Zwillinger+D[1].+Handbook+of+Differential+Equations+_3ed._+AP_+1997_

Handb ookofDi®eren tialEquations 3rdedition Daniel Zwillinger Academic Press, 1997 Contents Preface Introduction Introduction totheElectronic Version HowtoUseThisBook I.ADe¯nitions andConcepts 1De¯nition ofTerms..........................2 2Alternativ eTheorems ........................15 3Bifurcation Theory ..........................19 4ACaveatforPartial Di®eren tialEquations ............27 5Chaos inDynamical Systems ....................29 6Classi¯cation ofPartial Di®eren tialEquations ...........36 7Compatible Systems .........................43 8Conserv ation Laws..........................47 9Di®eren tialResultan ts........................50 10Existence andUniqueness Theorems ................53 11Fixed PointExistence Theorems ..................58 12Hamilton-Jacobi Theory .......................61 13Integrabilit yofSystems .......................65 14Internet Resources ..........................71 15InverseProblems ...........................75 16Limit Cycles .............................78 17Natural Boundary Conditions foraPDE..............83 18Normal Forms: Near-Iden tityTransformations ..........86 19Random Di®eren tialEquations ...................91 20Self-Adjoin tEigenfunction Problems ................95 21Stabilit yTheorems ..........................101 22Sturm-Liouville Theory .......................103 23Variational Equations ........................109 24WellPosedDi®eren tialEquations ..................115 25Wronskians andFundamen talSolutions ..............119 26Zeros ofSolutions ...........................123 I.BTransformations 27Canonical Forms ...........................128 28Canonical Transformations .....................132 29DarbouxTransformation .......................135 30AnInvolutory Transformation ....................139 31Liouville Transformation -1.....................141 32Liouville Transformation -2.....................144 33Reduction ofLinear ODEs toaFirstOrder System ........146 34Prufer Transformation ........................148 35Modi¯ed Prufer Transformation ...................150 36Transformations ofSecond Order Linear ODEs -1........152 37Transformations ofSecond Order Linear ODEs -2........157 38Transformation ofanODE toanIntegral Equation ........159 39Miscellaneous ODE Transformations ................162 40Reduction ofPDEs toaFirstOrder System ............166 41Transforming Partial Di®eren tialEquations ............168 42Transformations ofPartial Di®eren tialEquations .........173 IIExact Analytical Metho ds 43Introduction toExact Analytical Metho ds.............178 44Look-Up Technique ..........................179 45Look-Up ODE Forms.........................219 II.AExact Metho dsforODEs 46AnNthOrder Equation .......................224 47UseoftheAdjoin tEquation .....................226 48Autonomous Equations -Indep enden tVariable Missing .....230 49Bernoulli Equation ..........................235 50Clairaut's Equation ..........................237 51Computer-Aided Solution ......................240 52Constan tCoe±cien tLinear Equations ...............247 53ContactTransformation .......................249 54DelayEquations ...........................253 55Dependen tVariable Missing .....................260 56Di®eren tiation Metho d........................262 57Di®eren tialEquations withDiscon tinuities .............264 58Eigenfunction Expansions ......................268 59Equidimensional-in-x Equations ...................275 60Equidimensional-in-y Equations ...................278 61Euler Equations ............................281 62Exact FirstOrder Equations ....................284 63Exact Second Order Equations ...................287 64Exact NthOrder Equations .....................290 65Factoring Equations .........................292 66Factoring Operators .........................294 67Factorization Metho d........................300 68Fokker-Planc kEquation .......................303 69Fractional Di®eren tialEquations ..................308 70FreeBoundary Problems .......................311 71Generating Functions .........................315 72Green's Functions ..........................318 73Homogeneous Equations .......................327 74Metho dofImages ..........................330 75Integrable Combinations .......................334 76Integral Represen tation: Laplace's Metho d.............336 77Integral Transforms: Finite Intervals................342 78Integral Transforms: In¯nite Intervals...............347 79Integrating Factors ..........................356 80Interchanging Dependen tandIndep enden tVariables .......360 81Lagrange's Equation .........................363 82LieGroups: ODEs ..........................366 83Operational Calculus .........................379 84Pfa±an Di®eren tialEquations ....................384 85Reduction ofOrder ..........................389 86Riccati Equations ...........................392 87Matrix Riccati Equations ......................395 88Scale InvariantEquations ......................398 89Separable Equations .........................401 90Series Solution ............................403 91Equations Solvableforx.......................409 92Equations Solvablefory.......................411 93Superposition .............................413 94Metho dofUndetermined Coe±cien ts................415 95Variation ofParameters .......................418 96Vector Ordinary Di®eren tialEquations ...............421 II.BExact Metho dsforPDEs 97Backlund Transformations ......................428 98Metho dofCharacteristics ......................432 99Characteristic Strip Equations ...................438 100Conformal Mappings .........................441 101Metho dofDescen t..........................446 102Diagonalization ofaLinear System ofPDEs ............449 103Duhamel's Principle .........................451 104Exact Equations ...........................454 105Hodograph Transformation .....................456 106InverseScattering ...........................460 107Jacobi's Metho d...........................464 108Legendre Transformation ......................467 109LieGroups: PDEs ..........................471 110Poisson Formula...........................478 111Riemann's Metho d..........................481 112Separation ofVariables ........................487 113Separable Equations: StackelMatrix ................494 114Similarit yMetho ds..........................497 115Exact Solutions totheWaveEquation ...............501 116Wiener-Hopf Technique .......................505 IIIAppro ximate Analytical Metho ds 117Introduction toAppro ximate Analysis ...............510 118Chaplygin's Metho d.........................511 119Collocation ..............................514 120Dominan tBalance ..........................517 121Equation Splitting ..........................520 122FloquetTheory ............................523 123Graphical Analysis: ThePhase Plane ...............526 124Graphical Analysis: TheTangen tField...............532 125Harmonic Balance ..........................535 126Homogenization ............................538 127Integral Metho ds...........................542 128IntervalAnalysis ...........................545 129Least Squares Metho d........................549 130Lyapuno vFunctions .........................551 131Equiv alentLinearization andNonlinearization ...........555 132Maxim umPrinciples .........................560 133McGarv eyIteration Technique ...................566 134Momen tEquations: Closure .....................568 135Momen tEquations: ItoCalculus ..................572 136Monge's Metho d...........................575 137Newton's Metho d...........................578 138PadeAppro ximan ts.........................582 139Perturbation Metho d:Metho dofAveraging ............586 140Perturbation Metho d:Boundary LayerMetho d..........590 141Perturbation Metho d:Functional Iteration ............598 142Perturbation Metho d:Multiple Scales ...............605 143Perturbation Metho d:Regular Perturbation ............610 144Perturbation Metho d:Strained Coordinates ............614 145Picard Iteration ............................618 146Reversion Metho d..........................621 147Singular Solutions ..........................623 148Soliton-T ypeSolutions ........................626 149Stochastic Limit Theorems .....................629 150TaylorSeries Solutions ........................632 151Variational Metho d:EigenvalueAppro ximation ..........635 152Variational Metho d:Rayleigh-Ritz .................638 153WKB Metho d.............................642 IV.A Numerical Metho ds:Concepts 154Introduction toNumerical Metho ds.................648 155De¯nition ofTermsforNumerical Metho ds............651 156Available Software..........................654 157Finite Di®erence Formulas......................661 158Finite Di®erence Metho dology ....................670 159GridGeneration ...........................675 160Richardson Extrap olation ......................679 161Stabilit y:ODE Appro ximations ...................683 162Stabilit y:Couran tCriterion .....................688 163Stabilit y:VonNeumann Test....................692 164Testing Di®eren tialEquation Routines ...............694 IV.BNumerical Metho dsforODEs 165Analytic Continuation ........................698 166Boundary ValueProblems: BoxMetho d..............701 167Boundary ValueProblems: Shooting Metho d...........706 168Continuation Metho d........................710 169ContinuedFractions .........................713 170Cosine Metho d............................716 171Di®eren tialAlgebraic Equations ...................720 172Eigenvalue/Eigenfunction Problems .................726 173Euler's ForwardMetho d.......................730 174Finite Elemen tMetho d.......................734 175Hybrid Computer Metho ds.....................744 176InvariantImbedding .........................747 177Multigrid Metho ds..........................752 178Parallel Computer Metho ds.....................755 179Predictor-Corrector Metho ds....................759 180Runge-Kutta Metho ds........................763 181Sti®Equations ............................770 182Integrating Stochastic Equations ..................775 183Symplectic Integration ........................780 184UseofWavelets............................784 185WeightedResidual Metho ds.....................786 IV.CNumerical Metho dsforPDEs 186Boundary Elemen tMetho d.....................792 187Di®eren tialQuadrature .......................796 188Domain Decomp osition .......................800 189Elliptic Equations: Finite Di®erences ................805 190Elliptic Equations: Monte-Carlo Metho d..............810 191Elliptic Equations: Relaxation ...................814 192HyperbolicEquations: Metho dofCharacteristics .........818 193HyperbolicEquations: Finite Di®erences ..............824 194Lattice GasDynamics ........................828 195Metho dofLines ...........................831 196ParabolicEquations: Explicit Metho d...............835 197ParabolicEquations: Implicit Metho d...............839 198ParabolicEquations: Monte-Carlo Metho d............844 199Pseudosp ectral Metho d.......................851 Mathematical Nomenclature Errata Preface When I was a graduate student in applied mathematics at the California Institute of Technology, we solved many dierential equations (both ordinary dierentialequations and partial dierential equations). Given a dierential equation to solve, I would think of all the techniques I knew that might solve that equation. Eventually, the number of techniques I knew became so large that I began toforget some. Then, I would have to consult books on dierential equations to familiarize myself with a technique that I remembered only vaguely. This was a slow process and often unrewarding; I might spend twenty minutes reading abouta technique only to realize that it did not apply to the equation I was trying to solve. Eventually, I created a list of the dierent techniques that I knew. Each technique had a brief description of how the method was used and to what typesof equations it applied. As I learned more techniques, they were added to thelist. This book is a direct result of that list. At Caltech we were taught the usefulness of approximate analytic solutions and the necessity of being able to solve dierential equations numerically when exact or approximate solution techniques could not be found. Hence, approximateanalytical solution techniques and numerical solution techniques were also added to the list. Given a dierential equation to analyze, most people spend only a small amount of time using analytical tools and then use a computer to see whatthe solution \looks like." Because this procedure is so prevalent, this edition includes an expanded section on numerical methods. New sections on sympletic integration (see page 780) and the use of wavelets (see page 784) also have beenadded. In writing this book, I have assumed that the reader is familiar with dieren- tial equations and their solutions. The object of this book is not to teach novel techniques but to provide a handy reference to many popular techniques. All of the techniques included are elementary in the usual mathematical sense; becausethis book is designed to be functional it does not include many abstract methods of limited applicability. This handbook has been designed to serve as both a reference book and as a complement to a text on dierential equations. Eachtechnique described is accompanied by several references; these allow each topic to be studied in more detail. It is hoped that this book will be used by students taking courses in dierential equations (at either the undergraduate or the graduate level). It will introducethe student to more techniques than they usually see in a dierential equations xv xvi Preface class and will illustrate many dierent types of techniques. Furthermore, it should act as a concise reference for the techniques that a student has learned. This book should also be useful for the practicing engineer or scientist who solves dierential equations on an occasional basis. A feature of this book is that it has sections dealing with stochastic dier- ential equations and delay dierential equations as well as ordinary dierential equations and partial dierential equations. Stochastic dierential equations anddelay dierential equations are often studied only in advanced texts and courses; yet, the techniques used to analyze these equations are easy to understand and easy to apply. Had this book been available when I was a graduate student, it would have saved me much time. It has saved me time in solving problems that arose from my own work in industry (the Jet Propulsion Laboratory, Sandia Laboratories,EXXON Research and Engineering, The MITRE Corporation, BBN). Parts of the text have been utilized in dierential equations classes at the Rensselaer Polytechnic Institute. Students’ comments have been used to clarifythe text. Unfortunately, there may still be some errors in the text; I would greatly appreciate receiving notice of any such errors. Many people have been kind enough to send in suggestions for additional material to add and corrections of existing material. There are too many to name them individually, but Alain Moussiaux stands out for all of the checking he has performed. Thank you all! This book is dedicated to my wife, Janet Taylor. Boston, Mass. 1997 Daniel Zwillinger [email protected] CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 Introduction This book is a compilation of the most important and widely applicable methods for solving and approximating dierential equations. As a reference book, it provides convenient access to these methods and contains examples of their use. The book is divided into four parts. The rst part is a collection of trans- formations and general ideas about dierential equations. This section of the book describes the techniques needed to determine whether a partial dierential equation is well posed, what the \natural" boundary conditions are, and manyother things. At the beginning of this section is a list of denitions for many of the terms that describe dierential equations and their solutions. The second part of the book is a collection of exact analytical solution techniques for dierential equations. The techniques are listed (nearly) alpha- betically. First is a collection of techniques for ordinary dierential equations,then a collection of techniques for partial dierential equations. Those techniques that can be used for both ordinary dierential equations and partial dierential equations have a star ( ) next to the method name. For nearly every technique, the following are given: the types of equations to which the method is applicable the idea behind the method the procedure for carrying out the method at least one simple example of the method any cautions that should be exercised notes for more advanced users references to the literature for more discussion or more examples The material for each method has deliberately been kept short to simplify use. Proofs have been intentionally omitted. It is hoped that, by working through the simple example(s) given, the method will be understood. Enough insight should be gained from working the example(s)to apply the method to other equations. Further references are given for each method so that the principle may be studied in more detail or so more examples may be seen. Note that not all of the references listed at the end of a methodm a yb er e f e r r e dt oi nt h et e x t . The author has found that computer languages that perform symbolic manip- ulations (e.g., Macsyma, Maple, and Mathematica) are very useful for performing the calculations necessary to analyze dierential equations. Hence, there is a section comparing the capabilities of these languages and, for some exactanalytical techniques, examples of their use are given. xvii xviii Introduction Not all dierential equations have exact analytical solutions; sometimes an approximate solution will have to do. Other times, an approximate solution may be more useful than an exact solution. For instance, an exact solution in terms of a slowly converging innite series may be laborious to approximatenumerically. The same problem may have a simple approximation that indicates some characteristic behavior or allows numerical values to be obtained. The third part of this book deals with approximate analytical solution tech- niques. For the methods in this part of the book, the format is similar to that used for the exact solution techniques. We classify a method as an approximate method if it gives some information about the solution but does not give thesolution of the original equation(s) at all values of the independent variable(s). The methods in this section describe, for example, how to obtain perturbation expansions for the solutions to a dierential equation. When an exact or an approximate solution technique cannot be found, it may be necessary to nd the solution numerically. Other times, a numerical solution may convey more information than an exact or approximate analytical solution.The fourth part of this book is concerned with the most important methods for nding numerical solutions of common types of dierential equations. Although there are many techniques available for numerically solving dierential equations,this book has only tried to illustrate the main techniques for each class of problem. At the beginning of the fourth section is a brief introduction to the terms used in numerical methods. When possible, short Fortran or C programs 1have been given. Once again, those techniques that can be used for both ordinary dierential equations and partial dierential equations have a star next to the method name. This book is not designed to be read at one sitting. Rather, it should be consulted as needed. Occasionally we have used \ODE" to stand for \ordinary dierential equation" and \PDE" to stand for \partial dierential equation." This book contains many references to other books. Whereas some books cover only one or two topics well, some books cover all their topics well. The following books are recommended as a rst source for detailed understanding of the dierential equation techniques they cover; each is broad in scope and easyto read. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [3]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [4]Chester, C. R. Techniques in Partial Dierential Equations . McGraw{Hill Book Company, New York, 1970. [5]Collatz, L. The Numerical Treatment of Dierential Equations . Springer{ Verlag, New York, 1966. 1We make no warranties, express or implied, that these programs are free of error. The author and publisher disclaim all liability for direct or consequential damagesresulting from your use of the programs. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 Introduction xix [6]Gear, C. W. Numerical Initial Value Problems in Ordinary Dierential Equations . Prentice{Hall, Inc., Englewood Clis, NJ, 1971. [7]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [8]Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis . Interscience Publishers, Inc., New York, 1958. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 Introduction to the Electronic Version This third edition of Handbook of Dierential Equations is available both in print form and in electronic form. The electronic version can be used with any modernweb browser (such as Netscape or Explorer). Some features of the electronic version include Quickly nding a specic method for a dierential equation Navigating through the electronic version is performed via lists of meth- ods for dierential equations. Facilities are supplied for creating lists ofmethods based on lters . For example, a list containing all the dierential equation methods that have both a program and an example in the text can be created. Or, a list of dierential equation methods that containeither a table or a specic word can be created. It is also possible to apply boolean operations to lists to create new lists. Interactive programs demonstrating some of the numerical methods For some of the numerical methods, an interactive Java program is sup- plied. This program numerically solves the example problem described in the text. The parameters describing the numerical solution may be varied, and the resulting numerical approximation obtained. Live links to the internet The third edition of this book has introduced links to relevant web sites on the internet. In the electronic version, these links are active (clickingon one of them will take you to that site). In the print version, the URLs may be found by looking in the index under the entry \URL." Dynamic rendering of mathematics All of the mathematics in the print version is available electronically, both through static gifles and via dynamic Java rendering. xx How to Use This Book This book has been designed to be easy to use when solving or approximating the solutions to dierential equations. This introductory section outlines theprocedure for using this book to analyze a given dierential equation. First, determine whether the dierential equation has been studied in the literature. A list of many such equations may be found in the \Look-Up" section beginning on page 179. If the equation you wish to analyze is contained on oneof the lists in that section, then see the indicated reference. This technique is the single most useful technique in this book. Alternatively, if the dierential equation that you wish to analyze does not appear on those lists or if the references do not yield the information you desire, then the analysis to be performed depends on the type of the dierential equation. Before any other analysis is performed, it must be veried that the equation is well posed. This means that a solution of the dierential equation(s) exists, is unique, and depends continuously on the \data." See pages 15, 53, 101, and 115. Given an Ordinary Dierential Equation It may be useful to transform the dierential equation to a canonical form or to a form that appears in the \Look-Up" section. For somecommon transformations, see pages 128{162. If the equation has a special form, then there may be a specialized solution technique that may work. See the techniques on pages 275,278, and 398. If the equation is a {Bernoulli equation, see page 235. {Chaplygin equation, see page 511. {Clairaut equation, see page 237. {Euler equation, see page 281. {Lagrange equation, see page 363. {Riccati equation, see page 392. If the equation does not depend explicitly on the independent vari- able, see pages 230 and 411. If the equation does not depend explicitly on the dependent variable (undierentiated), see pages 260 and 409. xxi xxii How to Use This Book If one solution of the equation is known, it may be possible to lower the order of the equation; see page 389. If discontinuous terms are present, see page 264. The single most powerful technique for solving analytically ordinary dierential equations is through the use of Lie groups; see page 366. Given a Partial Dierential Equation Partial dierential equations are treated in a dierent manner from ordi- nary dierential equations; in particular, the typeof the equation dictates the solution technique. First, determine the type of the partial dierentialequation; it may be hyperbolic, elliptic, parabolic, or of mixed type (see page 36). It may be useful to transform the dierential equation to a canonical form, or to a form that appears in the \Look-Up" Section. For transformations, see pages 146, 166, 168, 173, 456, and 467. The simplest technique for working with partial dierential equations, which does not always work, is to \freeze" all but one of the inde- pendent variables and then analyze the resulting partial dierentialequation or ordinary dierential equation. Then the other variables may be added back in, one at a time. If every term is linear in the dependent variable, then separation of variables may work; see page 487. If the boundary of the domain must be determined as part of the problem, see the technique on page 311. See all of the exact solution techniques, which are on pages 428{508. In addition, many of the techniques that can be used for ordinary dif-ferential equations are also applicable to partial dierential equations. These techniques are indicated by a star with the method name. If the equation is hyperbolic, {In principle, the dierential equation may be solved using the method of characteristics; see page 432. Often, though, the calculations are impossible to perform analytically. {See the section on the exact solution to the wave equation on page 501. The single most powerful technique for analytically solving partial dierential equations is through the use of Lie groups; see page 471. Given a System of Dierential Equations First, verify that the system of equations is consistent; see page 43. Note that many of the methods for a single dierential equation may be generalized to handle systems. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 How to Use This Book xxiii By using dierential resultants, it may be possible to obtain a single equation; see page 50. The following methods are for systems of equations: {The method of generating functions; see page 315. {The methods for constant coecient dierential equations; see pages 421 and 449. {The nding of integrable combinations; see page 334. If the system is hyperbolic, then the method of characteristics will work (in principle); see page 432. See also the method for Pfaan equations (see page 384) and the method for matrix Riccati equations (see page 395). Given a Stochastic Dierential Equation A general discussion of random dierential equations may be found on page 91. To determine the transition probability density, see the discussion of the Fokker{Planck equation on page 303. To obtain the moments without solving the complete problem, see pages 568 and 572. If the noise appearing in the dierential equation is not \white noise," the section on stochastic limit theorems might be useful (see page 629). To numerically simulate the solutions of a stochastic dierential equa- tion, see the technique on page 775. Given a Delay Equation See the techniques on page 253. Looking for an Approximate Solution If exact bounds on the solution are desired, see the methods on pages 545, 551, and 560. If the solution has singularities that are to be recovered, see page 582. If the dierential equation(s) can be formulated as a contraction mapping, then approximations may be obtained in a natural way; see page 58. Looking for a Numerical Solution It is extremely important that the dierential equation(s) be well posed before a numerical solution is attempted. See the theorem on page 723 for an indication of the problems that can arise. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 xxiv How to Use This Book The numerical solution technique must be stable if the numerical so- lution is to approximate the true solution of the dierential equation; see pages 683, 688, and 692. It is often easiest to use commercial software packages when looking for a numerical solution; see page 654. If the problem is \sti," then a method for dealing with \sti" problems will probably be required; see page 770. If a low-accuracy solution is acceptable, then a Monte-Carlo solution technique may be used; see pages 810 and 844. To determine a grid on which to approximate the solution numeri- cally, see page 675. To nd an approximation scheme that works on a parallel computer, see page 755. Other Things to Consider Does the dierential equation undergo bifurcations? See page 19. Is the solution bounded? See pages 551 and 560. Is the dierential equation well posed? See pages 15 and 115. Does the equation exhibit symmetries? See pages 366 and 471. Is the system chaotic? See page 29. Are some terms in the equation discontinuous? See page 264. Are there generalized functions in the dierential equation? See pages 318 and 330. Are fractional derivatives involved? See page 308. Does the equation involve a small parameter? See the perturbation methods (on pages 586, 590, 598, 605, 610, and 614) or pages 538, 642. Is the general form of the solution known? See page 415. Are there multiple time or space scales in the problem? See pages 538 and 605. Always check your results! Methods Not Discussed in This Book There are a variety of novel methods for dierential equations and their solutions not discussed in this book. These include 1. Adomian’s decomposition method (see Adomian [1]) 2. Entropy methods (see Baker-Jarvis [2]) 3. Fuzzy logic (see Leland [5]) 4. Innite systems of dierential equations (see Steinberg [6]) 5. Monodromy deformation (see Chowdhury and Naskar [3])6.p-adic dierential equations (see Dwork [4]) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 How to Use This Book xxv References [1]Adomian, G. Stochastic Systems . Academic Press, New York, 1983. [2]Baker-Jarvis, J. Solution to boundary value problems using the method of maximum entropy. J. Math. and Physics 30 , 2 (February 1989), 302{306. [3]Chowdhury, A. R., and Naskar, M. Monodromy deformation approach to nonlinear equations | A survey. Fortschr. Phys. 36 , 12 (1988), 9399{953. [4]Dwork, B. Lectures on p-adic Dierential Equations . Springer{Verlag, New York, 1982. [5]Leland, R. P. Fuzzy dierential systems and Malliavin calculus. Fuzzy Sets and Systems 70 (1995), 59{73. [6]Steinberg, S. Innite systems of ordinary dierential equations with unbounded coecients and moment problems. J .M a t h .A n a l .A p p l .4 1 (1973), 685{694. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 xxvi How to Use This Book CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 2 I.A Denitions and Concepts 1. Denition of Terms Adiabatic invariant When the parameters of a physical system vary slowly under the eect of an external perturbation, some quantities are constant to any order of the variable describing the slow rate of change. Such a quantity is called an adiabatic invariant. This does not mean that these quantities are exactly constant but rather that their variation goesto zero faster than any power of the small parameter. Analytic A function is analytic at a point if the function has a power series expansion valid in some neighborhood of that point. Asymptotic equivalence Two functions, f(x)a n dg(x), are said to be asymptotically equivalent asx!x 0iff(x)=g(x)1a sx!x0,t h a ti s : f(x)=g(x)[ 1+o(1)] asx!x0. See Erd elyi [4] for details. Asymptotic expansion Given a function f(x) and an asymptotic se- riesfgk(x)gatx0, the formal seriesP1 k=0akgk(x), where thefakgare given constants, is said to be an asymptotic expansion off(x)i ff(x)−Pn k=0akgk(x)=o(gn(x)) asx!x0for everyn; this is expressed as f(x)P1 k=0akgk(x). Partial sums of this formal series are called asymptotic approximations tof(x). Note that the formal series need not converge. See Erd elyi [4] for details. Asymptotic series A sequence of functions, fgk(x)g, forms an asymp- totic series atx0ifgk+1(x)=o(gk(x)) asx!x0. Autonomous An ordinary dierential equation is autonomous if the in- dependent variable does not appear explicitly in the equation. For example, yxxx+(yx)2=yis autonomous while yx=xis not (see page 230). Bifurcation The solution of an equation is said to undergo a bifur- cation if, at some critical value of a parameter, the number of solutionsto the equation changes. For instance, in a quadratic equation with real coecients, as the constant term changes the number of real solutions can change from 0 to 2 (see page 19). Boundary data Given a dierential equation, the value of the depen- dent variable on the boundary may be given in many dierent ways. Dirichlet boundary conditions The dependent variable is pre- scribed on the boundary. This is also called a boundary con-dition of the rst kind. Homogeneous boundary conditions The dependent variable van- ishes on the boundary. Mixed boundary conditions A linear combination of the depen- dent variable and its normal derivative is given on the boundary, CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 1. Denition of Terms 3 or one type of boundary data is given on one part of the bound- ary while another type of boundary data is given on a dierent part of the boundary. This is also called a boundary condition of the third kind. Neumann boundary conditions The normal derivative of the de- pendent variable is given on the boundary. This is also called a boundary condition of the second kind. Sometimes the boundary data also include values of the dependent variable at points interior to the boundary. Boundary layer A boundary layer is a small region, near a boundary, in which a function undergoes a large change (see page 590). Boundary value problem An ordinary dierential equation, where not all of the data are given at one point, is a boundary value problem. For example, the equation y00+y= 0 with the data y(0) = 1,y(1) = 1 is a boundary value problem. Characteristics A hyperbolic partial dierential equation can be de- composed into ordinary dierential equations along curves known as char- acteristics. These characteristics are themselves determined to be the solutions of ordinary dierential equations (see page 432). Cauchy problem The Cauchy problem is an initial value problem for a partial dierential equation. For this type of problem there are initial conditions but no boundary conditions. Commutator IfL[
]a n dH[
] are two dierential operators, then the commutator of L[
]a n dH[
] is dened to be the dierential operator given by [L;H]: =LH−HL=−[H;L]:For example, the commutator of the operatorsL[
]=xd dxandH[
]=1+d dxis [L;H]= xd dx 1+d dx − 1+d dx xd dx =−d dx: See Goldstein [6] for details. Complete A set of functions is said to be complete on an interval if any other function that satises appropriate boundedness and smoothness conditions can be expanded as a linear combination of the original func-tions. Usually the expansion is assumed to converge in the \mean square," orL 2sense. For example, the functions fun(x)g:=fsin(nx);cos(nx)g are complete on the interval [0 ;1] because any C1[0;1] function, f(x), can be written as f(x)=a0+1X n=1 ancos(nx)+bnsin(nx) for some set offan;bng. See Courant and Hilbert [3, pages 51{54] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 4 I.A Denitions and Concepts Complete system The system of nonlinear partial dierential equa- tions:fFk(x1;:::;xr;y;p 1;:::;pr)=0jk=1;:::;sg, in one dependent variable,y(x), wherepi=dy=dxi, is called a complete system if each fFj;Fkg,f o r1j;kr, is a linear combination of the fFkg.H e r ef;g represents the Lagrange bracket. See Iyanaga and Kawada [8, page 1304]. Conservation form A hyperbolic partial dierential equation is said to be in conservation form if each term is a derivative with respect to some variable. That is, it is an equation for u(x)=u(x1;x2;:::;xn)t h a th a s the form@f1(u;x) @x1+


+@fn(u;x) @xn= 0 (see page 47). Consistency There are two types of consistency: Genuine consistency This occurs when the exact solution to an equation can be shown to satisfy some approximations that havebeen made in order to simplify the equation’s analysis. Apparent consistency This occurs when the approximate solution to an equation can be shown to satisfy some approximations thathave been made in order to simplify the equation’s analysis. When simplifying an equation to nd an approximate solution, the derived solution must always show apparent consistency. Even then, the approxi- mate solution may not be close to the exact solution, unless there is genuine consistency. See Lin and Segel [9, page 188]. Coupled systems of equations A set of dierential equations is said to be coupled if there is more than one dependent variable and each equation involves more than one dependent variable. For example, the system fy 0+ v=0;v0+y=0gis a coupled system for fy(x);v(x)g. Degree The degree of an ordinary dierential equation is the greatest number of times the dependent variable appears in any single term. For example, the degree of y0+(y00)2y+ 1 = 0 is 3, whereas the degree of y00y0y2+x5y= 1 is 4. The degree of y0=s i nyis innite. If all the terms in a dierential equation have the same degree, then the equation is called equidimensional-in- y(see page 278). Delay equation A delay equation, also called a dierential delay equa- tion, is an equation that depends on the \past" as well the \present." For example,y00(t)=y(t−) is a delay equation when >0. See page 253. Determined A truncated system of dierential equations is said to be determined if the inclusion of any higher order terms cannot aect the topological nature of the local behavior about the singularity. Dierential form A rst order dierential equation is said to be in dierential form if it is written P(x;y)dx+Q(x;y)dy=0 . Dirichlet problem The Dirichlet problem is a partial dierential equa- tion with Dirichlet data given on the boundaries. That is, the dependent variable is prescribed on the boundary. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 1. Denition of Terms 5 Eigenvalues, eigenfunctions Given a linear operator L[
] with bound- ary conditions B[
], there will sometimes exist nontrivial solutions to the equationL[y]=y(the solutions may or may not be required to also satisfyB[y] = 0). When such a solution exists, the value of is called an eigenvalue. Corresponding to the eigenvalue there will exist solutions fy(x)g; these are called eigenfunctions. See Stakgold [12, Chapter 7, pages 411{466] for details. Elliptic operator The dierential operatornX i;j=1aij@2 @xi@xjis an elliptic dierential operator if the quadratic form xTAx,w h e r eA=(aij), is positive denite whenever x6=0.I f t h efaijgare functions of some variable, say t, and the operator is elliptic for all values of tof interest, then the operator is called uniformly elliptic . See page 36. Euler{Lagrange equation Ifu=u(x)a n dJ[u]=R f(u0;u;x)dx, then the condition for the vanishing of the variational derivative of Jwith respect tou,J u= 0 is given by the Euler{Lagrange equation: @ @u−d dx@ @u0 f=0: Ifw=w(x)a n dJ=R g(w00;w0;w;x )dx, then the Euler{Lagrange equa- tion is @ @w−d dx@ @w0+d2 dx2@ @w00 g=0: Ifv=v(x;y)a n dJ=RR h(vx;vy;v;x;y )dxdy , then the Euler{Lagrange equation is@ @v−d dx@ @vx−d dy@ @vy h=0: See page 418 for more details. First integral: ODE When a given dierential equation is of order n and, by a process of integration, an equation of order n−1 involving an arbitrary constant is obtained, then this new equation is known as a rstintegral of the given equation. For example, the equation y 00+y=0h a s the equation ( y0)2+y2=Cas a rst integral. First integral: PDE A function u(x;y;z ) is called a rst integral of the vector eld V=(P;Q;R ) (or of its associated system:dx P=dy Q=dz R) if at every point in the domain Vis orthogonal to grad u, i.e., V
ru=P@u @x+Q@u @y+R@u @z=0: Conversely, any solution of this partial dierential equation is a rst integral ofV.N o t et h a ti f u(x;y;z ) is a rst integral of V,t h e ns oi s f(u). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 6 I.A Denitions and Concepts Frechet derivative, G^ ateaux derivative The G^ ateaux derivative of the operator N[
], at the \point" u(x), is the linear operator dened by L[z(x)] = lim !0N[u+z]−N[u] : For example, if N[u]=u3+u00+(u0)2,t h e nL[z]=3u2z+z00+2u0z0.I f , in addition, lim jjhjj!0jjN[u+h]−N[u]−L[u]hjj jjhjj=0 (as is true in our example), then L[u] is also called the Fr echet derivative ofN[
]. See Olver [11] for details. Fuchsian equation A Fuchsian equation is an ordinary dierential equation whose only singularities are regular singular points. Fundamental matrix The vector ordinary dierential equation y0= Ayfory(x), whereAis a matrix, has the fundamental matrix ( x)i f satises 0=A and the determinant of  is nonvanishing (see page 119). General solution Given annth order linear ordinary dierential equa- tion, the general solution contains all nlinearly independent solutions, with a constant multiplying each one. For example, the dierential equation y00+y= 1 has the general solution y(x)=1+Asinx+Bcosx,w h e r eA andBare arbitrary constants. Green’s function A Green’s function is the solution of a linear dier- ential equation, which has a delta function appearing either in the equation or in the boundary conditions (see page 318). Harmonic function A function(x) is harmonic if it satises Laplace’s equation:r2=0 . Hodograph In a partial dierential equation, if the independent vari- ables and dependent variables are switched, then the space of independent variables is called the hodograph space (in two dimensions, the hodograph plane) (see page 456). Homogeneous equation Used in two dierent senses: An equation is said to be homogeneous if all terms depend linearly on the dependent variable or its derivatives. For example, the equation yxx+xy= 0 is homogeneous whereas the equation yxx+xy=1i s not. A rst order ordinary dierential equation is said to be homogeneous if the forcing function is a ratio of homogeneous polynomials (seepage 327). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 1. Denition of Terms 7 Ill posed problems A problem that is not well posed is said to be ill posed. Typical ill posed problems are the Cauchy problem for the Laplace equation, the initial/boundary value problem for the backward heat equation, and the Dirichlet problem for the wave equation (see page115). Initial value problem An ordinary dierential equation with all of the data given at one point is an initial value problem. For example, theequationy 00+y= 0 with the data y(0) = 1,y0(0) = 1 is an initial value problem. Involutory transformation An involutory transformation Tis one that, when applied twice, does not change the original system; i.e., T2is equal to the identity function. L2function A function f(x) is said to belong to L2ifR1 0jf(x)j2dxis nite. Lagrange bracket IffFjgandfGjgare sets of functions of the inde- pendent variables fu;v;:::gthen the Lagrange bracket of uandvis dened to be fu;vg=X j@Fj @u@Gj @v−@Fj @v@Gj @u =−fv;ug: See Goldstein [6] for details. Lagrangian derivative The Lagrangian derivative (also called the ma- terial derivative) is dened byDF Dt:=@F @t+v
rF,w h e r e vis a given vector. See Iyanaga and Kawada [8, page 669]. Laplacian The Laplacian is the dierential operator usually denoted byr2(in many books it is represented as
). It is dened by r2= div(grad), whenis a scalar. The vector Laplacian of a vector is the dierential operator denoted by 45(in most books it is represented as r2). It is dened by45v= grad(div v)−curl curl v,w h e n vis a vector. See Moon and Spencer [10] for details. Leibniz’s rule Leibniz’s rule states that d dt Zg(t) f(t)h(t;)d! =g0(t)h(t;g(t))−f0(t)h(t;f(t)) +Zg(t) f(t)@h @t(t;)d: Lie algebra A Lie algebra is a vector space equipped with a Lie bracket (often called a commutator) [ x;y] that satises three axioms: [x;y] is bilinear (i.e., linear in both xandyseparately), the Lie bracket is anti-commutative (i.e., [ x;y]=−[y;x]), the Jacobi identity, [ x;[y;z]] + [y;[z;x]] + [z;[x;y]] = 0, holds. See Olver [11] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 8 I.A Denitions and Concepts Limit cycle A limit cycle is a solution to a dierential equation that is a periodic oscillation of nite amplitude (see page 78). Linear dierential equation A dierential equation is said to be linear if the dependent variable appears only with an exponent of 0 or 1. For example, the equation x3y000+y0+c o sx= 0 is a linear equation, whereas the equation yy0=1i s nonlinear . Linearize To linearize a nonlinear dierential equation means to ap- proximate the equation by a linear dierential equation in some region. Forexample, in regions where jyjis \small," the nonlinear ordinary dierential equationy 00+s i ny= 0 could be linearized to y00+y=0 . Linearizable Partial dierential equations that can be solved either by an appropriate inverse scattering scheme or by a transformation to a linear partial dierential equation are said to be linearizable. Lipschitz condition Iff(x;y) is a bounded continuous function in a domainD,t h e nf(x;y) is said to satisfy a Lipschitz condition in yinDif jf(x;y1)−f(x;y2)jKyjy1−y2j for some nite constant Ky, independent of x,y1,a n dy2inD. If, for some nite constant Kx,f(x;y) satises jf(x1;y)−f(x2;y)jKxjx1−x2j independent of x1,x2,a n dyinD,t h e nf(x;y) satises a Lipschitz con- dition inxinD. If both of these conditions are satised and K= max(Kx;Ky), thenf(x;y) satises a Lipschitz condition in D, with Lip- schitz constant K. This also extends to higher dimensions. See Coddington and Levinson [2] for details. Maximum principle There are many \maximum principles" in the literature. The most common is \a harmonic function attains its absolute maximum on the boundary" (see page 560). Mean value theorem This is a statement about the solution of Laplace’s equation. It states, \If r2u=0( i nNdimensions), then u(z)=R SudS=R SdS whereSis the boundary of a N-dimensional sphere centered at z." For example, in N= 2, we have, \In 2 dimensions, the value of a solution to Laplace’s equation at a point is the average of the values on any circle about that point." See Iyanaga and Kawada [8, page 624]. Metaparabolic equation A metaparabolic equation has the form L[u]+ M[ut] = 0, where u=u(x;t),L[
] is a linear dierential operator in xof degreen,M[
] is a linear dierential operator in xof degreem,a n dm<n . If, conversely, m>n , then the equation is called pseudoparabolic .S e e Gilbert and Jensen [5] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 1. Denition of Terms 9 Natural Hamiltonian A natural Hamiltonian is one having the form H=T+V,w h e r eT=1 2Pn k=1p2 kandVis a function of the position variables only (i.e., V=V(q)=V(q1;:::;qn)). Near identity transformation A near-identity transformation is a transformation in a dierential equation from the old variables fa,b,c;:::g to the new variables f,,γ;:::gvia a=+A(;;γ;::: ); b=+B(;;γ;::: ); c=γ+C(;;γ;::: ); ... wherefA;B;C;:::gare strictly nonlinear functions (i.e., there are no linear or constant terms). Very frequently fA;B;C;:::gare taken to be homogeneous polynomials (of, say, degree N) in the variables ;;γ;::: , with unknown coecients. For example, in two variables we might take A(;)=nX j=0Aj;n−jjn−j;B (;)=nX j=0Bj;n−jjn−j; for some given value of n(see page 86). Neumann problem The Neumann problem is a partial dierential equation with Neumann data given on the boundaries. That is, the normal derivative of the dependent variable is given on the boundary. See Iyanaga and Kawada [8, page 999]. Normal form An ordinary dierential equation is said to be in nor- mal form if it can be solved explicitly for the highest derivative; i.e., y(n)=G(x,y,y0;:::;y(n−1)). A system of partial dierential equa- tions (with dependent variables fu1;u2;:::;umgand independent variables fx;y1;y2;:::;ykg) is said to be in normal form if it has the form @ruj @xr=Fj x;y1;:::;yk;u1;:::;um;@u1 @x;:::;@r−1um @xr−1;:::;@u1 @y1;:::;@rum @ykr ; forj=1;2;:::;m . See page 86 or Iyanaga and Kawada [8, page 988]. Normal type An evolution equation is of normal type if it can be written in the form ut=un+h(u;u1;:::;um)w h e r en>m anduj=@ju=@xj. Nonlinear A dierential equation that is not linear in the dependent variable is nonlinear. Nonoscillatory The real solution y(x)o fyxx+f(x)y= 0 is said to be nonoscillatory in the wide sense in (0 ;1) if there exists a nite number c such that the solution has no zeros in [ c;1]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 10 I.A Denitions and Concepts Order of a dierential equation The order of a dierential equation is the greatest number of derivatives in any term in the dierential equation. For example, the partial dierential equation uxxxx =utt+u5is of fourth order whereas the ordinary dierential equation vx+x2v3+v= 3 is of rst order. Orthogonal Two vectors, xandy, are said to be orthogonal with respect to the matrix WifxTWy=0( o f t e n , Wis taken to be the identity matrix). Two functions, say f(x)a n dg(x), are said to be orthogonal with respect to a weighting function w(x)i f(f(x);g(x)) :=R f(x)w(x)g(x)dx= 0 over some appropriate range of integration. Here, an overbar indicates the complex conjugate. Oscillatory Consider the equation y00+f(x)y= 0 and the number of zeros it has in the interval [0 ;1]. If the number of zeros is innite, then the equation (and the solutions) are called oscillatory . Pade approximant AP a d  e approximant is a ratio of polynomials. The polynomials are usually chosen so that the Taylor series of the ratio is a prescribed function. See page 582. Particular solution Given a linear dierential equation, L[y]=f(x), the general solution can be written as y=yp+P iCiyiwhereyp,t h e particular solution, is any solution that satises L[y]=f(x). Theyiare homogeneous solutions that satisfy L[y] = 0, and thefCigare arbitrary constants. If L[
]i sa nnth order dierential operator, then there will be n linearly independent homogeneous solutions. Poisson bracket Iffandgare functions offpj;qjg, then the Poisson bracket offandgis dened to be [f;g]=X j@f @qj@g @pj−@f @pj@g @qj =X j@(f;g) @(qj;pj)=−[g;f]: The Poisson bracket is invariant under a change of independent variables. See Goldstein [6] or Olver [11] for details. Quasilinear equation Used in two dierent senses: A partial dierential equation is said to be quasilinear if it is linear in the rst partial derivatives. That is, it has the formPn k=1Ak(u;x)@u @xk= B(u;x) when the dependent variable is u(x)=u(x1;:::;xn)( s e e page 432). A partial dierential equation is said to be quasilinear if it has the formut=g(u)ux(n)+f(u;ux;yx(2);:::;ux(n−1))f o rn2. Radiation condition The radiation condition states that a wave equa- tion has no waves incoming from an innite distance, only outgoing waves. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 1. Denition of Terms 11 For example, the equation utt=r2umight have the radiation condition u(x;t)’A−exp(ik(t−x)) asx!−1 andu(x;t)’A+exp(ik(t+x)) asx!+1. This is also called the Sommerfeld radiation condition. See Butkov [1, page 617] for details. Riemann’s Pfunction Riemann’s dierential equation (see page 186) is the most general second order linear ordinary dierential equation with three regular singular points. If these singular points are taken to be a,b, andcand the exponents of the singularities are taken to be ;0;;0; γ;γ0(where+0++0+γ+γ0= 1), then the solution to Riemann’s dierential equation may written in the form of Riemann’s Pfunction as y(x)=P2 4abc γx 00γ03 5: Robbins problem An elliptic partial dierential equation with mixed boundary conditions is called a Robbins problem. See Iyanaga and Kawada [8, page 999]. Schwarzian derivative Ify=y(x), then the Schwarzian derivative of ywith respect to xis dened to be fy;xgy00 y00 −1 2y00 y02 =y000 y0−3 2y00 y02 : Ify=y(x)a n dz=z(x), thenfz;xg=fz;yg dy dx2 +fy;xg. Therefore, fx;yg=− dx dy2 fy;xg. Note also that fy;xgis the unique elementary function of the derivatives, which is invariant under homographic transfor-mations of x;t h a ti s ,fy;xg=n y; ax+b cx+do ,w h e r e(a;b;c;d ) are arbitrary constants with ad−bc= 1. See Ince [7, page 394]. Semi-Hamiltonian A diagonal system of equations having the form Ai(u)@tui=Bi(u)@xuiis called semi-Hamiltonian if the coecients satisfy Bi@uiAk=Ai@uiBkfori6=k. Semilinear equations A partial dierential equation is said to be semilinear if it has the form ut=ux(n)+f(u;ux;yx(2);:::;ux(n−1))f o r n2. Shock A shock is a narrow region in which the dependent variable under- goes a large change. Also called a \layer" or a \propagating discontinuity." See page 432. Singular point Given the homogeneous nth order linear ordinary dif- ferential equation y(n)+qn−1(x)y(n−1)+qn−2(x)y(n−2)+


+q0(x)y=0; the pointx0is classied as being an CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 12 I.A Denitions and Concepts Ordinary point: if each of thefqigare analytic at x=x0. Singular point: if it is not an ordinary point. Regular singular point: if it is not an ordinary point and ( x− x0)iqi(x) is analytic for i=0;1;:::;n . Irregular singular point: if it is not an ordinary point and not a regular singular point. The point at innity is classied by changing variables to t=x−1and then analyzing the point t= 0. See page 403. Singular solution A singular solution is a solution of a dierential equation that is not derivable from the general solution by any choice of the arbitrary constants appearing in the general solution. Only nonlinear equations have singular solutions. See page 623. Stability The solution to a dierential equation is said to be stable if small perturbations in the initial conditions, boundary conditions, orcoecients in the equation itself lead to \small" changes in the solution. There are many dierent types of stability that are useful. Stable A solution y(x) of the system y 0=f(y;x) that is dened forx> 0 is said to be stable if, given any >0, there exists a> 0 such that any solution w(x) of the system satisfying jw(0)−y(0)j<also satisesjw(x)−y(x)j<. Asymptotic stability The solution u(x) is said to be asymptoti- cally stable if, in addition to being stable, jw(x)−u(x)j!0a s x!1 . Relative stability The solution u(x) is said to be relatively stable ifjw(0)−u(0)j<implies thatjw(x)−u(x)j<u(x). See page 101 or Coddington and Levinson [2, Chapter 13] for details. Stefan problem A Stefan problem is one in which the boundary of the domain must be solved as part of the problem. For instance, when ajet of water leaves an orice, not only must the fluid mechanics equations be solved in the stream, but the boundary of the stream must also be determined. Stefan problems are also called free boundary problems (seepage 311). Superposition principle Ifu(x)a n dv(x) are solutions to a linear dierential equation (ordinary or partial), then the superposition principle states thatu(x)+v(x) is also a solution, where andare any constants (see page 413). Total dierential equation A total dierential equation is an equation of the form:P kak(x)dxk= 0. See page 384. Trivial solution The trivial solution is the identically zero solution. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 1. Denition of Terms 13 Turning points Given the equation y00+p(x)y= 0, points at which p(x) = 0 are called turning points. The asymptotic behavior of y(x)c a n change at these points. See page 645 or Wasow [13]. Weak solution A weak solution to a dierential equation is a function that satises only an integral form of the dening equation. For example,a weak solution of the dierential equation a(x)y 00−b(x) = 0 only needs to satisfyR S[a(x)y00−b(x)]dx= 0 where Sis some appropriate region. For this example, the weak solution may not be twice dierentiable everywhere. See Zauderer [14, pages 288{294] for details. Well posed problems A problem is said to be well posed if a unique, stable solution that depends continuously on the data exists. See page 115. Wronskian Given the smooth functions fy1;y2;:::;yng, the Wronskian is the determinant y 1y2::: yn y0 1y0 2::: y0 n ............ y(n−1) 1y(n−1) 2::: y(n−1) n If the Wronskian does not vanish in an interval, then the functions are linearly independent (see page 119). References [1]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [2]Coddington, E. A., and Levinson, N. Theory of Ordinary Dierential Equations . McGraw{Hill Book Company, New York, 1955. [3]Courant, R., and Hilbert, D. Methods of Mathematical Physics . Interscience Publishers, Inc., New York, 1953. [4]Erd elyi, A. Asymptotic Expansions . Dover Publications, Inc., New York, 1956. [5]Gilbert, R. P., and Jensen, J. A computational approach for constructing singular solutions of one-dimensional pseudoparabolic and metaparabolic equations. SIAM J. Sci. Stat. Comput. 3 , 1 (March 1982), 111{125. [6]Goldstein, H. Classical Mechanics . Addison{Wesley Publishing Co., Reading, MA, 1950. [7]Ince, E. L. Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [8]Iyanaga, S., and Kawada, Y. Encyclopedic Dictionary of Mathematics . MIT Press, Cambridge, MA, 1980. [9]Lin, C. C., and Segel, L. A. Mathematics Applied to Deterministic Problems in the Natural Sciences . The MacMillan Company, New York, 1974. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 14 I.A Denitions and Concepts [10]Moon, P., and Spencer, D. E. The meaning of the vector Laplacian. J. Franklin Institute 256 (1953), 551{558. [11]Olver, P. J. Applications of Lie Groups to Dierential Equations . No. 107 in Graduate Texts in Mathematics. Springer{Verlag, New York, 1986. [12]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. [13]Wasow, W. Linear Turning Point Theory , vol. 54. Springer{Verlag, New York, 1985. [14]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 2. Alternative Theorems 15 2. Alternative Theorems Applicable to Linear ordinary dierential equations. Idea It is often possible to determine when a linear ordinary dierential equation has a unique solution. Also, when the solution is not unique, it is sometimes possible to describe the degrees of freedom that make it non-unique. Procedure Alternative theorems describe, in some way, the type of solutions to expect from linear dierential equations. The most common alternativetheorems for dierential equations were derived by Fredholm. Suppose we wish to analyze the nth order linear inhomogeneous ordi- nary dierential equation with boundary conditions L[u]=f(x); B i[u]=0; fori=1;2;:::;n;(2.1) foru(x) on the interval x2[a;b]. First, we must analyze the homogeneous equation and the adjoint homogeneous equation. That is, consider the two problems L[u]=0; Bi[u]=0; fori=1;2;:::;n;(2.2) and L[v]=0; B i[v]=0; fori=1;2;:::;n;(2.3) whereL[
] is the adjoint of L[
], and thefB i[
]gare the adjoint boundary conditions (see page 95). Then Fredholm’s alternative theorem states that 1. If the system in (2.2) has only the trivial solution, that is u(x)0, then (a) the system in (2.1) has a unique solution. (b) the system in (2.3) has only the trivial solution. 2. Conversely, if the system in (2.2) has klinearly independent solutions, sayfu1;u2,:::;ukg,t h e n (a) the system in (2.3) has klinearly independent solutions, say fv1;v2,:::;vkg. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 16 I.A Denitions and Concepts (b) the system in (2.1) has a solution if and only if the forcing function appearing in (2.1), f, is orthogonal to all solutions to the adjoint system. That is ( f;vi): =Rb af(x)vi(x)dx=0f o r i=1;2;:::;k . (c) the solution to (2.1), if 2(b) is satised, is given by u(x)= u(x)+Pk j=icjuj(x) for arbitrary constants fcjg,w h e r eu(x)i s any solution to (2.1). Example 1 Given the ordinary dierential equation for u(x) u0+u=f(x); u(0) = 0;(2.4) we form the homogeneous system u0+u=0; u(0) = 0:(2.5) Because (2.5) has only the trivial solution, we know that the solution to equation (2.4) is unique. By the method of integrating factors (see page 356), the solution to (2.4) is found to be u(x)=Rx 0f(t)et−xdt. Example 2 Given the ordinary dierential equation for u(x) u0+u=f(x); u(0)−eu(1) = 0;(2.6) we form the homogeneous system u0+u=0; u(0)−eu(1) = 0:(2.7) In this case, (2.7) has the single non-trivial solution u(x)=e−x. Hence, the solution to (2.6) is notunique. To nd out what restrictions must be placed on f(x) for (2.6) to have a solution, consider the corresponding adjoint homogeneous equation v0−v=0; −ev(0) +v(1) = 0:(2.8) Since (2.8) has a single non-trivial solution, v(x)=ex, we conclude that (2.6) has a solution if and only if Z1 0f(t)etdt=0: (2.9) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 2. Alternative Theorems 17 If equation (2.9) is satised, then the solution of (2.6) will be given by u(x)=Ce−x+Zx 0f(t)et−xdt whereCis an arbitrary constant. Example 3 The solution(s) to xy00−(1 +x)y0+y= 0 depends on the boundary conditions as follows: 1. Withy(1) = 1,y0(1) = 2, the solution is y=3ex−1−(1 +x). 2. Withy(0) = 1,y0(0) = 2, there is no solution. 3. Withy(0) = 1,y0(0) = 1, there are innitely many solutions of the formy=C(ex−1−x)+1+x. Notes 1. Epstein [1, pages 83 and 111] discusses the Fredholm theorems in the general setting of a Banach space and a Hilbert space. 2. Interpretation of alternative theorems is usually straightforward when the underlying physics are understood. For example, the system −u00=f(x);0<x< 1u0(0) =a1;−u0(1) =a2 must satisfy the relationR1 0f(x)dx=a1+a2. This states that for a rod experiencing one-dimensional heat flow, a steady state is possible only if the heat supplied along the rod is removed at the ends. 3. A generalized Green’s function is a Green’s function (see page 318) for a dierential equation that does not have a unique solution. See Greenberg [2] for more details. 4. The Sturm{Liouville problem for u(x) on the interval x1xx2 −d dx p(x)du dx +q(x)u=f(x) (2.10) −p(x1)u0(x1)+r1u(x1)=0p(x2)u0(x2)+r2u(x2)=0 can be written as Zx2 x1h p(t)u02(t)+q(t)u2(t)i dt+r1u2(x1)+r2u2(x2) =Zx2 x1f(t)u(t)dt+g1u(x1)+g2u(x2): Hence, ifp(x) is positive, q(x),r1,a n dr2are non-negative and ifRx2 x1f(t)u(t)dt+g1u(x1)+g2u(x2) = 0, then there is a unique solution to (2.10). 5. See also Haberman [3, pages 307{314] and Stakgold [4, pages 82{90, 207{214, and 319{323]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 18 I.A Denitions and Concepts References [1]Epstein, B. Partial Dierential Equations: An Introduction . McGraw{Hill Book Company, New York, 1962. [2]Greenberg, M. D. Application of Green’s Functions in Science and Engineering . Prentice{Hall, Inc., Englewood Clis, NJ, 1971. [3]Haberman, R. Elementary Applied Partial Dierential Equations .P r e n t i c e { Hall, Inc., Englewood Clis, NJ, 1968. [4]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 3. Bifurcation Theory 19 3. Bifurcation Theory Applicable to Nonlinear dierential equations. Idea Given a nonlinear dierential equation that depends on a set of pa- rameters, the number of distinct solutions may change as the parameterschange. Points where the number of solutions change are called bifurcation points . Procedure Although bifurcations occur in all types of equations, we restrict our discussion to ordinary dierential equations. Consider the autonomous system dx dt=f(x;); (3.1) where xandfaren-dimensional vectors and is a set of parameters. Dene the Jacobian matrix by J(x;): =df dx=@fi @xj(x;)ji;j=1;:::;n : (3.2) Note thatJ(x;)zis the Frechet derivative off, at the point x(see page 6). Using the solution x(t;) of equation (3.1), the values of where one or more of the eigenvalues of Jare zero are dened to be bifurcation points. At such points, the number of solutions to equation (3.1) may change, and the stability of the solutions might also change. If any of the eigenvalues have positive real parts, then the correspond- ing solution is unstable. If we are concerned only with the steady-statesolutions of equation (3.1), as is often the case, then the bifurcation points will satisfy the simultaneous equations f(x;)=0; and det J=0: (3.3) Dene the eigenvalues of the Jacobian matrix dened in equation (3.2) to bef iji=1;:::;ng. We now presume that equation (3.1) depends on the single parameter . Suppose that the change in stability is at the point=b, where the real part of a complex conjugate pair of eigenvalues (1=2) pass through zero: <1(b)=0;=1(b)>0;<0 1(b)6=0; and, for all values of nearb,<i()<0f o ri=3;:::;n . Then, under certain smoothness conditions, it can be shown that a small amplitude periodic solution exists for nearb. Letmeasure the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 20 I.A Denitions and Concepts/./. /././. /././. /././. /././. /././. /././. /././. /././././. /././. /./././././././. /././. /././././. /././. /././././. /./././././././././././. /././././././././././././././././././././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././././././. /./././././. /././././. /./././. /././. /./././. /././. /././. /./. /././. /./. /././. /./. /./. /./. /./. /./. /./. /./. /./. /./. /. /./. /./. /. /./. /. /./. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /. /./. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /. /. /. /./. /. /./. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /. /. /. /./. /. /./. /. /. /. /. /./. /. /./. /. /. /./. /././. /././. /././././. /././././././././././././././././././././././././././././././. /./././. /./. /./. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./!R /#12 g/. /./. /. /. /././. /. /. /. /. /./. /. /. /. /./. /./. /. /. /. /. /./. /. /. /. /./. /./. /. /. /. /. /./. /. /. /. /./. /./. /. /. /. /. /./. /. /. /././. /. /./. /././. /././. /././. /././. /././. /. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /. /././. /././. /././. /././. /././. /././././. /././. /././. /. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././. /. /. /. /. /. /. /. /. /././. /././././././././././././././././././././././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /. /. /. /. /. /. /. /./././././././././././. /././././././././././. Figure 3.1: A bead on a spinning semi-circular wire. amplitude of the periodic solution. Then there are functions ()a n d (), dened for all suciently small, real , such that (0) =(0) = 0 and that the system with =b+() has a unique small amplitude solution of period T=2(1 +())==1(b). When expanded, we have ()=22+O(3). The sign of 2indicates where the oscillations occur, i.e., for<bor for>b. Example 1 The nonlinear ordinary dierential equation du dt=g(u)=u2−1u−2 (3.4) has steady-state solutions that satisfy g(u)=u2−1u−2=0 . T h e s e steady-state solutions have bifurcation points given by dg du=2u−1=0: Solving these last two equations simultaneously, it can be shown that the bifurcation points of the steady-state solutions are along the curve 4 2+ 2 1= 0. Further analysis shows that equation (3.4) will have two real steady-state solutions when 4 2+2 1>0, and it will have no real steady- state solutions when 4 2+2 1<0. Example 2 Consider a frictionless bead that is free to slide on a semi-circular hoop of wire of radius Rthat is spinning at an angular rate !(see gure 3.1). The equation for (t), the angle of the bead from the vertical, is given by d2 dt2+gsin R 1−!2R gcos =0; (3.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 3. Bifurcation Theory 21 wheregis the magnitude of the gravitational force. We dene the param- eterby=g=!2R. We will analyze only the case 0. The three possible steady solutions of equation (3.5) are given by for0;(t)=1=0; for1;(t)=2=c o s−1; for1;(t)=3=−cos−1: Therefore, for >1 (which corresponds to slow rotation speeds), the only steady solution is (t)=1.F o r1, however, there are three possible solutions. The solution (t)=1will be shown to be unstable for <1. To determine which solution is stable in a region where there are multi- ple solutions, a stability analysis must be performed. This is accomplished by assuming that the true solution is slightly perturbed from the givensolution, and the rate of change of the perturbation is obtained. If the perturbation grows, then the solution is unstable. Conversely, if the per- turbation decays (stays bounded), then the solution is stable (neutrally stable). First we perform a stability analysis for the solution (t)= 1. Dene (t)=1+(t); (3.6) whereis a small number and (t) is an unknown function. Using (3.6) in equation (3.5), and expanding all terms for 1, results in d2 dt2+g−1 =O(): (3.7) The leading order terms in equation (3.7) represent the Fr echet derivative of equation (3.5) at the \point" (t)=1, applied to the function (t). The solution of this dierential equation for (t), to leading order in ,i s (t)=Acost+Bsint; (3.8) whereAandBare arbitrary constants and =q g/parenleftbig−1 
.I f>1, then is real, and the solutions for (t) remain bounded. Conversely, if <1 thenbecomes imaginary, and the solution in (3.8) becomes unbounded astincreases. Hence, the solution (t)=1is unstable for <1. Now we perform a stability analysis for the solution (t)=2. Writing (t)=2+ (t) and using this form in equation (3.5) leads to the equation for (t): d2 dt2+g1−2  =O(): (3.9) The leading order terms in equation (3.9) represent the Fr echet derivative of equation (3.5) at the \point" (t)=2, applied to the function (t). The CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 22 I.A Denitions and Concepts/1 /#12/1 /#17 /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /./. /./././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./././. /. /./. /. /./. /./. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /./././././././././././././././././././././. /././././. /./././././././././././././././.SUS S /. /. /./. /./. /. /././. /. /./. /. /././. /. /./. /. /./. /./. /./. /. /./. /. /././. /./. /. /./. /./. /./. /. /./. /. /././. /. /./. /. /././. /. /./. /. /./. /././. /. /./. /. /././. /. /./. /. /./. /./. /./. /. /./. /././. /. /./. /. /./. /./. /./. /. /./. /. /././. /. /./. /. /././. /. /./. /./. /. /././. /. /./. /. /././. /. /./. /. /./. /./. /./. /. /./. /. /././. /./. /./. /./. /./. /. /./. /. /././. /./. /. /././. /./. /. /./. /././. /. /./. /././. /. /./. /./. /./. /./. /./. /././. /. /./. /./. /./. /./. /./. /. /././. /./. /. /././. /./. /. /./. /././. /. /./. /././. /. /./. /./. /././. /./. /. /././. /./. /./. /./. /./. /./. /././. /./. /././. /./. /./. /././. /./. /././. /./. /./. /././. /./. /././. /./. /./. /././. /./. /. /./././. /./././. /./. /./././. /././. /././. /././. /./././. /./. /./././. /./././. /././././. /./. /././././././. /././././././. /./././././. /./././././././. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././././././. /././././././. /././././././. /././././. /./././. /././././. /./. /./././. /././. /././. /././. /./././. /././. /././. /./. /./././. /././. /./. /./. /././. /./. /././. /./. /./. /././. /./. /././. /./. /./. /././. /./. /././. /./. /./. /././. /./. /. /././. /./. /./. /./. /./. /./. /././. /./. /. /././. /./. /. /./. /././. /. /./. /././. /. /./. /./. /./. /./. /./. /././. /. /./. /. /././. /./. /./. /. /././. /./. /. /././. /./. /. /./. /././. /. /./. /././. /. /./. /./. /./. /./. /. /./. /. /././. /./. /./. /./. /./. /. /./. /. /././. /. /./. /. /././. /./. /. /./. /. /././. /. /./. /. /././. /. /./. /. /./. /./. /./. /. /./. /././. /. /./. /. /./. /./. /./. /. /./. /. /././. /. /./. /. /././. /. /./. /./. /. /././. /. /./. /. /././. /. /./. /. /./. /./. /./. /. /./. /. /././. /./. /. /./. /./. /./. /. /./. /. /././. /. /./. /./. /./. /./. /. /./. /. /././. /. /./. /. /./.Figure 3.2: Bifurcation diagram for equation 3.6. A branch with the label \S" (\U") is a stable (unstable) branch. solution of this dierential equation for (t)i s (t)=Acost+Bsint, whereAandBare arbitrary constants and =q g/parenleftbig1−2 
.I f<1, then is real and the solutions for (t) remain bounded. Therefore, the solution (t)=2is stable for <1. In an exactly analogous manner, (t)=3is stable for<1. From what we have found, we can construct the bifurcation diagram shown in gure 3.2. In this diagram, the unstable steady solutions are in-dicated by a dashed line and the letter \U", and the stable steady solutions are indicated by the solid line and the letter \S". In words, this diagram states: For no rotation ( !=0o r=1), the only solution is (t)= 1=0 . As the frequency of rotation increases (and so decreases), the solu- tion(t)=1becomes unstable at the bifurcation point =1 . For<1, the are two stable solutions, (t)=2and(t)=3.I n this example, there is no way to know in advance which of these two solutions will occur (physically, the bead can slide up either side ofthe wire). The formula in (3.3) can be applied to equation (3.5) to determine the lo- cation of the bifurcation point without performing all of the above analysis. If we dene x 1=andx2=d dt, then equation (3.5) can be written as the system of ordinary dierential equations d dt x1 x2 =f(x)=x2 −gsinx1/parenleftbig 1−cosx1 
 ; which has the Jacobian matrix J=df dx=01 gcosx1+g /parenleftbig cos2x1−sin2x1
0 : CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 3. Bifurcation Theory 23 If>1, then no choice of ( x1;x2) will allow both fand detJto be zero simultaneously. For = 1, however, x1=x2=0m a k eb o t h fand detJ equal to zero. Hence, a bifurcation occurs at =1 . Example 3 Abelson [1] has developed a computer program in LISP that automat- ically explores the steady-state orbits of one-parameter families of period-ically driven oscillators. The program generates both textual descriptions and schematic diagrams. For example, consider Dung’s equation in the form ¨ x+0:1_x+x 3= pcost, where the parameter pis in the range [1 ;25] and only those solutions with−5_x5a n d−10¨x10 are considered. The program produced the graphical output shown in gure 3.3, along with the following textualdescription: The system was explored for values of pbetween 1 and 25, and 10 classes of stable periodic orbits were identied. Class A is already present at the start of the parameter range p= 1 with a family of order-1 orbits A 0.N e a rp=2:287, there is a supercritical-pitchfork bifurcation, and A0splits into symmetric families A1;0andA1;1, each of order 1. A1;0vanishes at a fold bifurcation near p=3:567.A1;1vanishes similarly. Class B appears around p=3:085 with a family of order-1 orbitsB0arising from a fold bifurcation. As the parameter pincreases,B0undergoes a period doubling cascade, reaching order 2 near p=4:876, and order 4 near p=5:441. Although the cascade was not traced past the order 4 orbit, there is ap- parently another period-doubling near p=5:52, and a chaotic orbit was observed at p=5:688. ... ClassJappears around p=2 3:96 as a family of order-5 orbits J0arising from a fold bifurcation. J0is present at the end of the parameter range at p= 25. This program is capable of recognizing the following types of bifur- cations: fold bifurcations, supercritical and subcritical flip bifurcations, supercritical and subcritical Niemark bifurcations, supercritical and sub- critical pitchfork bifurcations, and transcritical bifurcations. Notes 1. There are many dierent types of bifurcations. See gure 3.4 for diagrams of some of the following bifurcations: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 24 I.A Denitions and Concepts Figure 3.3: Graphical output generated automatically from the Bifurcation Interpreter in Abelson [1]. For Dung’s equation, the evolution of 10 classes of families of periodic orbits and their bifurcations has been traced. Thepvalues along the horizontal axis indicate the parameter value at which the bifurcations occur. (Reprinted with permission from Comp & Maths. With Appls. 20 , 8, Abelson, H., The bifurcation interpreter: A step towards the automatic analysis of dynamical systems, Copyright 1990, Pergamon Press.) Hopf bifurcation : a stable steady solution bifurcates into a stable oscillatory solution. That is, there are no stable steady solutionsin that particular region of parameter space. This occurs by having some of the eigenvalues of the Jacobian in (3.2) become CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 3. Bifurcation Theory 25/. /. /./. /. /. /./. /. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /./. /. /. /. /././././. /. /. /. /. /./. /././. /. /././. /. /././. /. /./. /. /././. /. /././. /. /././. /. /. /././././. /././././.fo ld /././. /./. /././. /././. /././. /././. /./././. /./././. /./././././. /././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /././././. /. /. /. /. /./. /././. /. /././. /. /././. /. /./. /. /././. /. /././. /. /././. /. /. /././././. /././././.transcr it ical /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././././././. /. /. /./. /. /. /./. /. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /././././. /. /. /. /. /./. /././. /. /././. /. /././. /. /./. /. /././. /. /././. /. /././. /. /. /././././. /././././.sup ercr it icalpit c hfor k /././. /./. /././. /././. /././. /././. /./././. /./././. /./././././. /././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /././././. /. /. /. /. /./. /././. /. /././. /. /././. /. /./. /. /././. /. /././. /. /././. /. /. /././././. /././././.su bc ri t icalpit c hfor k /././. /./. /././././. /./././. /././. /././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. Figure 3.4: Diagrams of some types of bifurcations. Unstable solutions are indicated by dashed lines; stable solutions are indicated by solid lines. purely imaginary. Fold bifurcation : on one side of the bifurcation point a stable and an unstable periodic point (of the same order) coexist. On the other side of the bifurcation point, both periodic points have vanished. Flip bifurcation (supercritical) : a stable periodic point of order n transitions to a stable periodic point of order 2 nand an unstable periodic point of order n. Flip bifurcation (subcritical) : an unstable periodic point of or- der 2nand a stable periodic point of order ntransition to an unstable periodic point of order n. Niemark bifurcation (supercritical) : a stable periodic transitions to an unstable periodic point and a stable limit cycle. Niemark bifurcation (subcritical) : a stable periodic point and unstable limit cycle transition to an unstable periodic point. Pitchfork bifurcation (supercritical) : a stable periodic point tran- sitions to two stable periodic points and an unstable periodic point, all of the same order. Pitchfork bifurcation (subcritical) : a stable periodic point and two unstable periodic points transition to an unstable periodic point. Transcritical bifurcation : a stable periodic point and an unstable periodic point exchange stabilities; on the other side of the bifurcation point, the extrapolated stable point is now unstable,and vice-versa. 2. For a dierential equation that is not autonomous, bifurcations can also occur from time-dependent solutions to other time-dependentsolutions. 3. For the general nite dimensional mapping, G(x), from R mto Rn, the Jacobian J(x): =@G @xneed not be square. In this case, the critical points (which include the bifurcation points) are in the set C,w i t h C:=fxjx2Rm;rankJ(x)<min(m;n)g: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 26 I.A Denitions and Concepts The regular points are Rm−C. The critical values are the values in the setG(C): =fyjy2Rn;y=G(x)f o rs o m e x2Cg. The regular values are Rn−G(C). 4. Sacks [8] describes the program POINCARE, which classies bifur- cation points and constructs representative phase diagrams for each type of behavior. The program is available directly from Sacks. 5. Numerical methods for computing bifurcations are described in Guck- enheimer et al. [3] and Jepson and Spence [6]. References [1]Abelson, H. The bifurcation interpreter: A step towards the automatic analysis of dynamical systems. Comp. & Maths. with Appls. 20 , 8 (1990), 13{35. [2]Guckenheimer, J. Patterns of bifurcations. In New Approaches to Nonlinear Problems in Dynamics , P. J. Holmes, Ed. SIAM, Philadelphia, PA, 1980, pp. 71{104. [3]Guckenheimer, J., Myers, M., and Sturmfels, B. Computing Hopf bifurcations. SIAM J. Numer. Anal. 34 , 1 (February 1997), 1{21. [4]Holodniok, M., and Kubcek, M. New algorithms for the evaluation of complex bifurcation points in ordinary dierential equations. A comparativenumerical study. Appl. Math. and Comp. 15 (1984), 261{274. [5]Iooss, G., and Joseph, D. D. Elementary Stability and Bifurcation Theory . Springer{Verlag, New York, 1989. [6]Jepson, A. D., and Spence, A. Numerical methods for bifurcation problems. In The State of the Art in Numerical Analysis , A. Iserles and M. J. D. Powell, Eds. Clarendon Press, Oxford, England, 1987, pp. 273{298. [7]Rand, R. H., and Armbruster, D. Perturbation Methods, Bifurcation Theory and Computer Algebra . No. 65 in Applied Mathematical Sciences. Springer{Verlag, New York, 1987. [8]Sacks, E. Automatic analysis of one-parameter planar ordinary dierential equations by intelligent numeric simulation. Articial Intelligence 48 (1991), 27{56. [9]Seydel, R. From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis . American Elsevier Publishing Company, New York, 1988. [10]Wood, E. F., Kempf, J. A., and Mehra, R. K. Bistab: A portable bifurcation and stability analysis package. Appl. Math. and Comp. 15 (1984), 345{355. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 4. A Caveat for Partial Dierential Equations 27 4. A Caveat for Partial Dierential Equations Idea To solve partial dierential equations correctly, a good understanding of the nature of the partial dierential equation is required. This requiresmore than a knowledge of the \physics" of the problem: a thorough under- standing of the type of partial dierential equation is needed. From Collatz [1, page 260]: That an investigation of the situation is absolutely essential is revealed even by quite simple examples; they show that formal calculation applied to partial dierential equations can leadto false results very easily and that approximate methods can converge in a disarmingly innocuous manner to values bearing no relation to the correct solution. Example Suppose we wish to solve the following wave equation (this example is from Collatz [1]) uxx=utt; u(x;0) = cosx;forjxj<= 2; @u(x;0) @t=c o sx;forjxj<= 2; u  2;t =s i nt;fort>0:(4.1.a-d) We will attempt to solve (4.1) by looking for a series solution of the form u(x;t)=1X n;m=0amnxmtn: (4.2) Using (4.2) in (4.1.a), we nd that am;n+2=(m+2 ) (m+1 ) (n+2 ) (n+1 )am+2;n: (4.3) To satisfy (4.1.b), we require ak;1= 0. To satisfy (4.1.c), we also require ak;0=( 0;k =o d d; (−1)q=(2q)!;k =e v e n=2q:(4.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 28 I.A Denitions and Concepts/./././. /./././. /./././. /./././. /./././. /./. /./. /./././. /./././. /./././. /./././. /././. /././. /./././. /./././. /./././. /./././. /. /. /./././. /./././. /./././. /./././. /./. /./. /./././. /./././. /./././. /./././. /./././. /./././. /./././. /./././. /. /. /./././. /./././. /./././. /././. /././. /./././. /./././. /./././. /./././. /./././. /./././. /./. /./. /./././. /./././. /././. /././. /./././. /./././. /. /. /./././. /./././. /./. /./. /./././. /./././. /./././. /./././. /. /. /./././. /././. /././. /./././. /./././. /./. /./. /././. /././. /. /./. /. /./. /.so lu t ion v alid h ere /#19/2/#19/2 /, /#19/2 tx /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /./././././. /././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./././. /./././././././././././././. /././././. /./././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /.Figure 4.1: Depiction of the characteristics and the range of validity of the solution found for equation 4.1. Evaluating equation (4.2) at x= 0 and using equations (4.3) and (4.4), we nd that u(0;t)=1X k=0a0;ktk=1X q=0(−1)q (2q)!t2q=c o st: (4.5) Now the conclusion in equation (4.5) is correct but only for 0t=2. This is because the characteristics (see page 432), t==2x, emanating from the points ( =2;0) and (−=2;0) do not allow u(0;t) to be determined directly for t>= 2. See gure 4.1 for a graphical representation of the characteristics of (4.1) and the region of validity for the solution in (4.5). References [1]Collatz, L. The Numerical Treatment of Dierential Equations . Springer{ Verlag, New York, 1966. [2]Rassias, J. M. Counter Examples in Dierential Equations and Related Topics . World Scientic, Singapore, 1991. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 5. Chaos in Dynamical Systems 29 5. Chaos in Dynamical Systems Applicable to Nonlinear dierential equations. Yields Information on whether or not a system is chaotic. Idea Chaos is a phenomenon that can appear in solutions to nonlinear dif- ferential equations. Chaos is easily dened and can be easily (numerically) found in some equations. Procedure For simplicity, we focus on deterministic systems modeled by coupled, autonomous, rst order, ordinary dierential equations of the form dxi dt=gi(x;q)f o ri=1;2;:::;n (5.1) where x=(x1;x2;:::;xn) is the state-space vector and q=(q1;q2;:::;qm) is a set of parameters. This equation determines a set of solutions, each specied by their initial values. We can specify the solution correspondingto the initial condition pbyx(t;p). Consider a set of initial conditions contained in a vanishing small volume V. Under the action of equation (5.1), the volume will change as a function oft. Precisely, dV dt=Z


Z V mX i=1@gi @xi! dx1


dxn: The summation term is the generalized divergence of gand is called the Lie derivative. Dissipative systems are characterized by contracting volumes;this is equivalent to dV=dt < 0. Conservative or Hamiltonian systems, in which equation (5.1) are Hamilton’s equations, obey Liouville’s theorem: dV=dt =0 . Any trajectory of a dissipative system as t!1 will approach a bounded region of phase space called an attractor. An attractor has zero volume in phase space. Attractors include points, limit cycles, and tori.For example, consider an unforced damped pendulum. The attractor for this is a point in phase space, the stable conguration with the pendulum hanging straight down. In this case, starting the pendulum swinging withslightly dierent initial conditions will lead to close paths in phase space and the same nal state. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 30 I.A Denitions and Concepts For nonlinear systems exhibiting chaos, the separation of two nearby trajectories increases exponentially with time. This is referred to as sensi- tive dependence on initial conditions . For dissipative systems, a stretching in one direction has to be accompanied by a more-than-compensatingcontraction in other directions, so that the volume of an arbitrary droplet of initial conditions will contract with time. The phase-space trajectories for a chaotic system asymptotically approach a strange attractor ,a na t t r a c t o r with a fractional dimension (i.e., a fractal). Lyapunov exponents are a measure of the rate of divergence (or conver- gence) of initially innitesimally separated trajectories. The ith Lyapunov exponent, i, can be found by considering the evolution of a vanishingly small set of initial conditions that form a hyperellipsoid. We dene i:= lim t!1 i(0)!01 ti(t) i(0) (5.2) wherei(t) is the length of the ith principal axis of the hyperellipsoid at timet,f o ri=1;2;:::;n . An attractor is chaotic if it has at least one positive Lyapunov exponent. The Lyapunov exponents can be determined by analyzing the linearized equations corresponding to equation (5.1). For illustrative purposes, wespecialize to n= 3 for the rest of this section. Consider the two close initial points: p 0=(x0;y0;z0)a n d p1=p0+x=(x0+x;y 0+y;z 0+z). We want to nd the evolution of the dierence a(t): =x(t;p1)−x(t;p0). Using Taylor series da1 dt=d[x1(t;p1)−x1(t;p0)] dt=d[g1(x(t;p0+x))−g1(x(t;p0))] dt @g1 @xx+@g1 @yy+@g1 @zz =@g1 @xa1+@g1 @ya2+@g1 @za3; where the partial derivatives are evaluated at x(t;p0). In general da dt=M(x)a=2 64@g1 @x@g1 @y@g1 @z @g2 @x@g2 @y@g2 @z @g3 @x@g3 @y@g3 @z3 75a; whereMis the Jacobian of the vector g. The Lyapunov exponents are related to the eigenvalues of the matrix M. In special situations, analytical methods can be used to obtain the Lyapunov spectra, while numerical methods must be used in general. Whenthere is a stationary solution given by dx dt=g(x)=0, the Jacobian matrix is time independent, and we can analytically obtain the (possibly complex) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 5. Chaos in Dynamical Systems 31/5/0 /7/5 /1/0/0 t /0 /1 x/./. /. /././. /. /./././. /./././././././././././././././././././././././. /./. /./././. /./. /./. /./. /./. /./. /. /./. /. /./. /. /./. /. /././. /./././. /. /././. /./. /. /./. /./. /. /./. /././. /./. /././././. /. /./././. /././. /./. /. /./. /./. /. /././. /. /. /./. /. /./. /. /./. /./. /./. /././././././././././. /./././. /././././. /./././. /./././././. /././././. /./././. /././././. /././././. /./././. /. /././. /./././. /. /./././././. /./././././././././././././././././././. /./. /./. /./. /./. /./. /./. /./. /. /./. /. /./. /. /./. /. /. /./. /./././. /./././. /. /. /. /./. /. /. /././. /./. /. /./. /./. /././. /././. /./. /. /./././. /./. /. /./. /. /. /././. /. /././. /. /././. /. /./. /./. /././././././././. /./././. /././. /./././. /././././././././././. /././././. /././././. /././././. /./././. /././././././. /././././. /./././././././././. /./././././././././././././././././. /./. /./. /./. /./. /./. /./. /./. 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/././././././././././././././././././. /././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /././././././././././././././././. /././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././././././././././././././././. /./././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././././././././././././././././. /././. /././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. 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Figure 5.1: Dung equation with Γ = 0 :20. 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/././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././. /. /././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././. /././. /././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././././././././././././././././././. /. /././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././. /. /./././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././././././././././././././././. /./././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././. /././. /././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. Figure 5.2: Dung equation with Γ = 0 :28. (Period 2 solution.) eigenvalues, from which the Lyapunov exponents may be found. In general, there are no stationary solutions and the equationsdx dt=gandda dt=M(x)a must be numerically solved simultaneously. See Wolf et al. [12] for a numerical technique for computing Lyapunov exponents. Example Consider the Dung equation: ¨ x+k_x−x+x3=Γc o s!t. This can be converted to an autonomous system as follows: dx dt=d dt2 4x y z3 5=2 4y −ky+x−x3+Γc o sz !3 5: (5.3) Figures 5.1{5.3 show the dierent behavior of this system ( x(t)v e r s u stand x(t)v e r s u sy(t)) whenk=0:3,!=1:2, and Γ takes on the values 0 :20, 0:28 and 0:50. For the numerical simulations shown, the initial conditions used were x0=( 1:3;0;0), and we began plotting the results when t=5 0 to remove any initial transients. From deeper analysis, it can be shown that the system has a period 1 (2, 4, 5, 2, 1) solution when Γ = 0 :20 (0.28, 0.29, 0.37, 0.65, 0.73). The solution is chaotic when Γ = 0 :50. A dierent set of parameters is shown in gure 5.4. This gure has a plot of the three Lyapunov exponents of equation (5.3) when !=1:0, k=0:5, and Γ is varied from 0 :2t o0:9. At low values of Γ, the system is periodic because the largest Lyapunov exponent is zero. The system follows CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 32 I.A Denitions and Concepts/7/5 /1/0/0 t /0 /1 /././././././. /. /. /. /. /. /. /. /. /. /././././././. /././././././. /./././././././././. /././././././././././././././. /. /. /. /. /././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././. /./././././. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /. /./. /. /. /. /. /. /. /. /. /././././././././././././././. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /././././. /././. /././. /././././././. /././. /./././././././. /./././. /./././././././. /. /. /. /. /. /. /. /. /././././././././././././. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /./. /. /. /. /. /. /. /. /././././././././. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./1 x/, /1 /0 /1/././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /. /./. /. /. /./. /. /./. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /./. /./././././././././././././././././././. /. /. /. /. /. /. /. /./././././. /././././././././././././././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./. /./. /././. /././. /./. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /././././././././././././././././././. /./././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././././././././././././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /./. /. /./. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /././././././././././././././././././././././././././././././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. Figure 5.3: Dung equation with Γ = 0 :50. (Chaotic solution.) Figure 5.4: The three Lyapunov exponents for Dung’s equation with !=1:0a n dk=0:5 when Γ is varied from 0 :2t o0:9. (From De Souza- Machado, S., Rollins, R. W., Jacobs, D. T., & Hartman, J. L. Studyingchaotic systems using microcomputer simulations and Lyapunov exponents. Amer. J. Physics 58 , 4, April 1990, 321{329.) a period doubling route to chaos at Γ 0:36, when the largest Lyapunov exponent becomes greater than zero. The system remains chaotic until the driving force gets very large (Γ >0:84) except for windows of periodicity, which occur throughout the chaotic regime. Notes 1. There are at least three scenarios in which the regular behavior of a system becomes chaotic. A standard route is via a series of period-doubling bifurcations. Two other routes to chaos that are fairly well understood are via intermittent behavior and through quasiperiodic solutions. 2. Many equations have been shown to be chaotic: Hale and Sternberg [4] have shown that the dierential delay equation dx(t) dt=ax(t)+bx(t−) 1+xn(t−)is chaotic for certain pa- rameter regimes. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 5. Chaos in Dynamical Systems 33C/2 G L RC/1 GN/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /./. /. /. /. /. /. /. /. /. /././. /././. /././. /././. /././. /././././. /././. /./././././. /././. /././. /././././. /././. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./././. /./././. /./././././. /./././. /././. /././. /././. /././. /. /././. /././. /././. /././. /. /. /. /. /. /. /./. /. /./. /. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /. /././. /. /./././././. /././. /././. /. /././. /./././././. /./././. /././. /././. /././. /././. /. /./. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /. /./. /. /. /. /././. /. /././././. /././././. /././././././././././. /././. /. /././././././. /././. /. /././. /././. /. /./. /. /./. /. /./. /./. /./././. /. /././././. /. /. /././. /. /. /././. /. /. /./. /. /. /./. /. /. /./././././././. /././././././././. /././. /./././. /./././. /././. /././. /. /./././. /. /././. /./. /././. /. /./. /././. /. /./. /. /././. /. /././. /. /./. /. /././. /. /. /. /. /. /. /././. /. /. /. /./././././././././. /./././././././. /./. /././././. /. /. /././. /. /. /./././. /././. /./././. /./. /. /././. /././. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /./././. /. /. /./. /./. /. /./././. /././././. /././. /././. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././. /./././. /./././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /, /1 /1 g/, g iv GN Figure 5.5: The canonical piecewise-linear circuit and the voltage-current characteristic of the nonlinear resistor GN. The equations dening the Lorenz attractor are _x=1 0y−10x; _y=−y−xz+2 8x; _z=xy−8 3z:(5.4) The R¨ ossler equations are _x=−(y+z); _y=x+ay; _z=b+xz−cz:(5.5) Whena=0:343,b=1:82, andc=9:75, this generates the \R¨ossler funnel." When a=0:2,b=0:2, andc=5:7, this generates \the simple R¨ ossler attractor." 3. For an autonomous electronic circuit to exhibit chaos, it must contain at least three energy storage devices. (Otherwise, the Poincar e{ Bendixson theorem states that the limiting set will be a point ora limit cycle, not a strange attractor.) A simple circuit with three energy storage devices that produces chaos is in Matsumoto [7]. The circuit given in Chua and Lin [2] (see gure 5.5) is almost as simple as that given by Matsumoto and can simulate (by choosing dierent values for the nonlinear resistor) dierent chaotic phenomenain a large three-dimensional state space. This circuit contains only six two-terminal elements: Five of them are linear resistors, capacitors, and inductors; and one element ( G N) is a three-segment, piecewise- linear resistor. 4. Dierent types of dynamical systems can have greater or lesser de- grees of randomness. A simple classication of the amount of ran- domness in dynamical systems is as follows: Ergodic systems : this is the \weakest" level of randomness, in which phase averages equal time averages. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 34 I.A Denitions and Concepts Mixing systems : here, no time averaging is required to reach \equilibrium." K-systems : systems with positive Kolmogorov entropy. This means that a connected neighborhood of trajectories must ex-hibit a positive average rate of exponential divergence. C-systems : every trajectory has a positive Lyapunov exponent. Bernoulli systems : these systems are as random as a fair coin toss. See Tabor [11] for details. 5. A technical denition of the Lyaponuv exponents is as follows: When A(t) is a bounded coecient matrix, consider the n-dimensional linear system y 0=A(t)y(t). Consider nlinearly independent solutions of this in the form yi=Y(t)pi,w h e r eY(t) is a fundamental solution matrix with Y(0) orthogonal, and the fpigform an orthonormal basis. The characteristic numbers are dened as i= lim t!1sup1 tlog (jjY(t)pijj): When the sum of the characteristic numbers is minimized, the or- thogonal basisfpigis called normal and thefigare the Lyapunov exponents . 6. There are many software packages for numerically computing Lya- punov exponents. See, for example, Parker and Chua [8] and Rollins [9]. 7. In this section we have focused on chaos appearing in coupled, rst- order, ordinary dierential equations. Chaos can also appear inpartial dierential equations and stochastic equations. 8. The papers by Ablowitz and Herbst [1], Lorenz [6], and and Yamaguti and Ushiki [13] describe and illustrate how numerical discretizationsof a dierential equation can lead to discrete equations exhibiting chaos. 9. By long-term integration of the equations governing the solar system on speical purpose computers, researchers have found that Pluto’s orbit is chaotic, the motion of the Jovian planet subsystem is chaotic, and the motion of comet Halley is chaotic. See, for example, Sussman and Wisdom [10]. References [1]Ablowitz, M. J., and Herbst, B. M. On homoclinic structure and numerically induced chaos for the nonlinear Schrodinger equation. SIAM J. Appl. Math. 50 , 2 (April 1990), 339{351. [2]Chua, L. O., and Lin, G.-N. Canonical realization of Chua’s circuit family. IEEE Trans. Circ. & Syst. 37 , 7 (July 1990), 885{902. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 5. Chaos in Dynamical Systems 35 [3]Geist, K., Parlitz, U., and Lauterborn, W. Comparison of dierent methods for computing Lyapunov exponents. Progr. Theoret. Phys 83 ,5 5 (1990), 875{893. [4]Hale, J. K., and Sternberg, N. Onset of chaos in dierential delay equations. J. Comput. Physics 77 (1988), 221{317. [5]Hassard, B., Zhang, J., Hastings, S. P., and Troy, W. C. A computer proof that the Lorenz equations have \chaotic" solutions. Appl. Math. Lett. 7, 1 (1994), 79{83. [6]Lorenz, E. N. Computational chaos|A prelude to computational insta- bility. Physica D 35 (1989), 299{317. [7]Matsumoto, T. Chaos in electronic circuits. Proc. IEEE , 8 (August 1987). [8]Parker, T. S., and Chua, L. O. Ecient solution of the variational equation for piecewise-linear dierential equations. Circuit Theory and Appl. 14, 4 (1986), 305{314. [9]Rollins, R. W. Chaotic Dynamics Workbench . Tech. rep., Physics Academic Software, AIP, New York, 1990. [10]Sussman, G. J., and Wisdom, J. Chaotic evolution of the solar system. Science 257 , 5066 (1992), 56{62. [11]Tabor, M. Chaos and Integrability in Nonlinear Dynamics . John Wiley & Sons, New York, 1989. [12]Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A. Determining Lyapunov exponents from a time series. Physica D 16 (1985), 285{317. [13]Yamaguti, M., and Ushiki, S. Chaos in numerical analysis of ordinary dierential equations. Physica D 3 (1981), 618{626. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 36 I.A Denitions and Concepts 6. Classication of Partial Dierential Equations Applicable to Partial dierential equations. Yields Knowledge of the type of equation under consideration. Procedure Most partial dierential equations are of three basic types: elliptic, hyperbolic, and parabolic. Elliptic equations are often called potential equations. They result from potential problems, where the potential might be temperature, voltage, or asimilar quantity. Elliptic equations are also the steady solutions of diusion equations, and they require boundary values in order to determine the solution. Hyperbolic equations are sometimes called wave equations, because they often describe the propagation of waves. They require initial conditions (where the waves start from) as well as boundary conditions (to describe how the wave and the boundary interact; for instance, the wave might be scattered or absorbed). These equations can be solved, in principle, by themethod of characteristics (see page 432). Parabolic equations are often called diusion equations because they describe the diusion and convection of some substance (such as heat).The dependent variable usually represents the density of the substance. These equations require initial conditions (what the initial concentration of the substance is) as well as boundary conditions (to specify, for instance,whether the substance can cross the boundary or not). The above classication is most useful for second order partial dieren- tial equations. For second order equations, only characteristic curves need to be considered. For equations of higher degree, characteristic surfaces must be considered, see Whitham [8, pages 139{141] or Zauderer [10, pages78{85 and 91{97] for more details. After two special cases, we specialize the rest of this section to second order partial dierential equations. Special Case 1 The most general second order linear partial dierential equation with constant coecients nX i;j=1aij@2u @xi@xj+nX i=1bi@u @xi+cu=d; may be placed in the form u11+


+unn+u=0; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 6. Classication of Partial Dierential Equations 37 if the equation is elliptic or may be placed in the form u11−u22−


−unn+u=0; if the equation is hyperbolic, for some value of . See Garabedian [3, pages 70{76] for details. Special Case 2 The (real valued) second order partial dierential equation in ndimen- sions nX i;j=1aij(x)@2u @xi@xj+f x;u;@u @x1;:::;@u @xn =0; (6.1) foru(x)=u(x1;:::;xn), whereaij=aji, may be classied at the point x0as follows. Let Abe the matrix ( aij(x0)). By means of a linear transformation, the quadratic form gTAgmay be reduced to the form 1g2 1+2g2 2+


+ng2 n: The values offig, which are the eigenvalues of A, determine the nature of the partial dierential equation (6.1). Because Ahas been assumed to be symmetric, all of the eigenvalues will be real. The classication at the point x0is then given by 1. If all of thefigare of the same sign, then equation (6.1) is elliptic atx0. 2. If any of thefigare zero, then equation (6.1) is parabolic at x0. 3. If none of the figare zero and they are not all of the same sign, then equation (6.1) is hyperbolic at x0. 4. If none of the figare zero and there are at least two that are positive and at least two that are negative, then equation (6.1) is ultrahyperbolic atx0. If an equation is parabolic along a smooth curve in a domain D,a n d the equation is hyperbolic on one side of the curve and elliptic on the other side of the curve, then the equation is of mixed type .T h es m o o t hc u r v e i s called the curve of parabolic degeneracy . Special Case 3 We further specialize here and restrict ourselves to second order equa- tions in two independent variables. Consider partial dierential equations of second order in two independent variables, of the form A(x;y)@2u @x2+B(x;y)@2u @x@y+C(x;y)@2u @y2=Ψ u;@u @x;@u @y;x;y ; (6.2) where Ψ need not be a linear function. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 38 I.A Denitions and Concepts If2 4B2−4AC > 0 B2−4AC=0 B2−4AC < 03 5at some point ( x;y), then equation (6.2) is2 4hyperbolic parabolic elliptic3 5 at that point. If an equation is of the same type at all points in the domain, then the equation is simply said to be of that type. Equation 6.2 can be transformed into a canonical form for each of the three types mentioned above. The procedures are as follows. Hyperbolic Equations For hyperbolic equations we look for a new set of independent variables =(x;y)a n d=(x;y) for which equation (6.2) may be written in the standard form u=(u;u;u;;): (6.3) Utilizing this change of variables, we can calculate ux=ux+ux; uy=uy+uy; uxx=uxx+2uxx+uxx+uxx+uxx; uxy=uxy+2u(xy+yx)+uxy+uxy+uxy; uyy=uyy+2uyy+uyy+uyy+uyy; to nd that equation (6.2) transforms into Au+Bu+Cu=(u;u;u;;); (6.4) where A=A2 x+Bxy+C2 y; B=Axx+B(xy+yx)+2Cyy; C=A2 x+Bxy+C2 y: SettingA=C= 0, we can nd the following partial dierential equations forand x y=−B+p B2−4AC 2A; x y=−B−p B2−4AC 2A:(6.5.a-b) These equations may be readily solved (in principle) by the method of characteristics. For example, to solve equation (6.5.a) we only need to solve −dy dx=−B+p B2−4AC 2A CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 6. Classication of Partial Dierential Equations 39 forQ(x;y)=R,w h e r eRis an arbitrary constant. Then will be given by =Q(x;y). Afterandare determined, then the original equation must be trans- formed into the new coordinates (see page 168). The resulting equation will then be in standard form. Note that another standard form for hyperbolic equations (in two in- dependent variables) is obtained from equation (6.3) by the change of variables =−;  =+: (6.6) This results in the equation u−u= u;u−u;u+u;1 2(+);1 2(−) : Example 1 Suppose we have the equation y2uxx−x2uyy=0: (6.7) We recognize this equation to be hyperbolic away from the lines x=0a n d y= 0. To nd the new variables and, we must solve the dierential equations in (6.5). For this equation, we have fA=y2,B=0 ,C=−x2g. Therefore (6.5) becomes x y=−x y;x y=x y; with the solutions =y2−x2,=y2+x2. In these new variables, equation (6.7) becomes u= 2(2−2)u− 2(2−2)u: (6.8) If the change of independent variable in (6.6) is made, then (6.8) becomes u−u=1 2u−1 2u: Parabolic Equations For parabolic equations, we look for a new set of variables =(x;y) and=(x;y) in which equation (6.2) can be written in one of the standard forms u=(u;u;u;;); (6.9.a) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 40 I.A Denitions and Concepts or u=(u;u;u;;): (6.9.b) Utilizing equation (6.4), we see that we need to determine andin such aw a yt h a t B=0=C; corresponding to (6.9.a) ; (6.10.a) or B=0=A; corresponding to (6.9.b) ; (6.10.b) IfA6= 0, then equation (6.10.a) corresponds to the single equation x y=−B 2A; (6.11.a) while, ifC6= 0, then equation (6.10.b) corresponds to the equation x y=−B 2C: (6.11.b) In either case, we have only to solve a single equation to determine .T h e variablecan then be chosen to be anything linearly independent of .A s before, once andare determined, then the equation needs to be written in terms of these new variables Example 2 Suppose we have the equation y2uxx−2xyuxy+x2uyy+uy=0: (6.12) SincefA=y2,B=−2xy,C=x2g, we nd that B2−4AC= 0 and so this equation is parabolic. In this case we choose to make B=C= 0. From equation (6.11.a) we must solvex y=x y, which has the solution =y2+x2. We choose =x. Using these values of and, we nd that (6.12) becomes u=2(+) −2u+1 −2u: Elliptic Equations For elliptic equations we look for a new set of variables =(x;y)a n d =(x;y) in which equation (6.2) can be written in the standard form u+u=(u;u;u;; ): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 6. Classication of Partial Dierential Equations 41 The easiest way in which to nd andis to determine variables =(x;y)a n d=(x;y) that satisfy (6.5) and then form =(+)=2, =(−)=2i(where, as usual, i=p−1). Note that in this case, the dierential equations in (6.5) are complex. However, since andare conjugate complex functions, the quantities andwill be real. Example 3 Suppose we have the equation y2uxx+x2uyy=0: We recognize this equation to be elliptic away from the lines x=0a n d y= 0. To nd the new variables and, we must solve the dierential equations in (6.5). For this equation, we have fA=y2,B=0 ,C=x2g. Therefore (6.5) becomes x y=−ix y;x y=ix y; with the solutions =y2−ix2,=y2+ix2. Formingandresults in =+ 2=y2; =− 2i=x2: In these new variables, equation (6.7) becomes u+u=−1 2u−1 2u: Notes 1. Equations of mixed type are discussed in Haack and Wendland [4] and Smirno [6]. 2. Given a partial dierential equation in the form of equation (6.1), the characteristic surfaces are dened by the characteristic equation nX i;j=1aij(x)@u @xi@u @xj =0: The solutions to this equation are the only surfaces across which u(x) may have discontinuities in its second derivatives. 3. The Notes section of the characteristics method (see page 432) de- scribes how to determine when a system of partial dierential equa- t i o n si sh y p e r b o l i c . 4. See also Farlow [2, pages 174{182 and 331{339], Moon and Spencer [5, pages 137{146], Stakgold [7, pages 467{482], and Young [9, pages 60{70]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 42 I.A Denitions and Concepts References [1]Bitsadze, A. V. Equations of the Mixed Type . The MacMillan Company, New York, 1964. [2]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [3]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [4]Haack, E. R., and Wendland, W. N. Lectures on Partial and Pfaan Dierential Equations . Pergamon Press, New York, 1972. Translated by E. R. Dawson and W. N. Everitt. [5]Moon, P., and Spencer, D. E. Partial Dierential Equations .D . C . Heath and Co., Lexington, MA, 1969. [6]Smirnoff, M. M. Equations of Mixed Type . Amer. Math. Soc., Providence, RI, 1978. [7]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. [8]Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. [9]Young, E. C. Partial Dierential Equations . Allyn and Bacon, Inc., Boston, MA, 1972. [10]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 7. Compatible Systems 43 7. Compatible Systems Applicable to Systems of dierential equations. Yields Knowledge of whether the equations are consistent. Procedure 1 The two equations f(x;y;z;p;q )=0a n d g(x;y;z;p;q )=0f o rz= z(x;y) (where, as usual, p=zxandq=zy) are said to be compatible if every solution of the rst equation is also a solution of the second equation, and conversely. These two equations will be compatible if ff;gg=0 ,w h e r e ff;gg:=@(f;g) @(x;p)+p@(f;g) @(z;p)+@(f;g) @(y;q)+q@(f;g) @(z;q); and where@(u;v) @(a;b)=juavaubvbj=uavb−vaubis the usual Jacobian. Procedure 2 The conditions for consistency of a system of simultaneous partial dif- ferential equations of the rst order, if the number of equations is an exact multiple of the number of dependent variables involved, is given in Forsyth[3, Part IV, pages 411{419]. To write the consistency conditions, let the unknown dependent variables be fz iji=1;:::;mg, let the independent variables befxjjj=1;:::;ng, and dene pij=@[email protected] e p r e s u m e the system has rmequations (with rn) and that these equations can be solved with respect to the pij.T h a ti s pij=@zi @xj=fij(fxlg;fzkg;fpg); fori=h1;mi,j=h1;ni,l=h1;ni,=h1;mi,=hr+1;ni. (Here we have introduced the notation ha;bito be the sequence of numbers a;a+ 1;a+2;:::;b .) Then, for consistency, the following conditions must be satised @fij @xa−@fia @xj+mX =1 fa@fij @z−fj@fia @z +mX s=1nX =r+1@fij @ps@fsa @x−@fia @ps@fsj @x +mX s=1nX =r+1mX =1@fij @ps@fsa @z−@fia @ps@fsj @z p =0;(7.1) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44 I.A Denitions and Concepts wherei=h1;mi,a=hj+1;ri,j=h1;r−1i,a n d mX s=1@fij @ps@fsa @pk−@fia @ps@fsj @pk+@fij @ps@fsa @pk−@fia @ps@fsj @pk =0; (7.2) wherei;k=h1;mi,a=hj+1;ri,;=hr+1;ni,j=h1;r−1i. Special Case 1 In the special case of m= 1, we have one dependent variable (which we callz)a n drequations. Let pj=@z=@xj=fj(z;x1;:::;xn;pr+1;:::;pn). In this case, equation (7.2) is automatically satised while equation (7.1) becomes dfj dxa−dfa dxj+nX =r+1@fj @pdfa dx−@fa @pdfj dx =0 fora=h1;j−1i,j=h1;ri, where we have denedd dxs=@ @xs+ps@ @z. Special Case 2 In the special case of r=n, the system of mnequations becomes pij=fij(z1;:::;zm,x1;:::;xn) and the consistency conditions become @fij @xa−@fia @xj+mX =1 fa@fij @z−fj@fia @z =0 fori=h1;mi,a=h1;j−1iandj=h1;ni. These are known as Mayer’s system of completely integrable equations. Special Case 3 Consider the special case of r=1 ,w i t hfF1=0 ,F2=0 ,:::,Fm=0g, where each Fj=pj−fj(z;x1;:::;xn;pr+1;:::;pn) is analytical in each of its arguments. A necessary and sucient condition for the set of equations to be consistent is that [ Fi;Fj] = 0, for all combinations of iandj. Here, [;] represents the usual Poisson bracket. Example Suppose we have the two following nonlinear partial dierential equa- tions forz(x;y): xzx=yzy;z (xzx+yzy)=2xy: (7.3) From (7.3) we identify f(x;y;z;p;q )=xp−yq; g (x;y;z;p;q )=z(xp+yq)−2xy: (7.4) Using (7.4) we can easily calculate @(f;g) @(x;p)=2xy;@(f;g) @(z;p)=−x2p; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 7. Compatible Systems 45 @(f;g) @(y;q)=−2xy;@(f;g) @(z;q)=xyp: Therefore, computing ff;gg, we nd it to be zero. Hence, the two equations in equation (7.3) have identical solution sets. Because the equations in (7.3) are compatible, we can combine them without changing the solution sets. Solving the equations in (7.3) simulta- neously for pandqto obtainfzx=p=y=z,zy=q=x=zg. These last two equations can be easily solved we obtain z2=C+2xy,w h e r eCis an arbitrary constant. Notes 1. Jacobi’s method (see page 464) takes a given partial dierential equa- tion and creates a compatible equation and then uses elimination between these two equations. 2. If it is known that a linear homogeneous ordinary dierential equa- tion of order nhas solutions in common with a linear homogeneous ordinary dierential equation of order m(withm<n ), then it is possible to determine a dierential equation of lower degree that has, as its solutions, these common solutions. If the linear homogeneousordinary dierential equations L 1[u]=0a n dL2[u] = 0 are dened by L1:=p0Dn+p1Dn−1+


