=0at Æ=0. Solution:Ç( ½, Æ)=1 2 Å Æ Ò Ó 0 È Í( È)¿0 ì È ½ 2 Å Æ ísinì È2+ ½2 4 Å Æ í ÙÈ, where¿0( î)isthezeroth Bessel function.Þß Refer ences :I.Sneddon (1951), B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 9.3.2-8. Domain: 0£ ½£ ï,0£ ¾£2 Ñ.Transv erse vibration ofacircular plate. Initial andboundary conditions forsymmetric transv erse vibrations ofacircular plate ofradius ï with clamped contour inthepolar coordinate system:Ç= Í( ½)at Æ=0, Î Ï Ç= Ð( Æ)at Æ=0;Ç=0 at ½= ï, Îµ Ç=0 at ½= ï. Solution:Ç( ½, Æ)=Ó âÀ=1 ¸¹Àcos( Åº2À Æ)+ »Àsin( Åº2À Æ) ¼ ðÀ( ½),ðÀ( ½)=Â0( ºÀ ï)¿0( ºÀ ½)-¿0( ºÀ ï)Â0( ºÀ ½), where the ºÀarepositi veroots ofthetranscendental equation (theprime denotes thederivative)¿0( º ï)Â ñ0( º ï)-Â0( º ï)¿ ñ0( º ï)=0, andthecoef®cients ¹Àand »Àaregivenby¹À=1òðÀ ò2 Ò ó 0 Í( ½) ðÀ( ½) ½ Ù½, »À=1Åº2À òðÀ ò2 Ò ó 0 Ð( ½) ðÀ( ½) ½ Ù½,òðÀ ò2=1 4 ï6¸ ðññÀ( ï) ¼2= ï2¿2 0( ºÀ ï)Â2 0( ºÀ ï).Þß Refer ence:B.M.Budak, A.A.Samarskii, andA.N.Tikhono v(1980). 9.3.3. Three- and ô-Dimensional Equations oftheFormõ2 öõ ÷2+ ø2 ù ùö=0 9.3.3-1. Three-dimensional case. Cauchy problem. Domain: - Ì< Ê< Ì,- Ì< ú< Ì,- Ì< î< Ì.Initial conditions areprescribed:Ç= Í( Ê, ú, î)at Æ=0, Î Ï Ç=0at Æ=0. Solution:Ç( Ê, ú, î, Æ)=1Ô2Ö ÑÅ Æ Ø3 Ò Ó -Ó Ò Ó -Ó Ò Ó -Ó Í( Ê+ È, ú+ ×, î+ û)così È2+ ×2+ û2 4 Å Æ -3 Ñ 4 í ÙÈ Ù× Ùû.Þß Refer ence:V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974). Page617 618 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.3.3-2. Three-dimensional case. Boundary value problem. Domain: 0£ Ê£ à1,0£ ú£ à2,0£ î£ à3(rectangular parallelepiped). Initial conditions:Ç= Í( Ê, ú, î)at Æ=0, Î Ï Ç= Ð( Ê, ú, î)at Æ=0. Boundary conditions:Ç= Î ü ü Ç=0at Ê=0, Ç= Î ü ü Ç=0at Ê= à1,Ç= Î èVè Ç=0at ú=0, Ç= Î èVè Ç=0at ú= à2,Ç= Î ý ý Ç=0at î=0, Ç= Î ý ý Ç=0at î= à3. Solution:Ç( Ê, ú, î, Æ)= ÎÎ Æ Òá1 0 Òá2 0 Òá3 0 Í( È, ×, û) Û ( Ê, ú, î, È, ×, û, Æ) Ùû Ù× ÙÈ +Ò á1 0 Ò á2 0 Ò á3 0 Ð( È, ×, û) Û ( Ê, ú, î, È, ×, û, Æ) Ùû Ù× ÙÈ, whereÛ ( Ê, ú, î, È, ×, û, Æ)=8Åà1 à2 à3 Ó âJþ =1 Ó âã=1 Ó â ÿ =11æ þã ÿ sin( ä þÊ)sin( å ãú)sin( ÿî) ´sin( ä þÈ)sin( å ã×)sin( ÿû)sin( æ þã ÿÅ Æ),ä þ = ÑÄà1, å ã= Ñ çà2, ÿ = Ñ à3, æ þã ÿ = ä2 þ + å2ã+ 2 ÿ . 9.3.3-3. -dimensional case. Cauchy problem. Domain: þ ={- Ì< Ê ÿ < Ì; =1, ËËË, }.Initial conditions areprescribed:= Í(x)at =0, Î Ï =0at =0, where x={ Ê1, ËËË, Ê þ }. Solution:(x, )=1Ô2Ö Ñ Ø þÒ (y)così|x-y| 4 - Ñ 4 í Ùy, where y={ ú1, , ú þ }and Ùy= Ùú1 Ùú þ .Þß Refer ence:V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974). 9.3.3-4. -dimensional case. Boundary value problem. Domain: ={0£ ÿ £ à ÿ ; =1,2, }( -dimensional rectangular parallelepiped). Initial conditions:= (x)at =0, = (x)at =0. Boundary conditions:= ü ü =0at ÿ =0, = ü ü =0at ÿ = à ÿ . Solution:(x, )= (y) Û (x,y, ) Ùy+ (y) Û (x,y, ) Ùy, Page618 9.3. TWO-DIMENSION ALNONST ATION ARYFOUR TH-ORDER EQUATIONS 619 whereÛ (x,y, )=2 þ 1 2 þ 1=1 2=1 =11æ 1, 2, , sin( 1 1)sin( 2 2) sin( ) ´sin( 1 1)sin( 2 2) sin( )sin æ 1, 2, , !#", 1= $ %1 1, 2= $ %2 2, , = $ % , æ 1, 2, , = 2 1+ 2 2+ &&&+ 2 . 9.3.4. Equations oftheForm '2 (' )2+ ø2 ù ù(+ * (= +( ,, -,)) 9.3.4-1. Domain: 0££ 1,0££ 2.Solution interms oftheGreen' sfunction. Weconsider boundary valueproblems inarectangular domain 0££ 1,0££ 2with thegeneral initial conditions . = (,)at !=0, / 0 . = 1(,)at !=0 andvarious homogeneous boundary conditions. The solution canberepresented interms ofthe Green' sfunction as. (,, !)= // ! 21 0 22 0 ( 3, 4) 5(,, 3, 4, !) 6 4 6 3+ 21 0 22 0 1( 3, 4) 5(,, 3, 4, !) 6 4 6 3 + 0 0 21 0 22 0 7( 3, 4, 8) 5(,, 9, 3, 4, :, !- 8) 6 4 6 3 6 8. Paragraphs 9.3.4-2 through 9.3.4-4 present theGreen' sfunctions forthree types ofboundary conditions. 9.3.4-2. The function anditssecond derivativesareprescribed atthesides ofarectangle:. = / ; ; . =0at=0, . = / ; ; . =0at= 1,. = / <=< . =0at=0, . = / <=< . =0at= 2. Green' sfunction:5(,, 3, 4, !)=4 1 2 =1 >=1sin( )sin( ? >)sin( 3)sin( ? >4)sin( @ > !)@ >,=$ A 1, ? >=$ B 2, @ >= C 2( 2+ ?2>)2+%. 9.3.4-3. The ®rst andthird derivativesareprescribed atthesides ofarectangle:/ ; . = / ; ; ; . =0at=0, / ; . = / ; ; ; . =0at= 1,/ < . = / <=<=< . =0at=0, / < . = / <=<=< . =0at= 2. Green' sfunction:5(,, 3, 4, !)=1 1 2 =0 >=0 D D >cos( )cos( ? >)cos( 3)cos( ? >4)sin( @ > !)@ >,= $ A 1, ? >= $ B 2, @ >= C 2( 2+ ?2>)2+%,D = E1forA=0, 2forA¹0. Page619 620 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.3.4-4. Mixedboundary conditions areprescribed atthesides ofarectangle:. = / ; ; . =0 at=0, . = / ; ; . =0 at= F1,/ < . = / <=<=< . =0at=0, / < . = / <=<=< . =0at= F2. Green' sfunction:5(,, 3, 4, !)=2F1 F2 G =1 G >=0D >sin( )cos( ? >)sin( 3)cos( ? >4)sin( @ > !)@ >,=$ AF1, ? >=$ BF2, @ >= C 2( 2+ ?2>)2+%,D >=E1forB=0, 2forB¹0. 9.3.5. Equations oftheForm '2 (' )2+ ø2 H'4 (' ,4+ '4 (' -4 I+ * (= +( ,, -,)) 9.3.5-1. Domain: 0££ F1,0££ F2.Solution interms oftheGreen' sfunction. Weconsider boundary valueproblems inarectangular domain 0££ F1,0££ F2with thegeneral initial conditions . = (,)at !=0, / 0 . = 1(,)at !=0 andvarious homogeneous boundary conditions. The solution canberepresented interms ofthe Green' sfunction as. (,, !)= // ! J 21 0 J 22 0 ( 3, 4) 5(,, 3, 4, !) 6 4 6 3+J 21 0 J 22 0 1( 3, 4) 5(,, 3, 4, !) 6 4 6 3 +J 0 0 J 21 0 J 22 0 7( 3, 4, 8) 5(,, 9, 3, 4, :, !- 8) 6 4 6 3 6 8. Paragraphs 9.3.5-2 through 9.3.5-4 present theGreen' sfunctions forthree types ofboundary conditions. 9.3.5-2. The function andits®rst derivativesareprescribed atthesides ofarectangle:. = / ; . =0at=0, . = / ; . =0at= F1,. = / < . =0at=0, . = / < . =0at= F2. Green' sfunction:5(,, 3, 4, !)=16F1 F2G =1 G >=1 4 ?4>KML NON( F1) P NON>( F2) Q2 L() P >() L( 3) P >( 4)sin( @ > !)@ >,@ >= C 2( 4+ ?4>)+%, L NON()= 62 L62, P NON>()= 62P >62. Here,L()= K sinh( F1)-sin( F1) Q K cosh( )-cos( ) Q - K cosh( F1)-cos( F1) Q K sinh( )-sin( ) Q,P >()= K sinh( ? >F2)-sin( ? >F2) Q K cosh( ? >)-cos( ? >) Q - K cosh( ? >F2)-cos( ? >F2) Q K sinh( ? >)-sin( ? >) Q, where the and ? >arepositi veroots ofthetranscendental equations cosh( F1)cos( F1)=1,cosh( ? F2)cos( ? F2)=1. Page620 9.4. FOUR TH-ORDER STATION ARYEQUATIONS 621 9.3.5-3. The function anditssecond derivativesareprescribed atthesides ofarectangle:= =0at =0, = =0at = 1,= =0at =0, = =0at = 2. Green' sfunction:( , , , , )=41 2 =1 =1sin( )sin( )sin( )sin( )sin( ) , = 1, = 2, = 2( 4+ 4)+ . 9.3.5-4. The ®rst andthird derivativesareprescribed atthesides ofarectangle: = =0at =0, = =0at = 1, = =0at =0, = =0at = 2. Green' sfunction:( , , , , )=11 2 =0 =0 cos( )cos( )cos( )cos( )sin( ) , = 1, = 2, = 2( 4+ 4)+ , = 1for=0, 2for¹0. 9.4. Four th-Or derStationar yEquations 9.4.1. Biharmonic Equation =0 Thebiharmonic equation isencountered inplaneproblemsofelasticity ( istheAiry stress function). Itisalsoused todescribe slow¯owsofviscous incompressible ¯uids ( isthestream function). Allsolutions oftheLaplace equation =0(seeSections 7.1and8.1)arealsosolutions ofthe biharmonic equation. 9.4.1-1. Two-dimensional equation. Particular solutions. Intherectangular Cartesian system ofcoordinates, thebiharmonic operator hastheform º 2= 44+2 42 2+ 44. 1 .Particular solutions:( , )= 3+ 2+ ! 2+ " 3+ 2+ #$ + %&2+ ' + ( + ),( , )=( cosh ( + sinh ( + ! cosh ( + " sinh ( )( cos ( + #sin ( ),( , )=( cos ( + sin ( + ! cos ( + " sin ( )( cosh ( + #sinh ( ),( , )= *2ln *+ *2+ !ln *+ ", *= +( - )2+( - #)2,( , )=( + + !)( "cosh ( + ,sinh ( )( cos ( + #sin ( ),( , )=( + + !)( "cosh ( + ,sinh ( )( cos ( + #sin ( ),( , )=( 2+ 2)( "cosh ( + ,sinh ( )( cos ( + #sin ( ),( , )=( 2+ 2)( "cosh ( + ,sinh ( )( cos ( + #sin ( ), where , , !, ", ,, , #, %, ', (,and )arearbitrary constants. Page621 622 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS TABLE 30 Particular solutions of the biharmonic equation in some orthogonal curvilinear coordinate systems; , , !, ", , #, and are arbitrary constants Transformation Particular solutions Polar coordinates *, -:= *cos -, = *sin - =( *2+ .+ *2- .+ ! * .+ " *- .)( cos -+ #sin -),= *2ln *+ *2+ !ln *+ " (at = 0) Bipolar coordinates , := %sinh cosh -cos , = %sin cosh -cos =1 cosh -cos / cosh( + 1) + sinh( + 1) + !cosh( - 1) + "sinh( - 1) 0( cos + #sin ) Degenerate bipolar coordinates 1, 2:= 112+ 22, = - 212+ 22 =1 ( 12+ 22)2/ cosh( 1)+ sinh( 1) + ! 1cosh( 1)+ " 1sinh( 1)0/ cos( 2)+ #sin( 2)0 2 . Fundamental solution: 3 3 ( , )=1 8 *2ln *, *= + 2+ 2. 3 . Particular solutions of the biharmonic equation in some orthogonal curvilinear coordinate sys- temsarelistedinTable30.465 Reference : N. N. Lebedev, I. P. Skal'skaya, and Ya. S. U¯yand (1972). 9.4.1-2. Two-dimensional equation. Various representations of the general solution. 1 . Various representations of the general solution in terms of harmonic functions:( , )= 11( , )+ 12( , ),( , )= 11( , )+ 12( , ),( , )=( 2+ 2) 11( , )+ 12( , ), where 11and 12are arbitrary functions satisfying the Laplace equation 1 7= 0( = 1,2).465 Reference : A. N. Tikhonov and A. A. Samarskii (1990). 2 . Complex form of representation of the general solution:( , )=Re/ 8 9(8)+ :(8)0, where9(8) and :(8) are arbitrary analytic functions of the complex variable8= + ;<;8= - ;<,;2= -1. The symbol Re[ ] stands for the real part of the complex quantity .465 Reference : A. V . Bitsadze and D. F. Kalinichenko (1985). 9.4.1-3. Two-dimensional boundary value problems for the upper half-plane. 1 . Domain: - =< < =,0 £ < =. The desired function and its derivative along the normal are prescribed at the boundary:= 0 at = 0, =9( ) at = 0. Solution:( , )= > - 9( ) ( - , ) ? , ( , )=1 22+ 2.465 Reference : G. E. Shilov (1965). Page 622 9.4. FOUR TH-ORDER STATION ARYEQUATIONS 623 2 .Domain: - =< < =,0£ < =.Thederivativesofthedesired function areprescribed atthe boundary: =9( )at =0, = :( )at =0. Solution:( , )=1 > - 9( ) @arctan A - B+ ( - ) ( - )2+ 2 C ? + 2 > - :( ) ? ( - )2+ 2+ !, where !isanarbitrary constant. Example. Letusconsider theproblem ofaslow(Stok es)in¯owofaviscous ¯uid intothehalf-plane through aslitof width 2 Dwith aconstant velocity Ethatmakesanangle Fwith thenormal totheboundary (theangle isreckoned from the normal counterclockwise). Withthestream function Gintroduced bytherelations HJI=- KMLKMNand HN= KMLK I( H&Iand HNarethe¯uid velocity components), theproblem isreduced tothespecial case oftheprevious problem withO( P)= Q Ecos Ffor| P|< D, 0 for| P|> D, R( P)= Q Esin Ffor| P|< D, 0 for| P|> D. Dean' ssolution:G( P, S)= E T U( P- D)cos F+ Ssin F Varctan W SP- D X- E T U( P+ D)cos F+ Ssin F Varctan W SP+ D X+ Y.465 Refer ence:I.Sneddon (1951). 9.4.1-4. Two-dimensional boundary value problem foracircle. Domain: 0£ *£ ,0£ -£2 Z.Boundary conditions inthepolar coordinate system:[=9( -)at *= \, ] ^ [= :( -)at *= \. Solution:[( *, -)=1 2 Z $ *2- \2)2@_>2 ` 0[ \- *cos( a- -)]9( a) ? a [ *2+ \2-2 \ *cos( a- -)]2-1 2 >2 ` 0 :( a) ? a*2+ \2-2 \ *cos( a- -) C.465 Refer ence:A.N.Tikhono vandA.A.Samarskii (1990). 9.4.1-5. Three-dimensional equation. Intherectangular Cartesian coordinate system, thethree-dimensional biharmonic operator isex- pressed as º 2= ]4] b4+ ]4] c4+ ]4]84+2 ]4] b2] c2+2 ]4] b2]82+2 ]4] c2]82. 1 d.Particular solutions intheCartesian coordinate system:[( b, c,8)= e f2+ g f+ h+ if, f= j( b- $2+( c- k)2+( l- m)2,[( b, c, l)= noe bsin( p b)+ gsin( p b)+ h bcos( p b)+icos( p b) qsin( r c)exp sut l j p2+ r2 v,[( b, c, l)=n e bsin( p b)+ gsin( p b)+ h bcos( p b)+icos( p b)qcos( r c)exps t l j p2+ r2v,[( b, c, l)=n e bsin( p b)+ gsin( p b)+ h bcos( p b)+icos( p b)qsinh( r c)exps t l j p2- r2v,[( b, c, l)=n e bsin( p b)+ gsin( p b)+ h bcos( p b)+icos( p b)qcosh( r c)exps t l j p2- r2v,[( b, c, l)=n e bsinh( p b)+ gsinh( p b)+ h bcosh( p b)+icosh( p b)qsinh( r c)sins l j p2+ r2v,[( b, c, l)=n e bsinh( p b)+ gsinh( p b)+ h bcosh( p b)+icosh( p b)qcosh( r c)coss l j p2+ r2v, where e, g, h,i, p,and rarearbitrary constants. Page623 624 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 2 d.Particular solutions inthecylindrical coordinate systems f=j b2+ c2v:[( f, w, l)= x y( r f)( e fcos w+ g fsin w+ h)( \1cos z w+ k1sin z w)( \2cosh r l+ k2sinh r l),[( f, w, l)= { y( r f)( e fcos w+ g fsin w+ h)( \1cos z w+ k1sin z w)( \2cosh r l+ k2sinh r l),[( f, w, l)= |}y( r f)( e fcos w+ g fsin w+ h)( \1cos z w+ k1sin z w)( \2cos r l+ k2sin r l),[( f, w, l)= ~ y( r f)( e fcos w+ g fsin w+ h)( \1cos z w+ k1sin z w)( \2cos r l+ k2sin r l),[( f, w, l)= x y( r f)( ecos z w+ gsin z w)( \1cosh r l+ k1sinh r l+ \2 lcosh r l+ k2 lsinh r l),[( f, w, l)= { y( r f)( ecos z w+ gsin z w)( \1cosh r l+ k1sinh r l+ \2 lcosh r l+ k2 lsinh r l),[( f, w, l)= |}y( r f)( ecos z w+ gsin z w)( \1cos r l+ k1sin r l+ \2 lcos r l+ k2 lsin r l),[( f, w, l)= ~ y( r f)( ecos z w+ gsin z w)( \1cos r l+ k1sin r l+ \2 lcos r l+ k2 lsin r l), where z=0,1,2, MM; e, g, h, \1, \2, k1, k2,and rarearbitrary constants; the x y( )and { y( ) aretheBessel functions; andthe | y( )and ~ y( )arethemodi®ed Bessel functions. 3 d.Particular solutions inthespherical coordinate systems f=j b2+ c2+ l2v:[( f)= e f2+ g f+ h+i f-1,[( f, )=s e f y+2+ g f y+ h f1- y+i f-1- yvMy(cos ),[( f, , w)=s e f y+2+ g f y+ h f1- y+i f-1- yvM y(cos )( \cos w+ ksin w), where z=0,1,2, MM; =0,1,2, MM, z; e, g, h,i, \,and karearbitrary constants; they( )aretheLegendre polynomials; andthe y( )aretheassociated Legendre functions de®ned byy( b)=1z!2 y y b y( b2-1) y, y( b)=(1- b2) 2 b y( b). 4 d.Fundamental solution: ( b, c, l)=-1 8 Z jb2+ c2+ l2. 5 d.Representations ofsolutions tothebiharmonic equation interms ofharmonic functions:[( b, c, l)= b 1( b, c, l)+ 2( b, c, l),[( b, c, l)=( b2+ c2+ l2) 1( b, c, l)+ 2( b, c, l), where 1and 2arearbitrary functions satisfying thethree-dimensional Laplace equation 3 =0 ( =1,2).Thecoef®cient bof 1inthe®rstformula canbereplaced by cor l.6 Refer ence:A.V.Bitsadze andD.F.Kalinichenk o(1985). 9.4.1-6. z-dimensional equation. 1 d.Particular solutions:[(x)= y, , =1 e b b b + y, =1 g b b+ y=1 h b +i,(x)= e f2+ g+ h f4- y+i f2- y, f2= y=1( b - )2,(x)=( e+ g f2- y) y=1 h b +i , 2= y=1( b - )2,(x)=exps t b yj y v y=1 b + y-1=1sin( b + p ), y= y-1=1 2,(x)= y=1 b + -1=1sin( b + p ) y=sinh( b ) , -1=1 2- y= 2=0, where the , , , ,, , , , , ,and arearbitrary constants. Page624 9.4. FOUR TH-ORDER STATION ARYEQUATIONS 625 2 d.Fundamental solution: (x)= ¡ ¢¡£ ¤( ¥ ¦2)|x|4- § 4 ¨ § ©2( ¥-2)( ¥-4)for ¥=3,5,6,7, ªMªMª; -1 8 ¨2ln|x| for ¥=4. For ¥=2,seeParagraph 9.4.1-1, Item 2 d.«6¬ Refer ence:G.E.Shilo v(1965). 3 d.Various representations ofsolutions tothebiharmonic equation interms ofharmonic functions:(x)= b ¯®1(x)+ ®2(x), °=1,2, ªMªMª, ¥;(x)=|x|2®1(x)+ ®2(x), |x|2= §²± =1 b2 ± , where ®1and ®2arearbitrary functions satisfying the ¥-dimensional Laplace equation ³§ ® ´=0 ( µ=1,2).«6¬ Refer ence:A.V.Bitsadze andD.F.Kalinichenk o(1985). 9.4.2. Equations oftheForm ¶ ¶ ·= ¸( ¹, º) Nonhomo geneous biharmonic equation. Itisencountered inplane problems ofelasticity and hydrodynamics. 9.4.2-1. Domain: - »< ¼< »,- »< ½< ». Solution: ¾ ( ¼, ½)= ¿ À -À ¿ À -À Á( Â, Ã) Ä Ä( ¼- Â, ½- Ã) Å Â Å Ã, Ä Ä( ¼, ½)=1 8 ¨( ¼2+ ½2)ln Æ ¼2+ ½2.«6¬ Refer ence:A.V.Bitsadze andD.F.Kalinichenk o(1985). 9.4.2-2. Domain: - »< ¼< »,0£ ½< ».Boundary value problem. Theupper half-plane isconsidered. Thederivativesareprescribed attheboundary:Ç È ¾ = É( ¼)at ½=0, Ç Ê ¾ = Ë( ¼)at ½=0. Solution:¾ ( ¼, ½)=1¨ ¿ À -À É( Â) Ìarctan Í ¼- Â½ Î+ ½( ¼- Â) ( ¼- Â)2+ ½2 Ï Å Â+ ½2¨ ¿ À -À Ë( Â) Å Â ( ¼- Â)2+ ½2 +1 8 ¨ ¿ À -À Å Â ¿ À 0 Ì1 2( Ð2 +- Ð2 -)- Ð2 -ln Ð+Ð- ÏÁ( Â, Ã) Å Ã+ Ñ, where Ñisanarbitrary constant,Ð2 +=( ¼- Â)2+( ½+ Ã)2, Ð2 -=( ¼- Â)2+( ½- Ã)2.«6¬ Refer ence:I.Sneddon (1951). Page625 626 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.4.2-3. Domain: 0£ ¼£ Ò1,0£ ½£ Ò2.The sides oftheplate arehinged. Arectangle isconsidered. Boundary conditions areprescribed: ¾ = Ç È È ¾ =0at ¼=0, ¾ = Ç È È ¾ =0at ¼= Ò1, ¾ = Ç ÊÊ ¾ =0at ½=0, ¾ = Ç ÊÊ ¾ =0at ½= Ò2. Solution: ¾ ( ¼, ½)= ¿ Ó1 0 ¿ Ó2 0Á( Â, Ã) Ô( ¼, ½, Â, Ã) Å Ã Å Â, whereÔ( ¼, ½, Â, Ã)=4Ò1 Ò2 À Õ×Ö =1 À ÕØ=11 ( Ù2 Ö + Ú2Ø)2sin( Ù Ö Û )sin( Ú Ø Ü)sin( Ù Ö Ý )sin( Ú Ø Þ),Ù Ö = ß àÒ1, Ú Ø= ß µÒ2. 9.4.3. Equations oftheForm á á â± ã â= ä( å, æ) 9.4.3-1. Homogeneous equation ( çº0). This equation describes theshapes oftwo-dimensional freetransv erse vibrations ofathinelastic plate; thefunction ¾ de®nes thede¯ection (transv ersedisplacement) oftheplate' smidplane points relati vetotheoriginal plane position and è= é1 ê4isthefrequenc yparameter .Here, ë ë= ë2is thebiharmonic operator and ëistheLaplace operator de®ned asë= ì í2í î2+í2í ï2 intheCartesian coordinate system,í2í ð2+1ð íí ð+1ð2í2í ñ2inthepolar coordinate system. 1 d.Particular solutions ( ò1, ò2, ò3, ò4, ó1,and ó2arearbitrary constants): ¾ ( Û , Ü)= ô6ò1sin( è1 Û )+ ó1cos( è1 Û ) õ ô6ò2sin( è2 Ü)+ ó2cos( è2 Ü) õ, é=( è2 1+ è2 2)2, ¾ ( Û , Ü)= ô6ò1sin( è1 Û )+ ó1cos( è1 Û ) õ ô6ò2sinh( è2 Ü)+ ó2cosh( è2 Ü) õ, é=( è2 1- è2 2)2, ¾ ( Û , Ü)= ô6ò1sinh( è1 Û )+ ó1cosh( è1 Û ) õ ô6ò2sin( è2 Ü)+ ó2cos( è2 Ü) õ, é=( è2 1- è2 2)2, ¾ ( Û , Ü)= ô6ò1sinh( è1 Û )+ ó1cosh( è1 Û ) õ ô6ò2sinh( è2 Ü)+ ó2cosh( è2 Ü) õ, é=( è2 1+ è2 2)2, ¾ ( , ö)= ô6ò1 ÷ Ö ( è )+ ò2 ø Ö ( è )+ ò3 ù Ö ( è )+ ò4 ú Ö ( è ) õcos(à ö), é= è4>0, ¾ ( , ö)= ô6ò1 ÷ Ö ( è )+ ò2 ø Ö ( è )+ ò3 ù Ö ( è )+ ò4 ú Ö ( è ) õsin(à ö), é= è4>0, where the÷ Ö ( Ý )andø Ö ( Ý )aretheBessel functions ofthe®rstandsecond kind, theù Ö ( Ý )andú Ö ( Ý ) arethemodi®ed Bessel functions ofthe®rstandsecond kind, = û Û 2+ Ü2,andà=0,1,2, üMüMü 2 d.General solution: ¾ ( Û , Ü)= ý1( Û , Ü)+ ý2( Û , Ü), where ý1and ý2arearbitrary functions satisfying theHelmholtz equationsë ý1+ þ é ý1=0, ë ý2- þ é ý2=0. Forsolutions tothese equations, seeSection 7.3. Page626 9.4. FOUR TH-ORDER STATION ARYEQUATIONS 627 9.4.3-2. Domain: 0£ Û £ ÿ1,0£ Ü£ ÿ2.Boundary value problem. Arectangle isconsidered. Boundary conditions areprescribed: ¾ = î î ¾ =0at Û =0, ¾ = î î ¾ =0at Û = ÿ1, ¾ = ïï ¾ =0at Ü=0, ¾ = ïï ¾ =0at Ü= ÿ2. Solution: ¾ ( Û , Ü)= 1 0 2 0 ç( Ý , Þ) ( , , Ý , Þ) Þ Ý , where( , , Ý , Þ, )=4ÿ1 ÿ2 =1 =1sin( )sin( )sin( Ý )sin( Þ) ( 2 + 2)2- , =ß ÿ1, =ß ÿ2. 9.4.3-3. Domain: 0£ £ ,0£ ö£2ß.Eigen value problem with º0. Theunkno wnanditsnormal derivativearezero ontheboundary ofacircular domain:= =0at = . Eigen values: = 4 4,=0,1,2, üMüMü,=1,2,3, üMüMü, where the arepositi veroots ofthetranscendental equation÷ ()ù ()-ù ()÷ ()=0. Numerical values ofsome roots: 01=3.196, 02=6.306, 03=9.439, 04=12.58; 11=4.611, 12=7.799, 13=10.96, 14=14.11; 21=5.906, 22=9.197, 23=12.40, 24=15.58, 31=7.144, 32=10.54, 33=13.79, 34=17.01. Eigen values:(c) ( , ö)= ù ( ) - ( ) cos( ),(s) ( ,)= ( ) - ( ) sin( ).! Refer ence:V.V.Bolotin (1978). 9.4.3-4. Domain: ( " #)2+( " $)2£1.Eigen value problem with º0. Theunkno wnanditsnormal derivativearezero ontheboundary ofanelliptic domain:= % % &=0on ( " #)2+( " $)2=1 ( #³ $). Eigen values andeigenfunctions (approximate formulas):01= 4 01 8 3#4+3$4+2#2$2, 01( ')= 0(01) 0(01 ')- 0(01) 0(01 '),(c) 11= 4 11 8 5#4+1$4+2#2$2, (c) 11( ',)= ()1(11) 1(11 ')- 1(11) 1(11 ') *cos,(s) 11= 4 11 8 1#4+5$4+2#2$2, (s) 11( ',)=( 1(11) 1(11 ')- 1(11) 1(11 ')*sin, Page627 628 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS where '= +( " #)2+( " $)2,and01=3.196and11=4.611aretheleast roots ofthetranscendental equations0() 1()+ 1() 0()=0,1() 1()- 1() 1()=0. Theaboveformulas were obtained with theaidofgeneralized (nonorthogonal) polar coordinates',de®ned by= # 'cos, = $,'sin(0£ '£1,0££2ß) andthevariational method. Themaximum error intheeigen value 1for -= +1-( $." #)2£0,86islessthan 1%. Theerrors in (c) 11and (s) 11for -£0,6donotexceed 2%. Inthelimit case -=0thatcorresponds toacircular domain, theformulas provide exact results.! Refer ence:L.D.Akulenk o,S.V.Nestero v,andA.L.Popo v(2001). 9.4.4. Equations oftheForm /4 0/ 14+ /4 0/ æ4= 2(1, æ) 9.4.4-1. Homogeneous equation ( º0). 1 3.Particular solutions:( , )=()4sin( )+ 5cos( )+ 6sinh( )+ 7cosh( )*exp 819 2 : ;sin 819 2 : ;,( <, :)=()4sin( <)+ 5cos( <)+ 6sinh( <)+ 7cosh( <)*exp 819 2 : ;cos 819 2 : ;,( <, :)=()4sin( <)+ 5cos( <)+ 6sinh( <)+ 7cosh( <)*exp 8-19 2 : ;sin 819 2 : ;,( <, :)=()4sin( <)+ 5cos( <)+ 6sinh( <)+ 7cosh( <)*exp 8-19 2 : ;cos 819 2 : ;, where4, 5, 6, 7,and arearbitrary constants. 2 3.General solution:( <, :)=Re(!=( >1)+ ?( >2)*. Here,=( >1)and ?( >2)arearbitrary analytic functions ofthecomple xvariables >1= <-19 2(1+ @) : and >2= <+19 2(1+ @) :.Thesymbol Re[4]stands fortherealpartofthecomple xquantity4.! Refer ence:A.V.Bitsadze endD.F.Kalinichenk o(1985). 3 3.Domain: - A< << A,0£ :< A.Boundary value problem. Theupper half-space isconsidered. Boundary conditions areprescribed:=0at :=0,% B ==( <)at :=0. Solution:( <, :)= C D -D E( F) G( <- F, H) I F, whereG( <, H)=1J K2 Larctan M1- < K2H N+arctan M1+ < K2H N O.P!Q Refer ence:G.E.Shilo v(1965). Page628 9.4. FOUR TH-ORDER STATION ARYEQUATIONS 629 9.4.4-2. Nonhomogeneous equation. Boundary value problems inarectangle. Weconsider problems inarectangular domain 0£ <£ R1,0£ H£ R2with different homogeneous boundary conditions. Thesolution canbeexpressed interms oftheGreen' sfunction asS( <, H)= C T1 0 C T2 0 U( F, V) G( <, H, F, V) I V I F. BelowaretheGreen' sfunctions fortwotypes ofboundary conditions. 1 3.Thefunction andits®rstderivativesareprescribed atthesides oftherectangle:S=% W S=0at <=0, S=% W S=0at <= R1,S=% X S=0at H=0, S=% X S=0at H= R2. Green' sfunction:G( <, H, F, V)=16R1 R2 D YZ =1 D Y[=1 \4 Z ] 4[ ^ Z ( <) _ [( H) ^ Z ( F) _ [( V) (\4 Z + ] 4[) ` ^ aba Z ( R1) _ aba [( R2) c2. Here,^ Z ( <)= `sinh(\ ZR1)-sin(\ ZR1) c `cosh(\ Z<)-cos(\ Z<) c - `cosh(\ ZR1)-cos(\ ZR1) c `sinh(\ Z<)-sin(\ Z<) c,_ [( H)= `sinh( ][R2)-sin( ][R2) c `cosh( ][H)-cos( ][H) c - `cosh( ][R2)-cos( ][R2) c `sinh( ][H)-sin( ][H) c, where the\ Z and ][arepositi veroots ofthetranscendental equations cosh(\ R1)cos(\ R1)=1,cosh( ]R2)cos( ]R2)=1. 2 3.Thefunction anditssecond derivativesareprescribed atthesides oftherectangle:S= dW W S=0at <=0, S= dW W S=0at <= R1,S= dXeX S=0at H=0, S= dXeX S=0at H= R2. Green' sfunction:G( <, H, F, V)=4R1 R2 D YZ =1 D Y[=11\4 Z + ] 4[sin(\ Z<)sin( ][H)sin(\ ZF)sin( ][V),\ Z = J fR1, ][= J gR2. 9.4.5. Equations oftheForm h4 ih j4+ h4 ih k4+ l i= m(j,k) 9.4.5-1. Particular solutions ofthehomogeneous equation (Uº0):S( <, H)= `)nsin( o <)+ pcos( o <)+ qsinh( o <)+ rcosh( o <) cexp( s H)sin( s H),S( <, H)= `)nsin( o <)+ pcos( o <)+ qsinh( o <)+ rcosh( o <) cexp( s H)cos( s H),S( <, H)= `)nsin( o <)+ pcos( o <)+ qsinh( o <)+ rcosh( o <) cexp(- s H)sin( s H),S( <, H)= `)nsin( o <)+ pcos( o <)+ qsinh( o <)+ rcosh( o <) cexp(- s H)cos( s H), where s=1t 2( o4+ u)1 v4; n, p, q, r,and oarearbitrary constants. Page629 630 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.4.5-2. Domain: 0£ <£ R1,0£ H£ R2.Boundary value problems. 1 w.Weconsider problems inarectan gulardomain with differenthomogeneo usboundaryconditions. Thesolution canbeexpressed interms oftheGreen' sfunction asS( <, H)= C T1 0 C T2 0 U( F, V) G( <, H, F, V) I V I F. BelowaretheGreen' sfunctions fortwotypes ofboundary conditions. 2 w.Thefunction andits®rstderivativesareprescribed atthesides oftherectangle:S= dW S=0at <=0, S= dW S=0at <= R1,S= dX S=0at H=0, S= dX S=0at H= R2. Green' sfunction:G( <, H, F, V)=16R1 R2 D YZ =1 D Y[=1\4 Z] 4[^ Z ( <) _ [( H) ^ Z ( F) _ [( V) (\4 Z + ] 4[+ u) ` ^ aba Z ( R1) _ aba [( R2) c2. Here,^ Z ( <)= `sinh(\ ZR1)-sin(\ ZR1) c `cosh(\ Z<)-cos(\ Z<) c - `cosh(\ ZR1)-cos(\ ZR1) c `sinh(\ Z<)-sin(\ Z<) c,_ [( H)= `sinh( ][R2)-sin( ][R2) c `cosh( ][H)-cos( ][H) c - `cosh( ][R2)-cos( ][R2) c `sinh( ][H)-sin( ][H) c, where the\ Z and ][arepositi veroots ofthetranscendental equations cosh(\ R1)cos(\ R1)=1,cosh( ]R2)cos( ]R2)=1. 3 w.Thefunction anditssecond derivativesareprescribed atthesides oftherectangle:S= dW W S=0at <=0, S= dW W S=0at <= R1,S= dXeX S=0at H=0, S= dXeX S=0at H= R2. Green' sfunction:G( <, H, F, V)=4R1 R2 D YZ =1 D Y[=11\4 Z + ] 4[+ usin(\ Z<)sin( ][H)sin(\ ZF)sin( ][V),\ Z = J fR1, ][= J gR2. 9.4.6. Stokes Equation (Axisymmetric FlowsofViscous Fluids) 9.4.6-1. Stokesequation forthestream function inthespherical coordinate system. TheStokesequation forthestream function intheaxisymmetric case iswritten asx2( x2 S)=0, x2º d2d y2+sin zy2 dd z M1 sin z dd z N. Itgoverns slowaxisymmetric ¯owsofviscous incompressible ¯uids, with Sbeing thestream function, yand zthespherical coordinates. Thecomponents ofthe¯uid velocity arerelated tothe stream function by { |=1y2sin z d Sd zand { }=-1ysin z d Sd y. Page630 9.4. FOUR TH-ORDER STATION ARYEQUATIONS 631 General solution ( n Z , p Z , q Z , r Z , ~ n Z , ~ p Z , ~ q Z ,and ~ r Z arearbitrary constants):S( y, z)= D YZ =0 n Zy Z + p Zy1- Z + q Zy Z +2+ r Zy3- Z; Z (cos z) + D YZ =2( ~ n Zy Z + ~ p Zy1- Z + ~ q Zy Z +2+ ~ r Zy3- Z;e Z (cos z),(1) where the Z ( )and Z ( )aretheGegenbauer functions ofthe®rstandsecond kind, respecti vely. These arelinearly related totheLegendre functions Z ( )and Z ( )by Z ( )= Z -2( )- Z ( ) 2 f-1, Z ( )= Z -2( )- Z ( ) 2 f-1( f³2). TheGegenbauer functions ofthe®rstkind arerepresented intheform ofa®nite powerseries as Z ( )=-1 ( f-1)! M II N Z -2M 2-1 2 N Z -1 =1×3 ..(2 f-3) 1×2 .. fL Z - f( f-1) 2(2 f-3) Z -2+ f( f-1)( f-2)( f-3) 2×4(2 f-3)(2 f-5) Z -4- ..O. Inparticular ,0( )=1, 1( )=- , 2( )=1 2(1- 2), 3( )=1 2 (1- 2),4( )=1 8(1- 2)(5 2-1), 5( )=1 8 (1- 2)(7 2-3). TheGegenbauer functions ofthesecond kind arede®ned as0( )=- , 1( )=-1, Z ( )=1 2 Z ( )ln1+ 1- + Z ( )at f³2, where thefunctions Z ( )areexpressed interms oftheGegenbauer functions ofthe®rstkind as Z ( )=-1 2 Z £ £1 2 Z +1 2Y(2 f-4 u+1) (2 u-1)( f- u)L1-(2 u-1)( f- u)f( f-1) O Z -2 +1( ); theseries start with 0or 1,depending onwhether fisoddoreven.Inparticular ,2( )=1 2 , 3( )=1 6(3 2-2), 4( )=1 24 (15 2-13), 5( )=1 120(105 4-115 2+16). For f³2,theGegenbauer functions ofthesecond kind assume in®nite values atthepoints= 1,which correspond to z=0and z= J.Therefore, ifphysically there arenosingularities intheproblem, then thequantities in(1)labeled with atilde must besetequal tozero. Inthe overwhelming majority ofproblems onthe¯owabout particles, drops, orbubbles, thestream function inthespherical coordinates isgivenbyformula (1)withn1= n0= p1= p0= q1= q0= r1= r0=0; ~ n Z = ~ p Z = ~ q Z = ~ r Z =0for f=2,3, .. Example 1.Intheproblem onthetranslational Stokes¯owabout asolid spherical particle, thefollowing boundary conditions areimposed onthestream function :=0at = , . =0at = , 1 2 2sin2 as , where istheradius oftheparticle andistheunperturbed ¯uid velocity atin®nity . Stokessolution:( , )=1 4 ( - )2 2+ sin2 . Here, only theterms for =2inthe®rstsum of(1)remain. Page631 632 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS Example 2.Intheproblem ontheaxisymmetric straining Stokes¯owabout asolid spherical particle, thefollowing boundary conditions areimposed onthestream function :=0at = , . =0at = , 1 2 3sin2cos as , where istheradius oftheparticle andistheshear coef®cient. Solution:( , )=1 2 3 33-5 2+3 2 22sin2cos . Here, only theterms for =3inthe®rstsum of(1)remain. Example 3.Solving theproblem ofthetranslational Stokes¯owabout aspherical drop (orbubble) isreduced tosolving theStokesequation outside andinside thedrop. Theboundary condition atin®nity isspeci®ed inexample 1.Conjugate boundary conditions aresetatthedrop surface; these conditions canbefound inthereferences cited belowandarenot written outhere. Hadamard±Rybczynski solution:( , )=1 4 2 2-3 +2+1 + +1 33sin2 for > ,( , )= 4( +1) 2 22-1sin2for < , where istheradius ofthedrop,theunperturbed ¯uid velocity atin®nity , theratio ofthedynamic viscosities ofthe ¯uids inside andoutside thedrop (thevalue =0corresponds toagasbubble and = toasolid particle). Example 4.Solving theproblem oftheaxisymmetric straining Stokes¯owabout aspherical drop (orbubble) isreduced tosolving theStokesequation outside andinside thedrop. The boundary condition atin®nity isspeci®ed inexample 2. Conjugate boundary conditions aresetatthedrop surface; these conditions canbefound inthereferences cited belowand arenotwritten outhere. Taylor solution:( , )=1 2 3 33-1 25 +2+1+3 2 +1 22sin2cos for > ,( , )=3 4 3+1 33 22-1sin2 cos for < , where isthedrop radius,theshear coef®cient, theratio ofthedynamic viscosities ofthe¯uids inside andoutside the drop (thevalue =0corresponds toagasbubble and = toasolid particle).! Refer ences :G.I.Taylor (1932), V.G.Levich (1962), J.Happel andH.Brenner (1965), A.D.Polyanin, A.M.Kutepo v, A.V.Vyazmin, andD.A.Kazenin (2001). 9.4.6-2. Stokesequation inthebipolar coordinate system. When studying axisymmetric problems ofa¯owabout twospherical particles (drops, bubbles), oneuses thebipolar coordinates , ;these arerelated tothecylindrical coordinates = ycos z,= ysin zby= sin cosh -cos , = sinh cosh -cos . Thegeneral solution oftheequation x2( x2 S)=0inthebipolar coordinate system hastheformS( , )=1 (cosh -cos )3 v2 ¡ ¢£ =0 ¤ £ +1(cos ) ¥ £ ( )+¡ ¢£ =0 ¦ £ +1(cos ) § £ ( ) ¨,¥ £ ( )= © £ cosh `( ª-1 2) «+ p £ sinh `( ª-1 2) «+ q £ cosh `( ª+3 2) «+ ¬ £ sinh `( ª+3 2) «,§ £ ( )= © £ cosh `( ª-1 2) «+ p £ sinh `( ª-1 2) «+ q £ cosh `( ª+3 2) «+ ¬ £ sinh `( ª+3 2) «, where the © £ , p £ , q £ , ¬ £ , © £ , p £ , q £ ,and ¬ £ arearbitrary constants andthe¤ £ ( ®)and¦ £ ( ®) aretheGegenbauer functions.! Refer ence:J.Happel andH.Brenner (1965). Page632 9.5. HIGHER -ORDER LINEAR EQUATIONS WITH CONST ANT COEFFICIENTS 633 9.4.6-3. Stokesequation intheoblate spheroidal coordinate system. When studying axisymmetric problems of¯owsabout spheroidal particles, oneuses theoblate spheroidal coordinates , ;these arerelated tothecylindrical coordinates = ¯cos °, = ¯sin °by= ±cosh sin , = ±sinh cos . Thesolution oftheequation ²2( ²2 ³)=0thatdescribes the¯owofa¯uid about aprolate spheroid inthedirection parallel tothespheroid axisisexpressed as³=1 2 ´ ±2cosh2 µsin2 ¶ ·1-[ ¸ ¹( ¸2+1)]-[( ¸2 0-1) ¹( ¸2 0+1)]arccot ¸ [ ¸0 ¹( ¸2 0+1)]-[( ¸2 0-1) ¹( ¸2 0+1)]arccot ¸0 º, ¸=sinh µ, ¸0=sinh µ 0. Here, ³isthestream function,´isthe¯uid velocity atin®nity , ±and ¸0aretheconstants related tothespheroid semiax es »and ¼( »> ¼)by ±= ½ »2- ¼2and ¸0= ¼.¹ ±.¾!¿ Refer ence:J.Happel andH.Brenner (1965). 9.5. Higher -OrderLinear Equations with Constant Coef®cientsÀThroughout Section 9.5thefollowing notation isused: x={ Á1, Â.Â.Â, Á £ },y={ H1, Â.Â.Â, H £ }, Ã={ Ä1, Â.Â.Â, Ä £ }, Å={ µ 1, Â.Â.Â, µ £ }, |x|= Æ Á2 1+ Ç.Ç.Ç+ Á2 £ ,| Ã|= Æ Ä2 1+ Ç.Ç.Ç+ Ä2 £ , Ã×x= Ä1 Á1+ Ç.Ç.Ç+ Ä £Á £ . 9.5.1. Fundamental Solutions. Cauc hyProblem 9.5.1-1. Domain: È £ ={- É< Á Ê< É; Ë=1, Â.Â.Â, ª}. Let beaconstant coef®cient linear differential operator such that Ì ÍÍ Á1, Â.Â.Â, ÍÍ Á £ Î º Ï ¢ÑÐ =0 » Ð 1, ÒÓÒÓÒ, ÐÕÔÍ ÐÍ Á Ð 1 1 Â.Â.ÂÍ Á ÐÕÔ£ , Ö= Ö1+ Ç.Ç.Ç+ Ö £ , where Ö1, Â.Â.Â, Ö £ arenonne gativeintegers, » Ð 1, ÒÓÒÓÒ, ÐÕÔ aresome constants, and ×istheorder ofthe operator .Ageneralized function (distrib ution) Ø Ø(x)= Ø Ø( Á1, Â.Â.Â, Á £ )thatsatis®es theequationÙÌ ÍÍ Á1, Â.Â.Â, ÍÍ Á £ ÚØ Ø(x)= Û(x), where Û(x)= Û( Á1) Â.Â.ÂÜÛ( Á £ )istheDirac delta function inthe ª-dimensional Euclidian space, is called thefundamental solution corresponding totheoperator Ù. Anyconstant coef®cient linear differential operator hasafundamental solution Ø Ø(x).The fundamental solution isnotunique Ðitisde®ned uptoanadditi veterm ³0(x)thatisanarbitrary solution ofthehomogeneous equation Ù Ý ÞÞ ß1, Â.Â.Â, ÞÞ ß Ô à³0(x)=0. Thesolution ofthenonhomogeneous equationÙÌ ÍÍ Á1, Â.Â.Â, ÍÍ Á £ Ú³= á(x) with anarbitrary right-hand sidehastheform³(x)= Ø Ø(x) â á(x), Ø Ø(x) â á(x)= ã ä ÔØ Ø(x-y) á(y) åy. Here, åy= å H1 Â.Â.Â,å H £ andtheconvolution Ø Ø â áisassumed tobemeaningful.¾!¿ Refer ences :G.E.Shilo v(1965), S.G.Krein (1972), L.HÈormander (1983), V.S.Vladimiro v(1988). Page633 634 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.5.1-2. Domain: 0£ æ< É,- É< Á Ê< É; Ë=1, Â.Â.Â, ç.Cauchy problem. Nowlet ÙÝ ÞÞ.è, ÞÞ ß1, Â.Â.Â, ÞÞ ß Ôà beaconstant coef®cient linear differential operator oforder éwith respect to æ.Then adistrib ution Ø Ø( æ,x)= Ø Ø( æ, Á1, Â.Â.Â, Á £ ),which isasolution oftheequationÙÌ ÍÍ æ, ÍÍ Á1, Â.Â.Â, ÍÍ Á £ ÚØ Ø( æ,x)=0 andsatis®es theinitial conditions*Ø Ø êê è=0=0, Í Ø ØÍ æ êêêê è=0=0, Â.Â.Â, Í ë-2Ø ØÍ æë-2 êêêê è=0=0, Í ë-1Ø ØÍ æë-1 êêêê è=0= Û(x), (1) iscalled afundamental solution oftheCauchy problem corresponding totheoperators Ù. Thesolution oftheCauchy problem forthelinear differential equationÙÌ ÍÍ æ, ÍÍ Á1, Â.Â.Â, ÍÍ Á £ Ú ì =0 (2) with thespecial initial conditionsìêê è=0=0, Í ìÍ æ êêêê è=0=0, Â.Â.Â, Í ë-2 ìÍ æë-2 êêêê è=0=0, Í ë-1 ìÍ æë-1 êêêê è=0= í(x) isgivenbyì ( æ,x)= Ø Ø( æ,x) â í(x), Ø Ø( æ,x) â í(x)º ã ä ÔØ Ø( æ,x-y) í(y) åy.¾!¿ Refer ence:S.G.Krein (1972). 9.5.1-3. Solution oftheCauchy problem forgeneral initial conditions. Ifthegeneral initial conditionsìêê è=0= í0(x), Í ìÍ æ êêêê è=0= í1(x), Â.Â.Â, Í ë-2 ìÍ æë-2 êêêê è=0= íë-2(x), Í ë-1 ìÍ æë-1 êêêê è=0= íë-1(x)(3) areset,thesolution ofequation (2)issought intheformì ( æ,x)= Ø Ø( æ,x) â î0(x)+ Í Ø Ø( æ,x)Í æ â î1(x)+ Ç.Ç.Ç+ Í ë-1Ø Ø( æ,x)Í æë-1 â îë-1(x). (4) Each term in(4)satis®es equation (2),andthefunctions îë-1, îë-2, Â.Â.