+pn−1D+pn; L2:=q0Dm+q1Dm−1+


+qm−1D+qm; whereDrepresentsd=dx and each of the functions fpi;qigdepends onx, dene the ordinary dierential equation R1[u]=0b y R1:=r0Dn−m+r1Dn−m−1+


+rn−m−1D+rn−m; where thefrigare dened by p0=r0q0; p1=r1q0+r0n−m 1 q0 0+q1 ; p2=r2q0+r1n−m−1 1 q0 0+q1 +r0n−m 2 q00 0+n−m 1 q0 1+q2 ; ... pn−m=rn−mq0+rn−m−11 1 q0 0+q1 +rn−m−22 2 q00 0+2 1 q0 1+2 0 q2 +:::; =rn−mq0+rn−m−1[q0 0+q1]+rn−m−2[q00 0+2q0 1+q2]+:::: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 46 I.A Denitions and Concepts Then the order of the operator L3:=L1−R1L2will be depressed as much as is possible (the order of L3will not exceed m−1). Note that only a nite number of rational operations and dierentiations are required to determine the frig. From the denition of L3,w e see that all solutions common to both L1[u]=0a n dt o L2[u]=0 will also be solutions to L3[u]=0 . I fL3is identically zero, then we have found a factorization of L1(see page 294). See Ince [4, pages 126{128] or Valiron [6, pages 320{322] for details. 3. Dierential resultants can also be used to derive consistency condi- tions. See Berkovich and Tsirulik [2] for details. 4. Wolf [7] describes an algorithm that determines if an overdetermined system of two equations for one function has any solution. An imple-mentation in FORMAC is mentioned. 5. See also Ames [1, pages 54{65] and Sneddon [5, pages 67{68]. References [1]Ames, W. F. ,E d . Nonlinear Partial Dierential Equations ,v o l .1 .A c a d e m i c Press, New York, 1967. [2]Berkovich, L. M., and Tsirulik, V. G. Dierential resultants and some of their applications. Dierentsial’nye Uravneniya 22 , 5 (May 1986), 750{757. [3]Forsyth, A. R. Theory of Dierential Equations . Dover Publications, Inc., New York, 1959. [4]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [5]Sneddon, I. N. Elements of Partial Dierential Equations . McGraw{Hill Book Company, New York, 1957. [6]Valiron, G. The Geometric Theory of Ordinary Dierential Equations and Algebraic Functions . Math Sci Press, Brookline, MA, 1950. [7]Wolf, T. An analytic algorithm for decoupling and integrating systems of nonlinear partial dierential equations. J. Comput. Physics 60 (1985), 437{ 446. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 8. Conservation Laws 47 8. Conservation Laws Applicable to Partial dierential equations. Yields Quantities that remain invariant during the evolution of the partial dierential equation. Procedure Given an evolution equation, which is a partial dierential equation of the form ut=F(u;ux;uxx;:::); (8.1) a conservation law is a partial dierential equation of the form @ @tT u(x;t) +@ @xX u(x;t) =0; (8.2) which is satised by all solutions of equation (8.1). We dene T(
)t ob e theconserved density andX(
)t ob et h e flux. An alternative statement of equation (8.2) is that Z T u(x;t) dx (8.3) is independent of t, for solutions of (8.1) such that the integral converges. More generally, a partial dierential equation of order min then independent variables x=(x1,x2,:::,xn) and a single dependent variable uis in conservation form if it can be written as nX i=1@ @xiFi(x;u;@u;@2u;:::;@m−1u)=0: (8.4) Here@jurepresents all jth order partial derivatives of uwith respect to x. Example 1 The Korteweg{de Vries equation ut=uxxx+uux (8.5) has an innite set of conservation laws. The rst few, in order of increasing rank, have the conserved densities T=u; T=u2; T=u3−3u2 x; T=5u4−60uu2 x−36uxuxxx; ... CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 48 I.A Denitions and Concepts To demonstrate, for instance, that T=u2is a conserved density, we compute @T @t=@(u2) @t=2uut=2uuxxx+2u2ux; where we have used the dening equation in (8.5) to replace the utterm. Now we must determine a flux Xsuch that equation (8.2) is satised. In this case, we nd X=u2 x−2uuxx−2 3u3. Example 2 The Schr¨ odinger equation −@2u @x2+V(x)u=i@u @t can be expressed in the form of equation (8.2) with T=i(x)u; X=(x)@u @x−0(x)u; where(x) is dened by 00(x)=V(x)(x). Notes 1. Conservation laws allow estimates of the accuracy of a numerical solution scheme (because the quantity in (8.3) must be invariant in time). 2. Not all partial dierential equations have an innite number of con- servation laws; there may be none or a nite number. 3. A conservation law for an evolution equation is called trivial if Tis, itself, thexderivative of some expression. If equation (8.1) has an innite sequence of nontrivial conservation laws, then the equation isformally integrable . Innite sequences of nontrivial conservation laws are given by Cavalcante and Tenenblat [2] for the following equations: Burgers, KdV, mKdV, sine{Gordon, sinh{Gordon. 4. If a given partial dierential equation is not written in conservation form, there are a number of ways of attempting to put it in a con-served form. Bluman er al. [1] have a short list of techniques. 5. If equation (8.4) is satised, then there exists an ( n−1)-exterior dierential form Fsuch that equation (8.4) can be written dF=0 . This implies that there is an ( n−2)-formsuch that F=d. This, in turn, means that there exists an antisymmetric tensor of rank n, , such that F i(x;u;@u;@2u;:::;@m−1u)=X i<jn(−1)j@ ij @xj+X 1j<i(−1)i−1@ ji @xj; fori=1;2;:::;n . CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 8. Conservation Laws 49 6. A computer program in REDUCE for determining conservation laws i sg i v e ni nI t oa n dK a k o[ 6 ] . I nG e r d t et al. [4] is the description of a computer program in FORMAC that determines conservation laws, determines Lie{B¨ acklund symmetries, and also attempts to determine when an evolution equation is formally integrable. 7. Torriani [10] shows how the terms appearing in the expression of the densities and the fluxes for the Korteweg-de Vries equation may be found by combinatorial methods. 8. El-Sherbiny [3] proves that unless a1=a2is a multiple root of order three of the algebraic equation a63−a52+a4−a3=0 ,t h e n the class of nonlinear evolution equations ut+ux+a1uux+a2uut+ a3uxxx+a4uxxt+a5uxtt+a6uttt= 0 with thefaigreal numbers has a nite number of conservation laws; otherwise, the class has an innite number of conservation laws. 9. See also Olver [9, Chapter 4, pages 246{291]. References [1]Bluman, G. W., Reid, G. J., and Kumei, S. New classes of symmetries for partial dierential equations. J. Math. Physics 29 , 4 (April 1988), 806{811. [2]Cavalcante, J. A., and Tenenblat, K. Conservation laws for nonlinear evolution equations. J. Math. Physics 29 , 4 (April 1988), 1044{1049. [3]El-Sherbiny, H. M. Conservation laws of a class of dierential equations II.Appl. Math. Lett. 6 , 4 (1993), 99{103. [4]Gerdt, V. P., Shvachka, A. B., and Zharkov, A. Y. Computer algebra applications for classication of integrable non-linear evolution equations. J. Symbolic Comp. 1 (1985), 101{107. [5]Ibragimov, N. H. Group theoretical nature of conservation theorems. Letters Math. Physics 1 (1977), 423{428. [6]Ito, M., and Kako, F. A REDUCE program for nding conserved densities of partial dierential equations with uniform rank. Comput. Physics Comm. 38(1985), 415{419. [7]LeVeque, R. J. Numerical Methods for Conservation Laws . Birkhauser, Basel, Switzerland, 1992. [8]Littinsky, E. Polynomial integrals of evolution equations. Comm. Math. Physics 121 (1989), 669{682. [9]Olver, P. J. Applications of Lie Groups to Dierential Equations . No. 107 in Graduate Texts in Mathematics. Springer{Verlag, New York, 1986. [10]Torriani, H. H. Conservation laws for the Korteweg-de Vries equation and the theory of partitions. Physics Letters 113A , 7 (6 January 1986), 345{348. [11]Vinokur, M. An analysis of nite-dierence and nite-volume formulations of conservations laws. J. Comput. Physics 81 (1989), 1{52. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 50 I.A Denitions and Concepts 9. Dierential Resultants Applicable to Two polynomial ordinary dierential equations. Yields One ordinary dierential equation in one independent variable. Idea Given two polynomial equations (in, say, xandy), the classical method of resultants is as follows: The equations can always be written as the system of linear equations Aw=0,w h e r eA=A(y)a n d w=w(x)6=0 . Because this system must have det A= 0, a polynomial equation only in ymay be determined. The technique for polynomial dierential equations is very similar. Procedure Resultants have classically been used to eliminate one variable between two polynomial equations. For example, suppose we have the two equations x3−3y2x2+x+5y2=0; x3+5y2x2−x+3y2=0:(9.1) These equations may be multiplied by powers of xto obtain the system of equations: x5−3y2x4+x3+5y2x2=0; x4−3y2x3+x2+5y2x =0; x3−3y2x2+x+5y2=0; x3+5y2x2−x+3y2=0; x4+5y2x3−x2+3y2x =0; x5+5y2x4−x3+3y2x2=0: This system can be written in matrix form as 2 66666641−3y 215y200 01−3y215y20 00 1 −3y215y2 00 1 5 y2−13y2 01 5 y2−13y20 15y2−13y2003 77777752 6666664x 5 x4 x3 x2 x 13 7777775=2 66666640 00 0 0 03 7777775: (9.2) This last equation is a 6 6 system of the form Aw=0. Because w6=0 (because, at least, the last component of wis non-zero), the determinant ofAmust vanish. Taking the determinant of the matrix in equation (9.2), we nd that ymust satisfy the equation 32y 2(289y8+1 6y4+1 )=0: (9.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 9. Dierential Resultants 51 All the dierent values of y, from the solutions of (9.1), must satisfy (9.3). Dierential resultants are the analogue of resultants applied to dier- ential systems. There are two steps analogous to multiplying the originalequations by powers of x. They are dierentiating one of the equations, multiplying one of the equations by some term that may involve the independent and/or the dependent variables. Although there are algorithms published on how to proceed in any given case, as in Mishina and Proskuryakov [3], they are generally written in the language of abstract algebra. Example Suppose we have the following two coupled dierential equations for fy(x);z(x)g A: 3yz+z−yx=0; B:−zx+z2+y2+y=0: We seek a single dierential equation involving only z(x). Note that we could solve equation (A) for y(x) (by integrating factors) and then substi- tute this result in equation (B), but this creates an algebraic mess. This,in turn, makes it dicult to obtain a single simple equation for z(x). If we form the equations fA;B;yA;yB;y xB;@xB;y@xAg, then we ob- tain the system 2 66666666400−10 0 3 zz 10 0 00 1 z 2−zx 3z00−10 z 0 11 0 00 z2−zx 0 00z2−zx11 0 0 0 0 1 2 002 zzx−zxx 00 0 12 2 zzx−zxx 03 7777777752 666666664y 2 y3 yx yyx y2yx y 13 777777775=2 6666666640 0 0 0 00 03 777777775: Taking the determinant of the matrix above, we conclude that z(x)i sa solution of the single ordinary dierential equation z 2 xx+(−16zx+1 2z2−3)zzxx+6 4z2z2 x+( 2 3−96z2)z2zx +( 3 6z4−17z2+2 )z2=0: Notes 1. This technique applies directly to systems of partial dierential equa- tions and to higher order equations. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 52 I.A Denitions and Concepts 2. There are specic technical requirements for when the classical method of resultants (when applied to polynomials) will work. There are similar requirements for when dierential resultants will work. See Mishina and Proskuryakov [3] for details. 3. Rubel [5] proves the following theorem, which indicates that elimina- tion is not always possible, at least for algebraic dierential equations (ADEs, see page 720): There exists a system of two ADEs, in the two dependent variablesuandvwhich possesses a real-valued Cn;mso- lutionu,von a certain open interval I, but which has no solutionu,vonIfor whichvsatises an ADE that does not involve uor any derivative of u. 4. By taking equations pairwise a system of, say, 10 equations in 10 dierent independent variables could, if fortunate, be reduced to a single equation in a single independent variable. 5. The two dierential equations considered do not both have to be polynomial for this reduction scheme to work. The two equations have only to be polynomials in one of the dependent variables (theone that will be removed). 6. Any linear second order ordinary dierential equation system can be interpreted as the resultant of the elimination of a dependent variablefrom a pair of conjugate rst order Hamilton’s equations. See Tolstoy [7] for details. References [1]Berkovich, L. M., and Tsirulik, V. G. Dierential resultants and some of their applications. Dierentsial’nye Uravneniya 22 , 5 (May 1986), 750{757. [2]Bocher, M. Introduction to Higher Algebra . The MacMillan Company, New York, 1907. [3]Mishina, A. P., and Proskuryakov, I. V. Higher Algebra . Pergamon Press, New York, 1965. [4]Ritt, J. F. Dierential Algebra . Dover Publications, Inc., New York, 1966. [5]Rubel, L. A. An elimination theory for systems of algebraic dierential equations. Houston J. Math. 8 , 2 (1982), 289{295. [6]Seidenberg, A. An elimination theory for dierential algebra. University of California Press 3 , 2 (1956), 31{66. [7]Tolstoy, I. Remarks on the linearization of dierential equations. J. Inst. Maths. Applics 20 (1977), 53{60. [8]Van Der Waerden, B. L. Modern Algebra . Frederick Ungar Publishing, New York, 1940. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 10. Existence and Uniqueness Theorems 53 10. Existence and Uniqueness Theorems Applicable to Dierential equations of all types. Yields Knowledge of whether a solution exists and, if so, if the solution is unique. Idea There are theorems available for many cases of interest. Procedure Corresponding to the diculty of the subjects involved, there are more theorems applicable to: ordinary dierential equations than partial dif- ferential equations, linear equations than nonlinear equations, and initial value problems than boundary value problems. In the following we indicate some of the simple theorems that are frequently useful. The last theorem is applicable to partial dierential equations; the rest are applicable to ordinary dierential equations. The rst and last twotheorems are for vector systems; the other theorems are for scalar equations. Theorem Consider the initial value problem: dx=dt=F(t;x)w i t h x(t 0)= x0,w h e r e x=x(t)=[x1(t)x2(t):::xn(t)]T. If each of the functions fFigandn @Fi @xjo are continuous in a region Rof (t;x) space containing the point x0, then there is an interval jt−t0j<hin which there exists a unique solution to the problem. Theorem Consider the initial value problem: y0=f(x;y)w i t hy(x0)= y0. Let the functions fbe continuous in some rectangle a<x<b , c<y<d containing the point ( x0;y0). Assume that f(x;y) satises a Lipschitz condition in y. Then, in some interval x0−h<x<x 0+h contained in a<x<b , there is a unique solution to the given problem. Theorem Consider the initial value problem: y0=f(x;y)w i t hy(x0)= y0. Let the functions fand@f=@y be continuous in some rectangle a<x<b ,c<y<d containing the point ( x0;y0). Then, in some intervalx0−h<x<x 0+hcontained in a<x<b , there is a unique solution to the given problem. Theorem Consider the initial value problem: y00=f(x;y;y0)w i t hy(x0)= y0,y0(x0)=y0 0. Let the functions f,fy,a n dfy0be continuous in an open region Rof three-dimensional ( x;y;y0) space. If the point CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 54 I.A Denitions and Concepts (x0;y0;y0 0)i si nR, then there exists some interval about x0for which there is a unique solution to the given problem. Theorem Consider the initial value problem: y(n)+p1(x)y(n−1)+


+pn−1(x)y0+p1(x)y=q(x); with y(x0)=y0;y0(x0)=y0 0; ::: y(n−1)(x0)=y(n−1) 0: If the functionsfpi(x)gandq(x) are continuous on the open interval a<x<b , then there exists a unique solution to the problem. Theorem Consider the initial value problem: x0=f(x;y;t );y0=g(x;y;t ) withx(t0)=x0,y(t0)=y0.I ffandgsatisfy a Lipschitz condition (with respect to xandy) in the regionfjt−t0jA,jx−x0jB, jy−y0jCg, then the problem has a unique solution in some interval a<t<b about the point t0. Theorem Consider the boundary value problem: x00=f(t;x;x0); 0<t< 1; x(0) =A; x (1) =B: Iffandfxare continuous and fx0, then there exists a unique solution. Theorem Consider the initial value problem y00+f(x;y;y0)=0; B1[y]=y0(a)+Ay(a)−C1=0; B2[y]=y0(b)+By(b)−C2=0;(10.1) wherefsatises a Lipschitz condition, and fyandfy0are bounded forxin the interval [ a;b] and for values of ( y;y0) of interest. Consider the two comparison equations u00 1+h1(x;u1;u0 1)=0;B 1[u1]=0;B 2[u1]=0; u00 2+h2(x;u2;u0 2)=0;B 1[u2]=0;B 2[u2]=0; withh1(x;y;y0)f(x;y;y0)h2(x;y;y0). We assume that the u1 andu2problems have unique solutions. Then there exists at least one solution to (10.1) in the given region, and every solution has thepropertyu 1(x)y(x)u2(x). (This theorem is one of the major results of the theory of dierential inequalities.) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 10. Existence and Uniqueness Theorems 55 Cauchy{Kowalewski Theorem If the vector u=u1u2::: unT satises ut=A(u)ux; u(0;x)=h(x); whereuk=uk(x;t),A(u) is an analytic matrix, and h(x)i sa n analytic function, then a neighborhood of t= 0 can be found in which there is a unique solution u,w i t he a c h ukbeing analytic. Example 1 The rst order initial value problem y0=jyj1=3;y (x0) = 0 (10.2) has a right-hand side that is not Lipschitz continuous at y= 0. This equation, in fact, has an innite number of solutions. Let x1andx2be any two numbers such that x1<x0<x2. Then the following function f(x)=8 >< >:−/parenleftbig2 3
3=2(x1−x)3=2;ifx<x 1; 0; ifx1<x<x 2;/parenleftbig2 3
3=2(x−x2)3=2; ifx2<x; is a solution to equation (10.2). Example 2 The nonlinear second order equation  u030 + 24(1−u)=0;u (0) = 1;u0(0) = 0; has at least three solutions: u(t)=1 ,u(t)=1−t2,a n du(t)=1+t2. Notes 1. Dierential equations with discontinuities (see page 264) and delay equations (see page 253) do not meet the requirements of the abovetheorems. They must be investigated separately. 2. It is often possible to determine when a linear ordinary dierential equation has a unique solution. When the solution is not unique, itis sometimes possible to describe the degrees of freedom that make it non-unique using alternative theorems (see page 15). 3. Fixed point theorems are a specic method that can be used to prove the existence of a solution (see page 58). The section on well posed dierential equations contains some results on existence anduniqueness (see page 115). 4. Bobisud and O’Regan [2] consider existence questions for some second order initial value problems of the form y 00+F(t;y;y0) = 0, where Fis allowed to be suitably singular. For example, F(t;y;y0)=t−1=2y−1=2 is allowed. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 56 I.A Denitions and Concepts 5. The existence of solutions to a dierential equation can be critically dependent on the size of the coecients in the equation. For example, Coddington and Levinson [3] show that the problem y00=−y0−(y0)3; y(0) =A; y (1) =B (A6=B) does not have a solution for small enough >0. 6. The classical problem −r2u=upin Ω;u =0 o n@Ω; where Ω is a bounded domain in RN, with smooth boundary @Ω, has the interesting existence property (see Peletier [8]): Ifp<N+2 N−2, then existence of a solution is assured for any domain Ω; IfpN+2 N−2, then there exists no solution in any star-shaped domain. Similar results are available for the equation ut=r2u+up; existence of a global positive solution depends on whether pis greater than 1+2=N(see Fujita [5]). 7. A classic result of Lewy [7] is that the equation −ux−iuy+2 (ix−y)uz=F(x;y;z ); whereF(x;y;z )i so fc l a s s C1,h a sn oH1-solution, no matter what open (x;y;z ) set is taken as the domain of existence. 8. Waterhouse [11] has the theorem: Theorem : Consider the homogeneous linear dierential equa- tion involving only derivatives of even order and even functions as coecients,/parenleftbig D2n+a1D2n−2+


+an
f=0w i t hai(x)= ai(−x) and having the symmetric homogeneous boundary con- ditionsB1(D)f(s)=


=Bn(D)f(s)=0=B1(D)f(−s)=


=Bn(D)f(−s)w i t hBi(D)=P jbijDj. If this boundary value problem has a non-trivial solution, and if each of the vectors ( bi0−bi1;bi2−bi3;:::)i si nt h es p a n of the vectors ( b10;b11;b12;:::)a n d(b20;b21;b22;:::), then this problem has a nontrivial solution that is either even or odd. 9. Agarwal and Sheng [1] provide necessary and sucient conditions for the existence and uniqueness of solutions of general nth order non- linear dierential equations satisfying Abel{Gontscharo boundary conditions. These are boundary conditions of the form y(i)(ai+1)= Ai+1for 0in−1w h e r e−1<aa1a2


anb<1. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 10. Existence and Uniqueness Theorems 57 References [1]Agarwal, R. P., and Sheng, Q. Abel{Gontscharo boundary value problems. Math. Comput. Modelling 17 , 7 (1993), 37{55. [2]Bobisud, L. E., and O’Regan, D. Existence of solutions to some singular initial value problems. J. Math. Anal. Appl. 133 (1988), 214{230. [3]Coddington, E. A., and Levinson, N. A boundary value problem for a nonlinear dierential equation with a small parameter. Proc. Amer. Math. Soc. 3 (1952), 73{81. [4]Feckan, M. A new method for the existence of solution of nonlinear dierential equations. J. Dierential Equations 89 (1991), 203{223. [5]Fujita, H. On the blowing up of solutions of the Cauchy problem for zzzref18refzzz. J. Fac. Sci. Univ. Tokyo Sect. A. Math 16 (1966), 105{113. [6]Levine, H. A. The role of critical exponents in blowup theorems. SIAM Review 32 , 2 (1990), 262{288. [7]Lewy, H. An example of a smooth linear partial dierential equation without solution. Annals of Math. 66 , 1 (July 1957), 155{158. [8]Peletier, L. A. Elliptic equations with nearly critical growth. In Equadi 1987, C. M. Dafermos, G. Ladas, and G. Papanicolaou, Eds., no. 118 in Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York, 1987, pp. 561{574. [9]Plum, M. Computer-assisted existence proofs for two-point boundary value problems. Computing 46 (1991), 19{34. [10]Redheffer, R. Dierential Equations . Jones and Bartlett Publishers, Boston, 1991. [11]Waterhouse, W. C. Some boundary value problems with even or odd solutions. SIAM Review 38 , 4 (December 1996), 645{646. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 58 I.A Denitions and Concepts 11. Fixed Point Existence Theorems Applicable to Dierential equations of all types. Yields A statement about the existence of the solution. Idea If the statement concerning the existence of a solution to a dierential equation can be interpreted as a statement concerning xed points in a Banach space, then a xed point theorem might be useful. Procedure The Schrauder xed point theorem states: LetXbe a non-empty convex set in a Banach space and let Ybe a compact subset of X. Suppose Y=f(X)m a p sX continuously into Y. Then there is a xed point x=f(x). By interpreting a given dierential equation as a continuous function in a Banach space, the above theorem indicates the existence of a solution. Example Suppose we wish to determine whether a solution exists to the nonlinear boundary value problem u00=−e−u(x); u(0) =u(1) = 0;(11.1) on the interval x2[0;1]. We rst note that the problem v00=−(x); v(0) =v(1) = 0; has the solution v(x)=Z1 0G(x;z)(z)dz; whereG(x;z) is the Green’s function (see page 321) G(x;z)=( (1−x)z;for 0zx; (1−z)x;forxz1: Hence, we can write equation (11.1) in the form of an equivalent integral equation u(x)=f(u(x))Z1 0G(x;z)e−u(z)dz: (11.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 11. Fixed Point Existence Theorems 59 To apply Schrauder’s xed point theorem to equation (11.2), we need to carefully dene the Banach space Band the sets XandY. If we dene B= space of continuous functions on (0 ;1); X=fu(x)j0u(x)1;u(x) is continuousg; Y=f(X); then we can apply the theorem. Note that in this example, Xis not compact but Yis. Note also that the bounds in Xwere derived after some analysis of equation (11.1). Finally, then, we conclude that equation (11.1) has a solution. Notes 1. In the example above we used a fairly standard linearization trick that can be described in more generality. Suppose that an expression D(f;g) (which could involve derivatives of fand/org) is linear in f. Suppose also that the linear dierential equation D(f;g)=0h a s a unique solution f=T[g]f o re a c hgin some function space. Then to nd a solution, in that function space, of the (possibly nonlinear) equationD(f;f) = 0 is equivalent to nding a xed point of the mappingT. Thus a particular nonlinear dierential equation can be studied by means of a more general linear dierential equation, together with a xed point problem. 2. Once a dierential equation has been formulated as a xed point statement, numerical methods that search for xed points in a func-tion space can be used. See, for example, Allgower [1]. 3. Interval techniques (see page 545) may also be used to bound the solution of a xed point statement. See Moore [7, Chapter 15, pages97{102] for details. 4. A contraction mapping is a functional iteration, say y n+1=N[yn], that converges to the solution of the xed point equation y=F[y]. The Picard iteration (see page 618) is such a mapping. 5. Another xed point theorem that is of use in dierential equations is Krasnoselskii’s theorem (see Franklin [3] for details): Consider the xed point equation x=f(x)+g(x)f o r xin a Banach spaceB. Let X be a non-empty closed convex set in B. Letf(x) map X continuously into a compact subset YX. Let g(x) be a contraction mapping on X (note that the range of g need not be compact). If it is assumed that y+g(x)2Xfor y2Yandx2X, then there is a xed point of x=f(x)+g(x). 6. Another xed point theorem that is of use in dierential equations is the Tihonov xed point theorem (see Iyanaga and Kawada [6, pages 542{543] for details): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 60 I.A Denitions and Concepts LetRbe a locally compact topological linear space, Aac o m p a c t convex subset of R,a n dTa continuous mapping sending Ainto itself. Then Thas xed points. 7. Existence theorems for solutions for dierential equations may be found on page 53. 8. See also Burton [2, Chapter 3, pages 164{196], Hale [4, Appendix, pages 171{172], Hartman [5, Chapter 12, pages 404{449], Smart [8,Chapter 6, pages 41{52], and Stakgold [9, pages 243{259]. References [1]Allgower, E. L. Application of a xed point search algorithm to nonlinear boundary value problems having several solutions. In Fixed Points: Algorithms and Applications , S. Karamardian, Ed. Academic Press, New York, 1977. [2]Burton, T. A. Perturbation and delays in dierential equations. SIAM J. Appl. Math. 29 , 3 (November 1975), 422{438. [3]Franklin, J. Methods of Mathematical Economics . Springer{Verlag, New York, 1980. [4]Hale, J. K. Oscillations in Nonlinear Systems . McGraw{Hill Book Company, New York, 1963. [5]Hartman, P. Ordinary Dierential Equations . John Wiley & Sons, New York, 1964. [6]Iyanaga, S., and Kawada, Y. Encyclopedic Dictionary of Mathematics . MIT Press, Cambridge, MA, 1980. [7]Moore, R. E. Interval Analysis . Prentice{Hall, Inc., Englewood Clis, NJ, 1966. [8]Smart, D. R. Fixed Point Theorems . Cambridge University Press, New York, 1974. [9]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 12. Hamilton{Jacobi Theory 61 12. Hamilton{Jacobi Theory Applicable to Conservative dynamical systems. Yields A reformulation of a system of ordinary dierential equations. Idea A change of variables may lead to more tractable equations. Procedure A conservative dynamical system has a Lagrangian Ldened by L= T−V,w h e r eT(V) is the kinetic (potential) energy. If the generalized coordinates in this system are q=(q1;q2;:::;qn), then the equations of motion are given by d dt@L @_qi −@L @qi=0;fori=1;2;:::;n; (12.1) where a dot denotes dierentiation with respect to t. The equations in (12.1) are called Lagrange’s equations. If we dene the generalized mo- menta bypi=@L @qiand the Hamiltonian by H=pT_q−L, then Lagrange’s equations become _qi=@H @pi; _pi=−@H @qi; @L @t=−@H @t:(12.2) These equations are called Hamilton’s equations. If we change from the (H;p;q) variables to the ( J;P;Q) variables via the canonical transforma- tion dened by the generating function S(P;q;t) (see page 132), then pi=@S @qi; Qi=@S @Pi; J(P;Q;t)=H p(P;Q;t);q(P;Q;t);t +@S @t:(12.3) In these new variables, Hamilton’s equations may be written _Qi=@J @Pi; _Pi=−@J @Qi:(12.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 62 I.A Denitions and Concepts If the canonical transformation is chosen so that J= 0, then (12.4) says thatPandQare constant. To have Jvanish identically, we require (from (12.3)) H@S @q1;@S @q2;:::;@S @qn;q1;q2;:::;qn;t +@S @t=0: This last equation is known as the Hamilton{Jacobi equation. The proce- dure is to solve the Hamilton{Jacobi equation for the generating function S, make a canonical change of variables using this generating function, and then solve Hamilton’s equation in these new coordinates. This will yield asolution to Lagrange’s equations. Example Suppose we want to solve the linear constant coecient ordinary dier- ential equation ¨q+!2q=0: (12.5) This dierential equation comes from the Hamiltonian H=1 2/parenleftbig p2+!2q2
, which, in turn, corresponds to the following Hamilton{Jacobi equation: 1 2"@S @q2 +!2q2# +@S @t=0: (12.6) To solve for S(q;t), we use separation of variables (see page 487), and look for a solution in the form S(q;t)=a(q)+b(t), for some unknown functions a(q)a n db(t). Using this form for Sin equation (12.6) and making the usual argument about which terms must depend upon which variables, we determine that a(q)a n db(t)m u s ts a t i s f y _b=−;da dq2 +!2q2=2; whereis a separation constant. Hence, S=−t+Rp 2−!2q2dq.I f we call=P, then we can compute from equation (12.3) Q=@S @P=−t+Z (2P−!2q2)−1=2dq=−t+1 !sin−1!qp 2P ; which may be inverted to yield q=p 2P !sinh !(t+Q)i , which is the solution to equation (12.5). Notes 1. Lagrange’s equations can be interpreted as the variational or Euler{ Lagrange equations for the functional J=R Ldt(see page 418). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 12. Hamilton{Jacobi Theory 63 2. The functions fandgare said to be in involution or to Poisson commute if the Poisson bracket [ f;g] is identically equal to zero. Liouville’s theorem states that a function Fis a rst integral of a system with Hamiltonian function Hif and only if HandFare in involution. See Abraham et al. [1, page 471] for details. 3. Poisson’s theorem states that the Poisson bracket of two rst integrals of a Hamiltonian system is again a rst integral. See Goldstein [2, Chapter 9, pages 273{317] for details. 4. Any function A(p;q) dened along the trajectories of equation (12.2) satises dA dt=[A;H ]=X j@A @qj@H @pj−@A @pj@H @qj where the square brackets denote the Poisson bracket. 5. A general form for a non-conservative system is often taken to be _qi=@C @pi+@D @qi _pi=−@C @qi+@D @pi(12.7) WhereC(p;q)a n dD(p;q) are called the conservative and dissipation functions. For D= 0, this reduces to equation (12.2). For C=0 , this becomes a gradient system. Any function A(p;q) dened along the trajectories of equation (12.7) satises dA dt=rA
rD+[A;C]: ChoosingA=CandA=D, we obtain the evolution equations for the conservative and dissipative functions dC dt=rC
rD; dD dt=rD
rD+[D;C]: Note thatr2Dequals the divergence of the vector eld of equation (12.7) and that the system is dissipative when r2D< 0. 6. Given the equations of motion: ¨ qi=fi(q;_q;t), the inverse problem of classical mechanics is to determine whether these equations are equivalent to the Euler{Lagrange equations based on a Lagrangian L. That is, a matrix w=w(q;_q;t) is desired so that wij(¨qj−fj)=d dt@L @_qi −@L @qi: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 64 I.A Denitions and Concepts The necessary and sucient conditions for the existence of wandL are called the Helmholtz conditions , they are @wij @_xk=@wik @_xj;wij=wji; Dwij=−1 2wik@fk @_xj−1 2wjk@fk @_xi 1 2D wik@fk @_xj−wjk@fk @_xi =wik@fk @xj−wjk@fk @xi withD=@ @t+P m _xm@ @xm+fm@ @_xm . See Hojman and Shepley [4]. 7. The KdV equation, ut=−uxxx+6uux, can be treated as a Hamilton- ian system, ut=fu;Hg, with the Hamiltonian and Poisson brackets dened by H=1 2Z u2(x)dxfu(x);u(y)g= −@3+4u@+2ux (x−y) 8. See also Haar [3, Chapter 6, pages 121{145] and Nayfeh [5, pages 179{189]. References [1]Abraham, R., Marsden, J. E., and Ratiu, T. Manifolds, Tensor Analysis, and Applications . Addison{Wesley Publishing Co., Reading, MA, 1983. [2]Goldstein, H. Classical Mechanics . Addison{Wesley Publishing Co., Reading, MA, 1950. [3]Haar, D. Elements of Hamiltonian Mechanics . Pergamon Press, New York, 1971. [4]Hojman, S. A., and Shepley, L. C. No Lagrangian? No quantization! J. Math. Physics 32 , 1 (Jan 1991), 142{146. [5]Nayfeh, A. H. Perturbation Methods . John Wiley & Sons, New York, 1973. [6]Sanz-Serna, J. M. Runge{Kutta schemes for Hamiltonian systems. BIT 28 (1988), 877{883. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 13. Integrability of Systems 65 13. Integrability of Systems Applicable to Systems of dierential equations. Yields Information about whether a Hamiltonian system is completely inte- grable. Idea The Painlev e test performs a singular point analysis, which gives infor- mation about integrability. Procedure An autonomous Hamiltonian system is called (Liouville) integrable if there exists another function Isuch that [H;I] = 0. This function must be functionally independent of H, it must exist globally and be single valued, and it must be a complex analytic function of its variables. If the Hamiltonian system has Ndegrees of freedom it is called com- pletely integrable if it possesses Nindependent single valued analytic rst integralsfIkgthat commute with respect to the Poisson bracket [In;Im]=NX i=1@In @qi@Im @pi−@In @pi@Im @qi =0: One of these rst integrals will be the Hamiltonian itself. Given a Hamiltonian system, there is no known systematic method for determining whether or not that system is integrable. Much recent work has focused on the Painlev et e s t . The test asserts that an equation is integrable if every ordinary dierential equation that arises as a similarityreduction of an integrable partial dierential equation has the Painlev e property; that is, it has no movable singularities except poles, perhaps after a transformation of variables. For the Painlev e test to be eective, it is necessary to determine the complete symmetry group of the dierential equation under consideration. If it passes the test, then it is believedthat the original partial dierential equation will be solvable by inverse scattering methods (see page 460). The Painlev e test also has applications in determining the stability of systems of ordinary dierential equations. Roughly speaking, a partial dierential equation is said to possess the Painlev e property if the only singularities of the general solution on arbi- trary non-characteristic surfaces are poles. Singular point analysis is used to determine if dierential equations have the Painlev e property. The test consists of substituting u(x)= 1X n=0un(x−x0)+p; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 66 I.A Denitions and Concepts for< 0, into the tested equation in the vicinity of a singular point x0 and investigating whether this expansion is compatible with the equation and contains a sucient number of undetermined coecients for the ap- proximation of a general solution. Example The motion of the Nparticle lattice is described by the Hamiltonian H(p;q)=1 2NX j=1p2 j+NX j=1eqj−qj+1(13.1) whereqN+1=q1(which corresponds to cyclic boundary conditions). If faj;bjgare dened by aj:=1 2e(qj−qj+1)=2;bj:=1 2pj; then the equations of motion are a0 j=aj(bj−bj+1);b0 j=2 (a2 j−1−a2 j): (13.2) If the following NNmatrices are dened: L=2 66666664b 1a10::: 0aN a1b2a2 00 0a2b3 00 ...... 00 0 bN−1aN−1 aN00 aN−1bN3 77777775 A=2 666666640−a 10::: 0aN a1 0−a2 00 0a2 00 0 ......... 00 0 ::: 0−aN−1 −aN 00 aN−1 03 77777775; then equation (13.2) may be written in the form dL dt=[A;L]=AL−LA: Note we also haved(Lk) dt=[A;Lk] for any positive integer k. From this it follows that the trace of the matrix Lkis constant. Hence, the tracesftr (L), tr (L2),:::,t r (Lk);:::gare rst integrals for (13.2). They turn out to be independent and in involution of each other. Hence, the Hamiltonian in equation (13.1) is completely integrable. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 13. Integrability of Systems 67 Notes 1. Several denitions of \integrability" are in use in the literature. For example, the PDE N(x;t;u )=0w i t h u(x;0) =f(x) is called com- pletely integrable if there is an integral equation for Kof the form K(x;y;t)+F(x;y;;t)+Z1 xK(x;z;t)H(z;y;t)dz=0 called the Gelfand{Levitan equation, such that FandHare uniquely determined from f(x) the solution of the PDE is given by u(x;t)=K(x;x;t). 2. Completely integrable PDEs are known to possess several remarkable properties including: the existence of soliton solutions (see page 626) the existence of an innite number of independent conservation laws (see page 47) a Lax representation (see page 460) B¨acklund transformations (see page 428) 3. In general, linear equations only have xed singularities while non- linear equations can have both xed and movable singularities. Consider the linear equation y00+p(x)y0+q(x)y= 0 which has the general solution y(x)=Ay1(x)+By2(x)w h e r eAandB are arbitrary constants. The location of the singularities of y(x) depend only on p(x)a n dq(x), not onAorB. The singularities of this equation are xed, since they do not depend upon the constants of integration. Consider the nonlinear equation y0+y2= 0 which has the general solutiony(x)=(x−x0)−1wherex0is an arbitrary constant. In this casey(x) has a singularity, a pole, which is movable since it depends on the constant of integration x0. 4. For rst order equations of the form y0=F(y;x), whereFis rational inyand analytic in x, the only equation which has no movable singularities other than poles is the Riccati equation y0=p0(x)+ p1(x)y+p2(x)y2. For second order equations of the form y00=F(y;y0;x), whereF is rational in yandy0and analytic in x, Painlev eet al. (see Ince [8]) showed that there are only 50 canonical equations which have no movable singularities except poles. Of these, 44 are integrable in terms of known functions (such as elliptic functions) and the re-maining 6 dened new transcendental functions, called the Painlev e transcendents (see page 128). 5. The three-particle Toda lattice has the Hamiltonian H= p2 1+p2 2+p2 3 2+ Vwith the potential energy V=ep1−p2+ep2−p3+ep3−p1. The sys- CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 68 I.A Denitions and Concepts Parameters Invariant b=2 (x2−2z)e2t b=0;=1 3 −rx2+1 3y2+2 3xy+x2z−3 4x4 e4t=3 b=1;r=0/parenleftbig y2+z2
e2t b=4;=1 4(1−r)z+rx2+y2−2xy+x2z−1 4x4 e4t b=1;=1/parenleftbig −rx2+y2+z2
e2t b=6−2; r=2−1(2−1)2 x2+y2−(4−2)xy+x2z−1 4x4 e4t Table 13.1: First integrals for the Lorenz equations. tems admits three independent integrals, for instance, the functions I1=p1+p2+p3 I2=p1p2+p2p3+p3p1−V I3=p1p2p3−p1ep2−p3−p2ep3−p1−p3ep1−p2 These integrals are in involution and they are independent. 6. Consider the Hamiltonian H=(p2 x+p2 y)=2+ V : For the H enon{Heiles potential V= 3y3+x2y, the system is integrable for = 1, 6, and 16. For the Holt potential V= 3y4=3+x2y−2=3, the system is integrable for = 1, 6, and 16. For the quartic potential V=ax4+bx2y2+cy4, the system is integrable if a:b:chave the ratios a:0:c, 1:2:1, 1:6:1, 1:12:16, 16:12:1, 1:6:8, or 8:6:1. 7. The Lorenz equations (see page 199) x0=(y−x) y0=−y−xz−rx z0=xz−bz have known rst integrals for several possible values of the parameters f;r;bg. For example, the rst integrals in table 13.1 are known. 8. Clarkson et al. [4] state that the only third-order semilinear par- tial dierential equations that are linearizable are equivalent to the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 13. Integrability of Systems 69 following six equations: ut=uxxx+γux; ut=uxxx+uux+γux; ut=uxxx+u2ux+γux; ut=uxxx−1 8u3 x+/parenleftbig eu+e−u
ux+γux; ut=uxxx−3 2uxu2 xx/parenleftbig 1+u2 x
−1−3 2P(u)(u2 x+1 )ux+γux; ut=uxxx−3 2u−1 xu2xx+u−1 x−3 2P(u)u2 x+γux; whereP(u) is the Weierstrass elliptic function and satises dP du2 =4P3−P−: 9. Clarkson et al. [4, page 1205] show that the PDE ut=uxx+h(u)ux,w h e r eh(u) is a rational function of u,c a n pass the Painlev e test only if h(u) is a linear function of u. ut=uxxx+(uuxx+u2 x)+3 2(−1)u2ux,w h e r eis a constant, can pass the Painlev e test only if =0 ,3/2,o r3 . 10. Hereman and Angenent [7] and Rand and Winternitz [13] describe Macsyma programs for determining whether a nonlinear ordinary dierential equation has the Painlev e property. (The dierential equation must be a polynomial in both the dependent and indepen- dent variables and in all derivatives.) 11. The only equations of the form uxt=f(u), wheref(u) is a linear combination of exponentials, which pass the Painlev e test are: the sine{Gordan equation uxt=s i nu, the Liouville equation uxt=eu, and the Bullough{Dodd equation uxt=eu−e−2u. 12. Polynomial potentials arise in many problems, particularly when truncated Taylor series are used to facilitate analytical study. It isuseful to examine the integrability of such potentials. In two dimen- sions there are only three independent integrable cubic potentials; they arex 3+3xy2+y3,2x3+xy2,a n d1 6x3+3xy2; see Cleary [5]. 13. The Mathematica package DSolveIntegrals can compute complete integrals of non-linear PDEs. For example, given yuy=u+x2u2 xthe integral is determined to be u=(−a2+4by−2alogx−log2x)=4. 14. The following equations are known to be completely integrable: sine{ Gordon equation, Do{Bullough, Ernst equation, axisymmetric sta-tionary Einstein{Maxwell equation. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 70 I.A Denitions and Concepts References [1]Albrecht, D. W., Mansfield, E. L., and Milne, A. E. Algorithms for special integrals of ordinary dierential equations. J. Phys. A: Math. Gen. 29(1996), 973{991. [2]B u c h l e r ,J .R . ,I p s e r ,J .R . ,a n dW i l l i a m s ,C .A . ,E d s . Integrability in Dynamical Systems . New York Academy of Sciences, New York, 1988. [3]Chang, Y. F., Tabor, M., and Weiss, J. Analytic structure of the Henon{Heiles Hamiltonian in integrable and nonintegrable regimes. J. Math. Physics 23 , 4 (April 1982), 531{538. [4]Clarkson, P. A., Fokas, A. S., and Ablowitz, M. J. Hodograph transformations of linearizable partial dierential equations. SIAM J. Appl. Math. 49 , 4 (August 1989), 1188{1209. [5]C l e a r y ,P .W . Nonexistence and existence of various order integrals for two- and three-dimensional polynomial potentials. J. Math. Physics 31 ,6 (June 1990), 1351{1355. [6]Gerdt, V. P., Zharkov, A. Y., Svinolupov, S. I., and Shabat, A. B. The use of computer algebra to investigate the integrability of non-linear evolution systems. U.S.S.R. Comput. Maths. Math. Phys. 28 , 6 (1988), 50{ 57. [7]Hereman, W., and Angenent, S. The Painleve test for nonlinear ordinary and partial dierential equations. Macsyma Newsletter (January 1989), 11{ 18. [8]Ince, E. L. Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [9]Mel’nikov, V. K. New method for deriving nonlinear integrable systems. J. Math. Physics 31 , 5 (May 1990), 1106{1113. [10]P r e l l e ,M .J . ,a n dS i n g e r ,M .F . Elementary rst integrals of dierential equations. Trans. Amer. Math. Soc. 279 , 1 (September 1983), 215{229. [11]Progrebkov, A. K. On the formulation of the Painleve test as a criterion of complete integrability of partial dierential equations. Inverse Prob. 5 (1989), L7{L10. [12]Ramani, A., Dorizzi, B., Grammaticos, B., and Bountis, T. Inte- grability and the Painleve property for low-dimensional systems. J. Math. Physics 25 , 4 (April 1984), 878{883. [13]Rand, D. W., and Winternitz, P. ODEPAINLEVE | a Macsyma package for Painleve analysis of ordinary dierential equations. Comput. Physics Comm. 42 (1986), 359{383. [14]Roekaerts, D., and Schwarz, F. Painleve analysis, Yoshida’s theorems and the direct methods in the search for integrable Hamiltonians. J. Phys. A: Math. Gen. 20 (1987), L127{L133. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 14. Internet Resources 71 14. Internet Resources Applicable to Many topics related to dierential equations. Procedure Much information about dierential equations is available through the internet. We list next some of these resources: Symbolic software packages 1. There are a multitude of commercial computer packages available for symbolically solving dierential equations (see page 240). Theseinclude AXIOM http://www.nag.co.uk:80 Derive http://www.derive.com Macsyma http://www.macsyma.com Maple http://www.maplesoft.com Mathematica http://www.wolfram.com REDUCE http://www.rrz.uni-koeln.de/REDUCE 2. The program CONVODE will symbolically solve ordinary and partial dierential equations across the internet. For example, sending depend y,x; CONVODE( {df(y,x,2)+4*y=0}, {y}, {x}, {}, {english}); [email protected] will have the solution ofy00+4y= 0 returned via email with comments in English (the de- fault is French). See http://www.physique.fundp.ac.be/physdpt/ administration/convode.html . 3. MathServ provides an interface between the user and Mathematica (a symbolic computational engine). Templates for twelve dierent types of ODEs are available; the user can speciy the functions appearingin them. The results are returned directly to your browser. See http://math.vanderbilt.edu/~pscrooke/detoolkit.html . Numerical software packages There are a multitude of commercial computer packages available for numerically solving dierential equations (see page 654). In particular, the Guide to Available Mathematical Software (GAMS) has a taxonomy ofsoftware classes, with many representatives of most classes. See http:// gams.nist.gov . This section lists a few packages that currently may be used freely for non-commercial purposes. Diffpack is a collection of C++ class libraries aimed at the numerical solution of partial dierential equations. The Dipack home page CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 72 I.A Denitions and Concepts int main() { real r0=.5, r1=1., x0=0., y0=0., theta0=0., theta1=1.; // parametersAnnulusMapping annulus(r0,r1,x0,y0,theta0,theta1); // annulus mappingMappedGrid mg(annulus); // MappedGrid for an annulusmg.update(); // create default variablesrealMappedGridFunction u(mg); // declare grid function on the gridu=1.; // initial condition u=1 MappedGridOperators op(mg); // difference operators and BCS u.setOperators(op); // associate with a grid functionreal t=0, dt=.005, a=1., b=1., nu=.1; // problem parametersfor( int step=0; step<100; step++ ) { // loop for number of time steps u.display("solution"); // print out the solutionu+=dt*((-a)*u.x()+(-b)*u.y()+nu*(u.xx()+u.yy())); // forward Euler stept+=dt;u.applyBoundaryCondition(0,BCTypes::dirichlet,BCTypes::allBoundaries,0.); // apply Boundary condition u=0 u.finishBoundaryConditions(); // fix up corners, periodic update }return 0; } Program 14.1: Overture program for a reaction diusion problem ishttp://www.oslo.sintef.no/avd/33/3340/diffpack . The code can be downloaded from http://www.oslo.sintef.no/diffpack/ pub1.4 or from Netlib at http://www.netlib.org . DsTool isA Dynamical System Toolkit with an Interactive Graphical Interface . It computes Poincar e sections and bifurcation diagrams and is easily extensible. It was created at Cornell University and runs under X windows. The program and documentation can be obtained via ftpfrommacomb.cam.cornell.edu in the/pub/dstool directory. KASKADE is a C++ package that solves elliptic partial dierential equations. It is an adaptive multilevel-code for linear scalar ellipticand parabolic problems in 1, 2, and 3 space dimensions. It includes examples for nonlinear methods used in obstacle, porous media, and Stefan problems. It can be obtained via ftpfromelib.zib-berlin.de in the directories /pub/kaskade/3.x and/pub/kaskade/Manuals/3.0 . Overture is a high level object oriented framework for solving PDEs on structured grids and overlapping grids using nite dierence and -nite volume methods. Overture is freely available and can be obtained fromhttp://www.c3.lanl.gov/~henshaw/Overture/Overture.html . For example, the entire program to solve the problem u t+aux+buy= (uxx+uyy) in an annulus A,w i t hu(t=0;A)=1a n du @A=0 , using forward Euler’s method, is in program 14.1. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 14. Internet Resources 73 Electronic journals The Electronic Journal of Dierential of Equations (EJDE) is dedi- cated to the rapid dissemination of high quality research in mathematics. Publications are available as PostScript, T EX, and DVI les. All topics related to dierential equations and their applications are considered forpublication. Research articles are refereed under the same standards as those used by the nest-quality printed journals. EJDE may be found at http://ejde.math.swt.edu . Other resources C*ODE*E is the acronym for the Consortium of ODE Experiments . Their goal is to share the rapidly growing wealth of computational in- struction techniques with teachers of dierential equations. The Con- sortium publishes a newsletter designed to provide a regular sourceof ideas, inspiration, and experiments for instructors of ODEs. The newsletter is available on-line and in print format. Their URL is http://www.math.hmc.edu/codee . IDEA is the acronym for Internet Dierential Equations Activities . This is an interdisciplinary eort to provide students and teachers with computer based activities for dierential equations in a wide variety of discplines. This is sponsored by the NSF. It includes aglossary of terms and many other features. Their URL is http:// www.sci.wsu.edu/idea . The American Mathematical Society maintains materials organized by mathematical subject classication at http://www.ams.org/mathweb/ mi-mathbyclass.html . In this classication, category 34 is \Or- dinary dierential equations" and category 35 is \Partial dieren- tial equations." The AMS Preprint Server for these categories maybe found at http://www.ams.org/preprints/34/msc34-page.html andhttp://www.ams.org/preprints/35/msc35-page.html . Los Alamos maintains a web site on \Exactly Solvable and Integrable Systems", see http://xxx.lanl.gov/archive/solv-int . The Norwegian University of Science and Technology maintains a \Conservation Laws Preprint Server" at http://www.math.ntnu.no/ conservation . The \Mathematics Archives," see http://archives.math.utk.edu , is supported by the NSF, the State of Tennessee, Calvin College,and the University of Tennessee, Knoxville. Their repository of links related to ordinary dierential equations and partial dierential equa- tions may be found at http://archives.math.utk.edu/topics/ ordinaryDiffEq.html andhttp://archives.math.utk.edu/topics/ partialDiffEq.html . CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 74 I.A Denitions and Concepts The Math/CS Department of Nebraska Wesleyan University has a dierential equations resource page documenting course materials (labs and projects) developed as part of an NSF/ILI grant. The URL ishttp://brillig.nebrwesleyan.edu/delabs . The Math Department at Oregon State University has developed a web-based study guide for several of its courses. The URL for the ODE home page is http://iq.orst.edu.mathsg/ode/ode.html . Note 1. The URLs in this section are subject to change. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 15. Inverse Problems 75 15. Inverse Problems Applicable to Inverse problems. Yields Information about parameters appearing in a dierential equation. Idea There are theorems that can be used to determine which inverse prob- lems may be solved. Procedure The eld of inverse problems is lled with specialized theorems that are useful for specic applications. Example 1 Consider the eigenvalue problem −u00+q(x)u=u; for 0x1; u(0) cos+u0(0) sin=0; u(1) cos+u0(1) sin=0;(15.1) whereis a complex parameter, q(x) is a real-valued function that is integrable on the interval [0 ;1], andandare values in the interval [0;). One common inverse problem consists of determining the function q(x) from the eigenvalues of equation (15.1). There are many dierent results in this area. For example Theorem Suppose that ( ;;q (x)) give rise to the eigenvalues fng and suppose that ( ;;q(x)) give rise to the eigenvalues fng.I f n=nforn=0;1;:::;q(x)=q(x)f o rx2(0;1 2); and=,t h e n q(x)=q(x) almost everywhere on the interval (0 ;1). Another typical theorem is the following: Theorem Let0<1<2<::: be the eigenvalues of the problem −y00+q(x)y=ywithy0(0) =y0() = 0, where q(x) is a real-valued continuous function. If n=n2forn=0;1;2;:::,t h e nq(x)=0 . Example 2 One common technique to show uniqueness for an inverse problem is to investigate a mapping between the solutions of two equations with dierentvalues for the parameter(s) of interest. We have, for example (see Rundell [11]): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 76 I.A Denitions and Concepts Theorem Letu(x)a n dv(x)s a t i s f y ut=uxx−a(x)u; ux(0;t)=0; vt=vxx−a(x)v; vx(0;t)=0; for 0x1a n d0t<T .I fu(0;t)=v(0;t), thenv(x;t)= u(x;t)+Rx 0K(x;s)u(s;t)ds,w h e r eK(x;s) satises the Goursat problem Kss−Ktt=(a(s)−a(x))K(x;s); for 0sx1; Ks(x;0) = 0 for 0x1; K(x;x)=1 2Zx 0(a(r)−a(r))dr for 0x1: In this case it is possible to show that ifRx 0K(x;s)f(s)ds=0f o rs o m e positive function f(x), thena=a. Notes 1. The numerical methods used to solve inverse problems tend to result in ill-conditioned systems. 2. If the spectrafigandfigare known for the following two problems (withH6=H): −y00+q(x)y=y; y0(0)−hy(0) = 0; y0(1)−Hy(1) = 0;−y00+q(x)y=y; y0(0)−hy(0) = 0; y0(1)−Hy(1) = 0; thenfq(x);h;H; Hgare all uniquely determined. See Rundell and Sacks [12]. References [1]Anger, G. Inverse Problems in Dierential Equations . Plenum Publishing Corp., New York, 1990. [2]Barnes, D. C. The inverse eigenvalue problem with nite data. SIAM J. Math. Anal. 22 , 3 (May 1991), 732{753. [3]Cannon, J. R., and Lin, Y. An inverse problem of nding a parameter in a semi-linear heat equation. J. Math. Anal. Appl. 145 , 2 (1990), 470{484. [4]Castillo, R. D. R. On boundary conditions of an inverse Sturm{Liouville problem. SIAM J. Appl. Math. 60 , 6 (December 1990), 1745{1751. [5]Colton, D., Ewing, R., and Rundell, W. ,E d s . Inverse Problems in Partial Dierential Equations . SIAM, Philadelphia, PA, 1990. [6]Eskin, G. Inverse spectral problem for the Schroedinger equation with periodic vector potential. Comm. Math. Physics 125 , 2 (1989), 263{300. [7]Hassan, A. A. M., and Abdel-Halim, I. H. Some inverse eigenvalue problems for the Laplacian operator, II. J. Inst. Math. Comput. Sci. Math. Ser. 2 , 2 (1989), 125{146. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 15. Inverse Problems 77 [8]Levitan, B. M., and Sargsjan, I. S. Sturm{Liouville and Dirac Operators . Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. [9]Pilant, M., and Rundell, W. Determining a coecient in a rst-order hyperbolic equation. SIAM J. Appl. Math. 51 , 2 (April 1991), 494{506. [10]Roy, D. N. G. Methods of Inverse Problems in Physics . CRC, Boca Raton, FL, 1990. [11]Rundell, W. The use of integral operators in undetermined coecient problems for partial dierential equations. Appl. Analysis 18 (1984), 309{ 324. [12]Rundell, W., and Sacks, P. E. Reconstruction techniques for classical inverse Sturm{Liouville problems. Math. of Comp. 58 , 197 (January 1992), 161{183. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 78 I.A Denitions and Concepts 16. Limit Cycles Applicable to Systems of nonlinear autonomous dierential equa- tions. Yields Knowledge of whether or not there exist limit cycles. Idea Knowing that limit cycles exist for a dierential system allows global characterizations of the dierential system. Procedure A non-constant solution of the systemdx dt=f(x) is called a cycle (or a limit cycle) if there is a positive number T(called the period of the cycle) such that x(t+T)=x(t) for allt. It is easy to show that inside of every cycle is at least one critical point (i.e., a point where f(x)=0,s e e page 526). In many systems it is not only true that there are nitely many cycles but also that all solutions tend to one of these cycles. This knowledge permits a concise characterization of the phase plane. Example 1 The nonlinear autonomous system dx dt=−y+x(1−x2−y2); dy dt=x+y(1−x2−y2) becomes, under the change of variables fx=rcos,y=rsing,t h e uncoupled system dr dt=r(1−r2);d dt=1: These new equations have the solution r(t)=1p 1+Ae−2t; (t)=t+B; whereAandBare arbitrary constants. Hence, the solution of the original system is x(t)=cos(t+B)p 1+Be−2t;y (t)=sin(t+B)p 1+Be−2t: This states that all solutions tend to the circle x2(t)+y2(t)=1a st!1 . Of course, in most circumstances it is not possible to construct explicitly the limit cycle. Generally theorems (such as those below) are used to prove the existence of a limit cycle. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 16. Limit Cycles 79 Example 2 The Van der Pol equation d2x dt2−/parenleftbig 1−x2
dx dt+x=0 with>0 has limit cycles. For this equation, there is negative damping for small values of xand positive damping for large values of x. Hence the value ofxincreases when xis small and it decreases when xis large. Notes 1. Given a limit cycle Γ and a positive number a, dene the annulus centered on Γt ob efxjdistance from xto Γ is less than agwhere the distance from xto Γ is dened to be min u2Γjx−uj. A cycle Γ is called isolated if there is a positive number afor which the annulus centered on Γ contains no other limit cycles. A cycle is non-isolated if every annulus centered of Γ contains at least one other limit cycle. The system dx dt=xsin/parenleftbig x2+y2
−y;dy dt=ysin/parenleftbig x2+y2
+x has innitely many isolated cycles whereas the system fx0=y,y0= −xghas innitely many non-isolated cycles. 2. Part of Hilbert’s 16th problem asked for the maximum number of limit cycles of the system fx0=A(x;y);y0=B(x;y)gwhereAand Bare polynomials. If AandBare polynomials of degree n,t h e n the maximum number is known as the Hilbert number or the Hilbertfunction,H n.I t i s k n o w n t h a t H0=0 ,H1=0 ,H24,H38, Hnn−1 2ifnis odd, and Hn<1. The example that demonstrates that H24 (found by Songling [12]) is x0=ax−y−10x2+( 5+b)xy+y2; y0=x+x2+( 8c−25−9b)xy; wherea=−10−200,b=−10−13,a n dc=−10−52.S e e a l s o J a m e s and Lloyd [4]. 3. Neto [8] has the two results: Theorem The equation x0=a2x2+a1x+a0,w h e r et h efaigare continuous functions on [0 ;1], has at most two closed solutions, if not all solutions in [0 ;1] are closed. and Theorem The equation x0=a3x3+a2x2+a1x+a0,w h e r et h e faigare continuous functions on [0 ;1], has at most three closed solutions. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 80 I.A Denitions and Concepts 4. Iff(x)a n dg(x) are continuous, have continuous derivatives, and satisfy the conditions: xg(x)>0f o rx6=0 , f(x) is negative in the interval a<x<b (witha<0a n db>0) and positive outside of this interval, R1 0f(x)dx=R0 −1f(x)dx=1, then every nontrivial solution of Li enard’s equation d2x dt2+f(x)dx dt+g(x) = 0 (16.1) is either a limit cycle or a spiral that tends toward a limit cycle as t!1 . See Birkho and Rota [1, pages 135{137] for details. 5. Li enard’s theorem states Iff(x)a n dg(x) are continuous and satisfy the conditions F(x): =Rx 0f(x)dxis an odd function, F(x) is zero only at x=0 ,x=a,x=−a,f o rs o m ea>0, F(x)!1 monotonically for x>a , g(x) is an odd function, and g(x)>0f o rx>0, then equation (16.1) has a unique limit cycle. For details, see Jordan and Smith [5]. Note that Van der Pol’s equation (see example 2) satises Li enard’s theorem and, hence, has a unique limit cycle. 6. Bendixson’s theorem states (see Simmons [11, pages 338{352]) If@F @x+@G @yis continuous and is always positive or always negative in a certain region of the phase plane, then the autonomous system dx dt=F(x;y);dy dt=G(x;y) has no limit cycles in that region. For example, the equation for the Lewis regulator d2x dt2+( 1−jxj)dx dt+x=0; which is equivalent to dx dt=F(x;y)=y;dy dt=G(x;y)=−x−(1−jxj)y; has@F @x+@G @y=jxj−1. Hence, the Lewis regulator has no limit cycles in the strip−1<x< 1. 7. Another statement of Bendixson’s theorem, regarding periodic solu- tions or limit cycles, can be stated as follows: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 16. Limit Cycles 81 Consider _x=f(x) in a simply connected domain D(in two dimensions). If the gradient of fis not identically zero over any subregion of Dand does not change sign in D,t h e nDcontains no closed trajectory. 8. The Levinson{Smith theorem states (see Hagedorn [3, page 143]) For the dierential equation x00+f(x;x0)x0+g(x) = 0 (16.2) if the following conditions are satised: xg(x)>0 for allx>0, R1 0g(x)dx=1, f(0;0)<0, there exists an x0>0 such that f(x;x0)0f o rjxj>x 0, for everyx0, there exists a constant M> 0, such that f(x;x0)−Mfor jxjx0, there exists an x1>x0such thatRx1 x0f(x;v(x))dx10Mx0, wherev(x) is any arbitrary positive and monotonically de- creasing function of x, then equation (16.2) has at least one limit cycle. 9. Sedaghat [9] shows that factorable planar systems (i.e., systems of the formx0=f(x)h(y)a n dy0=k(x)g(y)) do not have limit cycles. References [1]Birkhoff, G., and Rota, G.-C. Ordinary Dierential Equations .J o h n Wiley & Sons, New York, 1978. [2]Blows, T. R., and Lloyd, N. G. The number of limit cycles of certain polynomial dierential equations. Proc. Roy. Soc. Edinburgh 98A (1984), 215{239. [3]Hagedorn, P. Non-linear Oscillations . Clarendon Press, Oxford, England, 1982. [4]James, E. M., and Lloyd, N. G. A cubic system with eight small- amplitude limit cycles. IMA J. Appl. Mathematics 47 (1991), 163{171. [5]Jordan, D. W., and Smith, P. Nonlinear Ordinary Dierential Equations , second ed. Clarendon Press, Oxford, England, 1987. [6]Koditschek, D. E., and Narendra, K. S. Limit cycles of planar quadratic dierential equations. J. Dierential Equations 54 (1984), 181{195. [7]Kuang, Y. Finiteness of limit cycles in planar autonomous systems. Appl. Anal. 32 , 3{4 (1989), 253{264. [8]Neto, A. L. On the number of solutions of the equation zzzref11refzzz, zzzref12refzzz, for which x(0)=x(1) .Inventiones Mathematicae 59 (1980), 67{76. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 82 I.A Denitions and Concepts [9]Sedaghat, H. Geometric properties of factorable planar systems of dierential equations. SIAM Review 38 , 4 (December 1996), 660{665. [10]Shahshahani, S. Periodic solutions of polynomial rst order dierential equations. Nonlinear Analysis 5 , 2 (1981), 157{165. [11]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. [12]Songling, S. A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sinica 23 (1980), 153{158. [13]Yan-Qian, Y. et al. Theory of Limit Cycles ,v o l .6 6o f Translations of Mathematical Monographs . Amer. Math. Soc., Providence, RI, 1986. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 17. Natural Boundary Conditions for a PDE 83 17. Natural Boundary Conditions for a PDE Applicable to Partial dierential equations. Yields A proper set of boundary conditions. Idea Given a partial dierential equation it is not always clear what the \correct" boundary conditions are. This is especially true for nonlinear partial dierential equations. However, most partial dierential equations that arise in mathematical physics have been obtained from a variationalprinciple (see page 418). If we start with the variational principle, then \natural" boundary conditions will be generated while deriving the equation we started with.These boundary condition are, in a sense, the most appropriate bound- ary conditions for the original equation if there is no physical reason for imposing other conditions. Procedure The variational principle that is most often used is J=0 ,w h e r e  represents a variation and Jis a functional given by J[]=ZZ RL(;t;x)dtdx: HereL(
) is a linear or nonlinear functional and (x;t) is the unknown function to be determined. This variational principle states that the inte-gralJ[] should be stationary to small changes in .I fw el e th(x;t)b ea continuously dierentiable function, that is \small" in magnitude, then we can form J[+h]−J[]=ZZ Rn Ltht+Lxjhxj+Lo dtdx+O(jjhjj2); where subscripts on Ldenote partial derivatives. The variational principle requires that J:=J[+h]−J[] = 0, or that ZZ Rn Ltht+Lxjhxj+Lo dtdx=0: (17.1) IfRis assumed to be a parallelpiped, then let Dt(Dxj)d e n o t et h et w o parts of the boundary of Ron whicht(xj) is constant. By integration by CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 84 I.A Denitions and Concepts parts, equation (17.1) can be written as ZZ R −@ @tLt−@ @xjLxj+L hdtdx=0; (17.2) w h e r ew eh a v ea s s u m e dt h a t Lt Dt=0;Lxj Dxj=0: (17.3) Nowh(x;t) was assumed to be arbitrary, so from equation (17.2) we conclude that @ @tLt+@ @xjLxj−L=0: (17.4) We conclude that if we can write a given partial dierential equation in the form of equation (17.4) for some operator L(
), then equation (17.3) gives the \natural" boundary conditions. Example Given the partial dierential equation tt−2r2+2=0; (17.5) wherer2=PN j=1xjxj, we nd that L(;t;x)=1 22 t−1 22NX j=12 xj−1 222(17.6) makes equations (17.4) and (17.5) identical. Therefore, the \natural" boundary conditions for equation (17.5) are, using equation (17.6) in (17.3), t Dt=0;xj Dxj=0: (17.7) Equation (17.7) states that the partial dierential equation (17.5) requires both initial and boundary conditions. This was to be expected because equation (17.5) is a hyperbolic equation. For example, if N=1a n dRis the region [0 ;T][0;1), thenDt= ft=0g[ft=TgandDx1=fx1=0g[fx1=1g. Hence, the natural boundary conditions for equation (17.5) require that ft(0;x1),t(T;x1), x1(t;0),x1(t;1)gbe specied. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 17. Natural Boundary Conditions for a PDE 85 Notes 1. Finding the operator L(
) or, equivalently, nding the variational principleJ, is a non-trivial task in general. Also, very often one wants a vector variational principle that will encompass, simultane- ously, several separate equations. 2. See the section on variational equations (on page 418) for more ex- amples. 3. See also Kantorovich and Krylov [1, Chapter 4, pages 241{357] and Whitham [2]. References [1]Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis . Interscience Publishers, Inc., New York, 1958. [2]Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 86 I.A Denitions and Concepts 18. Normal Forms: Near-Identity Transformations Applicable to Systems of ordinary dierential equations. Yields A reformulation of the dierential equations. Idea Find a change of variables in the form of an innite series, so that the original system of dierential equations goes into a \normal" (or \simple" or \canonical") form. The normal form is the simplest member of an equiv-alence class of dierential equations, all exhibiting the same qualitative behavior. Normal forms are often useful for stability analyses. Procedure Start with the system x0=f(x) such that (without loss of generality) x=0is a critical point. Expand this system to obtain x0=Ax+H(x); where H(x)h a s strictly nonlinear functions (i.e., there are no linear or constant terms). IfH(x) has nonlinear terms of at least degree n, then make a near- identity transformation using polynomials of degree nwith unknown coef- cients. By appropriately choosing the unknown coecients in the near-identity transformation, the original dierential equations, when written in the new variables, will have increased the degree of the nonlinear terms by one. We can summarize the procedure as follows: We are given the system of ordinary dierential equations x 0=f(x)= Ax+H(x), which we wish to analyze near the point x=0. We make the near-identity transformation from xtouviax=u+ g(u), where g( ) is a strictly nonlinear function. This change of variables produces the new equation u0=[I+J]−1f(u+g(u)) =Au+K(u); (18.1) whereIis the identity matrix and J=@g @uis the Jacobian of the transformation. The function g( ) is chosen to eliminate the nonlinear terms in the equation for uthat are of least order. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 18. Normal Forms: Near-Identity Transformations 87 This procedure can be iterated. If the critical point is \hyperbolic" (all eigenvalues have non-zero real parts), then the nonlinear terms can always be removed (i.e., one order at a time). Also, the topological nature does not change. See Guckenheimerand Holmes [7, Section 3.3]. Example 1 Suppose we have the system of equations dx dt=x+y2; dy dt=y+xy: Dening x=xyT, this system has the form dx dt= 10 01 x+ y2 xy = 10 01 x+H(x); (18.2) where H(x) has quadratic nonlinearities. We now choose to make the near-identity change of variables (of second order) x=u+a02u2+a11uv+a20v2; y=v+b02u2+b11uv+b20v2;(18.3) whereuandvare functions of t. Combining equation (18.2) and equation (18.3) we nd du dt=u+( 1−a02)v2−a11uv−a20u2+ higher order terms ; dv dt=v−b02v2+( 1−b11)uv−b20u2+ higher order terms ;(18.4) where \higher order terms" means terms that are of order O(u3;u2v;uv2;v3). To eliminate the second order terms in equation (18.4), we take fa02=1 , a11=0 ,a20=0 ,b02=0 ,b11=1 ,b20=0g. With these values, the transformation in equation (18.3) becomes x=u+u2; y=v+uv so that the original dierential equations in (18.2) becomes du dt=u+ higher order terms ; dv dt=v+ higher order terms : This new system now has cubic nonlinearities. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 88 I.A Denitions and Concepts Example 2 The system of ordinary dierential equations for x(t)a n dy(t): x0=y+F(x;y); y0=G(x;y);(18.5) whereF() a n dG( ) are strictly nonlinear, has the normal form 0=1+D1r2+D2r4+D3r6+:::; r0=B1r3+B2r5+B3r7+:::; whereu=rcos,v=rsin,a n dfu;vgare related, via a near-identity transformation, to fx;yg. In this example, the linear equations are not sucient to determine the local behavior. Knowledge of B1is needed to determine stability (unless it is zero, in which case B2is needed, etc.). For example, if equation (18.5) has the form x0=y+Fxxx2 2+Fxyxy+Fyyy2 2+Fxxxx3 6+Fxxyx2y 2 +Fxyyxy2 2+Fyyyy3 6+:::; y0=Gxxx2 2+Gxyxy+Fyyy2 2+Gxxxx3 6+Gxxyx2y 2 +Gxyyxy2 2+Gyyyy3 6+:::; then we nd (see Takens [12] for details) 16B1=Gyyy+Gxxy+Fxyy+Fxxx+FyyGyy−FxxGxx−GxxGxy −GyyGxy+FxxFxy+FxyFyy: Example 3 The system of ordinary dierential equations for x(t)a n dy(t): x0=−y+F(x;y); y0=x+G(x;y);(18.6) whereF() a n dG( ) are strictly nonlinear, has the normal form u0=v+1X n=2bnun;v0=1X n=2anun; (18.7) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 18. Normal Forms: Near-Identity Transformations 89 wherefu;vgare related, via a near-identity transformation, to fx;yg.F o r example, if equation (18.6) has the form x0=−y+Fxxx2 2+Fxyxy+Fyyy2 2+Fxxxx3 6+Fxxyx2y 2 +Fxyyxy2 2+Fyyyy3 6+:::; y0=x+Gxxx2 2+Gxyxy+Fyyy2 2+Gxxxx3 6+Gxxyx2y 2 +Gxyyxy2 2+Gyyyy3 6+:::; then we nd that u0=v+1 2(Gxy+Fxx)u2+1 12(GxyGyy−FxxGyy +2FxyGxy+2Gxxy−FyyGxx−Gxx+2Fxxx)u3+:::; v0=1 2Gxxu2+1 6(3FxyGxx+Gxxx−FxxGxy)u3+:::;(18.8) whereCis an arbitrary constant. See Takens [12] for details. Another normal form for equation (18.6) is given by U0=V; U0=1X n=2anUn+1X n=2nbnUn−1; wherefU;Vgare related, via a near-identity transformation, to fx;yg.S e e Guckenheimer and Holmes [7] for details. Notes 1. Ifa26= 0, then the flow of the system in equation (18.7) is topologi- cally equivalent to the flow of the system fu0=v,v0=a2u2g, which can be integrated in terms of elliptic integrals. If a2=0 ,t h e no t h e r conclusions are possible; see Rand and Keith [11] for details. 2. To avoid computing the matrix inverse in equation (18.1), it is su- cient to expand ( I+J)−1intoI−J+J2−


+(−J)n−1if only the nonlinear terms of order nare to be removed. 3. The concept of normal forms does not require that the transforma- tions used be near-identity ones, but they are the ones most often used in practice. 4. The computations needed for this technique quickly become unman- ageable unless a computer algebra system is used. Macsyma programs for performing the necessary computations are given in Chow et al. [3] and in Rand and Keith [10]. 5. Abraham and Marsden [1, page 489] have the theorem CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 90 I.A Denitions and Concepts Consider the system described by the Lagrangian L=K− VwhereK=1 2P i;jmij_qi_qjandV=1 2P i;jcijqiqjand the matricesmijandcijare symmetric (this is no loss of generality) andmijis positive denite. Then there is a linear change of coordinates Qi=P jaijqjand _Qi=P jaij_qjsuch that the Lagrangian in the new coordinates is L=K−Vwhere K= 1 2P imi(_Qi)2,V=1 2P iciQiQi,a n dmi>0. The new coordinates fQ1;:::;Qn;_Q1;:::; _Qngare called normal modes and Lagrange’s equations become ¨Qi+2 iQi=0( f o ri=1;:::;n ) where2 i=−ci=mi. References [1]Abraham, R., and Marsden, J. E. Foundations of Mechanics: A Mathe- matical Exposition of Classical Mechanics . Addison{Wesley Publishing Co., Reading, MA, 1994. [2]Ashkenazi, M., and Chow, S.-N. Normal forms near critical points for dierential equations and maps. IEEE Trans. Circ. & Syst. 35 ,7( J u l y 1988), 850{862. [3]Chow, S.-N., Byron, B., and Wang, D. Computation of normal forms. J. Comput. Appl. Math. 29 , 2 (1990), 129{143. [4]Chua, L. O., and Kokubu, H. Normal forms for nonlinear vector elds| Part I: Theory and algorithm. IEEE Trans. Circ. & Syst. 35 , 7 (July 1988), 863{880. [5]Chua, L. O., and Oka, H. Normal forms for constrained nonlinear dierential equations|Part I: Theory. IEEE Trans. Circ. & Syst. 35 ,7 (July 1988), 881{901. [6]Freire, E., Gamero, E., and Ponce, E. An algorithm for symbolic com- putation of Hopf bifurcation. In Computers and Mathematics ,E .K a l t o f e n and S. M. Watt, Eds. Springer{Verlag, New York, 1990, pp. 109{118. [7]Guckenheimer, J., and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields . Springer{Verlag, New York, 1983. [8]Nayfeh, A. H. Method of Normal Forms . John Wiley & Sons, New York, 1993. [9]Rand, R. H., and Armbruster, D. Perturbation Methods, Bifurcation Theory and Computer Algebra . No. 65 in Applied Mathematical Sciences. Springer{Verlag, New York, 1987. [10]Rand, R. H., and Keith, W. L. Normal forms and center manifolds cal- culations on MACSYMA. In Applications of Computer Algebra , R. Pavelle, Ed. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1985. [11]Rand, R. H., and Keith, W. L. Determinacy of degenerate equilibria with linear part x’=y,y’=0 using MACSYMA. Appl. Math. and Comp. 21 (1987), 1{19. [12]Takens, F. Singularities of vector elds. Publ. Math. Inst. Hautes Etudes Sci. 43 (1974), 47{100. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 19. Random Dierential Equations 91 19. Random Dierential Equations Applicable to Dierential equations involving random terms. Idea While randomness can appear in dierential equations in many ways, most often it appears through \white noise" terms. Procedure Suppose that x(t) is a random process that satises the stochastic dierential equation dx(t)=a[x(t);t]dt+b[x(t);t]dw(t); (19.1) wherew(t) is a standard Wiener process. The Wiener process is a Gaussian random process that has a mean given by its starting point, E [ w(t)] = w0=w(t0), a variance of E (w(t)−w0)2 =t−t0, and a covariance of E[w(t)w(s)] = min(t;s). The sample paths of w(t) are continuous but not dierentiable. If we dene (x;t)=a(x;t)−1 2b(x;t)@b(x;t) @x; (19.2) then the solution to the stochastic dierential equation, x(t), can be shown to satisfy (see Gardiner [5]) x(t)=x(t0)+Zt t0[x(s);s]ds+ SZt t0b[x(s);s]dw(s): (19.3) whereSR represents the Stratonovich stochastic integral. Hence, an under- standing of stochastic integration is required to understand the solutions to stochastic dierential equations. Ifw(t) is a Wiener process and G(t;w(t)) is an arbitrary function, then the stochastic integral I=Rt t0G(s;w(s))dw(s) is dened as a limiting sum. Divide the interval [ t0;t]i n t onsub-intervals: t0t1


tn−1tn= t, and choose points figthat lie in each sub-interval: ti−1iti.T h e stochastic integral Iis dened as the limit of partial sums, I= limn!1Sn, withSn=Pn i=1G(i;w(i))[w(ti)−w(ti−1)]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 92 I.A Denitions and Concepts Consider, for example, the special case of G(t)=w(t). Then the expectation of Snis computed as E[Sn]=E"nX i=1w(i)[w(ti)−w(ti−1)]# =nX i=1[min(i;ti)−min(i;ti−1)] =nX i=1(i−ti−1): If we take i=ti+( 1−)ti−1(where 0<< 1), then E [ Sn]=Pn i=1(ti−ti−1)=(t−t0). Hence, the value of Sndepends on .F o r consistency, some specic choice must be made for the points fig. For the Ito stochastic integral (indicated byIR ), we choose i=ti−1 (i.e.,= 0 in the above). That is IZt t0G(s;w(s))dw(s) = ms-lim n!1(nX i=1G(ti−1;w(ti−1))[w(ti)−w(ti−1)]) ; (19.4) where ms-lim refers to the mean square limit. For the Stratonovich stochastic integral (indicated bySR ), we choose i=(ti+ti−1)=2 (i.e.,=1/2in the above). That is (see Schuss [7]) SZt t0G(w(s);x)dw(s) = ms-lim n!1(nX i=1G ti−1;wti+ti−1 2 [w(ti)−w(ti−1)]) :(19.5) The dierence in these two integrals can be seen in the evaluation ofRt t0w(s)dw(s). We nd thatIRt t0w(s)dw(s)= w2(t)−w2(t0)−(t−t0) =2 whileSRt t0w(s)dw(s)= w2(t)−w2(t0) =2. Notes 1. This book contains several sections for dealing with dierential equa- tions containing random terms: To determine the transition probability density, see the discus- sion of the Fokker{Planck equation on page 303. To obtain the moments without solving the complete problem, see pages 568 and 572. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 19. Random Dierential Equations 93 If the noise appearing in the dierential equation is not \white noise," the section on stochastic limit theorems might be useful (see page 629). To numerically simulate the solutions of a stochastic dierential equation, see the technique on page 775. 2. It can be shown that the Stratonovich integral has the usual proper- ties of integrals, such as the fundamental theorem of integral calculus: SZt t0f0(w(s))dw(s)=f(w(t))−f(w(t0)): 3. For arbitrary functions G, there is no connection between the Ito and Stratonovich integrals. However, when x(t) satises (19.1), then (see Gardiner [5, page 99]) SZt t0b[x(s);s]dw(s)= IZt t0b[x(s);s]dw(s)+1 2Zt t0b[x(s);s]@b[x(s);s] @xds: 4. The Black{Scholes PDE for option pricing is obtained using stochas- tic dierential equations (see Black and Scholes [1]). Let Srepresent the price of a share of stock, and assume Sfollows a geometric Brownian motion dS=Sdt +Sd! ,w h e r etis time,is a constant, andis the volatility constant. Let V(S;t) be the price of a derivative security whose payo is only a function of Sandt. Construct a portfolio consisting of Vand
shares of stock. The value Pof this portfolio is P=V+
S. The dierential of Pis given by dP=dV+
dS. Substituting for dV(using Ito’s lemma), and replacingdSby its assumed form results in dP=@V @t+S@V @S+1 22S2@2V @S2+
S + S@V @S+
S d!: The random component of the portfolio increment can be removed by choosing
=−@V @S. The concept of arbitrage says that dP=rPdt , whereris the (constant) risk-free bank interest rate. Combining the above results in the Black{Scholes PDE @V @t+rS@V @S+1 22S2@2V @S2−rV=0: References [1]Black, F., and Scholes, M. The pricing of options and corporate liabilities. J. Political Economy 81 (1973), 637{659. [2]Boyce, W. E. Random eigenvalue problems. In Probabilistic Methods in Applied Mathematics , A. T. Bharucha-Reid, Ed. Academic Press, New York, 1968, pp. 1{73. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 94 I.A Denitions and Concepts [3]Boyce, W. E. On a conjecture concerning the means of the eigenvalues of random Sturm{Liouville boundary value problems. Quart. Appl. Math. (1980), 241{245. [4]Day, W. B. Asymptotic expansions of eigenvalues and eigenfunctions of random boundary-value problems. Quart. Appl. Math. (July 1980), 169. [5]Gardiner, C. W. Handbook of Stochastic Methods . Springer{Verlag, New York, 1985. [6]Harlow, D. C., and Delph, T. J. The numerical solution of random intial- value problems. Math. and Computers in Simulation 33 (1991), 243{258. [7]Schuss, Z. Theory and Applications of Stochastic Dierential Equations . John Wiley & Sons, New York, 1980. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 20. Self-Adjoint Eigenfunction Problems 95 20. Self-Adjoint Eigenfunction Problems Applicable to Linear dierential operators. Yields Information that may be used to show completeness of a set of functions. Procedure Many of the dierential equations of mathematical physics are related to self-adjoint eigenfunction problems. As a special subcase, Sturm{Liouville equations are often self-adjoint eigenfunction problems. (Sturm{Liouvilleproblems are discussed in more detail on page 103.) LetL[
]b et h enth order linear operator dened by L[y]=p n(x)dny dxn+pn−1(x)dn−1y dxn−1+


+p0(x)y; where thefpi(x)gare complex valued and analytic and pn(x)6=0o nt h e intervalx2[a;b]. Denenboundary conditions by Bj[y]: =nX k=1 Mjkd(k−1)y dx(k−1)(a)+Njkd(k−1)y dx(k−1)(b) =0;j =1;:::;n; where thefMjk;Njkgare given complex constants. The problem we consider is L[y]=y; B [y]=0; (20.1) whereB[y] = 0 is a shorthand notation for fBj[y]=0jj=1;:::;ng.T h e system in equation (20.1) will always have the trivial solution, y(x)=0 . But, for certain values of , called eigenvalues , the system in equation (20.1) will have non-trivial solutions. Corresponding to the specic eigenvalue n will be one or more eigenfunctions , that is, non-trivial solutions to (20.1) when=n. We represent the complex conjugate of gby g. Dene the inner prod- uctoff(x)a n dg(x)b y(f;g)=Rb af(t)g(t)dtand the norm off(x)b y jjfjj:=p (f;f). If (f;g)=0 ,t h e n fandgare said to be orthogonal .I f ff1;f2;:::;fngare a set of functions with ( fi;fj)=0w h e n i6=j,t h e nt h e ffi(x)gare an orthogonal family . The adjoint operator to L[
], calledL[
], is dened by L[y]: =(−1)nd(n)[pn(x)y] dx(n)+(−1)n−1d(n−1)[pn−1(x)y] dx(n−1)+


+p0y: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 96 I.A Denitions and Concepts Letu(x) be a solution to the system fL[u]=0 ,B[u]=0g, and letv(x)b e a solution to the adjoint system fL[v]=0 ,B[v]=0g,w h e r efB[y]=0g is a shorthand notation for fB j[y]=0jj=1;:::;ngand theB i[
] are, for the moment, unspecied. Using the denitions of u(x)a n dv(x), we can calculate vL[u]−uL[v]=d dxJ(u;v); (20.2) whereJ(u;v) is called the bilinear concomitant and is dened by J(u;v)=nX m=1X j+k=m−1(−1)kdk dxk(pmu)djv dxj : (20.3) Integrating equation (20.2) results in Zb a(vL[u]−uL[v])dx=J(u;v)b a=J u(b);v(b) −J u(a);v(a) : (20.4) We now dene the B i[
] to be those boundary conditions for which the right-hand side of equation (20.4) vanishes. IfL=L,t h e nLis said to be formally self-adjoint .I fL=Land B=B,t h e nLis said to be self-adjoint .N o t e t h a t i f L[
] is formally self-adjoint, then n=2randL[
] must be of the form L[u]=dr dxr br(x)dru dxr +