Â, î0aredetermined successi velyfrom thelinear systemí0(x)= îë-1(x),í1(x)= îë-2(x)+ Í ë Ø Ø(0,x)Í æë â îë-1(x), ×××××××××××××××××××××××××××××××××××××××××××××××××××××××í Ê(x)= îë- Ê-1(x)+ Í ë Ø Ø(0,x)Í æë â îë- Ê(x)+ Ç.Ç.Ç+ Í ë+ Ê-1Ø Ø(0,x)Í æë+ Ê-1 â îë-1(x), Ë=2, Â.Â.Â, é-1. This system ofequations isobtained bysuccessi velydifferentiating relation (4)followed bysubsti- tuting æ=0andtaking intoaccount theinitial conditions (1)and(3).¾!¿ Refer ence:G.E.Shilo v(1965). *Thenumber ofinitial conditions canbelessthan ï(seeParagraph 9.5.4-1). Page634 9.5. HIGHER -ORDER LINEAR EQUATIONS WITH CONST ANT COEFFICIENTS 635 9.5.2. Elliptic Equations 9.5.2-1. Homogeneous elliptic differential operator . Aconstant coef®cient linear homogeneous differential operator oforder ËhastheformÙÊ Ì ÍÍ Á1, Â.Â.Â, ÍÍ Á £ Ú º ð ñ Ð 1, ÒÓÒÓÒ, ÐÕÔ ò óóÁ1 ô Ð 1Â.Â.Â ò óóÁ £ô ÐÕÔ , £ðöõ =1 Ö õ = ÷, where Ö1, Â.Â.Â, Ö £ arenonne gativeintegers. From nowon,weadopt thenotationò óóÁ1 ô Ð 1Â.Â.Â ò óóÁ £ô ÐÕÔ º ó Ð 1+ øÓøÓø+ ÐÕÔóÁ Ð 1 1 Â.Â.Â óÁ ÐÕÔ£ . Alinear homogeneous differential operator oforder ÷possesses thepropertyù ú ò û óóÁ1, Â.Â.Â, û óóÁ £ô= ûúù ú ò óóÁ1, Â.Â.Â, óóÁ £ô, û ¹0isanarbitrary constant . Alinear homogeneous differential operator ù úiscalled elliptic if,onreplacing in ù úthesymbolsüü ý 1, Â.Â.Â, üü ý Ô byvariables þ1, Â.Â.Â, þ £ ,oneobtains apolynomial ù ú( þ1, Â.Â.Â, þ £ )thatdoes notvanish if Ã¹0,i.e.,ù ú( þ1, Â.Â.Â, þ £ )º ð ñ Ð 1, ÒÓÒÓÒ, ÐÕÔþ Ð 1 1 Â.Â.Âÿþ ÐÕÔ£ ¹0 if| Ã|¹0. Alinear differential equationù ú ò óóÁ1, Â.Â.Â, óóÁ £ô ì º ð ñ Ð 1, ÒÓÒÓÒ, ÐÕÔ ó Ð 1+ øÓøÓø+ ÐÕÔìóÁ Ð 1 1 Â.Â.Â óÁ ÐÕÔ£ =0, £ð õ =1 Ö õ = ÷ (1) iscalled elliptic ifthelinear homogeneous differential operator ù úiselliptic. 9.5.2-2. Elliptic differential operator ofgeneral form. Ingeneral, aconstant coef®cient linear differential operator oforder ÷hastheform ú ò óóÁ1, Â.Â.Â, óóÁ £ô= ù ú ò óóÁ1, Â.Â.Â, óóÁ £ô+ ú -1ðöõ =0 ù õ ò óóÁ1, Â.Â.Â, óóÁ £ô, where ù úistheleading part oftheoperator and ù ( =0,1 Â.Â.Â, ÷)isalinear homogeneous differential operator oforder .Theoperator úissaidtobeelliptic ifitsleading part ù úiselliptic. Alinear differential equation ú ò óóÁ1, Â.Â.Â, óóÁ £ô ì =0 (2) iscalled elliptic ifthelinear differential operator úiselliptic. Alinear elliptic operator andalinear elliptic differential equation canonly beofeven order ÷=2 é,where éisapositi veinteger. Page635 636 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.5.2-3. Fundamental solution ofahomogeneous elliptic equation. Thefundamental solution ofthehomogeneous elliptic equation (1)with ÷=2 éisgivenby (x)=(-1) £ -1 2 4(2 ) £ -1(2 é- ç)! Ô | Ã×x|2 - £å £ù 2 ( Ã)if çisoddand2 ³ ç; (x)=(-1) £ -2 2 (2 ) £ (2 - ç)! Ô | Ã×x|2 - £ ln| Ã×x| å £ù 2 ( Ã)if çisevenand2 ³ ç; (x)=(-1) £ -1 2 2(2 ) £ -1 2 Ô ( £ -2 -1)( Ã×x) å £ù 2 ( Ã)if çisoddand2 < ç; (x)=(-1) £ 2( ç-2 -1)! (2 ) £ Ô | Ã×x|2 - £å £ù 2 ( Ã)if çisevenand2 < ç. Here, theintegration isperformed overthesurfaceofthe ç-dimensional sphere £ ofunitradius de®ned bytheequation | Ã|=1; Ã×x= þ1 Á1+ + þ £Á £ and ù 2 ( Ã)= ù 2 ( þ1, Â.Â.Â, þ £ ). Afundamental solution isanordinary function, analytic atanypoint x¹0;thisfunction is described, inaneighborhood oftheorigin ofcoordinates (as|x| 0),bytherelations (x)= û £ , |x|2 - £ if çisodd or çisevenand ç>2 ; £ , |x|2 - £ ln|x|if çisevenand ç£2 . Here, û £ , and £ , aresome nonzero constants. If2 > ç,thefundamental solution hascontinuous derivativesuptoorder 2 - ç-1inclusi veattheorigin. 9.5.2-4. Fundamental solution ofageneral elliptic equation. Thefundamental solution ofthegeneral elliptic equation (2)with ÷=2 isdetermined from the relation (x)= Ô ( Ã×x,- ç) å £ , (3) where ( , )=1 £ £ -1 2 +1 2 ! " - " #( - $, Ã)| $| å $, £ =2 £ % 2( ç &2). Here, thefunction#( , Ã)isafundamental solution oftheconstant coef®cient linear ordinary differential equation 2 òþ1 åå , Â.Â.Â, þ £åå ô #( , Ã)= ( ). If çisodd, thefundamental solution (3)canberepresented as (x)= ' £ ( ) ó £ -1ó £ -1#( , Ã) * å +, ' +=(-1) +-1 2 1´3 Â.Â.Â( ç-2) +(2 ) +-1 2.,.- Refer ences :I.M.Gel'f and, G.E.Shilo v(1959), S.G.Krein (1972). Page636 9.5. HIGHER -ORDER LINEAR EQUATIONS WITH CONST ANT COEFFICIENTS 637 9.5.3. Hyperbolic Equations Let ù üüè, üü ý 1, Â.Â.Â, üü ý (!beaconstant coef®cient linear homogeneous differential operator of order with respect to æ.Theoperator ùiscalled hyperbolic ifforanynumbers þ1, Â.Â.Â, þ +such that +/0=1 þ20=1,the th-order algebraic equationù( , þ1, Â.Â.Â, þ +)=0 with respect to has different realroots. Fundamental solution oftheCauchy problem for ³ ç-1: ( æ,x)=(-1) ++1 2 2(2 ) +-1( - ç-1)! 1=0( 2×x+ æ) - +-1[sign ( 2×x+ æ)] -1 |Ñ 3|sign( 2×Ñ 3) å 1if çisodd; ( æ,x)=2(-1) + 2 (2 ) +( - ç-1)! 1=0( 2×x+ æ) - +-1 |Ñ 3|sign( 2×Ñ 3)ln 4444 2×x+ æ2×x 4444 å 1if çiseven, where3= ù(1, 1, Â.Â.Â, 5+),|Ñ 3|= 6 7 8 38 1 92 + + 7 8 38 +92 , 2×Ñ 3= 1 8 38 1+ + 5+ 8 38 +, and å 1istheelement ofthesurface 3=0. Fundamental solution oftheCauchy problem for < ç-1: ( æ,x)=(-1) ++1 2 (2 ) +-1 :;=0 <( =- >)( ?×x+ æ) |Ñ @|sign( ?×Ñ @) å A ;if çisodd;B B( C,x)=(-1) = 2( D- E)! (2 F) = :;=0( ?×x+ C) >- =-1 |Ñ @|sign( ?×Ñ @) G A ;if Diseven.H.I Refer ences :I.M.Gel'f and, G.E.Shilo v(1959), S.G.Krein (1972). 9.5.4. Regular Equations. Number ofInitial Conditions intheCauc hyProblem 9.5.4-1. Equations with twoindependent variables (0£ C< J,- J< K< J). 1 L.Consider theconstant coef®cient linear differential equationM> NMC >= >-1Oú =0 P ú Q R MMK S M úNMC ú, (1) whereP ú( T)isapolynomial ofdegree ÷, R2=-1.Let U= U( A)bethenumber ofroots (taking into account their multiplicities) ofthecharacteristic equationV>- >-1Oú =0P ú( A) V ú =0 (2) whose realparts arenonpositi ve(orbounded above)forgivena A.If Uisthesame (uptoasetof measure zero) forall A W(- J, J),theequation (1)willbecalled regular with regularity index U. Classical equations such astheheat, wave,andLaplace equations areregular . Page637 638 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 2 L.IntheCauchy problem fortheregular equation (1),oneshould set Uinitial conditions ofthe formN XXZY=0= [0( K), MNMC XXXX\Y=0= [1( K), Â.Â.Â, M ]-2 NMC ]-2 XXXXZY=0= [ ]-2( K), M ]-1 NMC ]-1 XXXX\Y=0= [ ]-1( K). (3) Itshould beemphasized thattheregularity index Ucan,ingeneral, differfrom theequation orderEwith respect to C.Inparticular ,forthetwo-dimensional Laplace equation MY^Y N=- M _ _N,wehaveU=1and E=2;here, Hisreplaced by Candthe®rstboundary valueproblem intheupper half-planeC³0isconsidered. Fortheheat equation MY N= M _ _Nandthewaveequation MY^Y N= M _ _N,wehaveU= E=1and U= E=2,respecti vely. Example 1.Belowaretheregularity indices forsome fourth-order equations:`2 a` b2- c2 `4 a` d4=0 ( e=1, f=2); `2 a` b2+ c2 `4 a` d4=0 ( e=2, f=2);g`2` b2+ `2` d2 h2a=0 ( e=2, f=4); `4 a` b4- c2 `4 a` d4=0 ( e=3, f=4). 3 L.Thespecial solution B B= B B( C, K)thatsatis®es theinitial conditionsB BXXZY=0=0, M B BMC XXXXZY=0=0, Â.Â.Â, M ]-2 B BMC ]-2 XXXX\Y=0=0, M ]-1 B BMC ]-1 XXXXZY=0=<( K) (4) iscalled fundamental. Thefundamental solution canbefound byapplying theFourier transform inthespace variable toequation (1)(with N= B B)andtheinitial conditions (4). Example 2.Consider thepolyharmonic equation:g`2` b2+ `2` d2h i a=0. (5) Taking intoaccount therepresentation j2jk2=- lnm jjk o2,werewrite thecharacteristic equation (2)intheform ( p2- q2)i=0. Ithasonly onesolution whose realpartisnonpositi ve,speci®cally , p=-| q|.Considering themultiplicity oftheroot, we ®ndthattheregularity index eisequal to r. Inequation (5)with a= B B andtheinitial conditions (4)with e= r,weperform theFourier transform with respect to thespace variable, s ( b, q)= t u -u v wyx k B B ( b, d) z d. Asaresult, wearriveattheordinary differential equationgz2z b2- q2h i s =0 (6) andtheinitial conditionss {{}| =0=0, s ~| {{\| =0=0, , s (i-2) | {{\| =0=0, s (i-1) | {{\| =0=1. (7) Thebounded solution ofproblem (6),(7)isgivenbys ( b, q)= bi-1 ( r-1)!v-|x| | . Byapplying theinverseFourier transform, weobtain thefundamental solution ofthepolyharmonic equation intheformB B ( b, d)=1 2 t u -uv-wyx k s ( b, q) zq=1 2 bi-1 ( r-1)! t u -uv-wyx k-|x| |zq =1 2 bi-1 ( r-1)! gt u 0v-wyx k-x |zq+ t0 -uv-wyx k+x |zqh =1 2 bi-1 ( r-1)! g1b+ m d+1b- m dh=1 bi-1 ( r-1)! bb2+ d2. Page638 9.5. HIGHER -ORDER LINEAR EQUATIONS WITH CONST ANT COEFFICIENTS 639 4 L.Forgeneral initial conditions oftheform (3),thesolution ofequation (1)isdetermined onthe basis ofthefundamental solution from therelationN( C, K)= B B( C, K) 0( K)+ M B B( C, K)MC 1( K)+ + M ]-1 B B( C, K)MC ]-1 ]-1( K). (8) Each term in(8)satis®es equation (1),andthefunctions ]-1, ]-2, Â.Â.Â, 0arecalculated succes- sivelybysolving thelinear system[0( K)= ]-1( K),[1( K)= ]-2( K)+ M ] B B(0, K)MC ] ]-1( K), ............................................................[ ú( K)= ]- ú -1( K)+ M ] B B(0, K)MC ] ]- ú( K)+ + M ]+ ú -1 B B(0, K)MC ]+ ú -1 ]-1( K), ÷=2, Â.Â.Â, U-1. This system ofequations isobtained bysuccessi velydifferentiating relation (8)followed bysubsti- tuting C=0andtaking intoaccount theinitial conditions (3)and(4). Inthespecial case [0( K)= [1( K)= = [ ]-2( K)=0,oneshould set 0( K)= [ ]-1( K)and1( K)= = ]-1( K)=0in(8).H.I Refer ence:G.E.Shilo v(1965). 9.5.4-2. Equations with manyindependent variables (0£ C< J,x W =). Solving theCauchy problem fortheconstant coef®cient linear differential equationù Q MMC, MMK1, Â.Â.Â, MMK= S N=0 (9) with arbitrarily manyspace variables K1, Â.Â.Â, K=canbereduced tosolving theCauchy problem for anequation with onespace variable .Wetakeanauxiliary linear differential operatorù Q MMC, MM Sº ù Q MMC, 1 MM, Â.Â.Â, = MM S thatdepends ontwoindependent variables Cand sothattheCauchy problem fortheequationù Q MMC, MM S =0 (10) iswell posed. Then thefundamental solution oftheCauchy problem fortheoriginal equation (9)is givenbyB B( C,x)=: ( C, Ã×x,- D)G =. Here, ( C, , V)=1A= F =-1 2 +1 2 - ( , - )| | , =2 2( ¡2), where ( , )isthefundamental solution oftheCauchy problem fortheauxiliary equation (10). Ifthenumber ofspace variables isodd, onecanusethesimpler formula¢ ¢( ,x)=(-1) -1 2 £ -1 2 ¤! -1 2( -1)! ¥ ¦ § -1 -1 ( , ) ¨ © , = Ã×x.ª «¬ ® ¯ °The aboverelations hold forallequations forwhich theCauchy problem iswell posed.±.² Refer ences :I.M.Gel'f and, G.E.Shilo v(1959), S.G.Krein (1972). Page639 640 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.5.4-3. Stationary homogeneous regular equations (x W ³ ). Alinear differential operator ´ µ £ ¶¶ ·1, ¸¸¸, ¶¶ ·¦ ¤iscalled regular ifitishomogeneous andifthe gradient ofthefunction ´ µ( ¹1, ¸¸¸, ¹ )onthesetde®ned bytheequation ´ µ( ¹1, ¸¸¸, ¹ )=0is everywhere nonzero whene ver| º|¹0. The fundamental solution ofthelinear regular differential equation ´ µ £ ¶¶ ·1, ¸¸¸, ¶¶ ·¦ ¤»=0 generated bythelinear regular differential operator ´ µisexpressed as¢ ¢(x)= ¥ ¦ ¼ µ( º×x)´ µ( º) © , (11) where thefunction¼ µ( ½)isde®ned by¼ µ( ½)=(-1) -2 2 (2 ) ( ¾- )! ½ µ- ln| ½| if isevenand ¾³ ;¼ µ( ½)=(-1) +2 µ 2( - ¾-1)! (2 ) ½ µ- if isevenand ¾< ;¼ µ( ½)=(-1) -1 2 4(2 ) -1( ¾- )! ½ µ- sign ½ if isoddand ¾³ ;¼ µ( ½)=(-1) -1 2 2(2 ) -1 ¿( - µ-1)( ½) if isoddand ¾< . Theintegralin(11) isunderstood inthesense ofitsregularized value, i.e.,¢ ¢(x)=limÀ Á0 ¢ ¢À(x), ¢ ¢À(x)= ¥( Â)¦ ¼ µ( º×x)´ µ( º) ©( À), where ©( À)isthesetofpoints onasphere ofunitradius forwhich | ´ µ( º)|> Ã.±.² Refer ences :I.M.Gel'f and, G.E.Shilo v(1959), S.G.Krein (1972). 9.5.5. Some Special-T ypeEquations 1. Ä ÅÄ Æ= Ç È É Ê ÄÄ Ë Ì Å, Ç È( Í)= Î È Í È+ Ï Ï Ï+ Î1 Í+ Î0, Ê2=±1. Thecondition Re ´ ( ½)£ Ð< Ñisassumed tobemetforallreal ½. 1 Ò.Domain: - Ñ< Ó< Ñ.Cauchy problem. Aninitial condition isprescribed:»= Ô( Ó)at =0. (1) Solution:»( Ó, )= - ( Ó- , ) Ô( ) , ( Ó, )=1 2 -exp ÕÖ´ ( ×)- ØÙÓ × Ú ×. 2 Ò.The solution oftheCauchy problem with theinitial condition (1)forthenonhomogeneous equation Û» Û= ´ É Ø ÛÛÓÌ »+ Ü( Ó, ) isgivenby»( Ó, )= - ( Ó- , ) Ô( ) + Ý0 - ( Ó- , - Þ) Ü( , Þ) Þ, where thefunction ( Ó, )isde®ned inItem 1 Ò.±.² Refer ences :S.G.Krein (1972), V.S.Vladimiro v,V.P.Mikhailo v,A.A.Vasharin, etal.(1974). Page640 9.5. HIGHER -ORDER LINEAR EQUATIONS WITH CONST ANT COEFFICIENTS 641 2. ± =0, =1,2, 1 .General solution (tworepresentations): ( , )= -1 =0 ( + ), ( , )= -1 =0 ( + ), where the = ( )arearbitrary functions. 2 .Fundamental solution: ( , )= -1 ( -1)! ( + ). 3. ± 2 2 =0, =1,2, 1 .General solution (tworepresentations): ( , )= -1 =0 ( , ), ( , )= -1 =0 ( , ), where the = ( , )arearbitrary functions thatsatisfy theheat equations - =0. 2 .Fundamental solution: ( , )=1 2 ( -1)! -3 2exp - 2 4 . 3 .Domain: - < < .Cauchy problem. Initial conditions areprescribed: =0=0, =0=0, !"!"!, -2 -2 =0=0, -1 -1 =0= ( ). Solution: ( , )= # $ -$ ( %) ( - %, ) & %.')( Refer ence:G.E.Shilo v(1965). 4. 2 2± 2 2 =0, =1,2, 1 .General solution (tworepresentations): ( , )=-1 =0 *+ ( + )+ , ( - ) -, ( , )=-1 =0 *) ( + )+ , ( - ) -, where the = ( .)and , = , ( )arearbitrary functions. Page641 642 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 2 .Fundamental solution: ( , )=(-1)-1 4( -1)! /sign( - ) -1 =0(2 ) ( - )2- -20!( - 0-1)! +(-1)sign( + ) -1 =0(-2 ) ( + )2- -20!( - 0-1)! 1.')( Refer ence:G.E.Shilo v(1965). 5. 2 2+ 2 22 =0, =1,2, This isthepolyharmonic equation oforder with twoindependent variables. 1 .General solution (tworepresentations): ( , .)= -1 =0 32 ( , .),3= 4 2+ .2, ( , .)= -1 =0 ( , .), where the ( , .)arearbitrary harmonic functions ( 5 =0).Inthesecond relation, canbe replaced by . . 2 .Domain: - < < ,- < .< .Fundamental solution: ( , .)=122-1 * ( -1)! -232-2ln3,3= 4 2+ .2. 3 .Domain: - < < ,0£ .< .Boundary value problem. Boundary conditions areprescribed: 76 =0=0, . 76 =0=0, !"!"!, -2 .-2 76 =0=0, -1 .-1 86 =0= ( ). Solution: ( , .)= # $ -$ ( %) 9( - %, .) & %, 9( , .)=1( -1)! .2+ .2. SeealsoExample 2inParagraph 9.5.4-1.')( Refer ences :G.E.Shilo v(1965), L.D.Faddee v(1998). 6. 2 2+ 2 22 = :( , 2), =1,2, This isanonhomo geneous polyharmonic equation oforder with twoindependent variables. Particular solution: ( , .)=122[( -1)!]2 # $ -$ # $ -$ ;( %, <)[( - %)2+( .- <)2]-1ln[( - %)2+( .- <)2] & % & <. Thegeneral solution isgivenbythesumofanyparticular solution ofthenonhomogeneous equation andthegeneral solution ofthehomogeneous equation (seeequation 9.5.5.5, Item 1 ). Page642 9.5. HIGHER -ORDER LINEAR EQUATIONS WITH CONST ANT COEFFICIENTS 643 7. = > =0, == @? =1 2 2 ? . This isthepolyharmonic equation oforder Awith independent variables. For A=1,see Sections 7.1and8.1.For A=2,seeSubsection 9.4.1. For =2,seeequations 9.5.5.5 and9.5.5.6. 1 .Particular solutions: (x)= B-1DC =0 CE C (x), F=1,2, !"!"!, , where the C (x)arearbitrary harmonic functions ( 5 C =0). 2 .Fundamental solution for A³1and ³3: (x)= G H,B|x|2B-if isodd or isevenand >2 A;I,B|x|2B-ln|x|if isevenand £2 A. Here,H,B= J( K2) 2B( A-1)! 2(2- )(4- ) !"!"!(2 A- ),I,B= J( K2) 2B( A-1)! 2(2- )(4- ) !"!"!(2 A0-2- )(2 A0+2- )(2 A0+4- ) !"!"!(2 A- ), where A0= K2.Theexpression ofthecoef®cient I,Bcanbeobtained formally from theexpression ofH,Bbyremo ving themultiplier (2 A0- )equal tozero from thedenominator .')( Refer ence:G.E.Shilo v(1965). 8. > ? =0 ?= ?=0, =º 2 2+ 2 22. Particular solutions: ( , .)= B =1 ( , .), where the aresolutions oftheHelmholtz equations 5 - L =0andthe Lareroots ofthe characteristic equationB M =0 L =0.')( Refer ence:A.V.Bitsadze andD.F.Kalinichenk o(1985). 9. > @? =0 ? N ? []=0. Here, Oisanyconstant coef®cient linear differential operator with arbitrarily manyindependent variables 1, !"!"!, . Particular solutions: ( 1, !"!"!, )=B DC =1 P C C ( 1, !"!"!, ), where the C aresolution oftheequations O[ C ]- L C C =0the L C areroots ofthecharacteristic equationB M =0 L =0,andtheP C arearbitrary constants. Page643 644 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.6. Higher -OrderLinear Equations with Variab le Coef®cients 9.6.1. Equations Containing theFirstTime Deriv ative 9.6.1-1. Statement oftheproblem foranequation with twoindependent variables. Consider thelinear nonhomogeneous partial differential equation - O , [ ]=;( , ), (1) where O , isageneral linear differential operator oforder with respect tothespace variable ,O , [ ]º =0 ( , ) , (2) whose coef®cients = ( , )aresuf®ciently smooth functions ofboth arguments for ³0and1£ £ 2.Thesubscripts and indicate thattheoperator O , isdependent onthevariables and . Wesettheinitial condition = ( )at =0 (3) andthegeneral nonhomogeneous boundary conditionsJ(1)B[ ]º -1 =0 H(1)B ( ) = ,(1)B( )at = 1 ( A=1, !"!"!, Q),J(2)B[ ]º -1 =0 H(2)B ( ) = ,(2)B( )at = 2 ( A= Q+1, !"!"!, ),(4) where Q³1and ³ Q+1.Weassume thatboth setsoftheboundary formsJ(1)B[ ]( A=1, !"!"!, Q)andJ(2)B[ ]( A= Q+1, !"!"!, )arelinearly independent, which means thatforanynonzero RB= RB( ) thefollowing relations hold: CB=1 RB( )J(1)B[ ] S0, B= C +1 RB( )J(2)B[ ] S0. Inwhat follows,wedeal with thenonstationary boundary value problem (1),(3),(4). 9.6.1-2. The case ofgeneral homogeneous boundary conditions. The Green' sfunction. Thesolution ofequation (1)with theinitial condition (3)andthehomogeneous boundary conditionsJ(1)B[ ]=0at = 1 ( A=1, !"!"!, Q),J(2)B[ ]=0at = 2 ( A= Q+1, !"!"!, )(5) canbewritten as ( , )= # 21 ( .) 9( , ., ,0) & .+ # 0 # 21;( ., T) 9( , ., , T) & . & T. (6) Here, 9( , ., , T)istheGreen' sfunction thatsatis®es, for > T³0,thehomogeneous equation 9 - O , [ 9]=0 (7) Page644 9.6. HIGHER -ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS 645 with thespecial nonhomogeneous initial condition9=( - .)at = T (8) andthehomogeneous boundary conditionsJ(1)B[ 9]=0at = 1 ( A=1, !"!"!, Q),J(2)B[ 9]=0at = 2 ( A= Q+1, !"!"!, ).(9) Thequantities .and Tappear inproblem (7)±(9) asfreeparameters ( 1£ .£ 2),and( )isthe Dirac delta function. Itshould beemphasized thattheGreen' sfunction 9isindependent ofthefunctions;( , ), ( ),,(1)B( ),and ,(2)B( )thatcharacterize various nonhomogeneities oftheboundary valueproblem. Ifthe coef®cients ,H(1)B ,andH(2)B determining thedifferential operator (2)andboundary conditions (4) areindependent oftime ,then theGreen' sfunction depends only onthree arguments, 9( , ., , T)=9( , ., - T).')( Refer ence:Mathematical Encyclopedia (1977, Vol.1). 9.6.1-3. The case ofnonhomogeneous boundary conditions. Preliminary transformations. Tosolvetheproblem with nonhomogeneous boundary conditions (1),(3),(4),wechoose asuf®- ciently smooth ªtest functionº U= U( , )thatsatis®es thesame boundary conditions astheunkno wn function; thus,J(1)B[ U]= ,(1)B( )at = 1 ( A=1, !"!"!, Q),J(2)B[ U]= ,(2)B( )at = 2 ( A= Q+1, !"!"!, ).(10) Otherwise thechoice oftheªtest functionº Uisarbitrary andisnotlinkedtothesolution ofthe equation inquestion; there arein®nitely manysuch functions. Letuspass from V= V( W, X)tothenewunkno wn Y= Y( W, X)bytherelationV( W, X)= Y( W, X)+ Z( W, X). (11) Substituting (11) into(1),(3),and(4),wearriveattheproblem foranequation with amodi®ed right-hand side,[Y[X- \ ], ^[ Y]= _( W, X), _( W, X)= _( W, X)- [Z[X+ \ ], ^[ Z], (12) subject tothenonhomogeneous initial conditionY= `( W)- Z( W,0)at X=0 (13) andthehomogeneous boundary conditions a (1)b[ Y]=0at W= W1 ( c=1, deded, f), a (2)b[ Y]=0at W= W2 ( c= f+1, dedgd, h).(14) Thesolution ofproblem (12)±(14) canbefound using theGreen' sfunction byformula (6)inwhich oneshould replaceVby Y, _( W, X)by _( W, X),and `( W)by `( W)= `( W)- Z( W,0).Taking into account relation (11), for Vweobtain the representationV( W, X)= i ]2]1 `( j) k( W, j, X,0) lmj+ i ^ 0 i ]2]1 _( j, n) k( W, j, X, n) lmj lmn+ Z( W, X) - i ]2]1 Z( j,0) k( W, j, X,0) lmj- i ^ 0 i ]2]1 [Z[n( j, n) k( W, j, X, n) lmj lmn + i ^ 0 i ]2]1 k( W, j, X, n) \ o, p[ Z( j, n)] lmj lmn. (15) Changing theorder ofintegration andintegrating byparts with respect to n,we®nd, with reference totheinitial condition (8)fortheGreen' sfunction,i ^ 0 [Z[n k lmn= Z( j, X) q( W- j)- Z( j,0) k( W, j, X,0)- i ^ 0 Z( j, n) [k[n( W, j, X, n) lmn. (16) Page645 646 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS Wetransform theinner integralofthelastterm in(15) using theLagrange±Green formula [seeKamk e(1977)] toobtaini ]2]1 k \ o, p[ Z] lmj= i ]2]1 Z \ ro, p[ k] lmj+ s s[ Z, k] tt o= ]2o= ]1, (17)\ ro, p[ k]º u vxw =0(-1) w y wyj w z8{ w ( j, n) k |, s s[ }, k]º u-1vx~ =0 v+ = ~ (-1) y} yj yyj z7{ ~ +1( j, n) k |, where r , [ ]isthedifferential form adjoint with , [ ]of(2); }= }( j, n);and and arenonne gativeintegers. Using relations (16) and(17), werewrite solution (15) intheform( , )= 21 ( ) ( , , ,0) + 0 21 ( , ) ( , , , ) + 0 s s[ , ] 8=2=1 .(18) This formula wasderivedtaking intoaccount thefactthattheGreen' sfunction with respect to and satis®es theadjoint equation* + , [ ]=0. Forsubsequent analysis, itisconvenient torepresent thebilinear differential form s s[ , ]ass s[ , ]= -1 =0 [ ], [ ]= - -1C =0(-1) C C C ¡ C + +1( , ) ¢. (19) Note thatinthespecial case where operator (2)isbinomial,,[ ]= + 0( , ) , =const, thedifferential forms in(19) arewritten as£ £[ , ]= -1 =0(-1)- -1 - -1 - -1, [ ]= (-1)- -1 - -1 - -1. 9.6.1-4. The case ofspecial nonhomogeneous boundary conditions. Consider thefollowing nonhomogeneous boundary conditions ofspecial form thatareoften encoun- tered inapplications: ¥¤ ¥¤ = ¦(1) ¥¤ ( )at = 1 ( A=1, !"!"!, §), ¥¤ ¥¤ = ¦(2) ¥¤ ( )at = 2 ( A= §+1, !"!"!, ¨).(20) Without lossofgenerality ,weassume thatthefollowing inequalities hold:¨-1³ ©1> ©2> ª"ª"ª> © C , ¨-1³ © C +1> © C +2> ª"ª"ª> ©. TheGreen' sfunction satis®es thecorresponding homogeneous boundary conditions thatcanbe obtained from (20) byreplacing by andsetting ¦(1) ¥¤ ( )= ¦(2) ¥¤ ( )=0. *This equation canbederivedbyconsidering thecase ofhomogeneous initial andboundary conditions andusing arbitrariness inthechoice ofthetestfunction }= }( «, ¬);itshould betakenintoaccount thatthesolution itself must be independent ofthespeci®c form of },because }does notoccur intheoriginal statement oftheproblem. Byappropriately selecting thetestfunction, onecanalsoderivetheboundary conditions (21). Page646 9.6. HIGHER -ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS 647 Theadjoint homogeneous boundary conditions, with respect to(20), which must bemetbythe Green' sfunction with respect to and havetheform ® [ ]=0at = 1 ( © ¯¹ ©B, °= §+1, !"!"!, ¨; ±=1, !"!"!, §), ® [ ]=0at = 2 ( © ¯¹ © ², °=1, !"!"!, §; ±= §+1, !"!"!, ¨).(21) These conditions involvethelinear differential forms [ ]de®ned in(19). Foreach endpoint of theinterv alinquestion, theset{ © ¯}oftheindices intheboundary operators (21) together with the set{ © ²}oftheorders ofderivativesintheboundary conditions (20) makeupacomplete setof nonne gativeintegers from 0to ¨-1. Taking intoaccount thefactthatthetestfunction must satisfy theboundary conditions (20) andtheGreen' sfunction toconditions (21), werewrite solution (18) toobtain( , )= 21 ( ) ( , , ,0) + 0 21 ( , ) ( , , , ) - ³ ²=1 0 ¦(1) ¥¤ ( ) ¥¤ [ ] =1 + ²=³+1 0 ¦(2) ¥¤ ( ) ¥¤ [ ] =2 , (22) where the ¥¤ [ ]aredifferential operators with respect to ,which arede®ned in(19). IftheGreen' sfunction isknown,formula (22) canbeused toimmediately obtain thesolution ofthenonhomogeneous boundary value problem (1),(3),(20) forarbitrary( , ),( ), ¦(1) ¥¤ ( ) ( ±=1, !"!"!, §),and ¦(2) ¥¤ ( )( ±= §+1, !"!"!, ¨). 9.6.1-5. The case ofgeneral nonhomogeneous boundary conditions. Onsolving (4)forthehighest derivatives,wereduce theboundary conditions (4)tothecanonical form ¥¤ ¥¤ + ¥¤ -1µ´ =0 ¶(1)² ´ ( ) ´ ´ = ·(1) ¥¤ ( )at = 1 ( ±=1, !"!"!, §), ¥¤ ¥¤ + ¥¤ -1µ´ =0 ¶(2)² ´ ( ) ´ ´ = ·(2) ¥¤ ( )at = 2 ( ±= §+1, !"!"!, ¨),(23) where theleading terms indifferent boundary conditions aredifferent,¨-1³ ©1> ©2> ª"ª"ª> ©³, ¨-1³ ©³+1> ©³+2> ª"ª"ª> ©. Thesums in(23) donotcontain thederivativesoforders ©1, !"!"!, ©³(for = 1)and ©³+1, !"!"!, © (for = 2);thus,¶(1)² ´ ( )=0at ¸= © E( ¹=1, !"!"!, §),¶(2)² ´ ( )=0at ¸= © E( ¹= §+1, !"!"!, ¨). Itcanbeshownthatthesolution ofproblem (1),(3),(23) isgivenby( , )= 21 ( ) ( , , ,0) + 0 21 ( , ) ( , , , ) - ³ ²=1 0 ·(1) ¥¤ ( ) ¥¤ [ ] =1 + ²=³+1 0 ·(2) ¥¤ ( ) ¥¤ [ ] =2 , (24) where the ¥¤ [ ]aredifferential operators with respectto ,which arede®ned in(19). Relation (24) issimilar to(22) butcontains theGreen' sfunction satisfying themore complicated boundary conditions thatcanbeobtained from (23) bysubstituting for andsetting ·(1) ¥¤ ( )= ·(2) ¥¤ ( )=0. Page647 648 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 9.6.2. Equations Containing theSecond Time Deriv ative 9.6.2-1. The case ofhomogeneous initial andboundary conditions. Consider thelinear nonhomogeneous differential equation 2 2+ º( , ) - =0 ( , ) =( , ). (1) Wesetthehomogeneous initial conditions=0at =0, =0at =0(2) andthehomogeneous boundary conditions»(1)²[ ]=0at = 1 ( ±=1, !"!"!, §),»(2)²[ ]=0at = 2 ( ±= §+1, !"!"!, ¨),(3) where theboundary operators »(1)²[ ]and »(2)²[ ]arede®ned inParagraph 9.6.1-1. Thesolution ofproblem (1)±(3) canberepresented intheform*( , )= 0 21 ( , ) ( , , , ) . (4) Here, = ( , , , )istheGreen' sfunction; for > ³0,itsatis®es thehomogeneous equation 2 2+ º( , ) - =0 ( , ) =0 (5) with thespecial semihomogeneous initial conditions=0 at = , = ¼( - )at = (6) andthecorresponding homogeneous boundary conditions»(1)²[ ]=0at = 1 ( ±=1, !"!"!, §),»(2)²[ ]=0at = 2 ( ±= §+1, !"!"!, ¨).(7) Thequantities and appear inproblem (5)±(7) asfreeparameters ( 1£ £ 2),and ¼( )isthe Dirac delta function. One canverify bydirect substitution intotheequation andtheinitial andboundary conditions (1)±(3) thatformula (4)iscorrect, taking intoaccount theproperties (5)±(7) oftheGreen' sfunction. 9.6.2-2. The case ofnonhomogeneous initial andboundary conditions. Consider thelinear nonhomogeneous differential equation (1)with thegeneral nonhomogeneous initial conditions=0( )at =0, =1( )at =0(8) *Problem (1)±(3) isassumed tobewell posed. Page648 9.6. HIGHER -ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS 649 and thenonhomogeneous boundary conditions, reduced tothecanonical form (see Paragraph 9.6.1-5 ): ¥¤ ¥¤ + ¥¤ -1´ =0 ¶(1)² ´ ( ) ´ ´ = ·(1) ¥¤ ( )at = 1 ( ±=1, !"!"!, §), ¥¤ ¥¤ + ¥¤ -1´ =0 ¶(2)² ´ ( ) ´ ´ = ·(2) ¥¤ ( )at = 2 ( ±= §+1, !"!"!, ¨).(9) Introducing atestfunction = ( , )thatsatis®es thenonhomogeneous initial andboundary conditions (8),(9)andusing thesame lineofreasoning asinParagraph 9.6.1-3 forasimpler equation, wearriveatthesolution ofproblem (1),(8),(9)intheform( , )= 0 21 ( , ) ( , , , ) - 210( ) ½ ( , , , ) ¾=0 + 21 1( )+0( ) º( ,0) ¢¥( , , ,0) - ³ ²=1 0 ·(1) ¥¤ ( ) ¥¤ [ ]=1 + ²=³+1 0 ·(2) ¥¤ ( ) ¥¤ [ ]=2 , (10) where the ¥¤ [ ]aredifferential operators with respect to ,which arede®ned inrelations (19), Paragraph 9.6.1-3 .¿ À¥Á Â Ã®Ä ÅIfthecoef®cients ofequation (1)andthose oftheboundary conditions (9)aretime independent, i.e.,º= º( ), = ( ),¶(1)² ´ =const ,¶(2)² ´ =const, then insolution (10) oneshould set( , , , )= Æ ( , , - ), ( , , , ) =0=- Æ ( , , ). 9.6.3. Nonstationar yProblems with ManySpace Variab les 9.6.3-1. Equations with the®rst-order partial derivativewith respect to . Consider thefollowing linear differential operator with respect tovariables 1, !"!"!, :Ç x,[ ]º È 1, ÉÊÉÊÉ, ¥Ë ( 1, !"!"!, , ) 1+ ÌÊÌÊÌ+ ¥Ë 1 1 !"!"! ¥Ë. (1) The coef®cients È 1, ÉÊÉÊÉ, ¥Ë oftheoperator areassumed tobesuf®ciently smooth functions of1, !"!"!, and (and also bounded ifnecessary). The coef®cients ofthehighest derivatives areassumed tobeeverywhere nonzero. 1 Í.Cauc hyproblem ( ³0,x Î Ï).Thesolution oftheCauchy problem forthelinear nonhomo- geneous parabolic differential equation with variable coef®cients - Ç x,[ ]=(x, ) (2) under theinitial conditions=(x)at =0 (3) Page649 650 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS isgivenby(x, )= 0 Ð Ë(y, ) Ñ Ñ(x,y, , ) y + Ð Ë(y) Ñ Ñ(x,y, ,0) y, y= 1 !"!"! .(4) Here, Ñ Ñ= Ñ Ñ(x,y, , )isthefundamental solution oftheCauchy problem, which satis®es for > ³0 theequation Ñ Ñ - Ç x,[ Ñ Ñ]=0 (5) andthespecial initial conditionÑ Ñ = = ¼(x-y). (6) The quantities yand appear inproblem (5),(6)asfree parameters (y Î Ï),and ¼(x)isthe¨-dimensional Dirac delta function. Ifthecoef®cients È 1, ÉÊÉÊÉ, ¥Ë ofoperator (1)areindependent oftime ,then thefundamental solution depends ononly three arguments, Ñ Ñ(x,y, , )= Ñ Ñ(x,y, - ).Ifthecoef®cients of operator (1)areconstants, then Ñ Ñ(x,y, , )= Ñ Ñ(x-y, - ). 2 Í.Boundary value problems ( ³0,x Î Ò).Thesolutions oflinear boundary value problems ina spatial domain Òforequation (2)with initial condition (3)andhomogeneous boundary conditions forx Î Ò(these conditions arenotwritten outhere) aregivenbyformula (4)inwhich thedomain ofintegration Ïshould bereplaced by Ò.Here, by Ñ Ñwemean theGreen' sfunction thatmust satisfy ,apart from equation (5)andtheboundary condition (6),thesame homogeneous boundary conditions forx Î Òastheoriginal equation (2). Forboundary value problems, theparameter y belongs tothesame domain asx,i.e.,y Î Ò.Ó)Ô Refer ence:Mathematical Encyclopedia (1977, Vol.1). 9.6.3-2. Equations with thesecond-order partial derivativewith respect to Õ. 1 Í.Cauc hyproblem ( Õ³0,x Î Ï).Thesolution oftheCauchy problem forthelinear nonhomo- geneous differential equation with variable coef®cients 2 Ö Õ2- Ç x,[ Ö]=(x, Õ) (7) under theinitial conditionsÖ=(x)at Õ=0, Ö= ¦(x)at Õ=0(8) isgivenbyÖ(x, Õ)= × Ø 0 × Ù Ú Û(y, Ü) Ý Ý(x,y, Õ, Ü) Þy Þ Ü - × Ù Ú ß(y) à áá Ü Ý Ý(x,y, â, Ü) ã ä =0 Þy+ × Ù Ú å(y) Ý Ý(x,y, â,0) Þy. Here, Ý Ý= Ý Ý(x,y, â, Ü)isthefundamental solution oftheCauchy problemá2Ý Ýá â2- æx,Ø[ Ý Ý]=0,Ý Ý ççØ= ä=0, á Ý Ýá â ççççØ= ä= è(x-y), where yand Üplay theroleofparameters. Ifthecoef®cients é ê1, ëÊëÊë, ê Úofoperator (1)areindependent oftime â,then thefundamen- talsolution depends ononly three arguments, Ý Ý(x,y, â, Ü)= Ý Ý(x,y, â- Ü),and therelationìì äÝ Ý(x,y, â, Ü) çç ä =0=- ììØ Ý Ý(x,y, â)holds. Ifthecoef®cients ofoperator (1)areconstants, thenÝ Ý(x,y, â, Ü)= Ý Ý(x-y, â- Ü). Page650 9.6. HIGHER -ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS 651 2 í.Thesolution oftheCauchy problem forthemore complicated linear nonhomogeneous differ- ential equation with variable coef®cientsá2 îá â2+ ï(x, â) á îá â- æx,Ø[ î]= Û(x, â) with initial conditions (8)isexpressed asî(x, â)= × Ø 0 × ÙÚ Û(y, Ü) Ý Ý(x,y, â, Ü) Þy Þ Ü - × ÙÚ ß(y) à áá Ü Ý Ý(x,y, â, Ü) ãä =0 Þy+ × ÙÚ ð7å(y)+ ï(y,0) ß(y) ñ Ý Ý(x,y, â,0) Þy. (9) Here, Ý Ý(x,y, â, Ü)isthecorresponding fundamental solution oftheCauchy problem,á2Ý Ýá â2+ ï(x, â) á Ý Ýá â- æx,Ø[ Ý Ý]=0,Ý Ý ççØ= ä=0, á Ý Ýá â ççççØ= ä= è(x-y). 3 í.Boundary value problems ( â³0,x ò ó).Thesolutions oflinear boundary value problems ina spatial domain óforequation (7)with initial condition (8)andhomogeneous boundary conditions forx òá ó(these conditions arenotwritten outhere) aregivenbyformula (9)inwhich thedomain of integration ô õshould bereplaced by ó.Here, by Ý Ýwemean theGreen' sfunction thatmust satisfy , apart from equation (7)andtheinitial conditions (8),thesame homogeneous boundary conditions astheoriginal equation (7). 9.6.4. Some Special-T ypeEquations 1. ö ÷ö ø= ù(ø) ö ú ÷ö ûú+ðû ü(ø)+ ý(ø)ñ ö ÷ö û+ þ(ø)÷. Thetransformationî( ÿ, â)= ( , Ü)exp à × ( â) Þ â ã, = ÿ ( â)+ × å( â) ( â) Þ â, Ü= × ( â) õ( â) Þ â, where ( â)=exp à × ß( â) Þ â ã,leads tothesimpler constant coef®cient equationá á Ü= á õ á õ. 2. ö ÷ö ø = û2ú ö ú ÷ö ûú. Thetransformation =1 ÿ, = îÿ1-õleads totheconstant coef®cient equationá êá â ê= (-1) õá õ á õ. 3. ö ÷ö ø = ú =0 û ö ÷ö û . Thechange ofvariable =ln| ÿ|leads toaconstant coef®cient equation. Page651 652 HIGHER -ORDER PARTIAL DIFFERENTIAL EQUATIONS 4. ö ÷ö ø =( û2+ û+ )ú ö ú÷ö ûú. Thetransformationî( ÿ, â)= ( , â)| ÿ2+ ¥ÿ+ | õ-1 2, = × Þ ÿ ÿ2+ ¥ÿ+ leads toaconstant coef®cient equation. 5. öö ø± ú÷=0, =1,2, Here, isalinear differential operator ofanyorder with respect tothespace variable ÿwhose coef®cients candepend on ÿ. 1 í.General solution:î( ÿ, â)= õ-1 ê=0 â ê ê( ÿ, â), where the ê= ê( ÿ, â)arearbitrary functions that satisfy theoriginal equation with =1: (áØ- ) ê=0. 2 í.Fundamental solution:Ý Ýõ( ÿ, â)= âeõ-1 ( -1)! Ý Ý1( ÿ, â), where Ý Ý1( ÿ, â)isthefundamental solution oftheequation with =1. ¥Á Â Thelinear differential operator caninvolvearbitrarily manyspace variables. 6. ö2ö ø2± ú÷=0, =1,2, Here, isalinear differential operator ofanyorder with respect tothespace variable ÿwhose coef®cients candepend on ÿ. 1 í.General solution:î( ÿ, â)= õ-1 ê=0 â ê ê( ÿ, â), where the ê= ê( ÿ, â)arearbitrary functions that satisfy theoriginal equation with =1: (áØ Ø- ) ê=0. 2 í.Suppose thattheCauchy problem forthespecial case oftheequation with =1iswell posed ifonly oneinitial condition issetat â=0;thismeans that theconstant coef®cient differential operator issuch thattheequation with =1isregular with regularity index =1.Then the fundamental solution oftheoriginal equation canbefound bytheformulaÝ Ýõ( ÿ, â)= âeõ-1 ( -1)! Ý Ý1( ÿ, â), where Ý Ý1( ÿ, â)isthefundamental solution for =1. ¥Á Â Thelinear differential operator caninvolvearbitrarily manyspace variables. Page652 9.6. HIGHER -ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS 653 7. =0 [÷]=0. Here, isanylinear differential operator with arbitrarily manyindependent variables ÿ1, !"!"!, ÿõ. Particular solutions:î( ÿ1, !"!"!, ÿõ)= ! =1 " ( ÿ1, !"!"