+d dx b1(x)du dx +b0(x)u: (20.5) As we now record, self-adjoint operators have some very useful proper- ties. IfL[
] is self-adjoint, then The eigenvalues nof equation (20.1) are real. The eigenvalues are enumerable (with no cluster point). The eigenfunctions yn(x) corresponding to distinct eigenvalues are orthogonal. Iff(x) is any analytic function that satises the boundary conditions in equation (20.1) (i.e., Bj[f]=0 ,f o rj=1;:::;n ), then, on the interval [a;b], we have the representation f(x)=1X k=0(f;yk) (yk;yk)yk(x). That is, thefyk(x)gare complete. It is this last statement that is of par- ticular importance in solving dierential equations. The method suggestedby this statement, the method of eigenfunction expansions, is described on page 268. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 20. Self-Adjoint Eigenfunction Problems 97 Example 1 Suppose we have the linear dierential operator L[y]=d2 dx2 r2(x)d2y dx2 +d dx r1(x)dy dx +r0(x): (20.6) Because of the form of the operator, we know that L[
] will be formally self- adjoint (see equation (20.5)). For this operator, we can evaluate J(u;v)a t the upper and lower limits (from equation (20.3)) to nd J(u;v)b a=h v(r2u00)0−v0r2u00+r2v00u0−u(r2v00)0+r1(vu0−uv0)ib a: (20.7) To determine whether L[
] is self-adjoint or not, we need to specify B[y]. Because equation (20.6) is a fourth order operator, four boundary condi- tions are required. We will consider three separate cases: Case 1 IfB[y] is dened by B1[y]=y(a); B2[y]=y00(a); B3[y]=y(b); B4[y]=y00(b);(20.8) thenJ(u;v) can be evaluated and equation (20.7) can be simplied to yield r2v00u0+r1vu0b a: (20.9) If we choose B=B(i.e.,B i[y]=Bi[y]), then the quantity in (20.9) is identically zero. Hence, L[
], as dened by equations (20.6) and (20.8) is self-adjoint. Case 2 IfB[y] is dened by B1[y]=y(a); B2[y]=y0(a); B3[y]=y(b); B4[y]=y0(b);(20.10) thenJ(u;v) can be evaluated and equation (20.7) can be simplied to yield v(r2u00)0−v0r2u00b a: (20.11) Once again, if we choose B=B, then the quantity in (20.11) is identically zero. Hence, L[
], as dened by equations (20.6) and (20.10) is self-adjoint. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 98 I.A Denitions and Concepts Case 3 IfB[y] is dened by B1[y]=y(a); B2[y]=y0(a); B3[y]=y00(a); B4[y]=y000(a);(20.12) thenJ(u;v) can be evaluated and equation (20.7) can be simplied to yield v(r2u00)0−v0r2u00+r2v00u0−u(r2v00)0+r1(vu0−uv0) x=b: (20.13) If, in this case, we choose B=B, then the quantity in equation (20.13) does notvanish. IfB=B, then no information has been given at the boundary x=b, and the quantity in (20.13) is indeter- minate. Hence, L[
], as dened by equations (20.6) and (20.12), is not self-adjoint. An initial value problem can never be self-adjoint. Example 2 The operator L[y]=d dx a2(x)dy dx +a1(x)dy dx+a0(x); with the boundary conditions B1[y]=y(a); B2[y]=y0(b); is self-adjoint. See the section on Sturm{Liouville theory (page 103). Example 3 A third order linear ordinary dierential equation is formally self-adjoint if it has the form d2 dx2 P(x)dy dx +d dx P(x)d2y dx2 +d dx Q(x)y +Q(x)dy dx=0: (20.14) The general third order linear ordinary dierential equation A(x)d3y dx3+B(x)d2y dx2+C(x)dy dx+D(x)=0; will be formally self-adjoint if and only if B=3 2A0andD=1 2/parenleftbig C−1 3B0
0. The self-adjoint third order equation (20.14) has the rst integral P/parenleftbig 2yy00−(y0)2
+P0yy0+Qy2= constant: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 20. Self-Adjoint Eigenfunction Problems 99 Example 4 The general fourth order linear ordinary dierential equation A(x)y0000+B(x)y000+C(x)y00+D(x)y0+E(x)y=0; will be formally self-adjoint if and only if B=2A0andD=/parenleftbig C−1 2B0
0. Notes 1. Some of the conditions above can be relaxed, and the main results for self-adjoint operators will still be true. See, for instance, Coddingtonand Levinson [3, Chapter 7]. 2. For partial dierential equations there are many results analogous to those mentioned above for ordinary dierential equations. We enu- merate some of them for the Helmholtz equation in two dimensions: For the equation r 2+= 0, in a region R, with the boundary conditionsa+br
n= 0, given on the entire boundary of R(here nrepresents the unit normal): All the eigenvalues figare real. There are an innite number of eigenvalues. There is an eigen- value of least magnitude but no largest one. The eigenfunctions fi(x;y)gform a complete set: Any analytic function can be represented in the form f(x;y)=P iaii(x;y), for some set of constants faig. Eigenfunctions belonging to dierent eigenvalues are orthogonal. That isRR Rijdxdy =0 ,i fi6=j. An eigenfunction is related to it’s eigenvalue by the Rayleigh quotient =−H r
nds+RR Rjrj2dxdy RR R2dxdy: 3. Many other partial dierential equations have very similar properties. See Haberman [5, pages 214{219] for details. 4. Partial dierential equations can also be self-adjoint. The elliptic equationauxx+cuyy+dux+euy+fu=g(x;y) is said to be essentially self-adjoint when Nx=My,w h e r e N:=d−ax a;M :=e−cy c: In this case, an integrating factor is given by e,w h e r ex=N, y=M. Multiplying the original equation by this factor puts the equation in self-adjoint form. For example, the equation uxx+uyy+x2ux+y2uy+u=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 100 I.A Denitions and Concepts hasN=x2,M=y2, which leads to =1 3/parenleftbig x3+y3
. Multiplying the equation by eresults in the self-adjoint form of the equation: h e(x3+y3)=3uxi x+h e(x3+y3)=3uyi y+e(x3+y3)=3u=0: 5. See Birkho and Rota [1, Chapters 10{11], Butkov [2, Chapter 9, pages 332{404], Dunford and Schwartz [4], Ince [6, Chapters 9{11, pages 204{278], and Stakgold [7, Chapter 3]. References [1]Birkhoff, G., and Rota, G.-C. Ordinary Dierential Equations .J o h n Wiley & Sons, New York, 1978. [2]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [3]Coddington, E. A., and Levinson, N. Theory of Ordinary Dierential Equations . McGraw{Hill Book Company, New York, 1955. [4]Dunford, N., and Schwartz, J. Linear Operators, Part II: Spectral Theory . John Wiley & Sons, New York, 1958. [5]Haberman, R. Elementary Applied Partial Dierential Equations .P r e n t i c e { Hall, Inc., Englewood Clis, NJ, 1968. [6]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [7]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 21. Stability Theorems 101 21. Stability Theorems Applicable to Dierential equations of all types. Yields Knowledge of whether or not there are stable solutions. Idea There are theorems available for most cases of interest. Procedure There are many theorems that can be used to determine whether the solutions to a dierential equation are stable. For example, useful simple theorems include Theorem Consider the equation y0=Ay+f(t;y), whereAis a real constant matrix whose eigenvalues all have negative real parts. Let fbe real, continuous for small jyjandt0, and f(t;y)=o(jyj)a s jyj!0, uniformly for t0. Then the identically zero solution is asymptotically stable. Theorem If 1. Every solution of y0=Ayapproaches zero as t!1 , 2.jjf(z)jj=jjzjj! 0a sz!0, 3.jjf(z1)−f(z2)jjc1jjz1−z2jjforjjz1jjandjjz2jjless thanc2 wherec1!0a sc2!0, thenz=0is a stable solution of y0=Ay+f(y). Example Consider the equation y0=−2y+f(t). Using the second theorem the solutiony= 0 is stable for f(y)=ynwhenn>1. Notes 1. Stability is required if a dierential equation is to be well posed (see page 115). 2. Floquet theory and Lyapunov functions are two techniques that can determine whether an equation has stable or unstable solutions (seepages 523 and 551). 3. Note that solutions to the equation y 0=A(t)ycan be increas- ing even if all the eigenvalues of A(t) have negative real parts for any xed value of t. For example, consider the matrix A(t)=" −1 4(1+t)1 (1+t)2 −1 4−1 4(1+t)# . This matrix has the eigenvalues 1;2=−12i 4(1 +t), CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 102 I.A Denitions and Concepts yet the general solution to y0=A(t)yis given by y(t)= (1 +t)−3=4 −1 2(1 +t)1=4 + (1 +t)−3=4log(1 +t) (1 +t)1=4/parenleftbig 1−1 2log(1 +t)
 ; whereandare arbitrary constants. 4. There are many dierent technical denitions of stability. For the equation y0=f(t;y); (21.1) dened when tt0, the solution is said to be Stable if for each >0 there is a corresponding =()>0 such that any solution by(t) of equation (21.1) that satises the inequalityjby(t0)−y(t0)j<exists and satises the inequality jby(t)−y(t)j<for alltt0. A solution that is not stable is said to be unstable . Asymptotically stable if, in addition to the above stability re- quirements,jby(t)−y(t)j!0a st!1 , wheneverjby(t0)−y(t0)j is suciently small. Uniformly stable if for each>0 there is a corresponding = ()>0 such that any solution by(t) of equation (21.1) that satises the inequality jby(t0)−y(t0)j<for somet1t0exists and satises the inequality jby(t)−y(t)j<for alltt1. Uniformly asymptotically stable if, in addition to the require- ments for asymptotic stability, there is a 0>0, and for each > 0 a corresponding T=T()>0 such that ifjby(t1)−y(t1)j<0 for somet1t0,t h e njby(t)−y(t)j<for alltt1+T. Strongly stable if for each >0 there is a corresponding = ()>0 such that any solution by(t) of equation (21.1) that satises the inequality jby(t0)−y(t0)j<for somet1t0exists and satises the inequality jby(t)−y(t)j<for alltt0. References [1]Bellman, R. Stability Theory of Dierential Equations . McGraw{Hill Book Company, New York, 1953. [2]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [3]Coddington, E. A., and Levinson, N. Theory of Ordinary Dierential Equations . McGraw{Hill Book Company, New York, 1955. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 22. Sturm{Liouville Theory 103 22. Sturm{Liouville Theory Applicable to Second order linear ordinary dierential operators. Yields Information about whether an operator is self-adjoint. Procedure Many of the dierential equations of mathematical physics are Sturm{ Liouville equations. Sturm{Liouville equations arise naturally, for instance, when separation of variables (see page 487) is applied to the wave equation, the potential equation, or the diusion equation. The Sturm{Liouville operator, L, is dened by L:=1 s(x) −d dx p(x)d dx +q(x) ; (22.1) wherep,p0,q,a n dsare real and continuous and s(x)>0a n dp(x)>0o n the interval ( a;b). The Sturm{Liouville equation is dened by L[y(x)] =−y(x); (22.2) or, equivalently, −d dx p(x)dy dx +q(x)y+s(x)y=0; (22.3) forx2[a;b]. The parameter is an eigenvalue of the equation. Given a specic set of boundary conditions, there may be specic values of  for which equation (22.2) has a non-trivial solution. For dierent types of boundary conditions, dierent types of behavior are possible. Many facts are known about Sturm{Liouville systems: L, as dened by equation (22.1), is formally self-adjoint (see page 95), with the inner product, ( f;g)s:=R s(x)f(x)g(x)dx. Lis self-adjoint (see page 95) when {The boundary conditions are unmixed (or separated). That is, they are of the form 1y(a)+1y0(a)=0; 2y(b)+2y0(b)=0:(22.4) {The boundary conditions are periodic . That is, they are of the form y(a)=y(b); y0(a)=y0(b): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 104 I.A Denitions and Concepts When the boundary conditions are given as in equation (22.4), and, in addition, p(x)>0,q(x)>0,1=1>0,2=2>0, then {Lis a positive denite operator (i.e., ( Lu;u)>0, for allu6=0 ) . {The eigenvalues are simple (i.e., each eigenvalue has a single eigenfunction associated with it). When the operator Lis not self-adjoint then {Ifis a complex eigenvalue of L,t h e n is an eigenvalue of L, the adjoint ofL. {Eigenfunctions of Lare orthogonal to those of L. If the interval [ a;b] is nite and p(x)a n ds(x) are positive at the endpoints, then the problem is said to be regular . Otherwise, it is said to be singular . For singular Sturm{Liouville problems, problems are subdivided into two cases, the limit-circle case and limit-point case. Consider equation (22.2) when one of the endpoints is regular and the other singular. Dene thes-norm of a function u(x)b y jjujjs=(u;u)s=Zb as(x)ju(x)j2dx: If, for any particular complex number , the solution to equation (22.2) satises j jyjjs<1,t h e nLis said to be of the limit-circle type at innity. In this case, all solutions of equation (22.2) will satisfy jjyjjs<1,f o r any value of . j jyjjs=1,t h e nLis said to be of the limit-point type at innity. If both endpoints are singular, we introduce an intermediate point l, a<l<b and then classify Las being of the limit-point type or the limit-circle type at each endpoint according to the behavior of solutions in a<x<l and inl<x<b (the classication is independent of the choice ofl). For a given real , the problem in equation (22.2) is Oscillatory atx=aif and only if every solution has innitely many zeros clustering at a. Nonoscillatory atx=aif and only if no solution has innitely many zeros clustering at a. The classication is mutually exclusive for a xed but can vary with . IfLis in the limit-point case at innity, then there is the following completeness theorem: Theorem Ifg()=R1 0f(x)Ψ(x;)dx,t h e nf(x)=R1 −1g()Ψ (x;)d() for a (computable) density function (). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 22. Sturm{Liouville Theory 105 A completeness theorem is required for a proof that a separation of variables calculation (see page 487) has been done correctly. The following theorem and corollaries may help decide the type of the operatorL: Theorem LetMbe a positive dierentiable function, and let k1and k2be two positive constants such that for large x, q(x)−k1M(x); Z1 x(p(t)M(t))−1=2dt=1; jp1=2(x)M0(x)M−3=2(x)j<k2; thenLis in the limit-point case at innity. Corollary Ifq(x)−k,w h e r ekis a positive constant, andR1 np−1=2(t)dt=1(wherenis any nite number), then Lis in the limit-point case at innity. Corollary Ifp(x)=1f o r0 <x<1andq(x)−kx2for some positive constant k,t h e nLis in the limit-point case at innity. Example 1 The dierential equation and boundary conditions −(xy0)0=xy; u(1) = 0; u(2) = 0; correspond to the Sturm{Liouville operator in equation (22.1) with p(x)= x,q(x) = 0, and s(x)=x. This is a regular Sturm{Liouville problem on the interval [1 ;2]. The eigenvalues and eigenfunctions are readily computed (see Stakgold [6, page 423]. If we dene n=r2 n, then thernare determined from J0(rn) J0(2rn)=N0(rn) N0(2rn); and the corresponding eigenfunction is given by yn(x)=rnJ0(2rn)p 2p J0(rn)2−J0(2rn)2[J0(rn)N0(rnx)−J0(rnx)N0(rn)]: Example 2 The dierential equation with boundary conditions −(x2y0)0−u=0; u(1) = 0; u(e)=0; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 106 I.A Denitions and Concepts forx2[1;e] is a regular Sturm{Liouville problem with unmixed boundary conditions, so the eigenfunctions are complete. In this case we nd n=n22+1 2;yn=x−1=2sin(nlogx): 22.1 Classication of Sturm{Liouville Problems Pruess et al. [5] have devised a classication scheme and taxonomy for Sturm{Liouville problems on the interval ( a;b). They dene: Category 1: Problem (22.2) is nonoscillatory at x=aandx=b. The spectrum is simple, purely discrete, and bounded below. Category 2: Problem (22.2) is nonoscillatory at one endpoint. At the other endpoint, it is nonoscillatory for 2(−1;t0) and oscillatory for2(t0;1). The spectrum is simple and bounded below. The point spectrum (if any) is in (−1;t0)whereas (t0;1)is the continuous spectrum. Category 3: Problem (22.2) is nonoscillatory at one endpoint. At the other endpoint it is limit-circle and oscillatory. The spectrum is simple, unbounded both above and below, and purely discrete. Category 4: Problem (22.2) is nonoscillatory at one endpoint. At the other endpoint, it is limit-point and oscillatory. The spectrum is simple and purely continuous; the continuous spec- trum is the entire real line. Category 5: Problem (22.2) is limit-circle and oscillatory at x=a.I t i s limit-point and oscillatory at x=b. The spectrum is simple, unbounded both above and below, and purely discrete. Category 6: Problem (22.2) is limit-point and oscillatory at x=a.I t i s limit-point and oscillatory at x=b. The nature of the spectrum is unknown; a continuous spectrum is likely. Category 7: Problem (22.2) is limit-point and oscillatory at one endpoint (x=aorx=b). At the other endpoint, it is limit-circle and oscillatory. The spectrum is simple and purely continuous; the continuous spec- trum is the entire real line. Category 8: Problem (22.2) is limit-circle and oscillatory at one endpoint (x=aorx=b). At the other endpoint, it is nonoscillatory for 2(−1;t0) and oscillatory for 2(t0;1). The spectrum is simple; the point spectrum (if any) is unbounded below but bounded above by t0. The continuous spectrum is in (t0;1). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 22. Sturm{Liouville Theory 107 Category 9: Problem (22.2) is limit-point and oscillatory at one endpoint (x=aorx=b). At the other endpoint, it is nonoscillatory for 2(−1;t0) and oscillatory for 2(t0;1). The spectrum may be nonsimple. Category 10: Atx=aproblem (22.2) is nonoscillatory for 2(−1;t0) and oscillatory for 2(t0;1). Atx=b, it is nonoscillatory for 2(−1;t1) and oscillatory for 2(t1;1). The spectrum may be nonsimple. The point spectrum (if any) is in the interval (−1;min(t0;t1))and is bounded below. The continuous spectrum is in (min(t0;t1);1). Notes 1. For transformations of equation (22.3), see page 157. 2. The regular Sturm{Liouville equation, written in the form d2z dt2−r(t)z+z=0; with the boundary conditions z(0) =z(L) = 0, has the asymptotic eigenvalues and eigenfunctions zn(t)=r 2 Lsinn Lt +O1 n ; n=n22 L2+O(1) asn!1 . (See the Pr¨ ufer method on page 150.) 3. For the Sturm{Liouville equation L[y]=−(py0)0+qy−wy =0o n [a;1], deneandto be solutions satisfying f(a)=0 ,p0 x=a= 1gandf(a)=−1,p0 x=a=0g. The Titchmarsh{Weyl function m() is dened to be the functions fmg, dened on the upper and lower half planes, such thatR1 aj(x;)+m()(x;)j2dx<1for all strictly complex values of . 4. See also Birkho and Rota [1, Chapters 10{11], Coddington and Levinson [2, Chapters 7{12], Levitan and Sargsjan [4, Chapter 6,pages 139{182 and Chapter 12, pages 324{340], Stakgold [6, Chapter 7, pages 411{466], and Zauderer [7, pages 136{159]. References [1]Birkhoff, G., and Rota, G.-C. Ordinary Dierential Equations .J o h n Wiley & Sons, New York, 1978. [2]Coddington, E. A., and Levinson, N. Theory of Ordinary Dierential Equations . McGraw{Hill Book Company, New York, 1955. [3]Dunford, N., and Schwartz, J. Linear Operators, Part II: Spectral Theory . John Wiley & Sons, New York, 1958. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 108 I.A Denitions and Concepts [4]Levitan, B. M., and Sargsjan, I. S. Sturm{Liouville and Dirac Operators . Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. [5]Pruess, S., Fulton, C. T., and Xie, Y. The automatic classication of Sturm{Liouville problems. Appl. Math. and Comp. . (submitted for publication). [6]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. [7]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 23. Variational Equations 109 23. Variational Equations Applicable to Dierential equations that arise from variational principles. Yields A variational principle. Procedure Most dierential equations that arise in mathematical physics have been obtained from a variational principle. The variational principle that is most o f t e nu s e di s J=0 ,w h e r e represent a variation and Jis a functional given by J[u]=ZZ RL(x;@xj)u(x)dx: (23.1) Here,L(
) is a linear or nonlinear function of its arguments, and u(x)i st h e unknown function to be determined. This variational principle states that the integral J[u] should be stationary to small changes in u(x). If we let h(x) be a \small," continuously dierentiable function, then we can form J[u+h]−J[u]=ZZ R L(x;@xj)(u(x)+h(x))−L(x;@xj)u(x)/bracerightbig dx: (23.2) By integration by parts, equation (23.2) can often be written as J[u+h]−J[u]=ZZ RN(x;@xj)u(x)dx+O(jjhjj2); plus some boundary terms (see page 83). The variational principle requires thatJ:=J[u+h]−J[u] vanishes to leading order, or that N(x;@xj)u(x)=0: (23.3) Equation (23.3) is called the rst variation of equation (23.1) or the Euler{ Lagrange equation corresponding to equation (23.1). (This is sometimes called the Euler equation .) A functional in the form of equation (23.1) de- termines an Euler{Lagrange equation. Conversely, given an Euler{Lagrange equation, a corresponding functional can sometimes be obtained. Many approximate and numerical techniques utilize the functional asso- ciated with a given system of Euler{Lagrange equations. See, for example,the Rayleigh{Ritz method (page 638) and the nite element method (page 734). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 110 I.A Denitions and Concepts The following collection of examples assumes that the dependent vari- able in the given dierential equation has natural boundary conditions (see page 83). If the dependent variable did not have these specic boundary conditions, then the boundary terms that were discarded in going fromequation (23.2) to equation (23.3) would have to be satised in addition to the Euler{Lagrange equation. Example 1 The Euler{Lagrange equation for the functional J[y]=Z RF x;y;y0;:::;y(n) dx; (23.4) wherey=y(x)i s @F @y−d dx@F @y0 +d2 dx2@F @y00 −


+(−1)ndn dxn@F @y(n) =0: (23.5) For this equation the natural boundary conditions are given by y(x0)=y0;y0(x0)=y0 0; :::; y(n−1)(x0)=y(n−1) 0; y(x1)=y1;y0(x1)=y0 1; :::; y(n−1)(x1)=y(n−1) 1: Example 2 The Euler{Lagrange equation for the functional J[u]=ZZ RF(x;y;u;ux;uy;uxx;uxy;uyy)dxdy; (23.6) whereu=u(x;y)i s @F @u−@ @x@F @ux −@ @y@F @uy +@2 @x2@F @uxx +@2 @x@y@F @uxy +@2 @y2@F @uyy =0: (23.7) Example 3 The Euler{Lagrange equation for the functional J[u]=ZZ R" a@u @x2 +b@u @y2 +cu2+2fu# dxdy; (23.8) is @ @x a@u @x +@ @y b@u @y −cu=f: (23.9) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 23. Variational Equations 111 Example 4 For the 2mth order ordinary dierential equation (in formally self- adjoint form) mX k=0(−1)kdk dxk pk(x)dku dxk =f(x); u(a)=u0(a)=


=u(m−1)(a)=0; u(b)=u0(b)=


=u(m−1)(b)=0;(23.10) a corresponding functional is J[u]=Zb a mX k=0pk(x)dku dxk2 −2f(x)u(x)! dx: (23.11) Example 5 Consider the system of nsecond order ordinary dierential equations for the unknowns fuk(x)jk=1;:::;ng −nX k=1d dx pjk(x)duk dx +qjk(x)uk =fj(x); uj(a)=uj(b)=0;(23.12) forj=1;2;:::;n .I fpjk=pkj,qjk=qkj, if the matrixfpjkgis bounded and positive denite, and if the matrix fqjkgis bounded and non-negative denite, then a functional corresponding to equation (23.12) is J[u]=Zb a0 @nX j;k=1 pjk(x)duj dxduk dx+qjk(x)ujuk −nX j=1fj(x)uj(x)1 Adx: (23.13) Example 6 IfAij(x) is a symmetric and positive denite matrix, so that the partial dierential equation for u(x)=u(x1;:::;xm) −mX i;j=1@ @xi Aij@u @xj +C(x)u=f(x); (23.14) is elliptic in Ω, C(x)>0, and there are Dirichlet boundary conditions u @Ω=0; (23.15) then a corresponding functional is J[u]=Z Ω0 @mX i;j=1Aij@u @xi@u @xj+Cu2−2fu1 Adx; (23.16) where (23.16) is to be minimized over those functions that satisfy equation (23.15). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 112 I.A Denitions and Concepts Example 7 IfAij(x) is a symmetric and positive denite matrix, so that the partial dierential equation for u(x)=u(x1;:::;xm), −mX i;j=1@ @xj Aij@u @xj +C(x)u=f(x); (23.17) is elliptic in Ω, C(x)>0, and there are the boundary conditions 2 4mX i;j=1Aij@u @xjcos(;xi)+u3 5 @Ω=0; (23.18) whereis normal to @Ωa n dis a positive function on @Ω, then a corresponding functional is J[u]=Z Ω0 @mX i;j=1Aij@u @xi@u @xj+Cu2−2fu1 Adx+Z @Ωu2dS; (23.19) where (23.19) is to be minimized over those functions for which equation (23.18) is satised. Notes 1. Note that two dierent functionals can yield the same set of Euler{ Lagrange equations. For example, R Jdx =R (J+y+xy0)dx.T h e reason that R (y+xy0)dx= 0 is because the integrand is an exact dierential (i.e.,R (y+xy0)dx=R d(xy)). Hence, this integral is path independent; its value is determined by the boundary conditions. The Euler{Lagrange equations for the two functionalsRR uxxuyydxdy andRR (uxy)2dxdy are also the same. 2. If a dierential equation can be derived from a variational princi- ple, then admittance of a Lie group is a necessary condition to ndconservation laws by Noether’s theorem. 3. Even if the boundary conditions given with a dierential equation are not natural, a variational principle may sometimes be found. Consider J[u]=Z x2 x1F(x;u;u0)dx−g1(x;u) x=x1+g2(x;u) x=x2; whereg1(x;u)a n dg2(x;u) are unspecied functions. The necessary conditions for uto minimize J[u] are (see Mitchell and Wait [5]). @F @u−d dx@F @u0=0; @F @u0+@g1 @u x=x1=0;@F @u0+@g2 @u x=x2=0: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 23. Variational Equations 113 Ifg1andg2are identically zero, then we recover the natural boundary conditions. However, we may choose g1andg2to suit other boundary conditions. For example, the problem u00+f(x)=0; u0+u x=x1=0;u0+u x=x2=0 corresponds to the functional J[u]=Zx2 x11 2(u0)2−f(x)u dx+u2 2 x=x2−u2 2 x=x1: 4. This technique can be used in higher dimensions. For example, consider the functional J[u]=ZZ RF(x;y;u;ux;uy;uxx;uxy;uyy)dxdy +Z @RG(x;y;u;u;u;un)d; where@=@ and@=@n are partial dierential operators in the direc- tions of the tangent and normal to the curve @R. Necessary condi- tions forJ[u] to have a minimum are the Euler{Lagrange equations (given in equation (23.7)) together with the boundary conditions: @F @ux−@ @x@F @uxx y−@F @uy−@ @y@F @uyy x −@ @@F @uxx−@F @uyy xy+1 2@ @@F @uxy/parenleftbig x2 −y2 
 +1 2@ @x@F @uxy x−@ @y@F @uxy y +Gu−@ @@G @u+@2 @2@G @u=0; @G @un+@F @uxxy2 +@F @uyyx2 +@F @uxyxy=0;(23.20) wherex=dx dandy=dy d. See Mitchell and Wait [5] for details. 5. Mathematica has the package VariationalMethods which can deter- mine the Euler equations for a general integrand. 6. See also Butkov [1, pages 573{588], Collatz, [2, pages 540{541], Far- low, [3, pages 362{369], and Kantorovich and Krylov [4, Chapter 4, pages 241{357]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 114 I.A Denitions and Concepts References [1]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [2]Collatz, L. The Numerical Treatment of Dierential Equations . Springer{ Verlag, New York, 1966. [3]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [4]Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis . Interscience Publishers, Inc., New York, 1958. [5]Mitchell, A. R., and Wait, R. The Finite Element Method in Dierential Equations . John Wiley & Sons, New York, 1977. [6]Yourgrau, W., and Mandelstam, S. Variational Principles in Dynamics and Quantum Theory . Dover Publications, Inc., New York, 1979. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 24. Well Posed Dierential Equations 115 24. Well Posed Dierential Equations Applicable to Ordinary and partial dierential equations. Yields Knowledge of whether the equation is intrinsically well posed. Idea Before an attempt is made to determine or approximate the solution of a dierential equation, it should be checked to determine if the dierential equation problem is intrinsically well posed. Procedure A well posed dierential equation is one in which The solution exists. The solution is unique. The solution is stable (i.e., the solution depends continuously on the boundary conditions and initial conditions). If the dierential equation is not well posed, it is called an ill posed or improperly posed problem. For such problems, there may not be a solution, there may be more than one solution, or whatever solution is determined(by an approximate scheme) may be unrelated to the actual solution. For partial dierential equations, the third condition (concerning sta- bility) is generally the easiest to check. Example Consider the initial value problem for the unknown function u(x;t), utt=uxxxx; u(x;0) =g(x):(24.1) We will show that the solution to this problem is not stable. Suppose that equation (24.1) has a solution, say u0(x;t). Assume that is a xed number, much smaller than one in magnitude, and dene u1(x;t)b y u1(x;t)=u0(x;t)+eikxet; wherekandare also constants. At t=0 ,u1(x;0) diers from g(x)b ya quantity that has magnitude , an arbitrarily small amount. However, using u1(x;t) in equation (24.1), we determine that u1(x;t) will satisfy the equation if =k2. Therefore, at any xed value of t, sayt=T, there exists a solution u0(x;T) and an approximation to the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 116 I.A Denitions and Concepts solutionu1(x;T)=u0(x;T)+eikxek2T. The approximation satises the same dierential equation that the true solution satises. But because kis arbitrary, the approximate solution can be arbitrarily larger than the true solution by making karbitrarily large. Because two dierent expressions satisfy the same dierential equation and initially were arbitrarily close and are arbitrarily dierent in magnitude at any future time, we conclude that the problem is ill posed. Note that, with the proper boundary conditions and initial conditions, equation (24.1) would have a unique solution. But the solution would be unstable because the equation is intrinsically ill posed as an initial value problem. Hence, there would be, for instance, no easy way to numerically approximate the solution. Notes 1. For a discussion of existence and uniqueness theorems, see page 53. For a discussion of stability theorems, see page 101. 2. A standard example of an ill posed problem is Laplace’s equation with initial data. For example, the equation r2u= 0 with the initial data @u @y(x;0) =1 nsinnxhas the solution u(x;y)=1 n2sinnxsinhny.A s n!1 , the initial data are becoming arbitrarily small in magnitude whereas the solution (for y>0) is becoming arbitrarily large. 3. Certain classes of equations have been well studied. We can state For Laplace’s equation and elliptic equations in general, the Dirichlet problem is well posed. Also, the Neumann problemdoes not have a unique solution but is otherwise well posed. For the two-dimensional wave equation and hyperbolic equations in general, both are well posed as an initial value problem. Bothare, generally, ill posed as boundary value problems. For the heat equation and diusion equations in general, both are well posed when given Dirichlet data and the time variable is increasing; both are ill posed when the time variable is de- creasing. See Beck et al. [2] for numerical schemes related to a specic ill posed problem. 4. A backward heat equation (a parabolic equation with decreasing time) is ill posed. It may be made well posed, however, by requiring the solution to satisfy a suitable constraint. Typically, one asks fornon-negative solutions or for solutions that satisfy an a priori bound, which is obtained from physical considerations. 5. Payne [9] contains the following non-exhaustive list of methods that have been proposed and used in treating various types of improperly posed Cauchy problems: Function theoretic methods Eigenfunction methods Logarithmic convexity methods CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 24. Well Posed Dierential Equations 117 Weighted energy methods Lagrange identity methods Quasireversibility methods Restriction of data methods Numerical and programming methods Concavity methods Stochastic and probabilistic methods Method of generalized inverse in reproducing kernel spaces Comparison methods Payne [9] illustrates several of these methods on a backward heat equation. 6. As Fichera [4] shows, nding the correct boundary conditions for a degenerate problem (one in which the type changes) can be dicult in general. Fichera shows, for example, that the rst order equation foru(x;y) a(x;y)ux+b(x;y)uy+cu=f in the rectangle R=f−x;−yg,w h e naandb satisfy a(−;y)0;a (;y)0; b(x;−)0;b (x;)0; hasnoboundary conditions! However, the equation, −a(x;y)ux−b(x;y)uy+cu=f; inR, with the same conditions on aandb, requires that ube given on the entire boundary of R. 7. See also Garabedian [5, pages 450{457] and Zauderer [10, pages 103{ 113]. References [1]Argyros, I. K. On the cardinality of solutions of multilinear dierential equations and applications. Int. J. Math. &M a t h .S c i .9 , 4 (1986), 757{766. [2]B e c k ,J .V . ,B l a c k w e l l ,B . ,a n dS t .C l a i r ,J r . ,C .R . Inverse Heat Problems . Wiley, New York, 1985. [3]Buzbee, B. L., and Carasso, A. On the numerical computation of parabolic problems for preceding times. Math. of Comp. 27 , 122 (April 1973), 237{266. [4]Fichera, G. On a unied theory of boundary value problems for elliptic{ parabolic equations of second order. In Boundary Problems in Dierential Equations , R. E. Langer, Ed. University of Wisconsin Press, Madison, Wisconsin, 1960, pp. 97{120. [5]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 118 I.A Denitions and Concepts [6]Lavrent’ev, M. M., Romanov, V. G., and Shishatskii, S. P. Ill- Posed Problems of Mathematical Physics and Analysis . Amer. Math. Soc., Providence, RI, 1986. [7]Morozov, V. A. Methods for Solving Incorrectly Posed Problems . Springer{ Verlag, New York, 1984. [8]Pavlov, N. N. Smoothing of input data in the solution of ill-posed problems. U.S.S.R. Comput. Maths. Math. Phys. 29 , 5 (1989), 110{114. [9]P a y n e ,L .E . Improperly Posed Problems in Partial Dierential Equations . SIAM, Philadelphia, PA, 1975. [10]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 25. Wronskians and Fundamental Solutions 119 25. Wronskians and Fundamental Solutions Applicable to Linear ordinary dierential equations. Yields A formulation of a linear ordinary dierential equation as vector system Idea Annth order linear ordinary dierential equation can be written as a rst order ordinary dierential equation for a nelement vector. Procedure LetL[
] be the linear nth order ordinary dierential operator L[y]=dny dxn+a1(x)d(n−1)y dx(n−1)+


+an(x)y: The vector equation associated with the linear equation L[y]=0i sg i v e n by (see page 146) y0=A(x)y; (25.1) where y= yy0y00::: y(n−1)TandAis the matrix A=2 6666666401 0 0


0 00 1 0


0 00 0 1 0 ............ 00 0 0 1 −a n−an−1−an−2−an−3:::−a13 77777775: (25.2) Iffy 1;y2;:::;yngis any set of nsolutions to the equation L[y]=0 ,t h e n the matrix (x)=2 6664y 1y2


yn y0 1y0 2


y0 n ............ y(n−1) 1y(n−1) 2


y(n−1) n3 7775 is a solution matrix for equation (25.1). It is also called a fundamental solution . This matrix satises the dierential equation  0=A. The determinant of this matrix, det ( x), is called the Wronskian of L[y] = 0 with respect to fy1;y2;:::;yngand is denoted by W(y1;y2;:::;yn). Note that the Wronskian is a function of x. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 120 I.A Denitions and Concepts If (x) satises 0=A, thenj(x)j0=jjtrA(t), where trAdenotes the trace of the matrix A. Hence, det (x)=d e t (x0)e x pZx x0trA(s)ds : For the matrix in equation (25.2), we have tr A=−a1so that W(y1;:::;yn)(x)=e x p −Zx x0a1(s)ds W(y1;:::;yn)(x0): (25.3) This is sometimes called Liouville’s formula . From equation (25.3), we conclude that either W(x) vanishes for all values forx, or it is never equal to zero. If the Wronskian never vanishes, then the setfy1;y2;:::;yngis said to be linearly independent .A s e t o fn linearly independent solutions to L[y] = 0 is called a basis or a fundamental set. Alternately, given a set of nlinearly independent continuous functions, fy1;y2;:::;yng, it is possible to nd a unique homogeneous dierential equation of order n(with the coecient of y(n)being one) for which the set forms a fundamental set. This dierential equation is given by (−1)nW(y;y1;y2;:::;yn) W(y1;y2;:::;yn)=0: (25.4) Example 1 Given the second order linear ordinary dierential equation y00+y=0; (25.5) the setfsinx;cosxgforms a fundamental set because each element in this set satises equation (25.5) and also the Wronskian is given by W(sinx;cosx)=sinx cosx cosx−sinx=−1; which does not vanish. Because the Wronskian is constant, we have ver- ied thata 1(x) = 0 in equation (25.5) (the a1(x) term in this equation corresponds to the rst derivative term). Example 2 If we choose the two functions y1=s i nxandy2=x, we can determine the linear second order equation that has these solutions as its fundamental CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 25. Wronskians and Fundamental Solutions 121 set by constructing equation (25.4). Here, n= 2 so we nd (−1)2W(y;x;sinx) W(x;sinx)=yx sinx y 01c o sx y000−sinx xsinx 1c o sx; =(xcosx−sinx)y 00+(xsinx)y0−(sinx)y (xcosx−sinx); =y00+xsinx (xcosx−sinx)y0−sinx (xcosx−sinx)y: Notes 1. Given the linear partial dierential equation L[u]=nX i;j=1aij(x)@2u @xi@xj+nX i=1bi@u @xi+cu foru(x), let Γ = Γ( x;)=Γ (;x) be the geodesic distance between the points xand. (For a rectangular coordinate system, Γ( x;)= jjx−jj=p (x1−1)2+


+(xn−n)2.) A fundamental solution, S(x;), satisesL[S] = 0 and, near x=, has the form S=U Γm+ Vlog Γ +W,w h e r eU,V,a n dWare analytic functions and m= (n−2)=2. For example, for Laplace’s equation in ndimensions with n>2,r2u= 0, a fundamental solution is given by S=1 rn−2;withr=p (x1−1)2+


+(xn−n)2: See Garabedian [3, pages 152{153] for details. 2. The canonical form of a self-adjoint third order linear homogeneous dierential equation is y000+2Ay0+A0y= 0 (see pages 98 and 163). A fundamental set of solutions for this equation is fu2;uv;v2g,w h e r e u(x)a n dv(x) are any two linearly independent solutions of the second order dierential equation u00+1 2Au=0 . 3. Similar to the second example, it is possible to nd a single dierential equation whose solutions include the products of the solutions of twogiven linear homogeneous dierential equations; see Spigler [6]. 4. See also Boyce and DiPrima [1, pages 113{126], Coddington and Levinson [2, pages 67{84], Ince [4, pages 116{121], and Simmons [5, pages 76{80]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 122 I.A Denitions and Concepts References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Coddington, E. A., and Levinson, N. Theory of Ordinary Dierential Equations . McGraw{Hill Book Company, New York, 1955. [3]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [4]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [5]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. [6]Spigler, R. The linear dierential equation whose solutions are the products of solutions of two given dierential equations. J .M a t h .A n a l .A p p l .9 8 (1984), 130{147. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 26. Zeros of Solutions 123 26. Zeros of Solutions Applicable to Linear ordinary dierential equations. Yields Statements about the zeros of the solutions. Idea There are several standard theorems about the zeros of solutions of dierential equations. Procedure Consider the following equations: d dx p(x)dy dx +q(x)y= 0 (26.1) and d2y dx2+p(x)y=0 d2y dx2+q(x)y=0(26.2.a-b) and d dx p1(x)dy dx +q1(x)y=0 d dx p2(x)dy dx +q2(x)y=0:(26.3.a-b) 1. Consider the self-adjoint equation (26.1) in which p(x)>0a n dp(x) andq(x) are continuous. Sturm’s separation theorem states Theorem Letuandvbe linearly independent solutions of (26.1). Ifandare successive zeros of u,t h e nvhas one and only one zero in the interval ( ;). This has been extended by Makay [3] to be Theorem Consider the second order equation F(y00;y0;y;x)=0; (26.4) whereFis continuous. If the two conditions are satised Ifyis a solution of (26.4), then so is cy, for all real c. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 124 I.A Denitions and Concepts The solution of (26.4) as an initial value problems is unique. then the results of Sturm’s theorem apply to equation (26.4). 2. We have the following result about the interlacing of zeros: Theorem Letu(x)a n dv(x) be linearly independent solutions of equation (26.2.a) and assume u(x) has at least two zeros in the interval ( a;b). Then, if x1andx2are two consecutive zeros ofu(x), the function v(x) has one, and only one, zero in the interval (x1;x2). Theorem Letp(x) in equation (26.2.a) be continuous in ( a;b) with 0<mp(x)M. If the solution u(x) of (26.2.a) has two successive zeros x1andx2,t h e np Mx2−x1pm. 3. We have the following results about oscillatory solutions: Theorem Consider the self-adjoint equations in (26.3.a-b). If All the solutions of (26.3.a) are oscillatory as x!1 . q2(x)q1(x) are continuous functions, p2(x)p1(x)>0 are continuous functions, then all solutions of equation (26.3.b) are oscillatory. Theorem Ifp(x)(1 +)=4t2and>0, then all solutions to equation (26.2.a) are oscillatory. Theorem If all the solutions to equation (26.2.a) are oscillatory, and ifq(x)p(x), then all solutions of equation (26.2.b) are oscillatory. And we have the converse: Theorem Ifq(x)p(x) and some solutions to equation (26.2.b) are nonoscillatory, then some solutions of equation (26.2.a) must be nonoscillatory. 4. The Sturm comparison theorem is Theorem Consider the self-adjoint equations (26.3.a-b). Let p1(x)p2(x)>0a n dq1(x)q2(x) be continuous functions. Then between any two zeros of a nontrivial solution of equa-tion (26.3.a), there will be at least one zero of every nontrivial solution of (26.3.b). 5. Considering equation (26.1), let p(x)>0, and letpandqbe contin- uous on [0;1]. Then CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 26. Zeros of Solutions 125 Theorem IfR1 1dx p(x)andR1 1q(x)dxboth diverge, then every solution to equation (26.1) has innitely many zeros on the inter- val [1;1]. If, in addition,R1 0dx p(x)andR1 0q(x)dxboth diverge to +1, then every solution to equation (26.1) has innitely many zeros on the interval [0 ;1]. Theorem IfR1 adx p(x)converges and ifRx aq(s)dsis bounded by a constant for ax1 , then every non-trivial solution to equation (26.1) has at most a nite number of zeros on the interval [a;1]. 6. We also have the following nonoscillation results: Theorem If lim supx2p(x)=γand lim inf x2p(x)=γthen the solution of equation (26.2.a) is Nonoscillatory if γ<1/4 Oscillatory if1/4<γ Theorem For the equations in (26.2): If P(x)=xR1 xp(t)dt, Q(x)=xR1 xq(t)dt,0<Q(x)<P(x), and equation (26.2.a) is nonoscillatory in the wide sense, then equation (26.2.b) isnonoscillatory in the wide sense. Theorem Consider (26.2.a) and dene lim x!1sup/parenleftbig xR1 xp(s)ds
= Pand lim x!1inf/parenleftbig xR1 xp(s)ds
=Pthen A necessary condition that the solution to equation (26.2.a) be nonoscillatory is that P1/4andP1. A sucient condition that the solution to equation (26.2.a) be nonoscillatory is that P1/4. Notes 1. Makay’s [3] theorem applies to equations such as y00(y0)2+y3=0 . 2. For the eigenvalue problem L[u]=nu,l e tN()c o u n tt h en u m b e r of eigenvalues less than . In one dimension the asymptotics of N() can be easily determined because the nth eigenfunction has nzeros. For example, for the Schr¨ oedinger equation −r2 n+q(x) n=n n N(n+) 2+N(n−) 2=1 Z [n−q(x)]1=2 +dx+O1 n where [y]+( yify0 0i fy<0. The generalization of this formula to k dimensions is (see Newell [4]) N()=[1 +o(1)] 2kk=2Γ(k=2+1 )Z [n−q(x)]k=2 +dx+O1 n : CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 126 I.A Denitions and Concepts References [1]Banks, S. B. A note on the location of complex zeros of solutions of linear dierential equations. B u l l .A m e r .M a t h .S o c .1 8 , 1 (January 1988), 35{38. [2]Erbe, L. H., Kong, Q., and Zhang, B. G. Oscillation Theory for Functional Dierential Equations . Marcel Dekker, New York, 1994. [3]Makay, G. A simple proof for Sturm’s separation theorem. Amer. Math. Monthly (March 1992), 218{219. [4]Newell, G. F. Asymptotic distribution of eigenvalues for the multidimen- sional Schr odinger equation. J. Math. Physics 21 , 8 (August 1980), 2193{ 2201. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 128 I.B Transformations 27. Canonical Forms Applicable to The ordinary dierential equations: d2y dx2+2e x+fdy dx+p x2+2q x+r y=0; (27.1) d2y dx2+2 (e+fx)dy dx+(px2+2qx+r)y=0; (27.2) (+x)d2y dx2+(b+mx)dy dx+(c+nx)y=0; (27.3) d2y dx2=Fdy dx;y;x : (27.4) Idea Each of these equations has certain canonical forms. When approxima- tions and numerical values for these equations are reported in the literature, it is generally for the canonical forms. Procedure 1 By changing the dependent and independent variables from y=y(x) tov=v(z), via y(x)=zezv(z); x=z; for some choice of the constants f;;;g, equation (27.1) will take the form of one of the following four canonical forms: d2v dz2+A z2+2 z+B v=0; d2v dz2+A z2+2 z v=0; d2v dz2+A z2+1 v=0; d2v dz2+A z2v=0; whereAandBare constants. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 27. Canonical Forms 129 Procedure 2 By changing the dependent and independent variables from y=y(x) tov=v(z), via y(x)=ez+z2v(z); x=z+; for some choice of the constants f;;;;g, equation (27.2) will take the form of one the following four canonical forms: d2v dz2+/parenleftbig z2+J
v=0; d2v dz2−vz=0; d2v dz2+v=0; d2v dz2=0; whereJis a constant. Procedure 3 By changing the dependent and independent variables, equation (27.3) can be reduced to Weiler’s canonical form (this is also known as a Kummer equation) zd2v dz2+(b−z)dv dz−av=0: (27.5) The transformation used to produce equation (27.5) from equation (27.3) has several dierent forms depending on the numerical values of the coef- cients in equation (27.3), see Bateman [2] for details. Procedure 4 A critical point is called a moving critical point (or singularity) if its location depends on the initial conditions for the dierential equation (and so the location of the critical point is not xed solely by the coecients of the dierential equation). For example, the nonlinear dierential equation y00=(y0)22y−1 y2+1has the general solution y(x) = tan [log( Ax+B)], whereA andBare arbitrary constants. The initial conditions determine AandB and thus determine the location of the singularities of y(x). Given an ordinary dierential equation in the form of equation (27.4), ifF(y0;y;x) is rational in y0, algebraic in y, and analytic in x, and if all of the critical points are xed, then a change of variables of the form y(x)=az(x)+b cz(x)+d; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 130 I.B Transformations wherea,b,c,d,a n dware some functions of x, will transform the equation to one of 50 standard forms. Each of these 50 dierential equations is for the unknown function z(x). Of these standard forms, six have solutions in terms of the Painlev e transcendents and all the others have rst integrals that are equations of rst order or have elementary integrals. The equations that dene the six Painlev e transcendents are d2y dx2=6y2+x, d2y dx2=2y3+xy+, d2y dx2=1 y dy dx2 −1 xdy dx+1 x(y2+)+γy3+ y, d2y dx2=1 2y dy dx2 +3y3 2+4xy2+2 (x2−)y+ y, d2y dx2= 1 2y+1 y−1 dy dx2 −1 xdy dx+(y−1)2 x2 y+ y +γy x +y(y+1) y−1, d2y dx2=1 2 1 y+1 y−1+1 y−x dy dx2 − 1 x+1 x−1+1 y−x dy dx +y(y−1)(y−x) x2(x−1)2 +x y2+γ(x−1) (y−1)2+x(x−1) (y−x)2 . In the above equations, all of the parameters are assumed to be con- stant. Notes 1. The rst three transformations may be found in Bateman [2, pages 75{79]. 2. The transformations for equation (27.4) may be found in Ince [4, Chapter 14, pages 317{355]. 3. Even though the Painlev e equations do not have elementary solutions in general, some choices of the parameters will lead to equations solvable in terms of elementary functions. For example, y=−1=x is a solution of the second Painlev e equation when =1 ,a n d y=−1=x+3x2=(x3+ 4) is a solution of the same equation when =−2. See Airault [1] for details. References [1]Airault, H. Rational solutions of Painleve equations. Stud. Appl. Math. 61 (1979), 31{53. [2]Bateman, H. Partial Dierential Equations of Mathematical Physics .D o v e r Publications, Inc., New York, 1944. [3]Berkovitch, L. M. Canonical forms of ordinary linear dierential equations. Arch. Math. (Brno) 24 , 1 (1988), 25{42. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 27. Canonical Forms 131 [4]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [5]Irvine, D., and Savageau, M. A. Ecient solution of nonlinear ordinary dierential equations expressed in S-system canonical form. SIAM J. Numer. Anal. 27 , 3 (1990), 704{735. [6]Neuman, F. Transformation and canonical forms of functional{dierential equations. Proc. Roy. Soc. Edin. 115A (1990), 349{357. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 132 I.B Transformations 28. Canonical Transformations Applicable to A system of ordinary dierential equations that arise from a Hamiltonian. Yields A dierent system of ordinary dierential equations that arise from a dierent Hamiltonian. Procedure A Hamiltonian H(p;q), with p=(p1;:::;pn)a n d q=(q1;:::;qn), denes the system of ordinary dierential equations _pi=−@H @qi=−Hqi; _qi=@H @pi=Hpi; where a dot denotes dierentiation with respect to the independent variable t(see page 61). The fpi;qigare called the coordinates of the Hamiltonian. The transformation to the new system of coordinates fPi;Qigvia pi=pi(P;Q); qi=qi(P;Q);(28.1) is (commonly) said to be canonical if Hamilton’s equations remain in- variant. That is, there exists a new Hamiltonian K(P;Q) such that the equations _Pi=−KQi; _Qi=KPi;(28.2) are valid. Canonical transformations can be dened implicitly by a generating function . For instance, for almost arbitrary S(p;Q;t), a canonical trans- formation is given by Pi=−SQi; qi=−Spi; K(P;Q)=H(p;q)+St;(28.3.a-c) where equations (28.3.a) and (28.3.b) must be solved to obtain explicit expressions for q(P;Q),p(P;Q). Note that, for the Stterm, the derivative is taken with respect to the explicit dependence of Sont. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 28. Canonical Transformations 133 Other functional forms for the generating function are also possible. For example, a function of the form S(q;P;t) gives rise to the canonical transformation pi=Sqi; Qi=SPi; K(P;Q)=H(p;q)+St:(28.4.a-c) Example Given the Hamiltonian H=1 2/parenleftbig p2+a2(t)q2
; (28.5) Hamilton’s equations are f_p=−a2q,_q=pg, which can be combined to yield ¨q+a2q=0: (28.6) Hence, the Hamiltonian in (28.5) denes the second order ordinary dier- ential equation (28.6). Now consider the canonical transformation induced by the generating function S(q;P;t )=q2P. From equation (28.4) we nd p=2qP; Q=q2; K(Q;P)=1 2/parenleftbig p2+a2q2
=Q 2/parenleftbig 4P2+a2
: The equations corresponding to the new Hamiltonian are _P=−1 2/parenleftbig 4P2+a2
; _Q=4PQ:(28.7.a-b) Equation (28.7.a) is a nonlinear rstorder ordinary dierential equation for P(t). AfterP(t) is determined, equation (28.7.b) can be used to determine Q(t) by quadrature. Hence, this change of variable has changed a second order linear ordinary dierential equation into two successive rst order ordinary dierential equations. Notes 1. Canonical transformations are sometimes called contact transforma- tions. See page 249 for the correct denition of a contact transfor- mation. 2. Technically, and in more generality, a transformation of the 2 nvari- ablesfxj;pjjj=1;:::;ngto the 2nvariablesfXj;Pjjj=1;:::;ng CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 134 I.B Transformations is a canonical transformation if the dierential formPn j=1(PjdXj− pjdxj) is exact, i.e., there exists a function U=U(x;p) such that nX j=1(PjdXj−pjdxj)=dU: (28.8) 3. The section on Hamilton{Jacobi theory (see page 61) utilizes canon- ical transformations to derive the Hamilton{Jacobi equation. 4. Tolstoy [7] shows that any nonlinear ordinary dierential equation may be transformed, in principle, by a variable transformation into a linear dierential equation or a system of such equations. This is the reverse of the process that was seen in the example. 5. The set of all canonical transformations forms a group. 6. Fouling transformations are canonical transformations in which the pcoordinates in conguration space are preserved (i.e., P=p,Q= Q(p;q)). See Gelman and Saletan [4] for details. 7. A transformation, given by equation (28.1), which allows equation (28.2) to be written, and may or may not satisfy (28.8) is technically called a canonoid transformation . The lack of distinction between canonical and canonoid has occasionally led to ambiguity in the lit-erature. See Currie and Saletan [3] or Negri et al. [6] for details. 8. See also Caratheodory [1, Chapter 6, pages 79{101], Chester [2, pages 197{206], and Goldstein [5, Chapter 8, pages 237{272]. References [1]Caratheodory, C. Calculus of Variations and Partial Dierential Equa- tions of the First Order . Holden{Day, Inc., San Francisco, CA, 1965. [2]Chester, C. R. Techniques in Partial Dierential Equations . McGraw{Hill Book Company, New York, 1970. [3]Currie, D. G., and Saletan, E. J. Canonical transformations and quadratic Hamiltonians. Nuovo Cimento B 9 , 1 (1972), 143{153. [4]Gelman, Y., and Saletan, E. J. q-equivalent particle Hamiltonians. II: The two-dimensional classical oscillator. Nuovo Cimento B 18 , 1 (1973), 53{71. [5]Goldstein, H. Classical Mechanics . Addison{Wesley Publishing Co., Reading, MA, 1950. [6]Negri, L. J., Oliveira, L. C., and Teixeira, J. M. Canonoid transfor- mations and constants of the motion. J. Math. Physics 28 , 10 (Oct 1987), 2369{2372. [7]Tolstoy, I. Remarks on the linearization of dierential equations. J. Inst. Maths. Applics 20 (1977), 53{60. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 29. Darboux Transformation 135 29. Darboux Transformation Applicable to Linear second order ordinary dierential equations, a single equation or a system. Yields A reformulation of the problem. Procedure Given the equation y00=(f(x)+)y (29.1) fory(x), we say that the transformation z(x)=A(x;)y+B(x;)y0 is a Darboux transformation if z(x) satises a dierential equation of the form z00=(g(x)+)z: (29.2) For example, if w(x) is a known solution of equation (29.1), then a Darboux transformation is given by z=y0−yw0 w: (29.3) In this case, if ysatises equation (29.1), then z(x) satises equation (29.2) with f(x)=g(x)−2[logw(x)]00: That is to say, this transformation changes the potential function appearing in equation (29.1) from f(x)b yf=−2[logw(x)]00,w h e r ew(x)i sa n arbitrary solution of equation (29.1). The usefulness of this technique is that equation (29.2) might be easy to solve for z(x); theny(x)m a yb e found from equation (29.3) by a single integration. For the system of second order ordinary dierential equations y00=D(x)y; (29.4) whereD(x) is the matrix D(x)=2 6664d 11(x)d12(x)::: d 1n(x) d21(x)d22(x)::: d 2n(x) ............ d n1(x)dn2(x)::: dnn(x)3 7775; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 136 I.B Transformations we say that z(x)=A(x)y+B(x)y0; (29.5) whereAandBare matrices, is a Darboux transformation if zsatises an equation of the form z00=F(x)z; (29.6) whereF(x) is a new matrix. Sometimes Darboux transformations of this type can be used to decouple systems of dierential equations. See Humi [2] for details. Example 1 If the solution of the dierential equation y00=(f(x)+)y (29.7) is known for each value of (call the solutin y), andw(x)=y(x)i st h e solution when =, then the general solution of the dierential equation z00= w(x)d2 dx21 w(x) +− z (29.8) forz(x) is given by (see equation (29.3)) z=y0 −yw0(x) w(x); (29.9) for6=. In particular, if we take f(x) = 0 in equation (29.7), then y0(x)=Ax+Bwhen=0a n dy(x)=ep xfor6=0 . I fw et a k e =0 andw(x)=x, then equation (29.8) becomes z00=2 x2+ z; with the solution given by equation (29.9); that is, z(x)=ep x p −1 x : Example 2 This example is from Humi [2]. Suppose we wish to decouple a system of symmetric equations in the form of equation (29.4) with D(x)=u1(x)+d (x) d(x)u2(x)+ : CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 29. Darboux Transformation 137 If we apply a Darboux transformation, we can hope to obtain the form of equation (29.6) with F(x)g i v e nb y F(x)=v1(x)+ 0 0v2(x)+ : (29.10) If we choose B=Iin equation (29.5), then to obtain equation (29.6), we require A00+D0+AD=FA; 2A0+D=F: In our case, with D(x) given by equation (29.7) and F(x) given by equation (29.10) we require that the elements of the matrix A(x)s a t i s f y 2a0 12=2a0 21=−d; 2a0 11+u1(x)=v1(x); 2a0 22+u2(x)=v2(x):(29.11) It is a simple matter to integrate these equations to obtain a12(x)=a21(x)=c(x); a11(x)=1 2c1 2d(x)++I ; a22(x)=1 2c1 2d(x)−+I ; whereis an arbitrary constant and c(x)=−1 2Zx d(t)dt; I(x)=Zx c(t)[u2(t)−u1(t)]dt: This solution is valid if the consistency constraint u1+u2=2c2−d 2c0 +1 2d 2c2 +1 2c2(+I)2 (29.12) is satised. This constraint was derived in the solution of equation (29.11). Stated another way, we can choose dandu1−u2as arbitrary functions and then use equation (29.12) to compute the corresponding u1+u2for which the resulting system of equations can be decoupled by the use of a Darboux transformation. Note 1. See Ince [3, page 182] and Lamb [5, pages 38{41]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 138 I.B Transformations References [1]Darboux, G. C. R. Acad. Sci Paris (1882), 1456. [2]Humi, M. Factorization of systems of dierential equations. J. Math. Physics 27(Jan 1986), 76{81. [3]Ince, E. L. Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [4]Konopelchenko, B. G. On exact solutions of nonlinear integrable equations via integral linearising transforms and generalised Backlund{ Darboux transformations. J. Phys. A: Math. Gen. 23 (1990), 3761{3768. [5]Lamb, G. L. Elements of Soliton Theory . John Wiley & Sons, New York, 1980. [6]Levi, D. Toward a unication of the various techniques used to integrate nonlinear partial dierential equations: Backlund and Darboux transforma- tions vs. dressing method. Rep. Math. Phys. 23 , 1 (1986), 41{56. [7]Poplavskii, I. V. Generalized Darboux{Crum{Krein transformations. Theo. Math. Physics 69 , 3 (1986), 1278{1282. [8]Sall, M. A. Darboux transformations for non-Abelian and nonlocal equations of the Toda chain type. Theo. Math. Physics 53 (1982), 1092{ 1099. [9]Stanek, S., and Vosmansky, J. Transformations of linear second order ordinary dierential equations. Archivum Mathematicum (BRNO) 22 ,1 (1986), 55{60. [10]Zheng, W. M. The Darboux transformation and solvable double-well potential models for Schrodinger equations. J. Math. Physics 25 ,1( J a n 1984), 88{90. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 30. An Involutory Transformation 139 30. An Involutory Transformation Applicable to Nonlinear partial dierential equations of a certain form. Yields A reformation of the partial dierential equation. Idea Inverting the dependent and independent variables might lead to a more tractable equation. Procedure Suppose we have a partial dierential equation of the form  u;@ @x;@ @t := (u;ux;uxx;:::;ut;utt;:::)=0; (30.1) foru=u(x;t). We introduce the inverse transformation T=8 >< >:u0=x; x0=u; t0=t: Because applying Ttwice is equivalent to not applying T, the transforma- tion is involutory (i.e., T2=I= the identity). Noting that D0:=@ @x=1 @u0=@x0@ @x0 @0:=@ @t=@ @t0−@u0=@t0 @u0=@x0@ @x0; then, under T, equation (30.1) becomes (x;D0;@0)=0: (30.2) This transformation may be used to change classes of nonlinear equations with Dirichlet boundary conditions to linear form. For example, the class @u0 @t0−γ(u0)@ @x0 NX i=1i(u0;t0)D0ix0! =0; u0=Ψ 1(t0)o nx0= 1(t0); u0=Ψ 2(t0)o nx0= 2(t0); u0= (x0)a tt0=0;(30.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 140 I.B Transformations transforms, under T,t o @u @t+γ(u)@ @x NX i=1i(x;t)@iu @xi! =0; u= 1(t)o nx=Ψ 1(t); u= 2(t)o nx=Ψ 2(t); u=−1(x)att=0:(30.4) Example Given the equation and initial/boundary conditions @u0 @t0= /parenleftbig@u0 @x0
2@2u0 @x02; u0=0 onx0= 1(t0); u0=L onx0= 2(t0); u0= (x0)a tt0=0;(30.5) the transformed equation and initial/boundary conditions become @u @t=@2u @x2; u= 1(t)o nx=0; u= 2(t)o nx=L; u=−1(x)att=0:(30.6) Then equation (30.6) can be easily solved (by use of, say, Fourier trans- forms) to yield u(x;t)=2 L1X n=1exp −n22t L2 sinnx L"ZL 0−1()s i nn L d +n LZt 0expn22 L2 [1()−(−1)n2()]d : This last relation, can be implicitly solved for x=x(u;t); which (under T) is the solution to equation (30.5) (i.e., u0=u0(x0;t0)). Note 1. The Hodograph transformation is a dierent way in which the depen- dent and independent variables are interchanged (see page 456). Reference [1]Rogers, C. Inverse transformations and the reduction of nonlinear Dirichlet problems. J. Phys. A: Math. Gen. 17 (1984), L681{L685. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 31. Liouville Transformation { 1 141 31. Liouville Transformation { 1 Applicable to The general Sturm{Liouville problem −[p(x)y0]0+r(x)y=(x)y; foraxb; y0(a)+y(a)=0; y0(b)+y(b)=0:(31.1) Procedure The Liouville transformation (version 1) is to change the independent variable from x2[a;b]t ot2[0;]b y t=1 JZx a(x) p(x)1=2 dx; (31.2) whereJis dened by J=1 Zb a(x) p(x)1=2 dx; (31.3) and to change the dependent variable from y(x)t ou(t)b y u(t)=f(x)y(x)=[(x)p(x)]1=4y(x); (31.4) where we have dened f(x)=[(x)p(x)]1=4. With this change of variable, equation (31.1) becomes d2u dt2+[k2−q(t)]u=0; for 0t; u0(0) +hu(0) = 0; u0()+Hu()=0;(31.5) which is in Liouville normal form . The denitions of fk;q(t);h;Hgare as follows k2=J2; m(t)=r(x) (x); q(t)=ftt f+J2m(t); h=1 f2(a)[Jp(a)−f(a)ft(a)]; H=1 f2(b)[Jp(b)−f(b)ft(b)]: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 142 I.B Transformations Note thatq(t) may also be written as q(t)=r p+(p)−1=4d2 dt2[(p)1=4]; =r p+p 4"p0 p0 +0 0 +3 4p0 p2 +1 2p0 p0  −1 40 2# Example If we have the equation and boundary conditions −(xy0)0+1 xy=xy; forx2; y0()=0; y0(2)=0; then we identify p(x)=x; r (x)=1 x; (x)=x; a=; b =2;  =0; =0: A simple calculation results in J=1 ,t=x−,f(x)=px=pt+1 , m(t)=1 x2=1 (t+1)2,q(t)=3 4(t+1)2,k2=,h=−1 2,a n dH=−1 2(+1). Hence, we obtain u00+ −3 4(t+1 )2 u=0; for 0t; u0(0)−1 2u(0) = 0; u0()−1 2(+1 )u()=0:(31.6) Equation (31.6) is in Liouville normal form. Notes 1. The standard assumptions used with equation (31.1) are that on the interval [a;b]:pandqare real-valued, p>0,qdoes not vanish, and pandqhave continuous second derivatives. Boundedness conditions are also required for the new functions. 2. When= 0, the transformation t=Zx x0s jq(z)j p(z)dz; u(t)=[p(x)jq(x)j]1=4y(x);(31.7) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 31. Liouville Transformation { 1 143 when applied to equation (31.1), results in d2u dt2+[1+R(t)]u(t)=0; (31.8) where R(t)=p1=4jqj−3=4dp(x) dxd dx[p(x)jq(x)j]−1=4 x=x(t); and the plus (minus) sign is taken in equation (31.8) if q(x)>0 (q(x)<0). This is also called the Liouville transformation (see Eastham [4]). 3. The two dierent transformations, the one in equations (31.2) and (31.4), and the one in equation (31.7), are each sometimes called the Liouville{Green transformation. 4. See also Birkho and Rota [1, pages 265{267], Boyce [2, pages 20{21], Hille [5, page 340], Lakin and Sanchez [7, pages 36{41], and Valiron [8, page 511]. References [1]Birkhoff, G., and Rota, G.-C. Ordinary Dierential Equations .J o h n Wiley & Sons, New York, 1978. [2]Boyce, W. E. Random eigenvalue problems. In Probabilistic Methods in Applied Mathematics , A. T. Bharucha-Reid, Ed. Academic Press, New York, 1968, pp. 1{73. [3]Cassell, J. S. An extension of the Liouville{Green asymptotic formula for oscillatory second-order dierential equations. Proc. Roy. Soc .Edin. 100A (1985), 181{190. [4]Eastham, M. S. P. Asymptotic formulae of Liouville{Green type for higher- order dierential equations. J. London Math. Soc. 2 , 28 (1983), 507{518. [5]Hille, E. Lectures on Ordinary Dierential Equations . Addison{Wesley Publishing Co., Reading, MA, 1969. [6]Howard, H. C., and Maric, V. An extension of the Liouville{Green approximation. J. Math. Anal. Appl. 143 , 2 (1989), 548{559. [7]Lakin, W. D., and Sanchez, D. A. Topics in Ordinary Dierential Equations . Dover Publications, Inc., New York, 1970. [8]Valiron, G. The Geometric Theory of Ordinary Dierential Equations and Algebraic Functions . Math Sci Press, Brookline, MA, 1950. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 144 I.B Transformations 32. Liouville Transformation { 2 Applicable to The second order linear ordinary dierential equa- tion d2y dt2+m4(t)y= 0 (32.1) on the nite interval 0 tT,w h e r eis a constant and m(t)>0. Procedure The Liouville transformation (version 2) is to change the dependent and independent variables in equation (32.1) by x=1 JZt 0m2(z)dz; J=1 ZT 0m2(z)dz; w(x)=m(t)y(t): This transformation changes equation (32.1) into d2w dx2+ J2+Q(x) w=0; (32.2) for 0x,w h e r eQ(x) is dened by Q(x)=1 m(t)d2m(t) dx2=−J2 m(t)3d2 dt21 m(t) : (32.3) The inverse transformation, which takes equation (32.2) into equation (32.1), is given by t=JZx 0d [m()]2; J=TZ 0d [m()]2−1 ; wherem(x)=m(t) is any positive solution of the dierential equation d2m dx2=Q(x)m(x): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 32. Liouville Transformation { 2 145 Example Suppose we have (essentially) Airy’s equation d2y dt2+ty=0: (32.4) Comparing equation (32.4) to equation (32.1) shows that m(t)=t1=4. Using this value for m(t) produces J=2 3T3=2; x=t T3=2 ; w(x)=t1=4y(t): Under this change of variables, equation (32.4) becomes d2w dx2+4 92T3−5 361 x2 w=0: (32.5) For large values of x, an approximation to equation (32.5) might be ob- tained by discarding the second term in the parentheses. Notes 1. The function Q(x) dened in equation (32.3) will be a constant if and only ifm(t)=(t2+t+)−1=2. In this case, Q(x)=−J2(−42). 2. This transformation is useful when followed by some sort of asymp- totic analysis. When the magnitude of is large compared to Q(x), then the rst order approximation to equation (32.2) will be to dis- card theQ(x)t e r m . 3. See Magnus and Winkler [1, page 51]. Reference [1]Magnus, W., and Winkler, S. Hill’s Equation . Dover Publications, Inc., New York, 1966. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 146 I.B Transformations 33. Reduction of Linear ODEs to a First Order System Applicable to Linear ordinary dierential equations. Yields A rst order vector system. Idea By introducing variables to represent the derivatives in an nth order linear ordinary dierential equation, a rst order system of dierential equations may be obtained. Procedure Given the linear ordinary dierential equation dny dxn=an−1(x)d(n−1)y dx(n−1)+


+a1(x)dy dx+a0(x)y+b(x) (33.1) fory(x), introduce the variables fz1;z2;:::;zngdened by z1=dy dx;z 2=d2y dx2; :::; z n=dny dxn: Using these new variables, equation (33.1) may be written as d dxy=A(x)y+b(x); (33.2) where y= yy(1)::: y(n−1)T=yz 1z2::: zn−1T; b=00::: 0b(x)T; andAis the matrix 2 666666640100


0 0010


0 0001 0 ............ 0000 1 a 0(x)a1(x)a2(x)a3(x)::: an−1(x)3 77777775: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 33. Reduction of Linear ODEs to a First Order System 147 If the initial conditions for equation (33.1) were in the form y(x0)=c0;y0(x0)=c1;y00(x0)=c2;:::;y(n−1)(x0)=cn−1; then the initial condition for equation (33.2) is y(x0)= c0c1::: cn−1T. To solve an equation in the form of equation (33.2), see the section on vector ordinary dierential equations (page 421). Example Given the linear ordinary dierential equation with initial conditions d2y dx2+x2dy dx+( l o gx)y=s i nx; y(0) = 3;y0(0) = 4; it may easily be changed into the equivalent rst order system d dx y y0 = 01 −logx−x2 y y0 + 0 sinx ; or, equivalently, dy dt=A(x)y+b; where y=y y0 ,A=01 −logx−x2 ,a n d b=0 sinx . Notes 1. Many packaged computer programs require the input to be in the form of a rst order vector system. 2. The method of elimination is the opposite of the method presented here. In the method of elimination, a system of simultaneous equa- tions is converted into a single equation of higher order. See Finizioand Ladas [2, pages 162{170] for details. 3. See also Bronson [1, pages 185{192]. References [1]Bronson, R. Modern Introductory Dierential Equations . Schaum’s Outline Series. McGraw{Hill Book Company, New York, 1973. [2]Finizio, N., and Ladas, G. Ordinary Dierential Equations with Modern Applications . Wadsworth Publishing Company, Belmont, CA, 1982. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 148 I.B Transformations 34. Pr¨ ufer Transformation Applicable to Linear, homogeneous, second order dierential equa- tions. Yields An equivalent system of two rst order dierential equations. Idea This transformation changes an equation from Liouville normal form to two successive ordinary dierential equations. Procedure Suppose we have the Sturm{Liouville equation d dx P(x)du dx +Q(x)u=0; (34.1) dened on a<x<b ,w i t hP> 0;P2C1;andQcontinuous. If we think of this single second order equation as two rst order equations for the unknownsfu;u0g, then we can change the dependent variables from fu;u0gtoR(x)a n d(x)b y P(x)u0(x)=R(x)c o s(x); u(x)=R(x)s i n(x): (34.2) Using (34.2) in equation (34.1), we obtain two sequential rst order ordi- nary dierential equations for the unknowns R(x)a n d(x) d dx=Q(x)s i n2+1 P(x)cos2; dR dx=1 P(x)−Q(x) R(x)s i n2:(34.3.a-b) If equation (34.3.a) can be integrated, then equation (34.3.b) can be solved for R(x)=R(a)e x pZx a1 P(t)−Q(t) sin 2(t)dt : (34.4) Example If we have the linear second order homogeneous ordinary dierential equation xu00−u0+x3u=0; (34.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 34. Pr¨ ufer Transformation 149 then we can write equation (34.5) in Liouville normal form as d dx1 xu0 +xu=0; from which we can identify P(x)=1=x,Q(x)=x. Therefore, from equation (34.3.a), we have d dx=xsin2+1 1=xcos2 =x: This equation can be solved to yield (x)=x2 2+C,w h e r eCis an arbitrary constant. From equation (34.4), we then nd R(x)=R(a). Therefore, we conclude that u(x)=R(a)s i nx2 2+C =u(a)sin(x2=2+C) sin(a2=2+C) is the solution to equation (34.5). Notes 1. The Pr¨ ufer transformation is often used to obtain information about the zeros of u(x). 2. See also Birkho and Rota [5, pages 257{266]. References [1]Adamov a, D., Holrejs i, J., and Ulehla, I. The Atkinson{Prufer transformation and the eigenvalue problem for coupled systems of Schr odinger equations. J. Phys. A: Math. Gen. 17 (1984), 2621{2631. [2]Bailey, P. B. Sturm{Liouville eigenvalues via a phase function. SIAM J. Appl. Math. 14 (1966), 242{249. [3]Bellman, R., and Wing, G. M. An Introduction to Invariant Imbedding . SIAM, Philadelphia, PA, 1992. [4]Benson, D. C. A Prufer transformation for Lienard’s equation. SIAM J. Numer. Anal. 15 , 4 (July 1984), 656{669. [5]Birkhoff, G., and Rota, G.-C. Ordinary Dierential Equations .J o h n Wiley & Sons, New York, 1978. [6]Ulehla, I., and Horejsi, J. Generalized Prufer transformation and the eigenvalue problem for radial Dirac equations. Phys. Lett. A 113 , 7 (1986), 355{358. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 150 I.B Transformations 35. Modied Pr¨ ufer Transformation Applicable to Linear, homogeneous, second order ordinary dier- ential equations. Yields An equivalent system of two rst order ordinary dierential equations. Idea This transformation changes an equation from Liouville normal form to two successive ordinary dierential equations. Procedure Suppose we have an ordinary dierential equation in Liouville normal form u00+Q(x)u=0; (35.1) dened ona<x<b ,w i t hQ> 0. We dene the modied amplitude R(x) and the modied phase (x)b y u(x)=R(x) Q1=4sin(x); u0(x)=R(x)Q1=4cos(x):(35.2.a-b) Using equation (35.2) in equation (35.1), we determine the modied Pr¨ ufer system corresponding to equation (35.1) to be d dx=−Q1=2−1 4Q0 Qsin 2; 1 RdR dx=1 4Q0 Qcos 2:(35.3) The modied Pr¨ ufer transformation is usually used to obtain asymp- totic information about the solution to equation (35.1). Example Ifu(x) satises u00+ 1−M x2 u=0; (35.4) for 0<x<1, then the exact solution is u(x)=pxZn(x), whereZn(x) is a Bessel function and n=q M+1 4. Comparing equation (35.4) to CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 35. Modied Pr¨ ufer Transformation 151 equation (35.1), we identify Q(x)=1−M x2, so that equation (35.3) becomes d dx=−r 1−M x2+Msin 2 2(x3−Mx); 1 RdR dx=−Mcos 2 2(x3−Mx): ForM=O(1) andx1, the above expressions can be expanded to yield d dx’−1−1 2M x2+O1 x3 ; 1 RdR dx’O1 x3 ; which can be integrated (and then simplied) to yield (x)’1−x−M 2x+O1 x2 ; R(x)’R1+O1 x2 :(35.5) Using equation (35.5) and Q(x) in equation (35.2.a) provides an approxima- tion tou(x) for large values of x. This, in turn, provides an approximation to thenth Bessel function. Notes 1. The modied Pr¨ ufer transformation is often used with Q(x)=− q(x)w h e nis large in magnitude compared to q(x). 2. See also Birkho and Rota [1, pages 267{277]. References [1]Birkhoff, G., and Rota, G.-C. Ordinary Dierential Equations .J o h n Wiley & Sons, New York, 1978. [2]Hargrave, B. A. Numerical approximation of eigenvalues of Sturm{ Liouville systems. J. Comput. Physics 20 (1976), 381{396. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 152 I.B Transformations 36. Transformations of Second Order Linear ODEs { 1 Applicable to The second order linear ordinary dierential equa- tion y00+a(x)y0+b(x)y=0: (36.1) Transformation 1 If the dependent and independent variables in equation (36.1) are changed by t=Zx x0exp −Zr x0a(z)dz dr; w(t)=y(x); then equation (36.1) becomes d2w dt2+b(x(t)) exp −2Zx x0a(z)dz w=0: (36.2) Example For the ordinary dierential equation y00−3x 1−x2y0+7 1−x2y=0; the change of variables becomes t=x=p 1−x2and the equation corre- sponding to equation (36.2) isd2w dt2+7 (1+t2)2w=0 . Transformation 2 If in equation (36.1) the expression b0+2ab b3=2(36.3) is found to be a constant, then the change of independent variable given by z=CZp b(x)dx; (36.4) whereCis an arbitrary constant, will reduce equation (36.1) to an equation with constant coecients. Moreover, if the expression in equation (36.3)is not constant, then no change of independent variable alone will reduce equation (36.1) to an equation with constant coecients. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 36. Transformations of Second Order Linear ODEs { 1 153 Example Given the equation xy00+( 8x2−1)y0+2 0x3y=0; (36.5) we note that a(x)=8x−1=xandb(x)=2 0x2. Hence, the expression in equation (36.3) becomes b0+2ab b3=2=40x+4 0x2(8x−x−1) 203=2x3=320x3 203=2x3= constant: Therefore, if the independent variable is changed by z=CRp 20xdx, then equation (36.5), written in terms of z, will be a constant coecient dierential equation. A natural choice for CisC=2=p 20 so that the transformation becomes z=x2. Using this new variable in equation (36.5) results in the equation d2y dz2+4dy dz+5y=0; which has the solution y=e−2z(Acosz+Bsinz), whereAandBare arbitrary constants. Hence, the general solution to equation (36.5) is y=/parenleftbig Acosx2+Bsinx2
exp/parenleftbig −2x2
: Transformation 3 If the dependent variable is changed by y(x)=u(x)e x p −1 2Zx a(z)dz ; then equation (36.1) becomes u00+I(x)u=0; (36.6) where I(x)= b−1 4a2−1 2da dx : (36.7) Equation (36.6) is said to be the normal form for equation (36.1). The quantityI(x)i st h e invariant of equation (36.1). Two ordinary dierential equations that have the same normal form (i.e.,I(x) is the same) are said to be equivalent . This is because if y1(x) andy2(x)s a t i s f y y00 1+p1y0 1+q1y1=0; y00 2+p2y0 2+q2y2=0;(36.8) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 154 I.B Transformations and if both equations have the same invariant, then y1(x)=y2(x)e x p −1 2Zx p1(z)−p2(z) dz : (36.9) Conversely, if y1andy2are solutions to equation (36.8), and if y1(x)= f(x)y2(x)f o rs o m ef(x), then the invariants of the two equations in equa- tion (36.8) are the same. Example Suppose we wish to solve the equation d2y dx2−2 xdy dx+ a2+2 x2 y=0; (36.10) in whichais a constant. We nd that (comparing equation (36.10) with equation (36.1), and using equation (36.7)) I(x)= a2+2 x2 −1 44 x2−1 22 x2=a2: Now, we know the solution of d2v dx2+a2v= 0 (36.11) to bev(x)=Acosax+Bsinax,w h e r eAandBare arbitrary constants. Because equations (36.10) and (36.11) have the same invariant, one can betransformed into the other. Using equation (36.9), we nd y(x)=v(x)e x pZdx x =xv; and, hence, the solution of equation (36.10) is y(x)=Axcosax+Bxsinax. Transformation 4 If, instead of equation (36.1), both sides of y00+a(x)y0+b(x)y=c(x) (36.12) are multiplied by p(x)=e x pZx x0a(z)dz ; then equation (36.12) is put in the formally self-adjoint form d dx p(x)dy dx +q(x)y=r(x); (36.13) where q(x)=p(x)b(x); r(x)=p(x)c(x): See the method on page 157 for transformations of an equation in the form of equation (36.13). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 36. Transformations of Second Order Linear ODEs { 1 155 Transformation 5 Fernandez et al. [3] suggest transformatng equation (36.1) via y(x)=pxzexp/parenleftbig −R Q(x)dx
Y(z)w i t hx=x(z). This results in the equation Yzz+R(z)Y=0 ,w h e r e R(z)=(xz)2 b+xzzz 2(xz)3−3(xzz)2 4xz−a0 2−a2 4 : Example Suppose we wish to solve the equation (1−x2)y00−γxy0+y=0: Usinga=−γx=(1−x2),b==(1−x2), andx(z)=−coszresults in Yzz+R(z)Y=0w i t hR(z)=+(γ−1)2 4−(γ−1)(γ−3) 4s i n2z. Notes 1. Note that the invariant of the adjoint of equation (36.1) is equal to the invariant of equation (36.1). That is to say, invariants are preservedunder the operation of taking the adjoint. 2. If equation (36.6) has the two linearly independent solutions u(x)a n d v(x) and if we dene s(x): =u(x)=v(x), thenfs;xg=2I(x), where f;gdenotes the Schwarzian derivative. 3. Kamran and Olver [6] completely solve the equivalence problem, that is, determining when two second order linear dierential operators are the same under a change of variable. 4. See also Boyce and DiPrima [2, pages 141{143], Hill [4, pages 42{43], Ince [5, page 394], Murphy [7, pages 88{89], Piaggio [8, pages 91{92], and Rainville [9, pages 7{10 and 15{23]. References [1]Berkovich, L. M. Canonical forms of ordinary linear dierential equations. Arch. Math. (BRNO) 24 , 1 (1988), 25{42. [2]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [3]Fernandez, F. M., Pi ~neiro, A. L., and Moreno, B. Alternative factorization of eigenvalue problems in one dimension. J. Phys. A: Math. Gen. 27 (1994), 5013{5028. [4]Hill, J. M. Solution of Dierential Equations by Means of One-Parameter Groups . Pitman Publishing Co., Marsheld, MA, 1982. [5]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [6]Kamran, N., and Olver, P. J. Equivalence of dierential operators. SIAM J. Math. Anal. 20 , 5 (September 1989), 1172{1185. [7]Murphy, G. M. Ordinary Dierential Equations and Their Solution .D .V a n Nostrand Company, Inc., New York, 1960. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 156 I.B Transformations [8]Piaggio, H. T. H. An Elementary Treatise on Dierential Equations and Their Applications . G. Bell & Sons, Ltd, London, England, 1926. [9]Rainville, E. D. Intermediate Dierential Equations . The MacMillan Company, New York, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 37. Transformations of Second Order Linear ODEs { 2 157 37. Transformations of Second Order Linear ODEs { 2 Applicable to The second order linear ordinary dierential equa- tion in formally self-adjoint form L[y]: =d dx p(x)dy dx +q(x)y=0: (37.1) Transformation 1 If the independent variable in equation (37.1) is changed from xtosby s=Zdx p(x), and ifp(x)>0f o rx>x 0,a n dZ1 x0dx p(x)=1,t h e ne q u a t i o n (37.1) becomes d2y ds2+p(x)q(x)y=0: Note that, as x!1 ,w eh a v es!1 . See Courant and Hilbert [1, page 292]. Example For the ordinary dierential equation ( xy0)0+y= 0, we identify p(x)= x,q(x) = 1, andx0= 0. Hence, the change of variable s=l o gxresults in yss+esy=0 . Transformation 2 If the dependent variable in equation (37.1) is changed from y(x)t o w(x)b y w(x)=p p(x)y(x); then equation (37.1) becomes d2w dx2+" q p−1 2d dxp0 p −1 4p0 p2# w=0: Transformation 3 If the range of interest for equation (37.1) is x0<x<1and if the independent and dependent variables are changed by t=Zx x0s jq(z)j p(z)dz; u(t)=[p(x)jq(x)j]1=4y(x); CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 158 I.B Transformations then equation (37.1) becomes d2u dt2+[1+R(t)]u(t)=0; (37.2) where R(t)=p1=4jqj−3=4d dx[p(x)jq(x)j]−1=4 x=x(t); and the plus (minus) sign is taken in equation (37.2) if q(x)>0(q(x)<0). This transformation is sometimes called the Liouville{Green transfor- mation. This transformation is virtually identical to the Liouville trans- formation (see page 141). See Courant and Hilbert [1, page 292], Eastham[2], and Lakin and Sanchez [3, pages 36{41]. Transformation 4 If the independent and dependent variables are changed in equation (37.1) by y(x)=(x)w(t); t=Zx (z)dz; then equation (37.1) becomes  d dt p2dw dt +L[]w=0: (37.3) Note that the operator L[
] is dened by equation (37.1). If (z)i sc h o s e n to be (z)=1 p(z)2(z); then equation (37.3) simplies to1 p3d2w dt2+L[]w= 0. See Courant and Hilbert [1, page 292]. References [1]Courant, R., and Hilbert, D. Methods of Mathematical Physics . Interscience Publishers, Inc., New York, 1953. [2]Eastham, M. S. P. The Liouville{Green asymptotic theory for second- order dierential equations: A new approach and extensions. In Ordinary Dierential Equations and Operators , W. N. Everitt and R. T. Lewis, Eds., no. 1032 in Lecture Notes in Mathematics. Springer{Verlag, New York, 1983. [3]Lakin, W. D., and Sanchez, D. A. Topics in Ordinary Dierential Equations . Dover Publications, Inc., New York, 1970. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 38. Transformation of an ODE to an Integral Equation 159 38. Transformation of an ODE to an Integral Equation Applicable to Second order linear ordinary dierential equations. Yields An equivalent integral equation. Idea An ordinary dierential equation may sometimes be formulated as an integral equation. Procedure There is a standard transformation that will allow a linear second order initial value ordinary dierential equation to be written as a Volterra integral equation. Given the dierential equation with initial conditionsfory(x), d 2y dx2+A(x)dy dx+B(x)y=g(x); y(a)=; y0(a)=; an equivalent Volterra integral equation is y(x)=f(x)+Zx aK(x;)y()d; where f(x)=Zx a(x−)g()d+(x−a) A(a)+ +; K(x;)=(−x) B()−A0() −A(): There is also a standard transformation that will allow a linear second order boundary value ordinary dierential equation to be written as a Fredholm integral equation. Given the dierential equation and boundaryconditions for w(x), d 2w dx2+C(x)dw dx+D(x)w=j(x); w(a)=γ; w (b)=; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 160 I.B Transformations an equivalent Fredholm integral equation is w(x)=h(x)+Zb aH(x;)w()d; where h(x)=γ+Zx a(x−)j()d+x−a b−a" −γ−Zb a(b−)j()d# ; H(x;)=8 >>< >>:x−b b−ah C()−(a−) C0()−D()i ;forx>; x−a b−ah C()−(b−) C0()−D()i ;forx<: Example Ify(x) satises y00+y=x; y(0) = 0;y0(0) = 0; (38.1) theny(x) satises the following Volterra integral equation y(x)=x3 6+Zx 0(−x)y()d: (38.2) The solution to equation (38.1), y=x−sinx, satises equation (38.2). Notes 1. There are many other ways in which an ordinary dierential equation may be transformed into an integral equation. For example, if y(x) satises the nth order ordinary dierential equation y(n)(x)=f(x)+nX j=1Cj(x)y(j−1)(x) andu(x): =y(n)(x), thenu(x) satises the integral equation u(x)=F(x)+Zx aK(x;t)u(t)dt; K(x;t)=nX j=1Cj(x)(t−x)j−1 (j−1)!; whereF(x)i sf(x) plus a polynomial in ( x−a) generated by the initial conditions. See Squire [3, pages 223{227] for more details on this technique, as well as two other techniques. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 38. Transformation of an ODE to an Integral Equation 161 2. Bose [1] shows that every solution of the nth order linear homoge- neous dierential equation y(n)=an−1(x)y(n−1)+


+a0(x)y satises the integral equation y(x)=y(x0)+Zx x0h(u)du+Zx x0Zu x0G(u;v)a0(v)y(v)dv du; whereh(x) is the unique solution to h(n−1)=an−1(x)h(n−2)+


+a1(x)h; (38.3) h(x0)=y0(x0);h0(x0)=y00(x0);


;h(n−2)(x0)=y(n−1)(x0); andG(x;u) is the Green’s function associated with equation (38.3). 3. See also Jerri [2, pages 60{67]. References [1]Bose, A. K. An integral equation associated with linear homogeneous dierential equations. Int. J. Math. &M a t h .S c i .9 , 2 (1986), 405{408. [2]Jerri, A. J. Introduction to Integral Equations with Applications . Marcel Dekker, New York, 1985. [3]Squire, W. Integration for Engineers and Scientists . American Elsevier Publishing Company, New York, 1970. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 162 I.B Transformations 39. Miscellaneous ODE Transformations Applicable to Ordinary dierential equations. Procedure Many transformations have been developed for equations of specic forms. Transformation 1 Ify(x) is dened by the ordinary dierential equation d2y dx2=f(x)y; (39.1) and the dependent variable is changed by w()=p 0(x)y(x); (39.2) (for arbitrary =(x), orx=x()), then equation (39.1) becomes d2w d2= _x2f(x)+p _xd2 d2 _x−1=2 w; = _x2f(x)−1 2fx;g w;(39.3) where dots denote dierentiation with respect to ,a n dfx;gis the Schwarzian derivative of xwith respect to .I fw ec h o o s e (x)b y (x)=Zxp f(z)dz; (39.4) so thatw()=y(x)f1=4(x), then equation (39.3) becomes d2w d2=[ 1+()]w; (39.5) with ()=4ff00−5(f0)2 16f3=−1 f3=4d2 dx21 f1=4 : This is called the Liouville transformation by Olver [7, Chapter 6], and the Liouville{Green transformation by Lakin and Sanchez [6, pages 36{41]. Byneglecting() in equation (39.5) and solving for w(), we obtain the rst term in the WKB approximation (see page 642). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 39. Miscellaneous ODE Transformations 163 Example If we apply this transformation to Airy’s equation, y00=xy,f o rx>0, then we nd (using f(x)=x) (x)=Zxpzdz=2 3x3=2; w()=p 0(x)y(x)=x−1=4y(x): And so equation (39.5) becomes d2w d2− 1+5 362 w=0: This leads to the approximation w00−w= 0 when1 (which corresponds tox1). Transformation 2 This transformation removes the ( n−1)th derivative term in an nth order ordinary dierential equation. If y(x) satises (−1)n(py(n))(n)+L[y]=qy; (39.6) for 0x1, whereL[y] is a linear dierential operator of degree less than or equal to 2 n−2 and if the dependent and independent variables are changed from y(x)t ow(t)b y w(t)=(q2n−1p)1=4ny(x); t=1 KZx 0q p1=2n dx; K=Z1 0q p1=2n dx; then equation (39.6) is transformed into d2nw dt2n+H[w]=K2nw; whereH[w] is another linear dierential operator of degree less than or equal to 2n−2. See Boyce [1, page 21]. Transformation 3 The general third order linear homogeneous ordinary dierential equa- tion y000+p1(x)y00+p2(x)y0+p3(x)y=0; can be changed to the canonical form w000+2Aw0+(A0+b)w=0; (39.7) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 164 I.B Transformations by the change of variables w(x)=y(x)e x p −Zx x0p1(t)dt : If we write P2=p2−p2 1−p0 1; P3=p3−3p1p2+2p3 1−p00 1; thenA(x)a n db(x) may be written as A(x)=3 2P2; b(x)=P3−3 2P0 2: See Gregu s [3] for details. Transformation 4 The general fourth order linear homogeneous ordinary dierential equa- tion A(x)y0000+B(x)y000+C(x)y00+D(x)y0+E(x)y=0; fory(x) can be changed to the canonical form w0000+a(t)w00+b(t)w0+c(t)w=0; forw(t), by the transformation w(t)=(x)y(x);t =(x); wheref(x);(x)gare chosen to satisfy 03=e x p −1 2Zx x0B(z) A(z)dz : Notes 1. If the transformation given by equation (39.2) is applied to the equa- tion d2y dx2=[f(x)+g(x)]y; withdened by equation (39.4), then we obtain d2w d2= 1++g f w: 2. The dierential equation adjoint to equation (39.7) has the form: z000+2Az0+(A0−b)z= 0. Hence, the equation in equation (39.7) will be self-adjoint if and only if b(x)=0 . CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 39. Miscellaneous ODE Transformations 165 3. Olver [7, pages 190{192] proves that any one-dimensional, rst order Hamiltonian dierential operator can be put into constant coecient form by a suitable change of variables. 4. See also Hill [5, pages 44{45]. References [1]Boyce, W. E. Random eigenvalue problems. In Probabilistic Methods in Applied Mathematics , A. T. Bharucha-Reid, Ed. Academic Press, New York, 1968, pp. 1{73. [2]Gonzalez-Lopez, A. On the linearization of second-order ordinary dier- ential equations. Lett. Math. Phys. 17 , 4 (1989), 341{349. [3]Gregu s, M. Third Order Linear Dierential Equations . D. Reidel Publishing Co., Boston, MA, 1987. [4]Grissom, C., Thompson, G., and Wilkens, G. Linearization of second order ordinary dierential equations via Cartan’s equivalence method. J. Dif- ferential Equations 77 , 1 (1989), 1{15. [5]Hill, J. M. Solution of Dierential Equations by Means of One-Parameter Groups . Pitman Publishing Co., Marsheld, MA, 1982. [6]Lakin, W. D., and Sanchez, D. A. Topics in Ordinary Dierential Equations . Dover Publications, Inc., New York, 1970. [7]Olver, P. J. Darboux theorem for Hamiltonian dierential operators. J. Dierential Equations 71 (1988), 10{33. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 166 I.B Transformations 40. Reduction of PDEs to a First Order System Applicable to Nonlinear partial dierential equations. Yields A rst order system of partial dierential equations. Idea By introducing variables to represent the derivatives in a partial dier- ential equation, a rst order system may be obtained. Procedure Sometimes it is advantageous to reduce a partial dierential equation of high order for a single unknown function to a system of several rst order equations. This might be done, for instance, to utilize a specic numerical package that requires a partial dierential equation to be input as a rstorder system. This can always be done by introducing an appropriate set of derivatives as unknowns. The general procedure is to introduce new variables as the derivatives of the desired function and then \discover" relations among these functions. The following derivation for second order equations is from Garabedian [1]. Suppose we have the second order partial dierential equation, with boundary conditions u xx=G(x;y;u;ux;uy;uxy;uyy); u(0;y)=f(y); ux(0;y)=g(y);(40.1) for the unknown u(x;y). We introduce new variables, fu1;:::;u 8g, which are assumed to depend upon the new independent variables and,b y the denitions u1=x; u 4=ux;u 7=uxy; u2=y; u 5=uy;u 8=uyy; u3=u; u 6=uxx: If we then specify the new independent variables by requiring @u1 @=@u2 @;@u2 @=0; u1(0;)=0;u 2(0;)=; thenu1=x=andu2=y=. The purpose of introducing these new independent variables is to eliminate explicit dependence on xandy. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 40. Reduction of PDEs to a First Order System 167 With these new variables, equation (40.1) can be written as the system @u1 @=@u2 @;@u2 @=0;@u3 @=u4@u2 @;@u4 @=u6@u2 @; @u5 @=@u4 @;@u7 @=@u6 @;@u8 @=@u7 @: @u6 @=Gx@u2 @+u4Gu@u2 @+u6Gux@u2 @+Guy@u4 @+Guxy@u6 @+Guyy@u7 @:(40.2) Most of the above equations are consistency requirements; that is, ( ux)y= (uy)ximplies that ( u5)=(u4). The initial conditions for the variables fu1;:::;u 8gare given by u1(0;)=0; u2(0;)=; u3(0;)=f(); u4(0;)=g(); u5(0;)=f0(); u6(0;)=G(0;;f();g();f0();g0();f00()); u7(0;)=g0(); u8(0;)=f00():(40.3) Note that equation (40.2) is in the general form of a linear rst order system @uj @=8X k=1ajk(u1;:::;u 8)@uk @; forj=1;2;:::; 8. To convert the system in equation (40.2) back to the system in equation (40.1) may require the use of the boundary conditions in equation (40.3). Note 1. Systems of high order partial dierential equations can also be made into rst order systems by the introduction of enough terms. Forinstance, the system of equations for u(x;y)a n dv(x;y) F 1(x;y;u;ux;uy;v;vx;vy)=0 F2(x;y;u;ux;uy;v;vx;vy)=0 can be written as a rst order system, but the resulting system has 12 dependent variables. See Garabedian [1, pages 7{11] for details. Reference [1]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 168 I.B Transformations 41. Transforming Partial Dierential Equations Applicable to Partial dierential equations. Idea Changing variables in a partial dierential equation is a straightforward process. Procedure 1 The general procedure is simple: Construct a new function, which depends upon new variables, and then dierentiate with respect to the old variables to see how the derivatives transform. Procedure 2 If a dierential equation can be written in terms of coordinate-free expressions (e.g., in terms of the gradient operator), then a change ofvariables can be avoided by simply using the metric of the new coordinate system. This section contains representations of common coordinate-free expressions for an orthogonal coordinate system. Note that Moon and Spencer [4] list the metric coecients for 43 dierent orthogonal coordinate systems. (These consist of 11 general systems, 21 cylindrical systems, and11 rotational systems.) In an orthogonal coordinate system, let fa igdenote the unit vectors in each of the three coordinate directions, and let fuigdenote distance along each of these axes. The coordinate system may be designated by the metric coecientsfg11;g22;g33g, dened by gii=@x1 @ui2 +@x2 @ui2 +@x3 @ui2 ; (41.1) wherefx1;x2;x3grepresent rectangular coordinates. Using the metric coecients dened in equation (41.1), we dene g=g11g22g33. Whenrepresents a scalar and E=E1a1+E2a2+E3a3represents a vector, we have grad=r=a1pg11@ @u1+a2pg22@ @u2+a3pg33@ @u3; (41.2) divE=r
E =1pg@ @u1gE1 g11 +@ @u2gE2 g22 +@ @u3gE3 g33 ;(41.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 41. Transforming Partial Dierential Equations 169 curlE=rE=a1Γ1pg11+a2Γ2pg22+a3Γ3pg33; (41.4) r2=1pg@ @u1pg g11@ @u1 +@ @u2pg g22@ @u2 +@ @u3pg g33@ @u3 ; =1 h1h2h3@ @u1h2h3 h1@ @u1 +@ @u2h3h1 h2@ @u2 +@ @u3h1h2 h3@ @u3 ;(41.5) grad div E=r(r
E)=a1pg11@ @x1+a2pg22@ @x2+a3pg33@ @x3; (41.6) curl curl E=r(rE) =a1rg11 g@Γ3 @x2−@Γ2 @x3 +a2rg22 g@Γ1 @x3−@Γ3 @x1 +a3rg33 g@Γ2 @x1−@Γ1 @x2 ;(41.7) 45E= graddiv E−curl curl E =r(r
E)−r(rE) =a11pg11@ @x1+rg11 g@Γ2 @x3−@Γ3 @x2 +a21pg22@ @x2+rg22 g@Γ3 @x1−@Γ1 @x3 +a31pg33@ @x3+rg33 g@Γ1 @x2−@Γ2 @x1 ;(41.8) where  and Γ=( Γ 1;Γ2;Γ3) are dened by =1pg@ @x1 E1rg g11 +@ @x2 E2rg g22 +@ @x3 E3rg g33 ; Γ1=g11pg@ @x2(pg33E3)−@ @x3(pg22E2) ; Γ2=g22pg@ @x3(pg11E1)−@ @x1(pg33E3) ; Γ3=g22pg@ @x1(pg22E2)−@ @x2(pg11E1) :(41.9) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 170 I.B Transformations Operations for orthogonal coordinate systems are sometimes written in terms offhigfunctions, instead of the fgiigterms. Here, hi=pgii,s o thatpg=h1h2h3. For example Cylindrical Polar Coordinates x1=rcos; x 2=rsin; x 3=z g1=1;g 2=r2;g 3= 1 (41.10) Elliptic Cylinder Coordinates x1=u1u2;x 2=q (u2 1−c2)(1−u2 2);x 3=u3 g1=u2 1−c2u2 2 u2 1−c2;g 2=u2 1−c2u2 2 1−u2 2;g 3=1 Example 1 Suppose we have the equation fxx+fyy+xfy=0; (41.11) and we would like to transform the equation from the fx;ygvariables to thefu;vgvariables, where u=x; v =x y: Note that the inverse transformation is given by x=u,y=u=v. We deneg(u;v) to be equal to the function f(x;y)w h e nw r i t t e ni n the new variables. That is, f(x;y)=g(u;v)=g x;x y : (41.12) Now we create the needed derivative terms, carefully applying the chain rule. For example, by dierentiating equation (41.12) with respect to x, we obtain fx(x;y)=gu@ @x(u)+gv@ @x(v) =g1@ @x(x)+g2@ @xx y =g1+g21 y =g1+v ug2; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 41. Transforming Partial Dierential Equations 171 where we have used a subscript of \1" (\2") to indicate a derivative with respect to the rst (second) argument of the function g(u;v) (i.e.,g1(u;v)= gu(u;v)). Use of this \slot notation" tends to minimize errors. In a like manner, we nd fy(x;y)=gu@ @y(u)+gv@ @y(v) =g1@ @y(x)+g2@ @yx y =−x y2g2 =−v2 ug2: The second order derivatives can be calculated similarly: fxx(x;y)=@ @x(fx(x;y)) =@ @x g1+1 yg2 =g11+2v ug12+v2 u2g22; fxy(x;y)=@ @x −x y2g2 =−u2 v2g2−u3 v3g12−u2 v2g22; fyy(x;y)=@ @y −x y2g2 =2v3 u2g2+v4 u2g22: Finally, then, we can determine what equation (41.11) looks like in the new variables: 0=fxx+fyy+xfy = g11+2v ug12+v2 u2g22 +2v3 u2g2+v4 u2g22 +(u) −v2 ug2 =v2(2v−u2) u2gv+guu+2v uguv+v2(1 +v2) u2gvv: Example 2 As a simple example of using coordinate-free representations, consider the diusion equation in rectilinear coordinates: ut=(uxx+uyy+uzz): (41.13) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 172 I.B Transformations We recognize this to be the same as ut=r2u. Hence, using equation (41.10) in equation (41.5) we nd ut=r2u=@2u @r2+1 r@u @r+1 r2@2u @2+@2u @z2 : in cylindrical polar coordinates. Notes 1. A Macsyma program that will perform changes of variables in partial dierential equations is described in Steinberg [5]. 2. Mathematica has the package VectorAnalysis which can compute the divergence, curl, gradient, Laplacian, and the biharmonic opera- tor (r4) in 14 dierent coordinate systems. 3. The Laplacian ( r2) for 22 dierent coordinate systems is given start- ing on page 204. 4. See also Butkov [1, pages 34{39] and Moon and Spencer [3, Chapter 3]. References [1]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [2]Harper, D. Vector 33: A REDUCE program for vector algebra and calculus in orthogonal curvilinear coordinates. Comput. Physics Comm. 54 (1989), 295{305. [3]Moon, P., and Spencer, D. E. Field Theory for Engineers .D . V a n Nostrand Company, Inc., New York, 1961. [4]Moon, P., and Spencer, D. E. Field Theory Handbook . Springer{Verlag, New York, 1961. [5]Steinberg, S. Change of variables in partial dierential equations. Tech. rep., University of New Mexico, Albuquerque, New Mexico, 1983. Departmentof Mathematics preprint (AD-A214 702). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 42. Transformations of Partial Dierential Equations 173 42. Transformations of Partial Dierential Equations Applicable to Partial dierential equations. Procedure Many transformations have been developed for equations of specic forms. Euler Transformation Given the rst order partial dierential equation in two independent variables,F(x;y;z;p;q )=0( w i t h ,a su s u a l , p=zx,q=zy)a n dzxx6=0 the transformation 8 >>>>>>< >>>>>>:x=Z X y=Y z=XZX−Z p=X q=−ZY9 >>>>>>= >>>>>>;()8 >>>>>>< >>>>>>:X=z x Y=y Z=xzx−z P=x Q=−zy9 >>>>>>= >>>>>>;; (42.1) is known as the Euler transformation. Note that Z Y+zy= 0. Under this transformation, the original equation transforms into F(ZX;Y;XZX− Z;X;−ZY) = 0 (see Kamke [6, section 11.15, pages 100{101]). As an example, the equation G(xp−z;y;p;q ) = 0 becomes, under the Euler transformation, G(Z;Y;X;−ZY) = 0. As another example, the Clairaut partial dierential equation F=z−(xzx+yzy+f(zx;zy)) = 0 is transformed into F=Z−YZY+f(X;−ZY) = 0. Note that this latter equation is really an ordinary dierential equation for Z=Z(Y) (the variable Xacts as a parameter). Kircho Transformation Given the elliptic partial dierential equation div[K( ) grad ]=r
[K( )r ]=0; (42.2) for = (x), the Kircho transformation introduces the new dependent variable, ( x), dened by  =Z 0K(t)dt,w h e r e 0is some arbitrary reference value. This transforms equation (42.2) into Laplace’s equation r2 = 0; see Ames [1, pages 6{7 and 21{23]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 174 I.B Transformations Transformations of Parabolic Dierential Equations I The parabolic partial dierential equation ut=2uxx−ux+u; wheref;;gare constants, may be transformed into the simple diusion equationt=2xx, by means of the transformation (see Bateman [2, pages 75{79] or Farlow [3, page 58]) u(x;t)=(x;t)e x p 22x+ −2 42 t : (42.3) Transformations of Parabolic Dierential Equations II The nonlinear parabolic partial dierential equation ct=(D(c)cx)x may be transformed, via v(c;t)=D(c)cx, into the following equation with a simpler nonlinearity (see Hill [5, page 148]): D(c)vt=v2vcc: Transformation of Elliptic/Hyperbolic Equations The linear partial dierential equation (x)@2u @x2+(x)@u @x+γ(x)u=a@u @t+b@2u @t2(42.4) may be transformed into the equation c(X)@ @X1 c(X)@v @X =a@v @t+b@2v @t2: Through the transformation X=Zxdp j()jv(X;t)=u(x;t) u0(x); whereu0(x) is any nonzero \equilibrium" solution of (42.4), and c(X)i s a function completely determined by f(x);(x);γ(x)g. See Varley and Seymour [9]. Removing First Derivative Terms Linear elliptic equations and hyperbolic equations of second order, all of whose coecients of the derivative terms are constants, can be transformed so that the rst derivative terms no longer appear. For example, we presume that u(x) satises nX k=1k@2u @xk2+nX k=1bk@u @xk+c(x)u=0: (42.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 42. Transformations of Partial Dierential Equations 175 Note that scaling of the fxkgallows equation (42.5) to be written with eachfkge q u a lt o0 ,1 ,o r−1. If we presume that no kis equal to zero, and we dene w(x)=u(x)e x p" 1 2nX k=1bk k xk# ; thenw(x) satises (see Garabedian [4, pages 74{75]) nX k=1k@2w @xk2+ c(x)−1 4nX i=1b2 k ! w=0: Von Mises Transformation For fluid flow with constant viscosity, the Navier{Stokes equations (see page 179) sometimes take the form u@u @x+v@v @y=@2u @y2; @u @x+@v @y=0:(42.6.a-b) These are called the boundary layer equations . A standard procedure for analyzing the Navier{Stokes equations (and equations derived from them) is to introduce the stream function Ψ, dened by u=@Ψ @y;v =−@Ψ @x: With this denition, equation (42.6.b) is automatically satised. In the Von Mises transformation, Ψ and xare treated as the independent variables, instead ofyandx. This transforms equation (42.6.a) into @u @x=@ @Ψ u@u @Ψ : See Rosenhead [7], Schlichting [8], or von Mises [10]. Notes 1. If the boundary data are of the Neuman type, then the Kircho transformation may introduce nonlinearities in the boundary data for the  problem. 2. The Kircho transformation is frequently useful in free boundary problems (see page 311), where K( ) changes value across the (un- known) boundary. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 176 I.B Transformations References [1]Ames, W. F. ,E d . Nonlinear Partial Dierential Equations ,v o l .1 .A c a d e m i c Press, New York, 1967. [2]Bateman, H. Partial Dierential Equations of Mathematical Physics .D o v e r Publications, Inc., New York, 1944. [3]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [4]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [5]Hill, J. M. Solution of Dierential Equations by Means of One-Parameter Groups . Pitman Publishing Co., Marsheld, MA, 1982. [6]Kamke, E. Dierentialgleichungen Losungsmethoden und Losungen ,v o l .I I . Chelsea Publishing Company, New York, 1947. [7]Rosenhead, L. Laminar Boundary Layers . Clarendon Press, Oxford, England, 1963. [8]Schlichting, H. Boundary Layer Theory . McGraw{Hill Book Company, New York, 1955. [9]Varley, E., and Seymour, B. A method of obtaining exact solution to partial dierential equation with variable coecients. Stud. Appl. Math. 78 (1988), 183{225. [10]v o nM i s e s ,R . Bemerkungen zur hydrodynamik. Z. Angnew. Math. Mech. 7(1927), 425{431. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 178 II Exact Analytical Methods 43. Introduction to Exact Analytical Methods The methods in this section of the book are for the exact solution of dierential equations. The methods have been separated into two parts: Methods that can be used for ordinary dierential equations and, sometimes, partial dierential equations: When a method in this part can be used for a partial dierential equation, there is a star ( ) alongside the method name. Methods that can be used only for partial dierential equations. Because many of the common methods for partial dierential equations are also useful as methods for ordinary dierential equations, the rst part of this section should not be overlooked when attempting to nd the solution of a partial dierential equation. Listed below are, in the author’s opinion, those methods that are the most useful when solving ordinary dierential equations and partial dier- ential equations. These are the methods that might be tried rst. Most Useful Methods for ODEs Look-Up Technique (page 179) Look-Up ODE Forms (page 219) Computer-Aided Solution (page 240) Constant Coecient Linear Equations (page 247) Eigenfunction Expansions(page 268) Green’s Functions(page 318) Integral Transforms: Innite Intervals(page 347) Integrating Factors(page 356) Series Solution(page 403) Method of Undetermined Coecients(page 415) Most Useful Methods for PDEs Look-Up Technique (page 179) Eigenfunction Expansions(page 268) Green’s Functions(page 318) Integral Transforms: Innite Intervals(page 347) Method of Characteristics (page 432) Conformal Mappings (page 441) Lie Groups: PDEs (page 471) Separation of Variables (page 487) Similarity Methods (page 497) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 179 44. Look-Up Technique Applicable to Equations of certain forms. Yields A reference to the literature that may yield an analytical solution, an approximate analytical solution, or a numerical solution. Idea Many functions of mathematical physics have been well studied. If a dierential equation can be transformed to a known form, then informationabout the solution may be obtained by looking in the right reference. Procedure Compare the dierential equation that you are trying to analyze with the lists on the following pages. If the equation you are investigating appears, see the references cited for that equation. The equations listed in this section include Ordinary dierential equations (page 180) {First order equations {Second order equations {Higher order equations Partial dierential equations (page 189) {Linear equations {Second order nonlinearity {Higher order and variable order nonlinearities Systems of dierential equations (page 199) {Systems of ordinary dierential equations {Systems of partial dierential equations The Laplacian in dierent coordinate systems (page 204) Parametrized equations at specic values (page 205) Notes 1. Realize that the same equation may look dierent when written in dif- ferent variables. Some scaling of any given equation may be required to make it look like one of the forms listed. 2. Carslaw and Jaeger [36] have a large collection of exact analytical solutions for parabolic partial dierential equations. 3. In Kamke ([90] and [91]), Murphy [123], and Polyanin and Zaitsev [130] are long listings of ordinary dierential equations and partialdierential equations and their exact solutions. 4. The references follow the listings of dierential equations (page 209). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 180 II Exact Analytical Methods 5. A complete list of third-order polynomial evolution equations of not normal type with nontrivial Lie{B¨ acklund symmetries is in Fujimoto and Watanabe [60]. 44.1 Ordinary Dierential Equations 44.1.1 First Order Equations Abel equation of the rst kind (see Murphy [123, page 23]): y0=f0(x)+f1(x)y+f2(x)y2+f3(x)y3 Abel equation of the second kind (see Murphy [123, page 25]): [g0(x)+g1(x)y]y0=f0(x)+f1(x)y+f2(x)y2+f3(x)y3 Bernoulli equation (see page 235): y0=a(x)yn+b(x)y Binomial equation (see Hille [80, page 675]): (y0)m=f(x;y) Briot and Bouquet’s equation (see Ince [85, page 295]): xy0−y=a10x+a20x2+a11yx+a02y2+::: Clairaut’s equation (see page 237): f(xy0−y)=g(y0) Elliptic functions (see Gradshteyn and Ryzhik [69, page 917]): y0=p (1−y2)(1−k2y2) Euler equation (see Valiron [161, page 201]): y0=q ay4+by3+cy2+dy+e ax4+bx3+cx2+dx+e Euler equation (see Valiron [161, page 212]): y0+y2=axm Heisenberg equation of motion (see Iyanaga and Kawada [87, page 1083]): dA(t) dt=i h[H;A(t)] Jacobi equation (see Ince [85, page 22]): (a1+b1x+c1y)(xy0−y)−(a2+b2x+c2y)y0+(a3+b3x+c3y)=0 Lagrange’s equation (see page 363): y=xf(y0)+g(y0) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 181 L¨owner’s equation (see Iyanaga and Kawada [87, page 1345]): y0=−y1+(x)y 1−(x)y Riccati equation (see page 392): y0=a(x)y2+b(x)y+c(x) Unnamed equation (see Boyd [30]): y0=−pe−q=y Unnamed equation (see Goldstein and Braun [68, page 42]): g(y)y0=f(x)+h(x)GR f(x)dx−R g(y)dy Weierstrass function (see Rainville [131, page 312]): y0=p 4y3−g2y−g3 44.1.2 Second Order Equations Airy equation (see Abramowitz and Stegun [3, Section 10.4.1]): y00=xy Anger functions (see Gradshteyn and Ryzhik [69, page 989]): y00+y0 x+ 1−2 x2 y=x− x2sin Baer equation (see Moon and Spencer [119, page 156]): (x−a1)(x−a2)y00+1 2[2x−(a1+a2)]y0− p2x+q2 y=0 Baer wave equation (see Moon and Spencer [119, page 157]): (x−a1)(x−a2)y00+1 2[2x−(a1+a2)]y0− k2x2−p2x+q2 y=0 Bessel equation (see Abramowitz and Stegun [3, Section 9.1.1]): x2y00+xy0+(x2−n2)y=0 Bessel equation { modied (see Abramowitz and Stegun [3, Section 9.6.1]): x2y00+xy0−(x2+n2)y=0 Bessel equation { spherical (see Abramowitz and Stegun [3, Section 10.1.1]): x2y00+2xy0+ x2−n(n+1 ) y=0 Bessel equation { modied spherical (see Abramowitz and Stegun [3, Sec- tion 10.2.1]): x2y00+2xy0− x2+n(n+1 ) y=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 182 II Exact Analytical Methods Bessel equation { wave (see Moon and Spencer [119, page 154]): x2y00+xy0+ a2x4+b2x2−c2 y=0 B^ocher equation (see Moon and Spencer [119, page 127]): y00+1 2m1 x−a1+


+mn−1 x−an−1 y0 +1 4A0+A1x+


+Alxl (x−a1)m1(x−a2)m2


(x−an−1)mn−1 y=0 Confluent equation { general (see Abramowitz and Stegun [3, Section 13.1.35]): y00+2a x+2f0+bh0 h−h0−h00 h0 y0+"bh0 h−h0−h00 h0a x+f0 +a(a−1) x2+2af0 x+f00+(f0)2−a(h0)2 h# y=0 Coulomb wave functions (see Abramowitz and Stegun [3, Section 14.1.1]): y00+h 1−2 x−L(L+1) x2i y=0 Dung’s equation (see Bender and Orszag [20, page 547]): y00+y+ay3=0 Eckart equation (see Barut et al. [18]): y00+h  1++ (1+)2+γi y=0; =ex Ellipsoidal wave equation (see Arscott [13]): y00−(a+bk2sn2x+qk4sn4x)y=0 Complete elliptic integral (see Gradshteyn and Ryzhik [69, page 907]): d dxh x(1−x2)dy dxi −xy=0 Complete elliptic integral (see Gradshteyn and Ryzhik [69, page 907]): (1−x2)d dx xdy dx +xy=0 Emden equation (see Leach [102]): (x2y0)0+x2yn=0 Emden equation { modied (see Leach [102]): y00+a(x)y0+yn=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 183 Emden{Fowler equation (see Rosenau [137]): (xpy0)0xyn=0 Generalized Emden{Fowler equation (see Leach et al. [104]): y00+f(x)yn=0 Integrals of the error function (see Abramowitz and Stegun [3, Section 7.2.2]): y00+2xy0−2ny=0 Gegenbauer functions (see Infeld and Hull [86]): (1−x2)y00−(2m+3 )xy0+y=0 Halm’s equation (see Hille [80, page 357]): (1 +x2)2y00+y=0 Heine equation (see Moon and Spencer [119, page 157]): y00+1 2h 1 x−a1+2 x−a2+2 x−a3i y+1 4h A0+A1x+A2x2+A3x3 (x−a1)(x−a2)2(x−a3)2yi =0 Hermite polynomials (see Abramowitz and Stegun [3, Section 22.6.21]): y00−xy0+ny=0 Heun’s equation (see Ronveaux [134]): y00+h γ x+ x−1− x−ai y0+x−q x(x−1)(x−a)y=0 Hill’s equation (see Ince [85, page 384]): y00+(a0+2a1cos 2x+2a2cos 4x+:::)y=0 Hypergeometric equation (see Abramowitz and Stegun [3, Section 15.5.1]): x(1−x)y00+[c−(a+b+1 )x]y0−aby=0 Hyperspherical dierential equation (see Iyanaga and Kawada [87, page 1185]): (1−x2)y00−2axy0+by=0 Ince equation (see Athorne [14]): y00++cos 2t+γcos 4t (1+acos 2t)2y=0 Jacobi’s equation (see Iyanaga and Kawada [87, page 1480]): x(1−x)y00+[γ−(+1 )x]y0+n(+n)y=0 Kelvin functions (see Abramowitz and Stegun [3, Section 9.9.3]): x2y00+xy0−(ix2+2)y=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 184 II Exact Analytical Methods Kummer’s equation (see Abramowitz and Stegun [3, Section 13.1.1]): xy00+(b−x)y0−ay=0 Lagerstrom equation (see Rosenblat and Shepherd [138]): y00+k xy0+yy0=0 Laguerre equation (see Iyanaga and Kawada [87, page 1481]): xy00+(+1−x)y0+y=0 Lame equation (see Moon and Spencer [119, page 157]): y00+1 2h 1 x−a1+1 x−a2+1 x−a3i y0+1 4h A0+A1x (x−a1)(x−a2)(x−a3)i y=0 Lame equation (see Ward [167]): y00+(h−n(n+1 )k2sn2x)y=0 Lame equation { wave (see Moon and Spencer [119, page 157]): y00+1 2h 1 x+1 x−a+1 x−bi y0+1 4h (a2+b2)q−p(p+1)x+x2 x(x−a)(x−b)i y=0 Lane{Emden equation (see Seshadri and Na [147, page 193]): y00+2 xy0+yk=0 Legendre equation (see Abramowitz and Stegun [3, Section 8.1.1]): (1−x2)y00−2xy0+h n(n+1 )−m2 1−x2i y=0 Legendre equation { wave (see Moon and Spencer [119, page 155]): (1−x2)y00−2xy0−h k2a2(x2−1)−p(p+1 )−q2 x2−1i y=0 Lewis regulator (see Hagedorn [71, page 152]): y00+( 1−jyj)y0+y=0 Lienard’s equation (see Villari [163]): y00+f(y)y0+y=0 Liouville’s equation (see Goldstein and Braun [68, page 98]): y00+g(y)(y0)2+f(x)y0=0 Lommel functions (see Gradshteyn and Ryzhik [69, page 986]): x2y00+xy0+(x2−2)y=x+1 Magnetic pole equation (see Infeld and Hull [86]): y00−2 4m(m+1)+1 4− m+1 2 cosx sin2x+/parenleftbig +1 2
3 5y=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 185 Mathieu equation (see Abramowitz and Stegun [3, Section 20.1.1]): y00+(a−2qcos 2x)y=0 Mathieu equation { associated (see Ince [85, page 503]): y00+[ ( 1−2r)c o tx]y0+(a+k2cos2x)y=0 Mathieu equation { modied (see Abramowitz and Stegun [3, Section 20.1.2]): y00−(a−2qcosh 2x)y=0 Morse{Rosen equation (see Barut et al. [18]): y00+ cosh2ax+tanhax+γ y=0 Neumann’s polynomials (see Gradshteyn and Ryzhik [69, page 990]): x2y00+3xy0+(x2+1−n2)y=xcos2n 2+nsin2n 2 Painlev e transcendent { rst (see Ince [85, page 345]): y00=6y2+x Painlev e transcendent { second (see Ince [85, page 345]): y00=2y3+xy+a Painlev e transcendent { third (see Ince [85, page 345]): y00=1 y(y0)2−1 xy0+1 x(y2+)+γy3+ y Painlev e transcendent { fourth (see Ince [85, page 345]): y00=1 2y(y0)2+3y3 2+4xy2+2 (x2−)y+ y Painlev e transcendent { fth (see Ince [85, page 345]): y00= 1 2y+1 y−1 (y0)2−1 xy0+(y−1)2 x2 y+ y +γy x+y(y+1) y−1 Painlev e transcendent { sixth (see Ince [85, page 345]): y00=1 21 y+1 y−1+1 y−x (y0)2−1 x+1 x−1+1 y−x y0 +y(y−1)(y−x) x2(x−1)2 +x y2+γ(x−1) (y−1)2+x(x−1) (y−x)2 Painlev e{Ince { modied (see Abraham-Shrauner [2]): y00+yy0+y3 Parabolic cylinder equation (see Abramowitz and Stegun [3, Section 19.1.1]): y00+(ax2+bx+c)y=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 186 II Exact Analytical Methods Pinney equation (see Common et al. [50, page 908]): y00+f(x)y+cy−3 Poisson{Boltzmann equation (see Chambr e[ 3 9 ] ) : y00+k xy0=−ey P¨oschl{Teller equation { rst (see Barut et al. [18]): y00−h a2 (−1) sin2ax+(−1) cos2ax −b2i y=0 P¨oschl{Teller equation { second (see Barut et al. [18]): y00−h a2 (−1) sinh2ax+(−1) cosh2ax −b2i y=0 Polytropic dierential equation (see Iyanaga and Kawada [87, page 908]): (x2y0)0=−x2yn Rayleigh equation (see Birkho and Rota [24, page 134]): y00− 1−(y0)2 y0+y=0 Riccati{Bessel equation (see Abramowitz and Stegun [3, Section 10.3.1]): x2y00+ x2−n(n+1 ) y=0 Richardson’s equation (see Binding and Volkmer [23]): −y00=(sgnx+)y Riemann’s dierential equation (see Abramowitz and Stegun [3, Section 15.6.1]): y00+1−−0 x−a+1−−0 x−b+1−γ−γ0 x−c y0 +0(a−b)(a−c) x−a+0(b−c)(b−a) x−b+γγ0(c−a)(c−b) x−c y (x−a)(x−b)(x−c)=0 Spheroidal wave functions (oblate) (see Abramowitz and Stegun [3, Section 21.6.4]):  (1−x2)y00+ +c2x2−m2 1−x2 y=0 Spheroidal wave functions radial (see Abramowitz and Stegun [3, Section 21.6.3]):  (1 +x2)y00− −c2x2−m2 x2+1 y=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 187 Struve functions (see Abramowitz and Stegun [3, Section 12.1.1]): x2y00+xy0+(x2−2)y=4(x 2)+1 pΓ(+1 2) Symmetric top equation (see Infeld and Hull [86]): y00− M2−1 4+K2−2MK cosx sin2x+/parenleftbig +K2+1 4
 y=0 Tchebyche equation (see Abramowitz and Stegun [3, Section 22.6.9]): (1−x2)y00−xy0+n2y=0 Thomas{Fermi equation (see Bender and Orszag [20, page 25]): y00=y3=2x−1=2 Titchmarsh’s equation (see Hille [80, page 617]): y00+/parenleftbig −x2n
y=0 Ultraspherical equation (see Abramowitz and Stegun [3, Section 22.6.5]): (1−x2)y00−(2a+1 )xy0+n(n+2a)y=0 Van der Pol equation (see Birkho and Rota [24, page 134]): y00−(1−y2)y0+y=0 Wangerin equation (see Moon and Spencer [119, page 157]): y00+1 2h 1 x−a1+1 x−a2+2 x−a3i y0+1 4h A0+A1x+A2x2 (x−a1)(x−a2)(x−a3)2i y=0 Weber equation (see Moon and Spencer [119, page 153]): y00+ a2−b2 4x2 y=0 Weber functions (see Gradshteyn and Ryzhik [69, page 989]): y00+y0 x+ 1−2 x2 y=−1 x2[x++(x−)c o s] Whittaker’s equation (see Abramowitz and Stegun [3, equation 13.1.31]): y00+ −1 4+ x+1 4−2 x2 y=0 Whittaker{Hill equation (see Urwin and Arscott [159]): y00+(A+Bcos 2x+Ccos 4x)y=0 Unnamed equation (see Chrisholm and Common [45]): y00+(a0+a1y)y0+b0+b1y+b2y2+b3y3=0 Unnamed equation (see Gilding [65]): y00=−yp CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 188 II Exact Analytical Methods Unnamed equation (see Latta [101]): (1−x2)y00−2axy0+(b+cx2)y=0 Unnamed equation (see Leach et al. [103]): y00+yy0+y3=0 Unnamed equation (see Rubel [140]): xyy00+yy0−x(y0)2=0 Unnamed equation (see Setoyanagi [148]): y00+(axp+bxq)y=0 Unnamed equation (see Tsukamoto [158]): y00+eatyb=0 44.1.3 Higher Order Equations Products of Airy functions (see Abramowitz and Stegun [3, equation 10.4.57]): y000−4xy0−2y=0 Blasius equation (see Meyer [114, page 127]): y000+yy00=0 Falkner{Skan equation (see Cebeci and Keller [38]): y000+yy00+ 1−(y0)2 =0 Generalized hypergeometric equation (see Miller [117, page 271]):/parenleftbig xd dx+a1



/parenleftbig xd dx+ap
−d dx/parenleftbig xd dx+b1



/parenleftbig xd dx+bq
y=0 Laplace equations (see Valiron [161, pages 306{315]): (a0x+b0)y(n)+(a1x+b1)y(n−1)+


+(anx+bn)y=0 Sixth order Onsager equation (see Viecelli [162]): (ex(exyx)xx)xxx=f(x) Orr{Sommerfeld equation (see Herron [77]): 1 iR d2 dx2−22 y−h (f(x)−c) d2 dx2−2 −f00(x)i y=0 Unnamed equation (see Benguria and Depassier [21]): y000+y0=f(y) Unnamed equation (see Hershenov [78]): y000+ax y0+by=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 189 Unnamed equation (see Merkin [113]): y000+yy0+(y0)2=0 Unnamed equation (see Pfeier [129]): y000+q(x)y0+r(x)y=0 Unnamed equation (see Walker [164]): [(ry00)0−py0]0+qy=y Unnamed equation (see Watson [168, page 106]): y(m)=axy−m=2 44.2 Partial Dierential Equations 44.2.1 Linear Equations Biharmonic equation (see Kantorovich and Krylov [92, pages 595{615]): r4u=0 Linear Boussinesq equation (see Whitham [170, page 9]): utt−a2uxx=b2uxxtt Busemann equation (see Chaohao [42]): (1−x2)uxx−2xyuxy+( 1−y2)uyy+2a(xux+yuy)−a(a+1 )u=0 Chaplygin’s equation (see Landau and Lifshitz [99, page 432]): uxx+y2 1−y2=c2uyy+yuy=0 Diusion equation (see Morse and Feshback [122, page 271]): r
((x;t)ru)=ut Euler{Darboux equation (see Miller [116]): uxy+1 x−y(aux−buy)=0 Euler{Poisson{Darboux equation (see Ames [9, Section 3.3]): uxy+N x+y(ux+uy)=0 Helmholtz equation (see Morse and Feshback [122, page 271]): r2u+k2u=0 Klein{Gordon equation (see Morse and Feshback [122, page 272]): r2u−1 c2utt=2u Kramers equation (see Duck et al. [54]): Pt=Pxx−uPx+@ @u[(u−F(x))P] CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 190 II Exact Analytical Methods Lambropoulos’s equation (see Wilcox [171]): uxy+axux+byuy+cxyu +ut=0 Laplace’s equation (see Morse and Feshback [122, page 271]): r2u=0 Lavrent’ev{Bitsadze equation (see Chang [41]): uxx+( s g ny)uyy=f(x;y) Onsager equation (see Wood and Morton [172]): (ex(exuxx)xx)xx+B2uyy=F(x;y) Poisson equation (see Morse and Feshback [122, page 271]): r2u=−4(x) Schr¨oedinger equation (see Morse and Feshback [122, page 272]): −h2 2mr2u+V(x)u=ihut Spherical harmonics in three dimensions (see Humi [84]):h 1 sin@ @/parenleftbig sin@ @
+1 sin2@2 @2+l(l+1 )i Yl;m=0 Spherical harmonics in four dimensions (see Humi [84]): uxx+2 ( c o tx)ux+1 sin2x uyy+( c o ty)uy+1 sin2yuzz +(n2−1)u=0 Tricomi equation (see Manwell [110]): uyy=yuxx Wave equation (see Morse and Feshback [122, page 271]): utt=c2r2u Weinstein equation { generalized (see Akin [6]): r2u+p xn−1uxn−1+q xnuxn=0 44.2.2 Second Order Nonlinearity Benjamin{Bona{Mahony equation (see Avrin and Goldstein [15]): ut−uxxx+uux=0 Boussinesq equation (see Calogero and Degasperis [34, page 54]): utt−uxx−uxxxx+3 (u2)xx=0 Burgers equation (see Benton and Platzman [22]): ut+uux=uxx CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 191 Burgers equation { non-planar (see Sachdev and Nair [142]): ut+uux+Ju 2t= 2uxx Burgers equation { generalized (see Oliveri [128]): ut+uux−uxx+f(t)u=0 Ernst equation (see Calogero and Degasperis [34, page 62]): (<u)/parenleftbig urr+ur r+uzz
=u2 r+u2 z Fisher’s equation (see Kaliappan [89]): ut=Duxx+u−u2 Convective Fisher’s equation (see Sh¨ onborn et al. [151]): ut=1 2uxx+u(1−u)−uux Kadomtsev{Petviashvili equation (see Latham [100]): (ut+uxxx−6uux)xuyy=0 Generalized Kadomtsev{Petviashvili{Burgers equation (see Brugarino [31]):/parenleftbig ut+J 2tu+J1uux+J2uxx+J3uxxx
x+J4(t)uyy=0 Khokhlov{Zabolotskaya equation (see Chowdhury and Nasker [44]): uxt−(uux)x=uyy Korteweg{de Vries equation (KdV) (see Lamb [98, Chapter 4]): ut+uxxx−6uux=0 KdV equation { cylindrical (see Calogero and Degasperis [34, page 50]): ut+uxxx−6uux+u 2t=0 KdV equation { generalized (see Boyd [29]): ut+uux−uxxxxx =0 KdV equation { spherical (see Calogero and Degasperis [34, page 51]): ut+uxxx−6uux+u t=0 KdV equation { transitional (see Calogero and Degasperis [34, page 50]): ut+uxxx−6f(t)uux=0 KdV equation { variable coecient (see Nimala et al. [125]): ut+atnuux+btmuxxx=0 Korteweg{de Vries{Burgers equation (KdVB) (see Canosa and Gazdag [35]): ut+2uux−uxx+uxxx=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 192 II Exact Analytical Methods Kuramoto{Sivashinksy equation (see Michelson [115]): ut+r4u+r2u+1 2jr2uj2=0 Lin{Tsien equation (see Ames and Nucci [10]): 2utx+uxuxx−uyy=0 Regularized long-wave equation (RLW) (see Calogero and Degasperis [34, page 49]): ut+ux−6uux−utxx=0 Generalized shallow water wave equation (GSWW) (see Clarkson and Manseld [48]): uxxxt+auxuxt+butuxx−uxt−uxx=0 Thomas equation (see Rosales [135]): uxy+aux+buy+cuxuy=0 Unnamed equation (see Rosen [136]): utt+2uut−uxx=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 193 44.2.3 Higher Order/Variable Order Nonlinearities Ansph¨ aren equation (see Schief and Rogers [146]): /parenleftbigRu R2v2
u= R2Rv v2 v Generalized Benjamin{Bona{Mahony equation (see Goldstein and Wich- noski [67]): ut−r2ut+r
(u)) = 0 Benney equation (see Balmforth et al. [16]): ut+(un)x=−uxx−uxxx−uxxxx Born{Infeld equation (see Whitham [170, page 617]):/parenleftbig 1−u2 t
uxx+2uxutuxt−/parenleftbig 1+u2 x
utt=0 Boussinesq equation { modied (see Clarkson [46]): 1 3utt−utuxx−3 2u2 xuxx+uxxxx =0 Boussinesq equation { modied (see Clarkson [47]): utt−utuxx−1 2u2 xuxx+uxxxx =0 Buckmaster equation (see Hill and Hill [79]): ut=/parenleftbig u4
xx+/parenleftbig u3
x Generalized Burgers equation (see Sachdev et al. [141]): ut+unux+/parenleftbigj 2t+
u+/parenleftbig +γ x
un+1= 2uxx Generalized Burgers{Huxley equation (see Wang et al. [166]): ut−uux−uxx=u/parenleftbig 1−u
/parenleftbig u−γ
Cahn{Hilliard equation (see Novick-Cohen and Segel [126]): ut=r
h M(u)r @f @u−Kr2ui Calogero{Degasperis{Fokas equation (see Gerdt et al. [63]): uxxx−1 8u3 x+ux(Aeu+Be−u)=0 Caudrey{Dodd{Gibbon{Sawada{Kotera equation (see Aiyer et al. [5]): ut+uxxxxx +3 0uuxxx+3 0uxuxx+ 180u2ux=0 Clairaut’s equation (see Iyanaga and Kawada [87, page 1446]): u=xux+yuy+f(ux;uy) Inhomogenous nonlinear diusion equation (see Saied and Hussein [143]): xput=(xmunux)x CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 194 II Exact Analytical Methods Nonlinear diusion equation (see King [96]): @u @t=@ @x/parenleftbig u−4=3@u @x
Nonlinear diusion equation (see King [96]): @u @t=@ @x/parenleftbig u−2=3@u @x
Eckhaus partial dierential equation (see Kundu [97]): iut+uxx+2/parenleftbig juj2
xu+juj4u=0 Fisher equation { generalized (see Wang [165]): ut−uxx−m uu2 x=u(1−u) Fisher equation { generalized (see Kaliappan [89]): ut=uxx+u−uk Fisher equation { generalized (see Herrera et al. [76]): ut=uxx+up−u2p−1 Gardner equation (see Tabor [155, page 289]): ut=6 (u+a2u2)ux+uxxx Ginzburg{Landau equation (see Katou [94]): ut=( 1+ia)uxx+( 1+ic)u−(1 +id)juj2u Quintic Ginzburg{Landau equation (see Marcq et al. [111]): At=A+1Axx−3jAj2A−4jAj4A Hamilton{Jacobi equation (see page 61): Vt+H(t;x;Vx1;:::;Vxn)=0 Harry Dym equation (see Calogero and Degasperis [34, page 53]): ut=uxxxu3 Generalized axially symmetric Helmholtz equation (GASHE) (see Lown- des [109, page 96]): @2u @x2+@2u @y2+2 y@u @y+k2u=0 Generalized biaxially symmetric Helmholtz equation in ( n+ 1) variables (GASHEN) (see Lowndes [109, page 93]): Pn i=1@2u @xi2+@2u @y2+ y@u @y+k2u=0 Generalized biaxially symmetric Helmholtz equation (GBSHE) (see Lown- des [109, page 91]): @2u @x2+@2u @y2+2 x@u @x+2 y@u @y+k2u=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 195 Hirota equation (see Calogero and Degasperis [34, page 56]): ut+iau+ib(uxx−2ju2ju)+cux+d(uxxx−6juj2ux)=0 Kadomtsev{Petviashvili equation { modied (see Clarkson [46]): uxt=uxxx+3uyy−6u2 xuxx−6uyuxx KdV equation { deformed (see Dodd and Fordy [53]): ut+ uxx−2u3−3 2uu2 x +u2 x=0 KdV equation { generalized (see Rammaha [133]): ut+uux+pjujp−1ux=0 KdV equation { modied (mKdV) (see Calogero and Degasperis [34, page 51]): ut+uxxx6u2ux=0 KdV equation { modied modied (see Dodd and Fordy [53]): ut+uxxx−1 8u3 x+ux(Aeau+B+Ce−au)=0 KdV equation { Schwarzian (see Weiss [169]): ut ux+fu;xg= Klein{Gordon equation { nonlinear (see Matsuno [112]): r2u+up=0 Klein{Gordon equation { quasilinear (see Nayfeh [124, page 76]): utt−a2uxx+c2u=bu3 Kupershmidt equation (see Fuchssteiner et al. [59]): ut=uxxxxx +5 2uxxxu+25 4uxxux+5 4u2ux Liouville equation (see Matsuno [112]): r2u+eu=0 Liouville equation (see Calogero and Degasperis [34, page 60]): uxt=eu Molenbroek’s equation (see Cole and Cook [49, page 34]): r2=M2 1( 2 xxx+2xyxy+2 yyy+γ−1 2/parenleftbig 2 x+2 y−1
 xx+yy+y y) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 196 II Exact Analytical Methods Monge{Amp ere equation (see Moon and Spencer [121, page 171]): (uxy)2−uxuy=f(x;y;u;ux;uy) Monge{Amp ere equation (see Gilbarg and Trudinger [64]):u x1x1ux1x2::: ux1xn ux2x1ux2x2::: ux2xn ............ u xnx1uxnx2::: uxnxn=f(u;x;ru) Nagumo equation (see Zhi-Xiong and Ben-Yu [174]): u t=uxx+u(u−a)(1−u) Phi{four equation (see Calogero and Degasperis [34, page 60]): utt−uxx−u+u3=0 Plateau’s equation (see Bateman [19, page 501]): (1 +u2 x)uxx−2uxuyuxy+( 1+u2 y)uyy=0 Porous-medium equation (see Elliot, Herrero, King, and Ockendon [55]): ut=r
(umru) Generalized axially symmetric potential equation (GASPE) (see Lown- des [109, page 95]): @2u @x2+@2u @y2+2 y@u @y=0 Generalized biaxially symmetric potential equation (GBSPE) (see Lown- des [109, page 91]): @2u @x2+@2u @y2+2 x@u @x+2 y@u @y=0 Generalized biaxially symmetric potential equation in ( n+ 1) variables (GASPEN) (see Lowndes [109, page 92]): Pn i=1@2u @xi2+@2u @y2+ y@u @y=0 Rayleigh wave equation (see Hall [74]): utt−uxx=a(ut−u3 t) Sawada{Kotera equation (see Matsuno [112, page 7]): ut+4 5u2ux+1 5uxuxx+1 5uuxxx+uxxxxx =0 Schr¨oedinger equation { logarithmic (see Cazenave [37]): iut+r2u+ulogjuj2=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 197 Schr¨oedinger equation { derivative nonlinear (see Calogero and Degasperis [34, page 56]): iut+uxxi/parenleftbig juj2u
x=0 Schr¨oedinger equation { derivative nonlinear (see Hayashi and Ozawa [75]): i@t +@x =i@x(j j2 )+1j jp1−1+2j jp2−1 Schr¨oedinger equation { nonlinear (see Calogero and Degasperis [34, page 56]): iut+uxx2juj2u=0 Sine{Gordon equation (see Calogero and Degasperis [34, page 59]): uxx−uyysinu=0 Sine{Gordon equation { damped (see Levi et al. [105]): utt+ut−uxx+s i nu=0 Sine{Gordon equation { double (see Calogero and Degasperis [34, page 60]): uxt sinu+sin/parenleftbigu 2
 =0 Sine{Gordon { multidimensional (see Elzoheiry et al. [56]): urr+m−1 rur−utt=s i nu Sinh{Gordon equation (see Grauel [70]): uxt= sinhu Sinh{Poisson equation (see Ting et al. [156]): r2u+2sinhu=0 Strongly damped wave equation (see Ang and Dinh [12]): utt−r2u−r2ut+f(u)=0 Tzitzeica equation (see Schief [145]): uxy=eu−e−2u Unnamed equation (see Aguirre and Escobedo [4]): ut−r2u=up Unnamed equation (see Bluman and Kumei [25]): ut−@ @xh aux (u+b)2i =0 Unnamed equation (see Calogero [32]): uxt+uuxx+F(ux)=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 198 II Exact Analytical Methods Unnamed equation (see Calogero [33]): ut=uxxx+3 (uxxu2+3u2 xu)+3uxu4 Unnamed equation (see Daniel and Sahadevan [51]): ut=uxxx+u2uxx+3uu2 x+1 3u4ux Unnamed equation (see Fujita [61]): ut=r2u+eu Unnamed equation (see Fung and Au [62]): ut+uxxx−6u2ux+6ux=0 Unnamed equation (see Lin [107]): r2u+Ae−u=0 Unnamed equation (see Lindquist [108]): r
(jrujpru)=f Unnamed equation (see Roy and Chowdhury [139]): −iut+uxx+2juxj2u 1−uu=0 Unnamed equation (see Shivaji [150]): −r2u=exp u +u Unnamed equation (see Trubek [157]): r2u+Ku=0 Unnamed equation (see Yanagida [173]): r2u+Kjxjjujqu=0 Unnamed equation (see Utepbergenov [160]): z2uzz+r2u+a(z)u=0 Wadati{Konno{Ichikawa{Schimizu equation (see Calogero and Degasperis [34, page 53]): iut+h/parenleftbig 1+juj2
−1=2ui xx=0 Zoomeron equation (see Calogero and Degasperis [34, page 58]): @2 @t2−@2 @x2/parenleftbiguxt u
+2/parenleftbig u2
xt=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 199 44.3 Systems of Dierential Equations 44.3.1 Systems of ODEs Bonhoeer-van der Pol (BVP) oscillator (see Rajasekar and Lakshmanan [132]): x0=x−x3 3−y+I(t) y0=c(x+a−by) Brusselator (see Hairer et al. [73, page 112]): u0=A+u2v−(B+1 )u v0=Bu−u2v Full Brusselator (see Hairer et al. [73, page 114]): u0=1+u2v−(w+1 )u v0=uw−u2v w0=−uw+ Hamilton’s dierential equations (see Iyanaga and Kawada [87, page 1005]): dxi dt=Hpi(t;x;p) dpi dt=−Hxi(t;x;p) Jacobi elliptic functions (see Hille [80, page 66]): u0=vw v0=−uw w0=−k2uv Kowalevski’s top (see Haine and Horozov [72]): dm dt=mm+γl dγ dt=γm Lorenz equations (see Sparrow [152]): x0=(y−x) y0=rx−y−xz z0=xy−bz Lorenz equations { complex (see Flessas [58]): x0=(y−x) y0=rx−y−xz z0=−bz+1 2(xy+xy) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 200 II Exact Analytical Methods Lotka{Volterra equations (see Boyce and DiPrima [28, page 494]): u0=u(a−bv) v0=v(−c+du) Nahm’s equations (see Steeb and Louw [154]): Ut=[V;W ] Vt=[W;U ] Wt=[U;V] Toda lattice equation { relativistic (see Ohta et al. [127]): ¨xn= 1+_xn−1 c 1+_xn cexp(xn−1−xn) 1+( 1=c2)e x p (xn−1−xn) − 1+_xn c 1+_xn+1 cexp(xn−xn+1) 1+( 1=c2)e x p (xn−xn+1) Toda molecule equation { cylindrical (see Hirota and Nakamura [83]):/parenleftbig @rr+r−1@r
logVn−Vn+1+2Vn−Vn−1=0 Unnamed equation (see Steeb [153, page 57]): utt+c1jujn(uut)+c2jujmu=0 44.3.2 Systems of PDEs Ane Knizhnik{Zamolodchikov equation (see Cherednik [43]): @(z) @zi=kP jsij(z) zi−zj Beltrami equation (see Iyanaga and Kawada [87, page 1087]): fz=(z)fz Boomeron equation (see Calogero and Degasperis [34, page 57]): ut=b
vx vxt=uxxb+avx−2v[vb] Carleman equation (see Kaper and Leaf [93]): ut+ux=v2−u2 vt−vx=u2−v2 Cauchy{Riemann equations (see Levinson and Redheer [106]): ux−vy=0 uy+vx=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 201 Chiral eld equation (see Calogero and Degasperis [34, page 61]): (UUx)t+(UUt)x=0 Davey{Stewartson equations (see Champagne and Winternitz [40]): iut+uxx+auyy+bujuj2−uw=0 wxx+cwyy+d/parenleftbig juj2
yy=0 Dirac equation in 1 + 1 dimensions (see Alvarez et al. [7]): ut+vx+imu+2i/parenleftbig juj2−jvj2
u=0 vt+ux+imv+2i/parenleftbig jvj2−juj2
v=0 Dispersive long-wave equation (see Boiti et al. [27]): ut=(u2−ux+2w)x wt=( 2uw+wx)x Drinfel’d{Sokolov{Wilson equation (see Hirota et al. [82]): ut=3wwx wt=2wxxx+2uwx+uxw Euler equations (see Landau and Lifshitz [99, page 3]): @v @t+(v
grad)v=−1 gradP Fitzhugh{Nagumo equations (see Sherman and Peskin [149]): ut=uxx+u(u−a)(1−u)+w wt=u Gross{Neveu model (see Calogero and Degasperis [34, page 62]): iu(n) x=v(n)NX m=1 v(m)u(m)+u(m)v(m) iv(n) t=u(n)NX m=1 v(m)u(m)+u(m)v(m) Heisenberg ferromagnet equation (see Calogero and Degasperis [34, page 56]): st=ssxx Hirota{Satsuma equation (see Weiss [169]): ut=1 2uxxx+3uux−6wwx wt=−wxxx−3uwx CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 202 II Exact Analytical Methods Von K arman equations (see Ames and Ames [8]): r4u=E w2 xy−wxxwyy r4w=a+b[uyywxx+uxxwyy−2uxywxy] Kaup’s equation (see Dodd and Fordy [53]): fx=2fgc(x−t) gt=2fgc(x−t) KdV equation { super (see Kersten and Gragert [95]): ut=6uux−uxxx+3wwxx wt=3uxw+6uwx−4wxxx Klein{Gordon{Maxwell equations (see Deumens [52]): r2s−(jaj2+1 )s=0 r2a−r(r
a)−s2a=0 Landau{Lifshitz equation (see Barouch et al. [17]): Ut=U
Uxx+U
JU Matrix Liouville equation (see Andreev [11]):/parenleftbig UxU−1
t=U Maxwell’s equations (see Jackson [88, page 177]): r
D=4;rH=4 cJ r
B=0;rE+1 c@B @t=0 Reduced Maxwell{Bloch equations (see Calogero and Degasperis [34, page 59]): Et−v=0;q x+Ev=0 rx+!v=0;v x−!r−Eq=0 Nambu{Jona Lasinio{Vaks{Larkin model (see Calogero and Degasperis [34, page 62]): iu(n) x=v(n)NX m=1v(m)u(m) iv(n) t=u(n)NX m=1u(m)v(m) Navier’s equation (see Eringen and Suhubi [57]): (+2)rr
u−rr u=@2u @t2 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 203 Navier{Stokes equations (see Landau and Lifshitz [99, page 49]): ut+(u
r)u=−rP +r2u Pohlmeyer{Lund{Regge model (see Calogero and Degasperis [34, page 61]): uxx−uyysinucosu+cosu sin3u/parenleftbig v2 x−v2 y
=0 /parenleftbig vxcot2u
x=/parenleftbig vycot2u
y Vector Poisson equation (see Moon and Spencer [118]): 45A=−curlE Prandtl’s boundary layer equations (see Iyanaga and Kawada [87, page 672]): ut+uux+vuy=Ut+UUx+ uyy ux+vy=0 Sigma-model (see Calogero and Degasperis [34, page 61]): vxt+(vxvt)v=0 Massive Thirring model (see Calogero and Degasperis [34, page 62]): iux+v+ujvj2=0 ivt+u+vjuj2=0 Toda equation { 3 + 1-dimensional (see Hirota [81]): r2logVn−Vn+1+2Vn−Vn−1=0 Unnamed equation (see Salingaros [144]): ru=ku Veselov{Novikov equation (see Bogdanov [26]):/parenleftbig @t+@3 z+@3 z
v+@z(uv)+@z(vw)=0 @zu=3@zv @zw=3@zv Yang{Mills equation (see Calogero and Degasperis [34, page 62]): (UUt)t−(UUx)x=0 Anti-self-dual Yang{Mills equation (see Ablowitz et al. [1]): @ @x1 Ω−1@Ω @x1 +@ @x2 Ω−1@Ω @x2 =0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 204 II Exact Analytical Methods Zakharov equations (see Glassey [66]): iEt+Exx=NE Ntt−Nxx=@2 @x2(jEj2) 44.4 The Laplacian in Dierent Coordinate Systems For ease of recognizing an unknown Laplacian (i.e., r2) in a dierential equation, we have frequently used the indeterminates fx;y;zginstead of the more customary notation for a specic coordinate system. For detailson any of these coordinate systems, see Moon and Spencer [120]. 1. rectangular u xx+uyy+uzz 2. cylindrical polar urr+1 rur+1 r2u+uzz 3. elliptic cylinder1 cosh2x−cos2y[uxx+uyy]+uzz 4. parabolic cylinder1 x2+y2[uxx+uyy]+uzz 5. spherical urr+2 rur+1 r2u+cot r2u+1 r2sin2u 6. prolate spheroidal 1 sinh2x+s i n2y[uxx+c o t hxux+uyy+c o tyuy]+1 sinh2xsin2yuzz 7. oblate spheroidal 1 cosh2x−sin2y[uxx+t a n hxux+uyy+c o tyuy]+1 cosh2xsin2yuzz 8. parabolic1 x2+y2 uxx+1 x+uyy+1 yuy +1 x2y2uzz 9. conical uzz+2 zuz+1 z2(x2−y2)n (x2−b2)(c2−x2)uxx−x[2x2−(b2+c2)]ux+ (b2−y2)(c2−y2)uyy−y[2y2−(b2+c2)]uyo 10. logarithmic-cylinder ( x2+y2)[uxx+uyy]+uzz 11. tangent-cylinder ( x2+y2)2[uxx+uyy]+uzz 12. cardioid-cylinder ( x2+y2)3[uxx+uyy]+uzz 13. hyperbolic-cylinder 2p (x2+y2)[uxx+uyy]+uzz 14. rose-cylinder 2( x2+y2)3=2[uxx+uyy]+uzz 15. Cassinian-oval e−2xp e2x+2excosy+1[uxx+uyy]+uzz 16. inverse Cassinian-oval e−2x/parenleftbig e2x+2excosy+1
3=2[uxx+uyy]+uzz 17. Maxwell-cylinder/parenleftbig e2x+2excosy+1
−1[uxx+uyy]+uzz 18. bi-cylinder (cosh x−cosy)2[uxx+uyy]+uzz 19. inverse elliptic-cylinder(cosh2x−sin2y)2 (cosh2x−cos2y)[uxx+uyy]+uzz CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 205 20. log tan-cylinder (sinh22x+s i n22y)[uxx+uyy]+uzz 21. log cosh-cylinder (cosh2x−sin2y)2 (cosh2xsinh2x+ (sinhxcoshx+s i nycosy)2)4[uxx+uyy]+uzz 22. ellipsoidalp (x2−b2)(x2−c2) (x2−y2)(x2−z2)@xhp (x2−b2)(x2−c2)uxi +p (y2−b2)(c2−y2) (x2−y2)(y2−z2)@yhp (y2−b2)(c2−y2)uyi +p (b2−z2)(c2−z2) (x2−z2)(y2−z2)@zhp (b2−z2)(c2−z2)uzi 23. paraboloidals (x−b)(x−c) (x−y)(x−z)@xhp (x−b)(x−c)uxi +s (y−b)(y−c) (x−y)(z−y)@yhp (b−y)(c−y)uyi +s (z−b)(z−c) (z−x)(z−y)@zhp (b−z)(z−c)uzi 44.5 Parametrized Equations at Specic Values 1. Polyanin and Zaitsev [130, page 29] tabulate solvable cases of the Abel equation yy0−y=sx+Axm: m s arbitrary−2(m+1) (m+3)2 −715/4 −4 6 −5/2 12 −2 0 −2 2 −5/3−3/16 −5/3−9/100 −5/363/4 −7/5−5/36m s −1 0 −1/2−2/9 −1/2−4/25 −1/2 0 −1/2 20 0arbitrary 1/2−12/49 2−6/25 26/25 Solutions are also tabulated for the Abel equations yy0−y=sx+A/parenleftbig x1=2+A+γA2x−1=2
and yy0−y=sx+Axp+A2xq. 2. Polyanin and Zaitsev [130, pages 251{254] tabulate solvable cases of y00=A1xn1ym1+A2xn2ym2: Solvable two parameter families (arbitrary m1andm2) include fn1=0;n2=0g,fn1=−m−3;n2=−m2−3g,a n dfn1= −1/2(m1+3 );n2=−1/2(m2+3 )g. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 206 II Exact Analytical Methods Several solutions are tabulated where one or both of the Aiare specied. Solutions are also available for m1m2 n1n2 1−3arbitrary (n16=−2)0 −7−7 4 3 −5−5 2 0 −3−7 0 1 3 −4 0 0 1 −2−3−2 0 1 0 −2−1−2 −5 3−5 3−7 310 3 −4 3−10 3 −7 3 −2 3−4 3 0−2 3 2 0 1 −3 2−2−3 2−2 0 1 −7 5−7 5−8 513 5 −4 3−5 3−5 3−7 3 0 1 −3 5−7 5−12 5−13 5 0 1 −1 2−1 2−5 2−7 2 −1 3−5 3−8 3−10 3 −8 3−7 3 −8 3−4 3 0 0 1 2m1m2n1n2 0−2−3−2 0 1 −1−3−2 0 0 −2 3−3−7 3 0 0 −1 2−4−5 2 −3−7 2 −5 2 −2 −1 2 −5 3−7 6 −3 2−5 2 −3 2−2 −1 2 0 −4 34 3 0−2 −1 2 0 1 3−5 3−10 3−7 3 0 1 1−3−5 0 1 0 0−5−3 1 0 2 0 5−4 −3 −20 7−13 7 −12 7 −15 7−9 7 0 0 1 3 1−6−5 0 1 2−18 5−14 5 −12 5−11 5 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 207 3. Polyanin and Zaitsev [130, pages 304{306] tabulate solvable cases of the modied Emden{Fowler equation xy00−ky0=Axn+1ym: Solvable two parameter families include fk=n/2;m6=−1, n6=−2g,fk=−n+m+3 /m+1,m6=−1;n6=−2g,a n dfk= −2n+m+3 /1-m,m6=−1;n6=−2g. Solvable one parameter families (with n=−2) includefk= −1g,fm=−2g,fm=−1;k6=−1g,a n dfm=−1/2;k6=−1g. Solvable one parameter families (with n6=−2) include fm=−7;k=1 3(n−1)g fm=−7;k=1 5(n−3)g fm=−4;k=1 2ng fm=−4;k=1 3(n−1)g fm=−2;k=1 3(n−1)g fm=−5 2;k=1 2ng fm=−5 2;k=1 3(2n+1 )g fm=−5 3;k=−3n−7g fm=−5 3;k=1 2ng fm=−5 3;k=1 2(3n+4 )g fm=−5 3;k=1 3(n−1)g fm=−5 3;k=1 3(2n+1 )g fm=−5 3;k=1 4(n−2)g fm=−5 3;k=−1 4(3n+ 10)g fm=−5 3;k=1 7(6n+5 )g fm=−7 5;k=1 3(n−1)gfm=−7 5;k=−1 3(5n+ 13)g fm=−1;k=n+1g fm=−1;k=1 2ng fm=−1 2;k=−2n−5g fm=−1 2;k=1 2n)g fm=−1 2;k=1 2(3n+4 )g fm=−1 2;k=1 3(n−1)g fm=−1 2;k=1 3(2n+1 )g fm=−1 2;k=−1 3(2n+7 )g fm=−1 2;k=1 5(6n+7 )g fm=1 2;k=1 2ng fm=1 2;k=−1 3(2n+7 )g fm=2;k=−7n−15g fm=2;k=1 2ng fm=2;k=−1 3(n+5 )g fm=2;k=−1 6(7n+ 20)g 4. Polyanin and Zaitsev [130, pages 278{281] tabulate solvable cases of the Emden{Fowler equation y00=Axnym(y0)k: Solvable two parameter families include n=0 ,m=0 ,a n d fk=2n+m+3 /n+m+2;m6=−1;n6=−1g. Solvable one parameter families include {fk6=1;2;m=−1;n=−1g {fk6=3/2;m=−1 2;n=−1 2g {fk=3m+5 2m+3;m6=−3 2;n=−1 2g {fk=3m+5 2m+3;m6=−3 2;n=1g {fk=3n+4 2n+3;m=−1 2;n6=−3 2g {fk=3n+4 2n+3;m=1;n6=−3 2g{fk=3n+4 2n+3;m=−n−3;n6=−3 2g {fk=1;m6=−1;0;n=−1g {fk=2;m=−1;n6=−1;0g {fk=2;m6=−2;0;n=−1g {fk=3;m=−n−3;n=−1g Isolated points at which the solution is tabulated include CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 208 II Exact Analytical Methods kmn 1 2−1 2−5 2 1−15 8 −20 13 −5 4 0 2 3−1 2−7 64 5−5 2−1 2 1−2 1 −1−1 8 71−3 4 −1 26 5−1 2−2 35 41−1 2 0 9 7−1 21 1 13 10−1 2−5 227 20−1 2−2 318 13−1 2−7 27 5−7 41 −10 71 −2 31 −1 21 1 0 1 5 1 10 7−1 2−5 222 15−1 2−2 3kmn 3 2−2−1 2 1 −1 2−2 −1 2 1 1 2 −1 223 15−2 3−1 211 7−5 2−1 28 50 1 1−7 4 −10 7 −2 3 −1 2 1 5 21 13−7 2−1 233 20−2 3−1 217 10−5 2−1 212 71−13 8 −1 27 4−1 21 0 1 9 5−2 3−1 213 7−3 41 −1 21 2−1−1 1−2 11 5−1 2−5 27 3−7 6−1 2kmn 5 2−5 2−1 2 −15 81 −20 131 −5 41 0 1 3−5 2 −7 2−1 2 −10 3−5 3 −20 72 −5 2−1 2 −13 5−7 5 −7 3−5 3 −15 72 −2−2 −1 −1 2 1 −4 3−1 2 −7 6−1 2 −5 6−5 3 −1 2−5 2 −5 3 0−4 −5 2 −1 2 2 1−7 −4 −2 −5 3 −7 5 −1 2 0 2−5 3 3−7 5. Polyanin and Zaitsev [130, page 242] tabulate solvable cases of the Emden{Fowler equation y00=Axnym: Solvable one parameter families include n=0 ,n=−m−3, n=−1 2(m+3 ) ,m=0 ,a n dm=1 . Isolated points at which the solution is tabulated include: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 44. Look-Up Technique 209 mn −71 −73 −5/4−1/2 −2−2 −21 −5/3−10/3 −5/3−7/3 −5/3−5/6 −5/3−1/2 −5/3 1 −5/3 2mn −7/3−13/5 −7/3 1 −1/2−7/2 −1/2−5/2 −1/2−2 −1/2−4/3 −1/2−7/6 −1/2−1/2 −1/2 1 2−5 2−20/7 2−15/7 6. Solvable cases of the following equations are also tabulated in Polyanin and Zaitsev [130]: (y0)k=Ays+Bxr[130, page 106] (y0)k=Ays+Bex[130, page 107] (y0)k=Aey+Bxr[130, page 107] (y0)k=Aey+Bex[130, page 107] y00=(A1xn1ym1+A2xn2ym2)(y0)k[130, pages 314{319] y00=xnym(y0)k+xn−1ym+1(y0)k−1[130, pages 349{352] y00=A1xn1ym1(y0)k1+A2xn2ym2(y0)k2withk16=k2[130, page 367] y000=Axy(y0)γ(y00)[130, pages 529{535] y000=Aey(y0)γ(y00)[130, page 577] y000=Aye(y0)2(y00)[130, page 577] References [1]Ablowitz, M. J., Costa, D. G., and Tenenblat, K. Solutions of multidimensional extensions of the anti-self-dual Yang{Mills equation. Stud. Appl. Math. 77 (1987), 37{46. [2]Abraham-Shrauner, B. Hidden symmetries and linearization of the modied Painleve{Ince equation. J. Math. and Physics 34 ,1 0( O c t o b e r 1993), 4809. [3]Abramowitz, M., and Stegun, I. A. Handbook of Mathematical Functions . National Bureau of Standards, Washington, D.C., 1964. [4]Aguirre, J., and Escobedo, M. 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Modied equations, rational solutions, and the Painleve property for the Kadomtsev{Petviashvili and Hirota{Satsuma equations. J. Math. Physics 26 , 9 (Sept 1985), 2174{2180. [170] Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. [171] Wilcox, R. Closed-form solution of the dierential equation zzzref50refzzz zzzref51refzzz by normal-ordering exponential operators. J. Math. Physics 11, 4 (April 1970), 1235{1237. [172] W o o d ,H .G . ,a n dM o r t o n ,J .B . Onsager’s pancake approximation for the fluid dynamics of a gas centrifuge. J. Math. Physics 101 (1980), 1{31. Part 1. [173] Yanagida, E. Structure of radial solutions to zzzref4refzzz in zzzref5refzzz. SIAM J. Math. Anal. 27 , 3 (July 1996), 997{1014. [174] Zhi-Xiong, C., and Ben-Yu, G. Analytic solutions of the Nagumo equation. IMA J. Appl. Mathematics 48 , 2 (1992), 107{115. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 45. Look-Up ODE Forms 219 45. Look-Up ODE Forms Applicable to Ordinary dierential equations. Yields An idea of whether or not an ordinary dierential equation has a closed- form solution. Idea An experienced dierential equations practitioner can look at many second order ordinary dierential equations and readily guess whether or not there is a closed form solution because there are many familiar forms that often appear. Procedure Having a listing of familiar dierential equation forms will make it possible to recognize these forms. We have tabulated below many of the familiar forms that appear for second order ordinary dierential equations. In the listings below, ( ) represents a term that contains constants. Such a term may or may not be correlated with other terms of the form ( ). For example, equation 22.6.5 in Abramowitz and Stegun [1] is /parenleftbig 1−x2
y00−(2+1 )xy0+n(n+2)y=0; whereis a real constant and nis an integer. Isolating the xdependence, we list this equation as /parenleftbig 1−x2
y00+()xy0+()y=0 and disregard the fact that the hidden values have constraints on them and, in fact, are related. 45.1 Equations of the Form: y00+c(x)y=0 c(x) = ( ) [1, 22.6.10] c(x)=−x [1, 10.4.1] c(x)=()−x2[1, 22.6.20] c(x)=()+()x+()x2[1, 19.1.1] c(x)=()x()[1, 9.1.51] c(x)=()+() x2 [1, 9.1.49] c(x)=() x+() x2 [1, 9.1.50] c(x)=()−() x−() x2 [1, 14.1.1] c(x)=()−x2+() x2 [1, 13.1.1 and 22.6.8] c(x)=()e2x−( ) [1, 9.1.54] c(x)=() 1−x2+() +x2 4(1−x2)2 [1, 22.6.7] CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 220 II Exact Analytical Methods c(x)=() 1−x2+1 (1−x2)2 [1, 22.6.14] c(x)=() (1−x)2+() (1+x)2+() 1−x2 [1, 22.6.3] c(x)=() x+() x2+ ( ) [1, 22.6.17] c(x)=()+() sin2x[1, 22.6.8] c(x)=()+() sin2x 2+() cos2x 2[1, 22.6.4] 45.2 Equations of the Form: y00+b(x)y0+c(x)y=0 b(x)=−x,c(x) = ( ) [1, 22.6.21] b(x)=−2x,c(x) = ( ) [1, 22.6.19] b(x)=2x,c(x)=−()x [1, 7.2.2] b(x)=2x,c(x)=x2−( ) [1, 10.1.1] b(x)=2x,c(x)=()−x2[1, 10.2.1] b(x)=()−x,c(x) = ( ) [1, 22.6.15] b(x)=()x,c(x)=()+x()[1, 9.1.53] b(x)=() x,c(x) = ( ) [1, 9.1.52] b(x)=() ,c(x)=()−()c o sx [1, 20.1.1] 45.3 Equations of the Form: xy00+b(x)y0+c(x)y=0 b(x)=()−x,c(x) = ( ) [1, 13.1.1] b(x)=()+x,c(x)=()+() x[1, 22.6.16] 45.4 Equations of the Form: (1−x2)y00+b(x)y0+c(x)y=0 b(x)=() ,c(x)=()−()x2[1, 20.1.8] b(x)=−x,c(x) = ( ) [1, 22.6.9] b(x)=−x,c(x)=()−()x2[1, 20.1.7] b(x)=−2x,c(x) = ( ) [1, 22.6.13] b(x)=−2x,c(x)=()+() 1−x2 [1, 8.1.1] b(x)=−3x,c(x) = ( ) [1, 22.6.11 and 22.6.12] b(x)=()x,c(x) = ( ) [1, 22.6.5 and 22.6.6] b(x)=()+()x,c(x) = ( ) [1, 22.6.1 and 22.6.2] 45.5 Equations of the Form: x2y00+b(x)y0+c(x)y=0 b(x)=x,c(x)=x2−( ) [1, 9.1.1] b(x)=x,c(x)=()−x2[1, 9.6.1] b(x)=2x,c(x)=()+x2[1, 10.1.1] b(x)=2x,c(x)=()−x2[1, 10.2.1] 45.6 Equation of the Form: x(1−x)y00+b(x)y0+c(x)y=0 b(x)=()−()x,c(x) = ( ) [1, 15.5.1] Note 1. Realize that the same equation may look dierent when written in dif- ferent variables. Some scaling of any given equation may be required to make it look like one of the forms listed. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 45. Look-Up ODE Forms 221 Reference [1]Abramowitz, M., and Stegun, I. A. Handbook of Mathematical Functions . National Bureau of Standards, Washington, D.C., 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 222 II Exact Analytical Methods CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 224 II.A Exact Methods for ODEs 46. An Nth Order Equation Applicable to The equationdny dxn=f(x). Yields Two exact forms of the solution are available. Idea The explicit solution can be written analytically. Procedure The general solution of the ordinary dierential equation for y(x) dny dxn=f(x) can be found by integrating with respect to xa total ofntimes. This produces y(x)=Zx x0dxZx x0dx


Zx x0f(x)dx+C1(x−x0)n−1 (n−1)! +C2(x−x0)n−2 (n−2)!+


+Cn−1(x−x0)+Cn;(46.1) for anyx0,w h e r et h efCjgrepresent arbitrary constants. This solution can also be written as y(x)=1 (n−1)!Zx x0(x−t)n−1f(t)dt+C1(x−x0)n−1 (n−1)! +C2(x−x0)n−2 (n−2)!+


+Cn−1(x−x0)+Cn(46.2) in which there are no repeated integrals. Sometimes the form in equation (46.2) is more useful than the form in equation (46.1). Example The ordinary dierential equation y(4)=s i nx; y(0) = 0;y0(0) = 0; y00(0) = 0;y000(0) = 0 has the solution y(x)=Zx 0dxZx 0dxZx 0dxZx 0sinxdx: (46.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 46. An Nth Order Equation 225 This solution may also be written as y(x)=1 6Zx 0(x−t)3sintdt: (46.4) Sometimes it is easier to evaluate the expression in equation (46.4) (by expanding out ( x−t)3and integrating the four terms) to determine that y(x)=s i nx−x+x3 6 than it is to evaluate the expression in equation (46.3). Notes 1. When the answer is to be computed numerically, the solution rep- resented by equation (46.2) is more useful than the form in equa-tion (46.1). It is much easier to numerically approximate a one- dimensional integral than a multi-dimensional integral. 2. See Ince [1, page 42]. Reference [1]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 226 II.A Exact Methods for ODEs 47. Use of the Adjoint Equation Applicable to Linear dierential equations. Yields A linear dierential equation of lower order. Idea For every solution of the adjoint equation we can nd, we can reduce the order of the original equation by one. Procedure If we have the nth order linear dierential operator L[
]( s h o w no p e r a t - ing on the function u(x)) L[u(x)] =a0(x)dnu dxn+a1(x)dn−1u dxn−1+


+an−1(x)du dx+an(x)u; (47.1) then the adjoint of L[
]i sd e  n e dt ob e L[
], whereL[
] is given by (shown operating on the function w(x)) L[w(x)] =(−1)ndn dxn[a0(x)w]+(−1)n−1dn−1 dxn−1[a1(x)w]+


+(−1)1d dx[an−1(x)w]+(−1)0[an(x)w] (see page 95 for details). The bilinear concomitant ofL[
] is dened to be B(u;w)=n−1X k=0n−1X m=k(−1)m−ku(n−m−1)(akw)(m−k)(47.2) and satises the equation wL[u]−uL[w]=d dxB(u;w); (47.3) for allu(x)a n dw(x). Suppose we wish to solve the equation L[u]=f(x). If we can nd a solution to L[w] = 0 and call it w(x), then we have (substituting into equation (47.3)) wL[u]−uL[w]=d dxB(u;w); or w(x)f(x)=d dxB(u;w); CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 47. Use of the Adjoint Equation227 or B(u;w)=Zx w(x)f(x)dx: (47.4) Therefore, to nd u(x), we can solve equation (47.4) instead of L[u]=f(x). In other words, w(x) is an integrating factor for the equation L[u]=f(x). The original dierential equation, L[u]=f(x), is of degree nwhereas equation (47.4) is of degree n−1. Special Case Forn= 2 the adjoint equation is important enough to write separately. If the linear operator L[
]i sd e  n e db y L[u(x)] =R(x)u00+S(x)u0+T(x)u, then the adjoint is L[w(x)] =Rw00+( 2R0−S)w0+(R00−S0+T)w,a n d the bilinear concomitant is B(u;w)=uSw +u0Rw−u(Rw)0. Example Suppose we wish to solve the equation L[u] = 1, where L[u]=(x2−x)u00+( 2x2+4x−3)u0+8xu: The adjoint, in this case, is the operator L[w]=(x2−x)w00+(−2x2+1 )w0+( 4x−2)w; and the bilinear concomitant is given by B(u;w)=u(2x2+2x−2)w+u0(x2−x)w−u(x2−x)w0: (47.5) A solution to L[w] = 0, obtained by the method of undetermined coecients, is w(x)=x2. Using this solution in equation (47.4), we obtain (with f(x)=1 ) B(u;w)=Zx w(x)f(x)dx=Zx x2dx=x3 3+C; whereCis an arbitrary constant. Using w=w=x2in equation (47.5) produces B(u;w)=(x4−x3)u0+2x4u: Equating these last two equations yields a rst order equation for u: (x4−x3)u0+2x4u=x3 3+C: (47.6) Note that equation (47.6) is a rst order equation (the original dierential equation was of second order). Because equation (47.6) is a rst order linear CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 228 II.A Exact Methods for ODEs equation, it can be solved by the use of integrating factors. Multiplying by x−1 x3e2xand integrating results in (x−1)2e2xu(x)=Zxx−1 3e2x+Ce2xx−1 x3 dx =2x−3 12e2x+C 2x2e2x+D;(47.7) whereDis another arbitrary constant. Hence, the nal solution is u(x)=1 (x−1)22x−3 12+C 2x2+De−2x : (47.8) Notes 1. If an operator and its adjoint are identical, then the operator is said to be formally self-adjoint (see page 95). In this case, the adjoint method does not help to nd a solution of the original dierential equation. 2. Similar results hold for linear partial dierential equations. For the partial dierential operator L[u]=nX i;j=1aij(x)@2u @xi@xj+nX i=1bi(x)@u @xi+c(x)u; the adjoint operator is dened by M[w]=nX i;j=1@2(aijw) @xi@xj−nX i=1@(biw) @xi+cw: With this denition of the adjoint, we nd Z D wL[u]−uM[w] dx+Z @DB[u;w]dx1


cdxi


dxn=0; (47.9) whereB[u;w] is dened by B[u;w]=nX i=1(−1)i8 < :2 4bi−nX j=1@aij @xj3 5uw+nX j=1aij w@u @xj−u@w @xj9 = ;: In equation (47.9), dx1


cdxi


dxnindicates the product dx1


dxn with the factor dxiremoved. See Garabedian [1, pages 161{162] or Zauderer [5, pages 483{486] for details. 3. If the elliptic operator L[
]i sd e  n e db y L[u]=−r
(pru)+qu,t h e n wL[u]−uL[w]=r
(−pwru+purw): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 47. Use of the Adjoint Equation229 If the hyperbolic operator eL[
] is dened by eL[u]=utt+L[u], then weL[u]−ueL[w]=er
[−pwru+purw;wut−uwt]; whereer=[r;@=@t ] is the space{time gradient operator. If the parabolic operator bL[
] is dened by bL[u]=ut+L[u], then wbL[u]−ubL[w]=er
[−pwru+purw;uw ]; where the operator bL[
] is dened by bL[u]=−ut+L[u]. Each of the last three equations can be integrated to obtain an expression similar to equation (47.9). See Zauderer [5] for details. 4. See also Ince [2, pages 123{125], Kaplan [3, pages 448{453], and Valiron [4, pages 323{324]. References [1]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [2]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [3]Kaplan, W. Operational Methods for Linear Systems . Addison{Wesley Publishing Co., Reading, MA, 1962. [4]Valiron, G. The Geometric Theory of Ordinary Dierential Equations and Algebraic Functions . Math Sci Press, Brookline, MA, 1950. [5]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 230 II.A Exact Methods for ODEs 48. Autonomous Equations { Independent Variable Missing Applicable to Ordinary dierential equations of the form F(y(n);y(n−1);:::;y00;y0;y)=0 . Yields An ordinary dierential equation of lower order. Idea An autonomous equation is one left invariant under the transformation x!x+a. Any ordinary dierential equation in which the independent variable does not appear explicitly is an autonomous equation. Because we know something about the solution, we can reduce the order of the dierential equation. Procedure Given thenth order autonomous equation F(y(n),y(n−1),:::,y00,y0, y) = 0, change the dependent variable from y(x)t ou(y)=y0(x). The resulting ordinary dierential equation for u(y) will be of lower order. To nd how the higher order derivatives transform, consult table 48.1. Afterthe ordinary dierential equation of lower order has been solved for u(y), y(x) can be determined from integrating u(y)=y 0(x); i.e.,Zdy u(y)=x. Example Suppose we want to solve the nonlinear autonomous equation d2y dx2−dy dx=2ydy dx: (48.1) Because there are no explicit occurrences of xin equation (48.1), we recognize the equation to be autonomous. Therefore, we change variables in equation (48.1) by u(y)=dy dx. Using table 48.1, equation (48.1) transforms intoudu dy−u=2yuor udu dy−1−2y =0: (48.2) From equation (48.2), either u=0o rdu dy−1−2y=0 . I fu(y)=0 ,t h e n dy dx= 0 and so one solution to equation (48.1) is y(x)=A; (48.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 48. Autonomous Equations { Independent Variable Missing 231 yx=u; yxx =uuy; yxxx =uu2 y+u2uyy; yxxxx =uu3 y+4uyyuyu2+u3uyyy; yxxxxx =uu4 y+7uyyyuyu3+4u3u2 yyy+1 1u2u2 yuyy+u4uyyyy; yx(5) =uu4 y+7uy(3)uyu3+4u3u2 y(3)+1 1u2u2 yuyy+u4uy(4); yx(6) =uu5 y+1 1uy(4)uyu4+1 5u4u2 y(3)uyy+3 2u3u2 yuy(3) +34u3uyu2 yy+2 6u2u3 yuyy+u5uy(5); yx(7) =uu6 y+5 7u2u4 yuyy+ 122u3u3 yuyyy+3 4u4u3 yy+ 180u3u2 yu2yy +76u4u3 yy+1 5u5uy(3)2 + 192u4uyuyyuy(3) +26u5uyyuy(4)+1 6u5uyuy(5)+u6uy(6) Table 48.1: How to transform derivatives under the change of independent variable:u(y)=yx(x). (To simplify notation, we have dened yx(n)to be thenth derivative of ywith respect to x. Similarly for uy(n).) whereAis a constant. Conversely, if u(y)6= 0, then equation (48.2) requires that du dy−1−2y=0: (48.4) Equation (48.4) can be integrated to obtain u(y)=y2+y+B; (48.5) whereBis a constant. Using u(y)=dy dx, equation (48.5) can be written as dy dx=y2+y+B,s ot h a tZ dy y2+y+B=R dx, and therefore2 Dtan−1/parenleftbig2y+1 D
= x+C,w h e r eD2=4B−1a n dCis an additional constant. Inverting this last equation gives yexplicitly as a function of x y(x)=Etan(Ex+F)−1 2; (48.6) whereE=D=2a n dF=CE. Hence, the two solutions to equation (48.1) are given by equations (48.3) and (48.6). Notes 1. This method is derivable from Lie group methods (see page 366).2. Schwarz’s paper [4] describes a REDUCE program that will automat- ically determine rst integrals for an autonomous system of equations. 3. The easiest way to make the necessary transformation in an au- tonomous dierential equation is by replacing every occurrence of CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 232 II.A Exact Methods for ODEs d dxwithud dy. For instance, writing equation (48.1) in the form d dxd dx(y) −d dx(y)=2yd dx(y) leads immediately to equation (48.2) via ud dy ud dy(y) −ud dy(y)=2yud dy(y): 4. Sometimes it is advantageous to write a pair of rst order autonomous equations as a single rst order equation, by dividing the two equa-tions. For example, the non-linear predator{prey equations dx dt=ax−bxy;dy dt=−cy+dxy (48.7) can be written in the form dx dy=ax−bxy −cy+dxy: (48.8) Although equation (48.7) cannot be solved explicitly in nite terms, from equation (48.8) we can show that F(x;y): =dx+by−clogx− alogyis a constant on the solution curves fx(t);y(t)g. 5. It is straightforward to create a Macsyma program that will perform the necessary change of variables. Program 48.1 shows a terminalsession in which the input equation 1 yd2y dx2−1 y2dy dx2 −1+1 y3=0 is transformed into y3−udu dyy2+u2y−1=0: 6. Autonomous systems of ordinary dierential equations can have cen- ter manifolds , which are a classication of the solution surface. As a simple example, consider the system x0=Ax+f(x;y); y0=Bx+g(x;y); (48.9) whereAis a constant matrix all of whose eigenvalues are imaginary, Bis a constant matrix all of whose eigenvalues have negative real part, and the functions fandgand their rst derivatives vanish at the point ( 0;0). Then, there is a function hsuch that his an invariant manifold under equation (48.9). hand its rst derivatives vanish at ( 0;0). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 48. Autonomous Equations { Independent Variable Missing 233 DEPENDS(Y,X)$ AUTONOMOUS(EQN,Y,X):= BLOCK([NEW,A,U,MAX_DEGREE,J], DEPENDS(U,Y),MAX_DEGREE:DERIVDEGREE(EQN,Y,X),KILL(A),A[0]:Y,FOR J:1 THRU MAX_DEGREE DO ( A[J]:EXPAND( SUBST(U,DIFF(Y,X),DIFF(A[J-1],X)) ) ), FOR J:1 THRU MAX_DEGREE DO ( NEW: SUBST( A[J], DIFF(Y,X,J), NEW ) ), FACTOR(NEW) )$ EQN: DIFF( DIFF(Y,X)/Y, X) - 1 + 1/Y**3; 2 y( y ) xx x 1 ---- - ----- - 1 + -- y2 3 yy AUTONOMOUS(EQN,Y,X); 32 2 y- u uy+ uy - 1 y ----------------------- 3 y Program 48.1: Macsyma program to change variables. dy[1]= u[y[x]]; dy[2]= D[u[y[x]],x] /. y’[x]->u[y[x]];dy[n_]:= D[dy[n-1],x] /. y’[x]->u[y[x]] dy2[n_]:= dy[n] /. {u[y[x]]->u, u’[y[x]]->u’, u’’[y[x]]->u’’, u’’’[y[x]]->u’’’, u’’’’[y[x]]->u’’’’} Table[ {n,dy2[n]}, {n,1,5}] // ColumnForm Program 48.2: Mathematica program to change variables: u(y)=yx(x). The stability of the solution ( 0;0) is the same as that of the smaller system x0=Ax+f(x;h(x)). 7. The results in table 48.1 can be obtained with the Mathematica code in program 48.2. The output of that program is {1, u} {2, u u’} 22 {3, u u’ + u u’’} 3 2 3 (3) {4, u u’ + 4 u u’ u’’ + u u } 4 2 2 3 2 3 (3) {5, u u’ + 11 u u’ u’’ + 4 u u’’ + 7 u u’ u + 4 (4) uu } CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 234 II.A Exact Methods for ODEs 8. See Bender and Orszag [1, pages 24{25] and Rainville and Bedient [3, pages 268{269]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Man, Y. K. First integrals of autonomous systems of dierential equations and the Prelle{Singer procedure. J .P h y s .A :M a t h .G e n .2 7 (1994), L329{ L332. [3]Rainville, E. D., and Bedient, P. E. Elementary Dierential Equations . The MacMillan Company, New York, 1964. [4]Schwarz, F. A REDUCE package for determining Lie symmetries of ordinary and partial dierential equations. Comput. Physics Comm. 27 (1982), 179{186. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 49. Bernoulli Equation 235 49. Bernoulli Equation Applicable to Ordinary dierential equations of the form: y0+ P(x)y=Q(x)yn. Yields An exact solution of the given equation. Idea By a change of dependent variable, a Bernoulli equation (which is a nonlinear equation of the form y0+P(x)y=Q(x)yn,w h e r enis not equal to 1) can be transformed to a rst order linear equation. This linear equation can be solved by the use of integrating factors. Procedure Suppose we have the equation y0+P(x)y=Q(x)yn; (49.1) which we recognize to be a Bernoulli equation. To solve, we divide the equation by ynand change the dependent variable from y(x)t ou(x)b y u(x)=y(x)1−n: This changes equation (49.1) into the rst order linear dierential equation 1 1−nu0+P(x)u=Q(x): (49.2) An exact solution of equation (49.2) can be found by integrating factors (see page 356). The solution is given by u(x)=e x p (n−1)Zx P(t)dtZx exp (1−n)Zs P(t)dt Q(s)ds : (49.3) Example Suppose we have the equation y0+y=y3sinx: (49.4) To solve this equation, divide it by y3and then dene u(x)=y(x)−2so that equation (49.4) becomes −1 2u0+u=s i nx: (49.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 236 II.A Exact Methods for ODEs The solution to equation (49.5) (obtained by the method of integrating factors) is u(x)=Ae2x+2 5(cosx+2s i nx); whereAis an arbitrary constant. Using y(x)=u(x)−1=2, the nal solution is found to be y(x)= Ae2x+2 5(cosx+2s i nx)−1=2 : Notes 1. Ifn= 1, then the original equation is in the form of equation (49.2); and it can be solved directly by the use of integrating factors. 2. See also Boyce and DiPrima [1, page 28], Ince [2, page 22], Rainville and Bedient [3, pages 69{71], and Simmons [4, page 49]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [3]Rainville, E. D., and Bedient, P. E. Elementary Dierential Equations . The MacMillan Company, New York, 1964. [4]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 50. Clairaut’s Equation 237 50. Clairaut’s Equation Applicable to Dierential equations of the form: f(xy0−y)= g(y0). Yields An exact implicit solution. Sometimes a singular solution may also be obtained. Idea A solution of the dierential equation f(xy0−y)=g(y0)i sk n o w n . Procedure Given the equation f(xy0−y)=g(y0); (50.1) a general solution (for which y00= 0) is given implicitly by f(xC−y)=g(C); (50.2) whereCis an arbitrary constant. Equation (50.1) may also have a singular solution. If it does, it can be obtained by dierentiating equation (50.1)with respect to xto obtain y 00[f0(xy0−y)x−g0(y0)] = 0: (50.3) If the rst term in equation (50.3) is zero, then equation (50.2) is recovered. If the second term in equation (50.3) is zero, then equations (50.1) and (50.2) can be solved together to eliminate y0. The resulting equation for y=y(x) will have no arbitrary constants and so will be a singular solution. Example 1 Suppose we have the ordinary dierential equation (xy0−y)2−(y0)2−1=0: (50.4) Because equation (50.4) is of the same form as equation (50.1) (with f(x)= x2,g(x)=x2−1), a general solution can immediately be written down as (xC−y)2=C2+1o r y=Cxp C2−1; (50.5) whereCis an arbitrary constant. To nd the singular solution, we dierentiate equation (50.4) with respect toxto obtain y00[2(xy0−2)x−2y0]=0: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 238 II.A Exact Methods for ODEsC/2 G L RC/1 GN/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /./. /. /. /. /. /. /. /. /. /././. /././. /././. /././. /././. /././././. /././. /./././././. /././. /././. /././././. /././. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./././. /./././. /./././././. /./././. /././. /././. /././. /././. /. /././. /././. /././. /././. /. /. /. /. /. /. /./. /. /./. /. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /. /././. /. /./././././. /././. /././. /. /././. /./././././. /./././. /././. /././. /././. /././. /. /./. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /. /./. /. /. /. /././. /. /././././. /././././. /././././././././././. /././. /. /././././././. /././. /. /././. /././. /. /./. /. /./. /. /./. /./. /./././. /. /././././. /. /. /././. /. /. /././. /. /. /./. /. /. /./. /. /. /./././././././. /././././././././. /././. /./././. /./././. /././. /././. /. /./././. /. /././. /./. /././. /. /./. /././. /. /./. /. /././. /. /././. /. /./. /. /././. /. /. /. /. /. /. /././. /. /. /. /./././././././././. /./././././././. /./. /././././. /. /. /././. /. /. /./././. /././. /./././. /./. /. /././. /././. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /./././. /. /. /./. /./. /. /./././. /././././. /././. /././. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././. /./././. /./././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /./././././././././././././././. /././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /, /1 /1 g/, g iv GN Figure 50.1: Solution curves for the dierential equation in Example 2. If the second term is set equal to zero, then we nd y0=xy x2−1: (50.6) Using equation (50.6) in equation (50.4), we determine the singular solution to be x2+y2=1: (50.7) Note that equation (50.7) is not derivable from (50.5) for any choice of C. Example 2 For the dierential equation xy0−y=g(y0), withg(z)=5 2(z3−z), a set of solution curves is shown in gure 50.1. Because g(z) is a cubic, there are regions where there are three dierent solutions for a specied xandy. This is clearly shown in the gure. The singular solution to the above dierential equation can be easily shown to be y=( 5+2x)3=2=p 135. Notes 1. The singular solution obtained by this method turns out to be the locus of the solutions in equation (50.2). That is, the envelope of thesolutions in equation (50.2), for all possible values of the parameter C, will be the singular solution. See Ford [1, pages 16{18] for details. 2. A generalization of Clairaut’s equation is Lagrange’s equation (see page 363). 3. Clairaut’s partial dierential equation z=nX i=1xi@z @xi+f @z @x1;:::;@z @xn has the solution z=Pn i=1aixi+f(a1;a2;:::;an). See Kamke [3, section 13.8, page 123]. 4. See also Ince [2, pages 39{40] and Rainville and Bedient [4, pages 263{265]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 50. Clairaut’s Equation 239 References [1]Ford, L. R. Dierential Equations . McGraw{Hill Book Company, New York, 1955. [2]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [3]Kamke, E. Dierentialgleichungen Losungsmethoden und Losungen ,v o l .I I . Chelsea Publishing Company, New York, 1947. [4]Rainville, E. D., and Bedient, P. E. Elementary Dierential Equations . The MacMillan Company, New York, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 240 II.A Exact Methods for ODEs 51. Computer-Aided Solution Applicable to Some classes of ordinary dierential equations, most frequently rst and second order equations. Yields An exact solution. Idea Several of the popular computer algebra languages have a symbolic dierential equation solver. Procedure Find a computer system that runs any of the following commercial computer languages: AXIOM, Derive, FORMAC, Macsyma, Maple, Math- ematica, muMath, or REDUCE. Identify the routine that solves dierential equations automatically, and use that on your problem. For URLs of these software packages, see page 71. Nearly all of the symbolic algebra programs have a specialied interface that makes it easy to identify and use the dierential equation solver. This interface usually displays the output in a very attractive way; the ascii output shown below is less attractive but represents one output option. In each of the packages below a dierent package was asked to solve the simple dierential equations y00+4y=0a n dy0=xy2+y. Example 1 The following Macsyma session was run by Je Golden. Note that (c2), (c3), and (c4) are input lines (\command" lines) and that (d2), (d3), and (d4) are output lines (\display" lines). On the rst input line, the rst equation is dened to be eqn1 . On the second line, a solution is requested. Note that %k1and%k2are arbitrary constants in the solution that Macsyma found. The third input line denes the second equation tobeeqn2 , and the fourth line requests the solution (in this case %cis the arbitrary constant in the solution). Starting Macsyma math engine with no window system... This is Macsyma 421.0 for SGI (IRIX) computers.Copyright (c) 1982 - 1997 Macsyma Inc. All rights reserved.Portions copyright (c) 1982 Massachusetts Institute of Technology.All rights reserved.Type "DESCRIBE(TRADE_SECRET);" to see important legal notices.Type "HELP();" for more information. /usr/macsyma-421/system/init.lsp being loaded. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 51. Computer-Aided Solution 241 (c1) eqn1: ’diff(y,x,2) + 4*y = 0; 2 dy (d1) --- + 4 y = 0 2 dx (c2) ode(eqn1, y, x);/usr/macsyma-421/ode/ode.o being loaded. /usr/macsyma-421/ode/odeaux.o being loaded. /usr/macsyma-421/ode/ode2.o being loaded. (d2) y = %k1 sin(2 x) + %k2 cos(2 x)(c3) eqn2: ’diff(y,x) = x*y^2 + y; dy 2 (d3) -- - x y - y = 0 dx (c4) ode(eqn2, y, x); x %e (d4) y = ---------------- x % c-( x-1 )% e Example 2 The following MAPLE session was run by the author. Note that input lines begin with a greater than sign. On the rst input line, the rst equation is dened to be eqn1 . On the second input line, a solution is requested. Note that C1andC2are arbitrary constants in the solution that MAPLE found. The third input line denes the second equation to beeqn2 , and the fourth line requests the solution. |\^/| Maple V Release 3 (Zwillinger & Associates) ._|\| |/|_. Copyright (c) 1981-1994 by Waterloo Maple Software and the \ MAPLE / University of Waterloo. All rights reserved. Maple and Maple V<____ ____> are registered trademarks of Waterloo Maple Software. | Type ? for help. > eqn1:= diff(y(x),x$2)+4*y(x)=0; /2 \ |d | eqn1 := |----- y(x)| + 4 y(x) = 0 |2 | \d x / > dsolve( eqn1, y(x) ); CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 242 II.A Exact Methods for ODEs y(x) = _C1 cos(2 x) + _C2 sin(2 x) > eqn2:= diff(y(x),x)-x*y(x)^2-y(x)=0; /d \ 2 eqn2 := |---- y(x) | - x y(x) - y(x) = 0 \d x / > dsolve( eqn2, y(x) ); 1 ---- = - x + 1 + exp(- x) _C1 y(x) Example 3 The following Mathematica session was run by Alexei Bocharov. Note that thenth input line is denoted In[n]and thenth output line is denoted Out[n]. On the rst input line ( In[4] ), the rst equation is input and the solution is requested. Note that C[1] andC[2] are arbitrary constants in the solution that Mathematica found. The next input line denes thesecond equation and requests the solution. In[4]:= DSolve[y’’[x]+4y[x]==0,y[x],x] Out[4]= {{y[x] -> C[2] Cos[2 x] - C[1] Sin[2 x]}} In[5]:= DSolve[y’[x]==x*y[x]^2+y[x],y[x],x] 1 Out[5]= {{y[x] -> ----------------}} -x 1 - x - E C[1] Example 4 The following MuPAD terminal session was run by Paul Zimmermann. Note that input lines begin with the symbol >>. The rst command, setuserinfo(ode,1) , tells the system to prints comments. On the second input line, the rst equation is input and the solution is requested. Note thatC1,C2,a n dC3are arbitrary constants in the solutions that MuPAD found. The next input line denes the second equation and requests the solution. *----* MuPAD 1.4.0 --- Multi Processing Algebra Data Tool /| /| *----* | Copyright (c) 1992-97 by B. Fuchssteiner, Automath| *--|-* University of Paderborn. All rights reserved.|/ |/*----* ----------- Developers NSB Version --------------- >> setuserinfo(ode,1): >> solve(ode(y’(x)=x*y(x)^2-y(x), y(x))); CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 51. Computer-Aided Solution 243 Riccati equation Riccati method worked {1 } { 0, ----------------- } { x + C1 exp(x) + 1 } >> solve(ode(y’’(x)+4*y(x)=0, y(x))); linear ordinary differential equation of order 2with constant coefficients {C2 cos(2 x) + C3 sin(2 x)} Example 5 The following Derive terminal session was run by David Stoutemyer. Note that input and output lines begin with an octothorpe ( #) and are numbered consecutively. The input was entered in a one-line dialog boxthat had a Greek toolbar and other capabilities. #2: DSOLVE2(0, 4, 0, x, y) User #3: y COS(2 x) + c2 SIN(2 x) Simp(#2)#4: BERNOULLI_GEN(-1, x, 2, x, y) User 1x #5: --- = c #e - x + 1 Simp(#4) y Example 5 The following REDUCE terminal session was run by Winfried Neun. Note that all input lines are numbered. The rst command tells the system to load the ODE solver. On the second input line the rst equa- tion is input and the solution is requested. Note that arbconstant(1) {arbconstant(3) are arbitrary constants in the solutions that REDUCE found. The next input line denes the second equation and requests the solution. 1: load odesolve; (odesolve) 2: depend y,x;3: odesolve(df(y,x,2)+4*y=0,y,x);{y= - arbconst(2)*sin(2*x) + arbconst(1)*cos(2*x)} 4: odesolve (df(y,x)=x*y^2 +y,y,x); xx CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 244 II.A Exact Methods for ODEs 1 arbconst(3) - e *x + e {---=-------------------------} yx e Notes 1. A comparative dierential equation review of the languages AXIOM, Derive, Macsyma, Maple, Mathematica, MuPad, and REDUCE ismaintained by Postel and Zimmermann [13]. Presently, they have 54 equations that they have run though each of the above systems; the input and output les for each are available. 2. Moussiaux [12] has made available the program CONVODE , which sym- bolically solves ordinary and partial dierential equations across the internet. For example, sending depend y,x; CONVODE( {df(y,x,2)+4*y=0}, {y}, {x}, {}, {english}); [email protected] will have the solution ofy00+4y= 0 sent to you via email with comments in English (the de- fault is French). See http://www.physique.fundp.ac.be/physdpt/ administration/convode.html .N o t e t h a t CONVODE is based on REDUCE. 3. REDUCE can be used interactively over the web via the site http:// www.zib-berlin.de/Symbolik/reduce/testreduce.html . 4. MathServ provides an interface between the user and Mathematica (seehttp://math.vanderbilt.edu/~pscrooke/detoolkit.shtml ). Templates for twelve dierent types of ODEs are available; the usercan specify the functions appearing in them. 5. Packages that can handle a wider variety of dierential equations are constantly being created. See, for example, Chan [2], Kovacic [9],Schmidt [15], or Watanabe [20]. An example of the use of FORMAC may be found in Hanson et al. [5]. Shtokhamer [16] presents a Macsyma program that implements the Prelle{Singer algorithm and gives several examples. 6. All of the programs illustrated above and many others (such as the package by Hubbard and West [7]) can be run on a microcomputer (such as an IBM PC or a Macintosh). 7. Given a homogeneous linear dierential equation whose coecients are in a nite algebraic extension of Q[x], Singer’s [17] paper has a decision procedure to determine a basis for the Liouvillian solu- tions. Liouvillian functions are essentially those functions that can be built up from rational functions by algebraic operations, taking exponentials and by integration. In detail LetKbe a eld of functions. The function is a Liouvillian generator overKif it is: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 51. Computer-Aided Solution 245 {algebraic over K,t h a ti si fsatises a polynomial equation with coecients in K; {exponential over K, that is if there is a inKsuch that 0=0, which is an algebraic way of saying that =e x p; or {an integral over K, that is if there is a inKsuch that 0=, which is an algebraic way of saying that =R . LetKbe a eld of functions. An over-eld K(1;:::;n)o fK is called a eld of Liouvillian functions over Kif eachiis a Liouvillian generator over K. A function is Liouvillian over K if it belongs to a Liouvillian eld of functions over K. Then, some of the important theorems in this area are Theorem There is an algorithm that, given a second order linear dierential equation, y00+ay0+by=0w i t haandbrational functions of x, either nds two Liouvillian solutions such that every solution is a linear combination with constant coecients of these two solutions or proves that there is no Liouvilliansolution (except zero). Theorem There is an algorithm that, given a linear dierential equation of any order, the coecients of which are rational or algebraic functions: either nds a Liouvillian solution or provesthat there is none. Theorem LetAbe a class of functions containing the coe- cients of a linear dierential operator L,l e tgbe an element of A, and let us suppose that the equation L[y]=ghas an elementary solution over A. Then, either L[w] = 0 has an algebraic solution overA,o rybelongs toA. Theorem LetAbe a class of functions, that contains the co- ecients of a linear dierential operator L,l e tgbe an element ofA, and let us suppose that the equation L[y]=ghas a Liouvillian solution over A. Then either L[w] = 0 has a solution exp(R z(x)dx)w i t hzalgebraic over A,o rybelongs toA.a See Davenport et al. [4] for details. See also Bronstein [1]. References [1]Bronstein, M. The transcendental Risch dierential equation. J. Symbolic Comp. 9 , 1 (1990), 49{60. [2]Chan, W. C. A novel symbolic ordinary dierential equation solver. SIGSAM Bulletin 15 , 3 (August 1981), 9{14. [3]Char, B. W., Gedded, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B., and Watt, S. M. MAPLE V Library Reference Manual . Springer{ Verlag, New York, 1991. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 246 II.A Exact Methods for ODEs [4]Davenport, J. H., Siret, Y., and Tournier, E. Systems and Algorithms for Algebraic Computation . Academic Press, New York, 1988. [5]Hanson, J. H., Benander, A. C., and Benander, B. A. The computer generated symbolic solution of a system of linear rst order dierentialequations. Comp. & Maths. with Appls. 19 , 7 (1990), 7{12. [6]Hearn, A. C. REDUCE 2 user’s manual. Tech. Rep. UCP-19, University of Utah, Salt Lake City, 1973. Computational Physics Group Report. [7]Hubbard, J., and West, B. MacMath: A Dynamical Systems Software Package . Springer{Verlag, New York, 1991. [8]Inference Corp. 3916 S. Sepulveda Blvd., Culver City, CA, 90230. [9]Kovacic, J. J. An algorithm for solving second order linear homogeneous dierential equations. J. Symbolic Comp. 2 (1986), 3{43. [10]Macsyma .Mathematics Reference Manual . Macsyma, Arlington, MA, 1993. [11]Meyer, K. R., and Schmidt, D. Computer Aided Proofs in Analysis . Springer{Verlag, New York, 1990. [12]Moussiaux, A. CONVODE: A REDUCE package for solving dierential equations. J. Comput. Appl. Math. 48 , 1{2 (1993), 157{165. [13]Postel, F., and Zimmermann, P. Ar e v i e wo ft h eO D Es o l v e r so fA x i o m , Derive, Macsyma, Maple, Mathematica, MuPad, and Reduce. In Proceedings of the 5th Rhine Workshop On Computer Algebra (Saint-Louis, France, 1996), ISL. http://www.loria.fr/~zimmerma/ComputerAlgebra . [14]P r e l l e ,M .J . ,a n dS i n g e r ,M .F . Elementary rst integrals of dierential equations. Trans. Amer. Math. Soc. 279 , 1 (September 1983), 215{229. [15]Schmidt, P. Substitution methods for the automatic symbolic solution of dierential equations of rst order and rst degree. In EUROSAM 79 ,E .N g , Ed., Lecture Notes in Computer Science 1979. Springer{Verlag, New York, 1979, pp. 164{176. [16]Shtokhamer, R. Solving rst order dierential equations using the Prelle{ Singer algorithm. Tech. Rep. 88-09, University of Delaware, Newark, DE,1988. [17]Singer, M. F. Liouvillian solution of n-th order homogeneous linear dierential equations. Am. J. Math. 103 , 4 (1981), 661{682. [18]Tournier, E. Computer Algebra and Dierential Equations .A c a d e m i c Press, New York, 1990. [19]Watanabe, S. A technique for solving ordinary dierential equations using Riemann’s P{functions. In SYMSAC 81: Proceedings of the 1981 ACM Symposium on Symbolic and Algebraic Computation ,P .S .W a n g ,E d .A C M , New York, 1981, pp. 36{43. [20]Watanabe, S. An experiment towards a general quadrature for second order linear ordinary dierential equations by symbolic computation. InEUROSAM 1984 , J. Fitch, Ed., Lecture Notes in Computer Science. Springer{Verlag, New York, 1984, pp. 13{22. [21]Wolfram, S. The Mathematica Book , third ed. Cambridge University Press, New York, 1996. [22]Wooff, C., and Hodgkinson, D. muMATH: A Microcomputer Algebra System . Academic Press, New York, 1987. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 52. Constant Coecient Linear Equations 247 52. Constant Coecient Linear Equations Applicable to Homogeneous linear ordinary dierential equations with constant coecients. Yields An exact solution. Idea Linear constant coecient ordinary dierential equations have expo- nential solutions. The method of undetermined coecients can be used tosolve this type of equation after a polynomial has been factored. Procedure Given thenth order linear equation y(n)+an−1y(n−1)+


+a1y0+a0y=0; (52.1) where thefaigare constants, look for a solution of the form y(x)=Cex; (52.2) whereCis an arbitrary constant. Substituting equation (52.2) into equa- tion (52.1) yields exh n+an−1(n−1)+


+a1+a0i =0: (52.3) Hence, equation (52.2) is a solution of equation (52.1) if is a root of the characteristic equation , dened by n+an−1(n−1)+


+a1+a0=0: (52.4) If equation (52.4) has ndierent rootsfig, then the general solution to (52.1) is, by use of superposition, y(x)=Cnenx+Cn−1en−1x+


+C1e1x; where thefCigare arbitrary constants. If some of the roots of equation (52.4) are repeated (say 1=2=


=m), then the solution corre- sponding to these figis y(x)=(Cmxm−1+Cm−1xm−2+


+C2x+C1)e1x: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 248 II.A Exact Methods for ODEs Example Given the linear dierential equation y(7)−14y(6)+8 0y(5)−242y(4)+ 419y(3)−416y00+ 220y0−48y=0; (52.5) we substitute y(x)=exto nd the characteristic equation 7−146+8 05−2424+ 4193−4162+ 220−48 = 0; which factors as (−1)3(−2)2(−3)(−4) = 0: (52.6) The roots of equation (52.6) are f1;1;1;2;2;3;4g. The general solution to equation (52.5) is therefore y(x)=fC0+C1x+C2x2gex+fC3+C4xge2x+C5e3x+C6e4x; wherefC0;:::;C 6gare arbitrary constants. Notes 1. Using the transformation described on page 146, the system in equa- tion (52.1) can be written in the form y0=Ay,w h e r eAis an nnconstant matrix. Then the techniques for vectors ODEs (see page 421) may be used. 2. See Boyce and DiPrima [1, section 5.3, pages 263{268] and Simmons [2, pages 83{86]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 53. Contact Transformation 249 53. Contact Transformation Applicable to First order and (occasionally) second order ordinary dierential equations. Yields A reformulation, which may lead to an exact solution (sometimes in parametric form). Idea By changing variables, a dierent and sometimes easier dierential equation may be found. Procedure Given a relation between three variables (x;y;p )=0; (53.1) it will be a rst order ordinary dierential equation if dy−pdx=0 . I ft h e variables in equation (53.1) are changed by x=x(X;Y;P ); y=y(X;Y;P ); p=p(X;Y;P );(53.2) then the transformed equation ( X;Y;P ) = 0 will also be an ordinary dierential equation if dY−PdX = 0. If this is true, then equation (53.2) is acontact transformation . For example, the change of variables 8 >< >:x=P y=PX−Y p=X9 >= >;()8 >< >:X=p Y=px−y P=x9 >= >;(53.3) is a contact transformation. It is easy to show this: 0=dy−pdx =d(PX−Y)−XdP =PdX−dY: If the new dierential equation, ( X;Y;P ) = 0, can be solved, then the solution to (x;y;p ) = 0 may be determined by eliminating X,Y,a n dP from the original equation, using the solution found and the transformation rules. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 250 II.A Exact Methods for ODEs Example Suppose we have the nonlinear rst order ordinary dierential equation 2ydy dx2 −2xdy dx−y=0; (53.4) which we may write as 2yp2−2xp−y=0: We utilize the contact transformation in equation (53.3) to obtain, after some algebra, the new rst order ordinary dierential equation P+Y1−2X2 2X3−3X =0 o rdY dX+Y1−2X2 2X3−3X =0: (53.5) This dierential equation can be solved by integrating factors to obtain Y=C/parenleftbig 2X3−3X
1=3; (53.6) whereCis an arbitrary constant. Now that we have the solution of the transformed equation, we can nd the solution of the original dierential equation. UtilizingY=xX−yandP=xfrom equation (53.3), equations (53.5) and (53.6) can be written as x+(xX−y)1−2X2 2X3−3X =0; xX−y=/parenleftbig 2X3−3X
1=3:(53.7) NowXcan be eliminated between these two equations by, say, the method of resultants (see page 50). This produces the solution to equation (53.4) in the formf(x;y) = 0 (there are 21 algebraic terms in this representation). Alternately, we can obtain a parametric representation of the solution by solving equation (53.7) for x=x(X)a n dy=y(X) and then treating X as a parameter. Notes 1. Composing two contact transformations or taking the inverse of a contact transformation results in another contact transformation. Because the identity transformation is also a contact transformation, the set of all contact transformations forms an innite dimensionaltopological group. 2. This method is derivable from the method of Lie groups (see page 366), where it goes by the name of the extended group of transforma- tions . See Ince [4, pages 40{42] or Seshadri and Na [6, pages 18{20]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 53. Contact Transformation 251 3. The condition dy−pdx= 0 states that, if the point ( x;y) is on a curve, thenpshould be its tangent. The change of variables in this method gives a dierent parameterization of the same curve. In particular, if two curves touch in the old parameterization, then they also touch inthe new parameterization; hence the name of the transformation. 4. Some second order ordinary dierential equations also may be solved by this method. If R= dP dX=d2Y dX2and1 R=dp dx=d2y dx2,t h e nw em a y use the relation dP−RdX =dx−Rdp. 5. In more generality, a transformation of the 2 n+1 variablesfz;xj;pjj j=1;:::;ngto the 2n+ 1 variablesfZ;Xj;Pjjj=1;:::;ngis a contact transformation if the total dierential equation dz−p1dx1−p2dx2−


−pndxn=0 is invariant under the transformation; that is, if the equality (dZ−P1dX1−P2dX2−