!, ÿõ), where the aresolutions oftheequations [ ]- # =0,the # areroots ofthecharacteristic equation $ê=0 ê # ê=0,andthe" arearbitrary constants. Page653 Supplement A Special Functions andTheir Proper ties Throughout Supplement Aitisassumed that isapositive integer,unless otherwise speci®ed. A.1. Some Symbols and Coef®cients A.1.1. Factorials De®nitions andsome properties: 0!=1!=1, !=1×2×3 !"!"!( -1) , =2,3, !"!"!, (2 )!!=2×4×6 !"!"!(2 -2)(2 )=2 õ!, (2 +1)!!=1×3×5 !"!"!(2 -1)(2 +1)=2 õ+1% & ' (+3 2 ),!!= *(2 +)!! if =2 +, (2 ++1)!! if =2 ++1,0!!=1. A.1.2. Binomial Coef®cients De®nition:" , -= !+!( - +)!, where +=1, !"!"!, ," , .=(-1),(- /), +!= /( /-1) !"!"!( /- ++1)+!, where +=1,2, !"!"! General case:" 0 .= '( /+1)'( 1+1) '( /- 1+1),where '( 2)isthegamma function. Properties:"0.=1," , -=0for +=-1,-2, !"!"!or +> 3," 0+1.= /1+1" 0 .-1= /- 11+1" 0 .," 0 .+" 0+1.=" 0+1.+1," - -1 42=(-1) - 22 - 5 - 2 -=(-1) -(2 3-1)!! (2 3)!!,5 - 1 42=(-1) --1322 --1 5 --1 2 --2=(-1) --13(2 3-3)!! (2 3-2)!!,52 -+1-+1 42=(-1) -2-4 --15 - 2 -,5 - 2 -+1 42=2-2 -52 - 4 -+1,51 42-=22 -+1&5 - 2 -,5 -42-=22 -&5( --1) 42- . Page655 656 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES A.1.3. Pochhammer Symbol De®nition andsome properties ( +=1,2, !"!"!): ( /) -= /( /+1) !"!"!( /+ 3-1)= '( /+ 3)'( /)=(-1) - '(1- /)'(1- /- 3), ( /)0=1,( /) -+,=( /) -( /+ 3),,( 3),=( 3+ +-1)! ( 3-1)!, ( /)- -= '( /- 3)'( /)=(-1) - (1- /) -,where /¹1, !"!"!, 3; (1) -= 3!,(1 62) -=2-2 -(2 3)!3!,(3 62) -=2-2 -(2 3+1)!3!, ( /+ 7 +) -,=( /) 8,+ -,( /) 8,,( /+ 3) -=( /)2 - ( /) -,( /+ 3),=( /),( /+ +) - ( /) -. A.1.4. Bernoulli Number s De®nition:29 :-1= ; <-=0 = - 2 -3!. Thenumbers:=0=1,=1=-1 2,=2=1 6,=4=-1 30,=6=1 42,=8=-1 30,=10=5 66, !"!"!,=2 8+1=0for 7=1,2, !"!"! A.2. ErrorFunctions and Exponential Integral A.2.1. ErrorFunction and Complementar yErrorFunction De®nitions: erf 2=2% & > : 0exp(- ?2) @ ?, erfc 2=1-erf 2=2% & >;:exp(- ?2) @ ?. Expansion oferf 2intoseries inpowers of 2as 2 A0: erf 2=2% &; <,=0(-1), 22,+1 ( +)!(2 ++1)=2% &exp B- 22 C; <,=02, 22,+1 2 ++1)!!. Asymptotic expansion oferfc 2as 2 A D: erfc 2=1% &exp B- 22 C E F-1<8=0(-1) 8 B1 2 C822 8+1+ G B| 2|-2F-1 CIH, J=1,2, !"!"! A.2.2. Exponential Integral De®nition: Ei( 2)= > : -; 9 K? @ ? for 2<0, Ei( 2)=limLM+0 N >- L -; 9 K? @ ?+ > :L 9 K? @ ?PO for 2>0. Page656 A.3. SINEINTEGRAL AND COSINE INTEGRAL .FRESNEL INTEGRALS 657 Other integralrepresentations: Ei(- 2)=- 9- :>; 0 2sin ?+ ?cos ?22+ ?2 @ ?for 2>0, Ei(- 2)= 9- :>; 0 2sin ?- ?cos ?22+ ?2 @ ? for 2<0, Ei(- 2)=- 2 >; 1 9- :Kln ? @ ? for 2>0. Expansion intoseries inpowers of 2as 2 A0: Ei( 2)= QRRRRSRRRRT U+ln(- 2)+; <,=1 2,+× +!if 2<0,U+ln 2+ ; <,=1 2,+× +!if 2>0, where U=0.5772 !"!"!istheEuler constant. Asymptotic expansion as 2 A D: Ei(- 2)= 9- : -<,=1(-1),( +-1)!2,+ V -, V -< 3!2 -. A.2.3. Logarithmic Integral De®nition: li( 2)= QRRSRRT > : 0 @ ? ln ?=Ei(ln 2) if0< 2<1, limLM+0N >1- L 0 @ ? ln ?+ > : 1+ L @ ? ln ? Oif 2>1. Forsmall 2, li( 2)» 2 ln(1 6 2). Asymptotic expansion as 2 A1: li( 2)= U+ln|ln 2|+ ; <,=1ln, 2+× +!. A.3. Sine Integral and Cosine Integral. Fresnel Integrals A.3.1. Sine Integral De®nition: Si( 2)= > : 0sin ?? @ ?, si( 2)=- >;:sin ?? @ ?=Si( 2)- W2. Speci®c values: Si(0)=0,Si( D)= W2,si( D)=0. Properties: Si(- 2)=-Si( 2),si( 2)+si(- 2)=-W, lim: M-;si( 2)=-W. Page657 658 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES Expansion intoseries inpowers of 2as 2 A0: Si( 2)= ; <,=1(-1),+122,-1 (2 +-1)(2 +-1)!. Asymptotic expansion as 2 A D: si( 2)=-cos 2 E F-1<8=0(-1) 8(2 7)!22 8+1+ G B| 2|-2F-1 CIH+sin 2 E X-1<8=1(-1) 8(2 7-1)!22 8 + G B| 2|-2X CIH, where J, Y=1,2, !"!"! A.3.2. Cosine Integral De®nition: Ci( 2)=- >;:cos ?? @ ?= U+ln 2+ > : 0cos ?-1? @ ?, U=0.5772 !"!"! Expansion intoseries inpowers of 2as 2 A0: Ci( 2)=5+ln 2+ ; <,=1(-1), 22, 2 +(2 +)!. Asymptotic expansion as 2 A D: Ci( 2)=cos 2 E F-1<8=1(-1) 8(2 7-1)!22 8 + G B| 2|-2F CH+sin 2 E X-1<8=0(-1) 8(2 7)!22 8+1+ G B| 2|-2X-1 CH, where J, Y=1,2, !"!"! A.3.3. Fresnel Integrals De®nitions: Z ( 2)=1[ 2W > : 0sin ?[? @ ?= \2W > ] ^ 0sin ?2@ ?,5( _)=1[ 2W > ^ 0cos ?[? @ ?= \2W > ] ^ 0cos ?2@ ?. Expansion intoseries inpowers of _as _ A0:Z ( _)= \2W _ ` acb =0(-1) b_2 b +1 (4 d+3)(2 d+1)!,5( _)= \2W _ ` acb =0(-1) b_2 b (4 d+1)(2 d)!. Asymptotic expansion as _ e f:Z ( _)=1 2-cos _[ 2W _ g( _)-sin _[ 2W _ h( _),5( _)=1 2+sin _[ 2W _ g( _)-cos _[ 2W _ h( _),g( _)=1-1×3 (2 _)2+1×3×5×7 (2 _)4- ijiji,h( _)=1 2 _-1×3×5 (2 _)3+ ijiji. Page658 A.4. GAMMA AND BETAFUNCTIONS 659 A.4. Gamma and Beta Functions A.4.1. Gamma Function A.4.1-1. De®nition. Integralrepresentations. Thegamma function, '( k),isananalytic function ofthecomple xargument keverywhere, except forthepoints k=0,-1,-2, !"!"! ForRe k>0,'( k)= l` 0 mon-1 p- q rm. For-( s+1)<Re k<- s,where s=0,1,2, !"!"!,t( k)= l` 0 u p- q- v aw=0(-1) wx! ymon-1 rm. A.4.1-2. Some formulas. Euler formulat( k)=limv z ` s! snk( k+1) !"!"!( k+ s)( k¹0,-1,-2, !"!"!). Simplest properties:t( k+1)= k t( k), t( s+1)= s!, t(1)= t(2)=1. Symmetry formulas:t( k) t(- k)=- {ksin({ k), t( k) t(1- k)= {sin({ k),t |1 2+ k } t |1 2- k }= {cos({ k). Multiple argument formulas:t(2 k)=22n-1[{ t( k) t |k+1 2 },t(3 k)=33n-1 ~2 2{ t( k) t |k+1 3 } t |k+2 3 },t( s k)=(2{)(1-v) ~2svn-1 ~2v-1 b =0 t |k+ ds }. Fractional values oftheargument:t |1 2 }= {,t |-1 2 }=-2{, t |s+1 2 }= {2v(2 s-1)!!,t |1 2- s }=(-1)v2v {(2 s-1)!!. Asymptotic expansion (Stirling formula):t( k)= 2{ p-n kn-1 ~2 1+1 12 k-1+1 288 k-2+ ( k-3) (|arg| k<{). Page659 660 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES A.4.1-3. Logarithmic derivativeofthegamma function. De®nition: ( k)= rln t( k)rk= t n( k)t( k). Functional relations: ( k)- (1+ k)=-1k, ( k)- (1- k)=-{cot({ k), ( k)- (- k)=-{cot({ k)-1k, 1 2+ k - 1 2- k ={tan({ k), ( xk)=ln x+1x w-1ab =0 |k+ dx }. Integralrepresentations (Re k>0): ( k)= l` 0 p- q-(1+m)-n m-1 rm, ( k)=ln k+ l` 0 m-1-(1- p- q)-1 p- qn rm, ( k)=- + l1 01-mn-1 1-m rm, where =- (1)=0.5772 !"!"!istheEuler constant. Values forintegerargument: (1)=- , ( s)=- +v-1ab =1 d-1( s=2,3, !"!"!) A.4.2. Beta Function De®nition: ( _, )= l1 0 m ^-1(1-m) -1 rm, where Re _>0andRe >0. Relationship with thegamma function: ( _, )= t( _) t( )t( _+ ). A.5. Incomplete Gamma and Beta Functions A.5.1. Incomplete Gamma Function De®nitions (integralrepresentations): ( , _)= l ^ 0 p- qm-1 rm, Re >0,t( , _)= l`^ p- qm -1rm= t( )- ( , _). Page660 A.6. BESSEL FUNCTIONS 661 Recurrent formulas:( +1, )= ( , )- - ,( +1, )= ( , )+ - . Asymptotic expansions as 0:( , )= =0(-1) + !( + ),( , )= ( )- =0(-1) + !( + ). Asymptotic expansions as :( , )= ( )- -1- -1 =0(1- ) (- ) + | |- ,( , )= -1- -1 =0(1- ) (- ) + | |- -3 2 <arg <3 2. Integralfunctions related tothegamma function: erf =1 1 2, 2 ,erfc =1 1 2, 2 ,Ei(- )=- (0, ). A.5.2. Incomplete Beta Function De®nition: ( , )= 1 0 -1(1-) -1 !, where Re >0andRe ">0. A.6. Bessel Functions A.6.1. De®nitions and Basic Form ulas A.6.1-1. TheBessel functions ofthe®rstandthesecond kinds. TheBessel function ofthe®rstkind, # $( ),andtheBessel function ofthesecond kind, % $( )(also called theNeumann function), aresolutions oftheBessel equation2" &'& + " &+( 2- (2) "=0 andarede®ned bytheformulas# $( )= *) =0(-1) ) ( +2) $+2 ),! ( (+ ,+1), % $( )= # $( )cos (- #- $( ) sin (. (1) Theformula for % $( )isvalidfor (¹0, -1, -2, ././.(thecases (¹0, -1, -2, ././.arediscussed in what follows). Thegeneral solution oftheBessel equation hastheform 0 $( )= 11 # $( )+ 12 % $( )andis called thecylinder function. Page661 662 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES A.6.1-2. Some formulas. 2 ( 0 $( )= [ 0 $-1( )+ 0 $+1( )],!! 0 $( )=1 2[ 0 $-1( )- 0 $+1( )]= - 2 ( 0 $( )- 0 $431( ) 5,!![ $0 $( )]= $0 $-1( ), !![ - $0 $( )]=- - $0 $+1( ),61 !! 7 [ $# $( )]= $- # $- ( ), 61 !! 7 [ - $# $( )]=(-1) - $- # $+ ( ),#- ( )=(-1) # ( ), %- ( )=(-1) % ( ), =0,1,2, ././. A.6.1-3. TheBessel functions for (= - -1 2,where =0,1,2, ././.:#1 82( )= 92 sin ,#3 82( )= 92 61sin -cos 7, #-1 82( )= 92 cos ,#-3 82( )= 92 6 -1cos -sin 7,# +1 82( )= 92 sin - 2 [ 82] *) =0(-1) ) ( +2 ,)! (2 ,)!( -2 ,)!(2 )2 ) +cos - 2 [( -1) 82] *) =0(-1) ) ( +2 ,+1)! (2 ,+1)!( -2 ,-1)!(2 )2 ) +1,#- -1 82( )= 92 cos + 2 [ 82] *) =0(-1) ) ( +2 ,)! (2 ,)!( -2 ,)!(2 )2 ) -sin + 2 [( -1) 82] *) =0(-1) ) ( +2 ,+1)! (2 ,+1)!( -2 ,-1)!(2 )2 ) +1,%1 82( )=- 92 cos ,% +1 82( )=(-1) +1#- -1 82( ), %-1 82( )= 92 sin ,%- -1 82( )=(-1) # +1 82( ). A.6.1-4. TheBessel functions for (= - ,where =0,1,2, ././. Let (= beanarbitrary integer.Therelations#- ( )=(-1) # ( ), %- ( )=(-1) % ( ) arevalid. The function # ( )isgivenbythe®rst formula in(1)with (= ,and % ( )canbe obtained from thesecond formula in(1)byproceeding tothelimit ( .Fornonne gative , % ( ) canberepresented intheform% ( )=2 # ( )ln 2-1 -1 :) =0( - ,-1)!,! 2 -2 ) -1 *) =0(-1) ) 2 +2 ) ; ( ,+1)+ ; ( + ,+1),!( + ,)!, where ; (1)=- <, ; ( )=- <+ -1= ) =1 ,-1, <=0.5772 ././.istheEuler constant, ; ( )=[ln ( )]&isthe logarithmic derivativeofthegamma function. Page662 A.6. BESSEL FUNCTIONS 663 A.6.1-5. Wronskians andsimilar formulas:>( # $, #- $)=-2 sin( (), >( # $, % $)=2 ,# $( ) #- $+1( )+ #- $( ) # $-1( )=2sin( () , # $( ) % $+1( )- # $+1( ) % $( )=-2 . Here, thenotation >( ?, @)= ? @&- ?& @isused. A.6.2. Integral Representations and Asymptotic Expansions A.6.2-1. Integralrepresentations. Thefunctions # $and % $canberepresented intheform ofde®nite integrals (for >0): # $( )= A 0cos( sin B- ( B) !B-sin ( 0exp(- sinh- () !, % $( )= A 0sin( sin B- ( B) !B- 0( $DC+ - $DCcos () - sinh C!. For| (|<1 2, >0,# $( )=21+ $- $1 82 (1 2- () 1sin( ) !(2-1) $+1 82,% $( )=-21+ $- $1 82 (1 2- () 1cos( ) !(2-1) $+1 82. For (>-1 2,# $( )=2( +2) $1 82 (1 2+ () A 82 0cos( cos)sin2 $ !(Poisson' sformula) . For (=0, >0,#0( )=2 0sin( cosh) !, %0( )=-2 0cos( cosh) !. Forinteger (= =0,1,2, ././.,# ( )=1 A 0cos( - sin) !(Bessel' sformula) ,#2 ( )=2 A 82 0cos( sin)cos(2 ) !,#2 +1( )=2 A 82 0sin( sin)sin[(2 +1)] !. A.6.2-2. Integrals with Bessel functions: 0 E # $( ) != E+ $+1 2 $( F+ (+1) G( (+1) H F+ (+1 2, F+ (+3 2, (+1;- I2 4 , Re( F+ ()>-1, whereH( J, K, L;I)isthehyper geometric series (seeSection 10.9 ofthissupplement), M 0 I E % $(I) !I=-cos( () G(- () 2 $( F+ (+1) I E+ $+1H F+ (+1 2, (+1, F+ (+3 2,-I2 4 -2 $G( ()F- (+1 I E- $+1H F- (+1 2,1- (, F- (+3 2,-I2 4 , Re F>|Re (|-1. Page663 664 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES A.6.2-3. Asymptotic expansions as|I| N O:# $(I)= 92I Pcos Q4I-2 ( R- R 4 S TVU-1WX=0(-1) X( Y,2 Z)(2I)-2 X+ [(|I|-2U) \ -sin Q4I-2 Y R- R 4 S T U-1WX=0(-1) X( Y,2 Z+1)(2I)-2 X-1+ [(|I|-2U-1) \ ],^ _(I)= `2R aPsin Q4 a-2 Y R- R 4 S TU-1WX=0(-1) X( Y,2 Z)(2 a)-2 X+ [(| a|-2U) \ +cos Q4 a-2 Y R- R 4 S T U-1WX=0(-1) X( Y,2 Z+1)(2 a)-2 X-1+ [(| a|-2U-1) \ ], where ( Y, Z)=1 22 XZ!(4 Y2-1)(4 Y2-32) b/b/b[4 Y2-(2 Z-1)2]= c(1 2+ Y+ Z)Z!c(1 2+ Y- Z). Fornonne gativeinteger dandlarge a,eR a f2 g( a)=(-1) g(cos a+sin a)+ [( a-2),eR a f2 g+1( a)=(-1) g+1(cos a-sin a)+ [( a-2). A.6.2-4. Asymptotic forlarge Y( Y N O):f _( a) h1e 2 R Y Q i a 2 Y S _ , ^ _( a) h- `2R Y Q i a 2 Y S- _ , where ais®xed,f _( Y) h21 j3 32 j3c(2 k3)1Y1 j3, ^ _( Y) h-21 j3 31 j6c(2 k3)1Y1 j3. A.6.3. Zerosand Orthogonality Proper tiesofBessel Functions A.6.3-1. Zeros ofBessel functions. Each ofthefunctions f _( a)and ^ _( a)hasin®nitely manyrealzeros (forreal Y).Allzeros are simple, except possibly forthepoint a=0. Thezeros l Xof f0( a),i.e.,theroots oftheequation f0( l X)=0,areapproximately givenbyl X=2.4+3.13( Z-1) ( Z=1,2, b/b/b), with maximum error 0.2%. A.6.3-2. Orthogonality properties ofBessel functions. 1 m.Let n= n Xbepositi veroots oftheBessel function f _( n),where Y>-1and Z=1,2,3, b/b/b Then thesetoffunctions f _( n X ok p)isorthogonal ontheinterv al0£ o£ pwith weight o:q r 0 f _ sn X op t f _ sn u op t o v o= w0 if Z¹ x, 1 2 p2 yf z _( n X) {2=1 2 p2f2 _ +1( n X)if Z= x. Page664 A.6. BESSEL FUNCTIONS 665 2 m.Let n= n Xbepositi vezeros oftheBessel function derivative f z _( n),where Y>-1andZ=1,2,3, b/b/bThen thesetoffunctions f _( n X ok p)isorthogonal ontheinterv al0£ o£ pwith weight o:q r 0 f _ sn X op t f _ sn u op t o v o= | }'~0 if Z¹ x, 1 2 p2 s1- Y2n2Xt f2 _( n X)if Z= x. 3 m.Let n= n Xbepositi veroots ofthetranscendental equation n f z _( n)+ f _( n)=0,where Y>-1 and Z=1,2,3, b/b/bThen thesetoffunctions f _( n X ok p)isorthogonal ontheinterv al0£ o£ p with weight o:q r 0 f _ sn X op t f _ sn u op t o v o= | } ~0 if Z¹ x, 1 2 p2 s1+ 2- Y2n2Xt f2 _( n X)if Z= x. 4 m.Let n= n Xbepositi veroots ofthetranscendental equationf _( X ) ^ _( Xp)- f _( Xp) ^ _( X )=0 ( Y>-1, Z=1,2,3, b/b/b). Then thesetoffunctions _( X o)= f _( X o) ^ _( Xp)- f _( Xp) ^ _( X o), Z=1,2,3, b/b/b, satisfying theconditions _( Xp)= _( X )=0isorthogonal ontheinterv al p£ o£ with weight o:q r _( X o) _( u o) o v o= | }'~0 if Z¹ x, 222X f2 _( Xp)- f2 _( X )f2 _( X )if Z= x. 5 m.Let n= n Xbepositi veroots ofthetranscendental equationf z _( X ) ^z _( Xp)- f z _( Xp) ^z _( X )=0 ( Y>-1, Z=1,2,3, b/b/b) Then thesetoffunctions _( X o)= f _( X o) ^z _( Xp)- f z _( Xp) ^ _( X o), Z=1,2,3, b/b/b, satisfying theconditions z _( Xp)= z _( X )=0isorthogonal ontheinterv al p£ o£ with weight o:qr _( X o) _( u o) o v o= | }~0 if Z¹ x, 222X s1- Y222 t yf z _( p) {2yfz _( ) {2- s1- Y2p22 t if = x. A.6.4. Hankel Functions (Bessel Functions oftheThirdKind) TheHank elfunctions ofthe®rstkind andthesecond kind arerelated toBessel functions by(1) _( )= f _( )+ ^ _( ), (2) _( )= f _( )- ^ _( ), 2=-1. Asymptotics for N0:(1) 0( ) h2 ln , (1) _( ) h- c( Y) ( k2) _(Re Y>0),(2) 0( ) h-2 ln , (2) _( ) h c( Y) ( k2) _(Re Y>0). Asymptotics for| | N O:(1) _( ) h `2exp -1 2 Y-1 4 D(- <arg <2 ),(2) _( ) h `2exp- -1 2 Y-1 4 D(-2 <arg < ). Page665 666 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES A.7. Modi®ed Bessel Functions A.7.1. De®nitions. Basic Form ulas A.7.1-1. Themodi®ed Bessel functions ofthe®rst andthesecond kinds. Themodi®ed Bessel functions ofthe®rstkind, _( a),andthesecond kind, _( a)(also called the Macdonald function), oforder Yaresolutions ofthemodi®ed Bessel equationa2 z'zM M+ a zM-( a2+ Y2) =0 andarede®ned bytheformulas _( a)= u=0( a k2)2 u+ x!c( Y+ x+1), ( a)= 2 - - sin Y, (seebelowfor ( a)with Y=0,1,2, b/b/b). A.7.1-2. Some formulas. Themodi®ed Bessel functions possess theproperties- ( a)= ( a); - g( a)=(-1) gg( a), d=0,1,2, b/b/b 2 Y ( a)= a[ -1( a)- +1( a)], 2 Y ( a)=- a[ -1( a)- +1( a)],vva ( a)=1 2[ -1( a)+ +1( a)], vva ( a)=-1 2[ -1( a)+ +1( a)]. A.7.1-3. Modi®ed Bessel functions for Y= d 1 2,where d=0,1,2, b/b/b:1 j2( a)= `2asinh a, -1 j2( a)= `2acosh a,3 j2( a)= `2a -1asinh a+cosh a , -3 2( )= 2 -1cosh +sinh ,+1 2( )=1 2 ¢¡ £ ¤*¥ =0(-1) ¥ ( ¦+ §)!§!( ¦- §)!(2 ) ¥ -(-1) ¡-£ ¤:¥ =0( ¦+ §)!§!( ¦- §)!(2 ) ¥ ¨ ,- -1 2( )=1 2 ¢¡ £ ¤*¥ =0(-1) ¥ ( ¦+ §)!§!( ¦- §)!(2 ) ¥ +(-1) ¡-£ ¤:¥ =0( ¦+ §)!§!( ¦- §)!(2 ) ¥ ¨ ,© ª 1 2( )= 2 ¡-£, © ª 3 2( )= 2 «1+1 ¬¡-£,©+1 2( )= © - -1 2( )= 2 ¡-£ ¤*¥ =0( ¦+ §)!§!( ¦- §)!(2 ) ¥ . A.7.1-4. Modi®ed Bessel functions for Y= ¦,where ¦=0,1,2, // If Y= ¦isanonne gativeinteger,then©( )=(-1) +1 ( )ln 2+1 2 -1¤®=0(-1) ®« 2 ¬2 ®- ( ¦- ¯-1)!¯! +1 2(-1) ° ¤®=0 « ±2 ¬ +2 ® ²( ¦+ ¯+1)+ ²( ¯+1)¯!( ¦+ ¯)!; ¦=0,1,2, //, where ²( ³)isthelogarithmic derivativeofthegamma function; for ¦=0,the®rstsum isdropped. Page666 A.7. MODIFIED BESSEL FUNCTIONS 667 A.7.1-5. Wronskians andsimilar formulas:´( /µ, - µ)=-2±sin( Y), ´( ¶µ, ©µ)=-1±,/µ(±) - µ+1(±)- - µ(±) ·µ-1(±)=-2sin( Y)±, /µ(±) ©µ+1(±)+ ·µ+1(±) ©µ(±)=1±, where ´( ¸, ¹)= ¸ ¹ º£- ¸ º£ ¹. A.7.2. Integral Representations and Asymptotic Expansions A.7.2-1. Integralrepresentations. Thefunctions µ(±)and ©µ(±)canberepresented interms ofde®nite integrals:/µ(±)=± µ1 »22 µ ¼( Y+1 2) ½1 -1exp(-± ¾)(1-¾2) µ-1 »2 ¿¾(±>0, Y>-1 2),©µ(±)=½ ° 0exp(-±cosh¾)cosh( Y¾) ¿¾(±>0),©µ(±)=1 cos À1 2 Y Á ½ ° 0cos(±sinh¾)cosh( Y¾) ¿¾(±>0,-1< Y<1),©µ(±)=1 sin À1 2 Y Á ½ ° 0sin(±sinh¾)sinh( Y¾) ¿¾(±>0,-1< Y<1). Forinteger Y= ¦, (±)=1½ Â0exp(±cos¾)cos( ¦¾) ¿¾( ¦=0,1,2, //),© 0(±)=½ ° 0cos(±sinh¾) ¿¾=½ ° 0cos(± ¾)¾2+1 ¿¾(±>0). A.7.2-2. Integrals with modi®ed Bessel functions:½ £ 0± Ã /µ(±) ¿±=±Ã+ µ+1 2 µ( Ä+ Y+1) ¼( Y+1) Å « Ä+ Y+1 2, Ä+ Y+3 2, Y+1;±2 4 ¬, Re( Ä+ Y)>-1, whereÅ( Æ, Ç, È;±)isthehyper geometric series (seeSection 10.9 ofthissupplement),½ £ 0± Ã ©µ(±) ¿±=2 µ-1 ¼( Y)Ä- Y+1± Ã- µ+1Å « Ä- Y+1 2,1- Y, Ä- Y+3 2,±2 4 ¬ +2- µ-1 ¼(- Y)Ä+ Y+1± Ã+ µ+1Å « Ä+ Y+1 2,1+ Y, Ä+ Y+3 2,±2 4 ¬, Re Ä>|Re Y|-1. A.7.2-3. Asymptotic expansions as± É Ê:/µ(±)=¡£ 2 Ë± Ì 1+ Í ¤®=1(-1) ®(4 Î2-1)(4 Î2-32) //[4 Î2-(2 ¯-1)2]¯!(8±) ® Ï,©µ(±)= Ë 2± ¡-£ Ì 1+Í ¤®=1(4 Î2-1)(4 Î2-32) //[4 Î2-(2 ¯-1)2]¯!(8±) ® Ï. Theterms oftheorder of Ð(±-Í-1)areomitted inthebraces. Page667 668 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES A.8. AiryFunctions A.8.1. De®nition and Basic Form ulas A.8.1-1. TheAiry functions ofthe®rst andthesecond kinds. The Airy function ofthe®rst kind, Ai(±),andtheAiry function ofthesecond kind, Bi(±),are solutions oftheAiry equation Ñº'º£ £-± Ñ =0 andarede®ned bytheformulas Ai(±)=1Ë½ ° 0cos À1 3¾3+± ¾ Á ¿¾, Bi(±)=1Ë½ ° 0 Òexp À-1 3¾3+± ¾ Á+sin À1 3¾3+± ¾ ÁDÓ ¿¾. Wronskian: ´ ÔAi(±),Bi(±) Õ=1 Ö Ë. A.8.1-2. Connection with theBessel functions andthemodi®ed Bessel functions: Ai(±)=1 3 ± ÒØ×-1 »3( ³)-×1 »3( ³) Ó= Ë-1 Ù1 3± © 1 »3( ³), ³=2 3±3 »2, Ai(-±)=1 3 ± ÒÛÚ-1 »3( ³)+Ú1 »3( ³) Ó, Bi(±)= Ù1 3± Ò×-1 »3( ³)+×1 »3( ³) Ó, Bi(-±)= Ù1 3± ÒÛÚ-1 »3( ³)-Ú1 »3( ³) Ó. A.8.2. PowerSeries and Asymptotic Expansions A.8.2-1. Powerseries expansions as± É0: Ai(±)= È1 ¸(±)- È2 ¹(±), Bi(±)= 3Ò È1 ¸(±)+ È2 ¹(±) Ó,¸(±)=1+1 3!±3+1×4 6!±6+1×4×7 9!±9+ //= ° ¤*¥ =03 ¥À1 3 Á ¥±3 ¥ (3 §)!,¹(±)=±+2 4!±4+2×5 7!±7+2×5×8 10!±10+ //= ° ¤*¥ =03 ¥À2 3 Á ¥±3 ¥ +1 (3 §+1)!, where È1=3-2 »3Ö ¼(2 Ö3)»0.3550 and È2=3-1 »3Ö ¼(1 Ö3)»0.2588 . A.8.2-2. Asymptotic expansions as± É Ê. Forlargevalues of±,theleading terms ofasymptotic expansions oftheAiry functions are Ai(±) Ü1 2 Ë-1 »2±-1 »4exp(- ³), ³=2 3±3 »2, Ai(-±) Ü Ë-1 »2±-1 »4sin À³+Â4 Á, Bi(±) Ü Ë-1 »2±-1 »4exp( ³), Bi(-±) Ü Ë-1 »2±-1 »4cos À³+Â4 Á.ÝßÞ Refer ence:M.Abramo witz andI.Stegun(1964). Page668 A.9. D EGENERATE HYPERGEOMETRIC FUNCTIONS 669 TABLE A1 Special cases of the Kummer function à( Æ, Ç; ³)ÆÇ ³à Conventional notationÆ Æ± ¡ £ 1 2 2±1± ¡£sinh±ÆÆ+1 -± Æ±- á/â( Æ,±)Incomplete gamma functionâ( Æ,±)=½ £ 0 ¡- ã¾ á-1 ¿¾ 1 23 2-±2 Ë 2erf±Error function erf±=2Ë½ £ 0exp(-¾2) ¿¾ - ¦1 2±2 2 ¦! (2 ¦)! «-1 2 ¬- ä 2 å(±) Hermite polynomialsäå=(-1) å¡ £2 ¿å¿± å À¡-£2Á,¦= 0,1,2, //- ¦3 2±2 2 ¦! (2 ¦+1)! «-1 2 ¬- å ä 2 å+1(±) - ¦Ç± ¦! ( Ç)å æ( ç-1)å(±)Laguerre polynomialsæ( è)å(±)=¡£±- è¦! ¿å¿± å À¡-£± å+ èÁ,é= Ç-1, ( Ç)å= Ç( Ç+1) //( Ç+ ¦-1)Î+1 22 Î+1 2± ¼(1+ Î)¡ £ « ±2 ¬- µ× µ(±)Modi®ed Bessel functions× µ(±)¦+1 2 ¦+2 2± ¼« ¦+3 2 ¬¡ £ « ±2 ¬- å-1 2× å+1 2(±) A.9. Degenerate Hypergeometric Functions A.9.1. De®nitions and Basic Formulas A.9.1-1. The degenerate hypergeometric functions à( Æ, Ç;±) and ê( Æ, Ç;±). The degenerate hypergeometric functions à( Æ, Ç;±) and ê( Æ, Ç;±) are solutions of the degenerate hypergeometric equation± Ñº'º ë ë+( Ç-±) Ñº ë- Æ Ñ = 0. In the case Ç¹ 0,-1,-2,-3, //, the function à( Æ, Ç;±) can be represented as Kummer's series:à( Æ, Ç;±)= 1 + ° ì*í =1( Æ) í ( Ç) í î íï!, where ( Æ) í = Æ( Æ+ 1) ð/ð/ð( Æ+ ï- 1), ( Æ)0= 1. TableA1present ssomespecia lcaseswher e àcanbeexpresse dintermsofsimple rfunctions. The function ê( Æ, Ç; î) is de®ned as follows:ê( Æ, Ç; î)= ¼(1 - Ç)¼( Æ- Ç+ 1) à( Æ, Ç; î)+ ¼( Ç- 1)¼( Æ) î1- çà( Æ- Ç+ 1,2 - Ç; î). Page 669 670 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES A.9.1-2. Kummer transformation andlinear relations. Kummer transformation:à( Æ, Ç; î)= ñ ëà( Ç- ò, Ç;- î), ê( ò, Ç; î)= î1- çê(1+ ò- Ç,2- Ç; î). Linear relations for à: ( Ç- ò) à( ò-1, Ç; î)+(2 ò- Ç+ î) à( ò, Ç; î)- ò à( ò+1, Ç; î)=0,Ç( Ç-1) à( ò, Ç-1; î)- Ç( Ç-1+ î) à( ò, Ç; î)+( Ç- ò) îà( ò, Ç+1; î)=0, ( ò- Ç+1) à( ò, Ç; î)- ò à( ò+1, Ç; î)+( Ç-1) à( ò, Ç-1; î)=0,Ç/à( ò, Ç; î)- Ç/à( ò-1, Ç; î)- îà( ò, Ç+1; î)=0,Ç( ò+ î) à( ò, Ç; î)-( Ç- ò) îà( ò, Ç+1; î)- ò Ç/à( ò+1, Ç; î)=0, ( ò-1+ î) à( ò, Ç; î)+( Ç- ò) à( ò-1, Ç; î)-( Ç-1) à( ò, Ç-1; î)=0. Linear relations for ê:ê( ò-1, Ç; î)-(2 ò- Ç+ î) ê( ò, Ç; î)+ ò( ò- Ç+1) ê( ò+1, Ç; î)=0, ( Ç- ò-1) ê( ò, Ç-1; î)-( Ç-1+ î) ê( ò, Ç; î)+ îê( ò, Ç+1; î)=0,ê( ò, Ç; î)- ò ê( ò+1, Ç; î)- ê( ò, Ç-1; î)=0, ( Ç- ò) ê( ò, Ç; î)- îê( ò, Ç+1; î)+ ê( ò-1, Ç; î)=0, ( ò+ î) ê( ò, Ç; î)+ ò( Ç- ò-1) ê( ò+1, Ç; î)- îê( ò, Ç+1; î)=0, ( ò-1+ î) ê( ò, Ç; î)- ê( ò-1, Ç; î)+( ò- È+1) ê( ò, Ç-1; î)=0. A.9.1-3. Differentiation formulas andWronskian. Differentiation formulas:óóî à( ò, ô; î)= òô à( ò+1, ô+1; î),óóî ê( ò, ô; î)=- ò ê( ò+1, ô+1; î), óåóîå à( ò, ô; î)=( ò)å ( ô)å à( ò+ õ, ô+ õ; î),óåóîå ê( ò, ô; î)=(-1) å( ò)å ê( ò+ õ, ô+ õ; î). Wronskian: ö ( à, ê)= à ê ºë- à ºëê=- ¼( ô)¼( ò) î- çñ ë. A.9.1-4. Degenerate hyper geometric functions for õ=0,1,2, ð/ð/ð:ê( ò, õ+1; î)=(-1) å-1õ! ¼( ò- õ) ÷ à( ò, õ+1; î)ln î + ø ì:ù =0( ò) ù ( õ+1) ù úØû ( ò+ o)- û (1+ o)- û (1+ õ+ o) ü î ùo! ý+( õ-1)!¼( ò) å-1ìþù =0( ò- õ) ù (1- õ) ùî ù - åo!, where õ=0,1,2, ð/ð/ð(the lastsum isdropped for õ=0), û ( ÿ)=[ln ¼( ÿ)] ºisthelogarithmic derivativeofthegamma function,û (1)=- , û ( õ)=- + å-1ì*í =1 ï-1, where =0.5772 ð/ð/ðistheEuler constant. Page670 A.9. DEGENERA TEHYPERGEOMETRIC FUNCTIONS 671 If ô<0,then theformula( ò, ô; î)= î1- ( ò- ô+1,2- ô; î) isvalidforany î. For ô¹0,-1,-2,-3, ð/ð/ð,thegeneral solution ofthedegenerate hyper geometric equation can berepresented intheform= 1 ( ò, ô; î)+ 2 ( ò, ô; î), andfor ô=0,-1,-2,-3, ð/ð/ð,intheform= î1- ú1 ( ò- ô+1,2- ô; î)+ 2 ( ò- ô+1,2- ô; î) ü. A.9.2. Integral Representations and Asymptotic Expansions A.9.2-1. Integralrepresentations:( ò, ô; î)= ¼( ô)¼( ò) ¼( ô- ò) 1 0 ñ -1(1- ) - -1 ó (for ô> ò>0),( ò, ô; î)=1¼( ò) ø 0 ñ- -1(1+ ) - -1 ó (for ò>0, î>0), where ¼( ò)isthegamma function. A.9.2-2. Integrals with degenerate hyper geometric functions:( ò, ô; î) óî= ô-1ò-1 ( ò-1, ô-1; î)+ ,( ò, ô; î) óî=1 1- ò ( ò-1, ô-1; î)+ ,îå( ò, ô; î) óî= õ! å+1ì*í =1(-1) í +1(1- ô) íîå- í +1 (1- ò) í ( õ- ï+1)! ( ò- ï, ô- ï; î)+ ,îå ( ò, ô; î) óî= õ! å+1ìí =1(-1) í +1 îå- í +1 (1- ò) í ( õ- ï+1)! ( ò- ï, ô- ï; î)+ . A.9.2-3. Asymptotic expansion as| î| :( ò, ô; î)= ¼( ô)¼( ò) ñ î - ìå=0( ô- ò)å(1- ò)åõ! î- å+ , î>0,( ò, ô; î)= ( ô)( ô- ò)(- î)- å=0( )å( - ô+1)å!(- )- å+ , <0,( , ô; )= - å=0(-1) å( )å( - ô+1)å! - å+ ,- < < , where = ( --1). Page671 672 A. S PECIAL FUNCTIONS AND THEIR PROPERTIES A.10. Hypergeometric Functions A.10.1. De®nition and Some Formulas The hypergeometric function ( , , ; ) is a solution of the Gaussian hypergeometric equation( - 1) ! +[( + + 1) - ] + = 0. For ¹ 0,-1,-2,-3, " " ", the function ( , , ; ) can be expressed in terms of the hyperge- ometric series:( , , ; )= 1 + ø $# =1( ) # ( ) # ( ) # #%!, ( ) # = ( + 1) " " "( + %- 1), which certainly converges for | |<1. TableA2showssomespecia lcaseswher e canbeexpresse dintermofelementar yfunctions. A.10.2. Basic Properties and Integral Representations A.10.2-1. Some properties. The function possesses the following properties:( , , ; )= ( , , ; ),( , , ; )=(1 - ) &- '- (( - , - , ; ),( , , ; )=(1 - )- ' ) , - , ; - 1 *,+å+ å ( , , ; )=( )å( )å ( )å ( + , + , + ; ). If is not an integer, then the general solution of the hypergeometric equation can be written in the form= 1 ( , , ; )+ 2 1-&( - + 1, - + 1,2 - ; ). A.10.2-2. Integral representations. For > >0, the hypergeometric function can be expressed in terms of a de®nite integral:( , , ; )= ( )( )( - ) 1 0 (-1(1 - ) &- (-1(1 - ,)- ' +, where( ) is the gamma function. See M. Abramowitz and I. Stegun (1964) and H. Bateman and A. Erd Âelyi (1953, V ol. 1) for more detailed information about hypergeometric functions. A.11. Whittaker Functions The Whittaker functions - # , .( ) and / # , .( ) are linearly independent solutions of the Whittaker equation: ! + 0-1 4+1 2 %+ 11 4- 22 3-2 4 = 0. The Whittaker functions are expressed in terms of degenerate hypergeometric functions as- # , .( )= .+1 52 6- 52 11 2+ 2- %,1 + 2 2, 3,/ # , .( )= .+1 52 6- 52 11 2+ 2- %,1 + 2 2, 3. Page 672 A.10. HYPERGEOMETRIC FUNCTIONS 673 TABLE A2 Some special cases where thehyper geometric function ( , , ; ÿ) canbeexpressed interms ofelementary functions ÿ - 7 # =0(- ) # ( ) # ( ) # #%!,where =1,2, " " " - - - 8 7 # =0(- ) # ( ) # (- - 8) # #%!,where =1,2, " " " (1- )- '+1 21 2 2 1 2 0(1+ )-2 '+(1- )-2 ' 4+1 23 2 2(1+ )1-2 '-(1- )1-2 ' 2 (1-2 ) - 1 2 - 2 1 2 091,:1+ 2+ 32 '+ 1:1+ 2- 32 '4 1- 1 2 - 2 1:1+ 2+ 32 '-1+ 1:1+ 2- 32 '-1 2:1+ 2-1 2 2 -1 22 '-211+:1- 32-2 ' 1- 3 2 sin2sin[(2 -1) ] ( -1)sin(2 ) 2- 3 2 sin2sin[(2 -2) ] ( -1)sin(2 ) 1- 1 2 sin2cos[( 2 -1) ] cos +11 2 (1+ )(1- )- '-1+1 2 2 +1 ;1+:1- 2 <-2 '+1 2 2 1:1- ;1+:1- 2 <1-2 ' 1 21 23 2 21arcsin 1 2 13 2 - 21arctan 1 1 2 - 1ln( +1) 1 2 13 2 21 2 ln1+ 1- +1 + 8+1 + 8+ =+2 (-1) >( + 8+ =+1)!! =!( + 8)!( 8+ =)! +7+>+7+> ?(1- ) >+ @ +@+ @ A,=-ln(1- ), , 8, ==0,1,2, " " " Page673 674 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES A.12. Legendre Polynomials and Legendre Functions A.12.1. De®nitions. Basic Form ulas The Legendre polynomials B7= B7( )andtheLegendre functions C7( )aresolutions ofthe equation (1- 2) D ! E E-2 D E+ ( +1) D=0. TheLegendre polynomials B7( )andtheLegendre functions C7( )arede®ned bytheformulasB7( )=1!27 +7 +7( 2-1)7, C7( )=1 2 B7( )ln1+ 1- -7 >=118 B>-1( ) B7->( ). Thepolynomials B7= B7( )canbecalculated recursi velyusing therelationsB0( )=1, B1( )= , B2( )=1 2(3 2-1), " " ", B7+1( )=2 +1+1 B7( )- +1 B7-1( ). The®rstthree functions C7= C7( )havetheformC0( )=1 2ln1+ 1- , C1( )= 2ln1+ 1- -1, C2( )=3 2-1 4ln1+ 1- -3 2 . Thepolynomials B7( )havetheimplicit representationB7( )=2-7[7 52]>=0(-1) > F >7 F727-2> 7-2>, where [ G]istheintegerpartofanumber G. A.12.2. ZerosofLegendre Polynomials and theGenerating Function Allzeros of B7( )arerealandlieontheinterv al-1< <+1;thefunctions B7( )form anorthogonal system ontheinterv al-1£ £+1,withH+1 -1 B I( J) B>( J) K J= L0 if M¹ N, 2 2 M+1if M= N. Thegenerating function is 1O 1-2 PQJ+ P2= R SI=0 T I( J) P I(| P|<1). A.12.3. Associated Legendre Functions Theassociated Legendre functionsT >I( J)oforder Narede®ned bytheformulasT >I( J)=(1- J2) > U2 K >K J>T I( J), M=1,2,3, " " ", N=0,1,2, " " " Itisassumed byde®nition thatT0I( J)=T I( J). ThefunctionsT >I( J)form anorthogonal system ontheinterv al-1£ J£+1,withH+1 -1T >I( J)T > # ( J) K J= V W!X0 if M¹ %, 2 2 M+1( M+ N)! ( M- N)!if M= %. The functionsT >I( J)(with N¹0)areorthogonal ontheinterv al-1£ J£+1with weight (1- J2)-1,thatis,H+1 -1 T >I( J)T > # ( J) (1- J2) K J= V W!X0 if M¹ %, ( M+ N)!N( M- N)!if M= %. Page674 A.14. M ATHIEU FUNCTIONS 675 A.13. Parabolic Cylinder Functions A.13.1. De®nitions. Basic Formulas The Weber parabolic cylinder function Y Z( [) is a solution of the linear differential equation:\ ! ]^]+ _-1 4 [2+ `+1 2 a \= 0, where the parameter `and the variable [can assume arbitrary real or complex values. Another linearly independent solution of this equation is the function Y- Z-1( b,[); if `is noninteger, thenY Z(- [) can also be taken as a linearly independent solution. The parabolic cylinder functions can be expressed in terms of degenerate hypergeometric func- tions asY Z( [)= 21U2exp_-1 4 [2a c d _1 2 ad _1 2- Z 2 a e _- Z 2,1 2,1 2 [2a+ 2-1U2d _-1 2 ad _- Z 2 a [e _1 2- Z 2,3 2,1 2 [2aQf. For nonnegative integer `= M, we haveY I( [)= 2- IU2exp_-1 4 [2a g I_2-1U2[a, M= 0,1,2, h h h;g I( [)=(-1) Iexp_ [2a K IK [ Iexp_- [2a, whereg I( [) is the Hermitean polynomial of order M. A.13.2. Integral Representations and Asymptotic Expansions Integral representations:Y Z( [)= i2 j kexp_1 4 [2a HR 0 l Zexp_-1 2l2acos_ [l-1 2 k `a Klfor Re `>-1,Y Z( [)=1d(- `)exp _-1 4 [2a HR 0l- Z-1exp _- [l-1 2l2a Klfor Re `<0. Asymptotic expansion as | [| m n:Y Z( [)= [ Zexp _-1 4 [2ac o SI=0(-2) I_- Z 2 a I _1 2- Z 2 a IM!1[2 I+ p _| [|-2o-2affor |arg [|<3 k 4, where ( q)0= 1, ( q) I= q( q+ 1) h h h( q+ M- 1) for M= 1,2,3, h h h A.14. Mathieu Functions A.14.1. De®nitions and Basic Formulas A.14.1-1. Mathieu equation and Mathieu functions. The Mathieu functions ce I( J, r) and se I( J, r) are periodical solutions of the Mathieu equation\ s!st t+( q- 2 rcos2 J) \= 0. Such solutions exist for de®nite values of parameters qand r(those values of qare referred to as eigenvalues) .TheMathie ufunction sarelistedinTableA3. Page 675 676 A. S PECIAL FUNCTIONS AND THEIR PROPERTIES TABLE A3 The Mathieu functions ce I=ce I( J, r) and se I=se I( J, r) (for odd M, functions ce I and se Iare2 k-periodical, and for even M, they are k-periodical); de®nite eigenvalues q= q I( r) and q= ôuI( r) correspond to each value of parameter r. Mathieu functionsRecurrence relations for coef®cientsNormalization conditions ce2 I= R S>=0 v2 I 2>cos2 N J rv2 I 2= q2 Iv2 I 0;rv2 I 4=( q2 I-4)v2 I 2-2 rv2 I 0;rv2 I 2>+2=( q2 I-4 N2)v2 I 2>- rv2 I 2>-2, N³ 2(v2 I 0)2+ R S>=0(v2 I 2>)2 = w2if M= 0 1if M³ 1 ce2 I+1= R S>=0 v2 I+1 2>+1cos(2 N+1) J rv2 I+1 3=( q2 I+1-1- r)v2 I+1 1;rv2 I+1 2>+3=[ q2 I+1-(2 N+1)2]v2 I+1 2>+1 - rv2 I+1 2>-1, N³ 1 R S>=0(v2 I+1 2>+1)2= 1 se2 I=R S>=0 x2 I 2>sin2 N J, se0= 0 rx2 I 4=( ô2 I-4)x2 I 2;rx2 I 2>+2=( ô2 I-4 N2)x2 I 2>- rx2 I 2>-2, N³ 2 R S>=0(x2 I 2>)2= 1 se2 I+1= R S>=0x2 I+1 2>+1sin(2 N+1) J rx2 I+1 3=( ô2 I+1-1- r)x2 I+1 1;rx2 I+1 2>+3=[ ô2 I+1-(2 N+1)2]x2 I+1 2>+1 - rx2 I+1 2>-1, N³ 1 R S>=0(x2 I+1 2>+1)2= 1 A.14.1-2. Properties of the Mathieu functions. The Mathieu functions possess the following properties: ce2 I( J,- r)=(-1) Ice2 I y k 2- z, r {, ce 2 |+1( z,- r)=(-1) |se2 |+1 y k 2- z, r {, se2 |( z,- r)=(-1) |-1se2 | y k 2- z, r {, se 2 |+1( z,- r)=(-1) |ce2 |+1 y k 2- z, r {. Selecting suf®ciently large number Nand omitting the term with the maximum number in the recurrenc erelation s(indicate dinTableA3),wecanobtai napproximat erelation sforeigenvalues q| (or ô|) with respect to parameter r. Then, equating the determinant of the corresponding homo- geneous linear system of equations for coef®cientsv | > (orx |>) to zero, we obtain an algebraic equation for ®nding q|( r) (or ô|( r)). For ®xed real r¹ 0, eigenvalues q|and ô|are all real and different, while if r>0then q0< ô1< q1< ô2< q2< } } } if r<0then q0< q1< ô1< ô2< q2< q3< ô3< ô4< } } } The eigenvalues possess the propertiesq2 |(- r)= q2 |( r), ô2 |(- r)= ô2 |( r), q2 |+1(- r)= ô2 |+1( r). Tables of the eigenvalues q|= q|( r) and ô|= ô|( r) can be found in Abramowitz and Stegun (1964, Chapter 20). The solution of the Mathieu equation corresponding to eigenvalue q|(or ô|) has Mzeros on the interval 0 £ z< k( ris a real number). Page 676 A.16. ORTHOGON ALPOLYNOMIALS 677 A.14.1-3. Asymptotic expansions at r m0and r m n. Listed belowaretwoleading terms ofasymptotic expansions oftheMathieu functions ce|( z, r)and se|( z, r),aswell asofthecorresponding eigen values q|( r)and ô|( r),as r m0: ce0( z, r)=1O 2 y1- r 2cos2 z {, q0( r)=- r2 2+7 r4 128; ce1( z, r)=cos z- r 8cos3 z, q1( r)=1+ r; ce2( z, r)=cos2 z+ r 4 y1-cos4 z 3 {, q2( r)=4+5 r2 12; ce|( z, r)=cos M z+ r 4 ccos( M+2) zM+1-cos( M-2) zM-1 f, q|( r)= M2+ r2 2( M2-1)( M³3); se1( z, r)=sin z- r 8sin3 z, ô1( r)=1- r; se2( z, r)=sin2 z- rsin4 z 12, ô2( r)=4- r2 12; se|( z, r)=sin M z- r 4 csin( M+2) zM+1-sin( M-2) zM-1 f, ô|( r)= M2+ r2 2( M2-1)( M³3). Asymptotic results as r m n(- k j2< z< k j2):q|( r)»-2 r+2(2 M+1) Or+1 4(2 M2+2 M+1),ô|+1( r)»-2 r+2(2 M+1) Or+1 4(2 M2+2 M+1), ce|( z, r)» ~| r-1U4cos- |-1z cos2 |+1 exp(2 Orsin z)+sin2 |+1 exp(-2 Orsin z) , =1 2 z+ 4, se|+1( z, r)» |+1 r-1U4cos- |-1z cos2 |+1 exp(2 rsin z)-sin2 |+1 exp(-2 rsin z) , where the ~|and |some constants independent oftheparameter r. Refer ences :H.Bateman andA.ErdÂelyi(1955, Vol.3),M.Abramo witz andI.Stegun(1964). A.15. Modi®ed Mathieu Functions The modi®ed Mathieu functions Ce|( z, r)andSe|( z, r)aresolutions ofthemodi®ed Mathieu equation\ s!st t-( q-2 rcosh 2 z) \=0, with q= q|( r)and q= ô|( r)being theeigen values oftheMathieu equation (seeSection A.12). Themodi®ed Mathieu functions arede®ned as Ce2 |+ ( z, r)=ce2 |+ ( bz, r)= S$ =0 v2 |+ 2 + cosh[(2 + ) z], Se2 |+ ( z, r)=- bse2 |+ ( bz, r)= S =0 x2 |+ 2 + sinh[( 2 + ) z], where may beequal to0and1,andcoef®cientsv2 |+ 2 + andx2 |+ 2 + areindicated inSubsection A.12. Refer ences :H.Bateman andA.ErdÂelyi(1955, Vol.3),M.Abramo witz andI.Stegun(1964). A.16. Orthogonal Polynomials Allzeros ofeach oftheorthogonal polynomials |( z)considered inthissection arerealandsimple. Thezeros ofthepolynomials |( z)and |+1( z)arealternating. ForLegendre polynomials seeSection A.12. Page677 678 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES A.16.1.Laguerre Polynomials and Generaliz edLaguerre Polynomials A.16.1-1. Laguerre polynomials. TheLaguerre polynomials |= |( z)satisfy theequationz \ s!st t+(1- z) \ st+ \=0 andarede®ned bytheformulas|( z)=1! t |z | _ z |- ta=(-1) |! z |- 2z |-1+ 2( -1)2 2! z |-2+ } } }9. The®rstfour polynomials havetheform0=1, 1=- z+1, 2=1 2( z2-4 z+2), 3=1 6(- z3+9 z2-18 z+6). Tocalculate |( z)for ³2,onecanusetherecurrent formulas|+1( z)=1+1 (2 +1- z) |( z)- |-1( z) . Thefunctions |( z)form anorthonormal system ontheinterv al0< z< nwith weight- t: 0 - t|( z) ( z) z= w0if ¹ , 1if = . Thegenerating function is 1 1- exp y- Qz 1- {= S|=0 |( z) |, | |<1. A.16.1-2. Generalized Laguerre polynomials. Thegeneralized Laguerre polynomials |= |( z)( >-1)satisfy theequationz \ s!st t+( +1- z) \ st+ \=0 andarede®ned bytheformulas |( z)=1! z- t |z | _z |+- ta= |S=0 |- |+(- z) !. The®rsttwopolynomials havetheform 0=1, 1= +1- z. Tocalculate |( z)for ³2,onecanusetherecurrent formulas |+1( z)=1+1 (2 + +1- z) |( z)-( + ) |-1( z) . Thefunctions |( z)form anorthogonal system ontheinterv al0< z< nwith weight z- t: 0 z - t |( z) ( z) z= 0 if ¹ ,(+ |+1)|!if = . Thegenerating function is (1- )--1exp y- Qz 1- {= |=0 |( z) ¡ |, | ¡|<1. Page678 A.16. ORTHOGON ALPOLYNOMIALS 679 A.16.2. Cheb yshe vPolynomials and Functions A.16.2-1. Chebyshe vpolynomials. TheChebyshe vpolynomials ¢|= ¢|( z)satisfy theequation (1- z2) £ ¤!