−PndXn) =(dz−p1dx1−p2dx2−


−pndxn) holds identically for some nonzero function (x;p;z). See Iyanaga and Kawada [5, pages 286 and 1448] for details. 6. A contact transformation is also a canonical transformation (see page 132). The generating function of the canonical transformation, Ω, satises the three relations: Ω( x;z;X;Z )=0 ,@Ω @Xj+Pj@Ω @Z=0 ,a n d @Ω @xj+pj@Ω @z=0 . 7. Named contact transformations include (a) The Legendre transformation (see page 467) is given by Ω = Z+z+PxjXj,Z=P jpjxj−z,Xj=−pj,Pj=−xj,a n d =−1. (b) The Pedal transformation is given by Ω = Z2−zZ−PxjXj+PX2 j,Xj=−pjZ,pj=−2Xj−xj 2Z−z,a n d=Z 2Z−z. (c) The similarity transformation is given by Ω = ( Z−z)2−a2+P(Xj−x)j)2,Xj=xj−apj/parenleftbig 1+Pp2 j
−1=2,Pj=pj,Z= xj+a/parenleftbig 1+Pp2 j
−1=2,a n d=1 . 8. Some other contact transformations are 8 >>>< >>>:x=X−YP y=−Yp P2−1 p=Pp P2−19 >>>= >>>;()8 >>>< >>>:X=x−yp Y=yp p2−1 P=−pp p2−19 >>>= >>>;(53.8) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 252 II.A Exact Methods for ODEs 8 >>>>< >>>>:x=X−aP p 1+P2 y=Y+ap 1+P2 p=P9 >>>>= >>>>;()8 >>>>< >>>>:X=x+ap p 1+p2 Y=y−ap 1+p2 P=p9 >>>>= >>>>;: (53.9) 9. See also Bateman [1, pages 81{83], Carath eodory [2, Chapter 7, pages 102{120], and Chester [3, pages 206{207]. References [1]Bateman, H. Partial Dierential Equations of Mathematical Physics .D o v e r Publications, Inc., New York, 1944. [2]Caratheodory, C. Calculus of Variations and Partial Dierential Equa- tions of the First Order . Holden{Day, Inc., San Francisco, CA, 1965. [3]Chester, C. R. Techniques in Partial Dierential Equations . McGraw{Hill Book Company, New York, 1970. [4]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [5]Iyanaga, S., and Kawada, Y. Encyclopedic Dictionary of Mathematics . MIT Press, Cambridge, MA, 1980. [6]Seshadri, R., and Na, T. Y. Group Invariance in Engineering Boundary Value Problems . Springer{Verlag, New York, 1985. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 54. Delay Equations 253 54. Delay Equations Applicable to Ordinary dierential delay equations. Yields In many cases, an exact analytical solution. Idea There are several standard techniques for delay equations. Procedure The standard methods for solving delay equations are by the use of Laplace transforms Fourier transforms Generating functions General expansion theorems The method of steps For the rst two methods, the technique is the same as it is for ordinary dierential equations (see page 347). That is, the transform is taken of the delay equation; by algebraic manipulations the transform is explicitly determined; and then an inverse transformation is taken. See Example 1. For a delay equation with a single delay, the method of steps consists of solving the delay equation in successive intervals, whose length is the time delay. In each interval, only an ordinary dierential equation needs to besolved. See Example 2. The method of generating functions is frequently used when only in- tegral values of the variables are of interest. The technique is similarto the technique for integral transforms described above. For generating functions, the integration is replaced by a summation, and the \inverse transformation" is generally a dierentiation (see page 315 for more de-tails). See Example 3. The general expansion theorems are all of the same form; given a delay equation, the solution can be expressed as a sum over the roots of a transcendental equation called the characteristic equation . Example 1 Suppose we have the delay equation y0(t)+ay(t−1) = 0; (54.1) with the boundary conditions y(t)=y0 when−1t0; (54.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 254 II.A Exact Methods for ODEs whereais a constant. We dene the Laplace transform of y(t)t ob e Y(s)b yY(s)=R1 0e−sty(t)dt. Multiplying equation (54.1) by e−stand integrating with respect to tyields Z1 0e−sty0(t)dt+aZ1 0e−sty(t−1)dt=0: (54.3) The rst integral in equation (54.3) can be integrated by parts to yield Z1 0e−sty0(t)dt=sY(s)−y0: (54.4) The second integral in equation (54.3) can be evaluated by changing the variable of integration from ttou=t−1: aZ1 0e−sty(t−1)dt=aZ1 −1e−s(u+1)y(u)du =aZ1 0e−s(u+1)y(u)du+aZ0 −1e−s(u+1)y(u)du =ae−sY(s)+ay01−e−s s:(54.5) Utilizing equations (54.4) and (54.5) in equation (54.3) results in the alge- braic equation sY(s)−y0+ae−sY(s)+ay01−e−s s=0; which can be solved for Y(s): Y(s)=y0 s−ay0 s(s+ae−s): (54.6) If this formula for Y(s) is expanded as Y(s)=y0 s−y01X n=0(−1)nan+1e−nss−n−2; then an inverse Laplace transform may be taken term by term to conclude that y(t)=y0btc+1X n=0(−a)n(t−n+1 )n n!; (54.7) where the floor function, btc, is the greatest integer less than or equal to t. Another way of expressing the solution in equation (54.7) is by taking the inverse transform of Y(s), as dened in equation (54.6), directly, and CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 54. Delay Equations 255 using Cauchy’s theorem to evaluate the Bromwich contour integral. This results in y(t)=−ay0X resrt sr(1 +sr); (54.8) where the summation is over all roots of the equation s+ae−s=0: (54.9) All the roots of equation (54.9) will be simple unless a=e−1,w h e n there is a double root at s=−1. The solution in equation (54.8) can be approximated (for large t) by just using the srthat has the smallest real part. There exist theorems (see Pinney [15] for instance) that allowthe solution of equation (54.1) to be written in the form of equation (54.8) immediately. Example 2 In the method of steps, only a sequence of ordinary dierential equations need to be solved. To illustrate this method, consider equations (54.1) and(54.2). In the interval 0 y1, the solution satises y 0(t)+ay0=0; y(0) =y0:(54.10) The equation (54.10) has the solution y(t)=y0(1−at); for 0y1: (54.11) Now we solve for y(t) in the next interval of length one. Using equation (54.11) we nd that, in the interval 1 y2, the solution satises y0(t)+ay0[1−a(t−1)] = 0; y(0) =y0(1−a):(54.12) The equation (54.12) has the solution y(t)=y0 1−at+1 2a2(t−1)2 ; for 1y2: This process can be repeated indenitely. The solution obtained is identical to the solution in equation (54.7). Example 3 This example shows how generating functions may be used to solve delay equations. Consider equations (54.1) and (54.2). Dene the generating function associated with y(t), for 0t1, by Y(t;k)=1X p=0y(t+p)kp: (54.13) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 256 II.A Exact Methods for ODEs Once this generating function is known, y(t) may be obtained in either of the two ways y(t+p)=1 p!@p @kpY(t;k) k=0 =1 2iZ CY(t;k)k−p−1dk; whereCis a closed contour surrounding the origin in the k-plane and lying wholly within the region of analyticity in kofY(t;k). By dierentiating equation (54.13) with respect to t, multiplying by k, and redening p, we nd that Yt(t;k)=1X p=0y0(t+p)kp; kY(t;k)=1X p=1y(t+p+1 )kp:(54.14) If we now evaluate equation (54.1) when thas the value t+p, multiply by kp, and sum with respect to pfrom 1 to innity, we nd (using equation (54.14)) Yt(t;k)+a(kY(t;k)+y(t−1)) = 0 or, because 0t1, Yt(t;k)+akY(t;k)=−ay0: This equation is an ordinary dierential equation and can be readily solved to yield Y(t;k)=e−aktF(k)−y0 k; (54.15) whereF(k) is some unknown function. We can determine this function by a judicious use of the initial conditions. Evaluating equation (54.13) at t= 1, we nd kY(1;k)=k1X p=0y(1 +p)kp =1X p=0y(1 +p)kp+1 =y(0) +1X p=0y(p)kp =y(0) +Y(0;k):(54.16) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 54. Delay Equations 257 Evaluating equation (54.16) by use of equation (54.15) results in k e−akF(k)−y0 k =y0+ F(k)−y0 k ; or F(k)=y0 k(1−e−ak): This leads to the complete determination of the generating function Y(t;k)=y0 ke−akt 1−ke−ak−1 : Via some algebraic manipulations, we can obtain Y(t;k)=y01X p=0kpp+1X q=0(−a(p+t−q+1 ) )q q!; (54.17) so that the solution can be read o (compare equation (54.17) with equation (54.13)): y(t)=y0btc+1X q=0(−a)q(t−q+1 )q q!; where the floor function indicates the least integer. Notes 1. In the literature, equations of the form y0 h(t)=yh−1(t) are often called dierential{dierence equations , whereas equations of the form y0(t)=y(t−1) are called mixed dierential{dierence equations . Delay equations are also known as functional equations ,dierential{ delay equations ,dierential equations with deviating argument ,a n d equations with retarded arguments .Neutral dierential equations are dierential equations in which the highest order derivative of the unknown function is evaluated both at the present state tand at one of more past or future states. 2. The pantograph equation (see Buhmann and Iserles [4]) is _ x(t)= ax(t)+bx((t)) +c_x((t)). 3. The Cherwell{Wright dierential equation (see Iyanaga and Kawada [12, page 287]) is _ x(t)=(a−x(t−1))x(t). 4. Marsaglia et al. [13] numerically evaluate the following functions: Renyi’s function: [( x−1)y(x)]0=2y(x−1) Dickman’s function: xy0(x)=−y(x−1) Buchstab’s function: [ xy(x)]0=y(x−1) 5. Several authors have tried to analyze delay equations by replacing y(t−r) with the rst few terms of a Taylor series, say y(t−r)’y(t)−ry0(t)+1 2r2y00(t)−


+(−1)m1 m!rmy(m)(t): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 258 II.A Exact Methods for ODEs This is, in general, a bad idea as the approximations that are obtained are often unrelated to the original equation. See Driver [8, page 235] for more details. 6. The paper by Driver and Driver [7] gives explicit error bounds for the solution of x0(t)=bx(t−1) for a range of bvalues, when using the rst terms in an asymptotic expansion. For example, when x(t)=1 fort<0, andb=1 ,t h e nx(t)=xa(t)+g(t)w i t hxa(t)=1:13e0:567t andjg(t)j0:25e−1:47t. 7. The book by Pinney [15] contains a large compilation of delay equa- tions that have appeared in the literature. References are cited, and the (then) current knowledge of each of the equations is given. 8. The system of linear delay equations u0(t)=Au(t)+Bu(t−d); fortt0; u(t)=g(t); for−dtt0;(54.18) whered0 is the delay and AandBare constant square matrices has a solution of the form u(t)=cestif and only if sis a zero of the transcendental equation: det/parenleftbig Is−A−Be−ds
=0 . 9. As an example of the general expansion theorems, the equation au0(t)+bu(t)+cu(t−d)=0; wherea;b;c ,a n ddare all constant and dis positive, is satised by u(t)=X rpr(t)etsr; (54.19) wherefsrgare complex numbers satisfying asr+b+ce−dsr=0 , andpr(t) is a polynomial in tof degree less than the multiplicity sr(see Bellman and Cooke [3, page 55]). The sum in equation (54.19) is either nite or innite, with suitable conditions to ensureconvergence. In actuality, nding all the solutions to equation (54.15) is very dicult. This technique generalizes to higher order ordinary dierential equations and partial dierential equations, but the workin obtaining a solution becomes prohibitive unless numerical methods are used. 10. Delay equations are usually solved numerically. A survey of numerical techniques for solving delay equations may be found in Cryer [6]. Nieves’s paper [14] contains the description of a computer algorithmthat numerically approximates the solutions of functional equations with a minimal amount of user input. Virk’s paper [18] extends Runge{Kutta methods to delay-dierential equations (the methodhe presents is compromise between computational eciency and code complexity). 11. See also Saaty [16, Chapter 5, pages 213{261]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 54. Delay Equations 259 References [1]Al-Butib, A. N. One-step implicit methods for solving delay dierential equations. Int. J. Comp. Math. 16 (1984), 157{168. [2]Bakke, V. L., and Jackiewicz, Z. Stability analysis of linear multistep methods for delay dierential equations. Int. J. Math. &Math. Sci. 9 ,3 (1986), 447{458. [3]Bellman, R. E., and Cooke, K. L. Dierential{Dierence Equations . Academic Press, New York, 1963. [4]Buhmann, M., and Iserles, A. Stability of the discretized pantograph dierential equation. Math. of Comp. 60 , 202 (April 1993), 575{589. [5]Burton, T. A. Stability and Periodic Solutions of Ordinary and Functional Dierential Equations . Academic Press, New York, 1985. [6]Cryer, C. W. Numerical methods for functional dierential equations. InDelay and Functional Dierential Equations and Their Applications , K. Schmitt, Ed. Academic Press, New York, 1972, pp. 17{101. [7]D r i v e r ,B .K . ,a n dD r i v e r ,R .D . Simplicity of solutions of x’(t)=bx(t-1) . Journal of Mathematical Analysis and Applications 157 , 2 (15 May 1991), 591{608. [8]Driver, R. D. Introduction to Ordinary Dierential Equations .H a r p e r & Row Publishers, New York, 1978. [9]El’sgol’ts, L. E., and Norkin, S. B. Introduction to the Theory and Application of Dierential Equations with Deviating Arguments .A c a d e m i c Press, New York, 1973. [10]Feldstein, A., Iserles, A., and Levin, D. Embedding of delay equations into an innite-dimensional ODE system. J. Dierential Equations 117 (1995), 127{150. [11]H e r z ,A .V .M . Solutions of zzzref8refzzz approach the Kaplan{Yorke orbits for odd sigmoid g.J. Dierential Equations 118 (1995), 36{53. [12]Iyanaga, S., and Kawada, Y. Encyclopedic Dictionary of Mathematics . MIT Press, Cambridge, MA, 1980. [13]Marsaglia, G., Zaman, A., and Marsaglia, J. C. W. Numerical solution of some classicial dierential-dierence equations. Math. of Comp. 53, 187 (July 1989), 191{201. [14]Nieves, K. W. Automatic integration of functional dierential equations: An approach. ACM Trans. Math. Software 1 , 4 (Dec 1975), 357{368. [15]Pinney, E. Ordinary Dierence{Dierential Equations . University of California Press, Berkeley, CA, 1959. [16]Saaty, T. L. Modern Nonlinear Equations . Dover Publications, Inc., New York, 1981. [17]Torelli, L. Stability of numerical methods for delay dierential equations. J. Comput. Appl. Math. 25 (1989), 15{26. [18]V i r k ,G .S . IEE Proc. D 132 (1985), 119{123. [19]Weiner, R., and Strehmel, K. A type insensitive code for delay dier- ential equations basing on adaptive and explicit Runge{Kutta interpolation methods. Computing 40 , 3 (1988), 255{265. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 260 II.A Exact Methods for ODEs 55. Dependent Variable Missing Applicable to Ordinary dierential equations of the form G(y(n), y(n−1),:::,y00,y0,x)=0 . Yields An ordinary dierential equation of lower order. Idea If the dependent variable does not appear explicitly in an ordinary dierential equation, then the order of the ordinary dierential equationcan be reduced by 1. Procedure Suppose we have the nth order ordinary dierential equation G(y(n);y(n−1);:::;y00;y0;x)=0: (55.1) Notice that the variable y(x) does not appear explicitly in equation (55.1). If we dene p(x)=y0(x), then equation (55.1) becomes G(p(n−1);p(n−2);:::;p0;p;x)=0; (55.2) which is an ordinary dierential equation of order ( n−1) for the dependent variablep(x). After solving equation (55.2) for p(x),y(x) can be found by integrating p(x). Example Suppose we have the second order equation y00+y0=x: (55.3) Usingy0(x)=p(x), equation (55.3) can be written as p0+p=x: (55.4) Equation (55.4) can be solved by integrating factors (see page 356) to obtain p(x)=Ae−x+x−1; whereAis an arbitrary constant. Then p(x) can be integrated to obtain y(x) y(x)=Zx p(t)dt=B−Ae−x+x2 2−x; whereBis another arbitrary constant. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 55. Dependent Variable Missing 261 Notes 1. This solution technique can be derived from Lie group methods (see page 366). 2. See also Boyce and DiPrima [1, pages 111{112], Goldstein and Braun [2, pages 74{76], Ince [3, page 43], and Rainville and Bedient [4, pages266{268]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Goldstein, M. E., and Braun, W. H. Advanced Methods for the Solution of Dierential Equations . U.S. Government Printing Oce, Washington, D.C., 1973. NASA SP-316. [3]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [4]Rainville, E. D., and Bedient, P. E. Elementary Dierential Equations . The MacMillan Company, New York, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 262 II.A Exact Methods for ODEs 56. Dierentiation Method Applicable to Nonlinear ordinary dierential equations. Yields An explicit solution. Idea Sometimes dierentiating an ordinary dierential equation will result in an ordinary dierential equation that is easier to solve. Procedure Given an ordinary dierential equation, dierentiate it with respect to the independent variable. This will yield a new equation that may some-times factor (see page 292), or simplify in some other way. By considering each term in this new equation to be equal to zero, several possible solutions may be found. The general solution of each term must then be used in the original equation, possibly to constrain some of the parameters. Example Suppose that we have the nonlinear ordinary dierential equation 2yy00−(y0)2=1 3(y0−xy00)2: (56.1) If this equation is dierentiated with respect to x, the simplied result is y000/parenleftbig x2y00−xy0−3y
=0; from which we recognize that y000=0 o r x2y00−xy0−3y=0: (56.2) In the rst case, a candidate for the general solution is y(x)=ax2+bx+c: Using this form in the original equation, equation (56.1), we nd after some simplication that 3 ac=b2. Using this equation to determine c, a general solution to equation (56.1) is found to be y(x)=ax2+bx+b2 3a: (56.3) Another possibility is that the second expression in equation (56.2) is equal to zero. This second equation is an Euler equation (see page 281), and so the general solution is found to be y(x)=x3+ x: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 56. Dierentiation Method 263 Using this form in the original equation, equation (56.1), we nd after some simplication that = 0. Hence, two dierent solutions to equation (56.1) are given by y(x)=x3andy(x)= x: (56.4) Equations (56.3) and (56.4) contain three dierent solutions to equation (56.1). Notes 1. The above example is from Bateman [1, pages 66{67].2. This procedure is used to nd the singular solutions to Clairaut’s equation (see page 237). Reference [1]Bateman, H. Partial Dierential Equations of Mathematical Physics .D o v e r Publications, Inc., New York, 1944. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 264 II.A Exact Methods for ODEs 57. Dierential Equations with Discontinuities Applicable to Equations that contain discontinuous functions. Yields An exact solution. Idea Equations can be solved locally and then patched together at the points of discontinuity. Procedure The following discussion is limited to linear ordinary dierential equa- tions, but the general techniques apply to linear and nonlinear ordinary dierential equations and partial dierential equations. Suppose we have the equation an(x)y(n)+an−1y(n−1)+


+a1(x)y0+a0(x)y=b(x); (57.1) where thefai(x)gandb(x) may all be discontinuous. For example, a1(x) m a yl o o kl i k e a1(x)=( x if 0<x< 3; sinxif 3x<8: We presume that the fai(x)gandb(x) are discontinuous at only a nite number of points, say fx1;x2;:::;xmg, and that we wish to nd the solu- tion at the point xfwithx0<x1<


<xm<xf. Assume further that the initial datafy(x0);y0(x0);y00(x0);:::;y(n−1)(x0)gare all given. The general technique is to divide the interval from x0toxfintom intervals and solve equation (57.1) separately on each interval. Because the equation is continuous on these intervals, we can use any techniqueknown to us to nd the solution. Dene y j(x) to be the solution in the interval [xj;xj+1]. To determine yj(x) completely, we need to specify the value of fyj(xj), y0 j(xj),:::,y(n−1) j (xj)g. These can be determined from yj−1(x). Because an equation of nth order (which is what equation (57.1) is) must have continuous derivatives of all orders up to n−1, we simply match the values ofyj(x) and its derivatives to the values of yj−1(x) and its derivatives, all at the point xj. To illustrate this technique on equation (57.1), we would solve an(x)y(n) j+an−1y(n−1) j +


+a1(x)y0 j+a0(x)yj=b(x) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 57. Dierential Equations with Discontinuities265 in the interval [ xj;xj+1], forj=0;1;2;:::;m . To obtain the initial values for each equation we take 2 6664y 0(x0) y0 0(x0) ... y(n−1) 0 (x0)3 7775=2 6664y(x 0) y0(x0) ... y(n−1)(x0)3 7775; and then 2 6664y j(xj) y0 j(xj) ... y(n−1) j (xj)3 7775=2 6664yj−1(xj) y0 j−1(xj) ... y(n−1) j−1(xj)3 7775; forj=1;2;:::;m: Finally, the solution at x=xfwill be given by ym(xf). Example Suppose we want to determine the value of y(t)a tt=Twhen y00+f(t)y=0; andf(t)i sg i v e nb y f(t)=( −1f o r 0t<; 1f o rtT; given that y(0) = 1,y0(0) = 0. (Here, andTare xed constants.) To solve this problem, we break the interval from 0 to Tinto two intervals; interval I will be from 0 to while interval II will be from toT. In interval I, f(t) can be replaced by −1, so we solve y00 1−y1=0;y 1(0) = 1;y0 1(0) = 0: This equation has the solution y1(t)=c o s ht.I n i n t e r v a l I I , f(t)c a nb e replaced by 1, so we solve y00 2+y2= 0 (57.2) in the interval from toT.F o rt h e initial values ofy2(t), we use the nal values ofy1(t), that is, y2()=y1()=c o s h; y0 2()=y0 1() = sinh:(57.3) The solution of equations (57.2) and (57.3) is y2(t)=( s i ncosh+c o ssinh)s i nt+( c o scosh−sinsinh)c o st; and hence, the value of y(t)a tt=Tis given by y2(T)=( s i ncosh+c o ssinh)s i nT+( c o scosh−sinsinh)c o sT: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 266 II.A Exact Methods for ODEs Notes 1. When the discontinuities involve the dependent variable, then the problem is generally a free boundary problem. See Elliot and Ock- endon [3] or Fleishman [6] for a discussion. 2. If the discontinuity appearing in a linear dierential equation is a single delta function, which appears as a forcing function, then the solution will be a Green’s function (see page 318). 3. If the discontinuities include generalized functions (such as a delta function), then the solution may only exist in the weak sense. See Gear and sterby [7] for details. 4. There exist computer programs for numerically approximating dier- ential equations with discontinuities. See Enright et al. [4] or Gear and sterby [7]. 5. Fleishman [6] analyzes the equation _x=A(t)x+s g n ( x)+t(t), where \sgn" represents the signum function. 6. Das et al. [2] compare eight dierent approximations to a one-dimensional steady-state boundary value problem for a general symmetric second order ordinary dierential equation with discontinuous leading coef- cient. 7. See Leveque and Li [9] for methods for elliptic partial dierential equations. See also Boyce and DiPrima [1, Section 6.3.1, pages 304{309]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Das, B., Steinberg, S., Zhang, D., and Robey, T. Comparison of numerical solution methods for dierential equations with discontinuous coecients. Math. and Computers in Simulation 36 (1994), 57{75. [3]Elliot, C. M., and Ockendon, J. R. Weak and Variational Methods for Moving Boundary Problems . Pitman Publishing Co., Marsheld, MA, 1982. [4]Enright, W. H., Jackson, K. R., Norsett, S. P., and Thomsen, P. G. Eective solution of discontinuous IVPs using a Runge{Kutta formula pair with interpolants. Appl. Math. and Comp. 27 (1988), 313{335. [5]Filippov, A. F. Dierential Equations with Discontinuous Righthand Sides . Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. [6]Fleishman, B. A. Convex superposition in piecewise-linear systems. J. Math. Anal. Appl. 6 , 2 (April 1963), 182{189. [7]Gear, C. W., and sterby, O. O. Solving ordinary dierential equations with discontinuities. ACM Trans. Math. Software 10 , 1 (March 1984), 23{44. [8]Hajj, I. N., and Skelboe, S. Steady-state analysis of piecewise-linear dynamic systems. IEEE Trans. Circ. & Syst. CAS-28 , 3 (March 1981), 234{241. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 57. Dierential Equations with Discontinuities267 [9]Leveque, R. J., and Li, Z. The immersed interface method for elliptic equations with discontinuus coecients and singular sources. SIAM J. Numer. Anal. 31 , 4 (August 1994), 1019{1044. [10]Pan, H. H., and Hohenstein, R. M. A method of solution of an ordinary dierential equation containing symbolic functions. Quart. Appl. Math. (April 1981), 131{136. [11]Parker, T. S., and Chua, L. O. Ecient solution of the variational equation for piecewise-linear dierential equations. Circuit Theory and Appl. 14, 4 (1986), 305{314. [12]Stewart, D. A high accuracy method for solving ODEs with discontinuous right-hand side. Numer. Math. 58 (1990), 299{328. [13]Westreich, D. Numerical solution of the eigenvalue problem for discon- tinuous linear ordinary dierential equations. J. Inst. Maths. Applics 25 (1980), 147{160. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 268 II.A Exact Methods for ODEs 58. Eigenfunction Expansions Applicable to Linear dierential equations with linear boundary conditions. Yields An exact solution in terms of an innite series. Idea Any \well-behaved" function can be expanded in a complete set of eigenfunctions. In this method, we expand the dependent variable in a dierential equation as a sum of the eigenfunctions with unknown coe-cients. From the given equation and boundary conditions, equations can then be determined for the unknown coecients. Procedure We will describe the procedure for ordinary dierential equations, but the same procedure can be used for partial dierential equations (see Ex-ample 2). Assume that we want to solve the inhomogeneous linear ordinary dierential equation L[y]: =nX r=1pr(x)dry dxr=h(x); Bi[y]: =nX r=1 cirdry dxr(a)+dirdry dxr(b) =0;i =1;2;:::;n;(58.1.a-b) fory(x), wherex2[a;b]a n dfcir;dir;pr(x);h(x)gare all known. Let us suppose that we know a complete set of eigenfunctions fuk(x)g that satisfy the boundary conditions in equation (58.1) and are orthogonal with respect to some weighting function w(x). These could be obtained from a table (e.g., see table 77.1), or we might look for a set that is related to the dierential equation in (58.1). A common approach is to choose a set of eigenfunctions fukgthat satisfy H[uk]=kuk; Ri[uk]=0;i =1;2;:::;n;(58.2.a-b) whereH[
] is a linear operator related to L[
]i ns o m ew a y ,t h e Ri[
] are linear boundary conditions related to Bi[
]i ns o m ew a y ,a n d kis a constant (kis an eigenvalue of the ( H;fRig) system). The orthogonality condition requires that (uk;um): =Zb auk(x)um(x)w(x)dx=Nkkm=( 0f o rm6=k; Nkform=k: (58.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 58. Eigenfunction Expansions269 Frequently the operator H[
] is chosen to be the same as the operator L[
], and thefRigare chosen to be the same as the fBig. This is not required, nor must the degree of the dierential equation in (58.2.a) be n (which is the degree of the dierential equation in (58.1.a)). Because the presumed eigenfunctions are complete, we can write any \suciently smooth" function as a linear combination of these functions. In particular, we choose to represent y(x)a n dh(x)a s y(x): =1X k=1ykuk(x);h (x): =1X k=1hkuk(x): (58.4.a-b) Once thefykgare known, the problem is solved. The fhkgcan be de- termined, given h(x), by multiplying equation (58.4.b) by w(x)um(x)a n d integrating with respect to xfromatob. This calculation can be written as (h(x);um(x)) = 1X k=1hkuk(x);um(x)! ; =1X k=1hk(uk(x);um(x)); =1X k=1hk(Nkkm); =Nmhm; where we have utilized equation (58.3). If we take the fRigto be identical to thefBigthen, from equation (58.2.b), the boundary conditions for y(x) (in equation (58.1.b)) are automatically satised. Hence, only equation (58.1.a) needs to be satised. Using equation (58.4.a) in equation (58.1.a) results in L[y]=L"1X k=1ykuk(x)# =1X k=1ykL[uk] =h(x):(58.5) Thefykgcan now be determined from equation (58.5) by multiplying equation (58.5) by w(x)um(x) and integrating with respect to xfromato b. This produces 1X k=1yk(L[uk];um)=(h(x);um)=Nmhm;form=1;2;:::; (58.6) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 270 II.A Exact Methods for ODEs which is an innite system of linear algebraic equations. In principle, all of thefykgin equation (58.6) are coupled together. In practice, if a good choice was made for the eigenfunctions, then equa- tion (58.6) will simplify and ymcan be determined directly from equation (58.6). For instance, if H[
]i sc h o s e nt ob ee q u a lt o L[
]t h e nL[un]=nun (from equation (58.2)) and equation (58.6) becomesP1 k=1ykk(uk;um)= Nmhmor, by orthogonality, ym=hm=m. Example 1 Suppose we have the fourth order dierential equation and boundary conditions L[y]: =y0000+y00+y=h(x); y(0) = 0;y (1) = 0; y00(0) = 0;y00(1) = 0;(58.7) to solve for y(x) on the interval x2[0;1]. For this case we choose to use the eigenfunctions corresponding to the Sturm{Liouville operator (see page 103) H[u]=u00; u(0) = 0; u(1) = 0:(58.8) For the operator in equation (58.8), it is easy to determine that the eigen- functions are uk(x)=s i nkx, the eigenvalues are k=k,a n dt h e weighting function is w(x) = 1. Because this is a self-adjoint problem (see page 95), we know that these eigenfunctions are complete. Now that wehave a set of eigenfunctions, we observe that they satisfy the four boundary conditions given in equation (58.7). We writey(x) in terms of these eigenfunctions as y(x)=1X k=1yksinkx: (58.9) Using equation (58.9) in equation (58.7) and then multiplying by um(x) and integrating from x=0t ox= 1 results in Z1 0L[y(x)]um(x)dx=Z1 0L"1X k=1yksinkx# um(x)dx =1X k=1ykZ1 0L[sin(kx)]um(x)dx =Z1 0h(x)um(x)dx: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 58. Eigenfunction Expansions271 Equating the last two expressions, using um(x)=s i nmx and simplifying gives 1X k=1ykZ1 0/parenleftbig k44−k22+
sinkxsinmxdx = Z1 0h(x)s i nmxdx; or (sinceR1 0sinkxsinmxdx =1 2km) 1 2yk/parenleftbig k44−k22+
=Z1 0h(x)s i nkxdx: (58.10) Hence, solving equation (58.10) for ykand using this value in equation (58.9) results in the explicit solution y(x)=1X k=1 2R1 0h(x)s i nkxdx k44−k22+! sinkx: Ifandare such that k44−k22+= 0, for some value of k, then there will be no solution unlessR1 0h(x)s i nkxdx = 0. Even then, the solution will not be unique; this is because the dierential equation L[u]= 0, with the boundary conditions in equation (58.7), will have the solutionu(x)=Csinkx,w h e r eCis arbitrary. See the section on alternative theorems (page 15). Example 2 Suppose we want to solve the partial dierential equation t=xx; (x;0) =f(x); (0;t)=0; (1;t)=0;(58.11.a-d) for=(x;t). We can use the eigenfunctions in equation (58.8) to solve this problem. In this case, we expand (x;t)a s (x;t)=1X n=1an(t)s i nnx: (58.12) By using this representation for (x;t), the boundary conditions in equa- tion (58.11.b) and equation (58.11.c) are automatically satised. By multi- plying equation (58.12) by sin( mx) and integrating from x=0t ox=1 , we nd that an(t)=2Z1 0(z;t)s i nnzdz: (58.13) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 272 II.A Exact Methods for ODEs Using the boundary condition from (58.11.b) in equation (58.13) produces the initial values for the fan(t)g an(0) = 2Z1 0(z;0) sinnzdz =2Z1 0f(z)s i nnzdz: (58.14) Now, the correct procedure is to multiply the original equation, equation (58.11.a), by one of the eigenfunctions, sin mx, and integrate from x=0 tox= 1 to obtain Z1 0tsinmxdx =Z1 0xxsinmxdx: (58.15) After utilizing equation (58.12) for in equation (58.15), the resulting equation should be integrated by parts, using the information in equation (58.13). This results in a0 n(t)=−n22an(t); (58.16) where a prime denotes a derivative with respect to t. The solution of equation (58.16) is an(t)=an(0)e−n22t; = 2Z1 0f(z)s i nnzdz e−n22t;(58.17) where we have used equation (58.14). Combining equations (58.12) and (58.17), we determine the nal solution to equation (58.11) to be (x;t)=1X n=1 2Z1 0f(z)s i nnzdz e−n22tsinnx: Be aware that it would have been incorrect , when trying to obtain an ordinary dierential equation for an(t), to substitute equation (58.12) into equation (58.11.a) and then multiply by one of the eigenfunctions andperform the integration. Although this would have resulted in the same dierential equation and boundary conditions for a nin this example, it might not work in other cases (see the next example). The proper technique is to multiply the original equation by one of the eigenfunctions and then integrate by parts. Example 3 Consider solving Laplace’s equation in two dimensions in the unit square uxx+uyy=0; u(x;1) =u(0;y)=u(1;y)=0; u(x;0) =f(x): (58.18.a-c) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 58. Eigenfunction Expansions273 Since the functions fsinnygare complete on the interval [0 ;1], we choose to represent the solution to equation (58.18) in the form u(x;y)=1X n=1cn(x)s i nny; (58.19) from which we can deduce that cn(x)=2Z1 0u(x;y)s i nnydy: (58.20) From the boundary conditions on u(x;y)a tx=0a n da t x=1 ,w ea l s o nd thatcn(0) =cn(1) = 0. We will show that an incorrect answer is obtained if the fcngare determined in a naive way. If we substituted the assumed form of the solution (e.g., equation (58.19)), into the equation in (58.18.a), then we would nd uxx+uyy=1X n=1/parenleftbig c00 n−n22cn
sinny=0: Hence, by orthogonality, we would nd that c00 n−n22cn= 0. Solving this dierential equation with the boundary conditions on cn(e.g.cn(0) = cn(1) = 0), we would be led to cn(x)=0a n ds o u(x;y) = 0. This is clearly wrong . If, instead, the equation (58.18.a) is multiplied by 2 sin ny and inte- grated with respect to yfrom 0 to 1, then we obtain 0=Z1 02s i nny(uxx+uyy)dy =d2 dx2Z1 02u(x;y)s i nnydy +2uy(x;y)s i nny1 0 −2nu(x;y)c o sny1 0−n22Z1 02u(x;y)s i nnydy =c00 n+2nf(x)−n22cn; where we have integrated by parts twice, used equation (58.20) to substitute for the integral, and used the boundary conditions in equation (58.18.b-c). Solving this last equation for cn(x), we nd cn(x)=2nZ1 0G(x;t)f(t)dt; whereG(x;t) is the Green’s function G(x;t)=sinhnx<sinhn(1−x>) nsinhnand wherex>(x<) indicates the larger (smaller) of xandt. This second approach gives the correct solution to this problem. The reason that the rst approach would not work is that the series chosen to represent the solution does not have uniform convergence. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 274 II.A Exact Methods for ODEs Notes 1. Note that the solution in Example 2 would have been obtained in exactly the same form if separation of variables had been used (see page 487). 2. If the chosen eigenfunctions do not come from a self-adjoint operator, then it will be necessary to know the eigenfunctions of the adjoint operator. This is because the orthogonality condition will utilize the eigenfunctions of the adjoint operator. 3. Because the eigenfunctions we used in the examples were just sine functions, the expansions obtained here are identical to the results that would have been obtained from a Fourier sine series (see page 344). 4. To determine that a set of functions is complete, it is not necessary that they be derived from a self-adjoint operator. See Minzoni [6] for an example of a set of functions proved complete by using theorems from analysis. 5. See also Birkho and Rota [1, Chapter 11], Butkov [2, pages 304{318], and Farlow [4, Lesson 9, pages 64{71]. References [1]Birkhoff, G., and Rota, G.-C. Ordinary Dierential Equations .J o h n Wiley & Sons, New York, 1978. [2]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [3]Divis, Z. A note on the rate of convergence of Sturm{Liouville expansions. J. Approx. Theory 50 (1987), 200{207. [4]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [5]Kobayashi, M. Eigenfunction expansion: A discontinuous version. SIAM J. Appl. Math. 50 , 3 (June 1990), 910{917. [6]Minzoni, A. A. On the completeness of the functions zzzref32refzzz zzzref33refzzz for zzzref34refzzz and p(x) a zzzref35refzzz periodic function. Stud. Appl. Math. 75 (1986), 265{269. [7]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. [8]Titchmarsh, E. C. Eigenfunction Expansions Associated with Second{Order Dierential Equations . Clarendon Press, Oxford, England, 1946. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 59. Equidimensional-in-x Equations 275 59. Equidimensional-in-x Equations Applicable to Ordinary dierential equations of a certain form. Yields An autonomous ordinary dierential equation of the same order (which can then be reduced to an ordinary dierential equation of lower order). Idea An equidimensional-in- xequation is one in which the scaling of the x variable does not change the equation. By a change of independent variable, we can change an equation of this type into an autonomous equation. Procedure An equidimensional-in- xequation is one that is left invariant under the transformation x!ax,w h e r eais a constant. That is, if the original equation is an equation for y(x)a n dt h exvariable is replaced by the variableax0, then the new equation (in terms of yandx0) will be identical to the original equation (which is in terms of yandx). An equation of this type can be converted to an autonomous equation of the same order by changing the independent variable from xtotby the transformation x=et. Example Suppose we have the nonlinear second order ordinary dierential equa- tion xd2y dx2=2ydy dx: (59.1) First, we will show that this equation is equidimensional-in- x. Substituting ax0forxin equation (59.1) produces (ax0)d2y d(ax0)2=2ydy d(ax0); (59.2) or, multiplying equation (59.2) by the constant a x0d2y d(x0)2=2ydy dx0; which is identical to equation (59.1). Because we now know that equation (59.1) is equidimensional-in- x,w e change variables from y(x)t oy(t)b yx=et. Using table 59.1, we nd that ete−2t(ytt−yt)=2y(e−tyt); CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 276 II.A Exact Methods for ODEs yx=e−t(yt); yxx=e−2t(ytt−yt); yxxx=e−3t(yttt−3ytt+2yt); yxxxx =e−4t(ytttt−6yttt+1 1ytt−6yt); yxxxxx =e−5t(yttttt−10ytttt+3 5yttt−50ytt+2 4yt); yx(5)=e−5t(yt(5)−10yt(4)+3 5yttt−50ytt+2 4yt); yx(6)=e−6t(yt(6)−15yt(5)+8 5yt(4)−225yttt+ 274ytt−120yt); yx(7)=e−7t(yt(7)−21yt(6)+ 175yt(5)−735yt(4)+ 1624yttt−1764ytt+ 720yt): Table 59.1: How to transform derivatives under the change of dependent variable:x=et. (To simplify notation, dene yx(n)to be thenth derivative ofywith respect to x, and similarly for yt(n).) or ytt−yt=2yyt: (59.3) The equation in (59.3) is autonomous (there is no explicit tdependence). Hence, it can be reduced to an ordinary dierential equation of order one by the transformation u(y)=yt(t) (see page 230 for more information). Carrying out the details (equation (59.3) was the example in the section on autonomous equations), it is easy to derive that either y(t)i sac o n s t a n t for allt,o ry(t) satises y(t)=Etan(F+Et)−1 2; whereEandFare arbitrary constants. Changing the independent variable fromttoxwe have y(x)=Etan(F+Elogx)−1 2: Notes 1. This method is derivable from Lie group methods (see page 366). 2. It is straightforward to create a Macsyma program that will perform the necessary change of variables. Program 59.1 shows a terminalsession in which the input equation dy dx2 −yd2y dx2=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 59. Equidimensional-in-x Equations 277 (c1) DEPENDS(Y,X)$ (c2) EQUIDIMENSIONAL_IN_X(EQN,Y,X):= BLOCK([NEW,HOLD,J], DEPENDS([U],[T]),GRADEF(T, X, %E**(-T) ),NEW:SUBST( U, Y, EQN ),NEW:EV(NEW, DIFF),NEW:SUBST( %E**T, X, NEW), NEW:FACTOR(NEW), NEW)$ (c3) EQN: DIFF(Y,X)**2-Y*DIFF(Y,X,2); 2 (d3) (y ) - y y xx x (c4) EQUIDIMENSIONAL_IN_X(EQN,Y,X); -2t 2 (d4) - %e (u u - (u ) - u u ) tt t t Program 59.1: Macsyma program to change variables. is converted into the second order autonomous equation ud2u dt2−du dt2 −udu dt=0: This autonomous equation could then be reduced to a rst order equation (see page 230). 3. See Bender and Orszag [1, page 25]. Reference [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 278 II.A Exact Methods for ODEs 60. Equidimensional-in-y Equations Applicable to Ordinary dierential equations of a certain form. Yields An ordinary dierential equation of lower order. Idea An equidimensional-in- yequation is one in which the scaling of the y variable does not change the equation. This information can be used to lower the order of the equation by a change of the dependent variable. Procedure An equidimensional-in- yequation is one that is left invariant under the transformation y!ay,w h e r eais a constant. That is, if the original equation is an equation for y(x)a n dt h eyvariable is replaced by the variableay0, then the new equation (in terms of y0andx) will be identical to the original equation (which is in terms of yandx). An equation of this type can be converted to an equation of lower order by changing thedependent variable from y(x)t oe u(x). Example Suppose we have the equation (1−x)" yd2y dx2−dy dx2# +x2y2= 0 (60.1) to solve. We can tell by inspection that this equation is equidimensional- in-ybecause all of the yterms in equation (60.1) all appear to the same power. That is, the yterms in equation (60.1) are all quadratic, the terms being of the form fy2;y2 x;y2 xx;:::;yyx;yyxx;yxyxx;:::g. To formally show that equation (60.1) is equidimensional-in- y, substi- tuteay0foryin equation (60.1) to nd (1−x)" (ay0)d2(ay0) dx2−d(ay0) dx2# +x2(ay0)2=0: Or, because ais a non-zero constant, (1−x)" y0d2y0 dx2−dy0 dx2# +x2y02=0; (60.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 60. Equidimensional-in-y Equations 279 dy[0]= Exp[u[x]]; dy[1]= y[x] u’[x];dy[n_]:= D[dy[n-1],x]/. {y’[x]->y[x] u’[x]}dy2[n_]:= dy[n] /. {y[x]->y, u’[x]->u’, u’’[x]->u’’, u’’’[x]->u’’’, u’’’’[x]->u’’’’} Table[ {n,ddy2[n]}, {n,1,4}] // ColumnForm Program 60.1: Mathematica program to change variables: y(x)=eu(x). which has the same form as equation (60.1). Now, substituting eu(x)for y(x) in equation (60.1) produces (1−x)" y2 d2u dx2+du dx2! − ydu dx2# +x2y2=0; (60.3) where table 60.1 has been used to determine how the derivatives transform under this change of variable. For y6= 0, equation (60.3) becomes (1−x)d2u dx2+x2=0: (60.4) Note that equation (60.4) does not have any explicit ydependence. If it did have any such terms, then the original equation could not have been equidimensional-in- y. The solution to equation (60.3) is (see page 224) u(x)=ZxZwz2 z−1dz dw; =x3 6+x2 2+(x−1) log(x−1) +Ax+B; whereAandBare arbitrary constants. Hence, the solution of the original equation is y(x)=eu(x)=(x−1)(x−1)expx3 6+x2 2+Ax+B : Notes 1. This method is derivable from Lie group methods (see page 366). 2. Equidimensional-in- yequations are also called equations homoge- neous iny. 3. The results in table 60.1 can be obtained with the Mathematica code in program 60.1. The output of that program is: {1, y u’} 2 {2, y u’ + y u’’} 3 (3) {3, y u’ + 3 y u’ u’’ + y u } 4 2 2 (3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 280 II.A Exact Methods for ODEs y=eu; yx=yux; yxx=y(uxx+u2 x]; yxxx=y(uxxx+3uxuxx+u3 x]; yxxxx =y(uxxxx+4uxuxxx+3u2 xx+6u2 xuxx+u4 x]: yx(4)=y(ux(4)+4uxuxxx+3u2 xx+6u2 xuxx+u4 x); yx(5)=y(ux(5)+5uxux(4)+1 0uxxuxxx+1 0u2 xuxxx+1 5uxu2 xx+1 0u3 xuxx+u5 x); yx(6)=y(ux(6)+6uxux(5)+1 5uxxux(4)+1 5u2 xux(4)+1 0u2 xxx+2 0u3 xuxxx +1 5u3 xx+6 0uxuxxuxxx+4 5u2 xu2xx+1 5u4 xuxx+u6 x): Table 60.1: How to transform derivatives under the change of independent variable:y(x)=eu(x). (To simplify notation, dene yx(n)to be thenth derivative of ywith respect to x. Similarly for ux(n).) { 4 ,yu ’ +6yu ’ u ’ ’+3yu ’ ’ +4yu ’u + (4) yu } 4. See Bender and Orszag [1, page 27]. Reference [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 61. Euler Equations 281 61. Euler Equations Applicable to Linear ordinary dierential equations of the form a0xny(n)+a1xn−1y(n−1)+


+an−1xy0+any=0 . Yields An exact solution. Idea An equation of the above type can be turned into a linear constant co- ecient ordinary dierential equation by a change of independent variable.This new equation can be solved exactly. Procedure An Euler equation has the form a0xny(n)+a1xn−1y(n−1)+


+an−1xy0+any=0: (61.1) If the independent variable is changed from xtot(via the transformation x=et), then the resulting equation becomes a linear constant coecient ordinary dierential equation. This type of equation can be solved exactly.(Table 61.1 shows how the derivatives of ywith respect to xbecome derivatives of ywith respect to t.) Alternatively, a solution of the form y=x kcan be tried directly in equation (61.1). Example 1 Given the Euler equation x2yxx−2xyx+2y=0; we change variables by x=etto obtain ytt−3yt+2y=0: (61.2) The standard technique for solving a linear constant coecient ordinary dierential equation is to look for exponential solutions (see page 247). Usingy=etin equation (61.2), we nd the characteristic equation to be 2−3+2 = 0. The roots of this equation are =1a n d= 2. Therefore, the solution to equation (61.2) is y(t)=C1et+C2e2t; whereC1andC2are arbitrary constants. Writing this solution in the original variables, we determine the nal solution y(x)=C1x+C2x2: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 282 II.A Exact Methods for ODEs yx=e−t(yt); yxx=e−2t(ytt−yt); yxxx=e−3t(yttt−3ytt+2yt); yxxxx =e−4t(ytttt−6yttt+1 1ytt−6yt); yxxxxx =e−5t(yttttt−10ytttt+3 5yttt−50ytt+2 4yt): yx(5)=e−5t(yt(5)−10yt(4)+3 5yttt−50ytt+2 4yt); yx(6)=e−6t(yt(6)−15yt(5)+8 5yt(4)−225yttt+ 274ytt−120yt); yx(7)=e−7t(yt(7)−21yt(6)+ 175yt(5)−735yt(4)+ 1624yttt−1764ytt+ 720yt); Table 61.1: How to transform derivatives under the change of dependent variable:x=et(To simplify notation, dene yx(n)to be thenth derivative ofywith respect to x. Similarly for yt(n).) Example 2 Given the Euler equation x3y000−x2y00−2xy0−4y=0; (61.3) we usey=xkto nd the characteristic equation: k(k−1)(k−2)xk−k(k−1)xk−2kxk−4xk=0 or/parenleftbig k2+1
(k−4) = 0: This equation has the roots k=4a n dk=i. Hence, the general solution to equation (61.3) is y=C1x4+C2cos(logx)+C3sin(logx): Notes 1. This method is also applicable to the equation a0(Ax+B)ny(n)+a1(Ax+B)n−1y(n−1)+


+an−1(Ax+B)y0+any=0; which is only a trivial modication of an Euler equation. 2. Equations of the formdxp P(x)=dyp P(y),w h e r eP(x) is a polynomial of degree three or four, have also been called Euler equations (see Valiron [5, pages 201{202]). 3. Euler matrix dierential equations (in which the faigin equation (61.1) are all matrices) are discussed in J odar [3]. 4. See also Boyce and DiPrima [1, Section 4.4], Finizio and Ladas [2, pages 103{105], and Simmons [4, page 86]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 61. Euler Equations 283 References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Finizio, N., and Ladas, G. Ordinary Dierential Equations with Modern Applications . Wadsworth Publishing Company, Belmont, CA, 1982. [3]Jodar, L. Boundary value problems for second order operator dierential equations. Linear Algebra and Its Appls. 91 (1987), 1{12. [4]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. [5]Valiron, G. The Geometric Theory of Ordinary Dierential Equations and Algebraic Functions . Math Sci Press, Brookline, MA, 1950. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 284 II.A Exact Methods for ODEs 62. Exact First Order Equations Applicable to First order ordinary dierential equations. Yields An exact solution (generally implicit). Idea Some rst order ordinary dierential equations can be integrated di- rectly. Procedure If the given ordinary dierential equation has the form dy dx=N(x;y) M(x;y)(62.1) andN(x;y)a n dM(x;y) are such that @M @x+@N @y= 0 (62.2) then equation (62.1) is said to be an exact ordinary dierential equation. Such an equation can be solved exactly, though the answer may be in termsof an integral. The (implicit) solution will be of the form (x;y)=C; (62.3) whereCis an arbitrary constant. Motivating this is straightforward. Dierentiating equation (62.3) with respect to xand rearranging terms gives dy dx=−x y: (62.4) Comparing equation (62.4) to equation (62.1), we have x=−N; y=M; (62.5.a-b) and hence equation (62.2) is satised (because xy=yx). Conversely, if equation (62.2) is satised, then there is a such that equation (62.5) is satised. To solve equation (62.5) for , integrate equation (62.5.a) with respect toxand integrate equation (62.5.b) with respect to yfor (x;y)=−Z N(x;y)dx+f(y); (x;y)=Z M(x;y)dy+g(x);(62.6.a-b) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 62. Exact First Order Equations 285 wheref(y)a n dg(x) are unknown functions. Comparing equation (62.6.a) to equation (62.6.b) will determine f(y)a n dg(x). Knowing either of these, the full solution is then given by equation (62.6.a) or equation (62.6.b). Example Suppose we have the equation dy dx=3x2−y2−7 ey+2xy+1: (62.7) In equation (62.7) we identify N(x;y)=3x2−y2−7a n d M(x;y)=ey+2xy+1: Following our procedure, we nd Mx=−Ny=2yand so we know that we can solve equation (62.7) exactly. Integrating NandMwe nd (x;y)=−Z N(x;y)dx+f(y)=−(x3+y2x−7x)+f(y); (x;y)=Z M(x;y)dy+g(x)=(ey+y2x+y)+g(x):(62.8.a-b) Comparing equations (62.8.a) and (62.8.b), we deduce that x3−y2x+7x+f(y)=ey+y2x+y+g(x) or f(y)−(ey+y)=g(x)−(7x+x3): (62.9) From equation (62.9) we conclude that f(y)=ey+y+A; g (x)=7x−x3+A; (62.10.a-b) whereAis an arbitrary constant. Using either equation (62.10.a) in (62.8.a) or equation (62.10.b) in (62.8.b), we conclude (x;y)=−x3−y2+7x+ey+y+A: (62.11) The solution is then given by (x;y)=C,w h e r eCis an arbitrary constant. Therefore, −x3−y2+7x+ey+y=B (62.12) is the nal solution, where B:=A−Cis a nal arbitrary constant. Note that the solution in equation (62.12) is implicit. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 286 II.A Exact Methods for ODEs Note 1. See Boyce and DiPrima [1, pages 79{84], Rainville and Bedient [2, pages 29{33], and Simmons [3, pages 38{41]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Rainville, E. D., and Bedient, P. E. Elementary Dierential Equations . The MacMillan Company, New York, 1964. [3]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 63. Exact Second Order Equations 287 63. Exact Second Order Equations Applicable to Some nonlinear second order ordinary dierential equations of the form f(x;y;y0)y00+g(x;y;y0)=0 . Yields A rst integral (which will be a rst order ordinary dierential equa- tion). Idea Some second order ordinary dierential equations can be integrated once. Procedure The second order dierential equation F(x;y;y0;y00) = 0 (63.1) is said to be exact if it is the total dierential of some function; i.e., F= d=dx where=(x;y;y0). If equation (63.1) is exact, then =Cis a solution to equation (63.1), with Can arbitrary constant. Dierentiating =Cwith respect to x, we nd d dx=@ @x+@ @yy0+@ @y0y00: (63.2) Comparing equation (63.2) to equation (63.1), we conclude that, for equa- tion (63.1) to be exact, F(x;y;y0;y00)m u s th a v et h ef o r m F(x;y;y0;y00)=f(x;y;y0)y00+g(x;y;y0); (63.3) for some functions fandgwith f(x;y;y0)=@ @y0;g (x;y;y0)=@ @x+@ @yy0: (63.4.a-b) By dierentiating equation (63.4.a{b) with respect to x,y,a n dp, (using p:=dy=dx ), all dependence on can be eliminated between the two equations in equation (63.4) to obtain fxx+2pfxy+p2fyy=gxp+pgyp−gy; fxp+pfyp+2fy=gpp:(63.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 288 II.A Exact Methods for ODEs If the conditions in equation (63.5) hold, then equation (63.3) is exact. If equation (63.3) is exact, then we can integrate equation (63.4.a) (with respect top) to determine (x;y;y0)a s =h(x;y)+Z f(x;y;p )dp; (63.6) whereh(x;y) is, so far, an arbitrary function of integration. This function will be restricted when equation (63.6) is used in equation (63.4.b). Example Given the equation xyy00+x(y0)2+yy0=0; (63.7) which has the form of equation (63.3), we identify: f=xy,g=x(y0)2+ yy0=xp2+yp. It is easy to verify that equation (63.5) holds. Hence, equation equation (63.7) is exact. Equation (63.6) now becomes =h(x;y)+Z xydp =h(x;y)+xyp:(63.8) Using equation (63.8) in equation (63.4.b) yields g=xp2+yp=@ @x+@ @yy0 =(hx+yp)+(hy+xp)p:(63.9) Hence, ifhis constant, say h=D, then equation (63.9) will be satised. Therefore a rst integral of equation (63.7) is given by =C,o r C=(x;y;p ) =D+xyp =D+xydy dx:(63.10) In this example, the rst integral equation (63.10) can itself be integrated in closed form (this is often true). A solution to equation (63.7), obtained by solving the ordinary dierential equation in equation (63.10), is thus given by y2 2=(C−D)l o gx+E; whereEis another arbitrary constant. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 63. Exact Second Order Equations 289 Notes 1. The most general solution for h(x;y) in equation (63.9) is h=h(y− x). With this form for h, however, the rst integral cannot be integrated to yield an explicit solution. 2. Exact second order linear ordinary dierential equations have fac- torable operators (see page 294). 3. Given the dierential equation f(x;y;:::;y(n))=0; (63.11) denefi=@f @y(i). Then equation (63.11) will be exact if f0−df1 dx+d2f2 dx2−


+(−1)ndnfn dxn=0: (63.12) If the dierential equation (63.11) is exact, then a rst integral can be found by a repetitive sequence of steps: First, integrate the highest order term in fand call this result F1. Then, integrate the highest order term in fdx−dF1and call this result F2. Continue in this manner until fdx−dF1−dF2−


= 0. Then, a rst integral is given byF1+F2+


= constant. For example, given the nonlinear third order equation f=yy000−y0y00+y3y0=0; (63.13) we identify f3=y,f2=−y0,f1=−y00+y3,f0=y000+3y2y0and verify that equation (63.12) is satised. We then calculate F1=yy00, since the highest order term in fisyy000. Then,fdx−dF1=(−2y0y00+ y3y0)dx,a n ds ow et a k e F2=−(y0)2. Then,fdx−dF1−dF2=y3y0dx, and soF3=1 4y4. Finally, then, fdx−dF1−dF2−dF3=0 ,s ot h a t yy00−(y0)2+1 4y4= constant is a rst integral of equation (63.13). 4. See also Goldstein and Braun [1, page 93] and Murphy [2, pages 221{222]. References [1]Goldstein, M. E., and Braun, W. H. Advanced Methods for the Solution of Dierential Equations . U.S. Government Printing Oce, Washington, D.C., 1973. NASA SP-316. [2]Murphy, G. M. Ordinary Dierential Equations and Their Solution .D .V a n Nostrand Company, Inc., New York, 1960. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 290 II.A Exact Methods for ODEs 64. Exact Nth Order Equations Applicable to Linearnth order ordinary dierential equations. Yields A rst integral. Idea Some linear dierential equations can be integrated exactly without modifying the equation in any way. Procedure The linearnth order ordinary dierential equation Pn(x)dny dxn+Pn−1(x)dn−1y dxn−1+


+P1(x)dy dx+P0(x)y=R(x); (64.1) is said to be exact if it can be integrated once to yield Qn−1(x)dn−1y dxn−1+Qn−2(x)dn−2y dxn−2+


+Q1(x)dy dx+Q0(x)y=Z R(x)dx: (64.2) If equation (64.1) is exact, then the fQi(x)gmay be found from Qn−1=Pn; Qn−2=Pn−1−P0 n; Qn−3=Pn−2−P0 n−1+P00 n; ... Q0=P1−P0 2+P00 3−


+(−1)n−1P(n−1) n: A necessary and sucient condition for equation (64.1) to be exact can be found by dierentiating equation (64.2) with respect to xand comparing terms with equation (64.1). This condition is dnPn dxn−dn−1Pn−1 dxn−1+dn−2Pn−2 dxn−2−


+(−1)n−1dP1 dx+(−1)nP0=0: (64.3) Special Case The second order linear ordinary dierential equation P(x)y00+Q(x)y0+R(x)y=0 will be exact if and only if P00(x)−Q0(x)+R(x)=0 . CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 64. Exact Nth Order Equations 291 Example If we have the linear ordinary dierential equation of third order (1 +x+x2)d3y dx3+( 3+6x)d2y dx2+6dy dx=6x; (64.4) then we have P0=0 ,P1=6 ,P2= 3+6x,P3=1 +x+x2,a n dR(x)=6x. It is easy to verify that d3P3 dx3−d2P2 dx2+dP1 dx−P0=0; and so equation (64.4) is exact. Integrating equation (64.4) directly, we obtain (1 +x+x2)d2y dx2+( 2+4x)dy dx+2y=3x2+A; (64.5) whereAis an arbitrary constant. Now equation (64.5) is again exact, and so it can be integrated again to yield (1 +x+x2)dy dx+( 1+2x)y=x3+Ax+B; (64.6) whereBis an arbitrary constant. Finally, equation (64.6) is once again exact. It can be integrated to yield the general solution of equation (64.4) (1 +x+x2)y=x4 4+Ax2 2+Bx+C; whereCis an arbitrary constant. Note 1. See Ford [1, pages 77{78] and Murphy [2, pages 221{222]. References [1]Ford, L. R. Dierential Equations . McGraw{Hill Book Company, New York, 1955. [2]Murphy, G. M. Ordinary Dierential Equations and Their Solution .D .V a n Nostrand Company, Inc., New York, 1960. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 292 II.A Exact Methods for ODEs 65. Factoring Equations Applicable to Ordinary dierential equations and partial dieren- tial equations. Yields Equations of lower degree. Idea If a dierential equation can be factored into simple terms, then the solution to each of the factors is a solution to the original equation. Procedure Given a dierential equation, attempt to factor it. If this is possible, then solve each factor separately. Each of the solutions of the dierent factors will be a solution of the original dierential equation. Example The nonlinear ordinary dierential equation y0(y0+y)=x(x+y) (65.1) fory(x) may be factored into (y0+y+x)(y0−x)=0: (65.2) Solving each of the factors appearing in equation (65.2) separately, the solutions to equation (65.1) are given by y(x)=8 >< >:Ae−x+1−x; B+x2 2; whereAandBare constants. Notes 1. The complete solution to the original dierential equation may switch from one solution branch to another. 2. See Bateman [2, pages 97{98] and Fogiel [3, pages 1222{1229]. References [1]Argyros, I. K. On the cardinality of solutions of multilinear dierential equations and applications. Int. J. Math. &Math. Sci. 9 , 4 (1986), 757{766. [2]Bateman, H. Partial Dierential Equations of Mathematical Physics .D o v e r Publications, Inc., New York, 1944. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 65. Factoring Equations293 [3]Fogiel, M. The Dierential Equations Problem Solver . Research and Education Association, New York, 1978. [4]Klamkin, M. S. On soluble nth order linear dierential equations. J. Math. Anal. Appl. 84 (1981), 6{11. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 294 II.A Exact Methods for ODEs 66. Factoring Operators Applicable to Ordinary and partial dierential equations. Yields A sequence of lower order equations to solve. Idea If the operator representing a dierential equation can be \factored" into two or more operators, it may be easier to nd a solution. Procedure Suppose we wish to solve the dierential equation Q[u]=0f o rt h e quantityu(x), whereQ[
] is a dierential operator. When possible, \factor" the dierential equation Q[u]=0a sL[H[u]] = 0, where L[
]a n dH[
] are also dierential operators. Then solve the two equations: L[v]=0f o rv, and thenH[u]=v. Example 1 The fourth order partial dierential equation (r4−a2)u=0; (66.1) whereais a constant and r2is the usual Laplacian, may be factored as (r2−a)(r2+a)u=0: The general solution of equation (66.1), therefore, is given by the solution of the two successive second order dierential equations (r2−a)v=0; (r2+a)u=v:(66.2) Alternatively, equation (66.1) could have factored equation as (r2+a)(r2−a)u=0 so that the general solution of equation (66.1) can also be written as the solution of (r2+a)w=0; (r2−a)u=w:(66.3) Solving equation (66.2) or equation (66.3) as a sequence of two second order dierential equations may be easier than solving the fourth order equation (66.1) directly. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 66. Factoring Operators295 Example 2 If we want to solve the nonlinear ordinary dierential equation Q[u]=0 , where Q[u]=u2 xx−2uxuxx+2uux−u2=0 =(uxx−ux)2−(ux−u)2=0;(66.4) then we might factor the operator Q[
]a sQ[u]=L[H[u]], whereL[v]= v2 x−v2,a n dH[u]=ux−u. Therefore, the equation Q[u]=0c a nb es o l v e d by solving the sequence of rst order dierential equations L[v]=0;H [u]=v: The solution of L[v]=0i sv=Cex,w h e r eCis an arbitrary constant. The general solution of equation (66.4) can then be determined by solving H[u]=ux−u=v=Cex: (66.5) Equation (66.5) can be solved by the use of integrating factors (see page 356) to obtain the two possible forms of the solution u=8 >< >:(A+Cx)ex; Ce−x+Bex; whereAandBare also arbitrary constants. Example 3 The relativistic wave equation 1 c2@2 @t2−@2 @x2−@2 @y2−@2 @z2+m2c2 h2 =0 was factored by Dirac [4, Chapter 11] using hypercomplex algebra. If f1, 2,3,4grepresent four of the elements in this algebra that obey the relation+=2, then the factored equation is 1 cd dt−1d dx−2d dy−3d dz−4imc h 1 cd dt+1d dx+2d dy+3d dz+4imc h =0: The rst factor led to the correct relativistic theory for the electron, while the second factor led to Dirac’s prediction of the positron. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 296 II.A Exact Methods for ODEs Example 4 The formally self-adjoint homogeneous fourth order operator d2 dx2 P(x)d2y dx2d dx Q(x)dy dx +R(x)y may be factored into L[(x)L[y]], whereL[
] is the second order operator L[y]=d dx (x)dy dx +(x)y; wheref(x);(x);(x)gsatisfy (x)=0 2; (x)=20; (x)= 0 00+1 2γ ; andf(x);(x);γ(x);(x)gare any solution to P(x)=203; Q(x)=2000+2000+ 400−202+γ2 0; R(x)= 0(0000+γ00+0γ0+); with 4=2γ00+γ2. See Hill [9] for details. Notes 1. Note that the equation in example 2 can be directly factored as Q[u]= (uxx−2ux+u)(uxx−u). In this case, the factorization of the equation simplier than the factorization of the operator (see page 292). 2. It is not true that the number of distinct factorizations is limited by the order of the dierential equation. For example, the second order ordinary dierential equation (x2−x3)u00+( 2x2−4x)u0+( 6−2x)u=0; has the three distinct factorizations  xd dx−2 (x−x2)d dx+2x−3 u=0;  xd dx−3 (x−x2)d dx+x−2 u=0;  (x−x2)d dx+x−3 xd dx−2 u=0:(66.6) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 66. Factoring Operators297 3. The Laplacian in two dimensions admits the factorization: r2=@2 @x2+@2 @y2=@ @x−i@ @y@ @x+i@ @y =@ @z@ @z ; (66.7) wherei=p−1. Therefore, using z=x+iy, Laplace’s equation may be written as r2u=@2u @z@z= 0. This shows that the most general solution to Laplace’s equation in two dimensions is u=f(z)+ g(z), wheref(z)a n dg(z) are arbitrary functions. Also, because the biharmonic equation may be written as r4u=1 6@4u @2z@2z=0 , the general solution of the biharmonic equation is seen to be u= f(z)+g(z)+zh(z)+zj(z). The operators @=@z and@=@zare known asWirtinger derivatives . In two dimensions, solutions of Poisson’s equation may sometimes be found by use of Wirtinger derivatives. See Henrici [8, pages 300{302] for details. 4. It is possible to write down an \explicit" factorization of any nth order linear dierential equation. To do so, however, requires explicit knowledge of the nlinearly independent solutions. For example, if L[
] is the dierential operator L[u]=u00+p(x)u0+q(x)u; andu1;u2are any two linearly independent solutions of L[u]=0 , then L[u]=W(u1;u2) u1d dxu2 1 W(u1;u2)d dxu u1 ; whereW(u1;u2) is the Wronskian of u1(x)a n du2(x). In thenth order case, consider the dierential operator H[u]=u(n)+p1(x)u(n−1)+p2(x)u(n−2)+


+pn(x)u: Iffu1;u2;:::;ungarenlinearly independent solutions of H[u]=0 , then dene Wk(fork=1;2;:::;n ) to be the Wronskian of the rst klinearly independent solutions; that is, Wk:=W(u1;u2;:::;uk). Using this denition, we can write H[u]a s H[u]=Wn Wn−1d dxW2 n−1 Wn−1Wn





d dxW2 2 W1W3d dxW2 1 W0W2d dxu W1


 : See Rainville [12, pages 292{299] for details. 5. The factorization d dt−q(t)d dt+q(t) w=d2w dt2+wdq dt−q2 leads to the technique for solving Riccati equations (see page 392). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 298 II.A Exact Methods for ODEs 6. Dierential resultants can be used to analyze the factoring of opera- tors for linear dierential equations. See Berkovich and Tsirulik [1] for details. 7. Two dierential operators PandQare said to be permutable if P(Q)=Q(P). From Ince [10, page 131], we have IfPandQare permutable operators of orders mandnrespec- tively, they satisfy identically an algebraic relation of the form F(P;Q) = 0 of degree ninPand of degree minQ. For example, the operators P=d2 dx2−2 x2; Q=d3 dx3−3 x2d dx+3 x3; are permutable because PQ=QP. We can also nd the algebraic relationP3−Q2=0 ,o b s e r v e P(P(P(f))) =f000000−6 x2f0000+24 x3f000−72 x4f00+144 x5f0−144 x6f=Q(Q(f)): This example is due to Ince [10, page 131]. See also Gr¨ unbaum [7]. 8. Landau [11] gives a (surprising) factorization that depends on an arbitrary parameter a: y00−2 xy0+2 x2y=d dx−1 x(1 +ax)d dx−1+2ax x(1 +ax) y: 9. Schwarz [14] has developed an algorithm that will factor ordinary dierential equations. As an example, his program derives the fac- torization y00−3 4x2+5 2x3−1 4x4 y=d dx−3 2x+1 2x2+1 x−1 3 d dx+3 2x−1 2x2−1 x−1 3 y: References [1]Berkovich, L. M., and Tsirulik, V. G. Dierential resultants and some of their applications. Dierentsial’nye Uravneniya 22 , 5 (May 1986), 750{ 757. [2]Brownawell, W. D. On the factorization of partial dierential equations. Can. J. Math. 39 , 4 (1987), 825{834. [3]Chisholm, J. S. R., and Common, A. K. A class of second-order dierential equations and related rst-order systems. J. Phys. A: Math. Gen. 20 (1987), 5459{5472. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 66. Factoring Operators299 [4]Dirac, P. A. M. The Principle of Quantum Mechanics . Clarendon Press, Oxford, England, 1974. [5]Etgen, G. J., Jones, G. D., and Taylor, Jr., W. E. On the factorizations of ordinary linear dierential operators. Trans. Amer. Math. Soc. 297 , 2 (1986), 717{728. [6]Fordy, A. P., and Gibbons, J. Factorization of operators I. Miura transformations. J. Math. Physics 21 , 10 (Oct 1980), 2508{2510. [7]Grunbaum, F. A. Commuting pairs of linear ordinary dierential operators of orders four and six. Physica D 31 (1988), 424{433. [8]Henrici, P. Applied and Computational Complex Analysis , vol. 3. John Wiley & Sons, New York, 1986. [9]Hill, J. M. Solution of Dierential Equations by Means of One-Parameter Groups . Pitman Publishing Co., Marsheld, MA, 1982. [10]Ince, E. L. Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [11]Landau, E. Journal fur die reine und angewandte Mathematik 124 (1902), 115{120. [12]Rainville, E. D. Intermediate Dierential Equations . The MacMillan Company, New York, 1964. [13]Sandell, D. C., and Stein, F. M. Factorization of operators of second order linear homogeneous ordinary dierential equations. Two Year College Mathematics Journal 8 (1977), 132{141. [14]Schwarz, F. Ecient factorization of linear ODE’s. ACM-SIGSAM Bulletin 28 (1994), 9{17. [15]Weston, V. H. Factorization of the wave equation in higher dimensions. J. Math. Physics 28 , 5 (May 1987), 1061{1068. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 300 II.A Exact Methods for ODEs 67. Factorization Method Applicable to Eigenvalue/eigenfunction problems for homogeneous linear second order ordinary dierential equations. Yields An equation from which a single eigenfunction can be used to calculate additional eigenfunctions. Idea By \factoring" an ordinary dierential equation into a certain form, a ladder of eigenfunctions may be formed. Procedure Suppose we have the linear second order ordinary dierential equation d2y dx2+r(x;m)y+y=0; (67.1) wheremis an integer for which we would like to determine the eigenfunc- tionsfygcorresponding to a single value of the eigenvalue .W e d e n o t e the eigenfunction by y(;m) and suppress the xdependence. The equation in (67.1) is said to be factorizable if it is equivalent to each of Hm+1 +Hm+1 −y(;m)=L(;m +1 )y(;m); Hm −Hm +y(;m)=L(;m)y(;m);(67.2.a-b) whereL(;m) is a function and the Hm are dierential operators. Hm =k(x;m)d dx; For a factorizable equation, nding L(;m)a n dt h eHm is a dicult task. Also, not all equations in the form of equation (67.1) are factorizable. If equation (67.1) is factorizable and if y(;m) is a solution of equation (67.1), then (see notes) y(;m +1 )=Hm+1 −y(;m); y(;m−1) =Hm +y(;m);(67.3.a-b) are also solutions corresponding to the same value of , but dierent values ofm. Hence, given one solution of equation (67.1) (for a specic value of ), a ladder of solutions belonging to this value of may be formed by repeatedly iterating equation (67.3). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 67. Factorization Method 301 Example 1 The equation for the associated spherical harmonics may be put in the form d2y d2−m2−1 4 sin2+ +1 4 y=0: (67.4) This equation is factorizable, and we nd Hm = m−1 2 cotd dx; L(;m)=− m−1 22 ;(67.5) The eigenvalues of equation (67.4) are of the form =l(l+1 ) f o rl= m;m +1;:::. Some of the eigenfunctions of equation (67.4) are of the form yl l()=1
3
5


(2l+1 ) 2
2
4


(2l)1=2 sinl+1=2: All of the remaining eigenfunctions may be found from equation (67.3) and equation (67.5) to be given by ym−1 l()=1p (l+m)(l+1−m) m−1 2 cot+d d ym l(); ym+1 l()=1p (l+m+1 ) (l−m) m+1 2 cot−d d ym l(): Example 2 As another example, Legendre’s dierential equation (1−x2) (1−x2)y0 m0+m(m+1 )ym=0 has the factorizations Hm −Hm +ym=−m2ym; Hm+1 +Hm+1 −ym=−(m+1 )2ym; whereHm =( 1−x2)d dxmx. This factorization leads to the ladder of solutions:ym+1=Hm −ym. Notes 1. The results in equation (67.3) are straightforward to derive. For example, operating on equation (67.2.b) with Hm +results in Hm +Hm − Hm +y(;m)/bracerightbig =L(;m) Hm +y(;m)/bracerightbig : (67.6) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 302 II.A Exact Methods for ODEs Because this has the same form as equation (67.2.a), which is by hy- pothesis equivalent to equation (67.1), it must be that y=Hm +y(;m) is a solution of equation (67.1). In equation (67.3), we called this y(;m−1) because, when equation (67.6) is compared to equation (67.2.a), the parameter mis replaced by m−1. 2. The factorization method has been generalized to systems of equa- tions in Humi [4]. 3. The operators in equation (67.3) are sometimes called raising and low- ering operators. This method is sometimes called the ladder method . 4. Infeld and Hull [5] have a large list of equations to which this method applies. 5. The paper by Hermann [3] relates the technique in this section to Lie groups. Sattinger and Weaver [8, pages 49{54] also consider the relation to Lie groups. 6. See also Lamb [6, pages 38{41] and Morse and Feshback [7, pages 788{789]. References [1]Barut, A. O., Inomata, A., and Wilson, R. A new realization of dynamical groups and factorization method. J. Phys. A: Math. Gen. 20 (1987), 4075{4083. [2]Bessis, N., and Bessis, G. Algebraic recursive determination of matrix elements from ladder operator considerations. J. Phys. A: Math. Gen. 20 (1987), 5745{5754. [3]Hermann, R. Infeld{Hull factorization, Galois{Picard{Vessiot theory for dierential operators. J. Math. Physics 22 , 6 (June 1981), 1163{1167. [4]Humi, M. Factorization of systems of dierential equations. J. Math. Physics 27(Jan 1986), 76{81. [5]Infeld, L., and Hull, T. E. The factorization method. Rev. Mod. Physics 23, 1 (Jan 1951), 21{68. [6]Lamb, G. L. Elements of Soliton Theory . John Wiley & Sons, New York, 1980. [7]Morse, P. M., and Feshback, H. Methods of Theoretical Physics . McGraw{Hill Book Company, New York, 1953. [8]Sattinger, D. H., and Weaver, O. L. Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics . Springer{Verlag, New York, 1986. [9]Schrodinger, E. A method of determining quantum-mechanical eigenvalues and eigenfunctions. Proc. Roy. Irish Acad. A46 (1940), 9{16. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 68. Fokker{Planck Equation 303 68. Fokker{Planck Equation Applicable to Linear ordinary dierential equations with linearly appearing \white Gaussian noise" terms (a single dierential equation or as y s t e m ) . Yields A Fokker{Planck equation (which is a parabolic partial dierential equation) for the probability density of the solution. Idea If a dierential equation contains random terms, then the solution to the dierential equation can only be described statistically. The solutionto the Fokker{Planck equation is the probability density of the solution to the original dierential equation. Procedure Here we present the technique for constructing the Fokker{Planck equa- tion for a linear system of ordinary dierential equations depending on several white noise terms. Consider the linear dierential system for the m component vector x(t) d dtx(t)=b(t;x)+(t;x)n(t); x(t0)=y;(68.1.a-b) where(t;x) is a realmnmatrix and n(t) is a vector of nindependent white noise terms. That is, E[ni(t)] = 0; E[ni(t)nj(t+)] =ij();(68.2) whereE[
] is the expectation operator, ijis the Kronecker delta, and () is the delta function. The Fokker{Planck equation corresponding to equation (68.1.a) is given by @P @t=−mX i=1@ @xi(biP)+1 2mX i;j=1@2 @xi@xj(aijP); (68.3) whereP=P(t;x) is a probability density and the matrix A=(aij)i s dened by A(t;x)=(t;x)T(t;x). The initial conditions for equation (68.3) come from equation (68.1.b); they are P(t0;x)=mY i=1(xi−yi): (68.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 304 II.A Exact Methods for ODEs The solution of equations (68.3) and (68.4) is the probability density of the solution to equation (68.1). Any statistical information about x(t)t h a t could be ascertained from equation (68.1) can be derived from P(t;x). For example, the expected value of some function of xandt,s a yh(x;t), at a timet, can be calculated by E[h(x(t);t)] =Z1 −1h(x(t);t)P(t;x)dx: Special Case In the special case of one dimension, the stochastic dierential equation dx dt=f(x)+g(x)n(t); (68.5) withx(0) =z, corresponds to the Fokker{Planck equation @P @t=−@ @x(f(x)P)+1 2@2 @x2(g2(x)P); forP(t;x)w i t hP(0;x)=(x−z). Example Consider the Langevin equation x00+x0=N(t); (68.6) with the initial conditions x(0) = 0;x0(0) =u0; (68.7) whereN(t) satises E[N(t)] = 0; E[N(t)N(t+)] =():(68.8) From equation (68.8), we recognize that N(t) is a white noise term. There- fore, we can use the Fokker{Planck equation to determine the probability density ofx(t). Because equation (68.6) has second derivative terms, we rewrite equation (68.6) and equation (68.7) as the vector system (see page 146) d dt x u = u −u + 00 01 n1(t) n2(t) ; x u t=0=0 u0 :(68.9) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 68. Fokker{Planck Equation 305 The Fokker{Planck equation for P(t;x;u ), the joint probability density of xanduat timet,i s @P @t=−@ @x(uP)+@ @u(uP)+1 2@2P @u2; P(0;x;u)=(x)(u−u0):(68.10) In this example, we can solve equation (68.10) exactly by taking a Fourier transform in x(see page 350) and then using the method of characteristics (see page 432). We eventually determine P(t;x;u )=1 detDexp − x−x u−u D x−x u−uT! ; whereD= xxxu xuuu , and the parameters fx;u;xx;xu;uugare given by x=u0 /parenleftbig 1−e−t
; u=u0e−t; 2 xx=t 2−2 3/parenleftbig 1−e−t
+1 23/parenleftbig 1−e−2t
; 2 xu=1 2/parenleftbig 1−e−t
−1 22/parenleftbig 1−e−2t
; 2 uu=1 2/parenleftbig 1−e−2t
: The details of this calculation are presented in Schuss [7]. Notes 1. With a Fourier transform, the method of characteristics can often solve a Fokker{Planck equation in one dimension. 2. Because a Fokker{Planck equation and the equation for a Green’s function (see page 318) both have delta function forcing terms, thesolution techniques are similar. 3. Not all noise terms are white Gaussian noise (the requirements in equation (68.2) are very stringent). The book by Srinivasan and Vasudevan [8] has descriptions of several approximate techniques for other types of noise. 4. When the coecient of the noise term (i.e., g(x) in equation (68.5)) is small, then a singular perturbation problem generally results. 5. The solution of equation (68.1) is a Markov process; the density of its probability transition function is given by the solution to the Fokker{ Planck equation and its initial conditions. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 306 II.A Exact Methods for ODEs 6. Another name for the Fokker{Planck equation is the forward Kol- mogorov equation. 7. The solution of the Fokker{Planck equation in equation (68.3) (and its initial conditions in equation (68.4)) might be better representedbyP(t;x;t 0;y). The function P(t;x;t0;y) also satises the backward Kolmogorov equation, which is the adjoint of equation (68.3). This equation @P @t0=−mX i=1bi@P @yi−1 2mX i;j=1aij@2P @yi@yj; P(t0;x;t0;y)=(x−y);(68.11) has as its independent variables the \backward variables" ft0;yg. 8. When only moments of the probability density P(t;x) are required, the method of moments (see page 568) may sometimes be used tocalculate these moments without having to solve the Fokker{Planck equation. 9. Another equivalent form of equation (68.1.a) that often appears is dx(t)=b(t;x)dt+(t;x)dw(t); (68.12) where w(t) is a vector of independent standard Wiener processes (see page 91). 10. Consider a particle starting at yand randomly moving in a domain Ω. If the probability density of the location evolves according to @P @t=L[P]=−mX i=1bi(y)@P @yi+1 2mX i;j=1aij(y)@2P @yi@yj; (68.13) Then the expectation of the exit time w(y) is the solution of L[w]=−1i nΩ ,w i t h w=0o n@Ω. Then the probability u(y) that the exit occurs on the boundary segment Γ is the solution of L[u]=0i nΩw i t h u(y)=( 1f o r y2Γ 0f o r y2Ω=Γ. References [1]Chang, J. S., and Cooper, G. A practical dierence scheme for Fokker{ Planck equations. J. Comput. Physics 6 (1970), 1{16. [2]Dita, P. The Fokker{Planck equation with absorbing boundary. J. Phys. A: Math. Gen. 18 (1985), 2685{2690. [3]Gardiner, C. W. Handbook of Stochastic Methods . Springer{Verlag, New York, 1985. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 68. Fokker{Planck Equation 307 [4]Garrido, L., and Masoliver, J. On a class of exact solutions to the Fokker{Planck equations. J. Math. Physics 23 , 6 (June 1982), 1155{1158. [5]Harrison, G. W. Numerical solution of the Fokker Planck equation using moving nite elements. Num. Meth. Part. Di. Eqs. 4 (1988), 219{232. [6]Risken, H. The Fokker{Planck Equation . Springer{Verlag, New York, 1984. [7]Schuss, Z. Theory and Applications of Stochastic Dierential Equations . John Wiley & Sons, New York, 1980. [8]Srinivasan, S. K., and Vasudevan, R. Introduction to Random Dierential Equations and Their Applications . American Elsevier Publishing Company, New York, 1971. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 308 II.A Exact Methods for ODEs 69. Fractional Dierential Equations Applicable to Fractional dierential equations. Yields An exact solution. Idea There are two common ways to solve fractional dierential equations; using an integral transform or transforming to an ordinary dierential equation. Procedure There are two main methods for solving fractional dierential equations Transformation to an ordinary dierential equation Using the Laplace transform To transform to an ordinary dierential equation, care must be taken because the ordinary chain rule from calculus does not apply to fractional derivatives. Example 1 This example will convert a fractional dierential equation into an ordinary dierential equation. Suppose we wish to solve the fractional dierential equation d1=2f dx1=2+f= 0 (69.1) forf(x). To convert this to an ordinary dierential equation, we will dierentiate with respect to xone-half time. This will produce a new dierential equation that involvesd1=2f dx1=2. Eliminating this term between the new equation and equation (69.1), we will have determined an ordinarydierential equation. To dierentiate equation (69.1) with respect to xone-half time, we have to use the dierentiation rule (from Oldham and Spanier [3, page 155]) d 1−Q dx1−QdQ dxQf=df dx+C1xQ−2+C2xQ−3+


+CmxQ−m−1; where 0<Qm<Q +1 ,mis an integer and the fCigare arbitrary constants. Hence, dierentiating equation (69.1) one-half time results in df dx−C1x−3=2+d1=2f dx1=2=0: (69.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 69. Fractional Dierential Equations309 Eliminating the d1=2=dx1=2term between equations (69.1) and (69.2) re- sults in df dx−f=C1x−3=2; (69.3) which is an ordinary dierential equation for f(x). Equation (69.3) has the solution (obtained by use of integrating factors) f(x)=Dex−2C1pexerf(px)+1px ; (69.4) whereDis another arbitrary constant. If we now utilize equation (69.4) in equation (69.1), it turns out that DandC1are related by D=2C1p. This is because of the identities d1=2 dx1=2exerf(px)=ex;d1=2 dx1=21px=0; d1=2 dx1=2ex=1px+exerf(px); from Oldham and Spanier [3, pages 119 and 123]. Therefore, the solution of equation (69.1) is f(x)=D exerfc(px)−1px : Example 2 This example will solve a fractional dierential equation by use of Laplace transforms. Suppose we wish to solve the fractional dierential equation df dx+d1=2f dx1=2−2f=0: (69.5) The Laplace transform of equation (69.5) is sF(s)−f(0) +psF(s)−d−1=2f(0) dx−1=2−2F(s)=0; (69.6) whereF(s) is dened to be the Laplace transform of f(x); that is,F(s)=R1 0f(x)e−xsds. If we dene the constant CbyC=f(0)+d−1=2f(0)=dx−1=2, then the solution to equation (69.6) is given by F(s)=C (ps−1)(ps+2 )=C 3(ps−1)−C 3(ps+2 ); (69.7) and so the nal solution to equation (69.5) can be obtained by nding the inverse Laplace transform to equation (69.7), which is f(x)=C 3 2e4xerfc(2px)+exerfc(−px) : CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 310 II.A Exact Methods for ODEs Notes 1. Fractional dierential equations are also called extraordinary dier- ential equations . 2. One of many equivalent denitions for fractional derivatives is the following dq dxqf(x)=dn dxn1 Γ(n−q)Zx af(y) (x−y)q−n+1dy ; forn>q0. 3. Certain diusion problems can be reduced to the solution of a semi- dierential equation (one in which all the derivatives are either to aninteger order or a half integer order). See Oldham and Spanier [3, Chapter 11] for details. 4. A third technique for solving fractional dierential equations is by the use of power series (see page 403). For fractional dierential equations, a series of the form f(x)=x p1X k=0akxk=n is used, where p>−1,nis an integer, a06=0 ,a n dt h efaigare unknowns. 5. Erd elyi’s paper [1] contains several boundary value problems for or- dinary dierential equations that are solved by using fractional dif- ferential techniques. References [1]Erdelyi, A. Axially symmetric potentials and fractional integration. J. Soc. Indust. Appl. Math. 13 , 1 (March 1965), 216{228. [2]Nishimoto, K. Applications to the solutions of linear second order dierential equations of Fuchs type. In Fractional Calculus ,A .C .M c B r i d ea n dG .F . Roach, Eds. Pitman Publishing Co., Marsheld, MA, 1985, pp. 140{153. [3]Oldham, K. B., and Spanier, J. The Fractional Calculus . Academic Press, New York, 1974. [4]Ross, B. Fractional Calculus and Its Applications (Proceedings of the International Conference at the University of New Haven, June 1974) . No. 457 in Lecture Notes in Mathematics. Springer{Verlag, New York, 1975. [5]Wyss, W. The fractional diusion equation. J. Math. Physics 27 , 11 (1986), 2782{2785. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 70. Free Boundary Problems311 70. Free Boundary Problems Applicable to Systems of dierential equations in which the loca- tion of the boundary of the domain is one of the unknowns to be deter- mined. Idea Sometimes a similarity solution may be used to determine the location of the free boundary. In more dicult problems, a numerical techniquemay be required. Procedure In free boundary problems, a dierential equation must be solved in a domain whose size can vary. One of the unknowns to be determined is the size of the domain on which the equation is to be satised. Dierential equations of this type are most often solved numerically. In rare cases, an analytical solution may be obtained; these solutions are generally found by use of similarity methods (see page 497). Example Consider a mass of water in x0a tt i m et= 0. Initially, the water has the constant temperature TH>0. If a constant temperature TC<0i s maintained at the surface x= 0, then the boundary of freezing, x=s(t), will move into the fluid. The unknowns to solve for in this problem are thetemperature of the water w(x;t), the temperature of the ice u(x;t), and the location of the unknown boundary, x=s(t). See gure 70.1. The equations that describe the unknowns are u t=uxx; for 0<x<s (t);t0; wt=wxx; fors(t)<x<1;t0; u(0;t)=TC; w(x;0) =TH; u(s(t);t)=0; w(s(t);t)=0; ux(s(t);t)−wx(s(t);t)=s0(t):(70.1.a-g) Here we have dened the freezing boundary to be the curve along which the temperature is zero, and equation (70.1.g) represents the transfer of latent heat necessary to create the ice. The parameter is the latent heat of fusion times the density divided by the coecient of heat conduction. Now, we propose the similarity solution. Because diusion equations often have time scaling as the square of a distance, we assume that a CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 312 II.A Exact Methods for ODEs/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./. /././././././././././././././././././.so lidliquids t/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./././././././././././././././././././. /. /./././. /./././././././././././././././././././././././. /././././././././././././././././././. /. /./././. /./././././././././././././././././././././././. /././././././././././././././././././. /. /./././. /./././././././././././././././././././././././. /././././././././././././././././././. /. /./././. /././././././././././././././././././././././. /././././././././././././././././././. /. /./././. /././././././././././././././././././././././. /./././././././././././././././././. /. /./././. /././././././././././././././././././././././. /./././././././././././././././././. /. /./././. /./././././././././././././././././././././. /././././././././././././././././. /. /./././. /./././././././././././././././././././././. /././././././././././././././././. /. /./././. /././././././././././././././././././././. /./././././././././././././././. /. /./././. /./././././././././././././././././././. /././././././././././././././. /. /./././. /././././././././././././././././././. /./././././././././././././. /. /./././. /./././././././././././././././././. /./././././././././././././. /. /./././. /././././././././././././././././. /./././././././././././. /. /./././. /./././././././././././././././. /././././././././././. /. /./././. /././././././././././././././. /./././././././././. /. /./././. /./././././././././././././. /././././././././. /. /./././. /././././././././././././. /././././././. /. /./././. /././././././././././. /./././././. /. /./././. /./././././././././. /./././. /. /./././. /./././././././. /././. /. /./././. /./././././. /. /. /./././. /././././. /./././. /././. /./. /.Figure 70.1: This diagram illustrates the location of the freezing boundary for the system given in equation (70.1). solution to equation (70.1) can be found with u(x;t)=f()=fxp t ;w (x;t)=g()=gxp t ; (70.2) for some unknown functions f()a n dg(). Using these proposed forms in equation (70.1.g) shows that these forms are possible only if the freezingboundary is given by s(t)=p t; (70.3) for some value of . Using equations (70.2) and (70.3) in equation (70.1), we nd the equivalent system f00()+1 2f0()=0; for 0<<; g00()+1 2g0()=0; for<<1; f(0) =TC;f()=0; g(1)=TH;g()=0; f0()−g0()= 2:(70.4) The ordinary dierential equations in equation (70.4) may be solved to determine that f()=TC−THerf(=2) erf(=2); g()=TH erfc(=2)[erf(=2)−erf(=2)];(70.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 70. Free Boundary Problems313 wheresatises the transcendental equation TH erf(=2)+TC erfc(=2)=−p 2e2=4: Notes 1. In writing equation (70.1.a) and equation (70.1.b), we have assumed that the thermophysical parameters in both the ice and the water arethe same (i.e., the Stefan number, which is a ratio of these parameters, is equal to one). In reality, these parameters are dierent and a constant that cannot be scaled out must be introduced into either equation (70.1.a) or equation (70.1.b). 2. The example illustrated above is described in more detail in Crank [2, Chapter 3]. 3. Melting problems for a pure material are also known as Stefan prob- lems. 4. Another technique often used in free boundary problems is changing coordinates so that the free boundaries become xed in the new coordinate space. This is the idea behind the hodograph method (see page 456). 5. Free boundary problems often arise in hydrodynamics, when the flow over an airfoil is being computed. When the flow becomes supersonic, the type of governing equation changes from hyperbolic to elliptic and a dierent type of numerical scheme is required. Where the equationchanges type is not known a priori . 6. Some of the popular numerical techniques for solving free boundary problems go by the name of front tracking methods orfront xing methods . These techniques generally require that the location of the free boundary be approximately known before the computer code isrun. A better approach is to use enthalpy methods .T h e s e m e t h o d s d o not need initial information about the interfaces, and multiple fronts can also occur. 7. The paper by Hill and Dewynne [6] discusses several dierent approx- imation techniques applied to a single physical problem involving a free boundary. References [1]Charrier, P., and Tessieras, B. On front-tracking methods applied to hyperbolic systems of nonlinear conservation laws. SIAM J. Numer. Anal. 23, 3 (June 1986), 461{472. [2]Crank, J. Free and Moving Boundary Problems . Clarendon Press, Oxford, England, 1984. [3]Duncan, D. B. A simple and eective self-adaptive moving mesh for enthalpy formulations of phase change problems. IMA J. Num. Analysis 11(1991), 55{78. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 314 II.A Exact Methods for ODEs [4]Elliot, C. M., and Ockendon, J. R. Weak and Variational Methods for Moving Boundary Problems . Pitman Publishing Co., Marsheld, MA, 1982. [5]Furzeland, R. M. A comparative study of numerical methods for moving boundary problems. J. Inst. Maths. Applics 26 (1980), 411{429. [6]Hill, J. M., and Dewynne, J. N. On the inward solidication of cylinders. Quart. Appl. Math. 44 , 1 (April 1986), 59{70. [7]Marshall, G. A front tracking method for one-dimensional moving boundary problems. SIAM J. Sci. Stat. Comput. 7 , 1 (January 1986), 252{ 263. [8]Rubensten, L. I. The Stefan Problem . Amer. Math. Soc., Providence, RI, 1971. Translated by A. D. Solomon. [9]Womble, D. E. A front-tracking method for multiphase free boundary problems. SIAM J. Numer. Anal. 26 , 2 (April 1989), 380{396. [10]Wood, A. S. An ecient nite-dimensional scheme for multidimensional stefan problems. Int. J. Num. Meth. Eng. 23 (1986), 1757{1771. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 71. Generating Functions315 71. Generating Functions Applicable to Systems of dierential equations, where each equa- tion has a similar form. Yields An exact analytic solution. Idea Sometimes a single function can be used to contain the information in several equations. Procedure We illustrate the method as it applies to ordinary dierential equations. Suppose we have a system of ordinary dierential equations for fuk(t)g,a l l of the form d dtuN=f(uN−m;:::;uN;:::;uN+m;t); (71.1) forN=1;2;:::;1orN=1;2;:::;1. We might introduce the ordinary generating function G(s;t)=X kuk(t)sk; (71.2) or the exponential generating function H(s;t)=X kuk(t)sk k!: (71.3) Using equation (71.2) (or equations (71.3)) and (71.1), we can sometimes nd a partial dierential equation for G(s;t)( o rH(s;t)). After solving the partial dierential equation, we can determine the fuk(t)gfrom either uk(t)=1 k!d dsk G(s;t) s=0; or uk(t)=d dsk H(s;t) s=0: After we have solved for the fuk(t)g, we must then check that equation (71.2) (or equation (71.3)) converges for the values of tthat are of interest. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 316 II.A Exact Methods for ODEs Example The classic equations relating to service times are called the birth and death equations (see notes). For the special case of \constant death" and \linear birth," these equations have the form d dtP0(t)=−P0(t)+P1(t); d dtPN(t)=PN−1(t)−(+N)PN(t)+(N+1 )PN+1(t);(71.4) whereandare constants and N=1;2;:::;1. The initial conditions for equation (71.4) are PN(0) =Nj; (71.5) whereNjis the Kronecker delta and jis a given positive integer. The ordinary generating function is dened in this case by G(t;s)=1X k=0Pk(t)sk: (71.6) Dierentiating G(t;s) with respect to tleads to @G @t=1X k=0d dtPk(t) sk =[−P0(t)+P1(t)]s0 +1X k=1(Pk−1(t)−(+k)Pk(t)+(k+1 )Pk+1(t))sk =(s−1) P0+P1s+P2s2+


 +(1−s) P1+2P2s+3P3s2+


 =( 1−s) −G+@G @s :(71.7) The initial condition for G(t;s), from equations (71.5) and (71.6), becomes G(0;s)=sj: (71.8) The partial dierential equation in (71.7), with the initial condition in (71.8), can be solved by the method of characteristics (see page 432). The solution is G(t;s)=e−(1−s)(1−e−t)= 1−(1−s)e−tj: (71.9) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 71. Generating Functions317 Taking a Taylor series of equation (71.9) with respect to s(see equation (71.6)) allows all of the fPk(t)gto be found. For example P0(t)=e−(1−y)=(1−y)t; P1(t)=e−(1−y)=(1−y)t−1  y2+(t−2)y+ ; P2(t)=e−(1−y)=(1−y)t−2 2n 2y4+( 2t−42)y3 +[t(t−1)2−2(2t−3)]y2+( 2t−42)y+2o ;(71.10) wherey=e−t. Notes 1. In the birth and death equations (see Karlin and Taylor [2, page 135]),Pk(t) is the probability of kunnished jobs at time t.W ea l s o assume: Initially there are Mjobs to be nished, the average service time is, and the average number of new jobs spawned by an existing job isper unit time. 2. For the example given above, Laplace transforms (see page 350) could also have been used to solve equation (71.7) with equation (71.8). 3. Nonlinear systems of dierential equations can also be solved by this method. A classic application is to equations describing the aggregation of particles (see Feller [1, Chapter 17, pages 444{482]). 4. See Taylor and Karlin [4, pages 310{316 and 337{338]. References [1]Feller, W. An Introduction to Probability Theory and Its Applications .J o h n Wiley & Sons, New York, 1968. [2]Karlin, S., and Taylor, H. M. A First Course in Stochastic Processes . Academic Press, New York, 1975. [3]Letessier, J. The numerical resolution of birth and death Kolmogorov equations. Comp. & Maths. with Appls. 13 , 7 (1987), 595{600. [4]Taylor, H. M., and Karlin, S. An Introduction to Stochastic Modeling . Academic Press, New York, 1984. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 318 II.A Exact Methods for ODEs 72. Green’s Functions Applicable to Linear dierential equations with linear boundary conditions and initial conditions. Yields An exact solution, in the form of an integral or an innite series. Idea Initially, the solution of the linear dierential equation with a \point source" is determined. Then, using superposition, the \forcing function"(appearing in either the dierential equation or the boundary condition) is treated as a collection of point sources. Procedure Suppose we have the following linear dierential equation for u(x) L[u]=f(x); (72.1) with the linear homogeneous boundary conditions Bi[u]=0; (72.2) fori=1;2;:::;n . Suppose we can solve for G(x;z), whereG(x;z) satises L[G(x;z)] =(x−z); Bi[G(x;z)] = 0 and(x) is the usual delta function. Then the solution to equations (72.1) and (72.2) can be written as u(x)=Z G(x;z)f(z)dz; (72.3) integrated over some appropriate region. Conversely, suppose we want to solve the linear homogeneous dieren- tial equation L[v]=0; B[v]=h(x):(72.4) If we can solve L[g(x;z)] = 0; B[g(x;z)] =(x−z); CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 72. Green’s Functions319 forg(x;z), then the solution to equation (72.4) is given by v(x)=Z g(x;z)h(z)dz: BothG(x;z)a n dg(x;z) are called Green’s functions. The functions f(x) andh(x) are often referred to as \forcing functions." If, for example, f(x)0, then by equation (72.3) u(x)0. Green’s functions can be calculated once, then used repeatedly for dierent functions f(x)a n dh(x). Some Green’s functions are tabulated in table 72.1. To calculate the Green’s function G(x;z), we require: (a)L[G(x;z)] = 0; except at x=z: (b)Bi[G(x;z)] = 0: (72.5) (c) IfL[
]i sa nnth order ordinary dierential equation, then G(x;z) must be continuous (with its derivatives up to ordern−1) at x=z. (d)Zz+ z−L[G(x;z)]dx=1: The conditions on g(x;z) are very similar: (a)L[g(x;z)] = 0: (72.6) (b)B[g(x;z)] = 0: except at x=z; (c) IfL[
]i sa nnth order ordinary dierential equation, then g(x;z) must be continuous (with its derivatives up to order n−1) at x=z. (d)Zz+ z−B[g(x;z)]dx=1: Conditions (72.5.a,d) and (72.6.b,d) follow from the denition of the delta function. Conditions (72.5.c) and (72.6.c) follow from the denition of what a solution to an nth order dierential equation means; and conditions (72.5.b) and (72.6.c) follow from the dening equations for G(x;z)a n d g(x;z). Many methods can be used to construct a G(x;z)o rag(x;z)t h a t satises the above four requirements. We will illustrate two methods for constructing G(x;z) for the special case of a second order linear ordinary dierential equation. Then we illustrate the construction process for g(x;z) for a partial dierential equation. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 320 II.A Exact Methods for ODEs In the following, r=(x;y;z ),r0=(x0;y0;z0),R=jr−r0j,P2=(x− x0)2+(y−y0)2,a n dH(
) is the Heaviside function For the potential equation r2G+k2G=−4(r−r0), with the radiation condition (outgoing waves only), the solution is G=8 >< >:2i keikjx−x0jin one dimension, iH(1) 0(kP) in two dimensions, eikR Rin three dimensions, whereH(1) 0(
) is a Hankel function (also called a Bessel function of the third kind). For the diusion equation r2G−a2@G @t=−4(r−r0)(t−t0), with the initial condition G=0f o rt<t 0, and the boundary condition G=0a tr=1inNdimensions, the solution is G=4 a2 a 2p (t−t0)!N exp −a2jjr−r0jj2 4(t−t0) : For the wave equation r2G−1 c2@2G @t2=−4(r−r0)(t−t0), with the initial conditions G=Gt=0f o rt<t 0, and the boundary condition G=0a tr=1the solution is G=8 >>< >>:2cHh (t−t0)−jx−x0j ci for one space dimension, 2cp c2(t−t0)2−P2H (t−t0)−P c for two space dimensions, 1 RR c−(t−t0) for three space dimensions. Table 72.1: Green’s functions for common partial dierential equations. Special Case 1 Dene the general linear second order ordinary dierential equation with linear homogeneous boundary conditions by L[u]: =d dx p(x)du dx −s(x)u; B1[u]: =1u(a)+2u0(a)=0; B2[u]: =1u(a)+2u0(b)=0; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 72. Green’s Functions321 and suppose that we wish to solve L[u]=f(x). Ify1(x)a n dy2(x) are non-trivial (i.e., not identically equal to zero) and satisfy L[y1]=0;B 1[y1]=0; L[y2]=0;B 2[y2]=0; then we can write G(x;z)a s G(x;z)=(y1(x)y2(z) p(z)W(z)foraxz; y2(x)y1(z) p(z)W(z)forzxb; whereW(z)=y1(z)y2(z) y10(z)y20(z)is the Wronskian of y1(x)a n dy2(x)a tt h e pointx=z. Special Case 2 Suppose that L[
] is a self-adjoint operator, so that it has a complete set of orthogonal eigenfunctions (see page 103). Suppose further that we know the eigenvaluesfngand the eigenfunctions fngforfL;B 1;B2g.T h a ti s , L[n]=nn; B1[n]=0; B2[n]=0; thenG(x;z) is found to be G(x;z)=1X n=1n(x)n(z) nR 2n(x)dx: Example 1 Suppose we wish to solve y00=f(x); y(0) = 0;y(L)=0:(72.7) Using the rst method, we require the solutions y1(x)a n dy2(x)o f y00 1=0;y 1(0) = 0; y00 2=0;y 2(L)=0: The solutions to these equations are y1(x)=Ax; y 2(x)=B(x−L); CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 322 II.A Exact Methods for ODEs whereAandBare arbitrary constants. We compute the Wronskian to be W(z)=ABL . Therefore, G(x;z)=(x(z−L) Lfor 0xz; z(x−L) LforzxL:(72.8) Using the second method, we nd the eigenvalues and eigenfunctions to be n=n L;n(x)=s i nnx=s i nnx L ; so that G(x;z)=2L n1X n=1sinnx L sinnz L : (72.9) Using either of equations (72.8) or (72.9) for G(x;z), the solution to equa- tion (72.7) can be written as y(x)=ZL 0G(x;z)f(z)dz: (72.10) For example, using equation (72.8) in equation (72.10), the solution to (72.7) can be written as y(x)=ZL xx(z−L) Lf(z)dz+Zx 0z(x−L) Lf(z)dz: (72.11) Note the similarity between equation (72.11) and the form of the solution shown in the section on variation of parameters (see page 418). If, for example, f(x)=x3, then evaluation of equation (72.11) results in y(x)=x 20(x4−L4): The second method yields the same answer. For this example, the second method is equivalent to using nite Fourier series (see page 344). Example 2 Suppose we are given the parabolic partial dierential equation @2u @x2=1 a2@u @t(72.12) foru(x;t) with the initial and boundary conditions u(x;0) =h(x);u(1;t)=0: (72.13) We choose to write the solution as u(x;t)=Z1 −1g(x;t;z)h(z)dz; (72.14) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 72. Green’s Functions323 where the Green’s function g(x;t;z) satises @2g @x2=1 a2@g @t; g(x;0;z)=(z−x);g(1;t;z)=0: Taking a Fourier transform (in x) of the equation for g(x;t;z) results in dbg dt=−a2!2bg; bg(!;0;z)=1p 2ei!z;(72.15) wherebg(!;t;z) is dened to be the Fourier transform of g(x;t;z); that is, bg(!;t;z): =1p 2Z1 −1g(x;t;z)ei!xdx: Solving the ordinary dierential equation (72.15) results in bg(!;t;z)=1p 2ei!ze−a2!t: Using the inverse Fourier transform, we then have our solution g(x;t;z)=1p 2Z1 −1bg(!;t;z)e−i!xdx: By using the convolution theorem for Fourier transforms, we can determine that g(x;t;z)=1p 4a2te−(x−z)2=4a2t: This should be used in equation (72.14) to determine the solution to equa- tions (72.12) and (72.13). Notes 1. If zis in an-dimensional space, then the integrals appearing in equation (72.5.d) and equation (72.6.d) are nsingle integrals, each one over one of the coordinate axes. 2. Delta functions, in non-rectangular coordinate systems, are easily de- termined by a change of variables in the dening relation:R (z)dz= 1. In changing variables, the Jacobian of the transformation will thendivide the delta function terms. For example In a spherical coordinate system (denoted by the usual coor- dinatesr,,a n d) the delta function located at the point x 0=(r0;0;0)i sg i v e nb y (x−x0)=1 r2sin(r−r0)(−0)(−0); CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 324 II.A Exact Methods for ODEs forr06=0a n d06=0;. For a point source at r=r0and=0 , the representation (r−r0)()=2r2sinmay be used whereas a point source at the origin has the representation (r)=4r2. In a cylindrical coordinate system (denoted by the usual co- ordinates,,a n dz) the delta function located at the point x0=(0;0;z0)i sg i v e nb y (x−x0)=(−0)(−0)(z−z0) ; for0>0. A point source at the origin has the representation (z)() 2. 3. IfG(x;z) satises the problem adjoint to L[
] (see page 95), then G(x;z)=G(z;x). Therefore, if L[
] and its associated boundary conditions are self-adjoint and L[G(x;z)] =(x−z), thenG(x;z)= G(z;x). This is called the reciprocity principle . It can be observed in our example (see equation (72.9)). 4. When the operator is self-adjoint, the Green’s function is sometimes written in terms of the variables x<andx>instead ofxandz.W h e n this is done, x<(x>) represents the smaller (larger) of xandz.F o r example, (72.11) could have been written as G(x;z)=x<(x>−L) L. 5. Few analytic solutions of the Helmholtz equation r2G+k2 0n2(r)G=−(r−r0) are known when the index of refraction ,n(r), is variable. Solutions are known in the following cases: (point source) n=p 1+aTr+rTBr (point source, layered medium) n=z−1 (point source, layered medium) n=p A+Cz+Fz2 (line source) n=px (line source) n=p A+Bx+Cy+Dx2+Exy +Fy2 See Li et al. [8] for details. 6. As another example, the dierential equation with boundary condi- tions y00+k2y=f(x); y(0) = 0;y0(1) = 0 has the Green’s function G(x;z)=−cosk(1−x<)s i nkx> kcosk. 7. Consider the self-adjoint second order operator L[u]=(p(x)u0(x))0+ q(x)u(x), and consider the boundary conditions B1[u]: =a1u(a)+a2u0(a)=0; B2[u]: =b1u(b)+b2u0(b)=0:(72.16) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 72. Green’s Functions325 Dene(x)a n d (x) to be the solutions to L[]=r(x); B 1[]=0; L[ ]=r(x) ;B 2[ ]=0: Then, the Green’s function for the operator L−r, which satises the boundary conditions in equation (72.16), is given by G(x;z)= (x<) (x>) p(x)W(; ),w h e r eW(; ) represents the Wronskian. 8. There will not exist a Green’s function if the solution of the original problem is indeterminate. In this case, a generalized Green’s function will exist. As an example, consider the system y00=f(x); y(0) =y(1); y0(0) =y0(1): Ifu(x) is any solution to the above system, then so is u(x)+Cwhere Cis any constant. Because the solution of the original system is indeterminate, an ordinary Green’s function cannot be found. Seethe section on alternative theorems (page 15) or Farlow [5, pages 290{298] for details. 9. Sometimes, in such problems, the specic solution in which the Green’s function is symmetric in both xandzis chosen. This results in the modied Green’s function . See Stakgold [10, Chapter 1, pages 215{ 218] for details. 10. Fokker{Planck equations have delta function initial conditions. The methods used for solving these equations are the same as the methodsused for nding Green’s functions. 11. Some potential problems can be solved by assuming a continuum of sources. In these cases, the potential outside of the body, which is due to the presence of the body, is represented as the superposition of potentials due to point sources and dipoles lying entirely withinthe body. See Barshinger [1] for an example. 12. Butkovskiy’s book [3] has a comprehensive listing of Green’s func- tions. Any particular Green’s function problem is partitioned intoone of several separate disjoint groups labeled by a triple of integers: (r;m;n ). In this partitioning, rrepresents the dimension of the spatial domain, mis the order of the highest derivative with respect tot,a n dnis the order of the highest derivative with respect to the space variables. Over 500 problems are catalogued and solved. 13. See Butkov [2, Chapter 12, pages 503{552] and Zauderer [11, pages 353{449]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 326 II.A Exact Methods for ODEs References [1]Barshinger, R. The electrostatic eld about a thin oblate dielectric body of revolution. SIAM J. Appl. Math. 52 , 3 (June 1992), 651{675. [2]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [3]Butkovskiy, A. G. Green’s Functions and Transfer Functions Handbook . John Wiley & Sons, New York, 1982. Halstead Press. [4]Courant, R., and Hilbert, D. Methods of Mathematical Physics . Interscience Publishers, Inc., New York, 1953. [5]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [6]Greenberg, M. D. Application of Green’s Functions in Science and Engineering . Prentice{Hall, Inc., Englewood Clis, NJ, 1971. [7]Jordan, K. E., Richter, G. R., and Sheng, P. An ecient numerical evaluation of the Green’s function for the Helmholtz operator on periodic structures. J. Comput. Physics 63 , 1 (1986), 222{235. [8]Li, Y. L., Liu, C. H., and Franke, S. J. Three-dimensional Green’s functions for wave propogation in a linearly inhomogeneous medium|the exact analystical solution. J. Acoust. Soc. Am. 87 , 6 (June 1990), 2285. [9]Morse, P. M., and Feshback, H. Methods of Theoretical Physics . McGraw{Hill Book Company, New York, 1953. [10]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. [11]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 73. Homogeneous Equations 327 73. Homogeneous Equations Applicable to First order ordinary dierential equations of a cer- tain form. Yields An exact solution. Idea IfP(x;y)a n dQ(x;y) are homogeneous functions of xandyof the same degree, then, by the change of variable y=vx, the dierential equation y0=P(x;y)=Q(x;y) can be made separable. Procedure A function H(x;y) is called homogeneous of degree nifH(tx;ty )= tnH(x;y). In particular, a polynomial, P(x;y), of two variables is said to be homogeneous of degree nif every term of P(x;y) is of the form xjyn−j forj=0;1;:::;n . A homogeneous function of degree ncan be written as H(x;y)=xnH(1;y=x). Therefore, given an ordinary dierential equation of the form dy dx=P(x;y) Q(x;y); (73.1) whereP(x;y)a n dQ(x;y) are both homogeneous polynomials of degree n, we change variables by y=vxto obtain xdv dx+v=P(1;v) Q(1;v): Because this is a separable equation, it can be integrated to yield (see page 401)Zdv P(1;v) Q(1;v)−v=l o gx+C; whereCis an arbitrary constant. Example Suppose we have the ordinary dierential equation dy dx=2x3y−y4 x4−2xy3: (73.2) Because both the numerator and denominator of the right-hand side of equation (73.2) are homogeneous polynomials of degree four, we set y=vx to obtain xdv dx+v=2v−v4 1−2v3 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 328 II.A Exact Methods for ODEs or xdv dx=v+v4 1−2v3: This last equation is separable, and the solution is given by Zxdx x=Zv1−2v3 v+v4dv; logx=Zv1 v−3v2 1+v3 dv =l o gv−log(1 +v3)+l o gC(73.3) orx(1 +v3)=Cv,w h e r eCis an arbitrary constant. Substituting v=y=x in this yields the nal solution x3+y3=Cxy. Notes 1. Equation (73.1) may be made exact (see page 284) by multiplying by the integrating factor 1 =(Px−Qy). 2. This method is derivable from Lie group methods (see page 366).3. This method is contained in the method for scale invariant equations (see page 398). 4. Beware that the expression \homogeneous equation" has two entirely dierent meanings; see the denitions (page 6). 5. It may be simpler to think of homogeneous equations as ordinary dierential equations of the form dy=dx =f(y=x). This is equivalent to equation (73.1). 6. The equation dy dx=fa1x+b1y+c1 a2x+b2y+c2 (73.4) can always be made homogeneous or separable. Ifa1b26=a2b1, then the change of variables x=X+h; y=Y+k; changes equation (73.4) into the homogeneous equation dY dX=fa1X+b1Y a2X+b2Y ; whenhandksatisfy the equations:a1b1 a2b2h k =−c1 −c2 . Ifa1b2=a2b1, then the change of variables Y=x+b1 a1y= x+b2 a2yresults in the equation dY dx=1+b1 a1fa1Y+c1 a2Y+c2 : CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 73. Homogeneous Equations 329 7. See Boyce and DiPrima [1, pages 87{91], Ford [2, pages 40{45], Goldstein and Braun [3, pages 81{84], Ince [4, pages 18{20], and Simmons [5, pages 35{37]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Ford, L. R. Dierential Equations . McGraw{Hill Book Company, New York, 1955. [3]Goldstein, M. E., and Braun, W. H. Advanced Methods for the Solution of Dierential Equations . U.S. Government Printing Oce, Washington, D.C., 1973. NASA SP-316. [4]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [5]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 330 II.A Exact Methods for ODEs 74. Method of Images Applicable to Dierential equations with homogeneous boundary conditions and sources present. Yields An exact solution. Idea If we know the solution to a free space problem, then we can often use superposition to nd a solution in a nite domain with homogeneousboundary conditions. Procedure Given a problem with a point source present, solve the free space problem (i.e., disregarding the boundary conditions). By superposition, determine the solution when there are sources at dierent points of dierentstrengths. Choose the position and strengths of these sources so as to obtain the desired boundary conditions. The added sources cannot appear in the physical domain of the problem. Symmetry considerations tend to simplify the process of determining where the sources should go. Example 1 Suppose we wish to nd the potential, (x), outside of a grounded sphere of radius R, when there is a point source at position y(withjjyjj= >R ). The equations that represent this problem are r2=(x−y);  jjxjj=R=0; jjxjj=1=0;(74.1.a-c) in the region R<jjxjj<1(see gure 74.1). If the boundary condition at jjxjj=Ris ignored, then the problem r2Ψ=(x−y); Ψ jjxjj=1=0; has the solution (using Green’s functions, see table 72.1) Ψ=−1 4jjx−yjj: (74.2) If we place an additional source of strength Sat the point zand solve r2=(x−y)+S(x−z);  jjxjj=1=0;(74.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 74. Method of Images331/. /. /./. /. /. /. /. /. /./. /. /. /./. /. /./. /./. /. /./. /././. /./. /././. /././. /././././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././. /./././././. /././././. /././. /./././. /./././././././. /././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././././. /./././. /././. /./. /./. /./. /./. /./. /. /./. /. /./. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./#1E /=/0R/#0F z /#0F y/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././. /./././. /./././. /./././. /./././. /. /./. /./././././././. /././. /././. /./././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././. /./././. /././././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /.Figure 74.1: Equation (74.1) represents the potential outside of a grounded sphere of radius R, with a source point present. then we obtain (using equation (74.2) and superposition) =−1 4jjx−yjj−S 4jjx−zjj: (74.4) Note that the point zcannot be in the region R<jjxjj<1, because then equation (74.3) (whose solution we want to be the solution to equation(74.1)) will not satisfy equation (74.1.a). To determine the strength and location of the additional source ( Sand z), we calculate the potential at x=p,w h e r ejjpjj=R(i.e., on the surface of the sphere). We nd  x=p=−1 41 jjp−yjj+S jjp−zjj : F o rt h i st ob ez e r o( a n ds o= ), we require (after some vector algebra) S=−R4 4; z=R2 2y: Hence, =−1 41 jjx−yjj−R4 41 jjx−yR2=2jj (74.5) satises equation (74.3) and also equation (74.1.b). Because jjzjj<R (by virtue ofjjyjj=>R ) the point source, we added is not in the physical domain of the problem. Therefore, the solution to equation (74.1) is given by equation (74.5). Example 2 Suppose we wish to solve Laplace’s equation in the half plane: r2u=0; fory>0;−1<x<1; u(x;0) =f(x); u!0; asx2+y2!1:(74.6) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 332 II.A Exact Methods for ODEs/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /./././. /././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./././././././././././././././././././././. /././././././././././././././././././././.x y/#0F/#0F /#28 /#10/; /#11 /#29or igin al source/#28 /#10/; /, /#11 /#29im age sourceFigure 74.2: The original source and the image source for equation (74.8). The solution to (74.6) can be obtained by Green’s functions (see table 72.1): u(;)=−Z f(x)@G @y(x;0;;)dx; (74.7) where the Green’s function G(x;y;;) satises r2G=@2G @x2+@2G @y2=(x−)(y−); G(x;0;;)=0: (74.8.a-b) A solution to equation (74.8.a) is given by G(x;y;;)=1 2logp (x−)2+(y−)2: (74.9) But this does not satisfy equation (74.8.b). If we place an image source at (;−), having the opposite sign of the source at ( ;)t h e nG(x;y;;) will vanish along y= 0 by symmetry. See gure 74.2. Hence, the solution to (74.8) is G(x;y;;)=1 2logp (x−)2+(y−)2−1 2logp (x−)2+(y+)2: Using this is in equation (74.7), we obtain the solution to equation (74.6): u(;)=1 Z1 −1f(x)dx (x−)2+2: This solution is known as Poisson’s integral . Notes 1. The method of images is often used to solve Laplace’s equation in hydrodynamics and electrostatics. 2. The method of images can be used for diusion problems and hy- perbolic problems. See, for example, Butkov [1, pages 529{530 and 595{599] or Stakgold [5, pages 72{73 and 491{493]. 3. See also Jackson [3, pages 26{29], Kellog [4, pages 228{230], and Zauderer [6, pages 420{432]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 74. Method of Images333 References [1]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [2]Gautesen, A. K. Oblique derivative boundary conditions and the image method for wedges. SIAM J. Appl. Math. 48 , 6 (December 1988), 1487{1492. [3]Jackson, J. D. Classical Electrodynamics . John Wiley & Sons, New York, 1962. [4]Kellog, O. D. Foundations of Potential Theory . Dover Publications, Inc., New York, 1953. [5]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. [6]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 334 II.A Exact Methods for ODEs 75. Integrable Combinations Applicable to Systems of ordinary dierential equations. Yields One or more ordinary dierential equations that can be integrated exactly. Idea Sometimes, by combining pieces of a system of dierential equations, a combination of the dependent variables can be determined explicitly in terms of the independent variable. Procedure Integration of the system of ordinary dierential equations dxi dt=fi(t;x1;x2;:::;xn);fori=1;2;:::;n; is often accomplished by choosing integrable combinations . An integrable combination is a dierential equation that is derived from a system of dierential equations and is readily integrable. Example 1 Given the two equations dx dt=y anddy dt=x; (75.1) an integrable combination can be obtained by adding the two equations to obtain d(x+y) dt=x+y: This last equation can be integrated (treating x+yas a single variable) to yield x+y=Aet; (75.2) whereAis an arbitrary constant. For the equations in equation (75.1), another integrable combination may be obtained by subtracting the equa- tions. Integrating this new equation results in x−y=Be−t; (75.3) whereBis another arbitrary constant. The explicit solution for x(t)a n d y(t) may be obtained by combining equations (75.2) and (75.3). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 75. Integrable Combinations 335 Example 2 Suppose we have the nonlinear system of ordinary dierential equations dx dt=−3yz; dy dt=3xz; dz dt=−xy: Multiplying the rst equation by x, the second by 2 y, and the third by 3 z and adding, results in xdx dt+2ydy dt+3zdz dt=0: This last equation may be integrated to obtain x2+2y2+3z2=C,w h e r eC is an arbitrary constant. For this example, another integrable combination can be found by multiplying the rst equation by x, multiplying the second byy, and adding. This new dierential equation results in the additional relationx2+y2=D,w h e r eDis another arbitrary constant. Notes 1. Each linearly independent integrable combination yields a rst inte- gral of the original system. 2. See El’sgol’ts [1, pages 186{189]. References [1]El’sgol’ts, L. E. Dierential Equations and the Calculus of Variations . MIR Publishers, Moscow, USSR, 1970. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 336 II.A Exact Methods for ODEs 76. Integral Representation: Laplace’s Method Applicable to Linear ordinary dierential equations. Yields An integral representation of the solution. Idea Sometimes the solution of a linear ordinary dierential equation can be written as a contour integral. To nd such a representation, a lower order dierential equation may need to be solved. Procedure LetLz[
] be a linear dierential operator with respect to z, and suppose that the ordinary dierential equation we wish to solve has the form Lz[u(z)] = 0: (76.1) We look for a solution of equation (76.1) in the form u(z)=Z CK(z;)v()d; (76.2) for some function v()a n ds o m ec o n t o u r Cin the complex plane. The functionK(z;) is called the kernel . Some common kernels for Laplace’s method are Laplace kernel: K(z;)=ez: Euler kernel: K(z;)=(z−)N: We combine equations (76.2) and (76.1) to obtain Z CLz[K(z;)]v()d=0: (76.3) Now we must nd a linear dierential operator A[
], operating with respect to, such that Lz[K(z;)] =A[K(z;)]. AfterA[
] has been found, then equation (76.3) can be rewritten as Z CA[K(z;)]v()d=0: (76.4) Now we integrate equation (76.4) by parts. The resulting expression will be a dierential equation for v() with some boundary terms. The boundary terms determine the contour C, and the dierential equation determines v(). Knowing both v()a n dC, the solution to equation (76.1) is given by equation (76.2). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 76. Integral Representation: Laplace’s Method337 Special Case For the case where Lz[
] is a linear operator with polynomial coecients, the solution is easy to nd using the Laplace kernel. Let Lz[
] have the form Lz=NX r=0 MX s=0arszs! dr dzr; (76.5) where thefarsgare constants. Then dene the linear dierential operator M[
]b y M=NX r=0 MX s=0arsds ds! r: (76.6) Now dene M [
] to be the adjoint of M[
]. ThenLz[u(z)] = 0 will have a solution of the form u(z)=Z Cezv()d; ifv() satises M [v()] = 0; (76.7) andCis determined by h Pfez;v()gi C=0; (76.8) wherePfez;v()gis the bilinear concomitant of ezandv() (see page 226). Note the order of the original dierential operator in equation (76.5)wasNwhile the order of the dierential operators in equations (76.6) and (76.7) isM. Example Consider Airy’s equation u00−zu=0: (76.9) We assume that the solution of equation (76.9) has the form u(z)=Z Cezv()d; (76.10) for somev()a n ds o m ec o n t o u r C. Substituting equation (76.10) into equation (76.9), we nd Z C2v()ezd−zZ Cv()ezd=0: (76.11) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 338 II.A Exact Methods for ODEs The second term in equation (76.11) can be integrated by parts to obtain Z C2v()ezd− v()ez C+Z Cv0()ezd=0 or  v()ez C+Z Cez 2v()+v0() d=0: (76.12) We choose 2v()+v0() = 0 (76.13) and  v()ez C=0: (76.14) With these choices, equation (76.12) is satised. From equation (76.13) we can solve for v() v()=e x p −3 3 : (76.15) Using equation (76.15) in equation (76.14), we must choose the contour C so that  v()ez C= exp z−3 3 C=0; (76.16) for all real values of z. The only restriction that equation (76.16) places on Cis that the contour start and end in one of the shaded regions in gure 76.1. Finally, the solution to equation (76.9) can now be written u(z)=Z Ce(z−2=3)d: (76.17) Asymptotic methods can be applied to equation (76.17) to determine in- formation about u(z). For this example, we also could have used the general results in equa- tions (76.6){(76.8). Identifying equation (76.9) with the operator in equa-tion (76.5), we nd L z=d2 dz2−z; so that (from equation (76.6)) M=2−d d; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 76. Integral Representation: Laplace’s Method339/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /./. /./././././. /././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./././. /././././././././././././././././././. /././././././././././././././././././././.C /#18 /./././. /././. 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/././././././././. /././././././././. /./././././././././. /./. /./././././././././././././././././. /././././. /././././. /././././././././././././././././././././././. /./././././././. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /./. /./. /. /./. /. /././. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /./. /. /./. /./. /./. /./././././././././. /././././././././././././././././././././././././././././././././././././././././././././. /./././././././././././. /. /././. /. /././. /. /./././. /./././. /././. /././. /././././././. /./././. /././. /./././././././././././. /././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././././././. /././. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./././. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /././././././././. /././. /. /./././. /././././././././././././././././././././././././././././././././././././././. /./././. /./././. /././. /./././. /././. /. /./. /. /. /./. /. /./. /. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /. /./././. /./. /./.Figure 76.1: A solution to equation (76.9) is determined by any contour Cthat starts and ends in the shaded regions. All of the shaded regions extend to innity. One possible contour is shown. and also M =2+d d: So, we have to solve (from equation (76.7)) M [v()] =2v+v0=0: (76.18) Because this last equation is identical to equation (76.13), we nd the same v(). We compute the bilinear concomitant to be Pfez;v()g=v()d dez−ezd dv(); =/parenleftbig z+2
exp −z−3 3 ; and we nd the same contour Cas before (see (76.16)). Notes 1. Two linearly independent solutions of Airy’s equation are often taken to be Ai(x)=1 Z1 0cost3 3+xt dt; Bi(x)=1 Z1 0 exp −t3 3+xt cost3 3+xt dt: These solutions represent two dierent choices of the contour in equa- tion (76.17). 2. The Laplace equations (a0x+b0)y(n)+(a1x+b1)y(n−1)+


+(anx+bn)y=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 340 II.A Exact Methods for ODEs have solutions in the form of equation (76.2). Indeed, this was Laplace’s original example. See Davies [3, pages 342{367] or Valiron [6, pages 306-319] for details. 3. When the kernel of the transformation is some function of the product z, then this method is sometimes called the Mellin transformation. See Ince [4, pages 186{203 and 438{468] for details. 4. Sometimes a double integral is used to nd an integral representation. In this case, a solution of the form u(z)=RRK(z;s;t)w(s;t)dsdt is proposed. Details may be found in Ince [4, page 197]. As an example,the equation (x 2−1)d2y dx2+(a+b+1 )xdy dx+aby=0 has the two linearly independent solutions y(x)=Z1 0Z1 0exp xst−1 2(s2+t2) sa−1tb−1dsdt: 5. Equations of the form  xnF xd dx +G xd dx y=0; which are sometimes called Pfaan dierential equations, can also be solved by this method. See Bateman [2, Chapter 10, pages 260{264] or Ince [4, page 190] for details. 6. An application of this method to partial dierential equations may be found in Bateman [2, pages 268{275]. 7. The Mellin{Barnes integral representation for an ordinary dierential equation has the form u(z)=Z CK(z;)z"Qm j=1Γ(bj−)Qn j=1Γ( 1−aj+)Qq j=m+1Γ( 1−bj+)Qr j=n+1Γ(aj−)# d: In this representation, only the contour Cand the constants fai;bj;m;n;q;rg are to be determined (see Babister [1, pages 24{26] for details). References [1]Babister, A. W. Transcendental Functions Satisfying Nonhomogeneous Linear Dierential Equations . The MacMillan Company, New York, 1967. [2]Bateman, H. Dierential Equations . Longmans, Green and Co., 1926. [3]Davies, B. Integral Transforms and Their Applications , second ed. Springer{ Verlag, New York, 1985. [4]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 76. Integral Representation: Laplace’s Method341 [5]Olver, F. W. J. Asymptotics and Special Functions . Academic Press, New York, 1974. [6]Valiron, G. The Geometric Theory of Ordinary Dierential Equations and Algebraic Functions . Math Sci Press, Brookline, MA, 1950. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 342 II.A Exact Methods for ODEs 77. Integral Transforms: Finite Intervals Applicable to Linear dierential equations. Idea In order to solve a linear dierential equation, it is sometimes easier to transform the equation to some \space," solve the equation in that \space," and then transform the solution back. Procedure Given a linear dierential equation, multiply the equation by a kernel and integrate over a specied region (see table 77.1 on page 344 for a listingof common kernels and limits of integration). Use integration by parts to obtain an equation for the transform of the dependent variable. You will have used the \correct" transform (i.e., you have chosen the correct kernel and limits) if the boundary conditions given with the originalequation have been utilized. Now solve the equation for the transform of the dependent variable. From this, obtain the solution by multiplying by the inverse kernel and performing another integration. Table 77.1 also liststhe inverse kernel. Example 1 Suppose we have the boundary value problem for y=y(x) yxx+y=1; y(0) = 0;y(1) = 0:(77.1.a-c) Because the solution vanishes at both of the endpoints, we suspect that a nite sine transform might be a useful transform to try. Dene the nitesine transform of y(x)t ob ez(), so that z(): =Z 1 0y(x)s i nxdx: (77.2) (See \nite sine transform{2" in table 77.1). Now multiply equation (77.1.a) by sinxand integrate with respect to xfrom 0 to 1. This results in Z1 0yxx(x)s i nxdx +Z1 0y(x)s i nxdx =Z1 0sinxdx: (77.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 77. Integral Transforms: Finite Intervals343 If we integrate the rst term in equation (77.3) by parts, twice, we obtain Z1 0yxx(x)s i nxdx =yx(x)s i nxx=1 x=0−y(x)c o sxx=1 x=0 +2Z1 0y(x)s i nxdx:(77.4) Because we are interested only in =0;;2;::: (see table 77.1), the rst term on the right-hand side of equation (77.4) is identically zero. Because of the boundary conditions in equation (77.1.b-c), the second term on the right-hand side of equation (77.4) also vanishes. Because we have usedthe given boundary conditions to simplify certain terms appearing in the transformed equation, we suspect we have used an appropriate transform. If we had taken a nite cosine transform, instead of the one that we did, theboundary terms from the intergration by parts would not have vanished. Using equation (77.4), simplied, in equation (77.3) results in  2Z1 0y(x)s i nxdx +Z1 0y(x)s i nxdx =1−cos : Using the denition of z() (from equation (77.2)), this becomes 2z()+z()=1−cos  or z()=1−cos (1 +2): Now that we have found an explicit formula for the transformed function, we can use the summation formula (inverse transform) in table 77.1 to determine that y(x)=X =0;;2;:::2z()s i nx; =X =0;;2;:::21−cos (1 +2)sinx; =1X k=021−(−1)k (1 +2k2)ksinkx; =X k=1;3;5;:::4s i nkx (1 +2k2)k;(77.5) w h e r ew eh a v ed e  n e d k==. The exact solution of equation (77.1) is y(x)=1−cosx+cos 1−1 sin 1sinx. If this solution is expanded in a nite Fourier series, we obtain the repre- sentation in equation (77.5). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 344 II.A Exact Methods for ODEs Example 2 Suppose we have the following partial dierential equation for (r;t) (this corresponds to the temperature of a long circular cylinder whose surface is at a constant temperature) @2 @r2+1 r@ @r=1 @ @t; for 0r<1a n dt>0; (1;t)=0; fort>0; (r;0) = 0; for 0r<1:(77.6) Multiplying this equation by rJ0(pr) (wherepis positive and satises J0(p) = 0, see \nite Hankel transform{1" in table 77.1) and integrating with respect to rf r o m0t o1 ,w e n d p0J0 0(p)−p2=1 d dt; (77.7) where we have dened ( p;t)=R1 0(r;t)rJ0(pr)dr. This follows from the relation:Z1 0@2 @r2+1 r@ @r rJ0(pr)dr=p0J0 0(p)−p2(p;t). The initial condition in equation (77.6) is transformed to ( p;0) = 0. Using this, we can solve equation (77.7) to nd ( p;t)=0 pJ0 0(p) e−p2t−1 . Taking the inverse transform (and noting that J0 0(p)=−J1(p)), we arrive at the nal solution to equation (77.6) (r;t)=20X p e−p2t−1J0(pr) pJ1(p); where the summation is over all positive roots of J0(p)=0 . Table 77.1: Dierent transform pairs of the form v(k)=Z K(x;k)u(x)dx; u (x)=X kH(x;k)v(k): Finite cosine transform { 1 , (see Miles [5, page 86]) here landhare arbitrary, and the fkgsatisfyktankl=h. v(k)=Z1 0cos (xk)u(x)dx; u (x)=X k(2−k0)(2 k+h2)c o s(kx) h+l(2 k+h2)v(k): Finite cosine transform { 2 , (see Butkov [1, page 161]) this is the last transform with h=0 ,l=1 ,s ot h a t k=0;;2;::: . v(k)=Z1 0cos (xk)u(x)dx; u (x)=X k(2−k0)c o s(kx)v(k): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 77. Integral Transforms: Finite Intervals345 Finite sine transform { 1 , (see Miles [5, page 86]) here landhare arbitrary, and the fkgsatisfykcot(kl)=−h. v(k)=Z1 0sin (xk)u(x)dx; u (x)=X k2(2 k+h2)s i n(kx) h+l(2 k+h2)v(k): Finite sine transform { 2 , (see Butkov [1, page 161]) this is the last transform with h=0 ,l=1 ,s ot h a t k=0;;2;::: . v(k)=Z1 0sin (xk)u(x)dx; u (x)=X k2s i n(kx)v(k): Finite Hankel transform { 1 , (see Tranter [8, page 88]) here nis arbitrary and the fkgare positive and satisfy Jn(k)=0 . v(k)=Z1 0xJn(xk)u(x)dx; u (x)=X k2Jn(xk) J2 m+1(k)v(k): Finite Hankel transform { 2 , (see Miles [5, page 86]) here nandhare arbitrary and the fkgare positive and satisfy kJ0 n(ak)+hJn(ak)=0 . v(k)=Za 0xJn(xk)u(x)dx; u (x)=X k22 kJn(xk) f(h2+2 k)a2−m2gJ2n(ak)v(k): Finite Hankel transform { 3 , (see Miles [5, page 86]) here b>a , thefkgare positive and satisfy Yn(ak)Jn(bk)=Jn(ak)Yn(bk), and Zn(xk): =Yn(ak)Jn(xk)−Jn(ak)Yn(xk). v(k)=Zb axZn(xk)u(x)dx; u (x)=X k2 22 kJ2 n(bk)Zn(xk) J2n(ak)−J2n(bk)v(k): Legendre transform , (see Miles [5, page 86]) here k=0;1;2;:::. v(k)=Z1 −1Pk(x)u(x)dx; u (x)=X k2k+1 2Pk(x)v(k): Note 1. See Butkov [1, Chapter 5 and Section 8.5]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 346 II.A Exact Methods for ODEs References [1]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [2]Davies, B. Integral Transforms and Their Applications , second ed. Springer{ Verlag, New York, 1985. [3]Erdelyi, A. ,E d . Tables of Integral Transforms . McGraw{Hill Book Company, New York, 1954. In 3 volumes. [4]Magnus, W., Oberhettinger, F., and Soni, R. P. Formulas and Theorems for the Special Functions of Mathematical Physics . Springer{Verlag, New York, 1966. [5]Miles, J. W. Integral Transforms in Applied Mathematics . Cambridge University Press, New York, 1971. [6]Sneddon, I. N. The Use of Integral Transforms . McGraw{Hill Book Company, New York, 1972. [7]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. [8]Tranter, C. J. Integral Transforms in Mathematical Physics . Methuen & Co. Ltd., London, England, 1966. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 78. Integral Transforms: Innite Intervals347 78. Integral Transforms: Innite Intervals Applicable to Linear dierential equations. Idea In order to solve a linear dierential equation, it is sometimes easier to transform the equation to some \space," solve the equation in that \space," and then transform the solution back. Procedure Given a linear dierential equation, multiply the equation by a kernel and integrate over a specied region (see table 78.1 on page 349 for a listing of common kernels and limits of integration). Use integration by parts toobtain an equation for the transform of the dependent variable. You will have used the \correct" transform (i.e., you have chosen the correct kernel and limits) if the boundary conditions given with the original equation have been utilized. Now solve the equation for the transform of the dependent variable. From this, obtain the solution by multiplying by the inverse kernel and performing another integration. Table 78.1 also lists the inverse kernel. Warning After a solution is obtained by a transform method, it must be checked that the solution satises the requirements of the transform. For example, for a function to have a Laplace transform, it must be a L2function (i.e., square integrable). Example 1 Suppose we wish to nd the solution to the parabolic partial dierential equation ut=a2uxx (78.1) with the initial condition and boundary conditions given by u(x;0) = 0; u(0;t)=u0; fort>0; u(1;t)=0; fort>0;(78.2.a-c) whereaandu0are given constants. Because this problem is in a semi-innite domain (i.e., tvaries from 0 to1), we suspect that a Laplace transform in tmay be useful in nding CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 348 II.A Exact Methods for ODEs the solution. Let Lf
g denote the Laplace transform operator, and dene v(x;s): =Lfu(x;t)g=Z1 0e−stu(x;t)dt (78.3) to be the Laplace transform of u(x;t). We want to manipulate equation (78.1) into a form such that there are v(x;s) terms present. To obtain this form, multiply equation (78.1) by e−stand integrate with respect to tfrom 0t o1to obtainZ1 0e−stut(x;t)dt=a2Z1 0e−stuxx(x;t)dt: (78.4) The left-hand side of equation (78.4) can be integrated by parts while the x derivatives can be taken out of the integral in the right-hand side to obtain −u(x;t)e−st s1 0+Z1 0se−stu(x;t)dt=a2@2 @x2Z1 0e−stu(x;t)dt: If we assume that lim t!1e−stu(x;t) = 0 and use equation (78.2.a), then we obtain Z1 0se−stu(x;t)dt=a2@2 @x2Z1 0e−stu(x;t)dt: Finally, using the denition of v(x;s), from equation (78.3), we obtain sv(x;s)=a2@2 @x2v(x;s); (78.5) which is essentially an ordinary dierential equation in the independent variablex. The boundary conditions for this equation come from taking the Laplace transform of equation (78.2.b{c). We calculate v(0;s): =Lfu(0;t)g=Lfu0g=Z1 0e−stu0dt=u0 s; v(1;s): =Lfu(1;t)g=Lf0g=0: (78.6) Solving equation (78.5) with the boundary conditions in equation (78.6) results in v(x;s)=u0 se−xps=a: (78.7) A table of inverse Laplace transforms, when applied to equation (78.7), results in u(x;t)=L−1fv(x;s)g =1 2iZ+i1 −i1estv(x;s)ds =u0 1−erfx 2tpa ;(78.8) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 78. Integral Transforms: Innite Intervals349 which is the nal solution. Now that we have the solution, we must either verify that it solves the dierential equation and initial condition and boundary conditions that we started with (equation (78.1)), or we must verify that the stepswe performed in obtaining the solution are valid. In this case, it means verifying that lim t!1e−stu(x;t)=0a n dt h a t u(x;t) is square integrable. Because each of these are true, the solution found in equation (78.8) is correct. Example 2 Suppose we have the ordinary dierential equation d4y dx4=y+p(x) (78.9) fory(x), for−1<x<1, with the boundary conditions: y(1)=0 , y0(1) = 0. Because the equation is on a (doubly) innite domain, we try to use a Fourier transform in xto nd the solution. LetFf
g denote the Fourier transform operator, and dene z(!)=Ffy(x)g:=Z1 −1y(x)ei!xdx to be the Fourier transform of y(x). If we apply the operator Ff
g to equation (78.9) (by multiplying by ei!xand integrating with respect to x), we nd Z1 −1ei!xd4y dx4dx=Z1 −1ei!xydx+Z1 −1ei!xp(x)dx: Integrating by parts and using the given boundary conditions, this can be simplied to (i!)4z(!)=z(!)+Z1 −1ei!xp(x)dx: This last expression can be solved to yield z(!)=1 !4−1Z1 −1ei!xp(x)dx: (78.10) For any given p(x), the integral in equation (78.10) can be evaluated, and then an inverse Fourier transform can be taken to determine y(x)= F−1fz(!)g. Table 78.1: Dierent integral transform pairs of the form v()=Z K(x;)u(x)dx; u (x)=Zb aH(x;)v()d: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 350 II.A Exact Methods for ODEs Fourier transform , (see Butkov [3, Chapter 7]) v()=1p 2Z1 −1eixu(x)dx; u (x)=1p 2Z1 −1e−ixv()d: Fourier cosine transform , (see Butkov [3, page 274]) v()=r 2 Z1 0cos(x)u(x)dx; u (x)=r 2 Z1 0cos(x)v()d: Fourier sine transform , (see Butkov [3, page 274]) v()=r 2 Z1 0sin(x)u(x)dx; u (x)=r 2 Z1 0sin(x)v()d: Hankel transform , (see Sneddon [22, Chapter 5]) v()=Z1 0xJ(x)u(x)dx; u (x)=Z1 0J(x)v()d: Hilbert transform , (see Sneddon [22, pages 233{238]) v()=Z1 −11 (x−)u(x)dx; u (x)=Z1 −11 (−x)v()d: K{transform , (see Erd elyi [7] ) v()=Z1 0K(x)p xu(x)dx; u (x)=1 iZ+i1 −i1I(x)p xv()d: Kontorovich{Lebedev transform , (see Sneddon [22, Chapter 6]) v()=Z1 0Ki(x) xu(x)dx; u (x)=2 2Z1 0sinh()Ki(x)v()d: Kontorovich{Lebedev transform (alternative form), (see Jones [12]) v()=Z1 0H(2) (x)u(x)dx; u (x)=−1 2xZi1 −i1J(x)v()d: Laplace transform , (see Sneddon [22, Chapter 3]) v()=Z1 0e−xu(x)dx; u (x)=1 2iZ+i1 −i1exv()d: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 78. Integral Transforms: Innite Intervals351 Mehler{Fock transform of order m, (see Sneddon [22, Chapter 7]) v()=Z1 0sinh(x)Pm i−1=2(coshx)u(x)dx; u(x)=Z1 0tanh()Pm i−1=2(coshx)v()d: Mellin transform , (see Sneddon [22, Chapter 4]) v()=Z1 0x−1u(x)dx; u (x)=1 2iZ+i1 −i1x−v()d: Weber formula , (see Titchmarsh [24, page 75]) v()=Z1 apx[J(x)Y(a)−Y(x)J(a)]u(x)dx; u(x)=pxZ1 0J(x)Y(a)−Y(x)J(a) J2(a)+Y2(a)v()d: Weierstrass transform , (see Hirschman and Widder [10, Chapter 8]) v()=1p 4Z1 −1e(−x)2=4u(x)dx; u (x)=1p 4lim T!1ZT −Te(x−i)2=4v(i)d: Unnamed transform , (see Naylor [20]) v()=Z1 −1K0(j−xj)u(x)dx; u (x)=−1 2d2 dx2−1Z1 −1K0(j−xj)v()d: Unnamed transform , (see Titchmarsh [24, page 83]) v()=Z1 −1h Jip (ex)+J−ip (ex)i u(x)dx; u(x)=Z1 0Jip (ex)+J−ip (ex) 4 sinh/parenleftbig p
v()d: Notes 1. Note that many of the transforms in table 78.1 do not have a standard form. In the Fourier transform, for example, the twop 2terms might not be symmetrically placed as we have shown them. Also, a smallvariation of the K-transform is known as the Meijer transform (see Ditkin and Prudnikov [6, page 75]). 2. There are many tables of transforms available (see Bateman [7] or Magnus et al. [14]). It is generally easier to look up a transform than to compute it. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 352 II.A Exact Methods for ODEs 3. Transform techniques may also be used with systems of linear equa- tions. 4. If a function f(x;y) has radial symmetry, then a Fourier transform in bothxandyis equivalent to a Hankel transform of f(r)=f(x;y), wherer2=x2+y2. See Sneddon [22, pages 79{83]. 5. Integral transforms can be constructed by integrating the Green’s function for a Sturm{Liouville eigenvalue problem. This involves explicitly nding an integral representation of the delta function. For example, the relation ()=1 2Z1 −1eid (78.11) can be used to derive the Fourier transform. To see this, change to x−in equation (78.11), multiply by f() and integrate with respect toto obtain f(x)=1p 2Z1 −1eix1p 2Z1 −1f()eid d For more details, see Davies [5, pages 267{287], or Stakgold [23, Chapter 7, pages 411{466]. 6. Many of the transforms in table 78.1 have a convolution theorem , which describes how the transform of the product of two functions, is related to the transforms of the individual functions. For exam- ple, ifg(t) (respectively h(t),k(t)) has the Laplace transform G(s) (respectively H(s),K(s)), andG(s)=H(s)K(s), then g(t)=Zt 0h(t−)k()d: This is called a convolution product and is often denoted by g(t)= h(t)k(t). See Miles [16, Table 2.3, page 85]. 7. Most of the transforms in table 78.1 have simple formulae relating the transform of the derivative of a function to the transform of the function. For example, if G(s) is the Laplace transform of g(t), then Lfg(n)(t)g=snG(s)−g(n−1)(0) +sg(n−2)(0) +


+(−1)nsn−1g(0): 8. Two transform pairs that are continuous in one variable and discrete in the other variable, on an innite interval, are the Hermite trans- form u(x)=1X n=0vnHn(x)e−x2=2;vn=1 (2n)!pZ1 −1u(x)Hn(x)e−x2=2dx; whereHn(x)i st h enth Hermite polynomialand the Laguerre trans- form u(x)=1X n=0vnL n(x)n! Γ(n++1 );vn=Z1 0u(x)L n(x)xe−xdx; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 78. Integral Transforms: Innite Intervals353 whereL n(x) is the Laguerre polynomial of degree n,a n d0. See Haimo [9] for details. 9. Integral transforms are generally created for solving a specic dif- ferential equation with a specic class of boundary conditions. Forexample, the Mathieu integral transform (see Inayat-Hussain [11]) has been constructed for the two-dimensional Helmholtz equation in elliptic-cylinder coordinates. 10. The papers by Namias ([17] and [18]) on fractional order Fourier and Hankel transforms contain several examples of how the transformsmay be used to solve dierential equations. 11. Note that d r dxr=d dxx1 xd dxx2


1 xr−2d dxxr−11 xr−1d dx ="r−1Y i=11 xr−i−1d dxxr−i# d xr−1dx: (78.12) Then observe that the −transform, dened by g(x;)=Z[f(x);]=Z1 0


Z1 0f xY t1=r i eP tiY ti idti; f(x)=1 (2i)r−1Z(0+) −1


Z(0+) −1g xY t−1=r i; eP tiY t−i−1 idti; where=(1;:::;r−1)a n diruns from 1 to r−1 in each sum and product, can be used with (78.12) to obtain Zdru dxr;r =r xr−1dZ[u;r] dx−r−1X i=1Ci xr−i; wherer=(−1=r;−2=r;:::;−(r−1)=r). This transform can be applied, for example, to the equation y(r)+axy0+by=f(x)o rt o dr dxr+b1 xdr−1 dxr−1+


+br−1 xr−1d dx y+axy0+by=f(x): See Klyuchantsev [13] for details. 12. Classically, the Fourier transform of a function exists only if the func- tion being transformed decays quickly enough at 1. The Fourier transform can be extended, though, to handle generalized functions.For example, the Fourier transform of the nth derivative of the delta function is given by F/parenleftbig  (n)(t)
=(i!)n. Another way to approach the Fourier transform of functions that do not decay quickly enoughat either1or−1 is to use the one-sided Fourier transforms .S e e Chester [4] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 354 II.A Exact Methods for ODEs 13. Many of the transforms listed generalize naturally to ndimensions. For example, in ndimensions we have Fourier transform: v()=( 2)−n=2R Rnei /#18
xu(x)dx, u(x)=( 2)−n=2R Rne−i /#18
xv()d. Hilbert transform (see Bitsadze [2]): @f @xi=Γ(n=2) n=2Z Rn−1yi−xi jy−xjn(y)dy;i=1;2;:::;n−1; (y)=−Γ(n=2) n=2Z Rn−1(y−x)
rf jy−xjndy; 14. The name Bessel transform is given to an integral transform that in- volves a Bessel function. This class includes Hankel, K, Kontorovich{ Lebedev, and many other transforms. 15. Note that, for the Hilbert transform, the integrals in table 78.1 are to be taken in the principal value sense. 16. See also Abramowitz and Stegun [1, pages 1019{1030] and Butkov [3, Chapter 5, pages 179{220 and Section 8.5, pages 299{304]. References [1]Abramowitz, M., and Stegun, I. A. Handbook of Mathematical Functions . National Bureau of Standards, Washington, D.C., 1964. [2]Bitsadze, A. V. The multidimensional Hilbert transform. Soviet Math. Dokl. 35 , 2 (1987), 390{392. [3]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [4]Chester, C. R. Techniques in Partial Dierential Equations . McGraw{Hill Book Company, New York, 1970. [5]Davies, B. Integral Transforms and Their Applications ,s e c o n de d . Springer{Verlag, New York, 1985. [6]Ditkin, V. A., and Prudnikov, A. P. Integral Transforms and Operational Calculus . Pergamon Press, New York, 1965. Translated by D. E. Brown. English translation edited by I. N. Sneddon. [7]Erdelyi, A. ,E d . Tables of Integral Transforms . McGraw{Hill Book Company, New York, 1954. In 3 volumes. [8]Glaeske, H.-J. Operational properties of a generalized Hermite transfor- mation. Aequationes Mathematicae 32 (1987), 155{170. [9]Haimo, D. T. The dual Weierstrass{Laguerre transform. Trans. AMS 290 , 2 (August 1985), 597{613. [10]Hirschman, I. I., and Widder, D. V. The Convolution Transform . Princeton University Press, Princeton, NJ, 1955. [11]Inayat-Hussain, A. A. Mathieu integral transforms. J. Math. Physics 32 , 3 (March 1991), 669{675. [12]Jones, D. S. The Kontorovich{Lebedev transform. J. Inst. Maths. Applics 26(1980), 133{141. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 78. Integral Transforms: Innite Intervals355 [13]Klyuchantsev, M. I. An integral-transformation method of solving some types of dierential equations. Translated from Dierentsial’nye Uravneniya 23, 10 (October 1987), 1668{1679. [14]Magnus, W., Oberhettinger, F., and Soni, R. P. Formulas and Theorems for the Special Functions of Mathematical Physics . Springer{ Verlag, New York, 1966. [15]Marichev, O. I. Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables . John Wiley & Sons, New York, 1983. translated by L. W. Longdon, Halstead Press. [16]Miles, J. W. Integral Transforms in Applied Mathematics . Cambridge University Press, New York, 1971. [17]Namias, V. The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Maths. Applics 25 (1980), 241{265. [18]Namias, V. Fractionalization of Hankel transforms. J. Inst. Maths. Applics 26(1980), 187{197. [19]Nasim, C. The Mehler{Fock transform of general order and arbitrary index and its inversion. Int. J. Math. &Math. Sci. 7 , 1 (1984), 171{180. [20]Naylor, D. On an integral transform. Int. J. Math. &Math. Sci. 9 ,2 (1986), 283{292. [21]Oberhettinger, F., and Higgins, T. P. Tables of Lebedev, Mehler, and generalized Mehler transforms. Tech. rep., Boeing Scientic Research Laboratories, October 1961. Mathematical Note No. 246. [22]Sneddon, I. N. The Use of Integral Transforms . McGraw{Hill Book Company, New York, 1972. [23]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. [24]Titchmarsh, E. C. Eigenfunction Expansions Associated with Second{ Order Dierential Equations . Clarendon Press, Oxford, England, 1946. [25]T r a n t e r ,C .J . Integral Transforms in Mathematical Physics . Methuen & Co. Ltd., London, England, 1966. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 356 II.A Exact Methods for ODEs 79. Integrating Factors Applicable to Linear rst order ordinary dierential equations. Yields An exact equation that can then be integrated. Idea When a given equation is not exact, it may be possible to multiply the equation by a certain term so that it does become exact. The term that is used is called an integrating factor . Procedure Let us suppose that the nonlinear ordinary dierential equation M(x;y)dx+N(x;y)dy= 0 (79.1) is not exact (see page 284). It may be, however, that if equation (79.1) is multiplied by an integrating factor u(x;y), the resulting equation uMdx +uNdy =0 is exact. For this to be the case, we require @(uM)=@y=@(uN)=@x,o r u@M @y−@N @x =N@u @x−M@u @y: (79.2) In general, solving the partial dierential equation (79.2) for u(x;y)i s more dicult than solving the ordinary dierential equation (79.1). But, in certain cases, it may be easier. For example, 1. If1 N @M @y−@N @x =f(x), a function of xalone, then u(x;y)=u(x)= exp/parenleftbigRxf(z)dz
is an integrating factor for equation (79.1). 2. If1 M @M @y−@N @x =g(y), a function of yalone, then u(x;y)=u(y)= exp/parenleftbig −Ryg(z)dz
is an integrating factor for equation (79.1). Example Suppose we have the general linear rst order ordinary dierential equation y0+P(x)y=Q(x): (79.3) We recognize that the homogeneous equation corresponding to equation (79.3) isy0+P(x)y= 0. Written as dy+(P(x)y)dx= 0, we see that the rst case applies with f(x): =P(x) (because M=yP(x)a n dN=1) . CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 79. Integrating Factors357 Hence we have the integrating factor u(x)=e x p/parenleftbigRxP(z)dz
, and equation (79.3) can be written as (y0+P(x)y)e x pZx P(z)dz =Q(x)e x pZx P(z)dz ; or d dx yexpZx P(z)dz =Q(x)e x pZx P(z)dz ; and therefore (by integrating), we nd the solution to be y(x)=e x p −Zx P(z)dzZx Q(w)e x pZw P(z)dz dw: Special Case For a concrete illustration, the equation y0+1 xy=x2(79.4) hasfP(x)=1=x;Q (x)=x2g,s ot h a t u(x)=e x pZx1 zdz =e x p( l o gx) =x is an integrating factor. When equation (79.4) is multiplied by u(x)=x, we obtain xy0+y=x3; d(xy) dx=x3; xy=x4 4+C; ory=x3 4+C x,w h e r eCis an arbitrary constant. Notes 1. If equation (79.1) admits a one parameter Lie group with generators f;g(see page 366), then an integrating factor is given by u(x;y)= 1=(N−M). For example, the dierential equation y(y2−x)dx+ x2dy= 0 is invariant under the transformation fy0=e=2y,x0= exg. Therefore, the innitesimal operator of the group is described byf=1 2y,=xg. This leads to the integrating factor u= 2=3xy(x−2y2), which leads to the solution y=x=p 2x+C. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 358 II.A Exact Methods for ODEs 2. IfMx+Ny6= 0, and equation (79.1) is homogeneous (see page 327), then an integrating factor is given by u(x;y)=1=(Mx+Ny). For example, the dierential equation ( xy−2y2)dx−(x2−3xy)dy=0 is homogeneous and has the integrating factor u=1=xy2. This leads to the solutionx y−log(x2y3)=C. 3. IfM=M1(x)y−M2(x)ynandN= 1, then an integrating factor is given byu(x;y)=y−nexp/parenleftbig (1−n)=R M1dx
. 4. The dierential equation M1(x)M2(y)dx+N1(x)N2(y)dy=0h a s the integrating factor u=(M2N1)−1. 5. The dierential equation yf(xy)dx+xg(xy)dy=0 ,w h e nf6=g,h a s the integrating factor u=1=[xy(f−g)]. For example, the equation y(1−xy)dx−x(1 +xy)dy=0h a sff(z)=1−z,g(z)=−1−zgso that an integrating factor is given by u=1=2xy. This leads to the implicit solution yexy=Cx. 6. Given equation (79.1), if z=N−iMis an analytic function of xand y(i.e., the Cauchy{Riemann equations fNx=−My,Ny=Mxgare satised), then an integrating factor is given by 1 =(N2+M2). For example, the homogeneous equation /parenleftbig y2+2xy−x2
dy−/parenleftbig y2−2xy−x2
dx=0 has the integrating factor u=1=h 2/parenleftbig x2+y2
2i , which leads to the solutiony+x=C(x2+y2). 7. Sometimes an integrating factor of the form xkyncan be found (for specic values of kandn). This form of the integrating factor will always be adequate for dierential equations of the form xayb(pydx + qxdy )+xdye(rydx +sxdy ) = 0, wherefa,b,d,e,p,q,r,sgare constants. 8. The technique presented here also applies to linear ordinary dieren- tial equations of higher order. For example, the second order ordinary dierential equation pxd2y dx2+2xdy dx+3y=0 can be made exact (see page 287) by use of the integrating factor u(x)=px. Multiplying equation (8) bypxresults in xd2y dx2+2x3=2dy dx+3ypx=d dx xdy dx+( 2x3=2−1)y : Murphy [2, page 165] has a discussion of how to make second order ordinary dierential equations exact. 9. When the quasilinear partial dierential equation in two independent variables,M(x;y;u )ux=N(x;y;u )uy,h a sMx=Ny, then the solution is given implicitly by ( x;y;u ) = 0, where M=yand CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 79. Integrating Factors359 N=x. If, alternately, Mx6=Ny, then it may be possible to nd an integrating factor v(x;y) such that ( vM)x=(vN)y.F o r example, if ( Ny−Mx)=Mis a function of xalone, then v(x)= expZNy−Mx Mdx will be an integrating factor. 10. For example, the equation ux=yuyhas the integrating factor v(x)= ex. The solution can then be found to be u(x;y)=−Cy3e3x,w h e r e Cis an arbitrary constant. 11. See Boyce and DiPrima [1, pages 84{87], Murray [3, pages 22{27], Rainville and Bedient [5, pages 35{37 and 59{66], and Simmons [6, pages 42{46]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Murphy, G. M. Ordinary Dierential Equations and Their Solution .D .V a n Nostrand Company, Inc., New York, 1960. [3]Murray, J. D. Asymptotic Analysis . Springer{Verlag, New York, 1984. [4]Prelle, M. J., and Singer, M. F. Elementary rst integrals of dierential equations. Trans. Amer. Math. Soc. 279 , 1 (September 1983), 215{229. [5]Rainville, E. D., and Bedient, P. E. Elementary Dierential Equations . The MacMillan Company, New York, 1964. [6]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 360 II.A Exact Methods for ODEs 80. Interchanging Dependent and Independent Variables Applicable to Ordinary dierential equations. Yields A reformulation of the original equation. Idea Sometimes it is easier to solve an ordinary dierential equation by inter- changing the role of the dependent variable with the role of the independent variable. If this technique works, then the solution is given implicitly byx=x(y) instead of the usual y=y(x). Procedure Given the equation dy dx=f(x;y) to solve, it might be easier to solve the equivalent equation dx dy=1 f(x;y): This method can also be used for ordinary dierential equations with an order greater than 1. For these cases, table 80.1 can be used to de-termine how the derivatives fy x;yxx;:::gtransform into the derivatives fxy;xyy;:::g. Example 1 Suppose the solution is desired to the ordinary dierential equation dy dx=x x2y2+y5: Interchanging the dependent and independent variables in this equation produces dx dy=x2y2+y5 x=y2x+y5 x: (80.1) Equation (80.1) is now a Bernoulli equation with n=−1a n dc a nb es o l v e d exactly (see page 235). The solution is x(y)= Ae2y3=3−y3 2−3 41=2 ; (80.2) whereAis an arbitrary constant. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 80. Interchanging Dependent and Independent Variables 361 yx=x−1 y; yxx=−x−3 yxyy; yxxx=3x−5 yx2yy−x−4 yxyyy; yxxxx =−15x−7 yx3yy+1 0x−6 yxyyxyyy−x−5 yxyyyy yxxxxx = 105x−9 yx4yy−105x−8 yx2yyxyyy+1 0x−7 yx2yyy +1 5x−7 yxyyxyyyy−x−6 yxyyyyy Table 80.1: How higher order derivatives transform when the dependent and independent variables are switched. Example 2 The following formidable nonlinear ordinary dierential equation y00+xy(y0)3= 0 (80.3) becomes, after interchanging the dependent and independent variables, Airy’s equation d2x dy2=xy: Hence, the solution to equation (80.3) is given explicitely by x(y)=C1Ai(y)+C2Bi(y); whereC1andC2are arbitrary constants. Example 3 The nonlinear equation y00=(x−y)y03becomes, after interchanging variables,xyy=x−y. This equation has the solution x=y+Aey+Be−y. Notes 1. When this method is applied to partial dierential equations (and not ordinary dierential equations), then the method is called the hodograph transformation (see page 456). 2. See Bender and Orszag [1, Section 1.6] and Goldstein and Braun [2, page 107]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Goldstein, M. E., and Braun, W. H. Advanced Methods for the Solution of Dierential Equations . U.S. Government Printing Oce, Washington, D.C., 1973. NASA SP-316. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 362 II.A Exact Methods for ODEs [3]McAllister, B. L., and Thorne, C. J. Reverse Dierential Equations and Others That Can Be Solved Exactly . Tech. Rep. 11, University of Utah, Salt Lake City, 1952. Project Number NR-056-239 (Available through NTIS). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 81. Lagrange’s Equation 363 81. Lagrange’s Equation Applicable to Equations of the form y=xF dy dx +G dy dx . Yields An exact solution, sometimes given parametrically. Idea Equations of this form can be solved by quadratures. Procedure G i v e na ne q u a t i o no ft h ef o r m y=xFdy dx +Gdy dx ; (81.1) usepto represent dy=dx so that equation (81.1) can be written as y=xF(p)+G(p): (81.2) Now dierentiate equation (81.2) with respect to xto obtain dy dxp=F(p)+dp dxh xF0(p)+G0(p)i : (81.3) Equation (81.3) can be rewritten as dx dp=xF0(p) p−F(p) +G0(p) p−F(p) ; (81.4) which is now a linear dierential equation in xandp.I tc a nb es o l v e d b y the method of integrating factors (see page 356) to determine x=(p;C); (81.5) whereCis an arbitrary constant. Now there are two possibilities: Eliminatepbetween equations (81.2) and (81.5) to obtain the implict solution ( y;x;C )=0 . Use equation (81.5) in equation (81.2) to obtain the parametric solu- tion x=(P;C); y=(P;C)F(P)+G(P); wherePis a free parameter. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 364 II.A Exact Methods for ODEs Example 1 Suppose we have the equation y=2xdy dx−ady dx3 ; (81.6) whereais a constant. Comparing equation (81.6) to equation (81.1), we identifyF(p)=2p,G(p)=ap3. Hence, (81.5) becomes dx dp=−2x p+3ap: This last equation has an integrating factor of p2and so x=3a 4p2+C p2; (81.7) whereCis an arbitrary constant. Using equation (81.7) in equation (81.6), we can remove the xdependence to obtain y=a 2p3+2C p: Hence, a parametric solution of equation (81.6) is given by x=3a 4P2+C P2; y=a 2P3+2C P;(81.8) wherePcan have any value. By use of resultants (see page 50), the parameterPcan be removed from equation (81.8) to determine the implicit solution (27ay2−16x3)y2+1 6a2x(9ay2−4x3)C−128a3x2C2−64a4C3=0: IfCis taken to be zero, for example, then the explicit solutions y= 4 3p 3ax3=2andy= 0 are obtained. Example 2 If we have the equation y=2xdy dx−dy dx2 ; (81.9) then we make the identication fF(p)=2p;G(p)=−p2gso that equation (81.4) becomes dx dp=x −2 p +2; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 81. Lagrange’s Equation 365 or (using the integrating factor p2) x=2 3p2+C p2; (81.10) whereCis an arbitrary constant. Using equation (81.10) in equation (81.9) results in y=C p−p2 3: Hence, a parametric solution of equation (81.9) is given by x=2 3P2+C P2; y=C P+C P2;(81.11) wherePcan have any value. By use of resultants the parameter Pcan be removed from equation (81.11) to determine the implicit solution y2(4y−3x2)+6x(2x2−3y)C+9C2=0: Notes 1. Equation (81.1) is known as d’Alembert’s equation and also as an equation linear in xandy. 2. IfF1, then equation (81.1) is the same as Clairaut’s equation (see page 237). 3. The technique presented in this section is only an application of the more general technique of \solving for y" (see page 411). 4. See Ince [1, pages 38{39], Murphy [2, pages 65{66], and Valiron [3, pages 217{218]. References [1]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [2]Murphy, G. M. Ordinary Dierential Equations and Their Solution .D .V a n Nostrand Company, Inc., New York, 1960. [3]Valiron, G. The Geometric Theory of Ordinary Dierential Equations and Algebraic Functions . Math Sci Press, Brookline, MA, 1950. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 366 II.A Exact Methods for ODEs 82. Lie Groups: ODEs Applicable to Linear and nonlinear ordinary dierential equations. Yields Invariants and symmetries of a dierential equation. Often these can be used to solve a dierential equation. Idea By determining the transformation group under which a given dieren- tial equation is invariant, we can obtain information about the invariantsand symmetries of a dierential equation. Sometimes these can be used to solve a given dierential equation. Procedure A one parameter Lie group of transformations is a family of coordinate transformations of the form x=f(x;y;); y=g(x;y;);(82.1) such that= 0 gives the identity transformation. It is also required (for the transformations to form a group) that f(x;y;+)=f(x;y;), and f−1(x;y;)=f(x;y;−), with analogous formulae for g(x;y;). Equation (82.1) is called the global transformation group . Expanding (82.1) for small values of yields x=x+(x;y)+O(2); y=y+(x;y)+O(2); where (x;y)=@f @ =0; (x;y)=@g @ =0: (82.2) The quantities andare the innitesimal transformations of the group. Lie’s rst fundamental theorem states that knowledge of the innitesimalsf(x;y);(x;y)gis equivalent to knowing the functions ff;ggin (82.1). Annth order dierential equation G x;y;y 0;:::;y(n) = 0 (82.3) is said to be invariant under the group dened by equation (82.1) if the dierential equation G x;y;y0 ;:::;y(n)  =0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 82. Lie Groups: ODEs 367 is equivalent to equation (82.3) under the change of variables in (82.1). The innitesimal generator (also called the generator orinnitesimal operator ) associated with equation (82.1) is X=(x;y)@ @x+(x;y)@ @y. The prolongations ofXare dened by X(n)=@ @x+@ @y+nX l=1l@ @y(l); (82.4) where0=andl=D(l−1)−y(l)D(), forl=1;2;:::;n ,a n dt h e total derivative operator Dis dened by D:=@ @x+y0@ @y+y00@ @y0+:::. The dierential equation of nth order in equation (82.3), G= 0, will be invariant with respect to the one parameter group dened by (82.1) if X(n)G=0; (82.5) on the manifold G= 0 in the space of the variables fx;y;y0;:::;y(n)g. Note that equation (82.5) is quasilinear and the method of characteristicsmay be used to solve it. If the dierential equation G= 0 is invariant with respect to the group, then the subsidiary equations of equation (82.5) can be written as (see page432) dx =dy =d(y0) 1=


=d(y(n)) n: We can sometimes integrate two of these equations to obtain two integrals: u=u(x;y;y0;:::)a n dv=v(x;y;y0;:::). If the original equation, G=0 , is written in terms of these new variables, then the resulting dierential equation will be only of order n−1. Hence, we will have reduced the order of the given dierential equation. Special Case The condition for the equation F(x;y;y0;y00) = 0 to be invariant under the action of the group dened by equation (82.1) is that X(2)FjF=0=0 . WhenF=y00−f(x;y;y0), this determining equation becomes xx+( 2xy−xx)y0+(yy−2xy)y02−y03yy +(y−2x−3y0y)f−h x+(y−x)y0−y02i fy0 −fx−fy=0:(82.6) We emphasize that equation (82.6) is an identity in x,y,a n dy0. Because andcannot depend on y0, equation (82.6) separates into many simul- taneous equations for each type of y0term. Example 1 Given the class of second order ordinary dierential equations G(x;y;y0;y00)xy00−Fy x;y0 =0; (82.7) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 368 II.A Exact Methods for ODEs we ask if this dierential equation is invariant under the magnication group x=xe; y=ye:(82.8) If it is, then we should be able to reduce equation (82.7) to a sequence of rst order ordinary dierential equations. Using (82.8) in the denitions in equations (82.2) and (82.4), we can sequentially calculate (x;y)=x;  (x;y)=y; 0==y; 1=D(0)−y0D()=D(y)−y0D(x)=0 2=D(1)−y00D()=D(0)−y00D(x)=−y00; X(2)=x@ @x+y@ @y−y00@ @y00: ApplyingX(2)toG, we nd X(2)G= x@ @x+y@ @y−y00@ @y00h xy00−Fy x;y0i =x y00+y x2F1 +y −1 xF1 −y00(x) =0; whereF1denotes the derivative of Fwith respect to its rst argument. We conclude, then, that G= 0 is invariant under the magnication group. Now we form the subsidiary equations: dx x=dy y=dy0 0=dy00 −y00: From the rst equality,dx x=dy y, we nd that y=xis a constant; we write this asy=x=u. From the second equality,dy y=dy0 0, we nd that y0is a constant; we write this as y0=v. Now we will write the equation G= 0 in terms of the \constants" that parameterize the solution space: fu;vg. To change variables, we will need y00=dy0 dx=dv dx=dv dudu dx=dv duy0 x−y x2 =dv duv−u x: Hence, G=xy00−Fy x;y0 =(v−u)dv du−F(u;v)=0: (82.9) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 82. Lie Groups: ODEs 369 Finally, then, we have transformed the second order dierential equation G= 0 into a rst order dierential equation in terms of uandv.A f t e r this equation is solved for v=v(u), we then have a rst order equation for y(x) (usingu=y=xandv=y0). We now illustrate the above result with two special cases: 1. If we choose the special case F(u;v)=v−u(for which equation (82.7) becomes the linear equation x2y00−xy0+y= 0, with solutions y=xandy=xlogx), equation (82.9) becomes ( v−u)/parenleftbigdv du−1
=0 . The most general solution to this equation is v=u+C,w h e r eCis an arbitrary constant. Changing to our original variables, this becomes dy dx=y x+C. This equation has the solution y=Cxlogx+Dx,w h e r e Dis another arbitrary constant. 2. If we choose the special case F(u;v)=u2−v2(for which equation (82.7) becomes the nonlinear equation x3y00+x2(y0)2−y2=0 ) , equation (82.9) becomesdv du=−v−u. This rst order equation can be integrated to yield v=(u2−2u+2 )+Ce−u,w h e r eCis an arbitrary constant. In this case, we cannot integrate again to obtain y=y(x) in closed form. Example 2 For a given dierential equation, the dierent innitesimal generators will generate an r-dimensional Lie group ( Lr) The following four statements are equivalent (see Ibragimov [10, page 39]): 1. The second order ordinary dierential equation y00=f(x;y;y0) (82.10) can be linearized by a change of variables. 2. Equation (82.10) has the form y00=F3(x;y)y03+F2(x;y)y02+F1(x;y)y0+F0(x;y)=0 with coecientsfFi(x;y)gsatisfying the integrability conditions of the following over-determined system: @z @x=z2−F0w−F1z+@F0 @y+F0F2; @z @y=−zw+F0F3−1 3@F2 @x+2 3@F1 @y; @w @x=zw−F0F3−1 3@F1 @y+2 3@F2 @x; @w @y=−w2+F2w+F3z+@F3 @x−F1F3:(82.11) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 370 II.A Exact Methods for ODEs 3. Equation (82.10) admits the Lie algebra L8. 4. Equation (82.10) admits the Lie algebra L2with a basisfX1;X2g, such thatX1_X2= 0 (see the notes for the denition of the pseudoscalar product X1_X2=). Examples: Consider the equation y00=f(y0): (82.12) From the above, this will be linearizable if and only if f(y0)i sa polynomial of the third degree in y0. That is, if equation (82.12) has the form y00+A3y03+A2y02+A1y0+A0=0; where thefAigare constants, then it may be linearized. Consider the equation y00=f(y0) x: (82.13) From the above, this will be linearizable only if f(y0) is a polynomial of the third degree in y0. That is, equation (82.13) must have the form y00+1 x A3y03+A2y02+A1y0+A0 =0; where thefAigare constants. In this case, the integrability condi- tions in (82.11) become A2(2−A1)+9A0A3=0 3A3(1 +A1)−A2 2=0:(82.14) If we dene a=−A3andb=−A2, then we can solve equation (82.14) forA1andA2. We conclude: Equation (82.13) may be linearized if and only if it has the form: y00=1 x ay03+by02+ 1+b2 3a y0+b 3a+b3 27a2 : Consider the equation y00=F(x;y): (82.15) This matches the above form with F1=F2=F3=0a n dF0=F. In this case, the integrability conditions in equation (82.11) become zx=z2+Fw−Fy; zy=−zw; wx=zw; wy=−w2:(82.16) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 82. Lie Groups: ODEs 371 Translation in x x=x+ X=@xy=y Translation in y x=x X=@yy=y+ Scaling x=ex X=x@x+y@yy=ey Rotation in the ( x;y) planex=xcos−ysin X=−y@x+x@yy=xsin−ycos Table 82.1: Some common Lie group generators Using the rst two equations in (82.16) in the identity zxy=zyx, we nd the compatibility condition Fyy= 0. This is a necessary condition for the linearizability of equation (82.15). Notes 1. Lie group analysis is the most useful and general of all the techniques presented in this book. Some common generators are in table 82.1. Many of the other methods presented in this book can be derivedfrom the method of Lie groups. For example Equations with the dependent variable missing (see page 260) are invariant under the translation group fx =x;y=y+g. Equations with the independent variable explicitly missing (see page 230) are invariant under the translation group fx=x+ ;y=yg. Homogeneous equations (see page 327) are invariant under the ane groupfx=x;y=yeg. Scale invariant equations (see page 398) are invariant under the groupfx=xe;y=yepg. In Kumei and Bluman [13], it is shown that the hodograph transformation (see page 456) and the Legendre transformation(see page 467) are derivable from Lie group methods. Similarity solutions (see page 497) are all derivable from Lie group methods. Contact transformations (see page 249) and the Riccati transfor- mation (see page 392) are also derivable from Lie group methods. 2. Changing variables in an innitesimal generator is straightforward. Suppose we have the generator X=P n i=1bi@ @xi. To change vari- ables from the xi/bracerightbig coordinates to the fxi0gcoordinates (with xi0= xi0/parenleftbig xi
) we nd that X=Pn i=1/parenleftbig Xxi
@ @xi0. For example, consider the generator for scaling invariance: X=x@ @x+y@ @y. To change to CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 372 II.A Exact Methods for ODEs the variables u=y=xandv=xy, we form Xu= x@ @x+y@ @y u= x@ @x+y@ @yy x=0; Xv= x@ @x+y@ @y v= x@ @x+y@ @y xy=2xy=2v: Hence, we can write Xin the (u;v) coordinates as X=2v@ @v.( M a k - ing the further substitution b=1 2logv, we nd that X=@ @b.) 3. In the older literature, transformation groups were found and then classes of equations that were invariant under that group were deter- mined. This was what was done in the rst example in this section.For example, it can be shown that the most general second order dierential equation invariant under a group of the form x =f(x;)=x+(x)+O(2); y=g(x;)y=y+(x)y+O(2); has the form y00+0−2  y0+2−0 2 y=(A;B) s2; where  is an arbitrary function of its arguments, and fA;B;sgare dened by A(x;y)=sy; B(x;y)=(x−y)s; s(x)=e x p −Zx x0(t) (t)dt : See Hill [9, page 84] for details. 4. Recently, the procedure in the last note has been reversed: Given a dierential equation, nd a transformation group that leaves theequation invariant. To derive the transformation group, a set of par- tial dierential equations arising from the equation X (n)G=0m u s t be solved. For example, for the second order ordinary dierential equation ¨x=f(t;x;_x) to be invariant under the group x=x+ (t;x)+O(2); t=t+(t;x)+O(2); requires that the following equation (2 xt−tt)_x+( xx−2xt)_x2−xx_x3 +[ ( x−2t)−3x_x]f(t;x;_x)−ft(t;x;_x)− fx(t;x;_x) − t+( x−t)_x−x_x2 f_x(t;x;_x)=0 hold for all ( t;x;_x). See Aguirre and Krause [1] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 82. Lie Groups: ODEs 373 5. The analysis in this section can be obtained from the general results of Lie algebras. For example, if x(t) satises the equation ¨ x=f(x;_x), wherefis inC1, and the solution is analytic for all t, then the solution may be obtained from xt+=etΩx,w h e r e Ω=v@ @x +f(x;v)@ @v +@ @ ; and we have used xto denotex(). For example, for the dierential equation ¨x=1 ,w eh a v e f=1s ot h a tΩ =v@x+@v+@,a n d we can calculate Ωx=v; Ω2 x=Ωv=1; Ω3 x=Ω1=0; Ωk x=0; fork3 Using these calculations, we can then nd xt+=etΩx =1X k=0tkΩk  k!x =x+tv+t2 2; orx(t+)=x()+t_x()+t2=2. This also generalizes to higher dimensions. For example, the solution of the vector equation ¨x=f(x;_x) may be written as xt+=etΩx, where Ω=v
r x+f(x;v)
r v+@ @: 6. Note that an arbitrary function of xandy,F(x;y), can be for- mally expanded in terms of the generator, x,a n dyas F(x;y)=F(x;y)+@f @@ @x+@g @@ @y =0F(x;y)+


=F(x;y)+VF(x;y)+1 22V2F(x;y)+


=eVF(x;y): 7. If the parameter appearing in equation (82.1) had been an r- dimensional vector, then there would be rinnitesimal operators fX1;X2;:::;Xrg. Lie’s second fundamental theorem states that these operators generate an r-dimensional Lie group under commuta- tion [Xa;Xb]=Kc abXc,w h e r et h e K’s are called structure constants CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 374 II.A Exact Methods for ODEs No. Commutator Pseudoscalar Typied by I[X1;X2]=0X1_X26=0fX1=@x;X2=@yg II[X1;X2]=0X1_X2=0fX1=@y;X2=x@yg III [X1;X2]=X1X1_X26=0fX1=@y;X2=x@x+y@yg IV [X1;X2]=X1X1_X2=0fX1=@y;X2=y@yg Table 82.2: All possible cases for a two-dimensional Lie algebra and summation occurs over repeated indices. Lie’s third fundamental theorem relates the structure constants to one another. Ifr= 1 in the above, then the order of the original equation can be reduced by 1. If n2a n dr= 2, then the order of the original equation can be reduced by 2. If n3a n dr3, then it does not follow that the order of the original equation can be reduced by morethan 2. However, if the r-dimensional Lie algebra has a q-dimensional solvable subalgebra , then the order of the original equation can be reduced by q. See Bluman and Kumei [3] for details. 8. Given the two generators X 1=1@ @x+1@ @yandX2=2@ @x+2@ @y, the pseudoscalar product is X1_X2=12−21and the commutator is [X1;X2]=X1X2−X2X1. By a suitable choice of basis, any two-dimensional Lie algebra can be reduced to one of four types as shown in table 82.2. Hence, an algorithm for integrating second orderordinary dierential equations is given by (a) Calculate an admitted Lie algebra L r. (b) Compare rto 2: i. Ifr< 2, then the ODE cannot be completely integrated using Lie groups. ii. Ifr>2, then determine a sub-algebra L2Lr. (c) From the commutator and pseudoscalar product change the basis to obtain one of the four cases in table 82.2. (d) Integrate the resulting equation. (e) Rewrite the solution in the original variables. 9. The generators for some rst (second) order ordinary dierential equations are in table 82.3 (table 82.4). The Lie groups associated with some second order ordinary dierential equations are in table 82.5. 10. The semigroup approach to dierential equations starts with the evolution equation ut=Lu+Nu(whereLandNare constant coecient linear and nonlinear operators that do not depend on time) with the initial condition u(x;t0)=u0(x) and writes the solution as the nonlinear integral equation u(x;t)=e(t−t0)Lu0(x)+Zt t0e(t−t0)LN(u(x))d: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 82. Lie Groups: ODEs 375 Equation Generator y0=F(kx+ly)X=l@x−k@y y0=Fy x X=x@x+y@y y0=y x+Fy x X=x@y y0=F(x)yX=y@y Table 82.3: Generators for some classes of rst order ODEs Equation Generator y00=F(y;y0) X=@x y00=F(x;y0) X=@y y00=F(x;y−xy0)X=x@y y00=y03F y;y−xy0 y0 X=y@y x3y00=Fy x;y−xy0 X=x2@x+xy@y Table 82.4: Generators for some classes of second order ODEs Equation Lie groupL jLj y00=f(y;y0)f@xg 1 y00=f(y0)f@x;@yg 2 y00=f(y0) xf@y;x@x+y@yg 2 y00=Cy−3f@x;2x@x+y@y;x2@x+y2@yg 3 y00=Cey0f@x;@y;x@x+(x+y)@yg 3 y00=0f@x;@y;x@y;x@x;y@x;y@y;x2@x+xy@y;xy@x+y2@yg8 Table 82.5: Lie groups for some second order ODEs This representation of the solution is useful for proving existence and uniqueness of solutions and computing estimates of their magnitude, verifying dependence on initial and boundary data, as well as per- forming asymptotic analysis of the solution (see, e.g., Yosida [22]). 11. Using Lie groups to nd symmetries of dierential equations can be computationally intensive. Algorithms have been developed for CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 376 II.A Exact Methods for ODEs computerized handling of the calculations, see Azara [2] (for Maple), Bocharov and Bronstein [4], Champagne et al. [5] (for Macsyma), Eliseev et al. [7] (for REDUCE), or Head [8] (for muMATH). 12. It is also possible to nd discrete groups that transform solutions of ordinary dierential equations to other solutions, see Za tsev [23]. For example, the generalized Emden{Fowler equation y00=Axnym(y0)l is described by the parameters c=(n;m;l ). Under the discrete transformationfy=at,x=bug, the solution y=y(x;c)i sm a p p e d to the solution u=u(y;c0), where c0=(n;m; 3−l). Another such discrete transformation is given by fy=au−1=m,x=bt1=(n+1)gfor which c0= −n n+1;1 1−l;2m+1 m .Z a tsev [23] illustrates this method by writing the solution of y00=x−15=8ypy0in terms of the solutions tou00=6u2(which are elliptic functions). 13. Technically, a Lie group is a topological group (i.e., a group that is also a topological space), which is also an analytic manifold on which the group operations are analytic. The tangent space to that manifold is a Lie algebra, which is a linear vector space. See Sattingerand Weaver [16] for an algebraic approach to Lie groups. 14. Easily readable books that explain Lie groups more fully are Bluman and Kumei [3] and Stephani [20]. See also Ince [11, Chapter 4, pages 93{113]. and Olver [14]. 15. For the system of second order ordinary dierential equations ¨y a=!a(yi;_yi;t);a ; i =1;:::;N the generalization of equation (82.6) is (using the summation con- vention, () ;t@()=@t,a n d( );i@()=@yi) (see Stephani [20, page 95]): !a ;t+b!a ;b+/parenleftbig b ;t+_ycb ;c−_yb;t−_yb_yc;c
@!a @_yb +2!a/parenleftbig ;t+_yb;b
+!b/parenleftbig _yab−a ;b
+_ya_yb_yc;bc +2_ya_yc;tc−_yc_yba ;bc+_ya;tt−2_yba ;tb−a ;tt=0: 16. The Blaisus equation y000+yy00= 0 is invariant under the scaling y()=F()w h e r e==. Hence, ifF() is a solution, then so is F(). Consequently, the solution to the Blaisus equation with the boundary conditions fy(0) =y0(0) = 0;y0(1)=2gcan be solved by the sequence of two initial value problems F000+FF00=0F(0) =F0(0) = 0 F00(0) = 1 y000+yy00=0y(0) =y0(0) = 0 y00(0) = [2=F0(1)]3=2 This procedure is called exact shooting , see Klamkin [12]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 82. Lie Groups: ODEs 377 As another example, consider the generalized Emden{Fowler equa- tionN[u]=(tau0)0+ctbeu=0w i t hu0(0) = 0 and u(1)=0 (fora+b6=2 ) . I fU(t) is a solution of N[U]=0 ,t h e ns oi s u(t)=U/parenleftbig te=(b−a+2)
+. Hence, the original BVP can be solved by ndingUfromfN[U]=0;U(0) =U0(0) = 0gand then nding u fromfN[u]=0;u(0) =−U(1);u0(0) = 0g. References [1]Aguirre, M., and Krause, J. Innitesimal symmetry transformations of some one-dimensional linear systems. J. Math. Physics 25 , 2 (February 1984), 210{218. [2]Azara, J. L. R. A MAPLE program for the generation of the lie-series solution of systems of non-linear ordinary dierential equations. Comput. Physics Comm. 67 (1992), 537{542. [3]Bluman, G. W., and Kumei, S. Symmetries and Dierential Equations . Springer{Verlag, New York, 1989. [4]Bocharov, A. V., and Bronstein, M. L. Eciently implementing two methods of the geometrical theory of dierential equations: An experience in algorithm and software design. Acta Appl. Math. 16 (1989), 143{166. [5]Champagne, B., Hereman, W., and Winternitz, P. The computer calculation of Lie point symmetries of large systems of dierential equations. Comput. Physics Comm. 66 (1991), 319{340. [6]Dressner, L. Similarity Solutions of Nonlinear Partial Dierential Equa- tions . Pitman Publishing Co., Marsheld, MA, 1983. [7]Eliseev, V. P., Fedorova, R. N., and Kornyak, V. V. AR E D U C E program for determining point and contact Lie symmetries of dierential equations. Comput. Physics Comm. 36 (1985), 383{389. [8]Head, A. K. LIE, a PC program for Lie analysis of diferential equations. Comput. Physics Comm. 77 (1993), 241{248. [9]Hill, J. M. Solution of Dierential Equations by Means of One-Parameter Groups . Pitman Publishing Co., Marsheld, MA, 1982. [10]Ibragimov, N. H. ,E d . Symmetries, Exact Solution, and Conservation Laws , vol. 1 of Lie Group Analysis of Dierential Equations .C R C , B o c a Raton, FL, 1994. [11]Ince, E. L. Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [12]Klamkin, M. S. On the transformation of a class of boundary value problems into initial value problems. Amer. Math. Monthly 4 (1962), 43{47. [13]Kumei, S., and Bluman, G. W. When nonlinear dierential equations are equivalent to linear dierential equations. SIAM J. Appl. Math. 42 ,5 (October 1982), 1157{1173. [14]Olver, P. J. Applications of Lie Groups to Dierential Equations . No. 107 in Graduate Texts in Mathematics. Springer{Verlag, New York, 1986. [15]Ovsiannikov, L. V. Group Analysis of Dierential Equations .A c a d e m i c Press, New York, 1982. Translated by W. F. Ames. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 378 II.A Exact Methods for ODEs [16]Sattinger, D. H., and Weaver, O. L. Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics . Springer{Verlag, New York, 1986. [17]Schwarz, F. A REDUCE package for determining Lie symmetries of ordinary and partial dierential equations. Comput. Physics Comm. 27 (1982), 179{186. [18]Seshadri, R., and Na, T. Y. Group Invariance in Engineering Boundary Value Problems . Springer{Verlag, New York, 1985. [19]Steinberg, S. Lie series and nonlinear dierential equations. J. Math. Anal. Appl. 101 (1984), 39{63. [20]Stephani, H. Dierential Equations: Their Solution Using Symmetries . Cambridge University Press, New York, 1989. edited by M. MacCallum. [21]Winternitz, P. Lie groups and solutions of nonlinear dierential equations. InNonlinear Phenomena , K. B. Wolf, Ed., no. 189 in Lecture Notes in Physics. Springer{Verlag, New York, 1983, pp. 263{331. [22]Yosida, K. Functional Analysis . Springer{Verlag, New York, 1980. [23]Zatsev, V. F. On discrete-group analysis of ordinary dierential equations. Soviet Math. Dokl. 37 , 2 (1988), 403{406. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 83. Operational Calculus379 83. Operational Calculus Applicable to Ordinary and partial dierential equations. Yields A reformulation of the original dierential equation. Idea It may sometimes be easier to solve a dierential equation in a trans- formed space. Procedure Given an ordinary dierential equation, transform it to a eld of op- erators, solve the equation in that eld, and then transform back. In thiseld, ordinary functions, generalized functions, and dierential operators are all treated as objects in a single algebraic structure. The operator eld that is used has, among other elements, an identity operator (I), a dierentiation operator (often denoted by Dors)a n d an integration operator (often denoted by D −1). The operator D,w h e n applied to the operator corresponding to a function f(t), results in Dffg=ff0g+ff(0)g;: (83.1) The operator D−1, when applied to the operator corresponding to a func- tionf(t) results in D−1ffg=Zt 0f(u)du : The braces around the above expressions emphasize that they are operators in the eld. In many applications, the operator Dis formally treated as being a \large constant." There are tables of formulae describing how operators interact in their quotient eld. For example, because I D−= et/bracerightbig (83.2) we can calculate I (D−)2=I (D−)I (D−) = et/bracerightbig et/bracerightbig =Zt 0eue(t−u)du = tet/bracerightbig ; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 380 II.A Exact Methods for ODEs because the \product" of two operators is the operator corresponding to a convolution. The formula in equation (83.2) follows from equation (83.1) whenf(t)=et, because (D−) et/bracerightbig =/parenleftbig et/bracerightbig
+f1g− et/bracerightbig =I: It is easy to represent generalized functions and non-continuous func- tions in the eld. For example, a square wave of period 2 chas the operator representationI D(I+e−cD). Example 1 The following ordinary dierential equation for y(t) y00+y=0 has the operator representation /parenleftbig D2+1
fyg= 0 (83.3) orD2/parenleftbig 1+D−2
fyg=0:By applying D−2to the left of the above equa- tion, we obtain /parenleftbig 1+D−2
fyg=D−2f0g =At+B; whereAandBare arbitrary constants. This equation may be formally solved by \dividing" by the operator on the left and expanding terms. Wend fy(t)g=1 1+D−2(At+B) =/parenleftbig 1−D−2+D−4−



(At+B)/bracerightbig = (At+B)+ −At3 6−Bt2 2 + −At5 120−Bt4 24 +


 =fAsint+Bcostg:(83.4) Hence,y(t)=Asint+Bcost. Really, in this last calculation, there would be many more terms than those illustrated. For instance, when D−4is applied to ( At+B), we ob- tain −At5 120−Bt4 24 plus some terms of the form/parenleftbig C1t3+C2t2+C3t+C4
. When the form of the solution, with all these additional terms, is substi- tuted into the dening equation (83.3), these additional constants turn out to be zero. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 83. Operational Calculus381 Example 2 Consider the constant coecient linear ordinary dierential equation forz(t) z00+3z0+2z=f(t); z(0) = 1;z0(0) = 0: Because of the formula z(n)=Dnz−n z(n−1)(0) +Dz(n−2)(0) +


+Dn−1z(0)o (which parallels the rule for Laplace transforms), the equation for z(t)h a s the operator representation h D2fzg−Di +3h Dfzg−Ii +2fzg=ffg: This operator equation can be manipulated into fzg=D+3I D2+3D+2+ffg D2+3D+2 =2I D+1−I D+2+I D+1−I D+2 ffg = 2e−t/bracerightbig − e−2t/bracerightbig + e−t−e−2t/bracerightbig ffg; and hence, z(t)=2e−t−e−2t+Zt 0/parenleftbig e−u−e−2u
f(u)du; which is the same result that would be obtained by use of Laplace trans- forms. Notes 1. The operational calculus is also called the Heaviside calculus. 2. The operational calculus, at its simplest level, has a great similarity with Laplace transforms. One school of thought is that any integral transform creates an operational calculus. 3. It is sometimes dicult to justify the formal steps that are employed in using the operation calculus. One solution (see Erd elyi [3]) is to use a more precisely dened operator, such as the primary operator bDf(t)=f(t)+Zt 0e(t−)f()d; which has the inverse bD−1 g(t)=g(t)−Rt 0g()d. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 382 II.A Exact Methods for ODEs 4. Innite order dierential equations are often solved by techniques sim- ilar to those described above. For example, the ordinary dierential equations/parenleftbig d dx+1
−1y+(x−a)y=0a n d cosh/parenleftbig id dx
+H(x)−a y= 0 are innite order dierential equations for y(x)( h e r eH(x)r e p r e - sents the step function). Recent results (as well as the solutions to the two above equations) may be found in Dimitrov [2]. 5. The extension of this technique to partial dierential equations is straightforward. Using Dfor@ @xandD0for@ @t, a partial dierential equation can sometimes be written in the form P(D;D0)fyg=ffg. The \inversion" process will then proceed in two steps. For example, to obtain a particular solution of uxx−6uxt+9utt=1 2x2+3 6xt;a calculation analogous to the one in equation (83.4) might proceed as follows: fyg=1 P(D;D0)ffg =1 D2−6DD0+9D02/parenleftbig 12x2+3 6xt
=1 D2 1−3D0 D−2/parenleftbig 12x2+3 6xt
=1 D2 1+6D0 D+2 7D02 D2+:::! /parenleftbig 12x2+3 6xt
=1 D2/parenleftbig 12x2+3 6xt
+6 D3(36x) =/parenleftbig x4+6x3t
+/parenleftbig 9x4
=1 0x4+6x3t: 6. See Courant and Hilbert [1, Volume 2, pages 507{535] and Kaplan [5, pages 515{538]. References [1]Courant, R., and Hilbert, D. Methods of Mathematical Physics . Interscience Publishers, Inc., New York, 1953. [2]D i m i t r o v ,H .D . On the solutions of some linear operator non-polynomial dierential equations. J. Phys. A: Math. Gen. 15 (1982), 367{379. [3]Erdelyi, A. Operational Calculus and Generalized Functions . Holt, Rinehart and Winston, New York, 1962. [4]Glaeske, H.-J. Operational properties of a generalized Hermite transforma- tion. Aequationes Mathematicae 32 (1987), 155{170. [5]Kaplan, W. Operational Methods for Linear Systems . Addison{Wesley Publishing Co., Reading, MA, 1962. [6]Mikusinski, J. Operational Calculus , fth ed. Pergamon Press, New York, 1959. [7]Shtokalo, I. Z. Operational Calculus . Pergamon Press, New York, 1976. translated by V. Kumar. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 83. Operational Calculus383 [8]Yosida, K. Operational Calculus . Springer{Verlag, New York, 1984. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 384 II.A Exact Methods for ODEs 84. Pfaan Dierential Equations Applicable to Pfaan dierential equations. Yields Knowledge of whether the equation is integrable. Idea Pfaan dierential equations are partial dierential equations of the form f(x)
dx=nX i=1Fi(x1;x2;:::;xn)dxi=0: (84.1) For equations of this type, Ifn= 3, then a necessary and sucient condition that equation (84.1) be integrable is that f(x)
curlf(x)=0: Ifn4, then a necessary and sucient condition that equation (84.1) be integrable is that Fp@Fr @xq−@Fq @xr +Fq@Fp @xr−@Fr @xp +Fr@Fq @xp−@Fp @xq =0; wherep,q,a n drare any three of the integers 1 ;2;3;:::;n . There exist a number of techniques for integrating Pfaan equations. Example If we have the equation (y2+yz)dx+(xz+z2)dy+(y2−xy)dz=0; (84.2) then we identify n=3a n d f(x)=(y2+yz;xz +z2;y2−xy); so that curlf(x)=rf(x)=2 (−x+y−z;y;−y): Therefore f(x)
curlf(x) = 0, and there exists a solution to equation (84.2). The solution is, in fact, given by y(x+z)=C(y+z), whereCis any constant. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 84. Pfaan Dierential Equations 385 Procedure 1 If a Pfaan equation is integrable, then there exists an integrating factorsuch that d=nX i=1Fidxi: By appropriate manipulations of equation (84.1), it may be shown that  satises any of the equations −d =nX j=11 Fi@Fi @xj−@Fj @xi dxj; (84.3) fori=1;2;:::;n . Any one of these equations may be solved to determine an integrating factor. Alternatively, if two integrating factors can be found, sayand, then a solution to equation (84.1) is given by == constant. Example 1 The Pfaan dierential equation y(x2−y2−yz)dx+x(y2−x2−xz)dy+xy(x+y)dz=0 (84.4) can be shown to pass the integrability requirements. Substituting into equation (84.3) results in the three separate equations −d =2(x−y)(2x+2y+z) y(x2−y2−yz)dy−2(x+y) x2−y2−yzdz; =−2(x−y)(2x+2y+z) x(y2−x2−xz)dx−2(x+y) y2−x2−xzdz; =2dx x+dy y ;(84.5) forj=1;2;3. The last equation in (84.5) can be integrated to determine =1=(xy)2. Hence, multiplying equation (84.4) by 1 =(xy)2results in d=x2−y2−yz x2y dx+y2−x2−xz xy2 dy+x+y xy dz; which can be integrated to yield =x y+y x+x+y xy z+C; whereCis an arbitrary constant. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 386 II.A Exact Methods for ODEs Procedure 2 If an integrable Pfaan dierential equation is of the form Pdx+Qdy+ Rdz=0 ,w h e r e P,Q,a n dRare homogeneous functions of the same degree, then a solution may be found. First, dene Z=Px+Qy+Rz. Then, form Pdx +Qdy+Rdz−dZ Z+dZ Z= 0 (84.6) and integrate (we have addressed only the case of Z6= 0, although there are special techniques that can be used when Z=0 ) . Example 2 Given the Pfaan equation (yz+z2)dx−xzdy +xydz =0; we deneZ=xz(y+z). Forming equation (84.6) we obtain dZ Z−2(dy+dz) y+z=0; which can be immediately integrated to yield Z=C(y+z)2orxz= C(y+z), whereCis an arbitrary constant. Procedure 3 The Pfaan dierential equation Pdx+Qdy+Rdz= 0 can sometimes be solved by taking one variable, say z, as a constant. Then, the solution ofPdx+Qdy = 0 (because z= constant means that dz= 0) will be given byu(x;y)=c o n s t a n t . We take the \constant" in this last expression to be f(z). Dierentiating u(x;y)=f(z) and comparing to the original equation, we may sometimes obtain an ordinary dierential equation for f(z). Example 3 Given the Pfaan equation 2xdx+dy+( 1+2z2+2yz+2x2z)dz=0; we treatzas a constant to obtain 2 xdx+dy= 0, which has the solution x2+y= constant = f(z). This can be dierentiated to obtain 2xdx+dy+f0(z)dz=0: Comparing this to the original equation, we nd that f(z) satises the ordinary dierential equation: f0=1+2z2+2zf. Solving this equation to obtainf(z)=Ce−z2−z,w h e r eCis an arbitrary constant, we nd the solution to the original equation to be x2+y+z=Ce−z2: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 84. Pfaan Dierential Equations 387 Notes 1. Another name for a Pfaan dierential equation is a total dierential equation . 2. One way to solve Pfaan dierential equations in three dimensions is by the observation: if curl f(x)=0 ,t h e n f(x) must be the gradient of a scalar. Hence, the set of partial dierential equations fi(x)=@v(x) @xi;fori=1;:::;n; may be solvable for v(x). The solution to equation (84.1) would then be given implicitly by v(x)=c o n s t a n t . 3. If the Pfaan dierential equation is of the formPn i=1fi(xi)dxi=0 , then the integral surfaces are dened byPn i=1R fi(xi)dxi=C,w h e r e Cis an arbitrary constant. 4. Sometimes a Pfaan dierential equation can be reduced to a sys- tem of ordinary dierential equations. One such procedure is called Mayer’s method. See Carath eodory [1, pages 121{133] for details. 5. Given a system of mPfaan dierential equations in mdependent variablesfzjjj=1;2;:::;mgandnindependent variables fxkjk=1;2;:::;ng dzj=nX k=1Pjk(x;z)dxk;j =1;2;:::;m; the condition for complete integrability is given by @Pjk @xl+mX i=1@Pjk @ziPil=@Pjl @xk+mX i=1@Pjl @ziPik; forj=1;2;:::;m andk;l=1;2;:::;n . See Iyanaga and Kawada [6] for details on how this system may be solved. 6. Using the notation of exterior calculus, a total dierential equation is an equation of the form !=0 ,w h e r e !is a dierential 1-form, also called a Pfaan form,Pn i=1ai(x)dxion a manifold. See Zwillinger [9] for details. 7. See Ford [2, pages 135{141], Ince [5, pages 52{59], Moon and Spencer [7, pages 23{27], and Sneddon [8, pages 18{33]. References [1]Caratheodory, C. Calculus of Variations and Partial Dierential Equa- tions of the First Order . Holden{Day, Inc., San Francisco, CA, 1965. [2]Ford, L. R. Dierential Equations . McGraw{Hill Book Company, New York, 1955. [3]Griffiths, P. A., and Jensen, G. R. Dierential Systems and Isometric Imbeddings . Princeton University Press, Princeton, NJ, 1987. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 388 II.A Exact Methods for ODEs [4]Haack, E. R., and Wendland, W. N. Lectures on Partial and Pfaan Dierential Equations . Pergamon Press, New York, 1972. Translated by E. R. Dawson and W. N. Everitt. [5]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [6]Iyanaga, S., and Kawada, Y. Encyclopedic Dictionary of Mathematics . MIT Press, Cambridge, MA, 1980. [7]Moon, P., and Spencer, D. E. Partial Dierential Equations .D .C .H e a t h and Co., Lexington, MA, 1969. [8]Sneddon, I. N. Elements of Partial Dierential Equations . McGraw{Hill Book Company, New York, 1957. [9]Zwillinger, D. ,E d . Standard Mathematical Tables and Formulae ,3 0e d . CRC, Boca Raton, FL, 1995. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 85. Reduction of Order 389 85. Reduction of Order Applicable to Linear ordinary dierential equations. Yields A lower order dierential equation, if any non-trivial solution of the homogeneous equation is known. Idea For annth order linear ordinary dierential equation, any non-trivial solution of the homogeneous equation can be used to reduce the order ofthe equation by 1. For the special case of second order linear dierential equations, knowing any solution of the homogeneous equation allows the general solution to be found. Procedure We choose to illustrate the method for second order equations. If we have the general second order linear ordinary dierential equation y00+p(x)y0+q(x)y=r(x); (85.1) letz(x) be any non-trivial solution to the corresponding homogeneous equation; that is, z(x) satises z00+p(x)z0+q(x)z=0: (85.2) If we look for a solution of equation (85.1) in the form of y(x)=z(x)v(x), then we can obtain a solvable equation for v(x). Substituting y(x)= z(x)v(x) into equation (85.1) yields zv00+( 2z0+pz)v0+(z00+pz0+qz)v=r: (85.3) Becausez(x) satises equation (85.2), equation (85.3) becomes zv00+( 2z0+pz)v0=r: (85.4) If we now let w(x)=v0(x), then equation (85.4) becomes a rst order linear ordinary dierential equation for w(x). It can be solved by the use of integrating factors (see page 356). Example Given the second order linear dierential equation d2y dx2−2xdy dx+2y=3; (85.5) we recognize that z(x)=xis a solution of the homogeneous equation. Equation (85.4) becomes xd2v dx2+2 ( 1−x2)dv dx=3: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 390 II.A Exact Methods for ODEs This equation may be solved by recognizing that it is a linear rst order ordinary dierential equation in the unknown dv=dx . Hence, integrating factors can be used to nd dv=dx .A f t e rdv=dx is determined, it can be integrated directly to yield v(x)=3 2x+AZxet2 t2dt+B; whereAandBare arbitrary constants. Using the relationship y(x)= z(x)v(x), the general solution of equation (85.5) is y(x)=3 2+AxZxet2 t2dt+Bx: Notes 1. The general nth order linear ordinary dierential equation is treated in Finizio and Ladas [2, pages 108{116] and Rainville and Bedient [3, pages 127{129]. The general result is that Ifz(x) is a solution of the linear homogeneous equation z(n)+p1(x)z(n−1)+