¤ ¥ ¥- z £ ¤ ¥+ ¦2£=0 (1) andarede®ned bytheformulas¢|( z)=cos( ¦arccos z)=(-2) |¦! (2 ¦)! §1- z2 ¨ |¨ z | ©(1- z2) |-1 2 ª = ¦ 2[ | «2]¬=0(-1) ¬( ¦- -1)!!( ¦-2 )!(2 z) |-2 ¬( ¦=0,1,2, ® ® ®), where [ ¯]stands fortheintegerpartofanumber ¯. The®rstfour polynomials are¢0=1, ¢1= z, ¢2=2 z2-1, ¢3=4 z3-3 z. Therecurrent formulas:¢|+1( z)=2 z ¢|( z)- ¢|-1( z), ¦³2. Thefunctions ¢|( z)form anorthogonal system ontheinterv al-1< z<+1,with°+1 -1 ¢|( z) ¢ ¬( z)§1- z2 ¨ z= ±0 if ¦¹ , 1 2 ²if ¦= ¹0,²if ¦= =0. A.16.2-2. Chebyshe vfunctions ofthesecond kind. TheChebyshe vfunctions ofthesecond kind,³ 0( z)=arcsin z,³|( z)=sin( ¦arcsin z)=§1- z2¦ ¨ ¢|( z)¨ z( ¦=1,2, ® ® ®), justastheChebyshe vpolynomials, alsosatisfy thedifferential equation (1). Thegenerating function is 1- ¡Qz 1-2 ¡Qz+ ¡2= |=0 ¢|( z) ¡ |(| ¡|<1). A.16.3. Hermite Polynomial TheHermite polynomial ´|= ´|( z)satis®es theequation£ ¤!¤ ¥ ¥-2 z £ ¤ ¥+2 ¦ £=0 andisde®ned bytheformulas´|( z)=(-1) |exp µz2 ¶¨ |¨ z |exp µ- z2 ¶. The®rstfour polynomials are´0=1, ´1= z, ´2=4 z2-2, ´3=8 z3-12 z. Page679 680 A.SPECIAL FUNCTIONS AND THEIR PROPER TIES Therecurrent formulas:´|+1( z)=2 z ´|( z)-2 ¦ ´|-1( z), ¦³2. Thefunctions ´|( z)form anorthogonal system ontheinterv al- ·< z< ·with weight ¸- ¥2:° -exp µ- z2 ¶´|( z) ´ ¬( z)¨ z= 0 if ¦¹ ,§ ²2 |¦!if ¦= . TheHermite functions ¹|( z)areintroduced bytheformula ¹|( z)=exp µ-1 2 z2 ¶´|( z),where¦=0,1,2, ® ® ® Thegenerating function: exp µ- ¡2+2 ¡Qz ¶= |=0 ´|( z) ¡ |¦!. A.16.4. Jacobi Polynomials TheJacobi polynomials º , »|= º , »|( z)satisfy theequation (1- z2) £ ¤!¤ ¥ ¥+©½¼- ¾-( ¾+¼+2) z ª£ ¤ ¥+ ¦( ¦+ ¾+¼+1) £=0 andarede®ned bytheformulasº , »|=(-1) | 2 |¦!(1- z)- (1+ z)- »¨ |¨ z | ¿(1- z) + |(1+ z) »+ | À=2- | |¬=0 Á ¬|+ Á |- ¬|+ »( z-1) |- ¬( z+1) ¬, whereÁ Â Ãarebinomial coef®cients.ÄÅ References forSupplement: H.Bateman andA.ErdÂelyi(1953, 1955), M.Abramo witz andI.A.Stegun(1964). Page680 Supplement B Methods ofGeneraliz edandFunctional Separation ofVariab lesinNonlinear Equations ofMathematical Physics B.1. Introduction B.1.1. Preliminar yRemarks Separation ofvariables isthemost common approach tosolvelinear equations ofmathematical physics. This approach involvessearching forexactsolutions intheform oftheproduct offunctions depending ondifferent arguments (seeSection 0.4). Asfarasnonlinear equations with twoindependent variables , andadependent variable areconcerned, some ofthese equations alsohavesolutions with theform( , )= ( ) ( ) or ( , )= ( )+ ( ) thatarecalled multiplicatively andadditively separable,respecti vely.Wecallsuch solutions ordinary separ able solutions .Inparticular ,integrating afewclasses of®rst-order nonlinear partial differential equations isbased onsearching foradditi velyseparable solutions [e.g., seeAppell (1953), Kamk e (1965), Mark eev(1990), Zwillinger (1998), Polyanin, Zaitse v,andMoussiaux (2001)]. Overthelastdecade, more sophisticated, generalized andfunctional separ able solutions have been obtained foranumber ofsecond-order nonlinear equations ofmathematical physics. For example, Galaktiono vand Posashk ov(1989) and Galaktiono v,Posashk ov,and Svirshche vskii (1995) obtained generalized separable solutions with theforms ( , )= ( ) ( )+ ( )and( , )= ( ) ( )+ ( )forsome classes ofparabolic andhyperbolic equations with quadrat- icnonlinearities. InGalaktiono vandPosashk ov(1994), Galaktiono v(1995), andSvirshche vskii (1995), more complicated generalized separable solutions arepresented. Theresults ofGalaktiono v andPosashk ov(1994) andGalaktiono v(1995) arebased on®nding ®nite-dimensional subspaces thatareinvariant under appropriate nonlinear differential operators (inpractice, theauthors had to®nd asystem ofcoordinate functions inoneofthevariables bythemethod ofundetermined coef®cients). InGrundland andInfeld (1992), Miller andRubel (1993), Zhdano v(1994), andAndree v, Kaptso v,Pukhnache v,andRodiono v(1994), allnonlinear heat (diffusion) andwaveequations oftheform = ( )which admit functional separable solutions having theform( , )= ( ),where = ( )+ ( ),aredescribed. DoyleandVassiliou (1998) indicated all one-dimensional nonstationary heat equations = [ ( ) ]which admit solutions ofthe form ( , )= ( ), = ( )+ ( ).InZaitse vandPolyanin (1996), Polyanin andZhuro v(1998), Polyanin, Vyazmin, Zhuro v,andKazenin (1998), andPolyanin, Zhuro v,andVyazmin (2000), many nonlinear mathematical physics equations ofvarious types thatadmit generalized andfunctional separable solutions aredescribed (special attention waspaid toequations ofgeneral form which depend onarbitrary functions). *Sections B.1±B.5 were written with A.I.Zhuro v. Page681 682 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES Functional differential equations thatinvolveunkno wnfunctions (and their derivatives)with dif- ferent arguments arise when searching forordinary ,generalized, andfunctional separable solutions. Thecurrent supplement presents direct methods forandexamples ofconstructing such solutions and reviewsapplication ofthese methods tosolving various classes ofthesecond-, third-, fourth-, and higher -order partial differential equations (intotal, about 150nonlinear equations with solutions are described). Special attention ispaid toequations ofheat andmass transfer theory ,wavetheory ,and hydrodynamics, aswell asmathematical physics equations ofgeneral form thatinvolvearbitrary functions. Itshould benoted thatoften exact generalized andfunctional separable solutions cannot be obtained bygroup theoretic methods orother well-kno wnmethods. B.1.2. Simple Cases ofVariab leSeparation inNonlinear Equations Inisolated cases, theseparation ofvariables innonlinear equations iscarried outfollowing the same technique asinlinear equations. Speci®cally ,anexact solution issought intheform ofthe product orsum offunctions depending ondifferent arguments. Onsubstituting itintotheequation andperforming elementary algebraic manipulations, oneobtains anequation with thetwosides dependent ondifferent variables (forequations with twovariables). Then oneconcludes thatthe expressions oneach sidemust beequal tothesame constant quantity ,called aseparation constant. Belowweconsider speci®c examples. Example 1.Theheat equation with apowernonlinearity = (1) hasanexact solution intheproduct form= ( ) ( ). (2) Substituting (2)into(1)yields = +1( ) . Separating thevariables bydividing both sides by +1,weobtain +1= ( ) . Theleft-hand sidedepends of alone andtheright-hand sideon alone. This ispossible only if +1= , ( ) = , (3) where isanarbitrary constant (separation constant). Onsolving theordinary differential equations (3),weobtain asolution ofequation (1)with theform (2). Theprocedure forconstructing aseparable solution (2)ofthenonlinear equation (1)isidentical tothatused insolving linear equations [inparticular ,equation (1)with =0].Werefer tothecases ofsimilar separation ofvariables assimple separ able cases. Example 2.Thewaveequation with anexponential nonlinearity2 2= !#"$ (4) hasanadditi velyseparable solution= ( )+ ( ). (5) Onsubstituting (5)into(4)anddividing by !"% ,wearriveattheequation! - "% & '= ( ! ")( ) , whose left-hand sidedepends on alone andtheright-hand sideon alone. This ispossible only if! - "% & '= , ( ! ")( ) = , (6) Page682 B.1. INTR ODUCTION 683 where isanarbitrary constant. Solving theordinary differential equations (6)yields asolution ofequation (4)with the form (5). Example 3.Theheat equation inananisotropic medium with alogarithmic source *,+( ) -+ . *0/( .) . -= ln (7) hasamultiplicati velyseparable solution= ( ) ( .). (8) Onsubstituting (8)into(7),dividing by ,andtransposing individual terms oftheresulting equation, weobtain 1[+( ) ] - ln =-1[/( .) 1] 1+ ln . Theleft-hand side ofthisequation depends only on andtheright-hand only on ..Byequating both sides toaconstant quantity ,oneobtains ordinary differential equations for ( )and ( .). B.1.3. Examples ofNontrivial Variab leSeparation inNonlinear Equations Unlik elinear equations, thevariables innonlinear equations often separate differently .Weexemplify thisbelow. Example 4.Consider theequation with acubic nonlinearity =+( ) 2 2+ 2 - 3, (9) where+( )isanarbitrary function. Welook forexact solutions intheproduct form. Wesubstitute (2)into(9)anddivide the resulting equation by+( ) ( ) ( )toobtain + = & 2+ 2+[( )2- 2]. (10) Inthegeneral case, thisexpression cannot berepresented asthesum oftwofunctions depending ondifferent arguments. This howeverdoes notmean thatequation (9)hasnosolutions oftheform (2). 1 3.One canmakesure bydirect check that, for >0,thefunctional differential equation (10) hassolutions( )= exp 405 6 7, ( )=exp* 8+( ) 9 -, (11) where isanarbitrary constant. Solution (11) for makestheexpression insquare brack etsin(10) vanish, which allows separation ofvariables. 23.There isamore general solution ofthefunctional differential equation (10) for >0:( )= 1exp 4 6 7+ 2exp 4- 6 7,( )= !;: 3+8 1 2 8 ! 2 :9 -1 <2 , == 8+( ) 9 , where 1, 2,and 3arearbitrary constants. Thefunction = ( )makeseach oftheterms in(10) thatdepend on constant, namely ,& 2 > =const ,( )2- 2=const . Itisthiscircumstance thatmakesitpossible toseparate thevariables. Note thatthefunction = ( )satis®es theBernoulli equation = +( ) -4 1 2 3. 3 3.There isanother solution ofthefunctional differential equation (10) for <0:( )= 1sin4 6- 7+ 2cos4 6- 7,( )= !;:* 3+2 ( 2 1+ 2 2)8 ! 2 :9 --1 <2 , == 8+( ) 9 , where 1, 2,and 3arearbitrary constants. Thefunction = ( )makesboth terms in(10) thatdepend on constant. Note thatthefunction = ( )isdetermined bytheBernoulli equation = +( ) - ( 2 1+ 2 2) 3. Page683 684 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES Example 5.Consider thethird-order equation . 2 2+ 2 .2= ? 3 3+ @ 3 .3. (12) Welook foradditi veseparable solutions=+( )+/( .). (13) Substituting (13) into(12) yields/ 1+ & 2+ + / & 1#1= ?+ && 22+ @/ && 1#1#1. (14) This expression cannot berewritten asthesum oftwofunctions depending ondifferent arguments. Itisnotdif®cult toseethatthefunctional differential equation (14) issatis®ed if/ 1= 1 A B /( .)= 1 .+ 2,+( )= 3exp( 1 > ?)+ 4 (case 1),+ = 1 A B +( )= 1 + 2,/( .)= 3exp( 1 .> @)+ 4 .(case 2), where 1, 2, 3,and 4arearbitrary constants. Inboth cases, twoterms ofthefourin(14)vanish, which makesitpossible toseparate thevariables. Inaddition, equation (12) hasamore complicated solution oftheform (13):= 1 ! - C "+ @ED + 2 !#"1- )?D .+ 3, where 1, 2, 3,and Darearbitrary constants. Themechanism ofseparation ofvariables isdifferent here: both nonlinear terms ontheleft-hand sidein(14) contain terms which cannot berewritten inadditi veform butareequal inmagnitude and haveunlik esigns. Inadding, thetwoterms cancel, thus resulting inseparation ofvariables:/ 1+& 2 = 1 2 2D3 !"1- C "- 1 ?( )D)3 ! - C " ++ /& 1#1 =- 1 2 2D3 !"1- C "+ 2 @ED3 !"1/ 1+& 2+ + /& 1#1=- 1 ?( )D)3 ! - C "+ 2 @ED3 !"1= ?+&& 22+ @/&& 1#1#1. Example 6.Consider thesecond-order equation with acubic nonlinearity (1+ 2) 2 2+ 2 .2 -2 2 -2 . 2 = (1- 2). (15) Weseek anexact solution ofthisequation intheproduct form=+( )/( .). (16) Substituting (16) into(15) yields (1++2/2)(/ + &2++ / & 1#1)-2+ / F&/2(+ )2++2(/ 1)2 G= + /(1-+2/2). (17) This expression cannot berewritten asthesum oftwofunctions with different arguments. Nevertheless, equation (15) has solutions oftheform (16). One canmakesure bydirect check thatthefunctions+=+( )and/=/( .)satisfying the nonlinear ordinary differential equations (+ )2= H+4+ I+2+ , (/ 1)2= /4+( - I)/2+ H,(18) where H, I,and arearbitrary constants, reduce equation (17) toanidentity; toverify this, oneshould usetherelations+& 2=2 H+3+ I+and/& 1#1=2 /3+( - I)/thatfollowfrom (18).J KML NPORQ SBythevariable change T=4arctan equation (15) canbereduced toanonlinear heat equation with a sinusoidal source, U T= sin T. Theexamples considered aboveillustrate some speci®c features ofseparable solutions tonon- linear equations. Sections B.2andB.3outline fairly general methods forconstructing similar and more complicated solutions tononlinear partial differential equations. Page684 B.2. METHODS OFGENERALIZED SEPARA TION OFVARIABLES 685 B.2. Methods ofGeneraliz edSeparation ofVariab les B.2.1. Structure ofGeneraliz edSeparab leSolutions B.2.1-1. General form ofsolutions. Theclasses ofnonlinear equations considered. Tosimplify thepresentation, wecon®ne ourselv estothecase ofmathematical physics equations with twoindependent variables , andadependent variable (one oftheindependent variables canplay theroleoftime). Linear separable equations ofmathematical physics admit exact solutions intheform( , )= 1( ) 1( )+ 2( ) 2( )+ V)V)V+ W( ) W( ), (1) where the X= X( ) X( )areparticular solutions; thefunctions X( ),aswell asthefunctions X( ),with different numbers Yarenotrelated tooneanother . Also having exact solutions oftheform (1)aremanynonlinear partial differential equations with quadratic orpowernonlinearities 1( ) Z1( ) [1[ ]+ 2( ) Z2( ) [2[ ]+ V)V)V+ \( ) Z \( ) [ \[ ]=0, (2) where the [ X[ ]aredifferential forms thataretheproducts ofnonne gativeintegerpowers ofthe function anditspartial derivatives , , , , , ,etc. Wewillrefer to solutions (1)ofnonlinear equations (2)asgeneralized separ able solutions .Unlik elinear equations, innonlinear equations thefunctio ns X( )with different subscripts Yareusuallyrelated tooneanother [and tothefunctions ]( )].Subsections B.1.2 andB.1.3 giveexamples ofexact solutions (1)to nonlinear equations (2)forsome simple cases with ^=1or ^=2(for 1= 2=1)._ `#a b cPd eIfthe f( )and Z f( )in(2)areallconstant, then onecanseek solutions inthemore general form( , )= Wg\=1 \( h) \( i), h= j1 + j2 , i= k1 + k2 , where j1, j2, k1,and k2areconstants. Some solutions ofthissortarediscussed inSubsections B.7.1 andB.8.1. B.2.1-2. General form offunctional differential equations. Ingeneral, onsubstituting expression (1)intothedifferential equation (2),onearrivesatafunctional differential equation l 1( m) n1( o)+ l 2( m) n2( o)+ V)V)V+ l p ( m) n p ( o)=0 (3) forthe X( )and X( ).Thefunctionals l]( m)and n ]( o)depend only on and ,respecti vely,l]( m)º l] qr, 1, s1, sts1, u)u)u, W, sW, stsW v,n ]( o)º n ] qr, 1, s1, sts1, u)u)u, W, sW, stsW v.(4) Here, forsimplicity ,theformulas arewritten outforthecase ofasecond-order equation (2);for higher -order equations, theright-hand sides ofrelations (4)will contain higher -order derivatives of Xand ]. Page685 686 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES B.2.2. Solution ofFunctional Differential Equations byDifferentiation B.2.2-1. Description ofthemethod. 1 w.Assume that n p x 0.Wedivide equation (3)by n p anddifferentiate with respect to .This results inasimilar equation butwith fewerterms:y l 1( m) yn1( o)+ y l 2( m) yn2( o)+ V)V)V+ y l p -1( m) yn p -1( o)=0,y l]( m)= l]( m), yn ]( o)=[ n ]( o) z n p ( o)] s. Wecontinue theaboveprocedure until weobtain aseparable two-term equation{ l 1( m) {n1( o)+ { l 2( m) {n2( o)=0. (5) Three cases must beconsidered. Nonde generatecase:| { l 1( m)|+| { l 2( m)| x 0and| {n1( o)|+| {n2( o)| x 0.Then equation (5)is equivalent totheordinary differential equations{ l 1( m)+ | { l 2( m)=0, | {n1( o)- {n2( o)=0, where |isanarbitrary constant. Theequations { l 2=0and {n1=0correspond tothelimit case|= }. Twodegeneratecases :{ l 1( m)º0, { l 2( m)º0 ~ {n1,2( o)areany;{n1( o)º0, {n2( o)º0 ~ { l 1,2( m)areany. 2 w.The solutions ofthetwo-term equation (5)should besubstituted into theoriginal function- aldifferential equation (3)toªremo veºredundant constants ofintegration [these arise because equation (5)isobtained from (3)bydifferentiation]. 3 w.Thecase n p º0should betreated separately (since wedivided theequation by n p atthe®rst stage). Likewise, wehavetostudy allother cases where thefunctionals bywhich theintermediate functional differential equations were divided vanish._ `#a b cPd eThefunctional differential equation (3)canhappen tohavenosolutions._ `#a b cPd eAteach subsequent stage, thenumber ofterms inthefunctional differential equation canbereduced bydifferentiation with respect toeither or .Forexample, wecanassume atthe ®rststage that l p x 0.Ondividing equation (3)by l p anddifferentiating with respect to ,we again obtain asimilar equation thathasfewerterms. B.2.2-2. Examples ofconstructing exact generalized separable solutions. Belowweconsider speci®c examples illustrating theapplication oftheabovemethod toconstructing exact generalized separable solutions ofnonlinear equations. Example 1.Thetwo-dimensional stationary equations ofmotion ofaviscous incompressible ¯uid arereduced toa single fourth-order nonlinear equation forthestream function (seeequation 1inSubsection B.7.1), speci®cally , ( U )- ( U )= U U , U = 2 2+ 2 2. (6) Weseek exact separable solutions ofequation (6)intheform= ( )+ ( ). (7) Substituting (7)into(6)yields && 22- && ##= &&& 222 + &&& ### . (8) Differentiating (8)with respect to and ,weobtain& #&&& 222 - & 2&&& ### =0. (9) Nonde generatecase.If & 2 0and & # 0,weseparate thevariables in(9)toobtain theordinary differential equations &&& 222 = & 2, (10)&&& ### = & #, (11) which havedifferent solutions depending onthevalue oftheintegration constant . Page686 B.2. METHODS OFGENERALIZED SEPARA TION OFVARIABLES 687 1 .Solutions ofequations (10) and(11) for =0:( )= 1+ 2 + 3 2+ 4 3,( )= 1+ 2 + 3 2+ 4 3,(12) where and arearbitrary constants ( =1,2,3,4).Onsubstituting (12) into(8),weevaluate theintegration constants. Three cases arepossible:4= 4=0, , areanynumbers ( =1,2,3); =0, areanynumbers ( =1,2,3,4); =0, areanynumbers ( =1,2,3,4). The®rsttwosetsofconstants determine twosimple solutions (7)ofequation (6):= 1 2+ 2 + 3 2+ 4 + 5,= 1 3+ 2 2+ 3 + 4, where 1, E, 5arearbitrary constants. 2 .Solutions ofequations (10) and(11) for = 2>0:( )= 1+ 2 + 3 # + 4 - ,( )= 1+ 2 + 3 + 4 - .(13) Substituting (13) into(8),dividing by 3,andcollecting terms, weobtain3( - 2) + 4( + 2)- + 3( + 2) + 4( - 2)- =0. Equating thecoef®cients oftheexponentials tozero, we®nd3= 4= 3=0, 2= (case 1),3= 3=0, 2= , 2=- (case 2),3= 4=0, 2=- , 2=- (case 3). (The other constants arearbitrary .)These setsofconstants determine three solutions (7)ofequation (6):= 1 - + 2 + 3+ ,= 1 - + + 2 - - + 3,= 1 - - + 2 # - + 3, where 1, 2, 3,and arearbitrary constants. 3 .Solution ofequations (10) and(11) for =- 2<0:( )= 1+ 2 + 3cos( )+ 4sin( ),( )= 1+ 2 + 3cos( )+ 4sin( ).(14) Substituting (14) into(8)does notyield newrealsolutions. Degeneratecases. If & 2º0or & #º0,equation (9)becomes anidentity forany = ( )or = ( ),respecti vely. These cases should betreated separately from thenonde generate case. Forexample, if & 2º0,wehave ( )= + , where and arearbitrary numbers. Substituting this into(8),wearriveattheequation - && ##= &&& ### .Itsgeneral solution isgivenby ( )= 1exp(- )+ 2 2+ 3 + 4.Thus, weobtain another solution (7)ofequation (6):= 1 - + 2 2+ 3 + 4+ ( = , =0). Example 2.Consider thesecond-order nonlinear parabolic equation = 2 2+ ¡2 + ¢. (15) Welook forexact separable solutions ofequation (15) intheform= £( )+ ¤( ) ¥( ). (16) Substituting (15) into(16) andcollecting terms yields£ ¦- ¢+ ¤ ¦R¥= £ ¤ ¥#& 2+ ¤2 §)¥2¥#& 2+ ( ¥# )2 ¨. (17) Page687 688 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES Ondividing thisrelation by ¤2anddifferentiating with respect to and ,weobtain ( ¤ ¦ ¤2) ¦R¥# = ( £ ¤) ¦¥#&& 22. Separating thevariables, wearriveattheordinary differential equations¥#&& 22= © ¥# , (18) ( ¤ ¦ ¤2) ¦= ©( £ ¤) ¦, (19) where ©isanarbitrary constant. Thegeneral solution ofequation (18) isgivenby¥= ª 1 2+ 2 + 3 if ©=0,1 + 2 - + 3 if ©= 2>0,1sin( )+ 2cos( )+ 3if ©=- 2<0,(20) where 1, 2,and 3arearbitrary constants. Integrating (19) yields¤= + 1, £( )isany if ©=0,£= ¤+1 © ¤ ¦¤, ¤( )isanyif ©¹0,(21) where isanarbitrary constant. Onsubstituting solutions (20)and(21)into(17), onecanªremo veºtheredundant constants andde®ne thefunctions £and ¤.Belowwesummarize theresults. 1 .Solution for ¹- and ¹-2 := ¢( +2 ) 2( + )( + 1)+ 2( + 1)- ««+2 ¬-( + 3)2 2( +2 )( + 1)(corresponds to ©=0), where 1, 2,and 3arearbitrary constants. 2 .Solution for =- :=1)2 ¤ ¦¤+ ¤( 1 # + 2 - ) (corresponds to ©= 2>0), where thefunction ¤= ¤( )isdetermined from theautonomous ordinary differential equation& ¦'¦= )¢E2+4 241 2 2 ®, ¤= ®, whose solution canbefound inimplicit form. Inthespecial case 1=0or 2=0,wehave ¤= 1exp ¯1 2 )¢E2 2+ 2 ±°. 3 .Solution for =- :=-1)2 ¤ ¦¤+ ¤[ 1sin( )+ 2cos( )] (corresponds to ©=- 2<0). where thefunction ¤= ¤( )isdetermined from theautonomous ordinary differential equation& ¦'¦=- )¢E2+ 24( 2 1+ 2 2)2 ®, ¤= ®, whose solution canbefound inimplicit form. B.2.3. Solution ofFunctional Differential Equations bySplitting B.2.3-1. Preliminary remarks. Description ofthemethod. Asonereduces thenumber ofterms inthefunctional differential equation (3)bydifferentiation, redundant constants ofintegration arise. These constants must beªremo vedºatthe®nal stage. Furthermore, theresulting equation canbeofahigher -order than theoriginal equation. Toavoid these dif®culties, itisconvenient toreduce thesolution ofthefunctional differential equation to thesolution ofalinear functional equation ofastandard form andsolution ofasystem ofordinary differential equations. Thus, theoriginal problem splits into twosimpler problems. Belowwe outline thebasic stages ofthesplitting method. Page688 B.2. METHODS OFGENERALIZED SEPARA TION OFVARIABLES 689 Thecase ofevennumber ofterms inequation (3), ²=2 ³. 1 ´.Atthe®rst stage, wetreat equation (3)asapurely functional equation thatdepends ontwo variables µand ¶,where ·1( µ), ¸)¸)¸, · ¹( µ), º ¹+1( ¶), ¸)¸)¸, º2 ¹( ¶)areunkno wnquantities and thefunctions · ¹+1( µ), ¸)¸)¸, ·2 ¹( µ), º1( ¶), ¸)¸)¸, º ¹( ¶)areassumed tobeknown. Itcanbeshown(byinduction anddifferentiation) thatthefunctional equation (3)hasasolution depending on ³2arbitrary constants· »( µ)= ¼ »1 · ¹+1( µ)+ ¼ »2 · ¹+2( µ)+ ½)½)½+ ¼ »0¹P· 2 ¹( µ) ( ¾=1, ¸)¸)¸, ³),º ¹+ »( ¶)=- ¼1 »º1( ¶)- ¼2 »º2( ¶)- ½)½)½- ¼ ¹±»º ¹( ¶) ( ¾=1, ¸)¸)¸, ³),(22) where the ¼ »À¿arearbitrary constants. Note thatthere arealsoªdegenerateº solutions depending on fewerarbitrary constants (seeItem 2 ´inParagraph B.2.3-2). 2 ´.Atthesecond stage, wesubstitute the · »( µ)and º ¿( ¶)of(4)into (22). This results inan overdetermined system ofordinarydifferential equation sfortheunknownfunctio ns Á Â( )and Ã Ä( Å). Thecase ofoddnumber ofterms inequation (3), ²=2 ³-1. 1 ´.Ifthenumber orterms isodd( ²=2 ³-1),thefunctional equation (3)hastwodifferent solutions with ³( ³-1)arbitrary constants. Oneofthem canbeobtained from formulas (22)bysetting ·2 ¹º0 anddiscarding thelastterm with º2 ¹.The other solution canbeobtained from the®rst oneby renaming · »( µ) Æ º »( ¶). 2 ´.Further analysis foreach solution should beperformed following thesame scheme asinthe case ofevennumber ofterms in(3). B.2.3-2. Solutions ofsimple functional equations andtheir application. Belowwegivesolutions oftwosimple functional equations oftheform (3)that will beused subsequently forsolving speci®c nonlinear partial differential equations. 1 ´.Thefunctional equation·1 º1+ ·2 º2+ ·3 º3=0 (23) where the · »areallfunctions ofthesame argument andthe º »areallfunctions ofanother argument, hastwosolutions:·1= Ç1 ·3, ·2= Ç2 ·3, º3=- Ç1 º1- Ç2 º2,º1= Ç1 º3, º2= Ç2 º3, ·3=- Ç1 ·1- Ç2 ·2,(24) where Ç1and Ç2arearbitrary constants. 2 ´.Thefunctional equation·1 º1+ ·2 º2+ ·3 º3+ ·4 º4=0, (25) where the · »areallfunctions ofthesame argument andthe º »areallfunctions ofanother argument, hasasolution·1= Ç1 ·3+ Ç2 ·4, ·2= Ç3 ·3+ Ç4 ·4,º3=- Ç1 º1- Ç3 º2, º4=- Ç2 º1- Ç4 º2(26a) depending onfourarbitrary constants Ç È[seesolution (22)with ³=2, ¼11= Ç1, ¼12= Ç2, ¼21= Ç3, and ¼22= Ç4]. Equation (25) hasalsotwoªdegenerateº solutions·1= Ç1 ·4, ·2= Ç2 ·4, ·3= Ç3 ·4, º4=- Ç1 º1- Ç2 º2- Ç3 º3,º1= Ç1 º4, º2= Ç2 º4, º3= Ç3 º4, ·4=- Ç1 ·1- Ç2 ·2- Ç3 ·3(26b) involving three arbitrary constants. Page689 690 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES Example 3.Consider thenonlinear hyperbolic equationÉ2 ÊÉ 2= ÉÉ Ë Ê ÉÊÉ Ë ¡+ Ì( ) Ê+ Í( ), (27) where Ì( )and Í( )arearbitrary functions. Welook forgeneralized separable solutions with theformÊ( Ë, )= £( Ë) ¤( )+ Î( ). (28) Substituting (28) into(27) andcollecting terms yield¤2( £ £ Ï Ð) Ï Ð+ ¤ Î £ Ï&Ï Ð2Ð+( Ì ¤- ¤ Ï&Ï ¦'¦) £+ Ì Î+ Í- Î Ï&Ï ¦'¦=0. This equation canberepresented asafunctional equation (25) inwhichÑ 1=( £ £ Ï Ð) Ï Ð, Ñ 2= £ Ï&Ï Ð2Ð, Ñ 3= £, Ñ 4=1,Ò 1= ¤2, Ò 2= ¤ Î, Ò 3= Ì ¤- ¤Ï&Ï ¦'¦, Ò 4= Ì Î+ Í- ÎÏ&Ï ¦'¦.(29) Onsubstituting (29) into (26a), weobtain thefollowing overdetermined system ofordinary differential equations forthe functions £= £( Ë), ¤= ¤( ),and Î= Î( ): ( £ £Ï Ð)Ï Ð= 1 £+ 2, £Ï&Ï Ð2Ð= 3 £+ 4,Ì ¤- ¤Ï&Ï ¦'¦=- 1 ¤2- 3 ¤ Î, Ì Î+ Í- ÎÏ&Ï ¦'¦=- 2 ¤2- 4 ¤ Î.(30) The®rsttwoequations in(30) areconsistent only if1=6 2, 2= 2 1-4 0 2, 3=0, 4=2 2, (31) where 0, 1,and 2arearbitrary constants, andthesolution isgivenby£( Ë)= 2 Ë2+ 1 Ë+ 0. (32) Onsubstituting theexpressions (31) intothelasttwoequations in(30), weobtain thefollowing system ofequations for ¤( ) and Î( ):¤ Ï&Ï ¦'¦=6 2 ¤2+ Ì( ) ¤,Î Ï&Ï ¦'¦=[2 2 ¤+ Ì( )] Î+ ( 2 1-4 0 2) ¤2+ Í( ),(33) Relations (28), (32) andsystem (33) determine ageneralized separable solution ofequation (27). The®rstequation in(33)canbesolvedindependently; itislinear if 2=0andisintegrable inquadrature for Ì( )=const .Thesecond equation in(33) islinear in Î(for ¤known). Equation (27) does nothaveother solutions with theform (28) if Ìand Íarearbitrary function and £ Ó0, ¤ Ó0,andÎ Ó0.Ô ÕMÖ ×PØRÙ ÚItcanbeshownthatequation (27) hasamore general solution with theformÊ( Ë, Û)= £1( Ë) ¤1( )+ £2( Ë) ¤2( )+ ¤3( ), £1( Ë)= Ë2, £2( Ë)= Ë, (34) where thefunctions ¤ Ü= ¤ Ü( )aredetermined bytheordinary differential equations¤Ï&Ï1=6 ¤2 1+ Ì( ) ¤1,¤ Ï&Ï2=[6 ¤1+ Ì( )] ¤2,¤Ï&Ï3=[2 ¤1+ Ì( )] ¤3+ ¤2 2+ Í( ).(35) (The prime denotes thederivativewith respect to .)Thesecond equation in(35) hasaparticular solution ¤2= ¤1.Hence, itsgeneral solution canberepresented as(seePolyanin andZaitse v,1995)¤2= Ý1 ¤1+ Ý2 ¤1 Þ ß2à¤2 1. Thesolution obtained inExample 3corresponds tothespecial case Ý2=0. Example 4.Consider thenonlinear equationÉ2ÊÉ Ë Éà+ á ÉÊÉ Ë â2 - Ê É2ÊÉ Ë2= ã É3ÊÉ Ë3, (36) which arises inhydrodynamics (seeequations B.6.2.1, Item 3 äandB.7.2.1, Item 2 ä). Welook forexact solutions oftheformÊ= £(à) ¥( Ë)+ ¤(à). (37) Page690 B.2. METHODS OFGENERALIZED SEPARA TION OFVARIABLES 691 Substituting (37) into(36) yields£ Ï ¦R¥#Ï Ð- £ ¤ ¥#Ï&Ï Ð2Ð+ £2 §( ¥#Ï Ð)2- ¥2¥#Ï&Ï Ð2Ð ¨- ã £ ¥#Ï&Ï&Ï Ð2Ð2Ð=0. This functional differential equation canbereduced tothefunctional equation (25) bysettingÑ 1= £ Ï ¦, Ñ 2= £ ¤, Ñ 3= £2, Ñ 4= ã £,Ò 1= ¥Ï Ð, Ò 2=- ¥Ï&Ï Ð2Ð, Ò 3=( ¥Ï Ð)2- ¥2¥Ï&Ï Ð2Ð, Ò 4=- ¥Ï&Ï&Ï Ð2Ð2Ð. Onsubstituting these expressions into(26a), weobtain thesystem ofequations£ Ï ¦= å1 £2+ å2 ã £, £ ¤= å3 £2+ å4 ã £, ( ¥#Ï Ð)2- ¥2¥#Ï&Ï Ð2Ð=- å1 ¥#Ï Ð+ å3 ¥#Ï&Ï Ð2Ð, ¥#Ï&Ï&Ï Ð2Ð2Ð = å2 ¥#Ï Ð- å4 ¥#Ï&Ï Ð2Ð.(38) Itcanbeshownthatthelasttwoequations in(38) areconsistent only ifthefunction ¥anditsderivativearelinearly dependent,¥#Ï Ð= æ1 ¥+ æ2. (39) Thesixconstants æ1, æ2, å1, å2, å3,and å4must satisfy thethree conditionsæ1( å1+ æ2- å3 æ1)=0,æ2( å1+ æ2- å3 æ1)=0,æ2 1+ å4 æ1- å2=0.(40) Integrating (39) yields¥= ª æ3exp( æ1 Ë)- æ2æ1if æ1¹0,æ2 Ë+ æ3 if æ1=0,(41) where æ3isanarbitrary constant. The®rsttwoequations in(38) lead tothefollowing expressions for £and ¤:£= çèè éèèê å2 ãÝexp(- å2 ãà)- å1if å2¹0, -1å1à+ Ýif å2=0, ¤= å3 £+ å4 ã, (42) where Ýisanarbitrary constant. Formulas (41), (42) andrelations (40) allowusto®ndthefollowing solutions ofequation (36) with theform (37):Ê= Ë+ Ý1à+ Ý2+ Ý3 if å2= æ1=0, æ2=- å1;Ê= Ý1 - Ð+1à+ Ý2+ ã if å2=0, æ1=- å4, æ2=- å1- å3 å4;Ê= Ý1 -( Ð+ ë ì ¦)+ ã( + í)if å1= å3= æ2=0, å2= æ2 1+ å4 æ1;Ê= ã í+ Ý1 - Ð 1+ Ý2 - ì ë ¦+ ã( - í)if å1= å3 æ1- æ2, å2= æ2 1+ å4 æ1, where Ý1, Ý2, Ý3, í,and arearbitrary constants (these canbeexpressed interms ofthe å îand æ î). Theanalysis ofthesecond degenerate solution (26b) ofthefunctional equation (25) leads tothefollowing twomore general solutions ofthedifferential equation (36):Ê= Ëà+ Ý1+ ¤(à),Ê= £(à)- Ð- £Ï ¦(à) £(à)+ ã , where £(à)and ¤(à)arearbitrary functions, and Ý1and arearbitrary constants. B.2.4. Simpli®ed Scheme forConstructing Exact Solutions ofEquations with Quadratic Nonlinearities B.2.4-1. Description ofthesimpli®ed scheme. Toconstruct exact solutions ofequations (2)with quadratic orpowernonlinearities thatdonot depend explicitly on ï(all ð »constant), itisreasonable tousethefollowing simpli®ed approach. As Page691 692 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES before, weseek solutions intheform of®nite sums (1). Weassume thatthesystem ofcoordinate functions { Á »( ï)}isgoverned bylinear differential equations with constant coef®cients. Themost common solutions ofsuch equations areoftheformÁ »( ï)= ï », Á »( ï)= ñ ò ó±ô, Á »( ï)=sin( õ »±ï), Á »( ï)=cos( ö »±ï). (43) Finite chains ofthese functions (invarious combinations) canbeused tosearch forseparable solutions (1),where thequantities ÷ », õ »,and ö »areregarded asfreeparameters. Theother system offunctions { Ã »( Å)}isdetermined bysolving thenonlinear equations resulting from substituting (1) intotheequation under consideration. This simpli®ed approach lacks thegenerality ofthemethods outlined inSubsections B.2.2 andB.2.3. However,specifying oneofthesystems ofcoordinate functions, { Á »( ï)},simpli®es theprocedure of®nding exact solutions substantially .Thedrawback ofthisapproach isthatsome solutions oftheform (1)canbeoverlook ed.Itissigni®cant thattheoverwhelming majority of generalized separable solutions knowntodate, forpartial differential equations with quadratic nonlinearities, aredetermined bycoordinate functions (36) (usually with ø=2). B.2.4-2. Examples ofconstructing exact solutions ofhigher -order equations. Belowweconsider speci®c examples thatillustrate theapplication oftheabovesimpli®ed scheme toconstructing generalized separable solutions ofhigher -order nonlinear equations. Example 5.The equations oflaminar boundary layer ona¯atplate arereduced toasingle third-order nonlinear equation forthestream function (seeSchlichting 1981, Loitsyanskiy 1996):ÉÊÉÛ É2ÊÉ Ë ÉÛ- ÉÊÉ Ë É2ÊÉÛ2= ã É3ÊÉÛ3. (44) Welook forgeneralized separable solutions with theformÊ( Ë, Û)= Ë ù( Û)+ ú( Û), (45) which corresponds tothesimplest setoffunctions û1( Ë)= Ë, û2( Ë)=1with =2informula (1).Onsubstituting (45) into (44) andcollecting terms, weobtainË[( ùÏ)2- ù ùÏ&Ï- ã ùÏ&Ï&Ï]+[ ùÏ úÏ- ùúÏ&Ï- ã úÏ&Ï&Ï]=0. (The prime denotes thederivativewith respect to Û.)Tomeet thisequation forany Ë,oneshould equate both expressions in square brack etstozero. This results inasystem ofordinary differential equations for ù= ù( Û)and ú= ú( Û): ( ùÏ)2- ù ùÏ&Ï- ã ùÏ&Ï&Ï=0,ùÏ'ú#Ï- ùú#Ï&Ï- ã ú#Ï&Ï&Ï=0. Forexample, thissystem hasanexact solutionù=6 ãÛ+ Ý1, ú= Ý2Û+ Ý1+ Ý3 ( Û+ Ý1)2+ Ý4, where Ý1, Ý2, Ý3,and Ý4arearbitrary constants. Other generalized separable solutions ofequation (44) canbefound inPolyanin (2001b, 2001c) andSubsection B.6.1. Example 6.Consider the th-order nonlinear equationÉÊÉÛ É2 ÊÉ Ë ÉÛ- ÉÊÉ Ë É2 ÊÉÛ2= Ì( Ë) É üÊÉÛ ü, (46) where Ì( Ë)isanarbitrary function. Inthespecial case =3with Ì( Ë)= ã=const ,thisequation coincides with theboundary layer equation (44). Welook forgeneralized separable solutions oftheformÊ( Ë, Û)= û( Ë);ý+ ú( Ë), (47) which correspond tothesetoffunctions ù 1( Û)=ý, ù 2( Û)=1in(1).Onsubstituting (47) into(46) andrearranging terms, weobtain2 ý û[ ú#Ï Ð+ ü-2Ì( Ë)]=0. This equation ismetifú( Ë)=- ü-2Þ Ì( Ë)ß Ë+ Ý, û( Ë)isany, (48) where Ýisanarbitrary constant. (The other case û=0and úisanyisoflittle interest.) Formulas (47) and(48) de®ne an exact solution ofequation (46),Ê( Ë, Û)= û( Ë) ý- ü-2Þ Ì( Ë)ß Ë+ Ý, (49) which involvesanarbitrary function û( Ë)andtwoarbitrary constants Ýand . Note thatsolution (49)with =3and Ì( Ë)=const wasobtained byIgnato vich (1993) byamore complicated approach. Page692 B.3. METHODS OFFUNCTION ALSEPARA TION OFVARIABLES 693 B.3. Methods ofFunctional Separation ofVariab les B.3.1. Structure ofFunctional Separab leSolutions B.3.1-1. Functional separable solutions. Suppose anonlinear equation for þ= þ( ï, Å)isobtained from aseparable linear mathematical physics equation for ÿ= ÿ( ï, Å)byanonlinear change ofvariable þ= ( ÿ).Then, obviously ,the former hasexact solutions oftheformþ( ï, Å)= ( ÿ),where ÿ= È=1 Á È( ï) Ã È( Å). (1) Itisnoteworthy thatmanynonlinear partial differential equations thatarenotreduced tolinear equation haveexactsolutions oftheform (1)aswell. Wewillcallsuch solutions functional separ able solutions .Ingeneral, thefunctions Á È( ï), Ã È( Å),and ( ÿ)in(1)arenotknowninadvance and aretobeidenti®ed. Infunctional separation ofvariables, searching forsolutions intheforms þ= RÁ( ï)+ ( ) and þ= RÁ( ï) ( ) leads toequivalent results, because thetwoforms are functionally equivalent. Indeed, wehave RÁ( ï) ( ) = 1 RÁ1( ï)+ 1( ) ,where 1( ÿ)= ( ñ ),Á1( ï)=ln Á( ï),and 1( )=ln ( ). Inconstructing functional separable solutions with theform þ= RÁ( ï)+ ( ) , itisassumed that Á const and const. B.3.1-2. Various modi®cations. Belowwegivethree more general modi®cations ofsolution structure (1):þ( ï, )= ( ÿ), ÿ= =1 Á ( ) ( ), = 1 ï+ 2 , = 1 ï+ 2 ; (2)þ( ï, )= 1( ï) ( ÿ)+ 2( ï), ÿ= =1 Á ( ï) ( ); (3)þ( ï, )= 1( ) ( ÿ)+ 2( ), ÿ= =1 Á ( ï) ( ). (4) These canalsobeused to®ndexact solutions ofnonlinear mathematical physics equations. Thesolution structures of(1)±(4) coverallmost common types ofsolutions Ðtraveling wave, self-similar ,andadditi velyandmultiplicati velyseparable solutions (aswell asmanyinvariant solutions). Ingeneral, thefunctions Á ( ), ( ), Á ( ï), ( ), ( ÿ), ( ï),and ( )arenot knowninadvance andaretobedetermined intheanalysis. Note thatMiller andRubel (1993) studied functional separable solutions ofadifferent form (for astationary heatequation with anonlinear source); seealsoClarkson andKruskal (1989) andBurde (1994). B.3.2. Special Functional Separab leSolutions Tosimplify theanalysis, some ofthefunctions in(1)canbespeci®ed apriori andtheother functions willbede®ned intheanalysis. Wecallsuch solutions special functional separ able solutions . Page693 694 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES B.3.2-1. Solutions oftheform (1)with ÿlinear inoneoftheindependent variables. Consider functional separable solutions oftheform (1)inthespecial case where thecomposite argument ÿislinear inoneoftheindependent variables (e.g., in ï).Wesubstitute (1)into the equation under study andeliminate ïusing theexpression of ÿtoobtain afunctional differential equation with twoarguments. Inmanycases, thisequation canbesolvedbythemethods outlined inSection B.2. Example 1.Consider thenonstationary heat equation with anonlinear source = 2 2+ ( ). (5) Welook forfunctional separable solutions ofthespecial form= ( ), = û( ) + ù( ). (6) Thefunctions ( ), û( ), ù( ),and ( )aretobedetermined. Onsubstituting (6)into(5)andondividing by !,wehaveû "+ ù "= û2 # !