+pn(x)z=0 (85.6) and ify(x)=v(x)z(x), then the equation y(n)+p1(x)y(n−1)+


+pn(x)y=r(x) (85.7) transforms into v(n)+q1(x)v(n−1)+


+qn−1v0=r(x): This last equation may be reduced in order by dening w(x)=v0(x). 2. More generally, if fz1(x);:::;zp(x)gare linearly independent solu- tions of equation (85.6), then the substitution y(x)=z 1::: zpv z0 1::: z0 pv0 ......... z(p) 1::: z(p) pv(p) reduces equation (85.7) to a linear ordinary dierential equation of ordern−pforv(x). 3. See also Boyce and DiPrima [1, section 3.4, pages 127{131]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 85. Reduction of Order 391 References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Finizio, N., and Ladas, G. Ordinary Dierential Equations with Modern Applications . Wadsworth Publishing Company, Belmont, CA, 1982. [3]Rainville, E. D., and Bedient, P. E. Elementary Dierential Equations . The MacMillan Company, New York, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 392 II.A Exact Methods for ODEs 86. Riccati Equations Applicable to Ordinary dierential equations of the form y0= a(x)y2+b(x)y+c(x). Yields A reformulation as a linear second order ordinary dierential equation, or a second solution if one solution is already known. Idea A change of dependent variable can transform a Riccati equation to a linear second order ordinary dierential equation. Also, if one solution to a Riccati equation is known, then the other solution can be written down explicitly. Procedure 1 Suppose we have the Riccati equation y0=a(x)y2+b(x)y+c(x): (86.1) If the dependent variable in equation (86.1) is changed from y(x)t ow(x) by y(x)=−w0(x) w(x)1 a(x); (86.2) then we obtain the equivalent second order linear ordinary dierential equation w00−a0(x) a(x)+b(x) w0+a(x)c(x)w=0: (86.3) It might be easier to solve equation (86.3) than to solve equation (86.1) by other means. Procedure 2 Suppose we have the Riccati equation y0=a(x)y2+b(x)y+c(x); (86.4) and suppose further that one solution to this equation is already known to us, say, y(x)=z(x). Ify(x)=z(x)+u(x) is substituted in equation (86.4), then the solvable Bernoulli equation u0=(b+2az)u+au2 is obtained for u(x). To solve this equation, the new dependent variable v(x)=1=u(x) should be introduced and then integrating factors should be used (see pages 235 and 356). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 86. Riccati Equations 393 Example 1 Suppose we have the Riccati equation y0=exy2−y+e−x(86.5) to solve. By identifying a(x)=ex,b(x)=−1a n dc(x)=e−x, the change of variables in equation (86.2) becomes y(x)=−w0(x) w(x)e−x; (86.6) so that equation (86.5) becomes w00+w= 0, which could have been obtained directly from equation (86.3). The solution to this equation is w(x)=Asinx+Bcosx,w h e r eAandBare arbitrary constants. Using this solution in equation (86.6) leads to the general solution of equation(86.5) y(x)=−e −xAcosx−Bsinx Asinx+Bcosx : There should be only one arbitrary constant in the solution to equation (86.5), because it is a rst order ordinary dierential equation. In fact, this last equation may be written as y(x)=−e−xcosx−Csinx sinx+Ccosx ; w h e r ew eh a v ed e  n e d C=B=A (and assumed A6=0 ) . Example 2 Suppose we have the equation y0=y2−xy+ 1 (86.7) to solve. A solution to equation (86.7), obtained by inspection, is y(x)=x. We utilize this solution in forming y(x)=x+u(x); (86.8) and then (using equation (86.8) in equation (86.7)) the equation u0= u2+xuis obtained. This Bernoulli equation has the solution u(x)= ex2=2 A−Zx 0et2=2dt,w h e r eAis an arbitrary constant. Thus, the second solution to equation (86.7) is y(x)=x+ex2=2 A−Zx 0et2=2dt: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 394 II.A Exact Methods for ODEs Notes 1. The transformation in equation (86.2) is known as the Riccati trans- formation . 2. The identity d dx−q(x)d dx+q(x) u=u00+/parenleftbig q0−q2
u (86.9) shows that the dierential equation u00+p(x)u= 0 can be factored into the form of equation (86.9) if q0−q2=p, which is a Riccati equation. 3. See Bender and Orszag [1, Section 1.6], Boyce and DiPrima [2, pages 93{94 and 142{143], Goldstein and Braun [3, pages 45{36], Ince [4,pages 23{25 and 295], and Simmons [6, pages 62{63]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [3]Goldstein, M. E., and Braun, W. H. Advanced Methods for the Solution of Dierential Equations . U.S. Government Printing Oce, Washington, D.C., 1973. NASA SP-316. [4]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [5]R e i d ,W .T . Riccati Dierential Equations . Academic Press, New York, 1972. [6]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 87. Matrix Riccati Equations 395 87. Matrix Riccati Equations Applicable to Systems of quadratic ordinary dierential equations. Yields An exact solution. Idea There is an exact solution available for matrix Riccati dierential equa- tions. If a given system of ordinary dierential equations can be put in theform of a matrix Riccati equation, then the solution can be found. Procedure IfZ(t),A(t), andK(t) are allNNmatrices, then we can use the following theorem: IfZ(t) satises the following matrix Riccati dierential equation d dtZ=ZAZ +KZ+ZKT;Z (t=0 )=Z0; (87.1) thenZ(t) is explicitly given by Z(t)=Q(t) Z−1 0−Zt 0QT(s)A(s)Q(s)ds−1 QT(t); (87.2) whereQ(t) is dened to be the solution of d dtQ(t)=K(t)Q(t);Q (t=0 )=I; (87.3) Iis theNNidentity matrix, and the required matrix inverses are assumed to exist. If a given system of ordinary dierential equations can be placed in the form of equation (87.1), then the solution can be found from equation (87.2). Example Suppose we wish to solve the following system of coupled dierential equations for x(t)a n dy(t) dx dt=a(t)(y2−x2)+2b(t)xy+2cx; dy dt=b(t)(y2−x2)−2a(t)xy−2cy;(87.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 396 II.A Exact Methods for ODEs withx(0) =Dandy(0) =E. If we form the matrices Z=xy y−x , K=[c0 0c],Z0=DE E−D ,a n dA=h −a(t)b(t) b(t)a(t)i , then the equations in (87.4) are the same as those in equation (87.1). The solution for Q(t) from equation (87.3) is Q(t)=ectI. Therefore, the solution for Zis Z(t)=e2ct Z−1 0−Zt 0e2csA(s)ds−1 : If we dene (t)=Zt 0e2csa(s)ds; (t)=Zt 0e2csb(s)ds; then, by equating the corresponding entries of equation (87.2), we can nd fx(t);y(t)gin terms off(t);(t)g.W eh a v e x(t)=e2ct (t)(E2+D2)+D =
; y(t)=e2ct (t)(E2+D2)+D =
; where
=
( x) is dened by
(x)= 2(t)+2(t) [E2+D2]−2(t)E+2(t)D+1: Notes 1. Matrix Riccati equations arise naturally in a number of physical set- tings. For example, the gains in a Kalman{Bucy lter satisfy a matrix Riccati equation. Also, the deflection of a beam can be described by such equations. They also appear quite often in the context of control theory (see Jodar and Abou-Kandil [3]) and invariant embedding solutions (see page 747). 2. Kerner [7] shows that nonlinear dierential systems of arbitrary order _i=Xi(1;2;:::;k;t);fori=1;2;:::;k; may often be reduced to Riccati systems _xi=Ai+Bix+Cixx; fori=1;2;:::;n; nk; andA;B;C constant; and then to elemental Riccati systems _zi=Eizz; fori=1;2;:::;p; p (n)>n; where each Eiequals 0 or 1. His examples include ordinary dieren- tial equation systems that contain exponential functions and elliptic functions. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 87. Matrix Riccati Equations 397 3. Celletti and Francoise [2] study matrix dierential equations of the form _X=Y,_Y=−h(X)h0(X), wherehis a polynomial function. 4. Jodar and Navarro [4] write the solutions of the matrix dierential equationX(p)+Ap−1X(p−1)+


+A0X= 0 in terms of the matrix algebraic equation Yp+Ap−1Yp−1+


+A0=0 . References [1]Bittanti, S., Laub, A. J., and Willems, J. C. ,E d s . The Riccati Equation . Springer{Verlag, New York, 1991. [2]Celletti, A., and Francoise, J. P. Matrix-second order dierential equations and chaotic Hamiltonian systems. Z.-Angew.-Math.-Phys. 40 (1989), 925{930. [3]Jodar, L., and Abou-Kandil, H. A resolution method for Riccati dierential systems coupled in their quadratic terms. SIAM J. Appl. Math. 19, 6 (November 1988), 1425{1430. [4]Jodar, L., and Navarro, E. On complete set of solutions for polynomial matrix equations. Appl. Math. Lett. 3 , 1 (1990), 15{18. [5]Jones, R. A. Existence theorems for the matrix Riccati equation W’+WP(t)W+Q(t)=0 .Int. J. Math. &M a t h .S c i .1 (1978), 13{19. [6]Kenney, C. S., and Leipnik, R. B. Numerical integration of the dierential matrix Riccati equation. IEEE Trans. Automat. Control ,1 0 (1985), 962{970. [7]Kerner, E. H. Universal formats for nonlinear ordinary dierential equations. J. Math. Physics 22 , 7 (July 1981), 1366{1371. [8]Murty, K. N., Prasad, K. R., and Srinivas, M. A. S. Upper and lower bounds for the solution of the general matrix Riccati dierential equations. J. Math. Anal. Appl. 147 , 1 (1990), 12{21. [9]Rand, D. W., and Winternitz, P. Nonlinear superposition principles: A new numerical method for solving matrix Riccati equations. Comput. Physics Comm. 33 (1984), 305{328. [10]Razzaghi, M. A computational solution for the matrix Riccati equation using Laplace transforms. Int. J. Comp. Math. 11 (1982), 297{304. [11]R e i d ,W .T . Solutions of a Riccati matrix dierential equation as functions of initial values. J. Math. Mech. 8 (1959), 221{230. [12]R e i d ,W .T . Riccati Dierential Equations . Academic Press, New York, 1972. [13]Wilcox, R. M., and Harten, L. P. MACSYMA-generated closed-form solutions to some matrix Riccati equations. Appl. Math. and Comp. 14 (1984), 149{166. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 398 II.A Exact Methods for ODEs 88. Scale Invariant Equations Applicable to Ordinary dierential equations of a certain form. Yields An equidimensional-in- xordinary dierential equation of the same or- der (which can then be reduced to an ordinary dierential equation of lower order). Idea A scale invariant equation is one in which the equation is unchanged whenxandyare scaled in a certain way. When an equation is scale invariant, we can convert the equation into an equidimensional-in- xordi- nary dierential equation of the same order by a change of the dependent variable. This equidimensional-in- xordinary dierential equation can then be changed into an autonomous equation of lower order. Procedure A scale invariant equation is one that is left invariant under the trans- formationfx!ax;y!apyg,w h e r eaandpare constants. That is, if the original equation is an equation for y(x)a n dt h exvariable is replaced by the variable ax0and theyvariable is replaced by the variable apy0,t h e n the new equation (in terms of y0andx0) will be identical to the original equation (which is in terms of yandx). The way to determine the value ofpis to change variables and then see what value of pleaves the equation unchanged. A scale invariant equation can be converted to an equidimensional-in- x equation by the substitution for y y(x)=xpu(x): (88.1) By the techniques on page 275, this equidimensional-in- xequation may then be made autonomous, and then (after another transformation) theorder of the equation can be reduced. Example Suppose we have the nonlinear second order ordinary dierential equa- tion x2d2y dx2+3xdy dx=1 y3x4: (88.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 88. Scale Invariant Equations 399 To determine if this equation is scale invariant, and if so, what the value ofpis, we substitute ax0forxandapy0foryto obtain (ax0)2d2(apy0) d(ax0)2+3 (ax0)d(apy0) d(ax0)=1 (apy0)3(ax0)4 or apx02d2y0 dx02+3apx0dy0 dx0=a(−3p−4)1 y03x04: (88.3) Hence, if we choose pso thatp=−3p−4, then the form of equation (88.3) will be the same as the form of equation (88.2). So the equation is scale invariant, with the value p=−1. To make this equation equidimensional- in-x, we change variables by equation (88.1): y(x)=u(x)=x. Using this change of variables in equation (88.2) produces x2d2u dx2+xdu dx−u=1 u3: (88.4) Equation (88.4) is equidimensional-in- x, so we use the substitution x=et (see page 275) for d2u dt2−u=1 u3: (88.5) Equation (88.5) is autonomous, so we change the independent variable by v(u)=u0(t) (see page 230) for vdv du−u=1 u3: (88.6) The solution of equation (88.6) can be found by separating variables (see page 487) v(u)=r A−u2−1 u2; whereAis an arbitrary constant. To nd u(t), we must now solve du dt=v(u)=r A−u2−1 u2: (88.7) Equation (88.7) is a separable equation whose solution is u(t)=p coshB+ sinhBsin(2t+C); whereBandCare arbitrary constants. The last step is to recall that y(x)=u(x)=xand thatx=et. The nal solution is therefore y(x)=1 xp coshB+ sinhBsin(2 logx+C): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 400 II.A Exact Methods for ODEs Notes 1. This method is derivable from Lie group methods (see page 366). The innitesimal operator in this case is given by U=x@ @x+py@ @y. 2. A special case of this method (when p= 1) is the method for homo- geneous equations (see page 327). 3. Euler equations (see page 281) are scale invariant equations for any value of the parameter p. 4. Scale invariant equations are also called isobaric equations . 5. In Rosen’s paper [3], a change of variable is proposed, dierent from the one presented above, that often allows parametric solutions to beobtained. 6. See also Bender and Orszag [1, pages 25{26] and Goldstein and Braun [2, pages 81{84]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Goldstein, M. E., and Braun, W. H. Advanced Methods for the Solution of Dierential Equations . U.S. Government Printing Oce, Washington, D.C., 1973. NASA SP-316. [3]Rosen, G. Alternative integration procedure for scale-invariant ordinary dierential equations. Int. J. Math. &M a t h .S c i .2 (1979), 143{145. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 89. Separable Equations 401 89. Separable Equations Applicable to First order ordinary dierential equations. Yields An exact solution, often implicit. Idea First order ordinary dierential equations can be solved directly if the forcing term factors into a term involving only the independent variable and a term involving only the dependent variable. Procedure G i v e na ne q u a t i o no ft h ef o r m dy dx=f(y)g(x); (89.1) both sides can be formally multiplied by dx=f (y) and then integrated to obtain Zdy f(y)=Z g(x)dx: (89.2) The evaluation of equation (89.2) requires only that two integrals be eval- uated. An arbitrary constant of integration must be included to obtain the most general solution of equation (89.1). Example Suppose we have the equation dy dx=9x8+1 y2+1 to solve. Multiplying both sides of equation (89) by ( y2+1 )dxand then integrating results in Z (y2+1 )dy=Z (9x8+1 )dx: Evaluating the integrals yields y3 3+y=x9+x+C; whereCis an arbitrary constant. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 402 II.A Exact Methods for ODEs Notes 1. The solution obtained by this method will generally be implicit. 2. The formal procedure of multiplying equation (89.1) by dx=f (y)c a n be rigorously shown to give the correct answer. 3. See Boyce and DiPrima [1, pages 37{42], Ince [2, pages 17{18], and Simmons [3, pages 35{36]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [3]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 90. Series Solution403 90. Series Solution Applicable to Homogeneous linear ordinary dierential equations, most frequently second order dierential equations. Yields An innite series expansion of the two independent solutions. Idea If an innite series is substituted into a linear equation, the dierent coecients may be matched to obtain recurrences for the coecients of the series. Solving these recurrences results in an explicit solution. Procedure Given a homogeneous linear second order ordinary dierential equation in the form y00+P(x)y0+Q(x)y=0; (90.1) we search for a series solution around the point x= 0. There are four dierent cases to consider. Clearly, an expansion about any other point, x0, could be determined by changing the independent variable to t=x−x0and then analyzing the resulting equation near t=0 . 1. Ifx= 0 is an ordinary point of equation (90.1) (the denitions of ordinary points and singular points are given on page 11) then we may assume that P(x)a n dQ(x) have the known Taylor expansions P(x)=1X n=0Pnxn;Q (x)=1X n=0Qnxn; (90.2) in the regionjxj<,w h e r erepresents the minimum of the radii of convergence of the two series in equation (90.2). In this case, equation (90.1) will have two linearly independent solutions of the form y(x)=1X n=0anxn: (90.3) 2. Alternately, if x= 0 is a regular singular point of equation (90.1) then we may assume that P(x)a n dQ(x) have the known expansions P(x)=1X n=−1Pnxn;Q (x)=1X n=−2Qnxn; (90.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 404 II.A Exact Methods for ODEs in the regionjxj<. After determining the expansions in equation (90.4), we need to determine the roots to the indicial equation 2+(P−1−1) +Q−2=0; (90.5) which is obtained by utilizing y=xin equation (90.1), along with the expansions in equation (90.4), and then determining the coecient of the lowest order term. The two roots of this equationare called the exponents of the singularity . There are now several cases, depending on the values of the exponents of the singularity: (a) If 16=2and1−2is not equal to an integer, then equation (90.1) will have two linearly independent solutions in the forms y1(x)=jxj1 1+1X n=1bnxn! ; y2(x)=jxj2 1+1X n=1cnxn! :(90.6) (b) If1=2, then (calling =1) equation (90.1) will have two linearly independent solutions in the forms y1(x)=jxj 1+1X n=1dnxn! ; y2(x)=y1(x)l o gjxj+jxj1X n=0enxn:(90.7) (c) If1=2+M,w h e r eMis an integer greater than 0, then equation (90.1) will have two linearly independent solutions in the forms y1(x)=jxj1 1+1X n=1fnxn! ; y2(x)=hy1(x)l o gjxj+jxj21X n=0gnxn;(90.8) where the parameter hmay be equal to zero. The procedure in each of the four cases is the same: Substitute the given forms (i.e., the expansions in equation (90.3), (90.6), (90.7), or (90.8)) into the original equation (90.1) and equate the coecients of the xjandxjlogx terms for dierent values of j. This will yield recurrence relations for the unknown coecients. Solving these recurrence relations will determine the solution. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 90. Series Solution405 In the case of an ordinary point, there will be two unknown coecients that parameterize the series solutions in equation (90.3). These two co- ecients will generate the two linearly independent solutions of equation (90.1). Example 1 Given the equation y00+y=0; (90.9) we easily see that x= 0 is an ordinary point. Using equation (90.3) in equation (90.9) we nd (2a2+a0)+( 6a3+a1)x+( 1 2a4+a2)x2+::: +[ (n+1 ) (n+2 )an+2+an]xn+


=0: Hence, we must have an+2=−an (n+1)(n+2). Iterating this relation we nd a2m=(−1)m1 (2m)!;a 2m+1=(−1)m 1 (2m+1 ) !: (90.10) Hence, using equation (90.10) in equation (90.3), y(x)=a0 1−x2 2!+x4 4!−::: +a1 x−x3 3!+x5 5!−::: : (90.11) Of course, the exact solution to equation (90.9) is y(x)=a0cosx+a1sinx, which is what equation (90.11) has reproduced. Example 2 Given the equation y00+1+2x 2xy0−1 2x2y=0; (90.12) we easily see that x= 0 is a regular singular point. In this case we have (see equation (90.4)) P−1=1 2,Q−2=−1 2. Therefore, the indicial equation (from equation (90.5)) becomes 2−1 2−1 2=(−1) −1 2 =0: Because the roots 1=1 ,2=−1 2are unequal and do not dier by an integer, then we have case 2 (a). Using equation (90.6) in equation (90.12), for1= 1, and equating powers of xwe readily nd that X n1(n+1)(n)bnxn−1+1+2x 2x0 @1+X n1bnxn1 A−1 2x20 @x+X n1bnxn+11 A=0: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 406 II.A Exact Methods for ODEs Equating the coecients for dierent powers of x, we nd that b1=−2 5;bj+1=−2(j+1 ) 2j2+7j+5bj: Hence, one solution of equation (90.12) is of the form y1(x)=x 1−2 5x+4 35x2−::: : The other solution can be obtained by using 2=−1 2in equation (90.6) and equation (90.12). For this solution, we nd y2(x)=x−1=2 1−x+1 2x2−::: : The general solution of equation (90.12) is a linear combination of y1(x) andy2(x). Notes 1. This method is similar to the method of Taylor series (see page 632) but is dierent in that It allows for logarithmic terms to be present, as well as fractional powers. The recurrence relations are computed just once. The method applies only to linear ordinary dierential equa- tions. 2. The series solution in equations (90.3), (90.6), (90.7) and (90.8) will always converge in the region jxj<. 3. The series in equation (90.6) are sometimes called Frobenius series . For regular singular points, this method is sometimes called the method of Frobenius . 4. When the given linear ordinary dierential equation has an irregular singular point, then series solutions are dicult to obtain and they may be slowly convergent. Morse and Feshback [9, pages 667{674]discuss the canonical second order equations that have 1, 2, and 3 regular singular points, 1 regular and 1 irregular singular points, 1 and 2 irregular singular points. See Bender and Orszag [1, Chapter3]or Goldstein and Braun [6, Chapter 9, pages 251{279] for details. Often the WKB method (see page 642) is used to approximate the solution near an irregular singular point. 5. Understanding the nature of the singular points in an ordinary dif- ferential equation leads to an understanding of the types of boundaryconditions to be expected for that equation. For example, the ordi- nary dierential equation xy 0= 1 has the solution y=C+l o gx, whereCis an arbitrary constant. Only if y(x)i ss p e c i  e da ts o m e point other than x= 0 will it be possible to determine the constant C. The point x= 0 is a regular singular point of this equation. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 90. Series Solution407 6. This method extends easily to the general nth order homogeneous linear ordinary dierential equation at a regular singular point x0.I f the dierential equation is given by y(n)+qn−1(x) (x−x0)y(n−1)+qn−2(x) (x−x0)2y(n−2)+


+q0(x) (x−x0)ny=0; wherefq0(x);:::;qn−1(x)gare analytic at x0, then the indicial equa- tion foris given by ()n+qn−1(x0)()n−1+qn−2(x0)()n−2+


+q0(x0)()0=0; (90.13) where ()n:= ()(−1)


(−n+1 )a n d()0:= 1. If the nroots of equation (90.13) do not dier by integers, then there are nlinearly independent solutions of the form of equation (90.6). Otherwise, the forms in equation (90.7) and equation (90.8) must be generalized. See Bender and Orszag [1, Chapter 3] for details. 7. Series solutions can also be used to nd the solutions of partial dierential equations (see Collatz [3, pages 222{226 and 419{422] or Garabedian [5, Chapter 1, pages 1{17]), or to approximate the solution of nonlinear dierential equations, see Leavitt [8]. 8. Della Dora and Tournier [4] describe a computer package that will symbolically produce the series for singular points. The computer language Macsyma has the function SERIES that will compute the series expansion of a second order ordinary dierential equation. Program 90.1 shows a terminal session in which Airy’sequation (y xx+xy= 0) was input and the power series representation of the solution was obtained. Note that the function fff(n,i) is dened to be fff(n,i) =(n)i=n(n−1)


(n−i+1) in the Macsyma manual and that %k1and%k1are arbitrary constants that appear in the general solution. 9. When all of the singular points in an ordinary dierential equation are regular, then the equation is said to be of Fuchs’s type. A second order Fuchsian equation with 3 regular singular points can betransformed by a linear fractional transformation into the Riemann dierential equation: y 00+A1 x+A2 x−1 +A3 x2+A4 (x−1)2+A5 x(x−1) =0; where thefAigare constants. This equation can then be changed to a hypergeometric equation by a change of dependent variable. 10. See Boyce and DiPrima [2, Chapter 4, pages 187{256] and Ince [7, Chapter 16, pages 396{437]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 408 II.A Exact Methods for ODEs (c1) DERIVABBREV:TRUE; (c2) LOAD(SERIES)$(c3) DEPENDS(Y,X)$(c4) DIFF(Y,X,2) + X*Y = 0; (d4) y + x y = 0 xx (c5) NICEINDICES( SERIES(D4,Y,X) ); DIAGNOSIS: ORDINARY POINT inf inf ==== i 3 i ==== i 3 i\ ( -1 ) x \ ( -1 ) x (d5) y = %k2 x > ---------------- + %k1 > ---------------- /4 i/ 2 i==== fff (-, i) 9 i! ==== fff(-, i) 9 i!i=0 3 i=0 3 Program 90.1: Macsyma program to produce series solution. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [3]Collatz, L. The Numerical Treatment of Dierential Equations . Springer{ Verlag, New York, 1966. [4]Dora, J. D., and Tournier, E. Formal solutions of dierential equations in the neighborhood of singular points. In SYMSAC 81: Proceedings of the 1981 ACM Symposium on Symbolic and Algebraic Computation ,P .S .W a n g , Ed. pp. 25{29. [5]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [6]Goldstein, M. E., and Braun, W. H. Advanced Methods for the Solution of Dierential Equations . U.S. Government Printing Oce, Washington, D.C., 1973. NASA SP-316. [7]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [8]Leavitt, J. A. A power series method for solving nonlinear boundary value problems. Quart. Appl. Math. 27 , 1 (1969), 67{77. [9]Morse, P. M., and Feshback, H. Methods of Theoretical Physics . McGraw{Hill Book Company, New York, 1953. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 91. Equations Solvable for x 409 91. Equations Solvable for x Applicable to First order ordinary dierential equations that are of the rst degree in x; that is, equations of the form x=f(y;y0). Yields An exact solution, sometimes implicit. Idea Equations of the form x=f(y;y0) can be solved by nding a second equation involving x,y,a n dy0and then eliminating y0between the two equations. Procedure G i v e na ne q u a t i o no ft h ef o r m x=f y;dy dx ; (91.1) dene, as usual, p=dy dx, so that equation (91.1) may be written x=f(y;p): (91.2) Now dierentiate this with respect to yto obtain dx dy= y;p;dp dy or 1 p= y;p;dp dy (91.3) for some function . Now the ordinary dierential equation (91.3), for p=p(y), may sometimes be integrated to obtain F(y;p;C)=0; (91.4) for some function F,w h e r eCis an arbitrary constant. By elimination, the pmay sometimes be removed from equations (91.2) and (91.4) to determine y=y(x;C). In cases in which it cannot be removed, we obtain a parametric solution. Example Suppose we wish to solve the nonlinear ordinary dierential equation y=2xdy dx+ydy dx2 (91.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 410 II.A Exact Methods for ODEs fory(x). Solving equation (91.5) for xresults in x=−py 2+y 2p; (91.6) where we have used y0=p. Dierentiating equation (91.6) with respect to yand factoring results in either p=i(leading to the solution y=ix) or  1+1 p2 p+ydp dy =0: This equation may be integrated to yield py=C: (91.7) Solving equation (91.7) for pand using this in equation (91.5) results in the explicit solution 2xC−y2+C2=0: Note 1. See Piaggio [1, page 64]. Reference [1]Piaggio, H. T. H. An Elementary Treatise on Dierential Equations and Their Applications . G. Bell & Sons, Ltd, London, England, 1926. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 92. Equations Solvable for y 411 92. Equations Solvable for y Applicable to First order ordinary dierential equations that can be explicitly solved for y; i.e., equations of the form y=f(x;y0). Yields An exact solution, sometimes implicit. Idea Equations of the form y=f(x;y0) can be solved by nding a second equation involving x,y,a n dy0and then eliminating the y0term between the two equations. Procedure G i v e na ne q u a t i o no ft h ef o r m y=f x;dy dx ; (92.1) dene, as usual, p=dy dx, so that equation (92.1) may be written y=f(x;p): (92.2) Now dierentiate this with respect to xto obtain p=dy dx= x;p;dp dx ; (92.3) for some function . Now the ordinary dierential equation in (92.3), for p=p(x), may sometimes be integrated to obtain F(x;p;C)=0; (92.4) for some function F,w h e r eCis an arbitrary constant. By elimination, the pmay sometimes be removed from equations (92.2) and (92.4) to determine y=y(x;C). In cases in which it cannot be removed, we obtain a parametric solution. Example Suppose we wish to solve the nonlinear ordinary dierential equation x=ydy dx−xdy dx2 =yp−xp2(92.5) fory(x). Dierentiating equation (92.5) with respect to x, and using p=y0, results indp dx=px p2−1: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 412 II.A Exact Methods for ODEs This last equation may be integrated to determine 1 2x2=C+1 2p2−logp; (92.6) whereCis an arbitrary constant. Together, equations (92.5) and (92.6) constitute a parametric representation of the solution to equation (92.5): x=p 2C+p2−2l o gp y=x(1 +p2) p: In this representation, pis treated as a running variable. Notes 1. The technique used for Lagrange’s equation is a specialization of the present technique applied to a restricted class of equations (see page 363). 2. See Piaggio [1, page 63]. Reference [1]Piaggio, H. T. H. An Elementary Treatise on Dierential Equations and Their Applications . G. Bell & Sons, Ltd, London, England, 1926. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 93. Superposition413 93. Superposition Applicable to Linear dierential equations. Yields A set of linear dierential equations with \easier" initial conditions or boundary conditions. The sum of the solutions to these new equations willproduce the solution to the original equation. Idea By use of superposition, the solution to an inhomogeneous linear dier- ential equation may be determined in terms of simpler systems. Procedure Given a linear dierential equation with a forcing term, inhomogeneous initial conditions, or inhomogeneous boundary conditions, construct a set of equations with each equation having more homogeneous parts than theoriginal system. Solve each of these parts separately, and then combine them for the nal solution. Example Given the linear second order ordinary dierential equation L[y]=y00+a(x)y0+b(x)=f(x); (93.1) we choose y1(x)a n dy2(x) to be any linearly independent solutions of L[yi]=0 . I fC1andC2are any constants, then yc(x)=C1y1(x)+C2y2(x) is called the homogeneous solution or the complementary solution of equa- tion (93.1). We also dene yp(x) to be any solution to L[yp]=f(x). The functionyp(x) is called a particular solution . Any solution of equation (93.1) (there will be dierent solutions, de- pending on what initial conditions or boundary conditions are chosen with equation (93.1)) may be written in the form y(x)=yc(x)+yp(x); for some choice of C1andC2. Notes 1. In fluid dynamics, the influence of an obstacle in a flow can be simulated by a continuous superposition of sources. See, for instance, Homentcovschi [4]. 2. There also exist superposition principles for nonlinear equations. These are relations that allow new solutions, with arbitrary constants CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 414 II.A Exact Methods for ODEs in them, to be calculated from other solutions. For instance, if y1, y2,a n dy3are solutions of the Riccati equation (see page 392), then ywill also be solution if it satises y−y2 y−y1=Cy3−y2 y3−y1; whereCis an arbitrary constant. See Ince [5, pages 23{25] for details. 3. More generally, Lie and Scheers [7] showed that a necessary and sucient condition for a system of nrst order ordinary dierential equations to have a (nonlinear) superposition formula is that the system of equations be of the formdy dt=rX k=1fk(t)k(y) and that the vector elds Xk:=nX m=1m k(y)@ @ymgenerate a nite dimensional Lie algebra. Given a set of vector elds, Z=fX1;:::;Xrg, and a Lie bracket [;], a Lie algebra is generated by adding to Zall elements of the form [Xi;Xj]. This process is repeated with the new, potentially larger, setZuntil no new elements enter Z. The resulting Zis closed under the [ ;] operation and is a Lie algebra; it may contain a nite or an innite number of elements. 4. See also Boyce and DiPrima [1, Section 7.4 pages 352{357]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]del Olmo, M. A., Rodriguez, M. A., and Winternitz, P. Superposition formulas for rectangular matrix Riccati equations. J. Math. Physics 28 ,3 (March 1987), 530{535. [3]Harnad, J., Winternitz, P., and Anderson, R. L. Superposition principles for matrix Riccati equations. J. Math. Physics 24 , 5 (May 1983), 1062{1072. [4]Homentcovschi, D. Uniform asymptotic solutions of the potential eld around a thin oblate body of revolution. SIAM J. Appl. Math. 42 , 1 (February 1982), 44{65. [5]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [6]Jones, A. S. Quasi-additive solutions of nonlinear dierential equations. J. Austral. Math. Soc. (Series A) 42 (1987), 92{116. [7]Lie, S., and Scheffers, G. Vorlesungen uber Continuierlichen Gruppen mit geometrischen und anderen Anwendungen, Teubner, Leipzig, 1893. [8]Shnider, S., and Winternitz, P. Classication of systems of nonlinear ordinary dierential equations with superposition principles. J. Math. Physics 25, 11 (November 1984), 3155{3165. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 94. Method of Undetermined Coecients415 94. Method of Undetermined Coecients Applicable to Linear or nonlinear dierential equations, a single equation or a system. Yields An exact homogeneous solution, an exact particular solution, or both. Idea If the general form of the solution of a given dierential equation is known (or can be guessed), it can be substituted into the dening equa- tions with unknown coecients. Then the unknown coecients can bedetermined. Procedure Very often we can guess the form of a solution to a dierential equation. Or, we could just guess blindly. By having several unknown parameters in the assumed form of the solution, the solution should be able to t the dening equation(s). By forcing the guessed solution to satisfy the equation, we may be able to determine these unknown quantities. Example 1 Suppose we have the equation y00−2 x2y=7x4+3x3: (94.1) If we suspect that this equation has a power type solution for y(x), we might search for a solution in the form y(x)=axb; (94.2) whereaandbare unknowns to be determined. In this example, we presume thataandbare constants (in more complicated problems, the unknowns can be functions to be determined). We try to determine aandbby substituting our guess in the original equation for y(x). Using equation (94.2) in equation (94.1) yields axb−2(b2−b−2) = 7x4+3x3: (94.3) This equation must be satised for all values of x. There is no single set of values foraandbfor which this will be true. However, note the following: Ifb=6;a=1=4, then the left-hand side of equation (94.3) becomes 7x4. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 416 II.A Exact Methods for ODEs Ifb=5;a=1=6, then the left-hand side of equation (94.3) becomes 3x3. Ifb=−1, then the left-hand side of equation (94.3) becomes zero. Ifb= 2, then the left-hand side of equation (94.3) becomes zero. The rst two facts enable us to write the particular solution of equation (94.1) as yp(x)=1 4x6+1 6x5: The second two facts tell us that y(x)=x2andy(x)=1=xare both solutions to the homogeneous equation y00−2 x2y=0: Therefore, the complete solution to equation (94.1) is y(x)=1 4x6+1 6x5+Ax2+B x; whereAandBare arbitrary constants. Example 2 Suppose we have the partial dierential equation uxx=ut; u(0;t)=0; u(1;t)=0; u(x;0) = sinx(94.4) An appropriate guess for the form of the solution would be u(x;t)=f(t)s i nx; for some unknown function f(t). Using this guess in equation (94.4) results in the system f0+2f=0;f (0) = 1: Hence,f(t)=e−2t. Example 3 A guess for the form of the solution of the nonlinear equation ut=(uux)x(94.5) might be u(x;t)=f(t)+g(t)xp(94.6) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 94. Method of Undetermined Coecients417 for some functions f(t)a n dg(t) and some constant p. Using equation (94.6) in equation (94.5) leads to the choice p= 2. With this value, f(t) andg(t) can be determined so that u(x;t)=(C−6t)−1x2+(C−6t)−1=6: See Ames [1] for more details. Notes 1. In Table 3.1 of Boyce and DiPrima [2] is a description of general solution forms for a forced linear second order constant coecient dierential equation when the forcing function is a polynomial, a trigonometric function, an exponential function, or a combination of these terms. By utilizing this general form with unknown coecients,a solution may be obtained. 2. The reason that we suspected equation (94.1) to have a power type solution is that the homogeneous part of equation (94.1) is a Eulerequation. 3. See Boyce and DiPrima [2, Section 3.6.1, pages 146{155], Rainville and Bedient [3, pages 115{118], and Simmons [4, pages 87{90]. References [1]Ames, W. F. Ad hoc exact techniques for nonlinear partial dierential equations. In Nonlinear Partial Dierential Equations in Engineering ,W .F . Ames, Ed. Academic Press, New York, 1967. [2]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [3]Rainville, E. D., and Bedient, P. E. Elementary Dierential Equations . The MacMillan Company, New York, 1964. [4]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 418 II.A Exact Methods for ODEs 95. Variation of Parameters Applicable to Forced, linear ordinary dierential equations. Yields An integral representation of the particular solution. Idea If we know the solution to the homogeneous equation, we can write an expression for the particular solution. Procedure We illustrate the general technique for the linear ordinary dierential equation of second order. Suppose we have the equation y00+P(x)y0+Q(x)y=R(x); (95.1) and suppose that we know that fy1(x);y2(x)gare two linearly independent solutions to the homogeneous (unforced) equation y00+P(x)y0+Q(x)y=0: (95.2) That is, every solution of equation (95.2) is a linear combination of y1(x) andy2(x). We look for the particular solution of equation (95.1) in the form y(x)=v1(x)y1(x)+v2(x)y2(x); (95.3) wherev1(x)a n dv2(x) are to be determined. Dierentiating equation (95.3) with respect to xyields y0=(v1y0 1+v2y0 2)+(v0 1y1+v0 2y2): (95.4) We choose the second term in equation (95.4) to vanish, so that (v0 1y1+v0 2y2)=0: (95.5) If we now dierentiate equation (95.4) with respect to x, and use this expression (with equations (95.3), (95.4) and (95.5)) in equation (95.2)then we obtain v 0 1y0 1+v0 2y0 2=R(x): (95.6) Equations (95.5) and (95.6) constitute two algebraic equations for the two unknownsv0 1(x)a n dv0 2(x). Solving these two algebraic equations yields v0 1=−y2(x)R(x) W(y1;y2);v0 2=y1(x)R(x) W(y1;y2); (95.7) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 95. Variation of Parameters 419 whereW(y1;y2): =y1y0 2−y0 1y2is the usual Wronskian. The equations in (95.7) can be integrated and the results can be used in equation (95.3) for y(x)=−y1(x)Zy2(x)R(x) W(y1;y2)dx+y2(x)Zy1(x)R(x) W(y1;y2)dx: Example Suppose we have the equation y00+y=c s cx (95.8) to solve. The solutions to the homogeneous equation, y00+y= 0, are clearly y1(x)=s i nxandy2(x)=c o sx. Hence, we can compute the Wronskian to beW(y1;y2)=−1. Using this in equation (95.7) results in v1(x)=Z−cosxcscx −1dx= log(sinx); v2(x)=Zsinxcscx −1dx=−x: Hence, the particular solution to equation (95.8) is y(x)=s i nxlog(sinx)− xcosx. Notes 1. In Boyce and DiPrima [1, pages 156{162, 275{277, 391{393] or Finizio and Ladas [3, page 136] may be found the generalization of the analysis presented above for dierential equations of higher order. The result is Iffy1;y2;:::;yngform a fundamental system of solutions for the equation y(n)+an−1(x)y(n−1)+


+a1(x)y0+a0(x)y=0 and if the functions fu1;u2;:::;ungsatisfy the system of equa- tions y1u0 1+y2u0 2+


+ynu0 n=0; y0 1u01+y0 2u02+


+y0 nu0n=0; y00 1u01+y00 2u02+


+y00 nu0n=0; ... y(n−2) 1u0 1+y(n−2) 2u0 2+


+y(n−2) nu0n=0; y(n−1) 1u0 1+y(n−1) 2u0 2+


+y(n−1) nu0n=f(x); theny=u1y1+u2y2+


+unynis a particular solution of y(n)+an−1(x)y(n−1)+


+a1(x)y0+a0(x)y=f(x): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 420 II.A Exact Methods for ODEs 2. This last result could also have been obtained by applying variation of parameters to a system of linear rst order ordinary dierential equations. Suppose we have the system x0=P(t)x+g(t); x(t0)=x0;(95.9) where g(t) is a time-dependent vector and P(t) is a time-dependent matrix. Then the solution can be written as x(t)=Ψ (t)x0+Ψ (t)Zt t0Ψ−1(s)g(s)ds; where Ψ(t) is a fundamental matrix of the system. This means that Ψ(t) satises Ψ0=P(t)Ψ;Ψ(t0)=I; whereIis an identity matrix of appropriate size. See Boyce and DiPrima [1] or Coddington and Levinson [2, pages 87{88] for details. 3. If equation (95.9) is sti, that is P(t) has eigenvalues with widely separated positive and negative real parts (see page 770), then the fundamental matrix may become numerically singular for tt0.F o r example, the problem u0= 01 20 uhas the fundamental matrix  cosh(t−t0)1 sinh(t−t0) sinh(t−t0)c o s h(t−t0) .F o r(t−t0)16, this matrix is numerically singular even in 64-bit arithmetic. 4. See Ince [4, pages 122{123], Rainville and Bedient [5, pages 130{136], and Simmons [6, pages 90{93]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Coddington, E. A., and Levinson, N. Theory of Ordinary Dierential Equations . McGraw{Hill Book Company, New York, 1955. [3]Finizio, N., and Ladas, G. Ordinary Dierential Equations with Modern Applications . Wadsworth Publishing Company, Belmont, CA, 1982. [4]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [5]Rainville, E. D., and Bedient, P. E. Elementary Dierential Equations . The MacMillan Company, New York, 1964. [6]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 96. Vector Ordinary Dierential Equations 421 96. Vector Ordinary Dierential Equations Applicable to A system of constant coecient linear ordinary dierential equations. Yields An exact solution is obtained. Idea Very often a system of coupled equations with constant coecients can be transformed to a system of decoupled equations with constant coecients. Procedure Given a system of nordinary dierential equations with constant coef- cients, write the system as a vector ordinary dierential equation in the following form y0=Ay; y(t0)=y0; (96.1) where yis a vector of the unknowns and Ais a constant nnmatrix. Then determine the eigenvectors of A(i.e., those vectors xthat satisfy Ax=x for some non-zero value of ), and construct a diagonalizing matrix S whose columns are the eigenvectors of A. Then change variables by the transformation y=Su, so that equation (96.1) becomes ( Su)0=A(Su), or u0=S−1ASu: (96.2) By our choice of S, and assuming that Ahasnlinearly independent eigenvectors, the matrix S−1ASwill be diagonal. Hence, the equations in equation (96.2) will decouple and each row of equation (96.2) will be an ordinary dierential equation in one dependent variable ( ui). These equa- tions can be solved by the method applicable to linear constant coecient ordinary dierential equations (see page 247). Once uis known, then y can be recovered from y=Su. Example Suppose we have the system of equations dy1 dt=9y1+2y2; dy2 dt=y1+8y2: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 422 II.A Exact Methods for ODEs This system of equations can be written as a vector ordinary dierential equation as follows: d dty1 y2 =92 18y1 y2 ; (96.3) ory0=Ay,w h e r e y= y1y2TandA=[92 18]. The eigenvalues of A are=7a n d= 10 with the corresponding eigenvectors1−1Tand21T. Therefore, the diagonalizing matrix, S, whose columns are the eigenvectors of A,i sS=12 −11 . We will also need the inverse of S, which isS−1=h 1=3−2=3 1=31=3i . If we change variables by y=Su, then equation (96.3) attains the form of equation (96.2). Specically, we nd d dtu1 u2 =1=3−2=3 1=31=392 1812 −11u1 u2 ; = 70 01 0 u1 u2 :(96.4) Equation (96.4) can be expanded as du1 dt=7u1;du1 dt=1 0u2: Note that these last equations are decoupled and have constant coecients. The solutions to these equations are given by u1=Be7t;u 2=Ce10t; whereBandCare arbitrary constants. Therefore, using our original transformation, we obtain y=Su,o r y1 y2 =12 −11u1 u2 =12 −11Be7t Ce10t ; and therefore y1=Be7t+2Ce10t; y2=−Be7t+Ce10t:(96.5) The constants BandCmay be found by evaluating equation (96.5) at t=t0and using equation (96.1): y0=B1 −1 e7t0+C2 1 e10t0; = e7t02e10t0 −e7t0e10t0 B C :(96.6) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 96. Vector Ordinary Dierential Equations 423 Notes 1. Of course, some systems of equations that are not of rst order can also be reduced to the form of equation (96.1), see page 146. 2. Given the linear matrix dierential equation dR dt=B(t)R; R (t0)=I; whereRandBare square matrices, note that the determinent of R, jRjsatises djRj dt= trace(B)jRj;jRjt=t0=1: 3. For a similar technique applied to partial dierential equations, see page 449. 4. Given equation (96.1), a faster technique to nd the solution (analo- gous to the method for constant coecient linear equations on page 247) is to nd the eigenvalues figand eigenvectors fxigofAand then write the most general solution in the form y=nX i=1Cixieit; (96.7) where thefCigare unknown constants. For the example given, we can directly write the solution as x=C1x1e1t+C2x2e2t =C11 −1 e7t+C22 1 e10t; which is identical to equation (96.5). 5. This method is the same as \solving" the system in equation (96.1) by writing y=eAty0, where the exponential of a matrix is another matrix. See Coddington and Levinson [4, pages 67{77] or Moler andVan Loan [6] for details. 6. Similar results apply when Ais a function of t. The equation y 0= A(t)y,w i t h y(t0)=y0, has the solution y(t)=eB(t)y(t0), where B(t): =Rt t0A(t)dt, whenever BA=AB. 7. If the matrix Acannot be diagonalized (i.e., if Adoes not have n linearly independent eigenvectors), then Ahasgeneralized eigenvec- tors. If the vector z(m) isatises (A−iI)mz(m) i=0and (A− iI)m−1z(m) i6=0,t h e n z(m) iis called a generalized eigenvector of orderm. (Note that a generalized eigenvector of order 1 is a usual eigenvector). Given z(m) i, dene z(n−1) i =(A−iI)z(n) iforn= m;m−1;:::; 2, and dene yir=eit z(r) i+tz(r−1) i +tr−1 (r−1)!z(1) i CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 424 II.A Exact Methods for ODEs forr=1;2;:::;m . Then thefyirgwill be a collection of linearly independent vectors and all solutions of equation (96.1) will be of the formP iP rCiryir(as in equation (96.7)). See Campbell [3] for details. 8. An easier method to use when Adoes not have nlinearly independent eigenvectors is by the theorem of Leonard [5]: LetAbe a constant nnmatrix with characteristic polynomial p()=d e t (I−A)=n+cn−1n−1+


+c1+c0.T h e n eAt=x1(t)I+x2(t)A+x3(t)A2+


+xn(t)An−1,w h e r et h e xk(t), 1kn, are the solutions to the nth order scalar dierential equation x(n)+cn−1x(n−1)+


+c1x0+c0x=0 satisfying the following initial conditions: x1(0) = 1 x0 1(0) = 0 ... x(n−1) 1 (0) = 09 >>>>>= >>>>>;x 2(0) = 0 x0 2(0) = 1 ... x(n−1) 2 (0) = 09 >>>>>= >>>>>;


x n(0) = 0 x0 n(0) = 0 ... x(n−1) n (0) = 19 >>>>>= >>>>>;: 9. Nonhomogeneous systems of linear equations, of the form y 0=A(t)y+g(t); may also be analyzed (see Boyce and DiPrima [2, Chapter 7, pages 323{395]. The easiest method is a generalization of the method of variation of parameters (see page 418). Alternately, if the nonhomo-geneous system is of the form y 0=Ay+tu,w h e r eAis a constant matrix and uis an arbitrary vector, then the system may be re- written as d dsy t =Ay+tu 1 =Au 01y t ; which is now in the form of equation (96.1). 10. The solution of dX dt=AX+XB; X (0) =C; (96.8) whereA,B,C,a n dXareallmatrices is X(t)=eAtCeBt.S e e Bellman [1] for details. When AandBdepend ont,w eh a v e IfU(t) is a solution to U0=A(t)UwithU0(0) =IandV(t)i s a solution to V0=BT(t)VwithV0(0) =I, then the solution to (96.8) is given by X=UCVT. 11. For a review of eigenvalues and eigenvectors see Strang [7, Chapter 5, pages 171{230]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 96. Vector Ordinary Dierential Equations 425 References [1]Bellman, R. Introduction to Matrix Analysis . McGraw{Hill Book Company, New York, 1960. [2]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [3]Campbell, S. L. Singular Systems of Dierential Equations .P i t m a n Publishing Co., Marsheld, MA, 1980. [4]Coddington, E. A., and Levinson, N. Theory of Ordinary Dierential Equations . McGraw{Hill Book Company, New York, 1955. [5]Leonard, I. E. The matrix exponential. SIAM Review 38 , 3 (September 1996), 507{512. [6]Moler, C., and Van Loan, C. Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 20 , 4 (October 1978), 801{836. [7]Strang, G. Linear Algebra and Its Applications . Academic Press, New York, 1976. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 426 II.A Exact Methods for ODEs CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 428 II.B Exact Methods for PDEs 97. B¨ acklund Transformations Applicable to Nonlinear partial dierential equations. Yields If a B¨ acklund transformation can be found, then the solution of a non- linear partial dierential equation can be used to obtain either a dierentsolution to the same partial dierential equation, or to obtain a solution to a dierent nonlinear partial dierential equation. Idea From a solution of a nonlinear partial dierential equation, we can sometimes nd a relationship that will generate the solution of A dierent partial dierential equation (i.e., a B¨ acklund transforma- tion) The same partial dierential equation (i.e., an auto-B¨ acklund trans- formation) Procedure The rst step (which is extremely dicult) is to determine a B¨ acklund transformation between two partial dierential equations. There are var- ious methods described in the literature (see the references) that can be utilized for certain classes of equations. This transformation will utilize a solution of one of the partial dierential equations to determine a solutionto the other partial dierential equation. Example 1 Suppose we wish to determine solutions to the sine{Gordon equation uxt=s i nu: (97.1) An auto-B¨ acklund transformation is given by the pair of partial dierential equations vx=ux+2sinv+u 2 ; vt=−ut+2 sinv−u 2 :(97.2.a-b) That is, given a solution u(x;t) to equation (97.1), if v(x;t) satises equa- tion (97.2), then v(x;t) will also be a solution of equation (97.1). This may be veried by determining vxtboth by dierentiating equation (97.2.a) with CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 97. B¨ acklund Transformations 429 respect totand by dierentiating equation (97.2.b) with respect to x. This results in vxt=uxt+2s i nv−u 2 cosv+u 2 ; vxt=−uxt+2s i nv+u 2 cosv−u 2 :(97.3) Equating the two expressions in equation (97.3) results in equation (97.1), while adding them results in vxt=s i nv: Starting with the solution u(x;t) = 0 of equation (97.1), we can use the auto-B¨ acklund transformation to determine another solution; equation (97.2) becomes vx=2sinv 2;vt=2 sinv 2: This system of equations is easily solved to yield a new solution of the sine{Gordon equation tanv 4=Cexp t+x  : This solution may be used to determine another solution, and so on. Example 2 Suppose we wish to nd solutions to Burgers’s equation ut+uux=uxx: (97.4) Suppose that a solution of equation (97.4), w(x;t), is already known. If (x;t) is dened to be any solution of the following linear partial dierential equation t+w(x;t)x=xx; (97.5) andv(x;t) is dened by v(x;t)=−2x +w; (97.6) thenv(x;t) also satises Burgers’s equation. Hence, one solution of Burg- ers’s equation (i.e., w(x;t)) can be used to generate another solution. For example, a solution to equation (97.4) is clearly w(x;t) = 0. Using this in equation (97.5) results in t=xx. Each solution of this equation results in a new solution of (97.4). For example, one solution is (x;t)= e−x2=4t=p 4t. Using this in equation (97.6) results in the dierent solution to Burgers’s equation v(x;t)=x=t. This solution may be utilized to determine another solution, and the process can be repeated indenitely. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 430 II.B Exact Methods for PDEs Notes 1. The transformations in equations (97.5) and (97.6) with widentically equal to zero is the Cole{Hopf transformation (see Whitham [10, pages 97{98]) w6= 0 was rst found in Fokas [6] 2. The Cole{Hopf transformation may also be written as the set of partial dierential equations for the unknown v(x;t) vx=uv 2;vt=/parenleftbig 2ux−u2
v 4: 3. Sometimes a B¨ acklund transformation cannot be used to generate an innite sequence of new solutions; the solutions repeat after some point. See Chan and Zheng [4] for some techniques to nd new B¨acklund transformations when this occurs. 4. Sakovich [9] determines all evolution equations (equations of the form wt=f(wx;wxx;:::;wx:::x)) and all Klein{Gordon equations (equa- tions of the form wxy=f(w)) that admit a B¨ acklund autotransfor- mation (i.e., a mapping of the form =a[w], wherea[w] includes nite derivatives of w, that maps a solution of an equation to itself). Besides the linear equations, they include only the Liouville equation and the Burgers equation hierarchy. 5. The Miura transformation u=qx+q2connects the solution uof the KdV equation ut+uxxx+6uux= 0 and the solution qof the modied KdV equation qt+6q2qxqxxx=0 . 6. The transformation = log(2wxwy=w2) connects the solution of the Liouville equation xy=eto the solution wofwxy=0 . 7. An interesting linearization from Calogero [3] takes the Eckhaus equa- tion,i t+ xx+ j j4+2/parenleftbig j j2
x = 0, and makes the invertible change of variables (x;t)= (x;t)e x pZx −1j (x0;t)j2dx0 (x;t)=(x;t) 1+2Zx −1j(x0;t)j2dx0−1=2 to obtainit+xx=0 . References [1]Anderson, R. L., and Ibragimov, N. H. Lie{Backlund Transformation in Applications . SIAM, Philadelphia, PA, 1979. [2]Bluman, G. W., and Reid, G. J. Sequences of related linear PDEs. J. Math. Anal. Appl. 144 (1989), 565{585. [3]Calogero, F. Universal C-integable nonlinear partial dierential equation inN+1 dimensions. J. Math. Physics 34 , 7 (July 1993), 3197. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 97. B¨ acklund Transformations 431 [4]Chan, W. L., and Zheng, Y.-K. Backlund transformations for the Caudrey{Dodd{Gibbon{Sawada{Kotera equation and its zzzref10refzzz- modied equation. J. Math. Physics 30 , 9 (Sep 1989), 2065{2068. [5]Dodd, R. K., Eilbeck, J. C., and Morris, H. C. Solitons and Nonlinear Wave Equations . Academic Press, New York, 1982. [6]Fokas, A. Invariants, Lie{Backlund Operators and Bcklund Transforma- tions . PhD thesis, California Institute of Technology, Pasadena, CA, 1979. [7]Olver, P. J. Applications of Lie Groups to Dierential Equations . No. 107 in Graduate Texts in Mathematics. Springer{Verlag, New York, 1986. [8]Rogers, C., and Shadwick, W. F. Backlund Transformations and Their Applications . Academic Press, New York, 1982. [9]Sakovich, S. Y. On special B acklund autotransformations. J. Phys. A: Math. Gen. 24 (1991), 401{405. [10]Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 432 II.B Exact Methods for PDEs 98. Method of Characteristics Applicable to Systems of quasilinear partial dierential equations (i.e., one or more partial dierential equations linear in the rst derivatives of the dependent variables, with no higher order derivatives present). Yields If the initial data are not given along a characteristic, then an exact solution can be obtained (generally implicit). Idea A quasilinear partial dierential equation of hyperbolic type can be transformed into a set of ordinary dierential equations that dene the characteristics and a set of ordinary dierential equations that describe how the solution changes along any specic characteristic. Procedure Suppose we have the quasilinear partial dierential equation a1(x;u)ux1+a2(x;u)ux2+


+aN(x;u)uxN=b(x;u) (98.1) for the unknown u(x)=u(x1;x2;:::;xN). If we were to dierentiate u(x) with respect to the variable s, then we obtain du ds=@x1 @s ux1+@x2 @s ux2+


+@xN @s uxN: (98.2) If we dene @xk @s=ak(x;u); (98.3) fork=1;2;:::;N , then using equation (98.1) in equation (98.2) results in du ds=b(x;u): (98.4) To determine the solution of the partial dierential equation (98.1), we need to integrate the ordinary dierential equations given in equation (98.3) and (98.4). (Equation (98.3) may look like a partial dierential equation,but it is an ordinary dierential equation with respect to s.) To perform this integration, initial conditions are needed in sfor thefx kgand foru. Generally, the initial data for equation (98.1) will be given in the form g(x;u)=0; (98.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 98. Method of Characteristics 433 on some manifold in xspace. We identify this surface as correspond- ing tos= 0. If we think of xanduas depending on the variables fs;t1;t2;:::;tN−1g, then the variables ft1;t2;:::;tN−1gc a nb eu s e dt o parametrize the initial data in equation (98.5) (the examples will makethis clear). That is, x 1(s=0 )=h1(t1;t2;:::;tN−1); x2(s=0 )=h2(t1;t2;:::;tN−1); ... xN(s=0 )=hN(t1;t2;:::;tN−1); u(s=0 )=v(t1;t2;:::;tN−1):(98.6) Hence equation (98.6) supplies the initial conditions for the dierential equations in (98.3) and (98.4). After xanduare determined from equations (98.3), (98.4), and (98.6), then an implicit solution will have been obtained. If the fs;t1;t2;:::;tN−1g can be analytically eliminated, then an explicit solution will be obtained. It is not always possible to perform this elimination analytically. The physical picture of the construction of the solution is shown in gure 98.1. The solution uis determined by the ordinary dierential equation (98.4) along each characteristic. A characteristic is specied by the ftig values. The parameter srepresents scaled distance along a characteristic. When two characteristics cross, a shock is formed. Note that a shock cannot form if the equation (98.1) is linear; that is, eachfaigis only a function of xand not ofu. At a shock, extra conditions are required. (See Landau and Lifshitz [2, Chapter 9, pages 310{346]) for a discussion of the Rankine{Hugoniot adiabatic , which is used in fluid mechanics.) Example 1 Suppose we want to solve the quasilinear partial dierential equation ux+x2uy=−yu; u=f(y)o nx=0;(98.7.a-b) wheref(y) is a given function. Forming du=ds we have du ds=@x @s ux+@y @s uy: (98.8) Comparing equation (98.8) to equation (98.7), we take @x @s=1;@y @s=x2;du ds=−yu: (98.9.a-c) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 434 II.B Exact Methods for PDEs Figure 98.1: Depiction of the characteristics for a quasilinear equation. The initial data in equation (98.7.b) can be written parametrically as x(s=0 )=0; y(s=0 )=t1; u(s=0 )=f(t1):(98.10.a-c) That is, when s=0 ,w eh a v e u=f(y)a n dx= 0. The solution of (98.9.a) with (98.10.a) is x(s;t1)=s: (98.11) Therefore, equations (98.9.b) and (98.10.b) can be written as @y @s=s2;y (s=0 )=t1; with the solution y(s;t1)=s3 3+t1: (98.12) Finally, the equation for u(from equations (98.9.c), (98.10.c), and (98.12)) becomes du ds=−s3 3+t1 u; u (s=0 )=f(t1); with the solution u(s;t1)=f(t1)e x p −s4 12−st1 : (98.13) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 98. Method of Characteristics 435 Equations (98.11), (98.12), and (98.13) constitute an implicit solution of equation (98.7). In this case, it is possible to analytically eliminate the sandt1variables to obtain an explicit solution. From equation (98.11) we obtain s=x. Using this in equation (98.12) results in t1=y−x3 3. Using these two values in equation (98.13) results in the explicit solution u(x;y)=f y−x3 3 expx4 4−xy : Example 2 If we have the quasilinear partial dierential equation in three depen- dent variables ux+uy+xyuz=u2; u=x2ony=z;(98.14) then we can write equations (98.3), (98.4), and (98.6) as @x @s=1;@y @s=1;@z @s=xy; du ds=u2; x(s=0 )=t1;y(s=0 )=t2;z(s=0 )=t2;u(s=0 )=t2 1: The equations for xandycan be integrated to yield x=s+t1;y =s+t2: (98.15) Using these values for xandy, the equation for zbecomes @z @s=(s+t2)(s+t1); which can be integrated to yield z=s3 3+s2 2(t2+t1)+st2t1+t2: (98.16) The equation for ucan also be integrated to obtain u=t2 1 1−st2 1: (98.17) The equations in (98.15) (98.16), and (98.17) constitute an implicit solution to equation (98.14). The variables t1andt2can be eliminated to yield u=(x−s)2 1−s(x−s)2; z=−4s3 3−s2 2(x+y)+s(xy+1 )+y:(98.18.a-b) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 436 II.B Exact Methods for PDEs To actually evaluate u(x;y;z ) at some given value of x,y,a n dzrequires two steps. First, equation (98.18.b) must be solved for s, and then this value is utilized in equation (98.18.a). Alternatively, the method of resultants (see page 50) could be used to obtain a single polynomial equation in terms of x,y,z,a n du, alone. This results in an equation with many terms; the implicit solution given by equation (98.18) is more useful and more compact. Notes 1. This technique extends naturally to systems of partial dierential equations, with virtually no increase in complexity. This allows a single partial dierential equation of higher order (and hyperbolic type) to be analyzed. For example, the wave equation uxx=uttcan be written, in the variables fv:=ux,w:=utg, as the system of two quasilinear equations fvt=wx,wt=vxg: 2. The general quasilinear system of Nequations for the Nunknowns u=(u1;u2;:::;un) in the two independent variables fx;tghas the form NX j=1Aij(u;x)@uj @t+NX j=1aij(u;x)@uj @x+bi=0; fori=1;2;:::;N . This equation will be hyperbolic (and hence solvable by the method of characteristics) if there exist Nlinearly independent real-valued N-dimensional vectors fv(1),v(2);:::;v(N)g andNnon-zero real-valued two-dimensional vectors f(k);(k)gsuch that NX i;j=1v(k) ih Aij(k)−aij(k)i =0; fork=1;:::;N . See Whitham [4, Chapter 5, pages 113{142] for details and several examples using this formalism. 3. Referring to equation (98.1), it turns out that discontinuities in ru can propagate along characteristics, but discontinuities in ucannot. In fact, ifusatises a second order linear hyperbolic partial dieren- tial equation in xandy, and iffu;ux;uy,uxx,uxygare all continuous across a curve Cbutuyysuers a jump upon crossing C,t h e nCis necessarily a characteristic of the partial dierential equation. 4. Eliminating the fs;tgvariables at the end of the calculation will be possible, in principle, whenever the Jacobian of the transformation does not vanish; that is,@(u;x1;x2;:::) @(s;t1;t2;:::)6=0 . 5. An equivalent way of writing equation (98.3) is the form dx1 a1=dx2 a2=


=dxN aN; which are called the subsidiary equations . When one or more of the akare zero, this equation looks peculiar, but it should be interpreted CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 98. Method of Characteristics 437 to be the same as equation (98.3). This form is used in place of equation (98.3) in many older texts. This formulation has been used occasionally in this book. 6. See Farlow [1, Lesson 27, pages 205{212], Moon and Spencer [3, pages 27{29], and Zauderer [5, Chapter 3, pages 78{121]. References [1]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [2]Landau, L. D., and Lifshitz, E. M. Fluid Mechanics . Pergamon Press, New York, 1959. [3]Moon, P., and Spencer, D. E. Partial Dierential Equations .D .C .H e a t h and Co., Lexington, MA, 1969. [4]Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. [5]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 438 II.B Exact Methods for PDEs 99. Characteristic Strip Equations Applicable to Some partial dierential equations in two indepen- dent variables. Yields When the technique is applicable, an implicit solution. Idea This method appears to be a generalization of the method of charac- teristics, but it can in fact be derived from that method. The formulae presented here are handy to use directly. Procedure Given the partial dierential equation F(x;y;u;p;q )=0; (99.1) wherep=ux,q=uy, we search for a solution u=u(x;y). The technique is to solve the system of \strip equations" given by @x @s=Fp;@p @s=−Fx−pFu; @y @s=Fq;@q @s=−Fy−qFu; @u @s=pFp+qFq;(99.2) where we now consider fx;y;p;q;ugto all be functions of the two variables fs;tg. The equations in equation (99.2) are also called Charpit’s equations . The \initial" values for equation (99.2) (corresponding to s= 0) are given in terms of the other independent variable t. It will be possible to give initial values to all of the terms in equation (99.2) because the original equation (99.1) will have data with it that can be parameterized in terms oft. After we have determined fx;y;ugas functions offs;tg,w em u s ts o l v e the equations implicitly to obtain the nal solution in the form u=u(x;y). Example Suppose we have the nonlinear partial dierential equation uxuy−u=0; (99.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 99. Characteristic Strip Equations 439 with the initial data u=y2onx=0: (99.4) By comparing equation (99.3) with equation (99.1), we nd that F=pq−u. Hence, the equations in equation (99.2) can be written as @x @s=q;@p @s=p; @y @s=p;@q @s=q; @u @s=2pq:(99.5.a-e) The initial conditions for equation (99.5) are given by parameterizing equa- tion (99.4) in terms of the dummy variable t. One such parameterization (there are always innitely many) is x=0;y =t; u =t2: (99.6) To determine the initial conditions for pandq, we utilize the chain rule @u @t=pdx dt+qdy dt; which can be evaluated at s= 0 (using equation (99.6)) to yield 2t=p(0;t)
0+q(0;t)
1 orq(0;t)=2t. The original equation, (99.4), can be evaluated at s=0t o determine that p(0;t)=u(0;t)=q(0;t)=t=2. Now that we have the initial conditions for all ve variables appearing in equation (99.5), we can ndthe solution. Equations (99.5.b) and (99.5.d) can be integrated directly to yield p=1 2tes;q =2tes: Substituting these expressions in equations (99.5.a), (99.5.c), and (99.5.e) and integrating results in x=2t(es−1); y=1 2t(es+1 ); u=t2e2s:(99.7.a-c) Equations (99.7.a) and (99.7.b) can be inverted to produce sandtas functions of xandy: es=4y+x 4y−x;t =4y−x 4: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 440 II.B Exact Methods for PDEs Using these relations in equation (99.7.c) yields the nal answer u(x;t)=(x+4y)2 16: Notes 1. This method is sometimes called the Lagrange{Charpit method. 2. Frequently, inverting the variables at the end (i.e., nding s=s(x;y) andt=t(x;y)) is the only step that cannot be carried out analyti- cally. 3. The variable sreally species a characteristic, whereas trepresents distance along any single characteristic. 4. This technique works, as the example shows, even when the original equation is not quasilinear. That is, the method of characteristicscould not have been applied directly to equation (99.3). 5. See also Copson [1, pages 5{9], Garabedian [2, pages 24{31], Sneddon [3, pages 61{66], and Zauderer [4, pages 56{68]. References [1]C o p s o n ,E .T . Partial Dierential Equations . Cambridge University Press, New York, 1975. [2]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [3]Sneddon, I. N. Elements of Partial Dierential Equations . McGraw{Hill Book Company, New York, 1957. [4]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 100. Conformal Mappings 441 100. Conformal Mappings Applicable to Laplace’s equation ( r2u= 0) in two dimensions. Yields A reformulation of the original problem. Idea Laplace’s equation in two dimensions with a given boundary can be transformed to Laplace’s equation with a dierent boundary by a conformal map. The idea is to choose the conformal map in such a way that the new boundary makes the problem easy to solve. Procedure Given Laplace’s equation in the variables fx;yg(i.e.,r2u=uxx+uyy= 0), we dene the complex variable z=x+iy,w h e r ei=p−1. All of the boundaries of the original problem can now be described by values of z. Any analytic transformation between two complex variables, say = F(z), for which d=dz is never zero, is said to be conformal . It turns out that Laplace’s equation is invariant under a conformal map. That is, if =+i=F(z),uxx+uyy=0 ,a n dF(z) is a conformal map, then u+u=0 . In the new variables, f;g, the boundary might be very simple. If so, then Laplace’s equation can be solved in this new domain. Then the solution of Laplace’s equation in the original domain can be found by the change of variables induced by the conformal map. A commonly used conformal map is the Schwartz{Christoel transfor- mation . This maps a closed polygonal gure (with nvertices) into a half plane. The mapping is given by the solution of dz d=C(−1)1=−1(−2)2=−1


(−n)n=−1 (100.1) for appropriatef1;2;:::;ngandf1;2;:::;ng.T h efigare the interior angles of the polygon, and the figare the (complex valued) positions of the polygon’s vertices. After the dierential equation (100.1) is formulated, it must be solved. The unknown constant C, as well as the arbitrary constant resulting from the integration, will be determined when the figare prescribed. The resulting function =F(z) is the conformal map that maps the interior of the given polygonal gure into the half plane. See Trefethen [11] for a numerical implementation. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 442 II.B Exact Methods for PDEsx /= /, /1 x /=/1 /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /./././././. /././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./. /././././././././././././././././././. yzu /=/0 u /=/1 u /=/0x /././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /./././././. /././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./. /././././././././././././././././././. /#11/#10/#18 u /=/1u /=/0 Figure 100.1: The original domain for Laplace’s equation and the domain after a conformal mapping has been applied. Example 1 Suppose we have Laplace’s equation ( uxx+uyy= 0) to solve in the half planeH=f−1<x<1,0<y<1gwith the boundary conditions u(x;0) =( 0f o rjxj>1; 1f o rjxj1: Under the mapping =+i=F(z)=l o gz−1 z+1 =l o gx+iy−1 x+iy+1 ; (100.2) the half planeHis mapped into a strip of height in the (,) plane. See gure 100.1 for pictures of the two geometrical regions involved. In the (,) plane the boundary conditions become u(;0) = 0; u(;)=1: The solution to Laplace’s equation in this domain is simply u(;)==. To transform back to ( x,y) coordinates, the transformation in equation (100.2) must be inverted. After some algebra it can be shown that = argz−1 z+1 =t a n−12y x2+y2−1 ; so that u(x;y)=1 tan−12y x2+y2−1 : CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 100. Conformal Mappings 443/./././././././././././././. /././. /./././././././././././././././. /./././././././././././././. /././. /./././././././././././././././. /./././././././././././././. /././. /./././././././././././././././. /./././././././././././././. /././. /./././././././././././././././. /./././././././././././././. /././. /./././././././././././././././. /./././././././././././././. /././. /./././././././././././././././. /./././././././././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./. /././././././././././././././././././. yzx w /=/0w /=/0 /#0F /#0Fz/1 /= iaz/2 /=/0 w /=/1/#10/1 /= /, /1 /#10/2 /=/1 /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /./././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././. /./././././. /././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./. /./././././././././././././././././././#11/#10/#18w /=/0 w /=/1 w /=/0 /#0F /#0FFigure 100.2: The original domain for Laplace’s equation and the domain after the Schwartz{Christoel transformation has been applied. Example 2 Suppose we have Laplace’s equation ( r2w= 0) in the channel open on the right (see gure 100.2), with the boundary conditions w(x;0) = 0 for 0x<1 w(x;a)=0 f o r0x<1 w(0;y)=1 f o r0y1: The polygon in which this problem is being solved has vertices at z1=ia andz2= 0, with the corresponding interior angles 1=2==2. Using the Schwartz{Christoel transformation, we choose the vertices in the z plane to map to the vertices 1=−1a n d2=1i nt h e plane. The dierential equation (100.1) becomes dz d=C(+1 )1=2(−1)1=2 with the solution z=Ccosh−1+D,w h e r eDis an arbitrary constant. To determine the constants CandD, we must enforce that the vertices in thezplane mapped to the vertices in the plane. We have the two simultaneous equations: z1=ia=Ccosh−1(1)+D=Ccosh−1(−1) +D=Ci+D; z2=0=Ccosh−1(2)+D=Ccosh−1(1) +D=D; with the solution fD=0 ,C=a=g. Hence, the desired conformal mapping is =c o s h/parenleftbigz a
. The problem in the domain is now identical to the problem solved in Example 1. Notes 1. Conformal mappings are often used in hydrodynamics and electro- statics because, under a conformal mapping, lines of flow and equipo- tential lines are mapped into lines of flow and equipotential lines. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 444 II.B Exact Methods for PDEs 2. Conformal mappings are often used to obtain an orthogonal coordi- nate system inside of a two-dimensional body. This may be used, for instance, when a grid is required on which the solution to a partial dierential equation will be approximated numerically. 3. The mapping used in this method need not be conformal everywhere; it needs to be conformal only in the domain in which Laplace’s equation is being solved. (Very few maps are conformal everywhere.) 4. The Joukowski transformation ,g i v e nb y=z+a2=z, maps an ellipse into a circle or a circle into a strip. 5. Algebraic mappings, given by =z=,w i t h> 0, map a corner with angle to a corner with angle = . For instance, if =2, then a quarter plane ( ==2) is mapped to a half plane. 6. Numerical implementation of the Schwartz{Christoel transforma- tion can fail on some seemingly very simple polygons. Mapping a rectangle with an aspect ration of 20 to 1, or an other region with a similar degree of elongation, onto a half-plane may cause problems because the points in the transformed plane will be veryclose together. (This is known as the \crowding phenomenon.") 7. The Schwartz{Christoel transformation can also be used for doubly connected domains, see Iyanaga and Kawada [6, page 1156]. 8. Even when an analytic conformal map cannot be found, there are fast numerical techniques for nding an approximate conformal map. Riemann’s mapping theorem states that all bounded simply con- nected plane regions can be conformally mapped onto the unit disk, and all bounded doubly connected plane regions can be conformallymapped onto an annulus. Using Poisson’s formula (see page 478) exact solutions can be written down for these two geometries. See Fornberg [5] or Trefethen [12] for details. 9. Kober [8] has a large collection of conformal mappings, with the geo- metric regions in both the ( x;y)a n d(;) planes clearly illustrated. 10. Seymour [10] describes a computer package that permits real-time manipulation and display of conformal mappings of one complex plane onto another. 11. Ifr 2 x;yrepresents the Laplacian in fx;ygspace, then under the con- formal mapping =F(z) the operatorr2 x;yis mapped to the opera- torjF0(z)j2r2 ;. Hence, the biharmonic equation r4u:=r2 x;yr2x;yu= 0 becomesjF0(z)j2r2 ; jF0(z)j2r2 ; u=0 . 12. See also Farlow [3, Lesson 47, pages 379{388], Kantorovich and Krylov [7, Chapters 5 and 6, pages 358{615], and Levinson and Redheer [9, Chapter 5, pages 259{332]. References [1]Delillo, T. K. The accuracy of numerical conformal mapping methods: A survey of examples and results. SIAM J. Numer. Anal. 31 , 3 (June 1994), CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 100. Conformal Mappings 445 788{812. [2]D e L i l l o ,T .K . ,a n dE l c r a t ,A .R . A comparison of some numerical conformal mapping methods for exterior regions. SIAM J. Sci. Stat. Comput. 12, 2 (March 1991), 399{422. [3]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [4]Floryan, J. M., and Zemach, C. Schwarz{Christoel mappings: A general approach. J. Comput. Physics 72 (1987), 347{371. [5]Fornberg, B. A numerical method for conformal mapping. SIAM J. Sci. Stat. Comput. 1 (1980), 386{400. [6]Iyanaga, S., and Kawada, Y. Encyclopedic Dictionary of Mathematics . MIT Press, Cambridge, MA, 1980. [7]Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis . Interscience Publishers, Inc., New York, 1958. [8]Kober, H. Dictionary of Conformal Representations . Dover Publications, Inc., New York, 1952. [9]Levinson, N., and Redheffer, R. M. Complex Variables . Holden{Day, Inc., San Francisco, CA, 1979. [10]Seymour, H. R. Conform: A conformal mapping system. In SYMSAC ’86 (Proceedings of the 1986 ACM Symposium on Symbolic and Algebraic Computation) , B. W. Char, Ed. ACM, New York, 1986, pp. 163{168. [11]Trefethen, L. N. Numerical computation of the Schwarz{Christoel transformation. SIAM J. Sci. Stat. Comput. 1 , 1 (March 1980), 82{102. [12]Trefethen, L. N. Numerical Conformal Mapping . North{Holland Publishing Co., New York, 1986. [13]Walker, M. The Schwarz{Christoel Transformation and Its Applications {AS i m p l eE x p o s i t i o n . Dover Publications, Inc., New York, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 446 II.B Exact Methods for PDEs 101. Method of Descent Applicable to Partial dierential equations (most often, wave equations). Yields An exact solution. Idea For some partial dierential equations (in particular, some wave equa- tions) odd dimensional problems are \easier" than even dimensional prob-lems. Hence it is reasonable, when given a 2 n-dimensional problem, to instead solve a 2 n+ 1-dimensional problem and then \come down one dimension." Procedure Given a partial dierential equation in ndimensions for the quantity u(x)=u(x1;x2;:::;xn) L[u]=0; it might be easier to solve the n+ 1-dimensional problem L[v]+H[v]=0; forv(x;z)=v(x1;x2;:::;xn;z), whereH[
] is a dierential operator with respect toz. Then, when v(x;z)i sk n o w n , u(x) can be obtained by either (1) an appropriate integral over zor (2) taking vto be independent of z. Example Suppose we are given the two-dimensional wave equation utt=c2(uxx+uyy); (101.1) with the initial conditions u(0;x)=f(x);ut(0;x)=g(x); (101.2) where x=(x;y). We might choose to instead solve the three-dimensional wave equation vtt=c2(vxx+vyy+vzz); with the initial conditions v(0;x;z)=f(x);vt(0;x;z)=g(x): The three-dimensional wave equation has the well-known solution (see page 501) v(t;x;z)=ctM[g]+@ @t ctM[f] ; (101.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 101. Method of Descent 447 whereM[
] is a functional dened to be the average value of its argument on a circle of radius ct;t h a ti s , M[h(x;y;z )] :=1 4c2t2Z S(t)hdS =1 4c2t2Z 0Z2 0h(x+ctsincos;y+ctsinsin;z+ctcos) sindd;(101.4) whereS(t) is the surface of a sphere with origin at ( x;y;z ) and radius ct. To solve the two-dimensional wave equation (101.1), we merely utilize the fact that fandgare independent of the variable z. Performing some algebraic manipulations, equation (101.4) becomes M[h(x;y)] =1 2ctZZ (t)h(;)ddp c2t2−(x−)2−(y−)2; (101.5) where(t) is the interior of the circle: ( x−)2+(y−)2=c2t2. Using equation (101.5) in equation (101.3) results in the solution to equations (101.1) and (101.2). Notes 1. This method is also called Hadamard’s method of descent . 2. If the descent step was applied once again, the solution of the one- dimensional wave equation, wtt=c2wxx, could be obtained from equations (101.3) and (101.5). 3. Note that a line source, in three dimensions, might be viewed as a point source in two dimensions. 4. One reason that odd space dimensional problems are sometimes easier than even dimensional problems is Huygen’s principle. Huygen’s prin- ciple (see Chester [1, pages 154{156] or Garabedian [4, Section 6.3, pages 204{210]) states that the wave equation in an odd number ofspace dimensions depends only on the initial data (and its derivatives) on the perimeter of the domain of dependence. See the section on exact solutions of the wave equation (on page 501). 5. See also Copson [2, pages 95{96], Farlow [3, pages 187{188], Whitham [5, pages 219{235], and Zauderer [6, pages 226{232]. References [1]Chester, C. R. Techniques in Partial Dierential Equations . McGraw{Hill Book Company, New York, 1970. [2]C o p s o n ,E .T . Partial Dierential Equations . Cambridge University Press, New York, 1975. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 448 II.B Exact Methods for PDEs [3]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [4]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [5]Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. [6]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 102. Diagonalization of a Linear System of PDEs 449 102. Diagonalization of a Linear System of PDEs Applicable to A linear system of partial dierential equations in two independent variables, of the form ut+Aux=0 ,w h e r e Ais a constant matrix. Yields A set of uncoupled equations. Idea By diagonalizing the coecient matrix, the equations can be uncoupled and then solved. Procedure Given the linear system of dierential equations ut+Aux=0; (102.1) we change the dependent variables to decouple the system. If the ma- trixAisnnand has the eigenvectors fv1;v2;:::;vng(which we as- sume to be linearly independent), then we dene the matrix SbyS= v1v2:::vn . Changing variables in equation (102.1) by u=Sw results inSwt+ASwx=0 ,o r wt+wx=0; (102.2) where  = S−1ASis a diagonal matrix. The equations in (102.2) are now decoupled and can be solved separately for fw1(x;t);w2(x;t);:::;wn(x;t)g. After they have been found, umay be determined from u=Sw. Example Given the system of linear partial dierential equations in two indepen- dent variables @u1 @t+9@u1 @x+2@u2 @x=0; @u2 @t+@u1 @x+8@u2 @x=0;(102.3) we dene the vector u=[u1u2] and the matrix A=[92 18] so that equation (102.3) may be written in the form of equation (102.1). The eigenvalues of Aare=7a n d= 10 with the corresponding eigenvectors: v1=1−1Tandv2=21T. Hence, the matrix Sis given byS=21 1−1 , which has the inverse S−1=h 1=31=3 1=3−2=3i . Making the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 450 II.B Exact Methods for PDEs change of variables u=Swturns equation (102.1) into equation (102.2) with  dened by =S−1AS; = 1=31=3 1=3−2=3 92 18 21 1−1 ; =10 0 07 : The equations in (102.2) can then be separated to obtain @w1 @t+1 0@w1 @x=0; @w2 @t+7@w2 @x=0: These equations have the solution w1(x;t)=f(x−10t); w2(x;t)=g(x−7t); wherefandgare arbitrary functions of their arguments. Knowing wwe can determine u=Swto be u1(x;t)=2w1(x;t)+w2(x;t)=2f(x−10t)+g(x−7t); u2(x;t)=w1(x;t)−w2(x;t)=f(x−10t)−g(x−7t):(102.4) Knowing the general form of the solution, any initial conditions for u1(x;t) andu2(x;t) could be utilized. For example, if we had u1(x;0) = 3 sin 2x; u2(x;0) = 0;(102.5) then utilizing equation (102.4) in equation (102.5) produces 2f(x)+g(x)=3s i n2x; f(x)−g(x)=0; and sof(z)=g(z)=s i n2zand the nal solution can be written u1(x;t) = 2 sin(2x−20t) + sin(2x−14t); u2(x;t) = sin(2x−20t)−sin(2x−14t): Note 1. See Farlow [1, Lesson 29, pages 223{231] Reference [1]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 103. Duhamel’s Principle 451 103. Duhamel’s Principle Applicable to Linear parabolic and hyperbolic partial dierential equations. Yields An integral representation in terms of the solution of a more tractable partial dierential equation. Idea To solve a parabolic partial dierential equation with a time-varying source function and time-varying boundary conditions, only a parabolicpartial dierential equation with a constant source term and constant boundary conditions needs to be solved. Procedure Suppose we have the parabolic partial dierential equation for u(x;t) @ @tu(x;t)=L[u(x;t)] +F(x;t); u(y;t)=G(y;t); fort>0; u(x;0) =H(x);(103.1) whereL[
] is an elliptic operator in xandydenotes a point on the boundary. Note that equation (103.1) has a time-dependent source function F(x;t) and time-dependent surface conditions G(y;t). Instead of solving equation (103.1) for u(x;t), we choose to solve the parabolic partial dierential equation @ @tv(x;t;)=L[v(x;t;)] +F(x;); v(y;t;)=G(y;); fort>0; v(x;0;)=H(x);(103.2) forv(x;t;). Note that the variable of integration in equation (103.2) ist, while the source term and the surface conditions depend upon the parameter. Hence, the equation for v(x;t;) has (eectively) a constant source term and constant surface conditions. Thus, it should be easier to determinev(x;t;) than it was to determine u(x;t). Knowing the solution of equation (103.2), the solution to equation (103.1) can be written as u(x;t)=@ @tZt 0v(x;t−;)d: (103.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 452 II.B Exact Methods for PDEs This is easily derived from manipulations of the Laplace transforms of equation (103.1) and equation (103.2). See any of the references for details. Example Suppose we want to solve the equations describing the temperature of an initially cool, insulated rod with a temperature f(t) specied at one end ut=uxx; for 0<x< 1;0<t<1; u(0;t)=0; for 0<t<1; u(1;t)=f(t); for 0<t<1; u(x;0) = 0; for 0x1:(103.4) Instead of solving equation (103.4) for u(x;t)w es o l v e vt=vxx; for 0<x< 1;0<t<1; v(0;t;)=0; for 0<t<1; v(1;t;)=f(); for 0<t<1; v(x;0;)=0; for 0x1;(103.5) forv(x;t; ). By separation of variables (see page 487), the solution of equation (103.5) is found to be v(x;t; )=f()" x+2 1X n=1(−1)n ne−n22tsinnx# ; which, for notational convenience, we choose to write as v(x;t; )=f()g(x;t). Using equation (103.3), the solution for u(x;t) can then be written as u(x;t)=@ @tZt 0v(x;t−;)d =@ @tZt 0f()g(x;t−)d =@ @tZt 0f(t−T)g(x;T)dT =f(0)g(x;t)+Zt 0f0(t−T)g(x;T)dT;(103.6) where we dened T=t−in the above. If, for example, f(t)=e−t,t h e n equation (103.6) may be simplied to yield u(x;t)=x−e−t−1−2 1X n=1(−1)nsinnx n(1−n22)n n22e−n22t−e−to : CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 103. Duhamel’s Principle 453 Notes 1. The procedure for hyperbolic partial dierential equations is anal- ogous to the procedure for parabolic partial dierential equations. Consider, for example, the hyperbolic equation utt+L[u]=b(x;t); (whereL[
] is uniformly elliptic) with the boundary conditions u(x;0) =ut(x;0) = 0: Ifv(x;t; ) is dened to be the solution of vtt+L[v]=0; fort>; v(x;; )=0; vt(x;; )=b(x;); then we have u(x;t)=Rt 0v(x;t; )d. Using this formulation, it can be shown that the solution to utt−c2r2u=F(x;y;z;t ), is given by u(x;y;z;t )=1 4cZZZ 2+2+2c2t2F(;;;t−r=c) rddd; (103.7) wherer2=(x−)2+(y−)2+(z−)2. The integrand in equation (103.7) is called the retarded potential . 2. See Chester [1, pages 156{158], Courant and Hilbert [2, Volume 2, pages 202{204], Farlow [3, Lesson 14, pages 106{111], Sneddon [4, pages 278{282], and Zauderer [5, pages 159{165]. References [1]Chester, C. R. Techniques in Partial Dierential Equations . McGraw{Hill Book Company, New York, 1970. [2]Courant, R., and Hilbert, D. Methods of Mathematical Physics . Interscience Publishers, Inc., New York, 1953. [3]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [4]Sneddon, I. N. Elements of Partial Dierential Equations . McGraw{Hill Book Company, New York, 1957. [5]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 454 II.B Exact Methods for PDEs 104. Exact Equations Applicable to Quasilinear partial dierential equations. Yields An exact solution. Idea Some quasilinear partial dierential equations can be integrated di- rectly. Procedure Consider the quasilinear partial dierential equation M(x;y;u )ux=N(x;y;u )uy: (104.1) If this equation satises the exactness condition Mx=Ny, then an implicit solution to equation (104.1) will be given by (x;y;u ) = 0, where M=y;N =x: (104.2.a-b) To determine the function , integrate equation (104.2.a) to obtain =Z Mdy +g(x;u): (104.3) Then, using equation (104.2.b) we have Z Mxdy+gx(x;u)=N or (solving for gand integrating) g(x;u)=Z N−Z Mxdy dx+h(u); (104.4) whereh(u) is an arbitrary function. Using equation (104.4) in equation (104.3) results in the nal solution. Example Consider the equation yux=xuuy; for whichM=yandN=xu. This equation is exact because Mx=0= Ny. From equation (104.3), we have =Z Mdy +g(x;u)=1 2y2+g(x;u): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 104. Exact Equations 455 From equation (104.2.b), we have x=gx=N=xu,o rg=1 2x2u+ h(u). This leads to the general implicit solution: =1 2/parenleftbig y2+x2u
+h(u)=0: Choosing, for example, h(u)=1 2(au+b) results in the explicit solution u(x;y)=−b+y2 a+x2: Note 1. The above example is from Benton [1]. Reference [1]B e n t o n ,J r . ,S .H . The Hamilton{Jacobi Equation . Academic Press, New York, 1977. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 456 II.B Exact Methods for PDEs 105. Hodograph Transformation Applicable to Quasilinear partial dierential equations, a single equation, or a system of equations. Yields A new formulation of the original equations. Idea In a partial dierential equation, it may be easier to solve the equation with the dependent and independent variables switched. Procedure This procedure works on a quasilinear equation or a system of such equations. That is, every term of each equation must have one and only one rst derivative term, and there can be no higher order derivative termsin the equations. Consider the case of two dependent variables ( u;v) in two independent variables (x;y). Suppose L[u;v] = 0 represents the equation(s) to be solved foru(x;y)a n dv(x;y). This equation is transformed to the \hodograph" plane by writing x=x(u;v)a n dy=y(u;v) and transforming L[u;v]=0 into a new equation H[x;y] = 0. In this new equation, xandyare treated as the dependent variables. The solution obtained will, in general, be implicit. After the solution is obtained in the hodograph plane, the transformation must be checked to ensure that it is not singular. Example 1 Suppose we have a pair of nonlinear equations arising from gas dynamics (from Whitham [9, page 182]) vy+uvx+bvux=0; uy+uux+1 bvvx=0;(105.1) wherebis a constant. Because the equations in equation (105.1) are quasi- linear, the method of characteristics can be used to solve them. However, it is dicult to use that method directly. The hodograph transformation can be used on equation (105.1) by invertingu(x;y),v(x;y) to nd (see, e.g., Kaplan [6, pages 132{135], on how to change variables in this manner) xu=−vy=J; x v=uy=J; yu=vx=J; y v=−ux=J(105.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 105. Hodograph Transformation 457 whereJis the Jacobian of the transformation, J=uyvx−vyux. Using equation (105.2) in equation (105.1) results in the equations xu−uyu+bvyv=0; xv−uyv+1 bvyu=0:(105.3) Because the original equations were linear, the Jacobian factors out of the equations (assuming it never vanishes) and does not appear in (105.3). The equations in (105.3) are now quasilinear in the dependent variables (x;y). They may easily be solved by the method of characteristics; the details may be found in Whitham [9]. Example 2 An equation that arises in transonic small disturbance theory is xxx−yy=0: (105.4) Usinga:=xandb:=y, equation (105.4) can be written as the system of quasilinear equations: ay−bx=0;−aax+by=0: Using the hodograph transformation, these equations simplify to xb−ya=0;a yb−xa=0; withJ=xbya−ybxa. Combining these equations results in the familiar Tricomi equation: aybb−yaa=0 . Notes 1. The hodograph transformation is frequently used in fluid mechanics for problems with unknown boundaries. In many situations, the boundaries become xed in the hodograph plane. 2. The transformation will be non-singular if the Jacobian of the trans- formation,J, does not vanish in the region of interest. 3. Ames [1, pages 35{37] shows that the nonlinear equations ut−vx=0;vt−F2(u)ux=0; become, after applying the hodograph transformation, the linear equations: xv−yu=0;xu−F2(u)yv=0: 4. Whitham [9, page 617] shows how the Born{Infeld equation /parenleftbig 1−u2 t
uxx+2uxutuxt−/parenleftbig 1+u2 x
utt=0 may be linearized with the Hodograph transformation. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 458 II.B Exact Methods for PDEs 5. This technique can also be applied to ordinary dierential equations; a dierential equation for y(x) is inverted to become a dierential equation for x(y) (see page 360). 6. Given a PDE for u(x;t) Clarkson et al. [3] dene a pure hodograph transformation to be the change of independent variables f=t, =u(x;t)g. They dene an extended hodograph transformation to be the change of independent variables f=t,=Rx(u(z;t))dzg. Using these denitions they have: Theorem : The most general second-order, quasilinear PDE of the form ut=g(u)uxx+f(u;ux)w i t hdg=du6=0 , which may be transformed via an extended hodograph transformation to a semilinear partial dierential equation of the form S=S+G(S;S)i sg i v e nb y ut=g(u)uxx+gg00 g0−g0 2 u2 x+b0(u)ux where0d=du ,a n dg(u)a n db(u) are arbitrary functions. Theorem : The most general third-order, quasilinear PDE of the form ut=g(u)uxxx+f(u;ux;uxx)w i t hdg=du6= 0, which may be transformed via an extended hodograph transformation to a semilinear partial dierential equation of the form S=S+G(S;S;S)i sg i v e nb y ut=g(u)uxxx+Buux+Buxuxx +g00 g0−4g0 3g Bux+gg00 g0−g0 3 uxuxx whereBu@B=@u ,Bux@B=@ux,0d=du ,a n dg(u) andB(u;ux) are arbitrary functions. Theorem : The most general quasilinear PDE of the form ut=g(u)ux(n)+f(u;ux;


;ux(n−1))w i t hdg=du6=0 , which may be transformed via an extended hodographtransformation to a semilinear partial dierential equation of the form S =S(n)+G(S;S;


;S(n−1))i sg i v e nb y ut=g(u)ux(n)+g00 g0−n+1 ng0 g Bux +Buux+n−1X r=2Bux(r−1)ux(r)+gg00 g0−g0 n uxux(n−1) where0d=du ,a n dg(u)a n dB(u;ux;


;ux(n−2)) are arbitrary functions. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 105. Hodograph Transformation 459 7. The Harry Dym equation ut=(u−1=2)xxx, written in potential form (i.e., using u=vx)i svt=(v−1=2 x)xx. This equation is invariant under a pure hodograph transformation. That is, the transformed equation is w=(w−1=2 ). References [1]Ames, W. F. Ad hoc exact techniques for nonlinear partial dierential equations. In Nonlinear Partial Dierential Equations in Engineering ,W .F . Ames, Ed. Academic Press, New York, 1967. [2]Bergman, S. The hodograph method in the theory of compressible fluid. Tech. rep., Brown University, 1942. Supplement to Fluid Dynamics by von Mises and Friedrichs. [3]Clarkson, P. A., Fokas, A. S., and Ablowitz, M. J. Hodograph transformations of linearizable partial dierential equations. SIAM J. Appl. Math. 49 , 4 (August 1989), 1188{1209. [4]Crank, J. Free and Moving Boundary Problems . Clarendon Press, Oxford, England, 1984. [5]Crank, J., and Ozis, T. Numerical solution of a free boundary problem by interchanging dependent and independent variables. J. Inst. Maths. Applics 26(1980), 77{85. [6]Kaplan, W. Advanced Calculus . Addison{Wesley Publishing Co., Reading, MA, 1952. [7]Manwell, A. R. The Hodograph Equations: An Introduction to the Mathematical Theory of Plane Transonic Flow . Hafner, Darien, CT, 1971. [8]Siddiqui, A. M., Kaloni, P. N., and Chandna, O. P. Hodograph transformation methods in non-Newtonian fluids. J. Eng. Math. 19 (1985), 203{216. [9]Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 460 II.B Exact Methods for PDEs 106. Inverse Scattering Applicable to Nonlinear evolution equations, a single equation, or as y s t e m . Yields A reformulation into an inverse problem, which can sometimes result in an exact solution. Idea By rewriting the evolution equation, some natural eigenfunction prob- lems emerge. Procedure An evolution equation for u(t;x)=u(t;x1;:::;xm) may be written in the form ut=K(u); (106.1) whereK(
) denotes a nonlinear dierential operator in x.F o r a s y s t e m of equations, the uin equation (106.1) represents a vector of unknowns (u1;:::;un). The procedure is to write equation (106.1) in the Lax pair form (this is often the hardest part of the procedure) Lt=i[L;A]=i(LA−AL); (106.2) whereLandAare linear dierential operators in x, whose coecients are polynomials in uand its xderivatives. Here, Ltrefers to dierentiation ofu(and its derivatives) with respect to tin the expression for L.S e e Example 1 for how equation (106.2) is to be interpreted. Note that, if A were a Hamiltonian, then equation (106.2) would be a Heisenberg equation. A straightforward calculation now shows that it=(L−)(A−it); for arbitrary (t;x)a n d. If we assume that (t=0;x)a n d(t) are an eigenfunction{eigenvalue pair for L,t h a ti s L=; (106.3) and if the eigenfunctions fj(t;x)gevolve in time as it=A; (106.4) then the eigenvalues will be independent of time (i.e., t=0 ) . CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 106. Inverse Scattering 461 Hence, the time evolution of the eigenfunctions can be determined from equation (106.4). Using the eigenfunctions fj(t;x)g, an inverse problem must be solved; the operator Lmust be determined from knowledge of its eigenfunctions. Because Ldepends on u, this might lead to a solution for u. For some problems, the time evolution of the eigenfunctions can be used in the Gelfand{Levitan linear integral equation (see Faddeyev [7]), which may (sometimes) be solved to determine u(t;x). Given equation (106.1) and the initial conditions u(t=0;x), the pro- cedure can be summarized as Find the Lax pair representation of the evolution equation(s). Usingu(t=0;x), evaluate Latt= 0 and then determine the eigen- valuesfjgand the initial values of the eigenfunctions fj(0;x)g. These are the solutions to equation (106.3). Find the time evolution of the eigenfunctions by solving equation (106.4). Determineu(t;x) by solving an inverse problem; that is, using fj(t;x)g as the solutions to equation (106.3), determine Lfort>0. Note that thefj(t);xgare called the scattering data . Even if the last step cannot be carried out, useful information may be obtained from thescattering data. Example 1 For the KdV equation ut+uxxx−6uux=0; (106.5) a Lax pair is given by L=@2 @x2−u; A=−i 4@3 @x3−6u@ @x−3@u @x : (106.6.a-b) This may be veried by calculating, for an arbitrary function = (x), L(A( )) =i(3 uxxx+1 2 xuxx−3 uux+1 5 xxux−6 xu2 +1 0 xxxu−4 xxxxx ); A(L( )) =i(4 uxxx+1 2 xuxx−9 uux+1 5 xxux−6 xu2 +1 0 xxxu−4 xxxxx ); (LA−AL) =−i(uxxx−6uux) (106.7) Using equations (106.6.a) and (106.7), we then determine Lt=−ut; [L;A]=[LA−AL]=−i(uxxx−6uux):(106.8) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 462 II.B Exact Methods for PDEs When equation (106.8) is used in equation (106.2), the KdV equation (106.5) is the result. Example 2 For the sine{Gordon equation uxt=s i nu, the scattering equations (which determine the initial values of the eigenfunctions) for the vector eigenfunction Tmay be written as L   =i" @ @x1 2@u @x 1 2@u @x−@ @x#   =   ; whereas the evolution equations for the vector eigenfunction may be written as i@ @t  =A  =−1 4cosu sinu sinu−cosu  : Notes 1. The formulation of inverse scattering presented here is not the only possible formulation. There are other formulations, which may beeasier to carry out on specic problems. 2. The paper by Case and Kac [5] discusses a discrete inverse scattering problem; their problem illustrates many of the ideas from scattering theory without all of the mathematical diculties. 3. The KdV equation is the compatability condition of the linear system /parenleftbig @ 2 x+u−2
=0 /parenleftbig @t+4@3 x+6u@x+3ux
=0 whereis a spectral parameter. 4. The mKdV equation, ut+uxxx−6u2ux= 0 is the compatability condition of the system /parenleftbig @2 x+2u@x−2
=0 /parenleftbig @t+4@3 x+1 2u@2 x+6 (ux+u2)@x
=0 5. The Burgers equation, ut+uxx+2uux= 0 is the compatability condition of the system (@x−u−) =0 /parenleftbig @t+@2 x−2u@x
=0 References [1]Ablowitz, M. J., Kaup, D. J., Newell, A. C., and Segur, H. The inverse scattering transform | Fourier analysis for nonlinear problems. Stud. Appl. Math. 53 , 4 (Dec 1974), 249{315. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 106. Inverse Scattering 463 [2]Ablowitz, M. J., and Segur, H. Solitons and the Inverse Scattering Transform . SIAM, Philadelphia, PA, 1981. [3]Calogero, F., and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations .N o r t h { Holland Publishing Co., New York, 1982. [4]Calogero, F., and Nucci, M. C. Lax pairs galore. J. Math. Physics 32 , 1 (Jan 1991), 72{74. [5]Case, K. M., and Kac, M. A discrete version of the inverse scattering problem. J. Math. Physics 14 , 5 (1973), 594{603. [6]Eckhaus, W., and Harten, A. V. The Inverse Scattering Transformation and the Theory of Solitons . North{Holland Publishing Co., New York, 1981. [7]Faddeyev, L. D. The inverse problem in the quantum theory of scattering. J. Math. Physics 4 (1963), 72{104. [8]Ito, M. A REDUCE program for evaluating a Lax pair form. Comput. Physics Comm. 34 (1985), 325{331. [9]McLaughlin, J. R. Analytical methods for recovering coecients in dierential equations from spectral data. SIAM Review 28 , 1 (March 1986), 53{72. [10]Musetts, M., and Conte, R. Algorithmic method for deriving Lax pairs from the invariant Painleve analysis of nonlinear partial dierentialequations. J. Math. Physics 32 , 6 (June 1991), 1450{1457. [11]N u c c i ,M .C . Pseudopotentials, Lax equations, and Backlund transforma- tions for non-linear evolution equations. J. Phys. A: Math. Gen. 21 (1988), 73{79. [12]Tabor, M. Chaos and Integrability in Nonlinear Dynamics . John Wiley & Sons, New York, 1989. [13]Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 464 II.B Exact Methods for PDEs 107. Jacobi’s Method Applicable to First order partial dierential equations with three or more dependent variables. In the special case that the dependent vari-able appears explicitly in the equation, then it also applies to equations with two dependent variables. Yields An explicit solution if a certain step can be carried out. Idea Given a partial dierential equation for z(x)=z(x1;x2;:::;xn), if the set ofnrst derivativesfpi=@z=@xiji=1;2;:::;ngis explicitly known, thenz(x) may be found by integrating the Pfaan dierential equation: dz=p1dx1+