$! !+ ( ) !. (7) Weexpress from (6)interms of andsubstitute into(7)toobtain afunctional differential equation with twovariables and , - ù "+ ùû û "- û "û + û2 # !$! !+ ( ) !=0, which canbetreated asthefunctional equation (25) inSection B.2where% 1=- ù "+ ùû û ", % 2=- û "û, % 3= û2, % 4=1,& 1=1, & 2= , & 3= # !$! !, & 4= ( ) !. Substituting these expressions intorelations (26a) ofSection B.2yields thesystem ofordinary differential equations - ù "+ ùû û "= '1 û2+ '2,- û "û= '3 û2+ '4, # !$! !=- '1- '3 , ( ) !=- '2- '4 ,(8) where '1, '2, '3,and '4arearbitrary constants. Thesolution ofsystem (8)isgivenbyû( )= ( ) *1 +-2 ,4 " - '3'4 --1 .2 ,ù( )=- û( ) /0'1 1 2( ) 3 + '2 1 3 2( )+ *2 4,( )= *3 1exp 5-1 2 '3 2- '1 76 37+ *4,( )=- *3( '4 + '2)exp 5-1 2 '3 2- '1 76,(9) where *1, *2, *3,and *4arearbitrary constants. The dependence = ( )isde®ned bythelasttworelations in parametric form ( isconsidered theparameter). Inthespecial case '3= *4=0, '1=-1,and *3=1,thesource function canberepresented inexplicit form as( )=- ( '4ln + '2). (10) If '3¹0in(9),thesource function isexpressed interms ofelementary functions andtheinverseoftheerror function. Example 2.Consider themore general equation = 8( ) 2 2+ 9( ) + :( ) ( ). Page694 B.3. METHODS OFFUNCTION ALSEPARA TION OFVARIABLES 695 Welook forsolutions intheform (6).Inthiscase, only the®rsttwoequations insystem (8)willchange, andthefunctions( )and ( )willbegivenby(9). Example 3.Thenonlinear heat equation = /<;( ) 4+ ( ) hasalso solutions oftheform (6). Theunkno wnquantities aregoverned bysystem (8)inwhich # !$!must bereplaced by [ ;( ) !] !.Thefunctions2( )and =( )aredetermined bythe®rsttwoformulas in(9).One ofthetwofunctions ;( )and( )canbeassumed arbitrary andtheother isidenti®ed inthecourse ofthesolution. Thespecial case ( )=const yields;( )= *1 +2 > ?+( *2 + *3)+ > @. Example 4.Likewise, wecantreat the Ath-order nonlinear equation = B B+ ( ). Asbefore, welook forsolutions intheform (6). Inthiscase, thequantities22and # !$!in(8)must bereplaced by2 B and ( B)!,respecti vely.Inparticular ,for '3=0,apart from equations with logarithmic nonlinearities oftheform (10), we obtain other equations. Example 5.Forthe Ath-order nonlinear equation = B B+ ( ) , thesearch forexact solutions oftheform (6)leads tothefollowing system ofequations for2( ), =( ), ( ),and ( ): - = "+ =2 2 "= '1 2 B+ '2 2,- 2 "2= '3 2 B+ '4 2,( B)! !=- '1- '3 , ( )=- '2- '4 , where '1, '2, '3,and '4arearbitrary constants. Inthecase A=3,weassume '3=0and '1>0to®ndinparticular that ( )=- '2- '4arcsin ( C ). Example 6.Inaddition, searching forsolutions ofequation (5)with quadratically dependent on ,= ( ), =2( ) 2+ =( ), (11) also makessense here. Indeed, onsubstituting (11) into(5),wearriveatanequation thatcontains terms with 2anddoes notcontain terms linear in .Eliminating 2from theresulting equation with theaidof(11), weobtain - = "+ =2 2 "+22- 2 "2 +42 # !$! !-42 = # !$! !+ ( ) !=0. Tosolvethisfunctional differential equation with twoarguments, weapply thesplitting method outlined inSubsection B.2.3. Itcanbeshownthat, forequations (5),thisequation hasasolution with alogarithmic nonlinearity oftheform (10). B.3.2-2. Solution byreduction toequations with quadratic nonlinearities. Insome cases, solutions oftheform (1)canbesearched forintwostages. First, onelooks fora transformation thatwould reduce theoriginal equation toanequation with aquadratic (orpower) nonlinearity .Then themethods outlined inSection B.2areused to®ndsolutions oftheresulting equation. Sometimes, quadratically nonlinear equations canbeobtained using thesubstitutionsD( E)= E F (forequations with powernonlinearities ),D( E)= Gln E(forequations with exponential nonlinearities ),D( E)= HF (forequations with logarithmic nonlinearities ), where Gisaconstant tobedetermined. This approach isequivalent tospecifying theform ofthe function I( E)in(1)apriori. Page695 696 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES Example 6.Thenonlinear heat equation with alogarithmic source = 8 2 2+ J( ) ln + K( ) canbereduced bythechange ofvariable =+ !tothequadratically nonlinear equation = 8 2 2+ 8 ) -2 + J( ) + K( ), which admits separable solutions with theform=21( ) =1( )+22( ) =2( )+ =3( ), where21( )= 2,22( )= ,andthefunctions =>( )aredetermined byanappropriate system ofordinary differential equations. B.3.3. Differentiation Method B.3.3-1. Basic ideas ofthemethod. Reduction toastandard equation. Ingeneral, thesubstitution ofexpression (1)intothenonlinear partial differential equation under study leads toafunctional differential equation with three arguments Ðtwoarguments areusual,Land ,andthethird iscomposite, E.Inmanycases, theresulting equation canbereduced by differentiation toastandard functional differential equation with twoarguments (either Lor is eliminated). Twosolvethetwo-argument equation, onecanusethemethods outlined inSection B.2. B.3.3-2. Examples ofconstructing functional separable solutions. Belowweconsider speci®c examples illustrating theapplication ofthedifferentiation method for constructing functional separable solutions ofnonlinear equations. Example 7.Consider thenonlinear heat equation = /MJ( ) 4. (12) Welook forexact solutions with theform= ( ), =2( )+ =( ). (13) Onsubstituting (13) into(12) anddividing by !,weobtain thefunctional differential equation= "=2 # NONJ( )+(2 N)2 P( ), (14) whereP( )= J( ) # !$! !+ J !( ), = ( ). (15) Differentiating (14) with respect to yields2 # # NONONJ( )+2 N2 # NON[ J !( )+2 P( )]+(2 N)3P !=0. (16) This functional differential equation with twovariables canbetreated asthefunctional equation (23) ofSection B.2. This three-term functional equation hastwodifferent solutions. Accordingly ,weconsider twocases. Case 1.Thesolutions ofthefunctional differential equation (16)aredetermined from thesystem ofordinary differential equationsJ !+2 P=2 '1 J, P != '2 J,2 # # NONON+2 '1 2 N2 # NON+ '2(2 N)3=0,(17) where '1and '2arearbitrary constants. The®rsttwoequations (17) arelinear andindependent ofthethird equation. Their general solution isgivenbyJ= QR SRT + ,1 !( U1 + > !+ U2 +- > !) if '2 1>2 '2,+ ,1 !( U1+ U2 ) if '2 1=2 '2,+ ,1 ![ U1sin( C )+ U2cos( C )]if '2 1<2 '2, P= '1 J-1 2 J !, C= V| '2 1-2 '2|. (18) Page696 B.3. METHODS OFFUNCTION ALSEPARA TION OFVARIABLES 697 Substituting Pof(18) into(15) yields adifferential equation for = ( ).Onintegrating thisequation, weobtain= *1 1+ ,1 !| J( )|-3 .237+ *2, (19) where *1and *2arearbitrary constants. The expression of Jin(18) together with expression (19) de®ne thefunctionJ= J( )inparametric form. Without fullanalysis, wewillstudy thecase '2=0( C= '1)and '1¹0inmore detail. Itfollowsfrom (18) and(19) thatJ( )= U1 +2 ,1 !+ U2, P= '1 U2, ( )= *3( U1+ U2 +-2 ,1 !)-1 .2+ *2( *1= '1 U2 *3). (20) Eliminating yieldsJ( )= U2 *2 3*2 3- U1 2. (21) Thelastequation in(17) with '2=0hasthe®rstintegral2 # NON+ '1(2 N)2=const .Thecorresponding general solution is givenby2( )=-1 2 '1ln / W2W11 sinh250'1 XW2 +W3 6 4 forW1>0andW2>0;2( )=-1 2 '1ln /- W2W11 cos25'1 X-W2 +W3 6 4forW1>0andW2<0;2( )=-1 2 '1ln /- W2W11 cosh250'1 XW2 +W3 6 4forW1<0andW2>0;(22) whereW1,W2,andW3areconstants ofintegration. Inallthree cases, thefollowing relations hold: (2 N)=W1 +-2 ,1 Y+W2,2 # NON=- '1W1 +-2 ,1 Y. (23) Wesubstitute (20) and(23) intotheoriginal functional differential equation (14). Withreference totheexpression of in(13), weobtain thefollowing equation for == =( ):= "=- '1 U1W1 +2 ,1 Z+ '1 U2W2. Itsgeneral solution isgivenby=( )=1 2 '1ln U2W2W4exp(-2 '2 1 U2W2 )+ U1W1, (24) whereW4isanarbitrary constant. Formulas (13), (20)for ,(22), and(24)de®ne three solutions ofthenonlinear equation (12)with J( )oftheform (21) [recall thatthese solutions correspond tothespecial case '2=0in(18) and(19)]. Case 2.Thesolutions ofthefunctional differential equation (16)aredetermined from thesystem ofordinary differential equations2 # # NONON= '1(2 N)3,2 N2 # NON= '2(2 N)3,'1 J+ '2( J !+2 P)+ P !=0.(25) The®rsttwoequations in(25) areconsistent inthetwocases'1= '2=0 [ \2( )= U1 + U2,'1=2 '2 2 [ \2( )=-1'2ln| U1 + U2|.(26) The®rstsolution in(26) eventually leads tothetraveling wavesolution = ( U1 + U2 )ofequation (12) andthesecond solution totheself-similar solution oftheform = ] ( 2 ^ ).Inboth cases, thefunction J( )in(12) isarbitrary . Amore detailed analysis offunctional separable solutions (13) ofequation (12) canbefound inthereference cited below._a` Refer ence:P.W.DoyleandP.J.Vassiliou (1998). Example 8.One canlook formore complicated functional separable solutions ofequation (12) with theform= ( ), =2( b)+ =( ), b= + 8 ( 8=const ). Wesubstitute thisinto(12), divide theresulting functional differential equation by !,anddifferentiate with respect to to obtain - 82 # cdc+2 # # cdcdcJ( )+2 c2 # cdc[ J !( )+2 P( )]+(2 c)3P !=0, where thefunction P= P( )isde®ned by(15). This functional differential equation with twovariables band canbe treated asthefunctional equation (25) ofSection B.2. Thesolution of(25) isgivenbyrelations (26), thus representing a system ofordinary differential equations for J, P,and2. Page697 698 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES Example 9.Consider thenonlinear Klein±Gordon equation2 2- 2 2= ( ). (27) Welook forfunctional separable solutions inadditi veform:= ( ), =2( )+ =( ). (28) Substituting (28) into(27) yields= # "0"-2 # NON+ e( = ")2-(2 N)2 fK( )= g( ), (29) whereK( )= # !$! ^ !, g( )= 5 ( ) 6 ^ !. (30) Ondifferentiating (29) ®rstwith respect to andthen with respect to andondividing by = "2 N,wehave 2( = # "0"-2 # NON) K !+ e( = ")2-(2 N)2 fK # !$!= g # !$!. Eliminating = # "0"-2 # NONfrom thisequation with theaidof(29), weobtaine( = ")2-(2 N)2f( K # !$!-2 K7K !)= g # !$!-2 K !g. (31) This relation holds inthefollowing cases:K # !$!-2 K7K !=0, g # !$!-2 K !g=0 (case 1), ( = ")2= ' =+ U,(2 N)2=- '2+ U- *, g # !$!-2 K !g=( ' + *)( K # !$!-2 K7K !) (case 2 h),(32) where ', U,and *arearbitrary constants. Weconsider both cases. Case 1.The®rsttwoequations in(32) enable onetodetermine K( )and g( ).Integrating the®rstequation once yieldsK != K2+const .Further ,thefollowing cases arepossible:K= C, (33 8)K=-1 ^( + *1), (33 9)K=- Ctanh( C + *1), (33 :)K=- Ccoth( C + *1), (33 3)K= Ctan( C + *1), (33+) where *1and Carearbitrary constants. Thesecond equation in(32) hastheparticular solution g= K( ).Hence, itsgeneral solution inexpressed by(e.g., see Polyanin andZaitse v1995)g= *2 K( )+ *3 K( )1 37K2( ), (34) where *2and *3arearbitrary constants. Thefunctions ( )and ( )arefound from (30) as( )= U1 1 i( ) 37+ U2, ( )= U1 g( )i( ),wherei( )=exp /1 K( ) 374, (35) and U1and U2arearbitrary constants ( isde®ned parametrically). Letusdwell onthecase (33b). According to(34),g= '1( + *1)2+ '2+ *1, (36) where '1=- *3 ^3and '2=- *2areanynumbers. Substituting (33b) and(36) into(35) yields= U1ln| + *1|+ U2, = '1 U1( + *1)+ '2 U1 ( + *1)2. Eliminating ,wearriveattheexplicit form oftheright-hand sideofequation (27):( )= '1 U1 + j+ '2 U1 +-2j,where k= - U2U1. (37) Forsimplicity ,weset *1=0, U1=1,and U2=0anddenote '1= 8and '2= 9.Thus, wehave( )=ln| |, ( )= 8+ @+ 9+-2 @, K( )=-1 ^, g( )= 8l2+ 9 ^. (38) *Incase 2,equation (31) canberepresented asthefunctional equation considered inParagraph B.3.5-1. Page698 B.3. M ETHODS OF FUNCTIONAL SEPARATION OF VARIABLES 699 TABLE B1 Nonlinear Klein±Gordon equations m "0"on - m NON n = ( n ) admitting functional separable solutions of the form n = n ( ),=2( )+ =( p). Notation: ', *1, and *2are arbitrary constants; q= 1for >0and q= -1 for <0 No Right-hand side ( n ) Solution n ( ) Equations for =( p) and2( r) 1 8 n ln n + 9 n+ ! ( = ")2= *1 +-2Z+ 87=-1 2 8+ 9+ ', (2 N)2= *2 +-2Y- 82+1 2 8+ ' 28+ @+ 9+-2 @ ln| |( = ")2= 2 87=3+ ' =2+ *1 =+ *2, (2 N)2= -2 823+ '22- *1 2+ *2+ 9 3 8sin n + 9 )sin n ln tan n 4+ 2sin n 4-4arctan+ ! ( = ")2= *1 +2Z+ *2 +-2Z+ 9d=+ 8+ ', (2 N)2= - *2 +2Y- *1 +-2Y- 92+ ' 4 8sinh n + 9 )sinh n ln tanh n 4+ 2sinh n 2-2ln s s s s coth 2 s s s s ( = ")2= *1 +2Z+ *2 +-2Z- q 9d=+ 8+ ', (2 N)2= *2 +2Y+ *1 +-2Y+ q 92+ ' 5 8sinh n + 2 9 )sinh n arctan+ @ .2+cosh n 2-2ln s s s s tan 2 s s s s ( = ")2= *1sin2 =+ *2cos2 =+ q 9d=+ 8+ ', (2 N)2= - *1sin22+ *2cos22- q 92+ ' It remains to determine =( p) and2( r). We substitute (38) into the functional differential equation (29). Taking into account (28), we ®nd [ = # "0"=-( = ")2- 87=3- 9]-[2 # NON2-(2 N)2+ 823]+( = # "0"- 3 87=2)2- =(2 # NON+ 3 822)= 0. ( 39) Differentiating (39) with respect to pand ryields the separable equation* ( = # # "0"0"- 6 87= = ")2 N-(2 # # NONON+ 6 82 2 N) = "= 0, whose solution is determined by the ordinary differential equations= # # "0"0"- 6 87= = "= ' = ",2 # # NONON+ 6 82 2 N= '2 N, where 'is the separation constant. Each equation can be integrated twice, thus resulting in ( = ")2= 2 87=3+ ' =2+ *1 =+ *2, (2 N)2= -2 823+ '22+ *3 2+ *4,(40) where *1, *2, *3, and *4are arbitrary constants. Eliminating the derivatives from (39) with the aid of (40), we ®nd that the arbitrary constants are related by *3= - *1and *4= *2+ 9. So, the functions =( p) and2( r) are determined by the ®rst-order nonlinear autonomous equations ( = ")2= 2 87=3+ ' =2+ *1 =+ *2, (2 N)2= -2 823+ '22- *1 2+ *2+ 9. The solutions of these equations are expressed in terms of elliptic functions. Fortheothercasesin(33),theanalysi sisperforme dinasimila rway.TableB1present sthe®nalresult sforthecases (33a)±(33e). Case 2 . Integrating the third and fourth equations in (32) yields== (X U p+W1,2= (X U- * p+W2 if '= 0;==1 4 '( ' p+W1)2- U',2= -1 4 '( ' r+W2)2+ U- *'if '¹ 0;(41) whereW1andW2are arbitrary constants. In both cases, the function ( n ) in equation (27) is arbitrary. The ®rst row in (41) corresponds to the traveling wave solution n = n ( C r+ t p). The second row leads to a solution of the form n = n ( r2- p2)._a` References : A. M. Grundland and E. Infeld (1992), J. Miller and L. A. Rubel (1993), R. Z. Zhdanov (1994), V . K. Andreev, O. V . Kaptsov, V . V . Pukhnachev, and A. A. Rodionov (1994). Example 10. The nonlinear stationary heat (diffusion) equationm2 nm r2+ m2 nm u2= v( n ) isanalyze djustasthenonlinea rKlein±Gordo nequatio nconsidere dinExampl e9.The®nalresult sarelistedinTableB2; the traveling wave solutions n = n ( C r+ t p) and solutions of the form n = n ( r2+ u2), existing for any v( n ), are omitted._a` References : A. M. Grundland and E. Infeld (1992), J. Miller and L. A. Rubel (1993), R. Z. Zhdanov (1994), V . K. Andreev, O. V . Kaptsov, V . V . Pukhnachev, and A. A. Rodionov (1994). * To solve equation (39), one can use the solution of equation (25) in Section B.2 [see (26a)]. Page 699 700 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES TABLE B2 Nonlinear equations m NON n + mlww n = v( n )admitting functional separable solutions oftheform n = n ( x),x=2( r)+ =( u).Notation: y, *1,and *2arearbitrary constants; q=1for x>0, q=-1for x<0 No Right-hand side z( n ) Solution n ( x) Equations for2( r)and =( u) 1 8 n ln n + 9 n+ {(2 | N)2= *1 +-2Y+ 82-1 2 8+ 9+ y, ( =| w)2= *2 +-2Z+ 87=-1 2 8- y 28+ @+ 9+-2 @ ln| x|(2 | N)2=2 823+ y22+ *1 2+ *2, ( =| w)2=2 87=3- y =2+ *1 =- *2- 9 3 8sin n + 9 )sin n lntan n 4+2sin n 4-4arctan+ {(2 | N)2= *1 +2Y+ *2 +-2Y+ 92+ 8+ y, ( =| w)2= *2 +2Z+ *1 +-2Z+ 9d=- y 4 8sinh n + 9 )sinh n lntanh n 4+2sinh n 2-2ln sssscoth x 2 ssss(2 | N)2= *1 +2Y+ *2 +-2Y- q 92+ 8+ y, ( =| w)2=- *2 +2Z- *1 +-2Z- q 9d=- y 5 8sinh n +2 9 )sinh n arctan+ @ .2+cosh n 2-2lnsssstan x 2 ssss(2 | N)2= *1sin22+ *2cos22+ q 92+ 8+ y, ( =| w)2= *1sin2 =- *2cos2 =+ q 9d=- y B.3.4. Splitting Method. Reduction toaFunctional Equation with TwoVariab les B.3.4-1. Splitting method. Reduction toastandard functional equation. Thegeneral procedure forconstructing functional separable solutions, which isbased onthesplitting method, involvesseveralstages outlined below. 1 }.Substitute expression (1)intothenonlinear partial differential equation under study .This results inafunctional differential equation with three arguments Ðthe®rsttwoareusual, Land ~,andthe third iscomposite, E. 2 }.Reduce thefunctional differential equation toapurely functional equation with three argumentsL, ~,and Ewith theaidofelementary differential substitutions (byselecting andrenaming terms with derivatives). 3 }.Reduce thethree-ar gument functional differential equation bythedifferentiation method to thestandard functional equation with twoarguments (either Lor ~iseliminated) considered in Section B.2. 4 }.Construct thesolution ofthetwo-argument functional equation using theformulas givenin Subsection B.2.3. 5 }.Solvethe(overdetermined) system formed bythesolution ofItem 4 }andthedifferential substitutions ofItem 2 }. 6 }.Substitute thesolution ofItem 5 }intotheoriginal functional differential equation ofItem 1 }to establish therelations fortheconstants ofintegration anddetermine allunkno wnquantities. 7 }.Consider alldegenerate cases possibly arising duetoviolation ofassumptions adopted inthe previous analysis. Thesplitting method reduces solving thethree-ar gument functional differential equation to(i)solv- ingapurely functional equation with three arguments (byreducing ittoastandard functional equation with twoarguments) and(ii)solving asystem ofordinary differential equations. Thus, the initial problem splits intoseveralsimpler problems. Examples ofconstructing functional separable solutions bythesplitting method aregiveninSubsection B.3.5. Page700 B.3. METHODS OFFUNCTION ALSEPARA TION OFVARIABLES 701 B.3.4-2. Three-ar gument functional equations ofspecial form. Thesubstitution ofexpression (1)with =2intononlinear partial differential equation often leads tofunctional differential equations oftheform 1( ) 1( , )+ 2( ) 2( , )+ + ( ) ( , ) + +1( , )+ +2( , )+ + ( , )=0,(42) where ( )and ( , )arefunctionals dependent onthevariables and , ,respecti vely, ( )º , , , ), ( , )º , , , , , , . (43) (These expressions correspond toasecond-order equation.) Itisreasonable tosolveequation (42) bythesplitting method. Tothisend, wetreat (42) at the®rst stage asapurely functional equation, thus disre garding (43). Assuming that 1 0,we divide (42) by 1anddifferentiate with respect to toobtain asimilar equation butwith fewer terms: 2( ) (2) 2( , )+ + ( ) (2) ( , )+ (2) +1( , )+ + (2) ( , )=0, (44) where (2)= 1 + 1 .Wecontinue thisprocedure until wearriveatan equation independent of explicitly:( +1) +1( , )+ + ( +1) ( , )=0, (45) where ( +1) = ( ) ( ) + ( ) ( ) . Relation (45) canberegarded asanequation with twoindependent variables and .If( +1)( , )= ( ) ( )forall = +1, , ,then equation (45) canbesolvedusing the results ofSection B.2. B.3.5. Some Functional Equations and Their Solutions. Exact Solutions ofHeat and WaveEquations Inthissubsection, wediscuss severaltypes ofthree-ar gument functional equations thatarise most frequently infunctional separation ofvariables innonlinear equations ofmathematical physics. The results areused toconstruct exact solutions forsome classes ofnonlinear heat andwaveequations. B.3.5-1. The functional equation ( )+ !( )= ( ),where = ( )+ ( ). Here, oneofthetwofunctions ( )and ( )isprescribed andtheother isassumed unkno wn,also oneofthefunctions !( )and ( )isprescribed andtheother isunkno wn,andthefunction ( )is assumed unkno wn.* Differentiating theequation with respect to and yields =0.Consequently ,thesolution isgivenby ( )= " ( )+ #, !( )= " ( )- #+ $, ( )= " + $, (46) where ", #,and $arearbitrary constants. B.3.5-2. The functional equation ( %)+ !( )+ &( ) ( )+ ( )=0,where = ( )+ ( %). Differentiating theequation with respect to yields thetwo-argument equation! + & + & + =0. (47) *Insimilar equations with acomposite argument, itisassumed that '( () )const and *( +) )const . Page701 702 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES Such equations were discussed inSection B.2. Hence, thefollowing relations hold [see formulas (25) and(26a) inSection B.2]:! = "1 & + "2 ,& = "3 & + "4 , =- "1- "3 , =- "2- "4 ,(48) where "1, "2, "3,and "4arearbitrary constants. Byintegrating system (48) andsubstituting the resulting solutions intotheoriginal functional equation, oneobtains theresults givenbelow. Case 1.If "3=0in(48), thecorresponding solution ofthefunctional equation isgivenby =-1 2 "1 "4 2+( "1 #1+ "2+ "4 #3) - #2- #1 #3- #4,!=1 2 "1 "4 2+( "1 #1+ "2) + #2,&= "4 + #1,=- "1 + #3,=1 2 "1 "4 2-( "2+ "4 #3) + #4,(49) where the " and # arearbitrary constants and = ( )and = ( %)arearbitrary functions. Case 2.If "3¹0in(48), thecorresponding solution ofthefunctional equation is =- #1 #3 ,- -3 .+ / "2- "1 "4"3 0 - #2- #4- "1 "4"2 3,!= "1 #1"3 , -3 1+ / "2- "1 "4"3 0 + #2,&= #1 , -3 1- "4"3,= #3 ,- -3 - "1"3,= "4 #3"3 ,- -3 + / "1 "4"3- "20 + #4,(50) where the " and # arearbitrary constants and = ( )and = ( %)arearbitrary functions. Case 3.Inaddition, thefunctional equation hasthetwodegenerate solutions: = "1 + #1, != "1 + #2, &= "2, =- "1 - "2 - #1- #2, (51a) where = ( ), = ( %),and = ( )arearbitrary functions, "1, "2, #1,and #2arearbitrary constants; and = "1 + #1, != "1 + "2 &+ #2, =- "2, =- "1 - #1- #2, (51b) where = ( ), = ( %),and &= &( )arearbitrary functions, "1, "2, #1,and #2arearbitrary constants. The degenerate solutions (51a) and(51b) canbeobtained directly from theoriginal equation oritsconsequence (47) using formulas (26b) inSection B.2. Example 11.Consider thenonstationary heat equation with anonlinear source2 32 4= 22 32(2+ 5( 3). (52) Welook forexact solutions oftheform3= 3( 6), 6= '( ()+ *( 4). (53) Page702 B.3. METHODS OFFUNCTION ALSEPARA TION OFVARIABLES 703 Substituting (53) into(52) anddividing by 3 78yields thefunctional differential equation* 79= ' 7:7;<;+( ' 7;)2 3 7:78=8378+ 5( 3( 6))378. Werewrite itasthefunctional equation B.3.5-2 inwhich>( 4)=- * 79, ?( ()= ' 7:7;<;, @( ()=( ' 7;)2, A( 6)= 3 7:78=8B 3 78, C( 6)= >( 3( 6)) B 3 78. (54) Wenowusethesolutions ofequation B.3.5-2. Onsubstituting theexpressions of ?and @of(54) into(49)±(51), we arriveatoverdetermined systems ofequations for '= '( (). Case 1.Thesystem' 7:7;<;=1 2 D1D4 '2+(D1 E1+D2) '+E2, ( ' 7;)2=D4 '+E1 following from (49) andcorresponding toD3=0in(48) isconsistent inthecases'= F1 (+ F2 forD2=-D1 F2 1,D4=E2=0,E1= F2 1,'=1 4 D4 (2+ F1 (+ F2forD1=D2=0,E1= F2 1-D4 F2,E2=1 2 D4,(55) where F1and F2arearbitrary constants. The ®rst solution in(55) withD1¹0leads toaright-hand side ofequation (52) containing theinverse oftheerror function [theform oftheright-hand sideisidenti®ed from thelasttworelations in(49)and(54)]. Thesecond solution in(55) corresponds totheright-hand side 5( 3)= G1 3ln 3+ G2 3in(52). Inboth cases, the®rstrelation in(49) is,taking into account that >=- * 79,a®rst-order linear solution with constant coef®cients, whose solution isanexponential plus aconstant. Case 2.Thesystem' 7:7;<;= D1 E1D3 HJI3 K+ LD2- D1D4D3 M '+E2, ( ' 7;)2=E1HJI3 K- D4D3, following from (50) andcorresponding toD3¹0in(48) isconsistent inthefollowing cases:'= N O-D4 BD3 (+ F1 forD2=D1D4 BD3,E1=E2=0,'=-2D3ln| (|+ F1 forD1=1 2 D2 3,D2=D4=E2=0,E1=4D-2 3H-I3 P1,'=-2D3ln QQcos R1 2 S D3D4 (+ F1 T QQ+ F2 forD1=1 2 D2 3,D2=1 2 D3D4,E2=0,D3D4>0,'=-2D3ln QQsinh R1 2 S-D3D4 (+ F1 T QQ+ F2forD1=1 2 D2 3,D2=1 2 D3D4,E2=0,D3D4<0,'=-2D3ln QQcosh R1 2 S-D3D4 (+ F1 T QQ+ F2forD1=1 2 D2 3,D2=1 2 D3D4,E2=0,D3D4<0, where F1and F2arearbitrary constants. The right-hand sides ofequation (52) corresponding tothese solutions are represented inparametric form. Case 3.Traveling wavesolutions ofthenonlinear heat equation (52) andsolutions ofthelinear equation (52) with5 7U=const correspond tothedegenerate solutions ofthefunctional equation (51). Example 12.Likewise, onecananalyze themore general equation2 32 4= V( () 22 32(2+ W( () 2 32(+ 5( 3). (56) Itarises inconvectiveheat/mass exchange problems ( V=const and W=const ),problems ofheat transfer ininhomogeneous media ( W= V 7;¹const ),andspatial heat transfer problems with axial orcentral symmetry ( V=const and W=const B(). Searching forexact solutions ofequation (56) intheform (53) leads tothefunctional equation B.3.5-2 inwhich>( 4)=- * 79, ?( ()= V( () ' 7:7;<;+ W( () ' 7( (), @( ()= V( ()( ' 7;)2, A( 6)= 3 7:78=8B 3 78, C( 6)= >( 3( 6)) B 3 78. Substituting these expressions into(49)±(51) yields asystem ofordinary differential equations fortheunkno wns. Example 13.Equation (52) alsoadmits more complicated functional separable solutions with theform3= 3( 6), 6= '( X)+ *( 4), X= (+ V 4. Substituting these expressions intoequation (52) yields thefunctional equation B.3.5-2 again, inwhich ( (must bereplaced by X)>( 4)=- * 79, ?( X)= ' 7:7YZY- V ' 7Y, @( X)=( ' 7Y)2, A( 6)= 3 7:78=8B 3 78, C( 6)= >( 3( 6)) B 3 78. Further ,oneshould followthesame procedure ofconstructing thesolution asinExample 11.[ \] ^ _J` aInExamples 11±13, different equations were allreduced tothesame functional equation. This demonstrates theutility ofisolation andindependent analysis ofindividual types offunctional equations, aswell astheexpedience ofdeveloping methods forsolving functional equations with acomposite argument. Page703 704 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES B.3.5-3. The functional equation ( %)+ !( ) ( )+ &( ) ( )=0,where = ( )+ ( %). Differentiating with respect to yields thetwo-argument functional differential equation! + ! + & + & =0, (57) which coincides with equation (25) inSection B.2, uptonotation. Nonde generatecase.Equation (57) canbesolvedusing formulas (49) inSection B.2, justas wasthecase forequation (25). Inthisway,wearriveatthesystem ofordinary differential equations! =( "1 !+ "2 &) ,& =( "3 !+ "4 &) , =- "1 - "3 , =- "2 - "4 ,(58) where "1, "2, "3,and "4arearbitrary constants. Thesolution ofequation (58) isgivenby!( )= "2 #1 , 1 1+ "2 #2 , 2 1,&( )=( 1- "1) #1 , 1 1+( 2- "1) #2 , 2 1,( )= "3 #3 ,- 1 + "3 #4 ,- 2 ,( )=( 1- "1) #3 ,- 1 +( 2- "1) #4 ,- 2 ,(59) where #1, #2, #3,and #4arearbitrary constants and 1and 2areroots ofthequadratic equation ( - "1)( - "4)- "2 "3=0. (60) Inthedegenerate case 1= 2theterms, 2 1and,- 2 in(59) must bereplaced by , 1 1and,- 1 ,respecti vely.Inthecase ofpurely imaginary orcomple xroots, oneshould extract thereal (orimaginary) partoftheroots insolution (59). Onsubstituting (59) intotheoriginal functional equation, oneobtains conditions thatmust be metbythefreecoef®cients andidenti®es thefunction ( %),speci®cally ,#2= #4=0 b c ( %)=[ "2 "3+( 1- "1)2] #1 #3 ,- 1 .,#1= #3=0 b c ( %)=[ "2 "3+( 2- "1)2] #2 #4 ,- 2 .,"1=0 b c ( %)=( "2 "3+ 2 1) #1 #3 ,- 1 .+( "2 "3+ 2 2) #2 #4 ,- 2 ..(61) Solution (59), (61) involvesarbitrary functions = ( )and = ( %). Degeneratecase.Inaddition, thefunctional equation hasthetwodegenerate solutions: = #1 #2 , -1 ., != "2 #1 ,- -1 1, &= #1 ,- -1 1, =- #2 , -1 - "2 , where = ( ), = ( %),and = ( )arearbitrary functions, "1, "2, #1,and #2arearbitrary constants; and = #1 #2 , -1 ., &=- #1 ,- -1 1- "2 !, = "2 #2 , -1 , = #2 , -1 , where = ( ), = ( %),and != !( )arearbitrary functions, "1, "2, #1,and #2arearbitrary constants. Thedegenerate solutions canbeobtained immediately from theoriginal equation orits consequence (57) using formulas (26b) inSection B.2. Example 14.Forthe®rst-order nonlinear equation2 32 4= 5( 3) L 2 32(M2 + d( (), thesearch forexact solutions intheform (53) leads tothefunctional equation B.3.5-3 inwhich>( 4)=- * 79, ?( ()=( ' 7;)2, @( ()= d( (), A( 6)= 5( 3) 3 78, C( 6)=1 B 3 78, 3= 3( 6). Page704 B.3. M ETHODS OF FUNCTIONAL SEPARATION OF VARIABLES 705 B.3.5-4. Equation 1( )+ 2( )+ !1( ) e( )+ !2( ) ( )+ ( )= 0, = ( )+ ( ). Differentiating with respect to and dividing the resulting relation by e and differentiating with respect to , one arrives at the functional equation with two arguments and that is discussed in Section B.2 [see equation (3) and its solution (22)]. Example 15. Consider the following equation of steady-state heat transfer in an anisotropic inhomogeneous medium with a nonlinear source:22( f V( () 2 32( g+ 22+ f W( +) 2 32+ g= 5( 3). ( 62) The search for exact solutions in the form 3= 3( 6), 6= '( ()+ *( +), leads to the functional equation B.3.5-4 in which> 1( ()= V( () ' 7:7;<;+ V 7;( () ' 7;, > 2( +)= W( +) * 7:7hh+ W 7h( +) * 7h, ?1( ()= V( ()( ' 7;)2, ?2( +)= W( +)( * 7h)2,i( 6)= A( 6)= 3 7:78=8B 3 78, C( 6)= - 5( 3) B 3 78, 3= 3( 6). Here we con®ne ourselves to studying functional separable solutions existing for arbitrary right-hand side 5( 3). With the change of variable 6= j2, we look for solutions of equation (62) in the form3= 3( j), j2= '( ()+ *( +). ( 63) Taking into account that k lk ;= K m n 2land k lk h= o m p 2l, we ®nd from (62)q ( V ' 7;) 7;+( WZ* 7h) 7hr 3 7 l 2 j+ qV( ' 7;)2+ W( * 7h)2r j 3 7:7 lsl - 3 7 l 4 j3= 5( 3), 5( 3)= 5 R 3( j)T. ( 64) For this functional differential equation to be solvable we require that the expressions in square brackets be functions of j: ( V ' 7;) 7;+( WZ* 7h) 7h= t( j), V( ' 7;)2+ W( * 7h)2= u( j). Differentiating the ®rst relation with respect to (and +yields the equation ( t 7l Bj) 7l= 0, whose general solution ist( j)= F1 j2+ F2. Likewise, we ®nd u( j)= F3 j2+ F4. Here, F1, F2, F3, and F4are arbitrary constants. As a result, we have ( V ' 7;) 7;+( WZ* 7h) 7h= F1( '+ *)+ F2, V( ' 7;)2+ W( * 7h)2= F3( '+ *)+ F4. The separation of variables results in a system of ordinary differential equations for '( (), V( (), *( +), and W( +): ( V ' 7;) 7;- F1 '- F2= G1, ( WZ* 7h) 7h- F1 *= - G1,V( ' 7;)2- F3 '- F4= G2, W( * 7h)2- F3 *= - G2. This system is always integrable in quadrature and can be rewritten as ( F3 '+ F4+ G2) ' 7:7;<;+( F1 '+ F2+ G1- F3)( ' 7;)2= 0, V=( F3 '+ F4+ G2)( ' 7;)-2; ( F3 *- G2) * 7:7hh+( F1 *- G1- F3)( * 7h)2= 0, W=( F3 *- G2)( * 7h)-2.(65) Here, the equations for 'and *do not involve Vand Wand, hence, can be solved independently. Without full analysis of system (65), we note a special case where the system can be solved in explicit form. For F1= F2= F4= G1= G2= 0and F3= F¹ 0, we ®ndV( ()= vHxw ;, W( +)= yH{z h, '( ()= FH-w ;v |2, *( +)= FH-z hy }2, where v, y, |, and }are arbitrary constants. Substituting these expressions into (64) and taking into account (63), we obtain the ordinary differential equation for 3( j)3 7:7 lsl -1j 3 7 l =4F 5( 3). Syste m(65)hasothersolution saswell;theseleadtovariou sexpression sof V( ()and W( +).TableB3liststhecaseswhere these functions can be written in explicit form (the traveling wave solution, which corresponds to V=const and W=const, is omitted). In general, the solution of system (64) enables one to represent V( () and W( +) in parametric form.~ Reference : V . F. Zaitsev and A. D. Polyanin (1996), A. D. Polyanin and A. I. Zhurov (1998). Page 705 706 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES TABLE B3 Functional separable solutions oftheform 3= 3( j), j2= ( )+ ( ),forheat equations inananisotropic inhomogeneous medium with anarbitrary nonlinear source. Notation: F, v, y, |, }, ,and Garefreeparameters ( F¹0, |¹0, }¹0, ¹2,and G¹2) Heat equation Functions ( )and ( ) Equation for 3= 3( j)22 v + y = ()= 2-v(2- )2, = 2-y(2- )2 :lsl+4- (2- )(2- )1j l=4 () v w + y z h = () = v |2 -w, = y }2 -z h :lsl-1j l=4 () v w + y = ()=v |2 -w, = 2-y(2- )2 :lsl+ 2- 1j l=4 () v 2 + y 2 = ()= |ln| |, = }ln| |Equation (64); both expressions insquare brack etsareconstantv 2 2+ y 2 = () = | , = }ln| |Equation (64); both expressions insquare brack etsareconstant B.4. First-Or derNonlinear Equations B.4.1. Preliminar yRemarks For®rst-order partial differential equations with twoindependent variables, anexact solution= ( , , 1, 2) (1) thatdepends ontwoarbitrary constants 1and 2iscalled acomplete integral. Thegeneral integral (general solution) canberepresented inparametric form byusing thecomplete integral(1)andthe twoequations2= ( 1),1+ 2 ( 1)=0,(2) where isanarbitrary function andtheprime stands forthederivative.Fordetails, seeKamk e (1965), Courant andHilbert (1989), andPolyanin, Zaitse v,andMoussiaux (2001). The®rst-order equations with twoindependent variables considered belowarepurely illustrati ve. The book byPolyanin, Zaitse v,andMoussiaux (2001) presents manymore ®rst-order nonlinear equations thatadmit generalized separable solutions (without specifying themethod forobtaining them). B.4.2. Individual Equations 1. ± ( ) = ()+ (). Exact solution:= ¡( ) ¢ £ ( )+ ¤( ), where¡( )= ¥( ) ¦§1- ¢ ¥( )£ ¨-1 , ¥( )=exp ¦©¢ ª( )£ ¨,¤( )= «( ) ¦¬2+ ¢ ( )«( ) £ ¨, «( )= ¥( )exp ¦©¢ ¡( )£ ¨. Page706 B.4. FIRST-ORDER NONLINEAR EQUATIONS 707 2. + ( ) = ®2+ ()+ (). Exact solution:= ¡( )+ ¤( )exp ¦¬¯ ¢ £ ( ) ¨, ¤( )= 1exp ° ¢ ±²¯ ¡( )+ ª( ) ³£ ´, where 1isanarbitrary constant andthefunction ¡( )isdetermined bytheRiccati equation¡ µ= ¯ ¡2+ ª( ) ¡+( ). This equation isintegrable inquadrature foralotofspeci®c functions ª( )and( )[e.g., for( )º0 andany ª( )].Fordetails, seethebooks byKamk e(1977) andPolyanin andZaitse v(1995). 3. = () ¶+ () 2¶+1. Exact solutions:= ¡( ) ·+1+1¸+1 ¢ ( )¡( ) £ + 1, ¡( )= ¹ ¦2¸+1 ¢ ª( )£ + 2 ¨1 º2 . 4. = () » ¼ ½+ () »2¼ ½. Exact solutions:= ¡( ) ¾ ¿ À+1 Á¢ ( )¡( ) £ + 1, ¡( )= ¹ ¦2 Á¢ ª( )£ + 2 ¨1 º2 . 5. + ® Â Ã2 = () + (). Exact solution: Ä = ¡( Å) Æ+ ¢ ±ª( Å)- ¯ ¡2( Å) ³£ Å+ 1, ¡( Å)= ¢ ( Å)£ Å+ 2. 6. + ® Â Ã2 = () 2+ () + (). Exact solution: Ä = ¡( Å) Æ2+ ¤( Å) Æ+ Ç( Å), where thefunctions ¡( Å), ¤( Å),and Ç( Å)aredetermined bysolving thefollowing system ofordinary differential equations:¡ µ=-4 ¯ ¡2+ ( Å), (1)¤ µ=-4 ¯ ¡ ¤+ ª( Å), (2)Ç µ=- ¯ ¤2+( Å). (3) TheRiccati equation (1)canbeintegrated inquadrature fornumerous ( Å).Fordetails, seeKamk e (1977) andPolyanin andZaitse v(1995). Givenasolution ofequation (1),equations (2)and(3)are easy tointegrate, because theyarelinear intheunkno wns ¤and Ç. Page707 708 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 7. + ® Â Ã2 = È2+ ()+ (). 1 É.Exact solution for Ê=0: Ä = Ë( Å)( 1+ 2 Æ)+ Ë( Å) ¢ ±Ìª( Å)- ¯ 2 2 Ë2( Å) ³ £ ÅË( Å), Ë( Å)=exp ¦©¢ ( Å)£ Å ¨. 2 É.Exact solution for Ê¹0: Ä = ¡( Å)+ ¤( Å)exp Í¬¹ Æ Î ÊÏ ¯ Ð, ¤( Å)= Ñ1exp ° Ò Ó2 Ô<Õ( Ö)+ ×( Ö) Ø Ù Ö Ú. Thefunction Õ= Õ( Ö)isdetermined bytheRiccati equationÕ Û= Ô<Õ2+ ×( Ö) Õ+ Ü( Ö). This equation canbeintegrated inquadrature forvarious ×and Ü,inparticular ,for Ü( Ö)º0and arbitrary ×( Ö)andfor ×( Ö)ºconst and Ü( Ö)ºconst .Fordetails, seethebooks byKamk e(1977) andPolyanin andZaitse v(1995). 8. Ý1( Þ) ß à áà Þ â2 + Ý2( ã) ß à áà ã â2 = ä1( Þ)+ ä2( ã). This equation isencountered indifferential geometry instudying geodesic lines ofLiouville surfaces. Exact solutions: å = æ Ò ç Ü1( Ö)+ è1×1( Ö) Ù Ö æ Ò ç Ü2( é)- è1×2( é) Ù é+ è2. Thesigns before each oftheintegrals canbechosen independently ofeach other .êë Refer ences :P.Appell (1953), E.Kamk e(1965). 9. à áà Þ+ Ý ß à áà ã â= ä( Þ) ã+ ì( Þ). Exact solution:å = Õ( Ö) é+ Ò Óîí( Ö)- × ÍÕ( Ö) ï ØÙ Ö+ è1, Õ( Ö)= Ò Ü( Ö) Ù Ö+ è2. 10. à áà Þ+ Ý ß à áà ã â= ä( Þ)á+ ì( Þ). Exact solution:å =( è1 é+ è2) Õ( Ö)+ Õ( Ö) Ò Óîí( Ö)- × Íè1 Õ( Ö) ï Ø Ù ÖÕ( Ö), Õ( Ö)=exp ð§Ò Ü( Ö) Ù Ö ñ. 11. ò1 ß Þ, à áà Þ â+ ó ô õ ò2 ß ã, à áà ã â=0. Exact solution: å = Õ( Ö)+ ö( é). Thefunctions Õ= Õ( Ö)and ö= ö( é)aredetermined bysolving theordinary differential equations÷- ø ùË1 ÍÖ, Õ Ûúï= è, ÷ø ûË2 Íé, ö Û üï=- è, where èisanarbitrary constant. Page708 B.5. SECOND -ORDER NONLINEAR EQUATIONS 709 B.5. Second-Or derNonlinear Equations B.5.1. Parabolic Equations B.5.1-1. Equations oftheform ý þýÿ= ý2þý ú2+ Ë( Ö, , å ). 1. à áà = à2áà Þ2+ álná+[ Ý( Þ)+ ä( )]á. Exact solution with multiplicati veform:å ( Ö, )=expð è ÷ ÿ+ ÷ ÿÒ ÷- ÿÜ( ) Ù ñ Õ( Ö), where èisanarbitrary constant andthefunction Õ( )isdetermined bysolving theordinary differential equation Õ ÛÛú ú+ Ô<Õln Õ+ ×( Ö) Õ=0. 2. à áà = à2áà Þ2+ Ý( )álná+ ä( )á. 1 .Exact solution: å ( Ö, )=exp Ó( ) Ö+ ( ) Ø, where thefunctions ( )and ( )aregivenby( )= ÷ , ( )= ÷ + ÷ ÷- ( 2 ÷2 + Ü) , = , and and arearbitrary constants. 2 .Exact solution: å ( , )=exp ( ) 2+ ö( ) , where ( )and ö( )aregivenby( )= ÷ -4 ÷ â-1 , ö( )= ÷ + ÷ ÷- (2 + ) , = , and and arearbitrary constants. 3 .There arealsoexact solutions ofthemore general form å ( , )=exp 2( ) 2+ 1( ) + 0( ) , where thefunctions 2( ), 1( ),and 0( )aredetermined byasystem ofordinary differential equations thatcanbeintegrated.êë Refer ence:V.F.Zaitse v,A.D.Polyanin (1996). 3. = 2 2+ ( )ln+[ ( )2+ ì( )+s( )]. Exact solution: ( , )=exp 2( ) 2+ 1( ) + 0( ) , where thefunctions ( )( =1,2,3)aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients: 2=4 2 2+ 2+ , 1=4 2 1+ 1+ í, 0= 0+ 2 1+2 2+ ! (thearguments of , , í,and !arenotspeci®ed andtheprime denotes thederivativewith respect to ). Page709 710 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 4. = 2 2+[ "ln2+ ( )ln+ ( )]. Thechange ofvariable =exp #leads toanequation oftheform B.5.1.14,$#$= $2#$2+ $#$ %2 + ¸#2+ ( ) #+ ( ), which hasexponential andsinusoidal solutions in . B.5.1-2. Equations oftheform ý þýÿ= ý2þý ú2+ &', , , ý þý ú ï. 5. = ( ( %+ ( )ln. Exact solution: ( , )=exp ( ) 2+ ö( ) , where thefunctions ( )and ö( )aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients (thearguments of and arenotspeci®ed): ÿ=4 2+ ,ö ÿ=2 ( +1) + ö. Integrating successi velyyields( )= ÷ -4 ÷ %-1 , ö( )= ÷+2 ( +1) ÷ ÷- , = , where and arearbitrary constants. 6. = 2 2+[ ( )+ ( )] + ì( )ln+[ )( )+s( )]. Exact solution: ( , )=exp ( )+ ö( ) , where thefunctions ( )and ö( )aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients: ÿ= ( )+ í( ) *+ +( ), (1)ö ÿ= í( ) ö+ 2+ ( ) + !( ). (2) Integrating ®rst(1)andthen (2),weobtain ( è1and è2arearbitrary constants)( )= è1 ,( )+,( ) +( ),( ) ,,( )=expð ( ) + í( ) ñ,ö( )= -2 .( )+.( ) 2( )+ ( ) ( )+ !( ).( ) ,.( )=exp / 0 ( 1) 132. 7. 4= 5 2 2+[ (4)+ (4)] + 6(4)ln+[2 7(4)+ )(4)+s(4)]. Exact solution: ( , 1)=exp 2( 1)+ 8( 1)+ 9( 1) , Page710 B.5. SECOND -ORDER NONLINEAR EQUATIONS 711 where thefunctions ( 1), 8( 1),and 9( 1)aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients: :=4 ; 2+(2 + 0 ) + <,8 :=(4 ; + + 0 ) 8+2 + +,9 := 09+2 ; + ; 82+ 8+ !. 8. 4= 5 2 2+ / (4)+ (4) 2 + 6(4)ln+[2)(4)+s(4)]. Exact solution: ( , 1)=exp ( 1) 2+ 8( 1) , where thefunctions ( 1)and 8( 1)aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients: :=4 ; 2+(2 + 0 ) + +,8 := 08+2( ;+ ) + !. 9. 4= 5 2 2+ = %2 + (4)2+ (4)+ 6(4). Exact solution: ( , 1)= ( 1) 2+ 8( 1) + 9( 1), where thefunctions ( 1), 8( 1),and 9( 1)aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients: :=4 >*2+ ,8 :=4 >* 8+ ,9 :=2 ; + >?82+ 0 . 10. 4= 5 2 2+ = %2 + @+ ()+ (4). Exact solution inadditi veform: ( , 1)= ( )+ A B C : + B C : DB-C :( 1) E 1. Here, Aisanarbitrary constant andthefunction F( G)isdetermined bythenonlinear ordinary differential equation; F HIH J J+ >( F HJ)2+ KLF+ M( G)=0. Bychanging variable F HJ= ;> 8H J8thisequation isreduced tothesecond-order linear equation;28HIH J J+ ; K*8H J+ >LM( G) 8=0. 11. N ON 4= 5 N2ON P2+ = Q N ON P R2 + S(P) N ON P+ TO+ U(P)+ V( W). Exact solution inadditi veform: X ( G, Y)= F( G)+ Z [ \ : + [ \ :D[-\ :^] ( Y) E Y, Page711 712 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES where Zisanarbitrary constant andthefunction F( G)isdetermined bythesecond-order ordinary differential equation with variable coef®cients; F HIH J J+ _( F HJ)2+ M( G) F HJ+ ¸F+ `( G)=0. 12. N ON W= a N2ON P2+ = Q N ON P R2 + bO N ON P+ TO2+ S( W)O+ U( W). Theequation hasexact solutions oftheform X ( G, Y)= F( Y)+ c( Y)exp( d G), where disarootofthequadratic equation _Ld2+ Ked+ ¸=0. 13. N ON W= a N2ON P2+ S( W) Q N ON P R2 + U( W)O+ V( W). Exact solution: X ( G, Y)= F( Y) G2+ c( Y) G+ f( Y), where thefunctions F( Y), c( Y),and f( Y)aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients:FH :=4 M F2+ ` F, (1)cH :=(4 M F+ `) c, (2)fH := ` f+2 ; F+ M c2+ ] . (3) Equation (1)for FisaBernoulli equation, which iseasy tointegrate. After that, equations (2) and(3),which arelinear in cand f,areintegrated successi vely.Asaresult, we®ndF= [ g Q h1-4 D[ g M E YR-1 , i= D` E Y,c= h2exp j D (4 M F+ `) E Y3k,f= h3 [g+ [g D[-g(2 ; F+ M c2+ ] ) E Y, where h1, h2,and h3arearbitrary constants. Adegenerate solution with Fº0corresponds tothe limit case h1 l m.npo Refer ence:V.F.Zaitse v,A.D.Polyanin (1996). 14. N ON W= a N2ON P2+ S( W) Q N ON P R2 + = S( W)O2+ U( W)O+ V( W). 1 q.Exact solution: X ( G, Y)= F( Y)+ c( Y)exp rts G u- _ v, _<0, (1) where thefunctions F( Y)and c( Y)aredetermined bysolving thefollowing ®rst-order ordinary differential equations with variable coef®cients (thearguments of M, `,and ] arenotspeci®ed):FH := _LM F2+ ` F+ ] , (2)cH :=(2 _LM F+ `- ; _) c. (3) Equation (2)for F( Y)isaRiccati equation; itcanbereduced toasecond-order linear equation. Manysolutions ofequation (2)forvarious M, `,and ] canbefound inKamk e(1977) andPolyanin andZaitse v(1995). Whene verasolution ofequation (2)isknown,thesolution ofequation (3)for c( Y)canbe evaluated fromc( Y)= Zexp j- ; _?Y+ D (2 _LM F+ `) E Y3k, where Zisanarbitrary constant. Page712 B.5. SECOND -ORDER NONLINEAR EQUATIONS 713 2q.Exact solution ofamore general form: X ( G, Y)= F( Y)+ c( Y) whexp r'G u- _ v+ xexp r- G u- _ vLy, _<0, where thefunctions F( Y)and c( Y)aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients:FH := _LM r'F2+4 h x c2)+ ` F+ ] , (4)c H :=2 _LM F c+ ` c- ; _?c. (5) One canexpress Finterms of cfrom (5)andthen substitute into(4).Asaresult, oneobtains a second-order nonlinear equation for c;if M, `, ] =const ,thisequation isautonomous and, hence, admits reduction oforder . 3q.Exact solution ( Kisanarbitrary constant): X ( G, Y)= F( Y)+ c( Y)cos r'Gu _+ Kev, _>0, (6) where thefunctions F( Y)and c( Y)aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients:FH := _LM r'F2+ c2)+ ` F+ ] , (7)cH :=2 _LM F c+ ` c- ; _?c. (8) One canexpress Finterms of cfrom (8)andthen substitute into(7).Asaresult, oneobtains a second-order nonlinear equation for c;if M, `, ] =const ,thisequation isautonomous and, hence, admits reduction oforder .npo Refer ence:V.F.Zaitse v,A.D.Polyanin (1996). 15. N ON W= a N2ON P2+ S( W) Q N ON P R2 + = S( W)O N ON P+ b S( W)O2+ U( W)O+ V( W). Theequation hasexact solutions oftheform X ( G, Y)= F( Y)+ c( Y)exp( d G), where disarootofthequadratic equation d2+ _Ld+ K=0. 16. N ON W= a N2ON P2+ S(P) Q N ON P R2 + U(P) N ON P+ =O+ V(P)+ z( W). Exact solution inadditi veform: X ( G, Y)= F( G)+ Z [ { : + [ { : |[-{ :~} ( Y) Y, where Zisanarbitrary constant andthefunction ( )isdetermined bythefollowing second-order ordinary differential equation with variable coef®cients:; I + ( )( )2+ `( ) + _*+ ] ( )=0. 17. W= a 2 2+ S( W) Q R2 +[ U1( W)+ U0( W)] + V( W)+ z( W)2+ ( W)+s( W). Exact solution: X ( , Y)= ( Y) 2+ c( Y) + f( Y), where thefunctions ( Y), c( Y),and f( Y)aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients: :=4 2+(2 `1+ ] ) + } , (1)c :=(4 + `1+ ] ) c+2 `0 + , (2)f = ]f+2 + c2+ `0 c+ . (3) Equation (1)for ( Y)isaRiccati equation; itcanbereduced toasecond-order linear equation. Forsolutions ofRiccati equations, seeKamk e(1977) andPolyanin andZaitse v(1995). Whene vera solution ofequation (1)isknown,thesolutions ofequations (2)and(3)canbeobtained successi vely (theequations arelinear in cand f). Page713 714 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 18. W= a 2 2+ S Q, R+ + U( W). Exact solution inadditi veform: X ( , Y)= ( )+ Z [ { + [ { |[-{ `( Y) Y, where Zisanarbitrary constant andthefunction ( )isdescribed bythesecond-order ordinary differential equation I + ( , )+ _*=0. B.5.1-3. Equations oftheform = ( , Y) 2 2+ ` r', Y, X , v. 19. W= S( W) Q R+ U( W)ln. Exact solution: X ( , Y)=exp w( Y) 2+ c( Y) y, where thefunctions ( Y)and c( Y)aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients (thearguments of and `arenotspeci®ed): =4 2+ ` ,c =2( +1) + ` c. Integrating successi velyyields( Y)= [g Q h-4 | [g YR-1 , c( Y)= x [g+2( +1) [g | [-g Y, i= |` Y, where hand xarearbitrary constants. 20. W= S( W) 2 2+ j U( W)+ V( W) k +s( W)ln+[2z( W)+ ( W)]. Exact solution: X ( , Y)=exp w( Y) 2+ c( Y) y, where thefunctions ( Y)and c( Y)aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients: =4 2+(2 `+ ) + } , (1)c = *c+2( + ] ) + . (2) TheRiccati equation (1)forthefunction ( Y)canbereduced toasecond-order linear equation. Forsolutions oftheRiccati equation, seeKamk e(1977) andPolyanin andZaitse v(1995). On solving (1),onecandetermine thesolution ofthelinear equation (2)for c( Y). 21. W= S( W) Q R+ U( W)ln+ V( W). Exact solution: X ( , Y)=exp wp( Y) [- + c( Y) y, where thefunctions ( Y)and c( Y)aredetermined bytheordinary differential equations = d2( Y) 2+ `( Y) ,c = `( Y) c+ ] ( Y). Page714 B.5. SECOND -ORDER NONLINEAR EQUATIONS 715 Integrating yields( Y)= i( Y) jh- d2 |( Y) i( Y) Y k-1 , i( Y)=exp j |`( Y) Y k,c( Y)= x i( Y)+ i( Y) | ] ( Y)i( Y) Y, where hand xarearbitrary constants. 22. W= jS() k+ aln. Exact solution: X ( , Y)=exp wh [ + ( ) y, where hisanarbitrary constant andthefunction ( )isdetermined bytheordinary differential equation ( ) + ( )2+ =0. 23. W= jS() k+ aln+[ U()+ V( W)]. Exact solution inmultiplicati veform:X ( , Y)=exp jtZ [ + [ |[- ] ( Y) Y3k ( ), where Zisanarbitrary constant andthefunction ( )isdetermined bytheordinary differential equation ( ) + ln + `( ) =0. 24. W= S() 2 2+ U() + aln+[ V()+s( W)]. Exact solution inmultiplicati veform:X ( , Y)=exp j'Z [ + [ |[- ( Y) Y3k ( ), where Zisanarbitrary constant andthefunction ( )isdetermined bytheordinary differential equation( ) I + `( ) + ln + ] ( ) =0. 25. W= S() 2 2+ U , + + V( W). Exact solution inadditi veform: ( , )= ( )+ + |- ( ) , where isanarbitrary constant andthefunction ( )isdetermined bythefollowing second-order ordinary differential equation:( ) I + ¡( , )+ =0. Page715 716 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES B.5.1-4. Equations oftheform = 2 2+ ¢t, , , £. 26. W= 2 2+ ¤()+ + ¥. Exact solution: ( , )=( ¦?+ §) + ¨ + ©-1 ª ««0( ¬- ) ®( ) ¯ , where ¨, ©,and ¬0arearbitrary constants. 27. ° ±° ²= ± °2±° ³2+ ¤(²)±+ ´(²). 1 µ.Exact solution: ( ¬, )= ¶( )( ¨ ¬+ ©)+ ¶( )ª ¡( )¶( ) ¯ , ¶( )=exp ·ª ®( ) ¯ 3¸, where ¨and ©arearbitrary constants. 2 µ.Exact solution: ( ¬, )= ¹( )( ¬2+ ¨ ¬+ ©)+ ¹( )ª ¡( )¹( ) ¯ ,¹( )= ¶( ) ·t-2 ºª ¶( ) ¯ 3¸-1 , ¶( )=exp ·ª ®( ) ¯ 3¸, where ¨, ©,and arearbitrary constants. 28. ° ±° ²= ± °2±° ³2+ ¤(³)± ° ±° ³+ ´(²)±+ »(²). Exact solution: ( ¬, )= ¹( ) ¼( ¬)+ ½( ), where thefunctions ¹( ), ½( ),and ¼( ¬)aredescribed byordinary differential equations¹ ¾ = ¹2+ ¡( ) ¹,½ ¾ = ¿ ¹+ ¡( ) À?½+ ( ),º ¼ ¾I¾« «+ ®( ¬) ¼ ¾«= , where isanarbitrary constant. Integrating successi velyyields¹( )= Á( ) ·Â¨1- ª Á( ) ¯ 3¸-1 , Á( )=exp ·ª ¡( ) ¯ 3¸,½( )= ¨2 ¹( )+ ¹( )ª ( )¹( ) ¯ ,¼( ¬)= ©1ª ¯ ¬¶( ¬)+ ©2+ º ª ·ª ¶( ¬) ¯ ¬ ¸ ¯ ¬¶( ¬), ¶( ¬)=exp ·1º ª ®( ¬) ¯ ¬ ¸, where ¨1, ¨2, ©1,and ©2arearbitrary constants. Page716 B.5. SECOND -ORDER NONLINEAR EQUATIONS 717 29. ° ±° ²= ± °2±° ³2+ ¤(²) Ã ° ±° ³ 2 + ´(²) ° ±° ³+ »(²)±+s(²). Exact solution: ( ¬, )= ¹( ) ¬2+ ½( ) ¬+ Ä( ), where thefunctions ¹( ), ½( ), Ä( )aredetermined bysolving thefollowing system of®rst-order ordinary differential equations with variable coef®cients (the arguments of ®, ¡, ,and Åarenot speci®ed):¹ ¾ =2(2 ®+ º) ¹2+ ¹, (1)½ ¾ =(4 ® ¹+2 º ¹+ ) ½+2 ¡ ¹, (2)Ä ¾ =(2 º ¹+ ) Ä+ ® ½2+ ¡ ½+ Å. (3) Equation (1)for ¹= ¹( )isaBernoulli equation; itiseasy tointegrate. After that, onecan successi velyconstruct thesolutions ofequations (2)and(3);each equation islinear intheunkno wn function.ÆpÇ Refer ence:V.F.Zaitse v,A.D.Polyanin (1996). 30. ° ±° ²= ± °2±° ³2+ È Ã ° ±° ³ 2 + ¥±2+ ¤(²)±+ ´(²). 1 µ.Exact solution: ( ¬, )= ¹( )+ ½( )exp( É Ê ¬), Ê= Ã- §º+ ¦1 Ë2 , (1) where thefunctions ¹( )and ½( )aredetermined bysolving thefollowing ®rst-order ordinary differential equations with variable coef®cients (thearguments of ®and ¡arenotspeci®ed):¹ ¾ = §L¹2+ ® ¹+ ¡, (2)½ ¾ =( º Ê2¹+2 §L¹+ ®) ½. (3) Equation (2)for ¹= ¹( )isaRiccati equation; itcanbereduced toasecond-order linear equation. Thebooks byKamk e(1977) andZaitse vandPolyanin (1995) present manysolutions of equation (2)forvarious ®and ¡. Givenasolution ofequation (2),thesolution ofequation (3)for ½= ½( )isevaluated by½( )= exp ·ª( º Ê2¹+2 §L¹+ ®) ¯ 3¸, (4) where isanarbitrary constant. 2 µ.Exact solution ( ¨isanarbitrary constant): ( ¬, )= ¹( )+ ½( )cosh( Ê ¬+ ¨), Ê= Ã- §º+ ¦1 Ë2 , (5) where thefunctions ¹( )and ½( )aredetermined bysolving thefollowing ®rst-order ordinary differential equations with variable coef®cients (thearguments of ®and ¡arenotspeci®ed):¹ ¾ = §L¹2- ¦LÊ2½2+ ® ¹+ ¡, (6)½ ¾ =( º Ê2¹+2 §L¹+ ®) ½. (7) Onecanexpress ¹from (7)interms of ½andsubstitute theresulting ¹into(6).Asaresult, one arrivesatasecond-order nonlinear equation for ½(if ®, ¡=const, thisequation isautonomous and, hence, admits reduction oforder). Page717 718 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 3 µ.Exact solution ( ¨isanarbitrary constant): ( ¬, )= ¹( )+ ½( )sinh( Ê ¬+ ¨), Ê= Ã- §º+ ¦1 Ë2 , where thefunctions ¹( )and ½( )aredetermined bysolving thesystem of®rst-order ordinary differential equations¹ ¾ = §L¹2+ ¦LÊ2½2+ ® ¹+ ¡,½ ¾ =( º Ê2¹+2 §L¹+ ®) ½. 4 µ.Exact solution ( ¨isanarbitrary constant): ( ¬, )= ¹( )+ ½( )cos( Ê ¬+ ¨), Ê= Ã §º+ ¦1 Ë2 , (8) where thefunctions ¹( )and ½( )aredetermined bysolving thesystem of®rst-order ordinary differential equations¹ ¾ = §L¹2+ ¦LÊ2½2+ ® ¹+ ¡, (9)½ ¾ =(- º Ê2¹+2 §L¹+ ®) ½. (10) One canexpress ¹from (10) interms of ½andsubstitute theresulting ¹into(9). Thus, one arrivesatasecond-order nonlinear equation for ½(if ®, ¡=const, thisequation isautonomous and, hence, admits reduction oforder).ÆpÇ Refer ence:V.F.Zaitse v,A.D.Polyanin (1996). 31. ° ±° ²= ± °2±° ³2+ ¤(²) Ã ° ±° ³2 +[ ´1(²)³+ ´0(³)] ° ±° ³+ »(²)±+ Ì2(²)³2+ Ì1(²)³+ Ì0(²). Theequation hasanexact solution oftheform ( ¬, )= ¹( ) ¬2+ ½( ) ¬+ Ä( ), where thefunctions ¹( ), ½( ),and Ä( )aredetermined byasystem of®rst-order ordinary differential equations with variable coef®cients (thesystem isnotspeci®ed here). B.5.1-5. Equations oftheform Í ÎÍ = º ÍÍ« ¿ ®( ) Í ÎÍ« À+ ® ¢'¬, , , Í ÎÍ« £. 32. ° ±° ²= °° ³ Ã± Ï ° ±° ³ + ¤(²)±1±Ï. Thechange ofvariable Ð= Ñ leads toanequation oftheform B.5.1.29,ÒÐÒ= º Ð Ò2ÐÒ¬2+ ºÓ Ã ÒÐÒ¬ Ô2 + Ó®( Õ), which admits solutions with theform Ð= ¹( Õ) ¬2+ ½( Õ) ¬+ Ä( Õ). 33. ° ±° ²= Ö °° ³ Ã± Ï ° ±° ³ Ô+ ×(²)±+ ´(²)±1±Ï. Thechange ofvariable Ð= Ø Ñ leads toanequation oftheform B.5.1.29,ÒÐÒÕ= º Ð Ò2ÐÒ¬2+ ºÓ Ã ÒÐÒ¬ Ô2 + Ó®( Õ) Ð+ Ó Ù( Õ), which admits solutions with theform Ð= ¹( Õ) ¬2+ ½( Õ) ¬+ Ä( Õ).ÆpÇ Refer ence:V.F.Zaitse v,A.D.Polyanin (1996). Page718 B.5. SECOND -ORDER NONLINEAR EQUATIONS 719 34. ° ±° ²= Ö °° ³ Ã± Ï ° ±° ³ Ô+ È±1+Ï+ ×(²)±+ ´(²)±1±Ï. For Ú=0,seeequation B.5.1.33. Thechange ofvariable Ð= Ø Ñ leads toanequation oftheform B.5.1.30,ÒÐÒÕ= º Ð Ò2ÐÒ¬2+ ºÓ Ã ÒÐÒ¬ Ô2 + Ú ÓÐ2+ Ó®( Õ) Ð+ Ó Ù( Õ), which admits solutions with theformsÐ( ¬, Õ)= ¹( Õ)+ ½( Õ)exp( É Ê ¬),Ð( ¬, Õ)= ¹( Õ)+ ½( Õ)cosh( Ê ¬+ Û),Ð( ¬, Õ)= ¹( Õ)+ ½( Õ)sinh( Ê ¬+ Û),Ð( ¬, Õ)= ¹( Õ)+ ½( Õ)cos( Ê ¬+ Û), where thefunctions ¹( Õ)and ½( Õ)aredetermined byasystem of®rst-order ordinary differential equations; theparameter Êisarootofaquadratic equation and Ûisanarbitrary constant.ÆpÇ Refer ence:V.F.Zaitse v,A.D.Polyanin (1996). 35. ° ±° ²= Ö °° ³ Ã Ü Ý Þ ° ±° ³ Ô+ ×(²)+ ´(²) Ü±Ý Þ. Thechange ofvariable Ð= ß àÎleads toanequation oftheform B.5.1.27,ÒÐÒÕ= º Ð Ò2ÐÒ¬2+ Ê ®( Õ) Ð+ Ê Ù( Õ), which admits solutions with theform Ð= ¹( Õ) ¬2+ ½( Õ) ¬+ Ä( Õ). 36. ° ±° ²= Ö °° ³ Ã ÜÝ Þ ° ±° ³ Ô+ ×(³)+( È³+ á) Ü±Ý Þ. Thechange ofvariable Ð= ß àÎleads toanequation oftheform B.5.1.26,ÒÐÒÕ= º Ð Ò2ÐÒ¬2+ Ê ®( ¬) Ð+ Ê( Ú?¬+ §), which admits solutions with theform Ð= Ê( Ú?¬+ §) Õ+ ¹( ¬). 37. ° ±° ²= Ö °° ³ Ã Ü Ý Þ ° ±° ³ Ô+ È Ü Ý Þ+ ×(²)+ ´(²) Ü±Ý Þ. For Ú=0,seeequation B.5.1.35. Thechange ofvariable Ð= ß àÎleads toanequation oftheform B.5.1.30,ÒÐÒÕ= º Ð Ò2ÐÒ¬2+ Ú?Ð2+ Ê ®( Õ) Ð+ Ê Ù( Õ), which admits solutions with theformsÐ( ¬, Õ)= ¹( Õ)+ ½( Õ)exp( É â ¬),Ð( ¬, Õ)= ¹( Õ)+ ½( Õ)cosh( â ¬+ Û),Ð( ¬, Õ)= ¹( Õ)+ ½( Õ)sinh( â ¬+ Û),Ð( ¬, Õ)= ¹( Õ)+ ½( Õ)cos( â ¬+ Û), where thefunctions ¹( Õ)and ½( Õ)aredetermined byasystem of®rst-order ordinary differential equations; theparameter âisarootofaquadratic equation and Ûisanarbitrary constant.ÆpÇ Refer ence:V.F.Zaitse v,A.D.Polyanin (1996). Page719 720 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES B.5.1-6. Equations oftheform Í ÎÍ = ®( ¬, Õ, Ø) Í2ÎÍ«2+ Ù ã¬, Õ, Ø, Í ÎÍ« ä. 38. ° ±° ²= ×(³)Ö±+ È °2±° ³2. Exact solution:Ø( ¬, Õ)=1º ¿ ¹( ¬) Õ+ ½( ¬)- ÚÀ, where thefunctions ¹( ¬)and ½( ¬)aredetermined bytheordinary differential equations®( ¬) ¹ ¾I¾« «- ¹2=0, ®( ¬) ½ ¾I¾« «- ¹ ½=0. The ®rst equation canbetreated independently .The second equation hasaparticular solution½( ¬)= ¹( ¬),andhence, itsgeneral solution isgivenby½( ¬)= Û1 ¹( ¬)+ Û2 ¹( ¬)ª ¯ ¬¹2( ¬), where Û1and Û2arearbitrary constants. 39. ° ±° ²= ×(²) °° ³ Ã± Ï ° ±° ³ Ô+ ´(²)±1±Ï. Exact solution:Ø( ¬, Õ)=¿ ¹( Õ) ¬2+ ½( Õ)À1 Ë Ñ , where thefunctions ¹= ¹( ¬)and ½= ½( ¬)aredetermined bythe®rst-order ordinary differential equations¹ ¾ å=2( Ó+2)Ó ® ¹2, ½ ¾ å=2 ® ¹ ½+ Ó Ù. Integrating yields¹=1¶, ½= ¶- ÑÑ +2Ã æ+ Óª Ù¶ ÑÑ +2¯ ÕÔ, ¶= ç-2( Ó+2)Óª ® ¯ Õ, where æand çarearbitrary constants.ÆpÇ Refer ence:V.F.Zaitse v,A.D.Polyanin (1996). 40. ° ±° ²= °° ³ ·×(³) Ü èÞ ° ±° ³ ¸. Exact solution inadditi veform:Ø( ¬, Õ)=-1éln( éÕ+ Û)+1éln ·ê æ- é ë®( ë) ì ë+ ç í, where æ, ç,and Ûarearbitrary constants. B.5.1-7. Equations with three independent variables. 41. î ïî ð= îî ñ ò ×(ñ, ó) î ïî ñ í+ îî óòô(ñ, ó) î ïî ó í+ õïlnï. Exact solution inmultiplicati veform:Ø( ë, ö, Õ)=exp ãæ ß ÷ å ) ¼( ë, ö), where æisanarbitrary constant andthefunction ¼( ë, ö)satis®es thestationary equationøøëò ®( ë, ö) ø¼øëí+ øøöò Ù( ë, ö) ø¼øö í+ ù ¼ln ¼=0. Page720 B.5. SECOND -ORDER NONLINEAR EQUATIONS 721 42. = ( , ) + ( , ) + ( ) ln . Incomplete separable exact solution (thesolution isseparable inthespace coordinates and but notintime ): ( , , )= ( , ) ( , ). The functions ( , )and ( , )aredetermined from theone-dimensional nonlinear parabolic differential equations = ( , ) + ( ) ln + ( ) ,= ( , ) + ( ) ln - ( ) , where ( )isanarbitrary function. B.5.2. Hyperbolic Equations B.5.2-1. Equations oftheform 2 2= 2 2+ , , , . 1. 2 2= 2 2+ ! ln +[ ( )+ ( )] . Exact solution inmultiplicati veform: ( , )= ( ) ( ), where thefunctions ( )and ( )aredetermined bythesecond-order ordinary differential equations "#"$- %'&ln + ( )+ (=0, "#" + %'&ln + ( )- ()=0, where isanarbitrary constant. 2. 2 2= 2 2+ ! * +2 + , + ( ). Exact solution inadditi veform: ( , )= ( )+ -( .), .= + / , where /isanarbitrary constant andthefunctions = ( )and -= -( .)aredetermined bysolving thesecond-order ordinary differential equations "#"$- 01- ( )=0, (1) ( - /2) - "#" 232+ & - " 22+ 0)-=0. (2) Thegeneral solution ofequation (1)isgivenby( )= 1cosh( 4 )+ 2sinh( 4 )+14 5 0 ( 6)sinh[ 4( - 6)] 7 6if 0= 42>0,( )= 1cos( 4 )+ 2sin( 4 )+14 5 0 ( 6)sin[ 4( - 6)] 7 6 if 0=- 42<0, where 1and 2arearbitrary constants. Equation (2)canbesolvedwith thechange ofvariable 8( -)= -" 22,which leads toa®rst-order linear equation. Page721 722 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 3. 2 2= 2 2+ ! * +2 + , + ( ). Exact solution inadditi veform: ( , )= ( )+ ( ). Here,( )= 1cosh( 4 )+ 2sinh( 4 )if 0= 42>0,( )= 1cos( 4 )+ 2sin( 4 ) if 0=- 42<0, where 1and 2arearbitrary constants, andthefunction ( )isdetermined bytheordinary differential equation "#" + & "2+ 01+ ( )=0. 4. 2 2= 2 2+ ! * +2 + , + 9 2+ ( ) + ( ). Exact solution: ( , )= ( )+ ( )exp( / ), where /isaroot ofthequadratic equation &1/2+ 0/+ 4=0andthefunctions ( )and ( )are determined bythefollowing system ofsecond-order ordinary differential equations: "#"$= 4 2+ ( ) + ( ), (1) "#"$= %( 0/+2 4) + ( )+ /2(). (2) Inthespecial case ( )=const and ( )=const ,equation (1)isautonomous andhasparticular solutions oftheform =const and, hence, canbeintegrated inquadrature. Equation (2)islinear in ,andconsequently ,with =const ,itsgeneral solution isexpressed interms ofexponentials or sineandcosine. 5. 2 2= 2 2+ ( ) * +2 + ( ) + ( ). Exact solution: ( , )= ( ) 2+ ( ) + :( ), (1) where thefunctions ( ), ( ),and :( )aredetermined bysolving thefollowing system ofsecond- order ordinary differential equations with variable coef®cients (thearguments of , ,and arenot speci®ed): "#"$=4 2+ , (2) "#"$=(4 + ) , (3): "#"$= :+ 2+ +2 . (4) Equation (2)hasthetrivialparticular solution ( )º0;thecorresponding solution (1)islinear inthecoordinate . Equation (3)hasaparticular solution =Å ( ),where Å ( )isanynontri vialparticular solution ofequation (2).Hence, thegeneral solution ofequation (3)isgivenby( )= 1Å ( )+ 2Å ( )5 7 Å 2( ), where 1and 2arearbitrary constants. Ifthefunctions and proportional, then =-1 4 ; ( =const )isaparticular solution ofequation (2). Equation (4)linear in := :( ). Page722 B.5. SECOND -ORDER NONLINEAR EQUATIONS 723 6. 2 2= 2 2+ ( ) * +2 + ( )+ ( ). Exact solution inadditi veform: ( , )=1 2 < 2+ = + +5 0( - 6) ( 6) 7 6+ ( ). Here,<, =,and arearbitrary constants, andthefunction ( )isdetermined bysolving the second-order nonlinear ordinary differential equation "#" + ( ) "2+ ( )-<=0. 7. 2 2= 2 2+ ( ) * +2 + ! + ( )+ ( ). Exact solution inadditi veform: ( , )= ( )+ ( ). Here, thefunctions ( )and ( )aredetermined bysolving thesecond-order ordinary differential equations "#"$- &)- ( )=0, "#" + ( )( ")2+ &>+ ( )=0. Thegeneral solution ofthe®rstequation isgivenby( )= 1cosh( 4 )+ 2sinh( 4 )+14 5 0 ( 6)sinh[ 4( - 6)] 7 6if &= 42>0,( )= 1cos( 4 )+ 2sin( 4 )+14 5 0 ( 6)sin[ 4( - 6)] 7 6 if &=- 42<0, where 1and 2arearbitrary constants. 8. 2 2= 2 2+ ( ) * +2 + ! ( ) 2+ ( ) + ( ). 1 ?.Exact solution: ( , )= ( )+ ( )exp @ A- &, &<0, where thefunctions ( )and ( )aredetermined bysolving thefollowing second-order ordinary differential equations with variable coef®cients (thearguments of , ,and arenotspeci®ed): "#"$= & 2+ + , "#"$=(2 & + - &) . 2 ?.Exact solution ofamore general form: ( , )= ( )+ ( ) %<exp A- &+ =exp - A- & (, &<0, where<and =arearbitrary constants andthefunctions ( )and ( )aredeterm inedbythefollowing system ofsecond-order ordinary differential equations with variable coef®cients (thearguments of , ,and arenotspeci®ed): "#"$= & ( 2+4< = 2)+ + , "#"$=(2 & + - &) . 3 ?.Exact solution: ( , )= ( )+ ( )cos A &+ , &>0, (9) Page723 724 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES where isanarbitrary constant andthefunctions ( )and ( )aredetermined bythefollowing system ofsecond-order ordinary differential equations with variable coef®cients: "#"$= & 2+ 2)+ + , "#"$= 2 & + - & . 9. 2 2= 2 2+ * , ++ ( ). Exact solution inadditi veform: ( , )=1 2 < 2+ = + +5 0( - 6) ( 6) 7 6+ ( ). Here,<, =,and arearbitrary constants andthefunction ( )isdetermined bysolving the second-order nonlinear ordinary differential equation "#" + , "-<=0. 10. 2 2= 2 2+ * , ++ ! + ( ). Exact solution inadditi veform: ( , )= ( )+ ( ). Here, thefunctions ( )and ( )aredetermined bythesecond-order ordinary differential equations "#"$- &)- ( )=0, "#" + , "+ &>=0. Thegeneral solution ofthe®rstequation isgivenby( )= 1cosh( 4 )+ 2sinh( 4 )+14 5 0 ( 6)sinh[ 4( - 6)] 7 6if &= 42>0,( )= 1cos( 4 )+ 2sin( 4 )+14 5 0 ( 6)sin[ 4( - 6)] 7 6 if &=- 42<0, where 1and 2arearbitrary constants. 11. 2 2= 2 2+ * ,1 +. Exact solution inmultiplicati veform: ( , )= B C ( ), where /isanarbitrary constant andthefunction ( )isdetermined bythesecond-order linear ordinary differential equation "#"$= %D /2+ ( , /) (1. Page724 B.5. SECOND -ORDER NONLINEAR EQUATIONS 725 B.5.2-2. Equations oftheform 2 2= ( ) 2 2+ , , , . 12. 2 2= ( + E) F 2 2+ ( ), >0. Exact solution for G¹2: = ( 8), 8= %1 4 (2- G)2( + )2-( + H)2- I( I 2(2- I), where isanarbitrary constant andthefunction = ( 8)isdetermined bythegeneralized Emden±F owler equation"#" J>J-4 G2 84(1- I)I ( )=0. (1) Anumber ofexactsolutions toequation (1)forsome speci®c = ( )canbefound inPolyanin andZaitse v(1995). Inthespecial case G=1,thegeneral solution ofequation (1)isgivenby5 1+8 K( )-1 L27 = @8+ 2,K( )=5 ( ) 7 , where 1and 2arearbitrary constants. 13. 2 2= F 2 2+ ! F±1 + ( ), >0. Exact solution for G¹2: = ( .), .=1 4 (2- G)2( + )2- 2- I, where isanarbitrary constant andthefunction = ( .)isdetermined bytheordinary differential equation. "#" 232+< " 2- = ( )=0,where<= (4-3 G)+2 & 2 (2- G), ==1(2- G)2. For<¹1,thechange ofvariable .= 4 81 1- M( 4= @ 1)brings thisequation tothegeneralized Emden±F owler equation"#" J>J- 4 = (1-<)2 82 M-1 1- M ( )=0, whose solvable cases arepresented inPolyanin andZaitse v(1995). 14. 2 2= F 2 2+ F±1 ( ) . Exact solution for G¹2: = ( 8), 8= % 4 (2- G)2( + )2-4 4 2- I]1 L2, 4= @ 1, where isanarbitrary constant andthefunction = ( 8)isdetermined bytheordinary differential equation"#" J>J+2(2- G) %D(1- G)+ ( ) (18 " J=0. Thechange ofvariable N( )= 8 " Jleads toa®rst-order separable equation. Integrating thisequation yields thegeneral solution inimplicit form:5 7 G -2K( )+ 1=1(2- G)ln| 8|+ 2,K( )=5 ( ) 7 , where 1and 2arearbitrary constants. Page725 726 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 15. 2 2= F 2 2+ F±1 ( ) + ( ). Exact solution for G¹2: = ( 8), 8= %4 (2- G)2( + )2-4 4 2- I(1 L2, 4= @ 1, where isanarbitrary constant andthefunction = ( 8)isdetermined bytheordinary differential equation"#" J>J+2(2- G) %(1- G)+ ( ) (18 " J-1 4(2- G)2 ( )=0. 16. 2 2= O P Q 2 2+ ! O P Q + ( ), >0. Exact solution for /¹0: = ( 8), 8= %4 4 B-C - 4 /2( + )2(1 L2, 4= @ 1, where isanarbitrary constant andthefunction = ( 8)isdetermined bytheordinary differential equation"#" J>J+2( /- &) /18 " J+1 4 /2 ( )=0. (1) For &= /,thesolution ofequation (1)isgivenby5 1-2 4 /2K( )-1 L27 = @8+ 2,K( )=5 ( ) 7 , where 1and 2arearbitrary constants. For &¹1 2 /,thechange ofvariable .= 82 R- SCSCbrings (1)tothegeneralized Emden±F owler equation"#" 232+ 4(2 &- /)2 .4( SC- R) 2 R- SC ( )=0, whose solvable cases arepresented inPolyanin andZaitse v(1995). 17. 2 2= O P Q 2 2+ O P Q ( ) . Exact solution for /¹0: = ( 8), 8= %4 4 B-C - 4 /2( +<)2(1 L2, 4= @ 1. Here,<isanarbitrary constant andthefunction = ( 8)isdetermined bytheordinary differential equation"#" J>J+28 1-1 / ( ) " J=0, which bythechange ofvariable N( )= 8 " Jisreduced toaseparable ®rst-order equation. Integrating thisequation yields thegeneral solution inimplicit form:5 7 2K( )- / + 1=1 /ln| 8|+ 2,K( )=5 ( ) 7 , where 1and 2arearbitrary constants. 18. 2 2= O P Q 2 2+ O P Q ( ) + ( ). Exact solution for /¹0: = ( 8), 8= %4 4 B-C - 4 /2( + )2(1 L2, 4= @ 1, where isanarbitrary constant andthefunction = ( 8)isdetermined bytheordinary differential equation"#" J>J+28 1-1 / ( ) " J+1 4 /2 ( )=0. Page726 B.5. SECOND -ORDER NONLINEAR EQUATIONS 727 B.5.2-3. Other equations. 19. 2 2= ( ) * +. 1 ?.Exact solutions: ( , )=( 1 + 2)( 3 + 4)1 L2, ( , )=( 1 + 2) +5 S( - 6)( 1 6+ 2)2 ( 6) 7 6+ 3 + 4, where 1, 2, 3, 4,and arearbitrary constants. 2 ?.Exact solution: ( , )= ( ) 2+ ( ) + :( ), where thefunctions = ( ), = ( ),and := :( )aredetermined bythesystem ofordinary differential equations "#"$=6 ( ) 2, "#"$=6 ( ) ,: "#"$=2 ( ) :+ ( ) 2. 3 ?.Exact solution inmultiplicati veform: ( , )= T( ) U( ), where thefunctions T= T( )and U= U( )aredetermined bytheordinary differential equations ( isanarbitrary constant)T "#"$= ( ) T2, ( U U ") "= U. Thelatter equation isautonomous andhasaparticular solution U=1 6 2and, hence, isintegrable inquadrature. 20. 2 2= ( ) * ++ 2( ) 2+ 1( ) + 0( ). Exact solution: ( , )= ( ) 2+ ( ) + :( ), where thefunctions = ( ), = ( ),and := :( )aredetermined bythesystem ofordinary differential equations "#"$=6 ( ) 2+ 2( ), "#"$=6 ( ) + 1( ),: "#"$=2 ( ) :+ ( ) 2+ 0( ). B.5.3. Elliptic Equations B.5.3-1. Equations oftheform 2 2+ & 2 V2= , , , , V , &>0. 1. 2 2+ 2 2= ln +[ ( )+ ( )] . Exact solution inmultiplicati veform: ( , )= ( ) ( ), where thefunctions ( )and ( )aredetermined bytheordinary differential equations "#" - %Dln + ( )+ (=0, "#"V1V- %Dln + ( )- (1=0, where isanarbitrary constant. Page727 728 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 2. 2 2+ 2 2= ( ) ln +[ ( ) + ( )] . Exact solution inmultiplicati veform: ( , )= B- SV( ), where thefunction ( )isdetermined bytheordinary differential equation "#" = ( ) ln + % ( )- 2(. 3. 2 2+ 2 2= ( ) * +2 + ( ) + ( ). Exact solution: ( , )= ( ) 2+ ( ) + :( ). (1) Here, thefunctions ( ), ( ),and :( )aredetermined bythefollowing second-order ordinary differential equations with variable coef®cients (thearguments of , ,and arenotspeci®ed): "#" =4 2+ , (2) "#" =(4 + ) , (3): "#" = :+ 2+ -2 . (4) Ifasolution = ( )ofthenonlinear equation (2)isfound, then thefunctions = ( )and:= :( )canbedetermined successi velyfrom equations (3)and(4),which arelinear in and :. Bycomparing equations (2)and(3),onecanseethatequation (3)hasaparticular solution= ( ).Hence, thegeneral solution of(3)isgivenby(seePolyanin andZaitse v1995)( )= 1 ( )+ 2 ( )5 7 2( ), W0. Note thatequation (2)hasthetrivialparticu larsolutio n ( )º0,towhich there isacorrespo nding solution (1)linear inthecoordinate .Ifthefunctions and areproportional, then =-1 4 ; ( =const )isaparticular solution ofequation (2). 4. 2 2+ 2 2= ( ) * +2 + ! ( ) 2+ ( ) + ( ). 1 ?.Exact solution: ( , )= ( )+ ( )exp @ A- &, &<0, where thefunctions ( )and ( )aredetermin edbythefollowing secon d-orderordinarydifferential equations with variable coef®cients (thearguments of , ,and arenotspeci®ed): "#" = & 2+ + , "#" =(2 & + + &) . 2 ?.Exact solution ofamore general form: ( , )= ( )+ ( ) %<exp A- &+ =exp - A- & (, &<0, where thefunctions ( )and ( )aredetermined bythefollowing system ofsecond-order ordinary differential equations with variable coef®cients: "#" = & 2+4< = 2)+ + , "#" =2 & + + &>. 3 ?.Exact solution: ( , )= ( )+ ( )cos A &+ , &>0, where isanarbitrary constant andthefunctions ( )and ( )aredetermined bythefollowing system ofsecond-order ordinary differential equations with variable coef®cients: "#" = & 2+ 2)+ + , "#" =2 & + + &>.X'Y Refer ence:V.F.Zaitse v,A.D.Polyanin (1996). Page728 B.5. SECOND -ORDER NONLINEAR EQUATIONS 729 5. 2 2+ ! 2 2= 1 * , ++ 2 * , ++ 9 . Exact solution inadditi veform: ( , )= ( )+ ( ). Here, thefunctions ( )and ( )aredetermined bysolving thesecond-order ordinary differential equations "#" - 1 , "- 4 = ,&> "#"V1V- 2 , "V- 4 =- , where isanarbitrary constant. B.5.3-2. Equations oftheform % ( ) (+ V % ( ) V (= ( ). 6. * F ++ * ! Z += ( ). 1 ?.For G¹2and [¹2,there areexact solutions oftheform = ( .), .= %'&(2- [)22- I+ (2- G)2 2- \(1 L2. Here, thefunction = ( .)isdetermined bytheordinary differential equation"#" 232+ <. " 2= = ( ), (1) where<=4- G [ (2- G)(2- [), ==4 &(2- G)2(2- [)2. For [=4 ;G,oneobtains from (1)thefollowing exact solution totheoriginal equation with arbitrary = ( ):5 1+2 G2 &(2- G)4K( )-1 L27 = 2 @.,K( )=5 ( ) 7 , where 1and 2arearbitrary constants. 2 ?.Thechange ofvariable ]= .1- Mbrings (1)tothegeneralized Emden±F owler equation"#" ^>^= = (1-<)2 ]2 M 1- M ( ). (2) Alotofexact solutions toequation (2)with various = ( )canbefound inPolyanin andZaitse v (1995).X'Y Refer ence:V.F.Zaitse vandA.D.Polyanin (1996). 7. * O _ Q ++ * ! O ` a += ( ). For H b¹0,there areexact solutions oftheform = ( .), .= &>b2B- c+ H2B- dV1 L2, where thefunction = ( .)isdetermined bytheordinary differential equation"#" 232-1. " 2=< ( ),<=4 &>H2b2. (1) Thechange ofvariable ]= .2brings (1)tothegeneralized Emden±F owler equation"#" ^>^=1 4 < ]-1 ( ), whose solutions with ( )=( 4 + e)-1and ( )=( 4 + e)-2( 4, e=const )canbefound inPolyanin andZaitse v(1995).X'Y Refer ence:V.F.Zaitse vandA.D.Polyanin (1996). Page729 730 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 8. * F ++ * ! O ` a += ( ). For G¹2and b¹0,there areexact solutions oftheform = ( .), .= % &>b22- I+ (2- G)2B- dV(1 L2, where thefunction = ( .)isdetermined bytheordinary differential equation"#" 232+ GG-21. " 2=4 &>b2(2- G)2 ( ). 9. ( ) + ( ) = ln +[ 1( )+ 2( )] . Exact solution inmultiplicati veform: ( , )= ( ) ( ), where = ( )and = ( )aredetermined bytheordinary differential equations [ ( ) "] "=[ ln + 1( )+ ] , [ ( ) "V] "V=[ ln + 2( )- ] , where isanarbitrary constant. B.5.3-3. Other equations with twoindependent variables. 10. 2 2+ f gih( j) k+ l( j) m n kn o p=0. 1 q.Exact solution: r ( s, t)=( u s+ v) t- w xx0( s- y)( u y+ v)2 z( y) { y+ |1 s+ |2, where u, v, |1, |2,and s0arearbitrary constants. 2 q.Exact solution: r ( s, t)= }( s) t2+ ~( s) t+ ( s), where thefunctions }= }( s), ~= ~( s),and = ( s)aredetermined bytheordinary differential equations} #x x+6 z}2=0, (1)~ #x x+6 z} ~=0, (2) #x x+2 z} +2 } + z~2=0. (3) The nonlinear equation (1)canbetreated independently .For zºconst ,itssolution canbe expressed interms ofelliptic integrals. For z= x,aparticular solution of(1)is }=- 2 6 -x. Equations (2)and(3)canbesolvedsuccessi vely(these arelinear intheunkno wns). Because~= }( s)isaparticular solution ofequation (2),thegeneral solution of(2)isgivenby(seePolyanin andZaitse v1995)~( s)= |1 }( s)+ |2 }( s) w { s}2( s), where |1and |2arearbitrary constants. Page730 B.5. SECOND -ORDER NONLINEAR EQUATIONS 731 11. j n2kn j2+ o n2kn o2+ 9 j ±1n kn j+so ±1n kn o=h( k). For ¹2and ¹2,there isanexact solution oftheform r = r ( ), =g'(2- )2s2- + (2- )2t2- m1 2. Here, thefunction r = r ( )isdetermined bytheordinary differential equationu r# 3+ v r = z( r ), whereu=1 4 (2- )2(2- )2,v=1 4(2- )(2- ) D(3 -4 -4 +4)+21(2- )+2 e(2- )m.' Refer ence:V.F.Zaitse vandA.D.Polyanin (1996). 12. n2 n 2+ o n2 n o2+ 9 ±1 ( ) n n +so±1 ( ) n n o= l( ). For ¹2and ¹2,there isanexact solution oftheform r = r ( ), = (2- )2s2- + (2- )2t2- m1 2. 13. 2 2+ 2 2+ 9 +s = ( ). For ¡¹0,there isanexact solution oftheform r = r ( ), = ¢ ¡2- £x+ 2- d ¤ ¥1 2. Here, thefunction r = r ( )isdetermined bytheordinary differential equationu r# 3+ v r = z( r ), whereu=1 4 2¡2, v=1 4 ¡(3 ¡-21 ¡-2 e) ). 14. 2 2+ 2 2+ ¦ ( ) +s ( ) = §( ). For ¡¹0,there isanexact solution oftheform r = r ( ), = ¢ ¡2- £x+ 2- d ¤ ¥1 2. 15. 2 2+ 2 2+ ¦ ±1 +s = ( ). For ¹0and ¹2,there isanexact solution oftheform r = r ( ), = 2s2- + (2- )2- £ ¤¨1 2. Here, thefunction r = r ( )isdetermined bytheordinary differential equationu r# 3+ v r = z( r ), whereu=1 4 2(2- )2, v=1 4 (2- ) D (4-3 )+21 -2 e(2- ) ¨.' Refer ence:V.F.Zaitse vandA.D.Polyanin (1996). 16. 2 2+ 2 2+ ¦ ±1 ( ) +s ( ) = §( ). For ¹0and ¹2,there isanexact solution oftheform r = r ( ), = 2s2- + (2- )2- £ ¤¨1 2. Page731 732 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES B.5.3-4. Equations with three independent variables. 17. © ª+ © ª+ « © ¬ « « ª= ( ). For ¹2, ¹2,and ®¹2,there isanexact solution oftheformr = r ( ), 2=4 ¯ s2- (2- )2+ t2- (2- )2+ °2- ±²(2- ®)2 ³, where thefunction r ( )isdetermined bytheordinary differential equationr# 3+ u r = z( r ), u=2©1 2- +1 2- +1 2- ®ª-1.' Refer ence:A.D.Polyanin andA.I.Zhuro v(1998). 18. © ´ ª+ © ª+ « © ¬ µ ¶ « ª= ( ). For ·¹0, ¡¹0,and ¸¹0,there isanexact solution oftheformr = r ( ), 2=4© -x ·2+ - d ¤ ¡2+ - ¹1º²¸2ª, where thefunction r ( )isdetermined bytheordinary differential equationr# 3-1 r = z( r ). 19. © ª+ © ª+ « © ¬ µ ¶ « ª= ( ). For ¹2, ¹2,and ¸¹0,there isanexact solution oftheformr = r ( ), 2=4 ¯ s2- (2- )2+ t2- (2- )2+ - ¹1º²¸2 ³, where thefunction r ( )isdetermined bytheordinary differential equationr# 3+ u r = z( r ), u=2©1 2- +1 2- ª-1. 20. © ª+ © ª+ « © ¬ µ ¶ « ª= ( ). For ¹2, ¡¹0,and ¸¹0,there isanexact solution oftheformr = r ( ), 2=4 ¯ s2- (2- )2+ - d ¤ ¡2+ - ¹1º²¸2 ³, where thefunction r ( )isdetermined bytheordinary differential equationr# 3+ 2- 1 r = z( r ).' Refer ence:A.D.Polyanin andA.I.Zhuro v(1998). Page732 B.5. SECOND -ORDER NONLINEAR EQUATIONS 733 B.5.4. Equations Containing MixedDeriv atives B.5.4-1. Monge±Amp Áereequations. 1.© 2 ª2 = 2 2 2 2+ ( ). 1 q.Exact solutions:r ( s, t)= |1 t2+ |2 s t+ |2 2 4 |1 s2-1 2 |1 w x 0( s- y) z( y) { y+ |3 t+ |4 s+ |5,r ( s, t)=1s+ |1 © |2 t2+ |3 t+ |2 3 4 |2 ª-1 2 |2 w x 0( s- y)( y+ |1) z( y) { y+ |4 t+ |5 s+ |6, where |1, |2, |3, |4, |5,and |6arearbitrary constants. 2 q.Exact solutions for z( s)>0:r ( s, t)= » t w ¼ z( s) { s+ }( s)+ |1 t, where }( s)isanarbitrary function. 2.© 2 ª2 = 2 2 2 2+ ( ). 1 q.Exact solution:r ( s, t)= |1 t2- t w ½( s) { s+1 2 |1 w x( s- y) ½2( y) { y+ |2 s+ |3 t+ |4,½( s)=1 2 |1 w z( s) { s+ |5, where |1, ¾¾¾, |5,and arearbitrary constants. 2 q.Exact solution: r ( s, t)= }( s) t2+ ~( s) t+ ( s), where}( s)=1|1 s+ |2, ~( s)= |3 }( s)+ |4+ }( s) 2 |1 w z( s) { s [ }( s)]3-1 2 |1 w z( s) { s [ }( s)]2,( s)=1 2 w x( s- y)[ ~ ¿( y)]2}( y) { y+ |5 s+ |6, 3 q.Exact solutions cubic in t:r ( s, t)= |1 t3-1 6 |1 w x( s- y) z( y) { y+ |2 s+ |3 t+ |4,r ( s, t)= t3 ( |1 s+ |2)2-1 6 w x( s- y)( |1 y+ |2)2z( y) { y+ |3 s+ |4 t+ |5, where |1, ¾¾¾, |5,and arearbitrary constants. 4 q.Seethesolution ofequation B.5.4.5 inItem 2 qwith=1. Page733 734 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 3.© 2 ª2 = 2 2 2 2+ ( )2. 1 q.Exact solution quadratic in t: r ( s, t)= }( s) t2+ ¯À|1 w }2( s) { s+ |2³ t+1 2 |2 1 w x( s- y) }3( y) { y+ |3 s+ |4. Thefunction }= }( s)isdetermined bytheordinary differential equation} } #x x=2( } x)2-1 2 z( s). 2 q.Exact solutions quartic in t: r ( s, t)= |1 t4-1 12 |1 w x( s- y) z( y) { y+ |2 s+ |3 t+ |4,r ( s, t)= t4 ( |1 s+ |2)3-1 12 w x( s- y)( |1 y+ |2)3 z( y) { y+ |3 s+ |4 t+ |5, where |1, ¾¾¾, |5,and arearbitrary constants. 3 q.Seethesolution ofequation B.5.4.5 inItem 2 qwith=2. 4.© 2 ª2 = 2 2 2 2+ ( )2+ §( )+ Á( ). Exact solution: r ( s, t)= }( s) t2+ ~( s) t+ ( s), where thefunctions }= }( s), ~= ~( s),and = ( s)aredetermined bythesystem ofordinary differential equations} } #x x=2( } x)2-1 2 z( s),} ~ #x x=2 } x ~ x-1 2 ( s),} #x x=1 2( ~ x)2-1 2 Â( s). 5.© 2 ª2 = 2 2 2 2+ ( ) Ã. 1 q.Exact solutions: r ( s, t)= |1 t Ä+2 (+1)(+2)-1|1 w x( s- y) z( y) { y+ |2 s+ |3 t+ |4,r ( s, t)= t Ä+2 ( |1 s+ |2)Ä+1-1 (+1)(+2) w x( s- y)( |1 y+ |2) Ä+1z( y) { y+ |3 s+ |4 t+ |5, where |1, ¾¾¾, |5,and arearbitrary constants. 2 q.Exact solution: r ( s, t)= }( s) t Ä+2 2. Thefunction }= }( s)isdetermined bytheordinary differential equation(+2) } } #x x-(+2)2( } x)2+4 z( s)=0. 6.© 2 ª2 = 2 2 2 2+ ( )2Ã+2+ §( ) Ã. Exact solution: r ( s, t)= }( s) t Ä+2-1 (+1)(+2) w x( s- y) ( y)}( y) { y+ |1 s+ |2 t+ |3, Page734 B.5. SECOND -ORDER NONLINEAR EQUATIONS 735 where }= }( s)isdetermined bytheordinary differential equation (+1)(+2) } } #x x-(+2)2( } x)2+ z( s)=0. 7.© 2 ª2 = 2 2 2 2+ ( ) ´. 1 q.Exact solutions: r ( s, t)= |1 w x( s- y) z( y) { y+ |2 s-1|1 ·2 ¤+ |3 t+ |4,r ( s, t)= |1 £x+ ¤-1|1 ·2 w x( s- y) - £ ¿z( y) { y+ |2 s+ |3 t+ |4, where |1, |2, |3, |4, ,and arearbitrary constants. 2 q.Exact solution: r ( s, t)= }( s)exp¢1 2 · t ¥, where }= }( s)isdetermined bytheordinary differential equation} } #x x-( } x)2+4 ·-2 z( s)=0. 8.© 2 ª2 = 2 2 2 2+ ( ) 2´+ §( ) ´. Exact solution: r ( s, t)= }( s) ¤-1·2 w x( s- y) ( y)}( y) { y+ |1 s+ |2 t+ |3, where thefunction }= }( s)isdetermined bytheordinary differential equation} } #x x-( } x)2+ ·-2 z( s)=0. 9.© 2 ª2 = 2 2 2 2+ ( ) §(). Exact solution: r ( s, t)= |1 w x( s- y) z( y) { y-1|1 w ¤Å( t- ) ( ) { + |2 s+ |3 t+ |4, where |1, |2, |3, |4, ,andarearbitrary constants. B.5.4-2. Other equations with quadratic nonlinearities. 10. 2 ± 2 2= ( ). 1 q.Suppose r ( s, t)isasolution oftheequation. Then thefunctions r 1= » r¢ s, » t+ }( s) ¥+ |, where }( s)isanarbitrary function and |isanarbitrary constant, arealsosolutions oftheequation. 2 q.Exact solutions: r ( s, t)= » t ¯2 w z( s) { s+ |1³1 2 + }( s),r ( s, t)= |1 t2+ }( s) t+1 4 |1 ¯}2( s)-2 w z( s) { s³+ |2, where }( s)isanarbitrary function and |1and |2arearbitrary constants. 3 q.Exact solutions inimplicit form:w { rÆ 2 ½( Ç)+ ~( È)= » É+ }( Ç), where }( Ç)and ~( È)arearbitrary functions and ½( Ç)= Ê Ë( Ç) Ì Ç. Page735 736 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 11. 2 + () 2 2= §() + Á() +s(). Exact solution:È= Í( É) Ç+ Î( É), where thefunctions Í( É)and Î( É)aredetermined bythesystem ofordinary differential equationsË Í Í Ï#Ï¤1¤+( Í Ï¤)2= Ð Í+Â,Ë Í Î Ï#Ï¤1¤+ Í Ï¤ Î Ï¤= Ð Î+ Ñ. B.5.5. General Form Equations B.5.5-1. Equations oftheform Ò ÓÒ ¿= ½¢ Ç, Ô, È, Ò ÓÒ Õ, Ò2ÓÒ Õ2 ¥. 1. Ö ×Ö Ø= Ù Ú Û, Ö2×Ö Û2 Ü. Exact solution: Ý ( Þ, Ô)=( ß Þ+ à) Ô+ Í( Þ), where ßand àarearbitrary constants andthefunction Í( Þ)isdetermined bytheordinary differential equation á âÞ, Í Ï#ÏÕ Õ ã= ß Þ+ à. 2. Ö ×Ö Ø= Ù Ú Ö ×Ö Û, Ö2×Ö Û2 Ü. Exact solution: Ý ( Þ, Ô)= ß Ô+ à+ Í( ä Þ+ å Ô), where ß, à, ä,and åarearbitrary constants andthefunction Í( æ)isdetermined bytheordinary differential equationá âä Í Ïç, ä2Í Ï#Ïç>çã- å Í Ïç- ß=0, æ= ä Þ+ å Ô. 3. Ö ×Ö Ø=× Ù ÚØ,1× Ö2×Ö Û2 Ü. Exact solution inmultiplicati veform:Ý ( Þ, Ô)=( ß è é Õ+ à è-é Õ) ê1( Ô), ê1( Ô)=exp ëì á ( Ô, å2) í Ôïî,Ý ( Þ, Ô)=[ ßcos( å Þ)+ àsin( å Þ)] ê2( Ô), ê2( Ô)=exp ëì á ( Ô,- å2) í Ôïî, where ß, à,and åarearbitrary constants. 4. Ö ðÖ ñ=ð ò Úñ,1ð Ö2ðÖ Û2 Ü+ ó(ñ) ô õ ö+ ÷(ñ) ô±õ ö. Exact solution:Ý ( Þ, Ô)= è é Õê( Ô) ë$ß+ ì ø( Ô)ê( Ô) í Ôïî+ è-é Õê( Ô) ë$à+ ì ù( Ô)ê( Ô) í Ôïî,ê( Ô)=exp ëì á ( Ô, å2) í Ôïî, where ß, à,and åarearbitrary constants. Page736 B.5. SECOND -ORDER NONLINEAR EQUATIONS 737 5. Ö ðÖ ñ=ð ò1 Úñ,1ð Ö2ðÖ Û2Ü+ ô õ öò2 Úñ,1ð Ö2ðÖ Û2Ü+ ô±õ öò3 Úñ,1ð Ö2ðÖ Û2Ü. There aresolutions oftheformÝ ( Þ, Ô)= è é Õ ú( Ô)+ è-é Õ û( Ô). 6. Ö ðÖ ñ=ð ò Úñ,1ð Ö2ðÖ Û2 Ü+ ó(ñ)cos( ü Û)+ ÷(ñ)sin( ü Û). Exact solution:Ý ( Þ, Ô)=cos( å Þ) ê( Ô) ë$ß+ ì ø( Ô)ê( Ô) í Ôïî+sin( å Þ) ê( Ô) ëà+ ì ù( Ô)ê( Ô) í Ôïî,ê( Ô)=exp ëì á ( Ô,- å2) í Ôïî, where ß, à,and åarearbitrary constants. 7. Ö ðÖ ñ=ð ò1 Úñ,1ð Ö2ðÖ Û2 Ü+cos( ü Û)ò2 Úñ,1ð Ö2ðÖ Û2 Ü+sin( ü Û)ò3 Úñ,1ð Ö2ðÖ Û2 Ü. There aresolutions oftheformÝ ( Þ, Ô)=cos( å Þ) ú( Ô)+sin( å Þ) û( Ô). 8. Ö ðÖ ñ=ð ò Úñ, ó( Û)ð Ö2ðÖ Û2Ü. Exact solution inmultiplicati veform:Ý ( Þ, Ô)= ú( Þ)exp ëì á ( Ô, å) í Ôýî, where thefunction ú= ú( Þ)satis®es thelinear ordinary differential equationø( Þ) ú þ#þÕ Õ= å ú. 9. Ö ðÖ ñ=ð ò Úñ,1ð Ö2ðÖ Û2,ð Ö2ðÖ Û2± ÿ Ö ðÖ Û 2Ü. 1 .Exact solution inmultiplicati veform:Ý ( Þ, Ô)= exp ëÀå Þ+ ì á ( Ô, å2,0) í Ôýî, where isanarbitrary constant. 2 .Exact solution inmultiplicati veform:Ý ( Þ, Ô)=( ß è é Õ+ à è-é Õ) ú( Ô), where ßand àarearbitrary constants, andthefunction ú= ú( Ô)satis®es theordinary differential equation ú þ = ú á âÔ, å2,4 ß à å2ú2ã. 3 .Exact solution inmultiplicati veform:Ý ( Þ, Ô)=[ ßsin( å Þ)+ àcos( å Þ)] ú( Ô), where ßand àarearbitrary constants, andthefunction ú= ú( Ô)satis®es theordinary differential equation ú þ = ú á âÔ,- å2,- å2( ß2+ à2) ú2ã. Refer ence:Ph.W.Doyle(1996), thecase º0wasconsidered. 10. Ö ðÖ ñ=ð ò Úñ, Ö2ðÖ Û2, Ö ðÖ Û± Û Ö2ðÖ Û2,2ð±2 Û Ö ðÖ Û+ Û2Ö2ðÖ Û2 Ü. Exact solution inmultiplicati veform:Ý ( Þ, Ô)=( 2 Þ2+ 1 Þ+ 0) ú( Ô), where 0, 1,and 2arearbitrary constants, andthefunction ú= ú( Ô)satis®es theordinary differential equation ú þ = ú á âÔ,2 2 ú, 1 ú,2 0 úã. Refer ence:Ph.W.Doyle(1996), thecase º0wasconsidered. Page737 738 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES B.5.5-2. Equations oftheform 2 2= á âÞ, Ô, Ý , , Õ, 2 Õ2ã. 11. Ö2ðÖ ñ2=ò Ú Û, Ö ðÖ Û, Ö2ðÖ Û2 Ü+ ñ, Ö ðÖ ñ Ü+ ð. Exact solution inadditi veform:Ý ( Þ, Ô)= ú( Þ)+ û( Ô), where thefunctions ú( Þ)and û( Ô)aredetermined bysolving thesecond-order nonlinear ordinary differential equations ( isanarbitrary constant)á âÞ, ú þÕ, ú þ#þÕ Õ ã+ ú= ,û þ#þ - âÔ, û þ ã- û= . 12. Ö2ðÖ ñ2=ð ò ñ,1ð Ö ðÖ ñ,1ð Ö2ðÖ 2,ð Ö2ðÖ 2±ÿ Ö ðÖ 2Ü. 1 .Exact solution inmultiplicati veform:Ý ( Þ, Ô)=( ß è é Õ+ à è-é Õ) ú( Ô), where ßand àarearbitrary constants, andthefunction ú= ú( Ô)satis®es theordinary differential equation ú þ#þ = ú á âÔ, ú þ ú, å2,4 ß à å2ú2ã. 2 .Exact solution inmultiplicati veform:Ý ( Þ, Ô)=[ ßsin( å Þ)+ àcos( å Þ)] ú( Ô), where ßand àarearbitrary constants, andthefunction ú= ú( Ô)satis®es theordinary differential equation ú þ#þ = ú á âÔ, ú þ ú,- å2,- å2( ß2+ à2) ú2ã. 13. Ö2ðÖ ñ2=ð ò ñ, Ö2ðÖ 2, Ö ðÖ ± Ö2ðÖ 2,2ð±2 Ö ðÖ +2Ö2ðÖ 2 Ü. Exact solution inmultiplicati veform:Ý ( Þ, Ô)=( 2 Þ2+ 1 Þ+ 0) ú( Ô), where 0, 1,and 2arearbitrary constants, andthefunction ú= ú( Ô)satis®es theordinary differential equation ú þ#þ = ú á âÔ,2 2 ú, 1 ú,2 0 úã. 14. Ö2ðÖ ñ2=ð ò1 ñ,1ð Ö2ðÖ 2Ü+ ô õ öò2 ñ,1ð Ö2ðÖ 2Ü+ ô±õ öò3 ñ,1ð Ö2ðÖ 2Ü. There aresolutions oftheformÝ ( Þ, Ô)= è é Õ ú( Ô)+ è-é Õ û( Ô). 15. Ö2ðÖ ñ2=ð ò1 ñ,1ð Ö2ðÖ 2 Ü+cos( ü)ò2 ñ,1ð Ö2ðÖ 2 Ü+sin( ü)ò3 ñ,1ð Ö2ðÖ 2 Ü. There aresolutions oftheformÝ ( Þ, Ô)=cos( å Þ) ú( Ô)+sin( å Þ) û( Ô). Page738 B.6. THIRD -ORDER NONLINEAR EQUATIONS 739 B.6. Third-Or derNonlinear Equations B.6.1. Stationar yHydr odynamic Boundar yLayerEquations 1. Ö ðÖ Ö2ðÖ Ö ± Ö ðÖ Ö2ðÖ 2= Ö3ðÖ 3. Thesystem ofequations ofstationary laminar boundary layer ona¯atplate (Schlichting 1981, Loitsyanskiy 1996), 1 1 + 2 1 = 2 1 2, 1 + 2 =0, canbereduced tothisequation byintroducing thestream function inaccordance with therelations 1= !and 2=- "( and arethelongitudinal andtransv erse coordinates, 1and 2arethelongitudinal andtransv erse components ofthe ¯uid velocity ,and isthekinematic ¯uid viscosity). 1 .Suppose Ý = Ý ( Þ, #)isasolution ofthestationary hydrodynamic boundary layer equation. Then thefunctionÝ 1= 1 Ýâ2 Þ+ 3, 1 2 #+ ú( Þ)ã+ 4, where ú( Þ)isanarbitrary function and 1, 2, 3,and 4arearbitrary constants, isalsoasolution oftheequation. Refer ence:Yu.N.Pavlovskii (1961), L.V.Ovsyannik ov(1978). 2 .Exact solutions involving arbitrary functions:Ý ( Þ, #)= 1 #+ ú( Þ),Ý ( Þ, #)= 1 #2+ ú( Þ) #+1 4 1 ú2( Þ)+ 2,Ý ( Þ, #)=6 $ Þ+ 1#+ ú( Þ)+ 2 [ #+ ú( Þ)]2+ 3,Ý ( Þ, #)= ú( Þ)exp(- 1 #)+ $ 1 Þ+ 2,Ý ( Þ, #)= 1exp %- 2 #- 2 ú( Þ) &+ 3 #+ 3 ú( ')+ $ 2 '+ 4,(( ', #)=6 $ 1 '1 )3tanh *+ 2, *= 1 # '-2 )3+ ú( '),(( ', #)=-6 $ 1 '1 )3tan *+ 2, *= 1 # '-2 )3+ ú( '), where 1, 2, 3,and 4arearbitrary constants and ú( ')isanarbitrary function. The second solution isspeci®ed inZwillinger (1998) andthefourth and®fthwere obtained byIgnato vich(1993). 3 .Exact solution:(( ', #)= 'ø( #)+ù( #), (1) where thefunctionsø=ø( #)andù=ù( #)aredetermined bythesystem ofordinary differential equations (ø þ+)2-ø ø þ#þ+,+= $ø þ#þ#þ+,+,+, (2)ø þ+ù þ+-ø ù þ#þ+,+= $ù þ#þ#þ+,+,+. (3) Theorder ofequation (2)canbereduced bytwo.Assume thatasolutionø=ø( #)ofequation (2)is known.Then equation (3),which islinear inù,hastwolinearly independent particular solutionsù1=1,ù2=ø( #) Page739 740 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES The second particular solution isapparent from comparing equations (2)and(3). The general solution ofequation (2)canberepresented intheform (seeZaitse vandPolyanin 1995):ù( #)= 1+ 2ø+ 3 ø ì ûí #- ìø ûí # -,ø=ø( #), û=1 (ø þ +)2exp -1$ ìø í # -.(4) Itisnotdif®cult tocheck thatequation (2)hastheparticular solutionsø( #)=6 $( #+ )-1,ø( #)= . / +- 0 $,(5) where and 0arearbitrary constants. Withreference to(1)and(4),onecanseethatthe®rst solution in(5)leads tothethird solution inItem 2 with ú( ')=const .Substituting thesecond expression in(5)into(1)and(4)yields another solution. Refer ence:A.D.Polyanin (2001b, 2001c). 2. Ö ðÖ Ö2ðÖ Ö ± Ö ðÖ Ö2ðÖ 2= Ö3ðÖ 3+ ó(). Theequation oflaminar boundary layer with pressure gradient. 1 .Suppose (= (( ', #)isasolution oftheequation inquestion. Then thefunctions (Pavlovskii 1961)( 1= 1 ( 2', 1 #+ ú( ') 3+ , where ú( ')isanarbitrary function and isanarbitrary constant, arealsosolutions oftheequation. 2 .Nonviscous solutions (independent oftheviscosity $):(( ', #)= 1 # 42 ìø( ') í '+ 1 î1 )2 + ú( '),(( ', #)= 1 #2+ ú( ') #+1 4 1 4 ú2( ')-2 ìø( ') í ' î+ 2, where ú( ')isanarbitrary function and 1and 2arearbitrary constants. 3 .Exact solution forø( ')= 5 '+ :(( ', #)= ' 6( #)+ ( #), where thefunctions 6= 6( #)and = ( #)aredetermined bythesystem ofordinary differential equations ( 6 þ+)2- 6 6 þ#þ+,+= $ 6 þ#þ#þ+,+,++ 5, (1)6 þ+ þ+- 6 þ#þ+,+= $ þ#þ#þ+,+,++ . (2) The order oftheautonomous equation (1)canbereduced byone. Ifaparticular solution 6( #) ofequation (1)isknown,thecorresponding equation (2)canbereduced toasecond-order linear equation bythechange ofvariable 7( #)= þ+.If 6( #)= 1 8 5 #+ ,equation (2)canbeintegrated inquadrature, because itstwoparticular solutions areknownif =0,namely , 1=1and 2=11 2 8 5 #2+ #. 4 .Exact solution forø( ')= 5 . 9 ::(( ', #)= ú( ') . / +- 5 2 ; 02ú( ') . 9 :-/ +- $ 0 '+2 $ 02; #+2 $ 0;ln| ú( ')|, where ú( ')isanarbitrary function and 0isanarbitrary constant. Refer ence:A.D.Polyanin (2001b, 2001c). Page740 B.6. THIRD -ORDER NONLINEAR EQUATIONS 741 3. 2 ± 2 2= ( ) 2 2 + ( ) + ( ). This equation canbeused tomodel turbulent boundary layer . Exact solution: = ( ) + ( ), where thefunctions ( )and ( )aredetermined bythesystem ofordinary differential equations ( ) + -( )2+ =0, ( ) + - + =0. B.6.2. Nonstationar yHydr odynamic Boundar yLayerEquations 1. 2 + 2 ± 2 2= 3 3. This istheequation ofnonstationary laminar boundary layer ona¯atplate; and arethe longitudinal andtransv ersecoordinates and isthestream function (Schlichting 1981, Loitsyanskiy 1996). 1 .Suppose = ( , , )isasolution oftheequation isquestion. Then thefunction (see Vereshchagina 1973) 1= , + ( , ), + ( , ) + ( ), where ( , )and ( )arearbitrary functions, isalsoasolution oftheequation. 2 .Exact solutions: = !1 + ( , ), = !1 2+ ( , ) +1 4 !1 2( , )+ ( , ) , =6 " + !1+ ( , )+ !2 [ + ( , )]2+ ( , ) , = !1exp #- !2 - !2 ( , ) $+ !3 + !3 ( , )+ " !2 + ( , ) , =6 " !1 1 %3tanh &+ ( , ) , &= !1 + ( , ) 2 %3, =-6 " !1 1 %3tan &+ ( , ) , &= !1 + ( , ) 2 %3, where ( , )isanarbitrary function oftwoarguments and !1, !2, !3,and !4arearbitrary constants. 3 .Exact solution: ( , , )= '( , )+ (( , ), (3) where thefunctions '= '( , )and (= (( , )aredetermined bythesimpler equations with two variables2' + ) ' *2 - ' 2' 2= " 3' 3, (4)2( + ' ( - ' 2( 2= " 3( 3. (5) Page741 742 B. M ETHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES TABLE B4 Exact solutions of equation (4) NoFunction '= '( , ) (or general form of solution)Remarks (or determining equation) 1 '= ( ) ( ) is an arbitrary function 2 '= + !1+ ( ) ( ) is an arbitrary function,!1is any number 3 '=6 "+ ( )+ +( ) ( ) is an arbitrary function 4 '= !1exp#- , + , ( )$- +( )+ " , ( ) is an arbitrary function,!1, ,are any numbers 5 '= !1exp[- , + , ( )]+ 1, + !2- +( )+ " , ( ) is an arbitrary function,!1, !2, ,are any numbers 6 '= -+ !1exp[- , + , ( )] 1 + !2exp(- ,- )- +( )+ " ,-- ( ) is an arbitrary function,!1, !2,-, ,are any numbers 7 '= '( &), &= + , , ' ./.+( ' .)2- ' ' ./.= " ' ././. 8 '= -1 %2#10( &)-1 2 &$, &= -1 %2 3 4- 20 .+(0 .)2-0 0 ./.= "0 ././. Equation (4) is independent of (5). If a particular solution '= '( , ) of equation (4) is known, then the corresponding equation (5) can be reduced by the change of variable 2= 3 43 to the second-order linear equation 2 - ' 2 = " 22 2- ' 2. ( 6) Exac tsolution sofequatio n(4)arelistedinTableB4.Theordinar ydifferentia lequation sinthe last two rows are autonomous and, therefore, admit reduction of order. TableB5present ssolution sofequatio n(6)thatcorrespon dtothesolution sofequatio n(4) speci®e dinTableB4.Onecanseethatinthe®rstthreecasesthesolution sofequatio n(6)are expressed in terms of solutions to the classical heat equation with constant coef®cients. There are other three cases where equation (6) is reduced to a separable equation.4. Exact solution: ( , , )=#15( ) 6 71 8+ 9( ) 6 72 8$ 6 : + ( ) + ; ,5( )= !1exp ( " ,2- ; <1) + , ( ) ,9( )= !2exp ( " ,2- ; <2) + , ( ) , where ( ) is an arbitrary function and !1, !2, ;, <1, <2, and ,are arbitrary parameters. Page 742 B.6. T HIRD -ORDER NONLINEAR EQUATIONS 743 TABLE B5 Transformations of equation (6) for the corresponding exact solutions of equation (4) [the number in the ®rst column corresponds to the numbe roftheexactsolutio n '= '( , )inTableB4] No Transformations of equation (6) Resulting equation 1 2= =( >, ), >= + ? ( ) 3 @3 += " 32@3 A2 2 2=1++ B1 =( C, D), D=1 3( + !1)3+ !2,C=( + !1) + E ( )( + !1) + !3 3 @3 F= " 32@3 G2 3 2= >-3=( >, ), >= + ( ) 3 @3 += " 32@3 A2 4 2= 6 H I( J, ), J= - , + , ( )3 K3 += " ,232K3 H2+( " ,2- !1 , 6 H) 3 K3 H 7 2= =( &, ), &= + , 3 @3 += " 32@3 .2+# '( &)- ,$ 3 @3 .- ' .( &) = 82= -1 %2=( &, D), &= -1 %2, D=ln 3 @3 F= " 32@3 .2+0( &) 3 @3 .+ #1 -0 .( &) $L= 5 . Exact solution: ( , , )=5( ) exp( < + , )+ 9( ) exp(- < +- , )+ ( ) + ; ,5( )= !1exp ( " ,2- ; <) + , ( ) ,9( )= !2exp ( "-2,2- ; <-) +- , ( ) , where ( ) is an arbitrary function and !1, !2, ;, <,-, and ,are arbitrary parameters. 6 . Exact solution: ( , , )= =( C, ) C+ ( ) + ( ) , C= < + , , where ( ) and ( ) are arbitrary functions, <and ,are arbitrary parameters, and =( C, ) is a function satisfying the second-order linear parabolic equation = +# < ( )- , ( )$ = C= " ,22= C2-1, +( ). The transformation== 2( &, )-1, ( ), &= C- # < ( )- , ( )$ takes the last equation to the customary heat equation 2 = " ,222 &2. 7 . Exact solutions: = 6 M :2 + ( !1 6 : G+ !2 6-: G)+ ( , ) , C= + ( , ), = 6-M :2 +# !1sin( , C)+ !2cos( , C)$+ ( , ) , C= + ( , ), = !1 6-M :2Gsin( , C- 2 " ,2+ !2)+ ( , ) , C= + ( , ), Page 743 744 B. M ETHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES where ( , ) is an arbitrary function of two arguments and !1, !2, and ,are arbitrary constants.NPO Reference : A. D. Polyanin (2001b). 2. 2 + 2 ± 2 2= 3 3+ ( , ). The equation of nonstationary laminar boundary layer with pressure gradient.1. Suppose ( , , ) is a solution of the equation in question. Then the functions (see Vereshchag- ina 1973) 1= Q , Q + ( , ), + ( , ) + ( ), where ( , ) and ( ) are arbitrary functions, are also solutions of the equation. 2 . Nonviscous solution for any ( , ) (independent of the viscosity "): ( , , )= ; 2+ ( , ) +1 4 ; 2( , )+1 2 ; - ( , ) + ( ). where ( , ) and ( ) are arbitrary functions and ;is an arbitrary constant. Another nonviscous solution for any ( , ): ( , , )= ( , ) + ( , ), where ( , ) is an arbitrary function, and the function = ( , ) is determined by the ®rst-order equation + = ( , ). Nonviscous solutions for ( , )= ( ): ( , , )= Q 2 ( ) + !11 %2 + ( , ). 3 . Exact solutions for ( , )= 1( ) + 2( ): ( , , )= '( , )+ (( , ), where the functions '= '( , ) and (= (( , ) are determined from the simpler equations with two variables2' + ) ' *2 - ' 2' 2= " 3' 3+ 1( ), ( 1)2( + ' ( - ' 2( 2= " 3( 3+ 2( ). ( 2) Equation (1) is independent of (2). If '= '( , ) is a solution of equation (1), then the function'1= ' + ( ), R+ +( )witharbitrar y ( )isalsoasolutio nofequatio n(1).TableB6presents exact solutions of equation (1) for various 1= 1( ). The change of variable 2= 3 43 brings equation (2) to the second-order linear equation 2 - ' 2 = " 22 2- ' 2+ 2( ). ( 3) Letusdwel lonthe®rstsolutio nofequatio n(1)inTableB6:'( , )= ;( ) + ( ), where ; ++ ;2= 1( ). ( 4) Page 744 B.6. T HIRD -ORDER NONLINEAR EQUATIONS 745 TABLE B6 Exact solutions of equation (1) for various 1( ); ( ) is an arbitrary function Function1= 1( )Function '= '( , ) (or general form of solution)Determining equation (or determining coef®cients) Any'= ;( ) + ( ); ++ ;2= 1( )1( )=5 6- S + ,5>0,->0 '= 9 6-1 2 S + sin[ , + , ( )]+ +( ),'= 9 6-1 2 S + cos[ , + , ( )]+ +( ) 9= Q T2 UMS, ,= T S 2M1( )=5 6 S + ,5>0,->0 '= 9 61 2 S +sinh[ , + , ( )]+ +( ) 9= Q T2 UMS, ,= T S 2M1( )=5 6 S + ,5<0,->0 '= 9 61 2 S + cosh[ , + , ( )]+ +( ) 9= Q T2| U|MS, ,= T S 2M1( )=5 6 S + ,5is any,->0 '= ( ) 6 : - 5 6 S +-: 4 ,2( )+ +( ), ( )- " ,,= Q T S 2M1( )=5 -2'= -1 %2#10( &)-1 2 &$, &= -1 %2 3 4-5-20 .+(0 .)2-0 0 ./.= "0 ././.1( )=5 '= '( &), &= + , -5+ , ' ./.+( ' .)2- ' ' ./.= " ' ././. Exact solutions of the Riccati equation for ;= ;( ) with various 1( ) can be found in Polyanin and Zaitsev (1995). The substitution ;= +/Vbrings this equation to a second-order linear equation for( ): +W+- 1( ) = 0. In particular, if 1( )=const, we have;( )= < !1cos( < )- !2sin( < )!1sin( < )+ !2cos( < )for 1= - <2<0,;( )= < !1cosh( < )+ !2sinh( < )!1sinh( < )+ !2cosh( < )for 1= <2>0. On substituting solution (4) with arbitrary 1( ) into equation (3), we obtain 2 = " 22 2+# ;( ) + ( )$ 2 - ;( ) 2+ 2( ). ( 5) The transformation2=1X( ) =( C, D)+ 2( ) X( ) , D= X2( ) + !1,C= X( )+ ( ) X( ) + !2, X( )=exp ;( ) , takes (5) to the classical constant coef®cient heat equation = D= " 2= C2.Y Z\[ ] ^`_ a bTheordinar ydifferentia lequation sinthelasttworowsinTableB6(seethelast column) are autonomous and, hence, can be reduced in order.Y Z\[ ] ^`_ c bSuppose ( , , ) is a solution of the nonstationary hydrodynamic boundary layer equation with ( , )= 1( ) + 2( ). Then the function 1= ( + ( ), , )- +( ) , where +W++ 1( ) = 0, is also a solution of this equation. Page 745 746 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 4 .Exact solution for ( , )= ( ) 6 S + ,->0: ( , , )= ( , ) 6 : + ( , ) 6-: +1, ln| ( , )| - " , ,( , )=- 6 S + 2 ,2( , ) ( ) , ,= Q d e2 ", where f( , g)isanarbitrary function oftwoarguments. 5 h.Exact solutions for ( , g)= i( ) j k l,e>0:m( , n, g)= o1 pexp q1 2e g/r s t( u)sinh[ pn+ f( u, g)]+ vv g w f( u, g) x u,m( u, n, g)= o1 pexp q1 2e g/r s t( u)cosh[ pn+ f( u, g)]+ vv g w f( u, g) x u,t( u)=2w i( u) x u+ y1, p =d e2 ", where f( u, g)isanarbitrary function oftwoarguments. 6h.Exact solution for z( u, g)= i( u) j-k l,e<0:m( u, n, g)= o1 pexp q-1 2e g/r s t( u)sin[ pn+ f( u, g)]+ vv g w f( u, g) x u,m( u, n, g)= o1 pexp q-1 2e g/r s t( u)cos[ pn+ f( u, g)]+ vv g w f( u, g) x u,t( u)=2w i( u) x u+ y1, p =d e2 ". where f( u, g)isanarbitrary function oftwoarguments. 7h.Exact solution for z( u, g)= { jk |- }l:m( u, n, g)= f( u, g) j ~ - { 2e p 2f( u, g) j k |-~ - }l +1 pvv g wln| f( u, g)| x u- " pu+2 " p 2+ e n+1 pln| f( u, g)| , where f( u, g)isanarbitrary function oftwoarguments and p isanarbitrary constant. 8h.Exact solution for z( u, g)= z( g):m( u, n, g)=w ( , g) x + f( g) n+ t( g) u, = u+ pn, where f( g)and t( g)arearbitrary functions, and p arearbitrary parameters, and( , g)isafunction satisfying thesecond-order linear parabolic equationvv g+# f( g)- pt( g)$ vv = " p 2v2v 2-1 pf l( g)+1 pz( g). Thetransformation= 2( , g)-1 pf( g)+1 pw z( g) x g, = -w #P f( g)- pt( g) $ x g brings ittothecustomary heat equationv 2v g= " p 2v22v 2. 9h.Exact solution for z( u, g)= z( g):m( u, n, g)= y1 j-~ +~ (|,l)- {( g) f( u, g)- vv g w f( u, g) x u+ {( g) n+ " pu, {( g)=w z( g) x g+ y2, where f( u, g)isanarbitrary function oftwoarguments and y1, y2,and p arearbitrary constants. Page746 B.6. THIRD -ORDER NONLINEAR EQUATIONS 747 10h.Exact solution for z( u, g)= z( g):m( u, n, g)= f( u, g) j ~ + t( u, g) j-~ + ( u, g)+ {( g) n, where p isanynumber , f( u, g)isanarbitrary function oftwoarguments, andtheother functions are de®ned byt( u, g)= y " j2 ~2lf( u, g) u-w {( g) x g, {( g)=w z( g) x g+ y j2 ~2l;( u, g)=1 p{( g)ln| f( u, g)|+1 pvv g wln| f( u, g)| x u- " pu. 11h.Exact solutions for z( u, g)= z( g):m= j ~2l( y1 j ~ + y2 j-~ )+ vv g w f( u, g) x u+ w z( g) x g, = n+ f( u, g),m= j- ~2l# y1sin( p)+ y2cos( p)$+ vv g w f( u, g) x u+ w z( g) x g, = n+ f( u, g),m= y1 j-~ sin( p-2 " p 2g+ y2)+ vv g w f( u, g) x u+ w z( g) x g, = n+ f( u, g), where f( u, g)isanarbitrary function oftwoarguments and y1, y2,and p arearbitrary constants. 12 h.Exact solutions for z( u, g)= :m=- 6 " 3+ y2 2+ y1 + vv g w f( u, g) x u, = n+ f( u, g),m= u+ y1exp- " - 2 2+ y2 + vv g w f( u, g) x u, = n+ f( u, g), where f( u, g)isanarbitrary function oftwoarguments and y1, y2,and arearbitrary constants.P Refer ence:A.D.Polyanin (2001b). 3. 2 + 2 ± 2 2= 2 2 + (,). This equation describes the¯owofanon-Ne wtonian ¯uid inatwo-dimensional nonstationary boundary layer with apressure gradient. Here, misthestream function andthefunction z= z() depends ontherheological properties ofthe¯uid. Forpower-law¯uids, z= || -1. 1h.Theassertion ofItem 1h,equation 1inSubsection B.6.2, remains validforthisequation. 2h.The equation admits thenonviscous solutions presented inItem 2h,equation 2inSubsec- tionB.6.2, where z( u, g)must bereplaced by i( u, g). 3h.Exact solution for i( u, g)= i( g):m( u, n, g)= {( g) u+w 2( n, g) x n, where 2= 2( n, g)isafunction satisfying thesecond-order equationv 2v g- {( g) v 2v n= vv n z v 2v n + i( g). (1) Thetransformation2=( , g)+w i( g) x g, = n+w {( g) x g Page747 748 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES brings equation (1)tothesimpler equationvv g= vv z vv . (2) Equation (2)admits exact solutions with theforms [forany z= z( )]( , g)= ( ), = + pg equation p= z( )+ y;( , )= { + ( ), = + p equation p= z( + {)+ y;( , )= ( ), = ¡ equation1 2 -1 2 =[ z( )] , where {, , y,and ¢arearbitrary constants. The®rsttwoequations for = ( )canbesolvedin parametric form. 4 £.Exact solution for ¤( ¥, )= ¤( ):¦( ¥, §, )=w ( ¨, ) x ¨+ ©( ) §+ ª( ) ¥, ¨= §+ ¥, where ©( )and ª( )arearbitrary functions, isanarbitrary parameter ,and ( ¨, )isafunction satisfying thesecond-order nonlinear parabolic equation««+ ¬ ©( )- ª( ) ® ««¨= ««¨ z ««¨ - © ¯( )+ ¤( ). Thetransformation= °( , )- ©( )+ ± ¤( ) ² , = ¨- ± ¬³ ©( )- ª( ) ® ² leads tothesimpler equation«°«= «« ´¶µ · «°« ¸ ¹ º. Forexact solutions ofthisequation, seeItem 3 ». 5».Exact solution for ¼( ½, )= ¾( ) ½+ ¿( ):À( ½, Á, )= ÂPÃ( ) Á+ Ä( ) ÅL½+ ± Æ( Á, ) ² Á, where Ä( )isanarbitrary function, Ã= Ã( )isdetermined bytheRiccati equationÃ Ç È+ Ã2= ¾( ), and Æ= Æ( Á, )satis®es thesecond-order equationÉÆÉ= ÉÉÁ ´Êµ · ÉÆÉÁ ¹ º+ Â1Ã( ) Á+ Ä( ) Å ÉÆÉÁ- Ã( ) Æ+ ¿( ). ThetransformationÆ=1Ë( ) ´Ì( Í, Î)+ ± ¿( ) Ë( ) ² º, Î= ± Ë2( ) ² + Ï, Í= Á Ë( )+ ± Ä( ) Ë( ) ² + Ð, where Ë( )=exp´ ± Ã( ) ² º,leads tothesimpler equationÉÌ ÉÎ= ÉÉÍ ´Êµ · ÉÌ ÉÍ ¹ º. Forexact solutions ofthisequation, seeItem 3». Page748 B.7. FOUR TH-ORDER NONLINEAR EQUATIONS 749 B.7. Four th-Or derNonlinear Equations B.7.1. Stationar yHydr odynamic Equations (Navier±Stokes Equations) 1. Ñ ÒÑ Ó ÑÑ Ô( ÕÒ)± Ñ ÒÑ Ô ÑÑ Ó( ÕÒ)= Ö Õ ÕÒ, ÕÒ= Ñ2ÒÑ Ô2+ Ñ2ÒÑ Ó2. Thetwo-dimensional equations ofsteady-state motion ofaviscous incompressible ¯uid (stationary Navier±Stok esequations)× 1 Ø × 1Ø Ù+ × 2 Ø × 1Ø Ú=-1 ÛØÜØ Ù+ Ý Þ × 1,× 1 Ø × 2Ø Ù+ × 2 Ø × 2Ø Ú=-1 ÛØÜØ Ú+ Ý Þ × 2,Ø × 1Ø Ù+Ø × 2Ø Ú=0 arereduced totheequation under consideration. Tothisend, oneintroduces thestream function ßbytheformulas × 1= àáàâand × 2=- àáàãandeliminates thepressureÜ,using cross differentiation, from the®rsttwoequations. 1».Exact solutions inadditi veform:À( ½, Á)= ä1 Á3+ ä2 Á2+ ä3 Á+ ä4,À( ½, Á)= ä1 ½2+ ä2 ½+ ä3 Á2+ ä4 Á+ ä5,À( ½, Á)= ä1exp(- å Á)+ ä2 Á2+ ä3 Á+ ä4+ æ å ½,À( ½, Á)= ä1exp( å ½)- æ å ½+ ä2exp( å Á)+ æ å Á+ ä3,À( ½, Á)= ä1exp( å ½)+ æ å ½+ ä2exp(- å Á)+ æ å Á+ ä3, where ä1, ççç, ä5,and åarearbitrary constants. 2».Exact solutions:À( ½, Á)=( Ï ½+ Ð) è- é + æ å ½+ ä,À( ½, Á)= Ï è- é( + êLë)+ Ð( Á+ ì ½)2+ ä( Á+ ì ½)+ æ å( ì2+1) ½+ í,À( ½, Á)= ÂPÏsinh( î ½)+ Ðcosh( î ½) Åè- é + æå( î2+ å2) ½+ ä,À( ½, Á)= ÂPÏsin( î ½)+ Ðcos( î ½) Åè- é + æå( å2- î2) ½+ ä,À( ½, Á)= Ï è é + ï ë+ Ð è ð ë+ æ ñ Á+ æå ñ( î- ñ) ½+ ä, ñ= ò ó å2+ î2, where Ï, Ð, ä, í, ì, î,and åarearbitrary constants. 3 ».Exact solution:À( ½, Á)= ô( Á) ½+ õ( Á), where thefunctions ô= ô( Á)and õ= õ( Á)aredetermined bythesystem offourth-order ordinary differential equationsôÇ ôÇÇ - ô ôÇÇÇ = æ ôÇÇÇÇ , (1)õ Çô ÇÇ - ô õ ÇÇÇ = æ õ ÇÇÇÇ . (2) Integrating yields thesystem ofthird-order equations ( ôÇ )2- ô ôÇÇ = æ ôÇÇÇ + Ï, (3)õÇôÇ - ô õÇÇ = æ õÇÇÇ + Ð, (4) Page749 750 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES where Ïand Ðarearbitrary constants. Theorder oftheautonomous equation (3)canbereduced byone. Itisnotdif®cult toverify thatequation (1)hastheparticular solutionsô( Á)= Ã Á+ ö, (5)ô( Á)=6 æ( Á+ Ã)-1, (6)ô( Á)= Ã è- é + å æ, (7) where Ã, ö,and åarearbitrary constants. Ingeneral, equation (4)canbereduced bythechange ofvariable ÷= õÇtothesecond-order nonhomogeneous linear equationæ ÷ÇÇ + ô ÷Ç - ôÇ ÷+ Ð=0, where ÷= õÇ. (8) The corresponding homogeneous equation (with Ð=0)hastwolinearly independent particular solutions÷1= ø ôÇÇ if ôÇÇ ¹0,ô if ôÇÇ =0, ÷2= ÷1 ù Ë úÁ÷2 1, where Ë=exp û-1æ ù ô úÁ ¹.(9) (The ®rst solution isapparent from comparing equations (1)and(8)with Ð=0.)The general solutions ofequations (8)and(2)aregivenby÷= ä1 ÷1+ ä2 ÷2+ ä3 û ÷2 ù ÷1Ë úÁ- ÷1 ù ÷2Ë úÁ ¹, õ=ù ÷ úÁ+ ä4, ä3=- Ðæ.(10) Thegeneral solution ofequation (2)corresponding totheparticular solution (6)isrepresented asõ( Á)= ü ä1( Á+ Ã)3+ ü ä2+ ü ä3( Á+ Ã)-1+ ü ä4( Á+ Ã)-2, where ü ä1, ü ä2, ü ä3,and ü ä4arearbitrary constants (these areexpressed interms of ä1, ä2, ä3,and ä4). Thegeneral solutions of(2)corresponding totheparticular solutions (5)and(7)aredetermined from (9)and(10). 4».Exact solution ofamore general form:À( ½, Á)= ô( ý) ½+ õ( ý), ý= Á+ ì ½, where thefunctions ô= ô( ý)and õ= õ( ý)aredetermined bythesystem offourth-order ordinary differential equationsô Ç þô ÇÇ þ\þ- ô ô ÇÇÇ þ\þ\þ= æ( ì2+1) ô ÇÇÇÇ þ\þ\þ\þ, (11)õÇþôÇÇ þ\þ- ô õÇÇÇ þ\þ\þ= æ( ì2+1) õÇÇÇÇ þ\þ\þ\þ+4 ì æ ôÇÇÇ þ\þ\þ+2 ì ( ì2+1) ô ôÇÇ þ\þ. (12) Integrating yields thesystem ofthird-order equations ( ôÇ þ)2- ô ôÇÇ þ\þ= æ( ì2+1) ôÇÇÇ þ\þ\þ+ Ï, (13)õÇþôÇ þ- ô õÇÇ þ\þ= æ( ì2+1) õÇÇÇ þ\þ\þ+4 ì æ ôÇÇ þ\þ+2 ìì2+1 ù ô ôÇÇ þ\þ úý+ Ð, (14) where Ïand Ðarearbitrary constants. Theorder oftheautonomous equation (13) canbereduced byone. Itisnotdif®cult toverify thatequation (11) hastheparticular solutionsô( ý)= Ã ý+ ö, ý= Á+ ì ½,ô( ý)=6 æ( ì2+1)( ý+ Ã)-1,ô( ý)= Ã è- é þ+ å æ( ì2+1), where Ã, ö,and åarearbitrary constants. Ingeneral, equation (14) canbereduced bythechange ofvariable ÷= õÇþtoasecond-order nonhomogeneous linear equation.ÿ Refer ence:A.D.Polyanin (2001d). Page750 B.7. FOUR TH-ORDER NONLINEAR EQUATIONS 751 2. Ñ ÒÑ Ó ÑÑ Ô( ÕÒ)± Ñ ÒÑ Ô ÑÑ Ó( ÕÒ)= Ö Õ ÕÒ+ (Ó), ÕÒ= Ñ2ÒÑ Ô2+ Ñ2ÒÑ Ó2. This equation describes plane ¯owofaviscous incompressible ¯uid under theaction ofatransv erse force ( Àisthestream function). The case ô( Á)= Ãsin( å Á)corresponds toA.N.Kolmogoro v's model which isused todescribe subcritical andtranscritical (laminar -turb ulent) modes of¯ow. 1».Exact solution inadditi veform forarbitrary ( Á):À( , Á)=-1 2 æ ù 0( Á- ý)2 Ë( ý) úý+ ä1 è- é + ä2 Á2+ ä3 Á+ ä4+ æ å , Ë( ý)= è- é þù è é þ( ý) úý, where ä1, ä2, ä3, ä4,and åarearbitrary constants. Example. Inthecase (Ú)= cos( Ú),which corresponds to (Ú)= sin( Ú),itfollowsfrom theprevious formula with 1= 2= 4=0and =- Ý thatß(Ù,Ú)=- 2( 2+ Ý22) sin( Ú)+ Ý cos( Ú) + Ú- Ù, where and arearbitrary constants. This solution wasindicated byBelotserk ovskii andOparin (2000); itdescribes the ¯owwith aperiodic structure. 2».Exact solution inadditi veform for ( )= Ï è é + Ð è- é :( , )= ä1 è- é ë+ ä2 - Ïå3( ä2+ æ å) è é + Ðå3( ä2- æ å) è- é - æ å , where ä1and ä2arearbitrary constants. 3».Generalized separable solution forarbitrary ( ):( , )= ( ) + ( ), where thefunctions = ( )and = ( )aredetermined bythesystem offourth-order ordinary differential equations - = , (1) - = + ( ). (2) Integrating yields thesystem ofthird-order equations ( )2- = + , (3) - = +ù ( ) ú+ , (4) where and arearbitrary constants. Theorder oftheautonomous equation (3)canbereduced byone. Itisnotdif®cult toverify thatequation (1)hastheparticular solutions( )= + , (5)( )=6 ( + )-1, (6)( )= - + , (7) where , ,and arearbitrary constants. Ingeneral, equation (4)canbereduced bythechange ofvariable = tothesecond-order nonhomogeneous linear equation + - + =0, where = , =ù ( ) ú+ . (8) Page751 752 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES The corresponding homogeneous equation (with =0)hastwolinearly independent particular solutions:1= for ¹ + , for = + , 2= 1 ù ú2 1, where =exp û-1 ù ú !. (The ®rst solution isapparent from comparing equations (1)and(8)with =0.)The general solutions ofequations (8)and(2)aregivenby= "1 1+ "2 2+1 1 # 2 $ %-1 2 # 1 $ %, =# $ %+ "4. 3.1& ' (' ) '' &( *()±1& ' (' & '' )( *()= + * *(, *(=1& '' & , &' (' & !+1&2 '2(' )2. Theequations ofsteady-state ¯owofaviscous incompressible ¯uid (stationary Navier±Stok esequations) written inpolar coordinates ( -= .cos /, 0= .sin /)reduce tothisequation. Theradial andtangential components ofthe¯uid velocity are expressed interms ofthestream function 1bytheformulas 2 3=13 à54à6and 26=- à54à 3. 1».Exact solution inadditi veform:7( 8, 9)= "1 9+ "2 8 :1+2+ "3 82+ "4ln 8+ "5, where "1, ;5;5;, "5arearbitrary constants. 2».Exact solution:7( 8, 9)= <( 8) 9+ =( 8). Thefunctions <= <( 8)and == =( 8)aredetermined bythesystem ofordinary differential equations - < >L( <)+ <[L( <)] >= 8L2( <), (1) - = >L( <)+ <[L( =)] >= 8L2( =), (2) where L( <)= 8-1( 8 < >)>. Exact solution ofsystem (1)±(2):<( 8)= "1ln 8+ "2, =( 8)= "3 82+ "4ln 8+ "5 # ?@# 8 A( 8)$ 8 B$ 88+ "6,A( 8)=# 8(:2 CED)-1exp, "1 2 ln28 F$ 8, where G1, ;5;5;, G6arearbitrary constants. B.7.2. Nonstationar yHydr odynamic Equations 1.'' H( *()+' (' I '' J( *()±' (' J '' I( *()= + * *(, *(='2(' J2+'2(' I2. Thetwo-dimensional equations ofsteady-state ¯owofaviscous incompressible ¯uid (nonstationary Navier±Stok esequations) canbereduced tothisequation byintroducing astream function 7. 1».Exact solution:7( K,%, L)= M(%, L) K+ N(%, L), (1) where thefunctions M(%, L)and N= N(%, L)aredetermined from thesystem offourth-order one- dimensional equations O 3M OL O%2+ OM O% O 2M O%2- M O 3M O%3= O 4M O%4, (2)O 3N OL O%2+ ON O% O 2M O%2- M O 3N O%3= O 4N O%4. (3) Page752 B.7. F OURTH -ORDER NONLINEAR EQUATIONS 753 Equation (2) is independent of (3). Integrating (2) and (3) with respect to%yieldsO 2M OL O%+, OM O% F2 - M O 2M O%2= O 3M O%3+ <1( L), ( 4)O 2N OL O%+ OM O% ON O%- M O 2N O%2= O 3N O%3+ <2( L), ( 5) where <1( L) and <2( L) are arbitrary functions. Equation (5) is linear in N. The change of variableN=# P$ %- Q M+ Q R S%, whereP=P(%, L), M= M(%, L), ( 6) with Q= Q( L) satisfying the linear ordinary differential equationQ RR STS- <1( L) Q= <2( L), ( 7) brings (5) to the second-order homogenous linear equationOP OL= O 2P O%2+ M OP O%- OM O% P. ( 8) So, if a particular solution of equation (2) or (4) is known, then determining the function N reduces to solving the linear equations (7)±(8) followed by integrating in accordance with (6). TableB7listsexactsolution sofequatio n(2).Theordinar ydifferentia lequation sinthelasttwo rows, which determine a traveling wave solution and a self-similar solution, are autonomous and hence admit reduction of order. The general solution of the nonhomogeneous equation (7) can be found with the aid of the fundamental system of solutions for the corresponding homogeneous equation (with <2º 0). The necessary formulas and fundamental solutions of the homogeneous equation (7) that correspond to allexactsolution sofequatio n(2)listedinTableB7canbefoundinthehandbook sbyKam ke(1977) and Polyanin and Zaitsev (1995). Equation (8) for any function M= M(%, L) has the trivial solution,P= 0. The expressions in TableB7andrelatio n(6)withP= 0de®ne some exact solutions of the form (1). By analyzing nontrivial solutions of equation (8), one can obtain a wider class of exact solutions. TableB8liststransformation sthatsimplif yequatio n(8)forsomeofthesolution sofequatio n(2) [or(4)]giveninTableB7.Onecanseethatinthe®rsttwocases ,solution stoequatio n(8)are expressed in terms of solutions to the classical constant coef®cient heat equation. In the remainingthree cases, the equation reduces to a separable equation. 2». Exact solution of a more general form:7( K,%, L)= M( U, L) K+ N( U, L), U=%+ V K, where the functions M( U, L) and N= N( U, L) are determined from the system of fourth-order one- dimensional equationsO 3M OL OU2+ OM OU O 2M OU2- M O 3M OU3= ( V2+ 1) O 4M OU4, ( 9)O 3N OL OU2+ ON OU O 2M OU2- M O 3N OU3= ( V2+ 1) O 4N OU4+ 4 V O 3M OU3+2 VV2+ 1 , M O 2M OU2- O 2M OL OU F. ( 10) Integrating equations (9) and (10) with respect to UyieldsO 2M OL OU+, OM OU F2 - M O 2M OU2= ( V2+ 1) O 3M OU3+ <1( L), ( 11)O 2N OL OU+ OM OU ON OU- M O 2N OU2= ( V2+ 1) O 3N OU3+ A( U, L), ( 12) Page 753 754 B. M ETHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES TABLE B7 Exact solution of equations (2) and (4); W( L), X( L) are arbitrary functions and Y, Z, [are arbitrary constants NoFunction M= M(%, L) (or general form of solution)Function <1( L) in equation (4)Determining coef®cients (or determining equation) 1 M= W( L)%+ X( L)<1( L)= WR S+ W2N/A 2 M=6D\+ ]( S)+ XR S( L)<1( L)= 0 N/A 3 M= Yexp[- [%- [ X( L)]+ XR S( L)+ [ <1( L)= 0 N/A 4M= Yexp[- [%+ [ X( L)]+1[ L+ Z- X R S( L)+ [ <1( L)= 0 N/A 5 M= ^+ Yexp[- [%+ [ X( L)] 1+ Zexp(- [^ L)- X R S( L)+ [-^ <1( L)= 0^is an arbitrary constant 6 M= Y _- ` S sin[ [%+ [ X( L)]+ XR S( L)<1( L)= Z _-2 ` S^= [2, Z= Y2[2>0 7 M= Y _- ` S cos[ [%+ [ X( L)]+ XR S( L)<1( L)= Z _-2 ` S^= [2, Z= Y2[2>0 8 M= Y _ ` S sinh[ [%+ [ X( L)]+ XR S( L)<1( L)= Z _2 ` S^= [2, Z= Y2[2>0 9 M= Y _ ` S cosh[ [%+ [ X( L)]+ XR S( L)<1( L)= Z _2 ` S^= [2, Z= - Y2[2<0 10 M= X( L) _ a \- Y _ ` S-a \ 4 [2X( L)+ XR S( L)[ X( L)- [ <1( L)= Y _ ` S^= 2 [2 11 M= M( U), U=%+ [ L <1( L)= Y - Y+ [ MRR bcb+( MR b)2- M MRR bcb= MRRR bcbcb 12 M= L-1C2 de( U)-1 2 U f, U=% L-1C2<1( L)= Y L-2 3 4- Y-2 eR b+( eR b)2- e eRR bcb= eRRR bcbcb where <1( L) is an arbitrary function andA( U, L)= 4 V O 2M OU2-2 VV2+ 1 OM OL+2 VV2+ 1 g M O 2M OU2 h U+ i2( L) [ i2( L) is any]. Equation (12) is linear in N. The change of variableP= j kj btakes it to the second-order linear equation OP OL= ( V2+ 1) O 2P OU2+ M OP OU- OM OU P+ A( U, L). ( 13) So, if a particular solution of equation (9) or (11) is known, then determining the function N reduces to solving the linear equation (13). Scaling the independent variables by the formulasU=( V2+ 1) land L=( V2+ 1) m, one can reduce equation (9) to equation (2) in which nand Lmust bereplace dby land m,respect ively.Exac tsolution sofequatio n(2)arelistedinTableB7. Page 754 B.7. F OURTH -ORDER NONLINEAR EQUATIONS 755 TABLE B8 Transformations of equation (8) for the corresponding exact solutions of equation (4) [the number in the ®rst column corresponds to the numbe roftheexactsolutio n M= M( n, L)inTableB7] No Transformations of equation (8) Resulting equation 1 P=1o( S) p( q, m), m= r s2( L)h L,q= n s( L)+ r X( L) s( L)h L, s( L)=exp dr W( L)h Lf j tj u= j2tj v2 2P= l-3p( l, L), l= n+ X( L) j tj S= j2tj w2 3P= _ x y( z, L), z= - [ n- [ X( L)j {j S= [2j2{j x2+( [2- Y [ _ x) j {j x 11P=p( U, L), U= n+ [ Lj tj S= j2tj b2+ dM( U)- [f j tj b- MR b( U)p 12P= L-1 |2p( U, m), U= n L-1 |2, m=ln L j tj u= j2tj b2+ e( U)j tj b+ d1 - eR b( U)fp 3». Exact solution [special case of (1)]:}( K, n, L)= _-a \di( L) K+ ~( )f+ W( ) + X( ) n+ ( ),i( )= 1 ( ),( )=exp 2- g ( )h ,~( )= 2 ( )- 1 ( )g ( )h , where( ),( ), and ( ) are arbitrary functions and 1, 2, and are arbitrary parameters. 4 . Exact solution:}( , n, )= - ( ) + ( ) - +( ) +( ) n+ ( ),( )= 1exp ( 2+ 2) - g ( )h - g ( )h ,( )= 2exp ( 2+ 2) + g ( )h - g ( )h , where( ),( ), and ( ) are arbitrary functions and 1, 2, , and are arbitrary parameters. 5 . Exact solution:}( , n, )= - ( ) sin( )+ ( ) cos( ) +( ) +( ) n+ ( ), where( ),( ), and ( ) are arbitrary functions, and are arbitrary parameters, and the functions( ) and ( ) are determined by the nonautonomous system of linear ordinary differential equations = ( 2- 2)- ( ) + ( ) , = ( 2- 2)- ( ) - ( ) .(14) The general solution of system (14) is given by( )=exp ( 2- 2) - g h 1sin g h + 2cos g h ,( )=exp ( 2- 2) - g h 1cos g h - 2sin g h , Page 755 756 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES where=( )and=( ); 1and 2arearbitrary constants. Inparticular ,if= ( 2- 2) and= ,weobtain aperiodic solution( )= 1sin( )+ 2cos( ),( )= 1cos( )- 2sin( ). 6 .Exact solutions:}( , n, )= ( )exp( 1 + 1 n)+ ( )exp( 2 + 2 n)+( ) +( ) n+ ( ), where( ),( ),and ( )arearbitrary functions and 1, 1, 2,and 2arearbitrary parameters that satisfy oneofthetworelations 2 1+ 2 1= 2 2+ 2 2(®rst family ofsolutions ), 1 2= 2 1 (second family ofsolutions ), andthefunctions ( )and ( )aredetermined bytheordinary differential equations = ( 2 1+ 2 1)+ 1( )- 1( ) , = ( 2 2+ 2 2)+ 2( )- 2( ) . These equations areeasy tointegrate:( )= 1exp ( 2 1+ 2 1) + 1g ( )h - 1g ( )h ,( )= 2exp ( 2 2+ 2 2) + 2g ( )h - 2g ( )h . 7 .Exact solution:}( , n, )= 1sin( )+ 2cos( ) ¡( )sin( n)+ ( )cos( n) +( ) + ( ), where( )and ( )arearbitrary functions, 1, 2, ,and arearbitrary parameters, andthe functions ( )and ( )aredetermined bythenonautonomous system oflinear ordinary differential equations =- ( 2+ 2) - ( ) , =- ( 2+ 2) + ( ) .(15) Thegeneral solution ofsystem (15) isgivenby( )=exp - ( 2+ 2) 3sin g h + 4cos g h ,=( ),( )=exp - ( 2+ 2) - 3cos g h + 4sin g h , where 3and 4arearbitrary constants. 8 .Exact solution:}( , n, )= 1sinh( )+ 2cosh( ) ( )sin( n)+ ( )cos( n)+( ) + ( ), where( )and ( )arearbitrary functions, 1, 2, ,and arearbitrary parameters, andthe functions ( )and ( )aredetermined bythenonautonomous system oflinear ordinary differential equations = ( 2- 2) - ( ) , = ( 2- 2) + ( ) .(16) Thegeneral solution ofsystem (16) isgivenby( )=exp ( 2- 2) 3sin g h + 4cos g h ,=( ),( )=exp ( 2- 2) - 3cos g h + 4sin g h , where 3and 4arearbitrary constants. Page756 B.8. HIGHER -ORDER NONLINEAR EQUATIONS 757 9 .Exact solution:}( , n, )=p( q, )+( ) +( ) n, q= + n, where( )and( )arearbitrary functions, and arearbitrary parameters, andthefunctionp( q, ) isdetermined bythefourth-order linear equation¢3p¢ ¢q2+ ( )- ( ) ¢3p¢q3= ( 2+ 2) ¢4p¢q4. Thetransformation £ ( ¤, )= ¢2p¢q2, ¤= q- ¥ ( ¦)- ( ¦) § ¦ brings ittothecustomary heat equation¢ £¢¦= ¨( 2+ 2) ¢2 £¢¤2.©ª Refer ence:A.D.Polyanin (2001d). 2. « ¬« +1® « ¯« ° « ¬« ®±1® « ¯« ® « ¬« °= ± ²¬,¬= ²¯=1® «« ® ®« ¯« ® +1®2 «2¯« °2. The two-dimensional equations ofsteady-state ¯owofaviscous incompressible ¯uid written in polar coordinates arereduced tothisequation ( ³isthestream function). Exact solution:³( ´, µ, ¦)= ¶( ´, ¦) µ+ ·( ´, ¦). Thefunctions ¶= ¶( ´, ¦)and ·= ·( ´, ¦)satisfy thesystem ofequations L( ¶ )- ´-1¶ ¸L( ¶)+ ´-1¶[L( ¶)] ¸= ¨L2( ¶), (1) L( · )- ´-1· ¸L( ¶)+ ´-1¶[L( ·)] ¸= ¨L2( ·), (2) where thesubscripts ´and ¦denote partial derivatives;L( ¶)= ´-1( ´ ¶ ¸) ¸andL2( ¶)=LL( ¶). Fortheparticular solution ¶= ¹( ¦)ln ´+ º( ¦)ofequation (1),with ¹and ºarbitrary ,equation (2) canbereduced bythechange ofvariable £ =L( ·)toasecond-order linear equation. B.8. Higher -OrderNonlinear Equations B.8.1. Equations oftheForm » ¼» ½= ¾ ¿ À,½,¼, » ¼» À, Á Á Á, » Â ¼» ÀÂ Ã 1. « ¯« = Ä « Å ¯« ÆÅ+ Ç¯ln¯+ È()¯. 1 É.Exact solution:³( , ¦)=exp ÊTË Ì ÍÎ+ Ï Ì ÍÎ+ Ð Ë ÑÒ( Ó-1) Ì ÑÍÎ+ Ì ÍÎ ¥ Ì-ÍÎÔ¶( ¦)§ ¦Õ, where Ëand Ïarearbitrary constants. 2 É.Exact solution:³( , ¦)=exp ÊË Ì ÍÎ+ Ì ÍÎ ¥ Ì-ÍÎÔ¶( ¦)§ ¦Õ ¹( Ö), Ö= + × ¦, where Ëand ×arearbitrary constants andthefunction ¹= ¹( Ö)isdetermined from theautonomous ordinary differential equationÐ ¹(Ñ)v- × ¹ Øv+ Ò¹ln ¹=0, whose order canbereduced byone. Page757 758 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 2. « ¯« = Ä « Å ¯« ÆÅ+ Ç¯ln¯+[ È(Æ)+ Ù()]¯. Exact solution inmultiplicati veform:³( , ¦)=exp ÊÚ Ì ÍÎ+ Ì ÍÎ ¥ Ì-ÍÎ@·( ¦)§ ¦Õ ¹( Û), where Úisanarbitrary constant andthefunction ¹( ¦)isdetermined bytheordinary differential equationÐ ¹(Ñ)Ü+ Ò¹ln ¹+ ¶( Û) ¹=0. 3. « ¯« = Ä « Å ¯« ÆÅ+ È()¯ln¯+ Ù()¯. Exact solution:³( Û, ¦)=exp Ý¹( ¦) Û+ º( ¦) Þ. Thefunctions ¹( ¦)and º( ¦)aredetermined from¹( ¦)= Ë Ì ß, º( ¦)= Ï Ì ß+ Ì ß à Ì-ß(Ð Ë ÑÌ Ñß+ á) â ã, ä= à å â ã, where Ëand Ïarearbitrary constants. 4. æ çæ = è æ é çæ êé+ ë()çlnç+[ Ù()ê+ ì()]ç. Exact solution: í ( Û, ã)=exp Ýî( ã) Û+ º( ã) Þ. Thefunctions î( ã)and º( ã)aredetermined fromî( ã)= ï ð ß+ ð ß à ð-ß á â ã, ä= à å â ã,º( ã)= ñ ð ß+ ð ßà ð-ß( ò î ó+ ô) â ã, where ïand ñarearbitrary constants. 5. æ çæ = è æ é çæ êé+ õ ö æ çæ ê ÷2 + øç+ ë(). 1 ù.Exact solution: í ( Û, ã)= î2( ã) Û2+ î1( ã) Û+ î0( ã), where thefunctions î ú( ã)satisfy anappropriate system ofordinary differential equations. 2 ù.Exact solution:í ( Û, ã)= ï ð û@ü+ ð û@ü à ð-û@üÔå( ã) â ã+ ý( þ), þ= Û+ ÿ ã, where ïand ÿarearbitrary constants, andthefunction ý( þ)isdetermined bysolving theauton omous ordinary differential equationò ý(ó)+ ý 2- ÿ ý + ý=0. 6. æ çæ = è æ éçæ êé+ õ ö æ çæ ê÷2 + øç æ çæ ê+ ç2+ ë()ç+ (). Exact solution: í ( Û, ã)= î( ã)+ º( ã)exp( ÿ Û), Page758 B.8. HIGHER -ORDER NONLINEAR EQUATIONS 759 where ÿisaroot ofthequadratic equation ÿ2+ 5ÿ+ =0,andthefunctions î( ã)and º( ã)are determined from thesystem of®rst-order ordinary differential equationsî ü= î2+ å( ã) î+ á( ã), (1)º ü= ( 5ÿ+2 ) î+ å( ã)+ ò ÿó º. (2) FortheRiccati equation (1),seeKamk e(1977) andPolyanin andZaitse v(1995). Itisintegrable inquadrature if,forexample, (a) =0, (b) á( ã)º0, (c) å( ã)=const , á( ã)=const . Onsolving equation (1),onecanreadily solveequation (2),which islinear in º. 7. æ çæ = è æ é çæ êé+ ë() ö æ çæ ê ÷2 + õ ë()ç2+ ()ç+ ì(). 1 ù.Exact solution: í ( Û, ã)= î( ã)+ º( ã)exp Û - , <0, where thefunctions î( ã)and º( ã)aredetermined bysolving thefollowing ®rst-order ordinary differential equations with variable coef®cients (thearguments of å, á,and ôarenotspeci®ed):î ü= å î2+ á î+ ô, (1)º ü= 2 å î+ á+ ò - ó º. (2) Equation (1)isaRiccati equation for î= î( ã);itcanbereduced toasecond-order linear equation. Alotofexact solutions toequation (1)with various å, á,and ôcanbefound inKamk e (1977) andPolyanin andZaitse v(1995). Givenasolution ofequation (1),thecorresponding solution of(2)iscalculated byº( ã)= exp ò - óã+ à(2 å î+ á) â ã, where isanarbitrary constant. 2 ù.Exact solution ofamore general form:í ( Û, ã)= î( ã)+ º( ã)exp Û - + ( ã)exp - Û - , <0, (3) where thefunctions î( ã), ( ã),and ( ã)aredetermined bythefollowing system of®rst-order ordinary differential equations with variable coef®cients:î ü= å î2+ á î+ ô+4 å , (4) ü= 2 å î+ á+ ò - ó , (5) ü= 2 å î+ á+ ò - - ó . (6) Forequations ofevenorder with =2 ( =1,2, ),itfollowsfrom (5)and(6)thatthe functions ( ã)and ( ã)areproportional. Setting ( ã)= ï ( ã)and ( ã)= ñ ( ã),onecanrewrite solution (3)así ( Û, ã)= î( ã)+ ( ã) ïexp Û - + ñexp - Û - , <0, where thefunctions î( ã)and ( ã)aredetermined bythesystem ofordinary differential equationsî ü= å î2+4 ï ñ 2)+ á î+ ô, (7) ü= 2 å î+ á+(-1) ò . (8) Onexpressing îfrom (8)interms of andsubstituting theresult into (7),onearrivesata second-order nonlinear equation for ;if å, á, ô=const ,thisequation isautonomous andhence admits reduction oforder . Page759 760 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 3 ù.Exact solution:í ( Û, ã)= î( ã)+ ( ã)cos Û + ( ã)sin Û , >0, where thefunctions î( ã), ( ã),and ( ã)aredetermined byasystem ofordinary differential equations (notspeci®ed here). Forequations ofevenorder with =2 ( =1,2, ),there areexact solutions with the following form ( isanynumber):í ( Û, ã)= î( ã)+ ( ã)cos @Û + , >0. Thefunctions î( ã)and ( ã)aredetermined bythefollowing system of®rst-order ordinary differential equations with variable coef®cients:î ü= å î2+ 2)+ á î+ ô, (9) ü= 2 å î+ á+(-1) ò . (10) Onexpressing îfrom (10) interms of andsubstituting theresult into (9),onearrivesata second-order nonlinear equation for ;if å, á, ô=const ,thisequation isautonomous andhence admits reduction oforder . 8. æ çæ = è æ é çæ êé+ ë öê, æ çæ ê ÷+ (). Exact solution inadditi veform:í ( Û, ã)= ï ã+ ñ+ à á( ã) â ã+ î( Û). Here, ïand ñarearbitrary constants andthefunction î( Û)isdetermined from thenonlinear ordinary differential equationò î(ó)Ü+ å Û, î Ü - ï=0. 9. æ çæ = è æ é çæ êé+ ë öê, æ çæ ê ÷+ õç+ (). Exact solution inadditi veform:í ( Û, ã)= î( Û)+ ï ð ü+ ð üà ð- üá( ã) â ã. Here, ïisanarbitrary constant andthefunction î( Û)isdetermined from thenonlinear ordinary differential equationò î(ó)Ü+ å Û, î Ü + î=0. 10. æ çæ = èç æ é çæ êé+ ë()ç+ (). 1 ù.Exact solution:í ( Û, ã)= ä( ã) ïó-1 Ûó-1+ + ï1 Û+ ï0 + ä( ã) à á( ã)ä( ã) â ã, ä( ã)=exp @à å( ã) â ã, where ï0, ï1, , ïó-1arearbitrary constants. 2 ù.Exact solution:í ( Û, ã)= î( ã) Û ó+ ïó-1 Û ó-1+ + ï1 Û+ ï0 + î( ã) à á( ã)î( ã) â ã,î( ã)= ä( ã) - ò ! à ä( ã) â ã-1 , ä( ã)=exp @à å( ã) â ã, where ï0, ï1, , ïó-1,and arearbitrary constants. Page760 B.8. HIGHER -ORDER NONLINEAR EQUATIONS 761 11. = + 2+ ( ) + ( ). Exact solution: ( , )= ( ) ( )+ ( ), where thefunctions ( )and ( )aredetermined from thefollowing system of®rst-order ordinary differential equations ( isanynumber): = 2+ + ( ) , = + 2+ ( ) + ( ), andthefunction ( )isdetermined bythe th-order linear ordinary differential equation ( )+ = . 12. = + ( ) + ( ) + ( ). Exact solution: ( , )= ( ) ( )+ ( ), where thefunctions ( ), ( ),and ( )aredetermined bytheordinary differential equations = 2+ ( ) , = + ( ) + ( ), ( )+ ( ) = , where isanarbitrary constant. Integrating successi vely,for ( )and ( )weobtain( )= !( ) "$#- % !( ) & ('-1 , !( )=exp ")% ( ) & (',( )= * ( )+ ( ) % ( )( ) & , where #and *arearbitrary constants. B.8.2. Equations oftheForm +2 ,+ -2= . / 0,-, ,, + ,+ 0, 1 1 1, + 2 ,+ 02 3 1. 2 2= + ( ) + ln +[ ( )+ ( )] . Exact solution inmultiplicati veform: ( , )= ( ) ( ). Thefunctions ( )and ( )aredetermined bytheordinary differential equations 4 $- ln + ( )+ =0,( )+ ( ) + ln + ( )- =0, where isanarbitrary constant. Page761 762 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 2. 2 2= + 5 62 + 7 + ( ). 1 8.Exact solution: ( , )= 2( ) 2+ 1( ) + 0( ), where thefunctions 9( )satisfy anappropriate system ofordinary differential equations. 2 8.Exact solution: ( , )= ( )+ ( :), := + ; . Thefunctions ( )and ( :)aredetermined from theordinary differential equations 4 $- <- ( )= #, ( )=- ;2 4 =>=+ ?@ = A2+ < = #, where ;and #arearbitrary constants. 3. 2 2= + 5 62 + 7 + B 2+ ( ) + ( ). Exact solution: ( , )= ( )+ ( )exp( ; ), where ;isaroot ofthequadratic equation C;2+ <;+ D=0,andthefunctions ( )and ( )are determined from thesystem ofsecond-order ordinary differential equations 4 $= D 2+ ( ) + ( ), (1) 4 $=( <;+2 D) + ( )+ ; . (2) Inthespecial case ( )=const and ( )=const ,equation (1)isautonomous andhasparticular solutions oftheform =const and, hence, canbeintegrated inquadrature. Equation (2)islinear in ;therefore, for =const ,itsgeneral solution isexpressed interms ofexponentials orsineand cosine. 4. 2 2= + ( ) 5 62 + ( )+ ( ). Exact solution inadditi veform: ( , )=1 2 # 2+ * + + % 0( - E) ( E) & E+ ( ). Here, #, *,and arearbitrary constants, andthefunction ( )isdetermined bysolving the nonlinear ordinary differential equation( )+ ( ) ? A2+ ( )- #=0. 5. 2 2= + ( ) 5 62 + + ( )+ ( ). Exact solution inadditi veform: ( , )= ( )+ ( ). Thefunctions ( )and ( )aredetermined bysolving thenonlinear ordinary differential equations 4 $- - ( )=0,( )+ ( )( )2+ + ( )=0. Thegeneral solution ofthe®rstequation isgivenby( )= 1cosh( D )+ 2sinh( D )+1D % 0 ( E)sinh[ D( - E)] & Efor = D2>0,( )= 1cos( D )+ 2sin( D )+1D % 0 ( E)sin[ D( - E)] & E for =- D2<0, where 1and 2arearbitrary constants. Page762 B.8. HIGHER -ORDER NONLINEAR EQUATIONS 763 6. 2 2= 2 2 + ( ) 5 62 + ( ) 2+ ( ) + ( ). 1 8.Exact solution: ( , )= ( )+ ( )exp ?GF H- A, <0, where thefunctions ( )and ( )aredetermined bysolving thefollowing ®rst-order ordinary differential equations with variable coef®cients (thearguments of , ,and arenotspeci®ed): 4 $= C 2+ + , (1) 4 $= 2 C + +(-1) C. (2) Inthespecial case where , , areconstant, equation (1)hasparticular solutions oftheform=const .Here, thegeneral solution ofequation (2)isexpressed interms ofexponentials orsine andcosine. 2 8.Exact solution ofamore general form: ( , )= ( )+ ( ) #exp ?I H- A+ *exp ?- H- A, <0, where thefunctions ( )and ( )aredetermined bysolving thefollowing system ofsecond-order ordinary differential equations with variable coef®cients: 4 $= C( 2+4 # * 2)+ + , (3) 4 $=[2 C + +(-1) ] . (4) Onexpressing from (4)interms of andsubstituting theresult into (3),onearrivesata fourth-order nonlinear equation for ;if , , =const ,thisequation isautonomous andhence admits reduction oforder . 3 8.Exact solution ( <isanarbitrary constant): ( , )= ( )+ ( )cos ? H + < A, >0, where thefunctions ( )and ( )aredetermined bysolving thefollowing system ofsecond-order ordinary differential equations with variable coef®cients: 4 $= C ?I2+ 2)+ + , 4 $=2 C + +(-1) . 7. 2 2= + 5 , 6+ ( ). Exact solution inadditi veform: ( , )=1 2 # 2+ * + + % 0( - E) ( E) & E+ ( ). Here, #, *,and arearbitrary constants andthefunction ( )isdetermined bythenonlinear ordinary differential equation( )+ ?G , A- #=0. Page763 764 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 8. 2 2= + 5 , 6+ + ( ). Exact solution inadditi veform: ( , )= ( )+ ( ). Thefunctions ( )and ( )aredetermined bytheordinary differential equations 4 $- - ( )=0,( )+ ?I , A+ =0. Thegeneral solution ofthe®rstequation isgivenby( )= 1cosh( D )+ 2sinh( D )+1D % 0 ( E)sinh[ D( - E)] & Efor = D2>0,( )= 1cos( D )+ 2sin( D )+1D % 0 ( E)sin[ D( - E)] & E for =- D2<0, where 1and 2arearbitrary constants. 9. 2 2= + 5 ,1 6. Exact solution inmultiplicati veform: ( , )= J K ( ), where ;isanarbitrary constant andthefunction ( )isdetermined from thesecond-order linear ordinary differential equation 4 $= ; + ( , ;) . 10. 2 2= + ( ) + ( ). Exact solution: ( , )= ( ) ?)# + LLL+ #1 A+ ( ), where #1, MMM, #arearbitrary constants andthefunctions ( )and ( )aredetermined from the second-order ordinary differential equations 4 $= # ! 2+ ( ) , 4 $= # ! + ( ) + ( ). 11. 2 2= + 2+ ( ) + ( ). Exact solution: ( , )= ( ) ( )+ ( ), where thefunctions ( )and ( )aredetermined bythefollowing system ofsecond-order ordinary differential equations ( isanarbitrary constant): 4 $= 2+ + ( ) , 4 $= + 2+ ( ) + ( ). Thefunction ( )satis®es the th-order linear ordinary differential equation ( )+ = . Page764 B.8. HIGHER -ORDER NONLINEAR EQUATIONS 765 12. 2 2= + ( ) + ( ) + ( ). Exact solution: ( , )= ( ) ( )+ ( ), where thefunctions ( ), ( ),and ( )aredetermined bytheordinary differential equations 4 $= 2+ ( ) , 4 $=[ + ( )] + ( ), ( )+ ( ) = , where isanarbitrary constant. B.8.3. Other Equations 1. N 2 N± 2 N2= ( ) N. This isaspecial case ofequation B.8.3.3. Exact solution: ( , O)= ( ) JK P- ; -2% ( ) & + , where ( )isanarbitrary function; and ;arearbitrary constants. 2. N 2 N± 2 N2= ( ) 2 N2 + ( ). This isaspecial case ofequation B.8.3.3. Exact solution: ( , O)= ( ) JK P-1 2 ;2( ) ")% ( ) & + 1 ' J-K P- ;2 -2% ( ) & + 2, where ( )isanarbitrary function and 1, 2,and ;arearbitrary parameters. 3. N 2 N± 2 N2= Q 5 , , N, R R R, N 6. 1 8.If ( , O)isasolution oftheequation inquestion, then thefunction 1( , O)= ? , O+ ( ) A, where ( )isanarbitrary function, isalsoasolution oftheequation. 2 8.Lettheright-hand side oftheequation beindependent of explicitly .Then there areexact solutions oftheform = ( S), S= O+ ( ), where ( )isanarbitrary function and ( S)isasolution oftheordinary differential equationT? , U, MMM, ( )U A=0. 3 8.Lettheright-hand side oftheequation beindependent of and explicitly .Then there are exact solutions oftheform = + ( S), S= O+ ( ), where ( )isanarbitrary function, isanarbitrary constant, and ( S)isasolution oftheordinary differential equation T?G U, MMM, ( )U A+ 4 UU=0. Page765 766 B.METHODS OFGENERALIZED AND FUNCTION ALSEPARA TION OFVARIABLES 4. = Q( , V0, V1, R R R, V ), V W= XY= W(±1) Y+ WB!( Z± B)! Y± W Y Y, B=0,1, R R R, [. Exact solution inmultiplicati veform: ( , )=( 0+ 1 + LLL+ ) ( ), where 0, 1, MMM, arearbitrary constants andthefunction = ( )satis®es theordinary differential equation = T( , 0 , 1 , MMM, ).\^] Refer ence:Ph.W.Doyle(1996), thecase _a`)b º0wasconsidered. 5. c2 dc e2= dQ(e, V0, V1, R R R, V f), V W= fXY= W(±1) Y+ WB!( Z± B)! g Y± Wc Ydcg Y, B=0,1, R R R, [. Exact solution inmultiplicati veform:h( i, j)=( k0+ k1 i+ LLL+ k l i l) m( j), where k0, k1, MMM, k larearbitrary constants andthefunction m= m( j)satis®es theordinary differential equationm n4n o$o= m T( j, k0 m, k1 m, MMM, k l m). 6. Q 5g,1d c dcg, R R R,1d c fdcg f;1d c dc N, R R R,1d c p dc Np 6=0. Exact solution inmultiplicati veform:h( i, O)= q JK P m( i), where qand ;arearbitrary constants andthefunction m( i)isdetermined from the rth-order ordinary differential equationT?i, m ns tm, MMM, m( l)s tm; ;, MMM, ; u A=0. 7. Q 5g,1d c dcg, R R R,1d c fdcg f;1d c2dc N2, R R R,1d c2p dc N2p 6=0. 1 8.Exact solution:h( i, O)= v^qcosh( ; O)+ wsinh( ; O) xCm( i), where q, w,and ;arearbitrary constants andthefunction m( i)isdetermined bysolving therth-order ordinary differential equationT?Ii, m ns tm, MMM, m( l)s tm; ;2, MMM, ;2u A=0. 2 8.Exact solution:h( i, O)=v qcos( ; O)+ wsin( ; O)x m( i), where q, w,and ;arearbitrary constants andthefunction m( i)isdetermined bysolving therth-order ordinary differential equationT?Ii, m ns tm, MMM, m( l)s tm;- ;2, MMM,(-1)u ;2u A=0. 8. y1 5g, c dcg, R R R, c fdcg f 6+ y2 5 N, c dc N, R R R, c p dc Np 6= B d. Exact solution inadditi veform:h( i, O)= m( i)+ z( O). Thefunctions m( i)and z( O)aredetermined bytheordinary differential equations{ 1 ?Ii, m ns, MMM, m( l)s A- D m= k,{ 2 ?IO, z nP, MMM, z(u)P A- D z=- k, where kisanarbitrary constant. Page766 B.8. HIGHER -ORDER NONLINEAR EQUATIONS 767 9. y1 5g,1d c dcg, R R R,1d c fdcg f 6+ d Wy2 5 N,1d c dc N, R R R,1d c p dc Np 6=0. Exact solution inmultiplicati veform:h( i, O)= m( i) z( O). Thefunctions m( i)and z( O)aredetermined bytheordinary differential equationsm- 9 { 1 ?Ii, m ns tm, MMM, m( l)s tm A= k,z 9 { 2 ?IO, z nP tz, MMM, z(u)P tz A=- k, where kisanarbitrary constant. 10. y1 5g, c dcg, R R R, c fdcg f 6+ | } ~ y2 5 N, c dc N, R R R, c p dc Np 6=0. Exact solution inadditi veform:h( i, O)= m( i)+ z( O). Thefunctions m( i)and z( O)aredetermined bytheordinary differential equationsJ-K { 1 ?i, m ns, MMM, m( l)s A= k,J K { 2 ?O, z nP, MMM, z(u)P A=- k, where kisanarbitrary constant. 11. y1 g,1d c dcg, ,1d c fdcg f + y2 ,1d c dc , ,1d c p dc p = ln d. Exact solution inmultiplicati veform:h( i, )= m( i) z( ). Thefunctions m( i)and z( )aredetermined bytheordinary differential equations{ 1 i, m ns tm, , m( l)s tm - ln m= k,{ 2 , z n tz, , z(u)tz- ln z=- k, where kisanarbitrary constant. 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