+pndxn. Jacobi’s method determines the fpigfrom a given partial dierential equation. Procedure Let us presume that the given partial dierential equation for z= z(x)=z(x1;:::;xn), withn= 3, is of the form F(x;p)=0; (107.1) wherepi=@z=@xi. If we could nd two other equations, that have the same solution as equation (107.1), of the form fF2(x;p)=0 ,F3(x;p)=0g,t h e n we might be able to determine fp1=p1(x);:::;pn=pn(x)gby combining these three equations. Then we could nd z(x) by solving the Pfaan dierential equation (see page 384) dz=p1dx1+


+pndxn: (107.2) So, we need to determine fF2;F3gin such a way that their solutions are the same as the solution to equation (107.1). This requirement results in (see the section on compatible systems, page 43) [F;F 2]: =nX i=1@F @xi@F2 @pi−@F @pi@F2 @xi =0; [F;F 3]=0; [F2;F3]=0;(107.3) where [;] is the usual Poisson bracket. The characteristic equations for F2 (orF3), from equation (107.3), can be written as (see page 432) dx1 −@F @p1=dp1 @F @x1=


=dxn −@F @pn=dpn @F @xn: (107.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 107. Jacobi’s Method 465 (These are also known as the subsidiary equations.) Hence, the procedure is to solve equation (107.4) for F2(x;p)=0a n dF3(x;p)=0 . I tm u s tb e then be veried that [ F2;F3] = 0. Then solving fF=0;F1=0;F2=0gfor pi=pi(x) and integrating equation (107.2) results in a solution to equation (107.1). Example This example is from Piaggio [3, pages 162{170]. Suppose we have the following nonlinear partial dierential equation in three independent variables: 0=F(x;p)=2x1x3@z @x1+3x2 3@z @x2+@z @x22@z @x3; =2x1x3p1+3x2 3p2+p2 2p3:(107.5) The subsidiary equations in equation (107.4) can be written as dx1 −2x1x3=dp1 2x3p1=dx2 −3x2 3−2p2p3=dp2 0=dx3 −p2 2=dp3 2x1p1+6x3p2: (107.6) From the rst equality in equation (107.6) we have F2(x;p)=p1x1−A1=0; (107.7) whereA1is an arbitrary constant. From the fourth term in equation (107.6), we have F3(x;p)=p2−A2=0; (107.8) whereA2is another arbitrary constant. Clearly, [ F2;F3] = 0 for our chosen F2andF3. Combining equations (107.7) and (107.8) with the original equation, (107.5), we nd that p3=−1 A2 2(2A1x3+3A2x2 3): (107.9) In equations (107.7){(107.9) we have found expressions for the fpig. Hence, dz=p1dx1+p2dx2+p3dx3 =A1 x1dx1+A2dx2−1 A2 2(2A1x3+3A2x2 3)dx3; which can be integrated to yield the solution z=A1logx1+A2x2−1 A2 2(A1x2 3+A2x3 3)+A3; whereA3is another arbitrary constant. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 466 II.B Exact Methods for PDEs Notes 1. If the given partial dierential equation has only two independent variables and if the dependent variable zis explicit in the partial dierential equation, then we can transform the partial dierential equation into the form of equation (107.1). For example, if we haveF(x;y;z;p;q ) = 0 (where, as usual, p=@z=@x ,q=@z=@y ), suppose thatu(x;y;z ) = 0 is an integral of this equation. If we dene u 1= @u=@x ,u2=@u=@y ,u3=@u=@z , then we can write p=−u1=u3;q= −u2=u3. Using these denitions for pandqin the original equation yields an equation of the form f(x;y;z;u 1;u2;u3)=f(x;p)=0 . 2. Whenn>3, then the only change in the procedure is that we must now determinefF2;F3;:::;Fngand use these (with F)t os o l v ef o r thefpig. 3. When this method is specialized to two independent variables, it is often called Charpit’s method . See Chester [2, page 212, and Chapter 15, pages 315{337] or Piaggio [3] for details. 4. See also Ames [1, pages 54{57] and Sneddon [4, pages 69{73 and 78{80]. References [1]Ames, W. F. Ad hoc exact techniques for nonlinear partial dierential equations. In Nonlinear Partial Dierential Equations in Engineering ,W .F . Ames, Ed. Academic Press, New York, 1967. [2]Chester, C. R. Techniques in Partial Dierential Equations . McGraw{Hill Book Company, New York, 1970. [3]Piaggio, H. T. H. An Elementary Treatise on Dierential Equations and Their Applications . G. Bell & Sons, Ltd, London, England, 1926. [4]Sneddon, I. N. Elements of Partial Dierential Equations . McGraw{Hill Book Company, New York, 1957. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 108. Legendre Transformation 467 108. Legendre Transformation Applicable to Partial dierential equations in one dependent vari- able that are notof the form F(ux1;ux2;:::;uxn)=0 . Yields An alternative formulation of the original problem. Idea A surface in space may be described by a point or as an envelope of tangent planes. Changing variables from one representation to the other may facilitate nding a solution. After a solution is obtained, it can betransformed back to the original variables. Procedure We illustrate the technique for two independent variables; the notes show how the technique may be extended to nindependent variables. Given a function u(x;y), we change to the new variables w(;)b yt h e transformation w(;)+u(x;y)=x+y; (108.1) with the following denitions ux=; w=x; uy=; w=y: (108.2) From equation (108.1) and equation (108.2) it is easy to derive that uxx=Jw; uxy=uyx=−Jw; uyy=Jw; whereJis the Jacobian of the transformation. The Jacobian may be expressed as J=uxxuyy−(uxy)2=1 ww−(w)2: To be able to transform from the fu;x;ygvariables to thefw;;gvari- ables, the Jacobian must not vanish. If J6= 0, then the surface is said to bedevelopable . The solutions with J= 0 are said to be non-developable so- lutions. The non-developable solutions are not obtainable by the Legendretransformation. Summary For the partial dierential equation of at most second order in the variablesfu;x;yg, F(x;y;u;ux;uy;uxx;uxy;uyy)=0; (108.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 468 II.B Exact Methods for PDEs we make the Legendre transformation to obtain the new equation F(w;w;w+w−w;;;Jw ;−Jw;Jw)=0 (108.4) in the new variables fw;;g. Sometimes equation (108.4) is easier to solve than equation (108.3). After equation (108.4) is solved to determine w(;), we must change back to the original variables. Changing from the fw;;gvariables to the fu;x;ygvariables can be done (due to the implicit function theorem) but may be dicult. Example Consider the nonlinear partial dierential equation uxuy=x; (108.5) which we want to solve for u(x;y). The Legendre transformation of equa- tion (108.5) is (using the transformations in equations (108.1) and (108.2) or using equation (108.4) directly) w=: (108.6) This has the solution w(;)=1 22+f(); (108.7) wheref() is an arbitrary function of . We have now nished solving the dierential equation. Because we have the solution in terms of the new variables, all that remains is to transform to the old variables. This changeof variables will utilize the w(;) that was found. Using equation (108.6) and w =x(from (108.2)), we have x=: (108.8) Dierentiating equation (108.7) with respect to and usingy=w(from equation (108.2)) yields y=1 22+f0(): (108.9) Using equations (108.7){(108.9) in equation (108.1) produces the equation u=x+y−w(;)=2+f0()−f(): (108.10) Solving equation (108.8) for , and then substituting that result in equa- tions (108.9) and (108.10) produces y=x2 22+f0(); u=x2 +f0()−f():(108.11.a-b) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 108. Legendre Transformation 469 This is a parametric representation of the solution u(x;y). All of the developable solutions of equation (108.5) are completely characterized by equation (108.11). Given any f() we can, in principle, nd =(x;y) from equation (108.11.a). Using this value for in equation (108.11.b) then givesuas a function of xandy. To illustrate this, if we choose f()=A ; whereAis an arbitrary constant, then equation (108.11) becomes y=1 2x2−A1 2;u =x2−2A : (108.12.a-b) Solving equation (108.12.a) for and using this expression in equation (108.12.b) produces u(x;y)=p 2y(x2−2A): Now that we have an explicit solution, we must check that the Jacobian does not vanish. In this example, J6=0 . Notes 1. Observe that u=Dy+1 2Dx2+C; (108.13) whereCandDare constants, is also a solution to equation (108.5), but this solution is not contained in equation (108.11) for any f(). This is because the solution in equation (108.13) is non-developable (J=0 ) . 2. The Legendre transformation may be naturally extended to par- tial dierential equations in nvariables. The transformation (from u(x1;x2;:::;xn)t ow(1;1;:::;n)) and its inverse is given by u(x1;x2;:::;xn)=w(1;1;:::;n)+x11+x22+


+xnn; ux1=1;ux2=2;


;uxn=n; w1=x1;w2=x2;


;wn=xn: See Courant and Hilbert [3, Volume 2, pages 32{39] for more details. 3. Clairaut’s equation, u=xux+yuy+f(ux;uy), under the Legendre transformation, becomes the simple equation w=−f(;). 4. The Legendre transformation is an involutory transformation; that is, the Legendre transformation applied twice results in the originalequation. The Legendre transformation is also an example of a contact transformation (see page 249). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 470 II.B Exact Methods for PDEs 5. The Legendre transformation is used in mechanics when transforming from the Lagrangian formulation to the Hamiltonian formulation (or vice-versa). See Goldstein [5] for details. 6. The Legendre transformation is used in thermodynamics when trans- forming the fundamental equation from internal energy (canonical variables are specic volume and specic entropy) to the Gibbs func- tion (canonical variables are pressure and temperature), or to en- thalpy (canonical variables are pressure and specic entropy), or to the Helmholtz function (canonical variables are specic volume andtemperature). For more details of this application, see Kestin [6]. 7. If the Legendre transformation is applied to a partial dierential equa- tion of the form F(u x;uy) = 0, then the algebraic relation F(;)=0 results. Because w(;) cannot be determined from this equation, this class of equations cannot be solved by the use of the Legendre transformation. 8. See Ames [1, pages 37{40], Chester [2, pages 209{210], and Epstein [4, pages 65{68]. References [1]Ames, W. F. Ad hoc exact techniques for nonlinear partial dierential equations. In Nonlinear Partial Dierential Equations in Engineering ,W .F . Ames, Ed. Academic Press, New York, 1967. [2]Chester, C. R. Techniques in Partial Dierential Equations . McGraw{Hill Book Company, New York, 1970. [3]Courant, R., and Hilbert, D. Methods of Mathematical Physics . Interscience Publishers, Inc., New York, 1953. [4]Epstein, B. Partial Dierential Equations: An Introduction . McGraw{Hill Book Company, New York, 1962. [5]Goldstein, H. Classical Mechanics . Addison{Wesley Publishing Co., Reading, MA, 1950. [6]Kestin, J. A Course in Thermodynamics . Blaisdell Publishing Co., Waltham, MA, 1966. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 109. Lie Groups: PDEs 471 109. Lie Groups: PDEs Applicable to Linear and nonlinear partial dierential equations. Yields Similarity variables that may be used to decrease the number of inde- pendent variables in a partial dierential equation. Idea By determining the transformation group under which a given partial dierential equation is invariant, we can obtain information about the invariants and symmetries of that equation. This information, in turn, can be used to determine similarity variables that will reduce the numberof independent variables in the system. Procedure Some background material about Lie groups may be found in the section \Lie Groups: ODEs" (starting on page 366). We utilize terms that have been dened in that section. We illustrate the general technique on one partial dierential equation in two independent variables. Suppose we would like to solve the partial dierential equation N(u;x;y ) = 0 (109.1) foru(x;y). We rst determine a one parameter Lie group of transforma- tions, under which equation (109.1) is invariant; then we use this group to determine similarity variables. We suppose that the group has the form u=u+U(u;x;y )+O(2); x=x+X(u;x;y )+O(2); y=y+Y(u;x;y )+O(2):(109.2) We want this group to leave equation (109.1) invariant; that is, N(x;y;u)=0; (109.3) or, equivalently, u(x;y)=u(u;x;y ;): (109.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 472 II.B Exact Methods for PDEs Using the transformations in equation (109.2), the chain rule produces @x @x=1−(Xx+Xuux)+O/parenleftbig 2
; @x @y=−(Xy+Xuuy)+O/parenleftbig 2
; @y @x=−(Yx+Yuux)+O/parenleftbig 2
; @y @y=1−(Yy+Yuuy)+O/parenleftbig 2
:(109.5) From equation (109.5), it is conceptually easy (though algebraically in- tensive) to determine how derivatives in the fu;x;ygsystem transform to derivatives in the fu;x;ygsystem. For instance, @u @x=ux+/parenleftbig Ux+(Uu−Xx)ux−Yxuy−Xuu2 x−Yuuxuy
+O/parenleftbig 2
; @u @y=uy+/parenleftbig Uy+(Uu−Yy)uy−Xyux−Yuu2 y−Xuuyux
+O/parenleftbig 2
; @2u @x2=uxx+ −Yuuu2 xuy−Xuuu3 x−2Yuuxuxy−(3Xu+2Yyu)uxuy −Yuu2 y+(Xuu−2Yux)U2 x−2Yyuxy+(Uu−2XxYxx)uy +Uxx+( 2Uxu−Yxx)ux +O/parenleftbig 2
: (109.6) The group is then determined (i.e., fU;X;Ygare determined) by requiring equation (109.3) to be satised. After the group has been determined, a solution to equation (109.1) may be found from the invariant surface condition U(u;x;y )=X(u;x;y )@u @x+Y(u;x;y )@u @y; (109.7) which is just the rst order term of equation (109.4) when that equation is expanded for small values of . The solution of equation (109.7) leads to similarity variables that reduce the number of independent variables in the system. Note that equation (109.7) is quasilinear and that the subsidiary equations may be written as du U(u;x;y )=dx X(u;x;y )=dy Y(u;x;y ): (109.8) Example 1 Suppose we wish to analyze the heat equation uy=uxx: (109.9) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 109. Lie Groups: PDEs 473 We takeuy=uxxand substitute for the derivatives from equation (109.6). We also substitute uyforuxx(from equation (109.9)). This leads to a large expression that must equal zero. Equating to zero the coecients of fu;ux;uy;u2 x;u2y;uxy;uxuy;uxuxyg in this expression leads to eight simultaneous equations involving fU;X;Yg. The solution to these equations will determine the transformation group. Three of these equations are u2 xcoecient: Yu=0; uxuycoecient: Xu=0; uxuxycoecient: Uuu=0: These equations produce X(u;x;y )=X(x;y),Y(u;x;y )=Y(x;y)a n d U(u;x;y )=f(x;y)u+g(x;y), wherefandgare functions to be deter- mined. Using this simplication for fU;X;Yg, the other ve equations become Yx=0;f xx−fy=0; 2Xx−Yy=0;g xx−gy=0; Xy−Xxx+2fx=0:(109.10) If we take g= 0 (just to simplify the algebra), then the equations in equation (109.10) may be solved to determine the transformation group X=2c1y+4c2xy+c4+c5x; Y=4c2y2+2c5y+c6; U=−/parenleftbig c1x+c2(x2+2y)+c3
u;(109.11) wherefc1;:::;c 6gare arbitrary constants. Now that we have found a transformation group, similarity variables may be found. Special Case 1 If we take c1=c2=c4=c6= 0 in equation (109.11), then the subsidiary equations (from equation (109.8)) become du −c3u=dx c5x=dy 2c5y: Two solutions to these equations are constant =xpy; constant =u y; where=−c3=2c5. From these similarity variables, we propose a solution of the form =xpy;h ()=u y: (109.12) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 474 II.B Exact Methods for PDEs That is,u(x;y)=yh(x=py). Using this form in equation (109.9), we nd thath() satises the ordinary dierential equation h00=h−1 2h0.E v e r y solution to this equation will generate a solution to equation (109.9). Special Case 2 If we take c1=c2=c4=c5= 0 in equation (109.11), then the subsidiary equations (from equation (109.8)) become du −c3u=dx 0=dy c6: Two solutions to these equations are constant = x; constant =u ey; where=−c3=c6. From these similarity variables we propose a solution of the form =x; k ()=u ey: That is,u(x;y)=eyk(x). Using this form in equation (109.9), we nd thatk() satises the ordinary dierential equation k00−k=0 . E v e r y solution to this equation will generate a solution to equation (109.9). Example 2 Consider similarity solutions of Laplace’s equation in two dimensions: r2u=uxx+uyy= 0. To nd the Lie group of transformations that leaves this equation invariant, we consider the group dened in equation (109.2).After extensive algebra we nd that, to lowest order, fX;Y;Ugmay be expressed as X=d 1+d3x−d4y+d5(x2−y2)+2d6xy+ (cubic terms) ; Y=d2+d3y+d4x+2d5xy+d6(y2−x2) + (cubic terms) ; U=d7u+V(x;y);(109.13) whereV(x;y) is any solution to r2V=0a n dfd1;d2;:::;d 7gare arbitrary constants. The similarity solutions to r2u= 0 may now be determined from the subsidiary equations in (109.8). For simplicity, we will take V= 0, and investigate two possibilities for the other parameters in equation(109.13). Special Case 1 If we presume that the only non-zero parameters in equation (109.1) ared1,d2,a n dd7, then the subsidiary equations become du d7u=dx d1=dy d2: Using the equation specied by the second equality sign, we determine that d2x−d1yis constant. Using the equation specied by the rst equality CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 109. Lie Groups: PDEs 475 sign, we determine that ue−d7x=d1is constant. Hypothezing a solution of the formu(x;y)=ed7x=d1f(d2x−d1y), and then requiring that r2u=0 , leads to a constant coecient ordinary dierential equation for f: d2 1/parenleftbig d2 1+d2 2
f00−2d1d2d7f0+d2 7f=0: Special Case 2 If we presume that the only non-zero parameters in equation (109.1) ared3andd7, then the subsidiary equations become dx d3x=dy d3y=du d7u: These equations can be solved to determine that u(x;y)=ymg(),=y=x, wherem=d7=d3. By requiringr2u= 0 to hold, we nd the following ordinary dierential equation for g(): /parenleftbig 2+4
g00+2/parenleftbig m+2
g0+m(m−1)g=0: Notes 1. Lie group analysis is the most useful and general of all the techniques presented in this book. 2. There are other techniques for determining the group under which a given partial dierential equation is invariant. A list of techniques is given in Seshadri and Na [12]. 3. Ifu(x;y) is a solution of equation (109.9), then the following trans- formations also represent solutions: T1:(x!x+2cy u!ue−c(x+y2)) T2:8 >>>< >>>:x!x=(1−4cy) y!y=(1−4cy) u!up 1−4cyexp −cx2 1−4cy9 >>>= >>>; T 3:u!ecu T4:x!x+c T5:( x!ecx y!e2cy) T6:y!y+c:(109.14) These transformations were all obtained from the group in equation (109.11). For example, the similarity variable =x=pyin equation (109.12) is equivalent to transformation T5.I fu=m(x;y)i sa CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 476 II.B Exact Methods for PDEs solution of equation (109.9), then another solution is given by (using all of the transformations listed in (109.14)) u=1p1+4c2yexp c3−c1x+c2x2−c2 1y 1+4c2y me−c5(x−2c1y) 1+4c2y−c4;e−2c5y 1+4c2y−c6 : See Olver [9, pages 120{123] for details. 4. Using Lie groups to nd symmetries of partial dierential equations can be computationally intensive. Algorithms have been developed for computerized handling of the calculations. A computer packagein FORMAC is described in Fedorova and Kornyak [6], a Macsyma package is in Champagne et al. [4], a Maple package is in Manseld and Clarkson [8], and a REDUCE package is in Schwarz [11]. 5. A new technique for nding symmetries of partial dierential equa- tions that are neither point symmetries nor Lie{B¨ acklund symmetries may be found in Bluman et al. [3]. 6. The general equation of nonlinear heat conduction takes the form u t=(K(u)ux)x. For this equation, IfK(u) is constant, then the symmetry group is innite dimen- sional. IfK(u)=(au+b)−4=3,w i t ha6= 0, then there is a ve-parameter symmetry group. IfK(u)=(au+b)m,f o rm6=−4 3anda6= 0, then there is a four-parameter symmetry group. IfK(u)=ceau, then there is a four-parameter symmetry group. IfK(u) does not have one of the forms mentioned above, then there is a three-parameter symmetry group. 7. Olver [9] derives the complete symmetry group for many partial dierential equations, including the heat equation, wave equation,Euler equations, and Korteweg-de Vries equation. Ames and Nucci [1] studied the Burgers’s equation, Korteweg-de Vries equation (1 and 2 dimensions), Hopf equation, and Lin{Tsien equation. 8. Classical and nonclassical symmetries of the nonlinear heat equation u t=uxx+f(u) are considered in Clarkson and Manseld [5]. 9. The KdV equation, ut=uxxx+6uux, has the Lie point symmetries f@x;@t;−6t@x+@u;x@x+3t@t−2u@ug. 10. Burgers’s equation, ut−uux−uxx= 0, has the Lie point symmetries f@t;@x;t@x−@u;2t@t+x@x−u@u;t2@t+tx@x−(x+tu)@ug. 11. The section on similarity methods (beginning on page 497) shows how to nd similarity variables of a specic form. The techniques inthis section are, of course, much more general and will determine all possible similarity variables. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 109. Lie Groups: PDEs 477 References [1]Ames, W. F., and Nucci, M. C. Analysis of fluid equations by group methods. Journal of Engineering Mathematics 20 (1985), 181{187. [2]Bluman, G. W., and Kumei, S. Symmetries and Dierential Equations . Springer{Verlag, New York, 1989. [3]Bluman, G. W., Reid, G. J., and Kumei, S. New classes of symmetries for partial dierential equations. J. Math. Physics 29 , 4 (April 1988), 806{811. [4]Champagne, B., Hereman, W., and Winternitz, P. The computer calculation of Lie point symmetries of large systems of dierential equations.Comput. Physics Comm. 66 (1991), 319{340. [5]Clarkson, P. A., and Mansfield, E. L. Symmetry reductions and exact solution of a class of nonlinear heat equations. Physica D 70 (1993), 250{288. [6]Fedorova, R. N., and Kornyak, V. V. Determination of Lie{Backlund symmetries of dierential equations using FORMAC. Comput. Physics Comm. 39 (1986), 93{103. [7]Hill, J. M. Solution of Dierential Equations by Means of One-Parameter Groups . Pitman Publishing Co., Marsheld, MA, 1982. [8]Mansfield, E. L., and Clarkson, P. A. Application of the dierential algebra package diffgrob2 to classical symmetries of dierential equations. J. Symbolic Computation 11 (1994). [9]Olver, P. J. Applications of Lie Groups to Dierential Equations . No. 107 in Graduate Texts in Mathematics. Springer{Verlag, New York, 1986. [10]Reiman, A. Computer-aided closure of the Lie algebra associated with a nonlinear partial dierential equation. Comp. & Maths. with Appls. 7 ,7 (1981), 387{393. [11]Schwarz, F. Automatically determining symmetries of partial dierential equations. Computing 34 (1985), 91{106. [12]Seshadri, R., and Na, T. Y. Group Invariance in Engineering Boundary Value Problems . Springer{Verlag, New York, 1985. [13]Steinberg, S. Applications of the Lie algebraic formulas of Baker, Campbell, Hausdor, and Zassenhaus to the calculation of explicit solutions of partial dierential equations. J. Dierential Equations 26 (1977), 404{ 434. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 478 II.B Exact Methods for PDEs 110. Poisson Formula Applicable to Laplace’s equation ( r2u= 0) in two dimensions withu(x) prescribed on a circle; that is, the Dirichlet problem in a disk. Yields An exact solution, given by an integral. Idea A simple extension of the Cauchy integral formula (from complex vari- able theory) allows the solution for Laplace’s equation in a circle to be written down analytically. Procedure Ifu(r;) satises r2u=urr+1 rur+1 r2u=0; for 0<r<R; u(R;)=f();for 0<2; (110.1) thenu(r;)f o r0<r<R is given by u(r;)=1 2Z2 0R2−r2 R2−2Rrcos(−)+r2f()d: (110.2) This is known as the Poisson formula for a circle. Example If we have r2u=0;u (R;)=s i n; then u(r;)=1 2Z2 0R2−r2 R2−2Rrcos(−)+r2sind  =r Rsin; where the integral was carried out by using the method of residues. Notes 1. By use of conformal mappings (see page 441), Laplace’s equation in two dimensions for a non-circular region can often be changed to solving Laplace’s equation in a circular region. Poisson’s formula canbe used for this new problem, and then the mapping can be used to nd the solution for the original geometry. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 110. Poisson Formula 479 2. The solution to equation (110.1) could also have been obtained by the use of Fourier series (see page 344). Using this technique, the solution to equation (110.1) becomes u(r;)=a0 2+1X n=1r an (ancosn+bnsinn); (110.3) wherefan;bngare dened by an=1 Z −f()c o snd; b n=1 Z −f()s i nnd: (110.4) Note that this same solution would have been obtained by utilizing separation of variables. Farlow [2, Lesson 33, pages 262{269] andYoung [6, pages 273{285] show that the Poisson formula in equation (110.2) may be derived from the solution in equations (110.3) and (110.4). 3. The Neumann problem for a disk r 2v=0;@v @n(R;)=g(); (110.5) may be converted to the Dirichlet problem (equation (110.1)) if we dene f()=Z 0g()d; v(x;y)=Z(x;y) (uydx−uxdy);(110.6) see Young [6, pages 273{285] for details. Note that the periodicity requirement of f() requires that g()s a t i s f yZ2 0g()d= 0. This must be satised if there is to exist any solution to equation (110.5). This requirement is related to the alternative theorems on page 15. (Note that the solution to equation (110.5) is indeterminate with respect to a constant.) 4. The solution to the exterior problem r2w=0; w(R;)=f();w bounded at r=1(110.7) is given by w(r;)=−1 2Z2 0R2−r2 R2−2Rrcos(−)+r2f()d; (110.8) which is valid for rR. See Kantorovich and Krylov [4, pages 572{575] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 480 II.B Exact Methods for PDEs 5. Other exact solutions to Laplace’s equation are also known. For example, Ifr2u= 0 in a sphere of radius one and u(1;;)=f(;), then u(r;; )=1 4Z 0Z2 0f(;)1−r2 (1−2rcosγ+r2)3=2sin dd; (110.9) where cosγ:= cosc o s+s i n sin  cos(−). Ifr2u= 0 in the half plane, y0, andu(x;0) =f(x), then u(x;y)=1 Z1 −1f(t)y (x−t)2+y2dt: (110.10) Ifr2u= 0 in the half space, z0, andu(x;y;0) =f(x;y), then u(x;y;z )=z 2Z1 −1Z1 −1f(;) [(x−)2+(y−)2+z2]3=2dd: (110.11) Ifr2u= 0 in the annulus, 0 <ar1, andu(1;)a n d u(a;) are given, then an explicit solution is given by Villat’s integration formula. See Iyanaga and Kawada [3, page 1450] for details. 6. See also Churchill [1, Chapter 11, pages 242{258] and Levinson and Redheer [5, page 360]. References [1]Churchill, R. V. Complex Variables and Applications . McGraw{Hill Book Company, New York, 1960. [2]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [3]Iyanaga, S., and Kawada, Y. Encyclopedic Dictionary of Mathematics . MIT Press, Cambridge, MA, 1980. [4]Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis . Interscience Publishers, Inc., New York, 1958. [5]Levinson, N., and Redheffer, R. M. Complex Variables . Holden{Day, Inc., San Francisco, CA, 1979. [6]Young, E. C. Partial Dierential Equations . Allyn and Bacon, Inc., Boston, MA, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 111. Riemann’s Method 481 111. Riemann’s Method Applicable to Linear hyperbolic equations of the second order in two independent variables. Yields An exact solution in terms of the solution to the adjoint equation. Idea The solution of a non-characteristic initial value problem in two di- mensions can be found if the adjoint equation with specied boundary conditions can be solved. Procedure Suppose we have the hyperbolic partial dierential equation L[u]=uxy+a(x;y)ux+b(x;y)uy+c(x;y)u=f(x;y); (111.1) whereu(x;y) is specied on the boundary Γ, which is not a characteristic (see gure 111.1). Note that any linear hyperbolic equations of second order in two independent variables can be written in the form of equation(111.1). We wish to nd u(S)=u(;), whereSrepresents an arbitrary point and is indicated in gure 111.1. If we assume that the initial curve Γ is monotonically decreasing, then we can write the solution as u(;)=1 2R(P;;)u(P)+1 2R(Q;;)u(Q) −ZQ PB[u(x;y);R(x;y;;)] +ZZ Df(x;y)R(x;y;;)dxdy;(111.2) where B[u;v]= avu+1 2vuy−1 2vyu dy+ −bvu+1 2vux−1 2vxu dx; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 482 II.B Exact Methods for PDEs/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /./././././. /././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./. /./././././././././././././././././././,x yD /#0FS /#0F P/#0F Q /././. /././. /././. /././. /././././. /././. /././. /./././. /./. /./././. /. /./. /./././. /././. /././. /././././. /./././././././. /././././. /./././././. /./././././. /./././. /./././. /./. /././. /./. /././././. /./././././. /./././././. /./././././. /./././. /./././. /././././././././././././././././././././. /./././././././././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././././. /././././. /./././. /./././. /./././. /./././. /././. /././. /././. /././. /././. /././. /./. /./. /././. /./. /./. /./. /././. /./. /./. /. /./. /./. /./. /./. /./. /./. /./. /. /./. /./. /./. /. /./. /./. /. /./. /./. /. /./. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /././. /./././././. /././././././. /. /./././././././. /./././././././././. /./. /. /./././././././. /./././././././././././. /./././. /. /./././././././. /././././././././././././. /././././. /. /./././././././. /./././././././././././././. /./././././. /. /./././././././. /././././././././././././././. /././././././. /. /./././././././. /./././././././././././././././. /./././././././. /. /./././././././. /././././././././././././././././. /././././././././. /. /./././././././. /./././././././././././././././././. /./././././././././. /. /./././././././. /././././././././././././././././././. /./././././././././. /. /./././././././. /././././././././././././././././././. /././././././././././. /. /./././././././. /./././././././././././././././././././. /././././././././././. /. /./././././././. /./././././././././././././././././././. /./././././././././././. /. /./././././././. /./././././././././././././././././././. /./././././././././././. /. /./././././././. /././././././././././././././././././././. /./././././././././././. /. /./././././././. /././././././././././././././././././././. /././././././././././././. /. /./././././././. /././././././././././././././././././././. /././././././././././././. /. /./././././././. /./././././././././././././././././././././. /././././././././././././. /. /./././././././. /./././././././././././././././././././././. /././././././././././././. /. /./././././././. /./././././././././././././././././././././. /./././././././././././././. /. /./././././././. /./././././././././././././././././././././. /./././././././././././././. /. /././././././.Figure 111.1: Domain in which equation (111.1) is solved. (note thatB[u;v] includes the dierential terms dxanddy)a n dR(x;y;;) is the Riemann function dened by Rxy−aRx−bRy+(c−ax−by)R=0; R(;y;;)=e x pZy a(;)d ; R(x;;;)=e x pZx b(;)d ; R(;;;)=1:(111.3) In this formulation, PSis a horizontal segment and QSis a vertical segment that contain the domain the dependence D. The derivation of this formula is more detailed than the format of this book allows. See Garabedian [6, pages 127{135] for a full description. A simple motivation for the Riemann function is given in Kreith [9]. Example 1 Suppose we have the partial dierential equation 2w−2w=0; w(;1) =f(); w(;1) =g();(111.4.a-c) where−1<<1,1<<1,a n df()a n dg() are given functions. If we change variables in equation (111.1) from fw;;gtofu;x;ygby u(x;y)=w(;); x=; y = ; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 111. Riemann’s Method 483/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./. /./././././././././././././././././././#11 /#10/#0B/#3E /0/#0B/#3C /0 /./. /./. /. /./. /./././. /././. /./././././././././. /././././. /./././. /././././. /./././. /././././. /././. /././. /././././././. /././././././././././././././. /./././././././././././. /./././././././././././././././././././././././././././././././././././././././. /./././. /././././. /./././. /././. /././. /././. /././. /./. /././. /./. /./. /./. /./. /./. /./. /./. /./. /./. /./. /. /./. /./. /. /./. /./. /. /./. /./. /. /./. /. /./. /. /. /. /./. /././. /././././. /./././././. /./././././././. /./././././././. /././././././././. /././././././././. /./././././././././. /./././././././././. /././././././././././. /././././././././././. /./././././././././././. /./././././././././././. /././././././././././././. /./././././././././././. /././././././././././././. /././././././././././././. /./././././././././././././. /././././././././././././. /./././././././././././././. /./././././././././././././. /./././././././././././././. /./././././././././././././. /./././././././././././././. /./././././././././././././. /././././././././././././././. /./././././././././././././. /././././././././././././././. /. /. /. /./. /././. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /./. /. /. /./. /./. /. /. /. /. /./. /. /. /. /./. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /././././././././././././././. /./././././././././././././. /././././././././././././././. /./././././././././././././. /./././././././././././././. /./././././././././././././. /./././././././././././././. /./././././././././././././. /./././././././././././././. /././././././././././././. /./././././././././././././. /././././././././././././. /././././././././././././. /./././././././././././. /././././././././././././. /./././././././././././. /./././././././././././. /././././././././././. /././././././././././. /./././././././././. /./././././././././. /././././././././. /././././././././. /./././././././. /./././././././. /./././././. /././././. /././. /.Figure 111.2: Domain in which equation (111.4) is solved. (see the transformation on page 168), then equation (111.4.a) becomes uxy−1 2xuy=0: (111.5) The boundary conditions in equation (111.4) transform to u s;1 s =f(s); sux s;1 s +1 suy s;1 s =g(s);(111.6) where−1<s<1. By manipulations of equation (111.6), we can derive u s;1 s =f(s); ux s;1 s =1 2 f0(s)+1 sg(s) ; uy s;1 s =1 2 sg(s)−s2f(s) :(111.7) The domain in which equations (111.5) and (111.7) are to be solved is shown in gure 111.2. To solve equations (111.5) and (111.7), we use Riemann’s method. Comparing equation (111.5) to equation (111.1) we determine a=0 , b=−1=2x,c=0 ,f= 0. Hence, the solution (from equation (111.2)) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 484 II.B Exact Methods for PDEs becomes u(;)=1 2R(P;;)u(P)+1 2R(Q;;)u(Q) −ZQ P1 2Ruy−1 2Ryu dy− −1 2xRu+1 2Rux−1 2Rxu dx :(111.8) All that remains is to nd the Riemann’s function. From equation (111.3), R(x;y;;) satises Rxy+1 2xRy=0; R(;y;;)=1; R(x;;;)=r  x; R(;;;)=1:(111.9.a-d) Because equation (111.9.a) can be integrated directly with respect to xand then with respect to y, the general solution to equation (111.9) is easily seen to be of the form R(x;y;;)=M(x;;)+K(y;;)px; (111.10) for someM(x;;)a n ds o m e K(y;;). Using equation (111.10) in the boundary conditions in equation (111.9), the solution is found to be R(x;y;;)=r  x: (111.11) Using equation (111.11) in equation (111.8), we can nd u(;) and hence, w(;) for any values of and. Example 2 The Riemann’s function for the partial dierential equation uxy=1 4k2u; (111.12) (whenkis a constant) is R(x;y;;)=I0 kp (x−)(y−) ; whereI0is the usual modied Bessel function of order zero. Hence, the solution to equation (111.12) with the boundary conditions ux= (x)w h e n y=0; uy=(x)w h e n x=0; is given by u(x;y)=Zy 0I0 kp x(y−) ()d+Zx 0I0 kp y(x−) ()d: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 111. Riemann’s Method 485 Notes 1. Numerical techniques based on this method are called Godunov meth- ods, after Godunov [7]. A comparison of some of these methods can be found in Woodward and Colella [11]. 2. Essentially, the Riemann’s function is a type of Green’s function, the connection is made in Zauderer [12, pages 485{492]. What we have called the Riemann’s function is sometimes called a Green’s function or a Riemann{Green function. 3. If the operator L[u] in equation (111.1) is self-adjoint, then we have the reciprocity principle: R(x;y;;)=R(;;x;y). 4. Numerical methods for solving hyperbolic equations that use the Rie- mann’s function are generally referred to as Godunov-type methods. A comparison of some Godunov-type methods with more classicialmethods may be found in Woodward and Colella [11]. 5. Copson [3, pages 77{88] suggests that the Riemann’s function may often have the form R(x;y;;)=1X k=0Gkk (k!)2; where  = ( x−)(y−). When this is the case, then only the coecientsfGkgmust be found. Copson [3, pages 77{88] gives several examples of this approach. 6. The technique presented here may be extended to higher order equa- tions, for which the Riemann tensor must be determined. See Courantand Hilbert [4, Volume II, pages 450{461]. 7. See also Bateman [1, pages 280{285], Chester [2, pages 222{231], Davis [5, pages 75{79], and Sneddon [10, pages 119{122]. References [1]Bateman, H. Dierential Equations . Longmans, Green and Co., 1926. [2]Chester, C. R. Techniques in Partial Dierential Equations . McGraw{Hill Book Company, New York, 1970. [3]Copson, E. T. Partial Dierential Equations . Cambridge University Press, New York, 1975. [4]Courant, R., and Hilbert, D. Methods of Mathematical Physics . Interscience Publishers, Inc., New York, 1953. [5]Davis, J. L. Finite Dierence Methods in Dynamics of Continuous Media . The MacMillan Company, New York, 1986. [6]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [7]Godunov, S. K. Finite dierence methods for numerical computations of discontinuous solutions of equations of fluid dynamics. Mat. Sb. (1959), 271{295. In Russian. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 486 II.B Exact Methods for PDEs [8]Iraniparast, N. Green{Riemann functions for a class of hyperbolic focal point problems. SIAM J. Math. Anal. 20 , 2 (March 1989), 408{414. [9]Kreith, K. Establishing hyperbolic Green’s functions via Leibniz’s rule. SIAM Review 33 , 1 (March 1991), 101{105. [10]Sneddon, I. N. Elements of Partial Dierential Equations . McGraw{Hill Book Company, New York, 1957. [11]Woodward, P., and Colella, P. The numerical simulation of two- dimensional fluid flow with strong shocks. J. Comput. Physics 54 (1984), 115{173. [12]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 112. Separation of Variables 487 112. Separation of Variables Applicable to Most often, linear homogeneous partial dierential equations. Yields An exact solution, generally in the form of an innite series. Idea We look for a solution to a partial dierential equation by separating the solution into pieces, where each piece deals with a single dependentvariable. Procedure For linear homogeneous partial dierential equations, try to represent the solution as a sum of terms in which each term factors into a product of expressions, each expression dealing with a single independent variable. For nonlinear equations, try to represent the solution as a sum of suchexpressions. In all cases, not only must the equation admit a solution of the proposed form, but the boundary conditions must also have the right form. In more detail, suppose that L[u] = 0 is a linear partial dierential equation for u(x) that has the form L[u]=P iLi[u], where the Li[u] are dierential operators. We look for a solution of this partial dierentialequation in the form u(x)=u(x 1;x2;


;xn)=X1(x1)X2(x2):::Xn(xn); where the functions fX1;X2;:::;Xngare to be determined. By using the above form in the original equation and reasoning about which terms depend upon which variables, we can often reduce the original partial dierential equation into an ordinary dierential equation for each of the fXig. In carrying this out, arbitrary constants will be introduced. After the resulting ordinary dierential equations are solved, the arbitrary constants can generally be found by physical reasoning. Because superposition can be used in linear equations, any number of terms (of the form shown above) will also be a solution of the original equation. Also, if each of these terms is multiplied by some constant and then added together, the resulting expression will also be a solution. Hence,the nal solution will frequently be a sum or an integral. This sum will have unknown constants in it due to the constants allowed in the superposition. These constants will be determined from the initialconditions and/or the boundary conditions. The only time that we can be sure that we have found the most general solution to a given ordinary dierential equation by this technique is when CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 488 II.B Exact Methods for PDEs there exists a \completeness theorem" for each of the ordinary dierential equations that we have found. Example 1 Suppose we wish to solve the heat equation in a circle @u @t=r2u1 r@ @r r@u @r +1 r2@2u @2; (112.1) foru(t;r; ). We try to separate variables in equation (112.1) by proposing a solution of the form u(t;r; )=T(t)R(r)(): (112.2) Substituting equation (112.2) into equation (112.1) and simplifying yields 1 rRd dr rdR dr +1 r2d2 d2−1 TdT dt=0: (112.3) By the assumption made implicitely in equation (112.2), only the third term in equation (112.3) has any dependence on the variable t. Because the other terms cannot have any tdependence, it must be that the third term also has no tdependence. Therefore, this term must be equal to some (unknown) constant; that is, 1 TdT dt=−= some unknown constant : (112.4) The minus sign in equation (112.4) is taken for convenience later. Using equation (112.4) in equation (112.3) and simplifying, we nd r Rd dr rdR dr +r2+1 d2 d2=0: (112.5) The third term in equation (112.3) is the only one that could depend on , but we easily see that it cannot depend on because the rst two terms in equation (112.5) could not cancel out any dependence. Therefore, we must conclude that 1 d2 d2=−= another unknown constant : (112.6) Using equation (112.6) in equation (112.5), we nd rd dr rdR dr +(−+r2)R=0: (112.7) Note that we have, at this point, found ordinary dierential equations that describe each of the terms in the solution proposed in equation (112.2). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 112. Separation of Variables 489 But, in doing so, we have introduced two arbitrary constants; and . Solving the ordinary dierential equations in equations (112.4), (112.6), and (112.7) yields T(t)=Ae−t; ()=Bsin(p)+Ccos(p); R(r)=DJp(p r)+EYp(p r);(112.8) wherefA;B;C;D;Egare arbitrary constants and fJ;Ygare Bessel func- tions. By superposition, the most general solution to equation (112.2) can now be written as u(t;r; )=Z1 −1dZ1 −1d e−th B(;) sin(p)+C(;)c o s (p)i h D(;)Jp(p r)+E(;)Yp(p r)i ; (112.9) wherefB;C;D;Egmay depend on and. Now physical reasoning and the initial conditions and boundary conditions must be used to evaluate fB;C;D;Eg. For example, if the heat equation in (112.1) is being solved in the entire circle, then it must be that the solution is periodic in with period 2 . That is,u(t;r; )=u(t;r; +2). This constraint (which is equivalent to ()= (+2)), placed on equation (112.8), restrictspto be an integer. Hence, in this case, the most general solution has the form (using n2=) u(t;r; )=Z1 −1d1X n=0e−th B(;n2)s i nn+C(;n2)c o sni h D(;n2)Jn(p r)+E(;n2)Yn(p r)i : If the point r= 0 was included in the domain of the original problem, then we would require E(;n2)0 becauseYn(r) is unbounded at r=0 . Likewise, only those values of 0 will be physically realistic. Hence, in this case, we nd u(t;r; )=Z1 0d1X n=0e−th B(;n2)s i nn+C(;n2)c o sni Jn(p r): (112.10) More conditions could be placed on the coecients depending on the exact form of the initial conditions and boundary conditions. Example 2 Suppose we have the nonlinear equation f(x)u2 x+g(y)u2 y=a(x)+b(y) (112.11) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 490 II.B Exact Methods for PDEs to solve. We might propose a solution of the form u(x;y)=(x)+ (y): (112.12) Using equation (112.12) in equation (112.11) results in the equation f(x)[0(x)]2−a(x)=g(y)[ 0(y)]2−b(y): (112.13) The left-hand side of equation (112.13) must be independent of x(because the right-hand side is); hence, we can set f(x)[0(x)]2−a(x)== some constant ; (112.14) and then g(y)[ 0(y)]2−b(y)=: (112.15) Solving equations (112.14) and (112.15), we have determined that a solution to equation (112.11) is given by v(x;y)=Zx x0s a()+ f()d+Zy y0s b()− g()d+; (112.16) whereis another arbitrary constant. The solution in (112.16) may not be the most general solution to equation (112.11). For nonlinear equations, it is very dicult to determine whether the most general solution has been found. Notes 1. Note that the solution in equation (112.10) could also have been obtained by use of Fourier series (see page 344). The form of the solution in equation (112.10) (i.e., the e−tterm) suggests that a Laplace transform might also be an appropriate way to analyze equa-tion (112.1). 2. Carslaw and Jaeger [4] have the decompositions (similar to equation (112.9)) for many heat conduction problems. 3. If the equation L[u] = 0 can be separated into ordinary dierential equations when u(x)= u1(x1)u2(x2)


un(xn) R(x)andR6= 1, then the equation is said to be Rseparable. 4. Moon and Spencer [11] list 11 common orthogonal coordinate systems in which both Laplace’s equation and Helmholtz’s equation separate. These coordinate systems are rectangular, circular cylinder, ellip- tic cylinder, parabolic cylinder, spherical, prolate spheroidal, oblatespheroidal, parabolic, conical, ellipsoidal, and paraboloidal. Also included are the exact decompositions that are obtained (similar to CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 112. Separation of Variables 491 (112.9)). The above analysis is repeated for 21 dierent cylindrical coordinate systems that are obtained by translating an orthogonal map in a direction perpendicular to the plane of the map. The above analysis is again carried out for 10 dierent rotational coordinatesystems that are obtained by twirling an orthogonal map in a plane about an axis. In each of these 31 coordinate systems, Laplace’s equation or Helmholtz’s equation separates (or is Rseparable). 5. A necessary and sucient condition for a system with 2 degrees of freedom, with the Hamiltonian H= 1 2(p2 x+p2 y)+V(x;y), to be separable in elliptic, polar, parabolic, or cartesian coordinates is that the expression (Vyy−Vxx)(−2axy−b0y−bx+d) +2Vxy(ay2−ax2+by−b0x+c−c0) +Vx(6ay+3b)+Vy(−6ax−3b0) vanishes for some constants ( a;b;b0;c;c0;d)6=( 0;0;0;c;c; 0). The values of these constants determine in which of the above four co- ordinate systems the dierential equations separate. For 3 degrees of freedom, a similar expression has been devised that determines inwhich of 11 dierent coordinate systems the equations separate. For more details, see Marshall and Wojciechowski [9]. 6. The equation ( +V(x))u=utt−uxx+V(x)u=0( w h e r e is the D’Alembert operator) can be non-trivially separated if and only if the function V(x) is given (up to an equivalence relation) by one of the following 12 forms (here m;m 1;m2are arbitrary real parameters andm26=0 ) : (a)V=(m1+m2sinx)c o s−2x (b)V=(m1+m2sinhx)c o s h−2x (c)V=(m1+m2coshx) sinh−2x (d)V=m1ex+m2e2x (e)V=m1+m2x−2 (f)V=m(g)V=mx (h)V=mx−2 (i)V=msin−2x (j)V=msinh−2x (k)V=mcosh−2x (l)V=mex (See Zhdanov et al. [13].) Using these forms for V(x), there are 8 inequivalent forms of ( +V)u= 0 that can be non-trivially separated. These forms and the number of coordinate systems in which they separate are: (a) 2systems: u+mxu =0 (b) 9systems: u+mx−2u=0 (c) 4systems: u+(m1+m2cosx)s i n−2xu=0 (d) 4systems: u+(m1+m2sinhx)c o s h−2xu=0 (e) 11systems: u+(m1+m2coshx) sinh−2xu=0 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 492 II.B Exact Methods for PDEs (f) 6systems: u+(m1+m2ex)exu=0 (g) 6systems: u+(m1+m2x−2)u=0 (h) 11systems: u+mu=0 7. The Hartree{Fock approximation is a technique for approximating the eigenfunctions u(x) and eigenvalues of the partial dierential equation −r2u+f(x)u=u; (112.17) whenf(x) is a prescribed function. The technique consists of ap- proximating f(x)b y f(x)’f1(x1)f2(x2)


fn(xn): Iff(x) has the form shown above, then equation (112.17) can be solved by separation of variables. The solution will be of the form u(x)=u1(x1)u2(x2)


un(xn); =1+2+


+n: In the Hartree{Fock approximation, a variational principle is used to determine what the \best" ffj(xj)gare. See Fischer [6] for details. 8. Miller [10] contains a group theoretical approach to the method of separation of variables. For many linear dierential equations, theseparated solutions are easily related to the Lie algebra generated by the equation. 9. See Boyce and DiPrima [3, Chapter 10, pages 513{580]. References [1]Arscott, F. M., and Darai, A. Curvilinear co-ordinate systems in which the Helmholtz equation separates. IMA J. Appl. Mathematics 27 (1981), 33{70. [2]Blum, E. K., and Reid, G. J. On the numerical solution of three- dimensional boundary value problems by separation of variables. SIAM J. Numer. Anal. 25 , 1 (Februrary 1988), 75{90. [3]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [4]Carslaw, H. S., and Jaeger, J. C. Conduction of Heat in Solids . Clarendon Press, Oxford, England, 1984. [5]Doyle, P. W. Separation of variables for scalar evolution equations in one space dimension. J. Phys. A: Math. Gen. 29 (1996), 7581{7595. [6]Fischer, C. F. Approximate solution of Schr odinger’s equation for atoms. In Numerical Integration of Dierential Equations and Large Linear Systems , J. Hinze, Ed. Springer{Verlag, New York, 1982, pp. 71{81. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 112. Separation of Variables 493 [7]Hainzl, J. On a general concept for separation of variables. SIAM J. Math. Anal. 13 , 2 (March 1982), 208{225. [8]Kaufman, L., and Warner, D. D. Algorithm 685: A program for solving separable elliptic equations. ACM Trans. Math. Software 16 , 4 (Dec 1990), 323{351. [9]Marshall, I., and Wojciechowski, S. When is a Hamiltonian system separable? J. Math. Physics 29 , 6 (June 1988), 1338{1346. [10]Miller, Jr., W. Symmetry and Separation of Variables . Addison{Wesley Publishing Co., Reading, MA, 1977. [11]Moon, P., and Spencer, D. E. Field Theory for Engineers .D . V a n Nostrand Company, Inc., New York, 1961. [12]Moon, P., and Spencer, D. E. Field Theory Handbook . Springer{Verlag, New York, 1961. [13]Zhdanov, R. Z., Revenko, I. V., and Fushchych, W. I. Orthogonal and non-orthogonal separation of variables in the wave equation zzzref1refzzz. J. Phys. A: Math. Gen. 26 (1993), 5959{5972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 494 II.B Exact Methods for PDEs 113. Separable Equations: St¨ackel Matrix Applicable to Helmholtz’s or Laplace’s equation in some orthog- onal coordinate systems. Yields An exact solution, generally in the form of an innite series. Idea If certain conditions hold, then it is possible to separate variables in an orthogonal coordinate system for Helmholtz’s equation or for Laplace’s equation. Procedure Suppose we have an orthogonal coordinate system in the variables fu1;u2;u3gwith the metric fgiig. As usual, we dene g=g11g22g33. Assume that the St¨ ackel matrix Sis dened by S=2 411(u1) 12(u1) 13(u1) 21(u2) 22(u2) 23(u2) 31(u3) 32(u3) 33(u3)3 5 in which each row only contains functions of one variable. Dene the determinant of Sto bes s= 111213 212223 313233; and note that the cofactors of the elements in the rst column are given by M 11= 2223 3233M 21=− 1213 3233M 31= 1213 2223; If the following relations hold g ii=s Mi1pg s=f1(u1)f2(u2)f3(u3);(113.1) then the Helmholtz equation r2W+2W= 0 separates with the solution given byW=W1(u1)W2(u2)W3(u3), where thefWigare dened by 1 fid dui fidWi dui +Wi3X j=1jij=0; (113.2) with1=2,a n d2and3arbitrary. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 113. Separable Equations: St¨ ackel Matrix 495 Example In parabolic coordinates f;; g, we have the metric coecients g11= g22=2+2andg33=22. Hence,pg=(2+2). The Laplacian in parabolic coordinates is given by r2=1 2+2@2 @2+1 @ @+@2 @2+1 @ @ +1 22@2 @ 2: With this form, it would appear unlikely that the Helmholtz equation r2W+2W= 0 would separate. But, note that the St¨ ackel matrix S=2 42−1−−2 21−−2 00 13 5; from which we nd s=2+2,M11=M21=1 ,a n dM31=−2+−2, satises the equations in (113.1) (when we take f1=,f2=,f3=1 ) . From this we conclude that the Helmholtz equation does separate in par- abolic coordinates. The separation equations (corresponding to equation (113.2)) are 1 d d dW1 d +W1 12−2−3 2 =0 1 d d dW2 d +W2 12+2−3 2 =0 d2W3 d 2+3W3=0; whereW=W1()W2()W3( ). Notes 1. The St¨ ackel matrix is not unique. 2. Not all orthogonal coordinate systems allow separation. 3. All cylindrical coordinate systems in which the Helmoltz equation separates has a St¨ ackel matrix of the form S=2 40 1213 0 2223 10 13 5: 4. For every rotational coordinate system, the Helmholtz equation sep- arates with a St¨ ackel matrix of the form S=2 4111213 212223 0013 5: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 496 II.B Exact Methods for PDEs 5. Necessary and sucient conditions for separation of the Laplace equa- tion (r2W= 0) are gii gjj=Mj1 Mi1pg gii=f1(u1)f2(u2)f3(u3)Mi1: References [1]Boyer, C. P., Kalnins, E. G., and Miller, Jr, W. Stackel-equivalent integrable Hamiltonian systems. SIAM J. Math. Anal. 17 (1986), 778{797. [2]Eisenhart, L. P. Separable systems of Stackel. Annals of Math. 35 (1934), 284{305. [3]Kalnins, E. G., Benenti, S., and Miller, Jr, W. Integrability, Stackel spaces, and rational potentials. J. Math. Physics 38 , 5 (May 1997), 2345{ 2365. [4]Kalnins, E. G., and Miller, W. Dierential{Stackel matrices. J. Math. and Physics 26 (1995), 1560{1565. [5]Kalnins, E. G., and Miller, W. Generalized{Stackel matrices. J. Math. and Physics 26 (1995), 2168{2173. [6]Kalnins, E. G., and Miller, Jr, W. The general theory of R-separation for Helmholtz equations. J. Math. Physics 24 (1983), 1047{1053. [7]Moon, P., and Spencer, D. E. Field Theory Handbook . Springer{Verlag, New York, 1961. [8]Stackel, P. Uber die integration der Hamilton{Jacobischen dierentialge- ichung mittels separation der variabeln . Habilitationschrift, Halle, 1891. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 114. Similarity Methods 497 114. Similarity Methods Applicable to Linear or nonlinear partial dierential equations, and also systems of dierential equations. Yields An equation with one fewer independent variables. Idea Sometimes the number of independent variables in a partial dierential equation can be reduced by taking algebraic combinations of the indepen-dent variables. Procedure The idea of this method is to nd new independent variables (called similarity variables ) that are combinations of the old independent variables. The dierential equation, when written in the new variables, will notdepend on all of the new variables. One technique for discovering the correct new variables is to choose temporary variables to be a parameter to some (unknown) power times the old variables. After writing the equation in terms of the temporary vari- ables, the powers can be found by requiring homogeneity in the parameter.New variables are then constructed from the old variables in such a way that the parameter does not enter. Example 1 Suppose the following linear partial dierential equation @u @t+u 2t=@2u @z2; (114.1) foru(t;z) is to be simplied from being a function of the two independent variablesft;zgto being a function of only one independent variable. We dene the temporary variables u0,z0,t0and the parameter by u=u0; t=t0m; z=z0n;(114.2) for some unknown values of nandm. In these temporary variables, equation (114.1) becomes @u0 @t01−m+u0 2t01−m=@2u0 @(z0)21−2n: (114.3) For the parameter to be eliminated from equation (114.3), we require that the exponents of in each term of equation (114.3) all be the same. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 498 II.B Exact Methods for PDEs That is, 1−m=1−2n. This equation has the solution m=2n. At this point we know that there are similarity solutions of equation (114.1) but still must determine what they are. Using m=2nin (114.2), the change of variables becomes u=u0; t=t02n; z=z0n:(114.4) Combining the original independent variables ft;zg, we form a new inde- pendent variable fgwhose transformation from the old variables to the temporary variables does not depend on : :=zp t=z0 p t0: Now we have to propose the similarity solution. We look for a solution of the form u(t;z)=vzp t =v(): (114.5) When the form in equation (114.5) is used in equation (114.1), we obtain 2d2v d2+dv d−v=0; (114.6) which is now an ordinary dierential equation. Every solution of equation (114.6) will generate a solution of equation (114.1). Example 2 Consider the following nonlinear partial dierential equation: @u @t+u 2t+u@u @z=@2u @z2(114.7) foru(t;z). This equation diers from equation (114.1) by the uuzterm. We wish to simplify this equation from being a function of the two indepen-dent variablesft;zgto being a function of only one independent variable. After we do this, we will nd a solution for the = 0 case. We dene the temporary variables u 0,z0,t0, and the parameter by equation (114.2). In these temporary variables, equation (114.7) becomes @u0 @t01−m+u0 2t01−m+u0@u0 @z02−n=@2u0 @(z0)21−2n: (114.8) For the parameter to be eliminated from equation (114.8), we require that the exponents of in each term of equation (114.8) all be the same. That is, 1−m=2−n=1−2n: (114.9) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 114. Similarity Methods 499 These equations have the unique solution: n=−1,m=−2. At this point, we know that there is a similarity solution of equation (114.7). Using n=−1,m=−2 in equation (114.2) changes the variables to fu=u0, t=t0−2,z=z0−1g. Combining the original independent variables ft;zg, we form a new independent variable fgwhose transformation from the old variables to the temporary variables does not depend on : :=zp t=z0 p t0: Combining the original dependent variable fugwith the original indepen- dent variablesft;zg, we can form a new dependent variable fwgwhose transformation from the old variables to the temporary variables does not depend on: w=t zu=t0 z0u0: (114.10) Now we have to propose the similarity solution. By solving equation (114.10) for u, we are led to the assumption u(t;z)=z twzp t =z tw(): (114.11) When the form in equation (114.11) is used in equation (114.7), we obtain 2d2w d2+/parenleftbig 4+2−22w
dw d+( 1−2w)w=0: (114.12) If we dene g()b yg()=w(), then equation (114.12) becomes 2d2g d2+(−2g)dg d=0: (114.13) Every solution of this ordinary dierential equation will lead to similarity solutions of equation (114.7). In the special case of = 0 (when equation (114.7) becomes the identical to equation (114.1)), the general solution toequation (114.13) is given by g()=A+Berf p 4 ; whereAandBare arbitrary constants. This results in the solution u(t;z)=1p t A+Berfzp 4t to equation (114.1). Note that this similarity solution could nothave been obtained from equation (114.6), because the scalings in equations (114.5) and (114.11) are dierent. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 500 II.B Exact Methods for PDEs Notes 1. In general, a partial dierential equation may have some similarity solutions and some solutions that are not similarity solutions. 2. This method is sometimes called the method of one parameter groups , due to the single parameter that was used in equation (114.2). This method is derivable from Lie group methods (see page 471). 3. To solve a dierential system (dierential equation(s) with boundary condition(s)), the boundary conditions as well as the equation(s) mustadmit the similarity variable. 4. This method also applies to systems of ordinary dierential equations. If du dx=f(x;u) is a system of rst order ordinary dierential equations foru=(u1;:::;un), and if there exists a one parameter group of symmetries of the system, then there is a change of variables ( y;w)= (x;u), which takes the system intodw dy=g(y;w1;:::;wn−1). Hence, the original system reduces to a system of n−1 ordinary dierential equations for ( w1;:::;wn−1) together with the quadrature wn(y)=R gn(y;w1(y);:::;wn−1(y))dy. 5. For some systems, there are natural similarity variables. For example, in a two-dimensional problem with radial symmetry, the variable r (wherer2=x2+y2) should be a similarity variable if the original equations were written in terms of xandy. Similarly, in a radially symmetric three-dimensional problem, the variable (where2= x2+y2+z2) should be a similarity variable. 6. For diusion equations, similarity solutions are often of the form f(x=p t)o rtf(x=p t). 7. The partial dierential equation F/parenleftbig tx;u;ut x;ux t
=0f o ru(x;t)h a s the similarity variable w=tx. Considering u=u(w), we nd the equivalent ordinary dierential equation F(w;u;uw;uw)=0 . 8. See also Ames [1, pages 135{141] and Seshadri and Na [6, pages 39{ 42]. References [1]Ames, W. F. Ad hoc exact techniques for nonlinear partial dierential equations. In Nonlinear Partial Dierential Equations in Engineering ,W .F . Ames, Ed. Academic Press, New York, 1967. [2]Dressner, L. Similarity Solutions of Nonlinear Partial Dierential Equa- tions . Pitman Publishing Co., Marsheld, MA, 1983. [3]Kevorkian, J., and Cole, J. D. Perturbation Methods in Applied Mathematics . Springer{Verlag, New York, 1981. [4]King, J. R. Exact similarity solutions to some nonlinear diusion equations. J. Phys. A: Math. Gen. 23 (1990), 3681{3697. [5]Roseneau, P., and Schwarzmeier, J. L. Similarity solutions of systems of partial dierential equations. Comput. Physics Comm. 27 (1982), 179{186. [6]Seshadri, R., and Na, T. Y. Group Invariance in Engineering Boundary Value Problems . Springer{Verlag, New York, 1985. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 115. Exact Solutions to the Wave Equation 501 115. Exact Solutions to the Wave Equation Applicable to Then-dimensional wave equation. Yields An explicit solution in terms of an integral. Idea An exact formula is available for the n-dimensional wave equation utt=r2u. Procedure Then-dimensional wave equation @2u @t2=r2u=@2u @x12+


+@2u @xn2; (115.1) with the initial data (we use x=(x1;:::;xn)) u(0;x)=f(x);ut(0;x)=g(x); (115.2) has two dierent (but similar) forms of the solution, depending on whether nis even or odd. When nis odd the solution is given by u(t;x)=1 1
3


(n−2)@ @t@ t@t(n−3)=2 tn−2![f;x;t] +@ t@t(n−3)=2 tn−2![g;x;t] ;(115.3) where![h;x;t] is dened to be the average of the function h(x)o v e rt h e surface of an n-dimensional sphere of radius tcentered at x.T h a ti s , ![h;x;t]=1 n(t)Z h(0;)dΩ; wherej−xj2=t2,n(t) is the surface area of the n-dimensional sphere of radiust,a n ddΩ is an element of area. (Note that n(t)=2n=2tn−1=Γ/parenleftbign 2
.) Whennis even the solution to equation (115.1) and equation (115.2) is given by u(t;x)=1 2
4


(n−2)@ @t@ t@t(n−2)=2Zt 0![f;x;]n−1dp t2−2 +@ t@t(n−2)=2Zt 0![g;x;]n−1dp t2−2 ; (115.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 502 II.B Exact Methods for PDEs where![h;x;t] is dened as above. Because the expression in equation (115.4) is integrated over , the values of fandgmust be known everywhere in the interior of then-dimensional sphere. Special Case 1 Whenn= 1, the above formulae produce the D’Alembert solution (see Chester [1, pages 17{23]) of the equation utt=c2uxx: u(x;t)=1 2[f(x−ct)+f(x+ct)] +1 2cZx+ct x−ctg()d: (115.5) Special Case 2 Whenn= 2, the above formulae produce the Parseval solution u(x;t)=1 2@ @tZZ R(t)f(x1+1;x2+2)p t2−2 1−2 2d1d2 +1 2ZZ R(t)g(x1+1;x2+2)p t2−2 1−2 2d1d2; whereR(t) is the regionf(1;2)j2 1+2 2t2g. Special Case 3 Whenn= 3, the above formulae produce the Poisson solution (also known as the Kircho solution) u(x;t)=@ @t t![f;x;t] +t![g;x;t]; where ![h;x;t]=1 4Z2 0Z 0h(x1+tsincos;x2+tsinsin;x3+tcos) sindd: Example A string stretched in the shape of a sine wave and then released from rest will have the displacement u(x;t), where utt=uxx; u(x;0) = sinx; ut(x;0) = 0: By virtue of equation (115.5), this has the solution u(x;t)=1 2 sin(x−t)+ sin(x+t) . CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 115. Exact Solutions to the Wave Equation 503 Notes 1. The solutions given in equation (115.3) and equation (115.4) may be derived from one another by the method of descent (see page 446). 2. The name \D’Alembert solution" is also applied to the solution of the wave equation in a semi-innite domain vtt=c2vxx; v(0;t)=0; for 0<t<1; v(x;0) =f(x);for 0x<1; vt(x;0) =g(x);for 0x<1: This equation has the solution (see Farlow [2, page 143], page 143) v(x;t)=( 1 2[f(x+ct)+f(x−ct)] +1 2cRx+ct x−ctg()d; forxct; 1 2[f(x+ct)−f(ct−x)] +1 2cRx+ct ct−xg()d; forx<ct: 3. Consider the inhomogeneous wave equation @2u @t2−@2u @x2−@2u @y2−@2u @z2=F(t;x;y;z ); with the homogeneous initial conditions: u(0;x;y;z )=0;ut(0;x;y;z )=0: The solution is given by u(t;x;y;z )=1 4ZZZ tF(t−;;; ) ddd; with=p (x−)2+(y−)2+(z−)2. 4. Another useful formula is for the solution of @2u @t2=@2u @x2+@2u @y2+@2u @z2+u; u(0;x;y;z )=f(x;y;z ); ut(0;x;y;z )=g(x;y;z ); whereis an arbitrary constant. The solution is given by u(t;x;y;z )=@ @t t![f;x;t]+Zt 02![f;x;]I(t2−2)d +t![g;x;t]+Zt 02![g;x;]I(t2−2)d; whereI(a): =I0 0(pa)=paandI0is the usual modied Bessel func- tion. 5. See Farlow [2, Lessons 17 and 18, pages 129{145] and Garabedian [3, pages 191{210]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 504 II.B Exact Methods for PDEs References [1]Chester, C. R. Techniques in Partial Dierential Equations . McGraw{Hill Book Company, New York, 1970. [2]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [3]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 116. Wiener{Hopf Technique 505 116. Wiener{Hopf Technique Applicable to Linear partial dierential equations on an innite interval that have dierent types of boundary data on dierent parts of theinterval. Yields An exact solution. Idea In some linear partial dierential equations, we would like to take a Fourier transform but cannot because the boundary data type changes along the boundary. The Wiener{Hopf technique is to take a Fourier transform anyway and allow part of the data to be \missing." Solving the problem (using Liouville’s theorem), we determine the \missing" data and the solution simultaneously. Procedure Sometimes a linear partial dierential equation has a form amenable to a Fourier transform, but the boundary conditions would seem to preclude it. For example, the reduced wave equation r2+k2= 0 (116.1) in two dimensions may suggest the use of a Fourier transform in x.B u t ,i f the boundary conditions are given by, say, @(x;0) @y=0 f o rx0; (x;0) is continuous for x<0;(116.2) then it is not clear how to take such a transform. Generally, we would require@=@y to be known for all x, before we could take a Fourier transform. The solution technique is to assume that@=@y is known for allxand then take a Fourier transform. The quantity @=@y forx<0 will be determined when the nal solution is determined. The solution procedure uses Liouville’s theorem, one form of which is IfE(z) is an entire function (i.e., E(z) is analytic in the nite jzjplane) and if E(z) is bounded by a constant as jzj!1 , thenE(z) is identically constant. (See, e.g., Levinson and Redheer [4].) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 506 II.B Exact Methods for PDEs The dicult part of the solution procedure will turn out to be the \factorization" step. That is, given the functions A(!);B(!);C(!)( a l l analytic in the strip <=!< ), nd functions  +(!), Ψ−(!) satisfying A(!)+(!)+B(!)Ψ−(!)+C(!)=0; (116.3) where Equation (116.3) holds in the strip: <=!< . +(!) is analytic in the upper-half plane: <=!. Ψ−(!) is analytic in the lower-half plane: =!< . We will continue to use the following standard notation: a subscript of \+" (\−") indicates a function that is analytic in the upper (lower) half plane<=!(=!< ). Example Suppose we have the linear partial dierential equation exterior to the half line (y=0;x0) xx+yy−x=0; (116.4) with the boundary conditions !0a sr=p x2+y2!1; =e−xony=0;x0:(116.5.a-b) Dene the Fourier transform of (x;y)b y (!;y)=1p 2R1 −1(x;y)ei!xdx. If we assume that x!0a sr!1 , then equation (116.4) can be Fourier transformed (by multiplying by ei!xand integrating with respect to x)t o yield d2 dy2−(!2−i!) = 0: (116.6) If we extend the denition of (x;0) in equation (116.5.b) to be (x;0) =( e−xforx0; u(x)f o rx<0;(116.7) whereu(x) is unknown, then we can transform equation (116.7) to nd (!;0) =U(!)+1p 21 1−i!; (116.8) whereU(!) is the Fourier transform of u(x) on the semi-innite interval; that is, U(!)=1p 2Z0 −1u(x)ei!xdx: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 116. Wiener{Hopf Technique 507 The solution of equation (116.6) (which is an ordinary dierential equa- tion iny) using equation (116.8), which vanishes as jyj!1 ,i s (!;y)= U(!)+1p 21 1−i! exp −jyjp !2−i! ; (116.9) where the square root branch is specied by <p !2−i!0. Once we determine U(!), we can (in principle) invert equation (116.9) by taking an inverse Fourier transform. This would yield (x;y). Finding U(!) is the hard part of the calculation. Because the solution of the original problem (and its derivatives) must be continuous across y=0( f o rx<0), we dene a function f(x)b y f(x): =y(x;0+)−y(x;0−); =( 0f o rx<0; v(x)f o rx>0;(116.10.a-b) where 0+(0−) indicates a vanishingly small quantity that is greater (less) than zero and v(x) is an unknown function. Taking the Fourier transform of equation (116.10.b) produces F(!): =1p 2Z1 −1f(x)ei!xdx =1p 2Z1 0v(x)ei!xdx;(116.11) whereas the Fourier transform of equation (116.10.a) produces F(!)=y(!;0+)−y(!;0−) =−2 U(!)+1p 21 1−i!p !2−i!;(116.12) where the solution in equation (116.9) has been used. Using our subscript convention and the denition in equation (116.11), we note that F(!)= F+(!), where, for instance, we could take =1=3. We now assume thatU(!)=U−(!), for, say, =2=3. This places a constraint on u(x) that has to be veried at the end of the calculation. By algebraic manipulations of equation (116.12), we can obtain (this step should not be trivialized, it is the hardest step in the calculation) −F+(!) 2p!−p−ip(1−i!) +=U−(!)p !−i+p!−i−p−2ip 2(1−i!) − (116.13) If we dene E(!) to be the left-hand side of equation (116.13), then E(!) is entire. This is because the left-hand side and the right-hand side CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 508 II.B Exact Methods for PDEs of equation (116.13) overlap in the strip <=!< ,a n dt h e s et w o functions are analytic in their respective half planes. Hence, one side of equation (116.13) supplies the analytic continuation of the other side. If we now assume that F+(!)!0a sj!j!1 in=!> , !U−(!)!0a sj!j!1 in=!< , thenE(!)!0a sj!j!1 . By Liouville’s theorem we can conclude that E(!)0 and so from equation (116.13) U(!)=U−(!)=−1p!−ip!−i−p−2ip 2p1−i! : Using this in equation (116.9) and taking an inverse Fourier transform yields(x;y). Notes 1. The Wiener{Hopf method was originally formulated for the solution of integral equations. 2. The problem in equations (116.1) and (116.2) is analyzed in more detail in Carrier et al. [1, pages 376{386]. The same problem, with an incident oblique wave, is solved in Davies [2, pages 288{307]. References [1]Carrier, G. F., Krook, M., and Pearson, C. E. Functions of a Complex Variable . McGraw{Hill Book Company, New York, 1966. [2]Davies, B. Integral Transforms and Their Applications , second ed. Springer{ Verlag, New York, 1985. [3]Heins, A. E. The scope and limitations of the method of Wiener and Hopf. Communications on Pure and Applied Mathematics 9 (1956), 447{466. [4]Levinson, N., and Redheffer, R. M. Complex Variables . Holden{Day, Inc., San Francisco, CA, 1979. [5]Noble, B. Methods Based on the Wiener{Hopf Technique . Pergamon Press, New York, 1958. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 510 III Approximate Analytical Methods 117. Introduction to Approximate Analysis Sometimes an exact solution cannot be obtained for a dierential equation and an approximate solution must be found. Other times, an approximate solution may convey more information than an exact solution. There are essentially two types of approximations: Those that give an approximation over a range of the independent variable Those that give an approximation only near a single point Approximations of the second type are more common. This section of the book is not broken up into methods for ordinary dierential equations and methods for partial dierential equations because most of the methods can be used for either type of dierential equation. Listed below are, in the author’s opinion, those methods that are the most useful when approximating the solution to ordinary dierential equa- tions and partial dierential equations. These are the methods that might be tried rst. Most Useful Methods Collocation (page 514) Dominant Balance (page 517) Graphical Analysis: The Phase Plane (page 526) Least Squares Method (page 549) Lyapunov Functions (page 551) Newton’s Method (page 578) Perturbation Method: Method of Averaging (page 586) Perturbation Method: Boundary Layer Method (page 590) Perturbation Method: Regular Perturbation (page 610) WKB Method (page 642) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 118. Chaplygin’s Method 511 118. Chaplygin’s Method Applicable to An initial value problem for a single rst order ordinary dierential equation. Yields Improved upper and lower bounds on the solution. Idea Using an upper and lower bound on the solution, a set of tighter bounds can be constructed. Procedure For an equation of the form y0=f(x;y),y(x0)=y0, the method is derived from the following theorem (due to Chaplygin): Theorem : If the dierential inequalities u0(x)−f(x;u(x))<0; v0(x)−f(x;v(x))>0;(118.1) hold forx>x 0,w i t hu(x0)=y0andv(x0)=y0,t h e n u(x)<y(x)<v(x) (118.2) holds for all x>x 0. The procedure is to determine (or \guess") a u(x)a n dav(x)t h a t satisfy equation (118.1). Then there are two dierent techniques available for computingfu1(x);v1(x)g, such that u(x)<u1(x)y(x)v1(x)<v(x): (118.3) For each of the two techniques, the functions fu1(x);v1(x)gwill be dierent. The functions obtained, fu1(x);v1(x)g, will also satisfy equation (118.1), and the process may be iterated. Special Case 1 LetKbe the Lipschitz constant of the function f(x;y). Then, if fu1(x);v1(x)gare dened by u1(x)=u(x)+Zx x0e−K(x−t)[f(t;u(t))−u0(t)]dt; v1(x)=v(x)−Zx x0e−K(x−t)[v0(t)−f(t;v(t))]dt; then equation (118.3) will be satised. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 512 III Approximate Analytical Methods Special Case 2 For this technique, it must be true that @2f=@y2is of constant sign in the region of interest. Once this has been established, dene fM(x),N(x), cM(x),bN(x)gby M(x)y+N(x)=f(x;u(x)) +f(x;v(x))−f(x;u(x)) v(x)−u(x)(y−u(x)); cM(x)y+bN(x)=f(x;u(x)) +fy(x;u(x))(y−u(x)): (118.4) (Note that both sides of each equation are linear in the indeterminate y.) Then dene u1(x) to be the solution of y0=M(x)y+N(x);y (x0)=y0: (118.5) and denev1(x) to be the solution of y0=cM(x)y+bN(x);y (x0)=y0: (118.6) With these denitions for u1(x)a n dv1(x), equation (118.3) will be satised. Note that the equations (118.5) and (118.6) can be solved by the use ofintegrating factors (see page 356). Example Suppose we wish to bound the solution to the equation y0=y2+x2;y (0) = 0; whenxis in the range [ 0 ;1=p 2] . First, observe that u(x)=x3=3a n dv(x)=1 1x3=30 satisfy the con- ditions of Chaplygin’s theorem, so that equation (118.2) holds. Using the rst technique, we recognize that K=p 2 in the region of interest, so that the functions u1(x)=x3 3+1 9Zx 0t6e−p 2(x−t)dt; v1(x)=11 30x3−Zx 0t2 10−121 900t6 e−p 2(x−t)dt; (118.7) satisfy the constraint in equation (118.3). Using the second technique, we note that@2f=@y2= 2 and so we can use the results in equations (118.4), (118.5), and (118.6). It is straightforward to calculate M(x)=7 10x3;cM(x)=2 3x3; N(x)=x2−11 90x6;bN(x)=x2−1 9x6: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 118. Chaplygin’s Method 513 Solving equations (118.5) and (118.6), we nd u1(x)=ex4=6Zx 0 z2−1 9z6 e−z4=6dz; v1(x)=e7x4=40Zx 0 z2−11 90z6 e−7z4=40dz:(118.8) Notes 1. The above example is from Mikhlin and Smolitskiy [4]. The exact solution is given by y(x)=x Y−3=4(x2=2)−J−3=4(x2=2) J1=4(x2=2)−Y1=4(x2=2)=1 3x3+1 63x7+2 2079x11+O/parenleftbig x15
2. The approximations in equation (118.7) may be expanded about x= 0t oo b t a i n u1(x)=1 3x3+1 63x7+O/parenleftbig x8
;v 1(x)=1 3x3+O/parenleftbig x4
: 3. The approximations in equation (118.8) may be expanded about x= 0t oo b t a i n u1(x)=1 3x3+1 63x7+2 2079x11+O/parenleftbig x15
; v1(x)=1 3x3+1 63x7+O/parenleftbig x11
: 4. Another useful inequality (see McNabb [3]) is the following: Ifu(t),v(t), andf(t;w) satisfy sucient smoothness conditions on [a;b], ifu(a)<v(a), and ifu0−f(t;u)<v0−f(t;v)f o r a<tb,t h e nu(t)<v(t)o n[a;b]. 5. This procedure can be implemented numerically. 6. See also Lakshmikantham and Leela [2, pages 64{69] and Mikhlin and Smolitskiy [4, pages 9{12]. References [1]Fabry, C., and Habets, P. Upper and lower solutions for second-order boundary value problems with nonlinear boundary conditions. Nonlinear Analysis 10 , 10 (1986), 985{1007. [2]Lakshmikantham, V., and Leela, S. Dierential and Integral Inequalities . Academic Press, New York, 1969. [3]McNabb, A. Comparison theorems for dierential equations. J. Math. Anal. Appl. 119 (1986), 417{428. [4]Mikhlin, S. G., and Smolitskiy, K. L. Approximate Methods for Solutions of Dierential and Integral Equations . American Elsevier Publishing Company, New York, 1967. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 514 III Approximate Analytical Methods 119. Collocation Applicable to Ordinary and partial dierential equations. Yields An approximation to the solution, valid over an interval. Idea An approximation to the solution with some free parameters is pro- posed. The free parameters are determined by forcing the approximation to exactly satisfy the given equation at some set of points. Procedure Suppose we are given the dierential equation N[y]=0; (119.1) fory(x)i ns o m er e g i o n R, with the boundary conditions B[y]=0; (119.2) on some portion of the boundary of R. We choose an approximation to y(x) that has several parameters in it, say y(x)’w(x;), whereis a vector of parameters. This approximation is chosen in such a way that it satises the boundary conditions in equation (119.2). The unknown parameters aredetermined by requiring the approximation to satisfy equation (119.1) at some collection of points. Example Suppose we wish to approximate the solution to the ordinary dierential equation N[y]=y00+y+x=0; y(0) = 0;y(1) = 0;(119.3) by the method of collocation. We choose to approximate the exact solution by y(x)’w(x)=1x(1−x)+2x(1−x2): Note that w(x) satises the boundary conditions for y(x). Using this approximation, we nd N[w(x)] =−1(2−x+x2)−2(5x+x3)+x: Now, we must choose the collocation points. We choose the two points x=1=3a n dx=2=3. Requiring N[w(x)] to be zero at these two points CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 119. Collocation 515 results in the simultaneous equations −48 271−46 272−1 3=0; −48 271−98 272−2 3=0: The solution to these equations is 1=9=416,2=9=52. Hence, our approximation to the solution of equation (119.3) is y(x)’9 416x(1−x)+9 52x(1−x2): (119.4) Note that the exact solution to equation (119.3) is y(x)=sinx sin 1−x.T h e maximum dierence between the approximate solution in equation (119.4)and the exact solution in the range 0 <x< 1, occurs at x’0:7916 where the error is approximately 0.00081. Notes 1. This method is an example of a weighted residual method . 2. This method is often implemented numerically. 3. There are many choices for the form of the approximation to use. An increasingly popular technique is to use sinc functions; see, for example, Carlson et al. [2]. 4. Ascher et al. [1] contain a review of numerical implementations of the colocation method. References [1]Ascher, U., Christiansen, J., and Russell, R. D. Collocation software for boundary{value ODEs. ACM Trans. Math. Software 7 , 2 (June 1981), 209{222. [2]Carlson, T. S., Dockery, J., and Lund, J. A sinc-collocation method for initial value problems. Math. of Comp. 66 , 217 (January 1997), 215{235. [3]Collatz, L. The Numerical Treatment of Dierential Equations . Springer{ Verlag, New York, 1966. [4]Hanke, M. On a least-squares collocation method for linear dierential- algebraic equations. Numer. Math. 54 (1988), 79{90. [5]Houstis, E. N., Christara, C. C., and Rice, J. R. Quadratic-spline collocation methods for two-point boundary value problems. Internat. J. Numer. Methods Eng. 26 , 4 (1988), 935{952. [6]Houstis, E. N., Mitchell, W. F., and Rice, J. R. Collocation software for second-order elliptical partial dierential equations. ACM Trans. Math. Software 11 , 4 (Dec 1985), 379{412. [7]Hussaini, M. Y., Kopriva, D. A., and Patera, A. T. Spectral collocation methods. Appl. Num. Math. 5 (1989), 177{208. [8]Lie, I. The stability function for multistep collocation methods. Numer. Math. 57 , 8 (1990), 779{787. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 516 III Approximate Analytical Methods [9]Madsen, N. K., and Sincovec, R. F. Algorithm 540: PDECOL general collocation software for partial dierential equations. ACM Trans. Math. Software 5 , 3 (Sept 1979), 326{351. [10]Sakai, M. A collocation method for a singular boundary value problem. Congr. Numer. 62 (1988), 171{179. [11]Wright, K., Ahmed, A. H. A., and Seleman, A. H. Mesh selection in collocation for boundary value problems. IMA J. Num. Analysis 11 ,1 (January 1991), 7{20. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 120. Dominant Balance 517 120. Dominant Balance Applicable to Linear and nonlinear dierential equations. Yields An approximation to the solution valid in a region. Idea A dierential equation with many terms in it might be well determined by only a few of those terms. Procedure If there are Mterms in a dierential equation, try solving the dierential equation in a region by only considering 2 (or 3, or 4, :::,o rM−1) terms to be important in that region. Discard all the other terms and solve this dierential equation with fewer terms. After a solution is obtained, check that the discarded terms are actually smaller than the terms that wereretained. Example Suppose we have the equation y00−2 x3=2y0=3 16x2; (120.1) and we would like to nd an approximate solution as x!0. To determine the solution uniquely in this region, we must specify some information abouty(x)a sx!0. In this example, we choose the condition: y!0a s x!0. There are three dierent two-term balances of equation (120.1) that we can take; that is, the rst two terms in equation (120.1) can be takenapproximately equal, the rst and third terms can be taken approximately equal, or the second and third terms can be taken approximately equal. These possibilities yield the following two term balances: y 00−2 x3=2y0’0; which requires that jy00j3 16x2; (120.2) or y00’3 16x2; which requires that jy00j2y 0 x3=2; (120.3) or −2 x3=2y0’3 16x2; which requires that jy00j3 16x2: (120.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 518 III Approximate Analytical Methods We will investigate each of these in turn. The solution to equation (120.2) is y1(x)=A+BZ exp −4 x1=2 dx; whereAandBare arbitrary constants. Note that this solution violates the condition in equation (120.2) because jy00 1j=2jBj x3=2exp −4 x1=2 3 16x2asx!0: Therefore equation (120.2) is an inconsistent balance . The solution to equation (120.3) is y2(x)=−3 16logx+Cx+D; whereCandDare arbitrary constants. But this solution cannot satisfy y!0a sx!0, so it must also be discarded. The solution to equation (120.4) is y3(x)=−3 16px; where we have already used the fact that y!0a sx!0. For this solution, the condition in equation (120.4) is satised, because jy00j=3 32x3=23 16x2asx!0: Hence, we have found a consistent balance . We conclude that y(x)−3 16pxasx!0: Notes 1. Even if a consistent balance has been found, the solution associated with that balance may be unrelated to the true solution of the dier- ential equation(s). This is because a consistent balance has apparent consistency but not necessarily genuine consistency .A n o t h e r s e t o f words that express the same ideas are honest methods and dishonest methods . See Keller [2] or Lin and Segel [4, pages 188{189] for more details. 2. See Bender and Orszag [1, pages 83{88]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 120. Dominant Balance 519 References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Keller, J. B. Wave propagation in Random Media ,v o l .1 3o f Proc. Sympos. Appl. Math. Amer. Math. Soc., Providence, RI, 1960. [3]Levinson, N. Asymptotic behavior of solutions of non-linear dierential equations. Stud. Appl. Math. 48 (1969), 285{297. [4]Lin, C. C., and Segel, L. A. Mathematics Applied to Deterministic Problems in the Natural Sciences . The MacMillan Company, New York, 1974. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 520 III Approximate Analytical Methods 121. Equation Splitting Applicable to Dierential equations. Yields An exact solution but usually not the most general form of the solution. Idea By equating two parts of a dierential equation to a common term, we may be able to nd a fairly general solution to the given dierential equation. Procedure Separate a dierential equation into two (or more) terms such that a general solution is available for one of the terms. Use the other term(s) torestrict this general solution. Example 1 Suppose we have, from fluid dynamics, the stream function form of the boundary layer equations to solve for ( x;y): yxy−xyy=yyy: (121.1) We split this equation by choosing both the right and the left-hand sides of this equation to be identically equal to zero. That is, we break equation (121.1) into the two simultaneous equations yxy−xyy=0; yyy=0:(121.2.a-b) Any solution of equation (121.2) is also a solution of equation (121.1). Note that the converse is nottrue: A solution to equation (121.1) may not satisfy equation (121.2.a) or equation (121.2.b). Hence, the solution that is obtained from equation (121.2) will not be the most general solution. The general solution to equation (121.2.b) can be easily found because it is essentially an ordinary dierential equation in the independent variable y: (x;y)=a(x)y2+b(x)y+c(x); (121.3) for arbitrary coecient functions a(x),b(x), andc(x). Using equation (121.3) in equation (121.2.a), we conclude that (2ay+b)(2ya0+b0)−(a0y2+b0y+c0)(2a)=0 (121.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 121. Equation Splitting 521 must hold for all values of xandy. Hence,a(x),b(x), andc(x)c a nb e restricted by equating the coecients of y2,y1,a n dy0in equation (121.4) to zero. This results in coecient of y2:4aa0−2aa0=0; (121.5) coecient of y1:( 2ab0+ba0)−2ab0=0; (121.6) coecient of y0:bb0−2ac0=0: (121.7) Now we solve the equations appearing in equations (121.5), (121.6), and (121.7). Equation (121.5) can be valid only if a(x)i sac o n s t a n t ,s a y A. Then equation (121.6) is valid for any b(x) and equation (121.7) can be rewritten as (b2)0−4Ac0=0: (121.8) Equation (121.8) can be integrated to determine c(x)=b2(x) 4A+D,w h e r e Dis an arbitrary constant of integration. Now, using what we have found, the solution in equation (121.3) becomes (x;y)=Ay2+b(x)y+b(x)2 4A+D ; (121.9) for arbitrary A,D,a n db(x). Example 2 Basarab-Horwath et. al [2] present a method, which uses equation split- ting, for nding solutions of the d’Alembert equation u@2 @x02−@2 @x12−


−@2 @xn2 u=F(u): ChoosingP(w) to be an arbitrary polynomial and =−1;0;1, they make the change of variable u= (w), where  satises the dierential equation  00+0P0 P =F(): Then, using equation splitting, they arrive at the two partial dierential equations w=P0 P =@w @x02 −@w @x22 −


−@w @xn2 : They demonstrate their method by nding solutions of u=s i nu. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 522 III Approximate Analytical Methods Notes 1. Example 1 is from Ames [1, pages 59 and 65{69]. 2. Note that for the equations in (121.2) we could have found the general solution of equation (121.2.a) and then used equation (121.2.b) to restrict it. The general solution of equation (121.2.a) is ( x;y)= F(y+G(x)), whereFandGare arbitrary functions. Using this solution in equation (121.2.b) and determining conditions on Fand Gresults in the solution in equation (121.9). 3. See also Goldstein and Braun [3, page 109] and Whitham [4, page 421]. References [1]Ames, W. F. Ad hoc exact techniques for nonlinear partial dierential equations. In Nonlinear Partial Dierential Equations in Engineering ,W .F . Ames, Ed. Academic Press, New York, 1967. [2]Basarab-Horwath, P., Fushchich, W., and Serov, M. A simple method of nding solutions of the nonlinear d’Alembert equation. J. Phys. A: Math. Gen. 25 (1992), L871{L877. [3]Goldstein, M. E., and Braun, W. H. Advanced Methods for the Solution of Dierential Equations . U.S. Government Printing Oce, Washington, D.C., 1973. NASA SP-316. [4]Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 122. Floquet Theory 523 122. Floquet Theory Applicable to Linear ordinary dierential equations with periodic coecients and periodic boundary conditions. Yields Knowledge of whether all solutions are stable. Idea If a linear dierential equation has periodic coecients and periodic boundary conditions, then the solutions will generally be a periodic func-tion times an exponentially increasing or an exponentially decreasing func- tion. Floquet theory will determine if the solution is exponentially increas- ing (and so \unstable") or exponentially decreasing (and so \stable"). Procedure Suppose we have an nth order linear ordinary dierential equation whose coecients are periodic with common period T. The general tech- nique is to write the ordinary dierential equation as a rst order vector system of dimension n(see page 146), and then solve this vector ordinary dierential equation for any set of nlinearly independent conditions, for 0tT. This yields a propagator matrix B, such that y(t+mT)=Bmy(t), wherem=1;2;:::. Hence, to determine the stability of the original prob- lem, we need only determine the eigenvalues of B. If any of the eigenvalues are larger than one in magnitude, then the solution is \unstable." As an example of the general theory, we consider second order linear ordinary dierential equations of the form y00+q(t)y=0; (122.1) whereq(t) is periodic with period T, i.e.,q(t+T)=q(t). We can write equation (122.1) as a vector ordinary dierential equation in the form y(t)= y(t) y0(t) ; y0= 01 −q(t)0 y; where y(0) = y(0) y0(0) is known in principle. We now dene u(t)a n dv(t) to be the solutions of u(t) u0(t) = 01 −q(t)0 u(t) u0(t) ; u(0) u0(0) = 1 0 ; (122.2) and  v(t) v0(t)0 = 01 −q(t)0 v(t) v0(t) ; v(0) v0(0) = 0 1 : (122.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 524 III Approximate Analytical Methods Then, by superposition, y(t)=A(t)y(0) = u(t)v(t) u0(t)v0(t) y(0). Equiva- lently, y(T)=By(0), whereB=A(T). Hence, y(2T)=By(T)=B2y(0), y(3T)=B3y(0), etc. The eigenvalues of Bare needed to determine stability. By the usual calculation, will be an eigenvalue of Bif and only ifjB−Ij= 0. We calculate, jB−Ij=u(T)−v (T) u 0(T)v0(T)− = 2−[u(T)+v0(T)] + [u(T)v0(T)−u0(T)v(T)] =2−
+1;(122.4) where we have dened
= u(T)+v0(T), and we set u(T)v0(T)−u0(T)v(T) equal to one because the Wronskian of equation (122.1) is identically equal to one. Solving equation (122.4) for , we determine that =1 2
q 1 4
2−1, and so we conclude Ifj
j<2, then, for both values of ,w eh a v ejj1a n ds oa l lo f the solutions to equation (122.1) are stable. Ifj
j>2, then there is least one value of withjj>1a n ds ot h e solutions to equation (122.1) are unstable. Example Suppose we have the equation y00+f(t)y=0; (122.5) wheref(t) is a square wave function of period T f(t+T)=f(t)=( −1f o r 0t<T= 2; 1f o rT=2tT:(122.6) Note that f(t) is notcontinuous. This does not change any of the analysis. We can solve equation (122.5) and equation (122.6) by using f(t)=−1 and solving forfu(t);v(t)gin the interval 0 t<T = 2. Then we set f(t) = 1 and solve for fu(t);v(t)gin the interval T=2<tT, using as initial conditions the values calculated when we took f(t)=−1. See the section on solving equations with discontinuities (page 264). The solutions of equations (122.2) and (122.3) are found to be (for T=2<tT) u(t) = (sinhsin+c o s hcos)s i nt+ (sinhcos+c o s hsin)c o st; and v(t)=( c o s hsin+ sinhcos)s i nt+( c o s hcos−sinhsin)c o st; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 122. Floquet Theory 525 where=T=2. From these equations, we determine
to be
=u(T)+v0(T)=2c o s hcos: (122.7) The conclusion is that the solutions to equation (122.5) will be stable or unstable depending on whether the magnitude of
, as given by equation (122.7), is greater than or smaller than 2. Dierent values of Twill give dierent conclusions. For example, IfT=17 orT=e2,t h e nj
j>2 and some unstable solutions to equation (122.5) exist. IfT=1 orT=,t h e nj
j<2 and all to the solutions to equation (122.5) are stable. Notes 1. Mathematicians call this technique Floquet theory, whereas physicists call it Bloch wave theory. Solid state physicists use this technique to determine band gap energies. 2. Note that the periodicity of f(t) in equation (122.5) does not,b yi t s e l f , insure that y(t) has a periodic solution. If, however, f(t) is periodic and has mean zero, then equation (122.5) will have a periodic solution of the same period. 3. The linear system y0=B(t)yis said to be noncritical with respect to Tif it has no periodic solution of period Texcept the trivial solution y=0. Otherwise, the system is said to be critical. 4. See also Coddington and Levinson [1, pages 78{81], Kaplan [3, pages 472{490], Lukes [5, Chapter 8, pages 162{179], and Magnus and Winkler [6, pages 3{10]. References [1]Coddington, E. A., and Levinson, N. Theory of Ordinary Dierential Equations . McGraw{Hill Book Company, New York, 1955. [2]Hassan, H. S. Floquet solutions of nonlinear ordinary dierential equations. Proc. Roy. Soc. Edin. Sect. A 106 , 3{4 (1987), 267{275. [3]Kaplan, W. Operational Methods for Linear Systems . Addison{Wesley Publishing Co., Reading, MA, 1962. [4]Kuchment, P. Floquet Theory for Partial Dierential Equations ,v o l .6 0o f Operator Theory Advances and Appliations . Birkhauser, Basel, Switzerland, 1993. [5]Lukes, D. L. Dierential Equations: Classical to Controlled .A c a d e m i c Press, New York, 1982. [6]Magnus, W., and Winkler, S. Hill’s Equation . Dover Publications, Inc., New York, 1966. [7]Sleeman, B. D., and Smith, P. D. Double periodic Floquet theory for a second order system of ordinary dierential equations. Quart. J. Math. Oxford Ser. 37 , 147 (1986), 347{356. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 526 III Approximate Analytical Methods 123. Graphical Analysis: The Phase Plane Applicable to Two coupled autonomous rst order ordinary dif- ferential equations or an autonomous second order ordinary dierentialequation. Yields A graphical representation of the solution. Idea The qualitative features of the solution of two coupled autonomous rst order ordinary dierential equations may be ascertained from the phaseplane. Procedure Suppose we have the set of two coupled autonomous rst order ordinary dierential equations dx dt=f(x;y);dy dt=g(x;y): (123.1) Astincreases,x(t)a n dy(t) will describe a path in ( x;y) space. This will not be the case at those points ( x0;y0), where f(x0;y0)=0;g(x0;y0)=0: At these points, the value does not change with t:x(t)=x0andy(t)=y0. These points are called critical points . (They are also called equilibrium points orsingular points ). To analyze the motion near a single critical point, we linearize equation (123.1) about that point. By a linear change of variables, we can place the critical point at the origin ( x;y)=( 0;0). Near a critical point at the origin, equation (123.1) can be written as dx dt=ax+by+bf(x;y); dy dt=cx+dy+bg(x;y);(123.2) wherebf(x;y)=o(jxj+jyj)a n dbg(x;y)=o(jxj+jyj)a sx!0,y!0. We assume that a,b,c,dare real numbers and they are not all equal to zero. If we discard the bfandbgterms in equation (123.2) and look for solutions of the form x(t)=Aet;y(t)=Bet; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 123. Graphical Analysis: The Phase Plane 527/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. 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/. /. /. /. /. /. /./. /. /. /. /. /. /. /./. /. /. /. /./. /./. /././. /./. /. /././. /. /. /./. /. /. /./. /. /. /././. /./. /./. /././. /./././. /./././././. /./././././././././././././. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././. /./. /././. /. /./././././././. /./. /./. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /. /./. /. /. /. /. /. /. /./. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /./. /. /. /. /. /. /./. /. /. /. /. /. /./. /. /. /. /. /./. /./. /. /./. /./././. /. /./. /. /./. /. /./. /. /././. /./././. /././././././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././././. /./. /. /./. /. /. /./. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /./. /. /. /./. /././. /./. /./. /./. /./. /. /././././././././././././././././././././././#28e/#29Figure 123.1: The dierent types of behavior in the phase plane: (a) and (c) are nodes, (b) is a saddle point, (d) is a center, and (e) is a spiral. then we nd that must be an eigenvalue of the matrix ab cd .T h a t i s , must satisfy 2−(a+d)+(ad−bc)=0: (123.3) There are ve dierent types of behavior that can be observed near the critical point (0 ;0), based on the roots of equation (123.3). If the roots of equation (123.3) are Real, distinct, and of the same sign, then the critical point is called anode. (See gure 123.1.a for a typical picture.) Note that the symmetry axes are determined by the eigenvectors of the 2 2 matrix shown above. Real, distinct, and of opposite signs, then the critical point is called asaddle point . (See gure 123.1.b for a typical picture.) Real and equal, then the critical point is again a node. (See gure 123.1.c for a typical picture.) Pure imaginary, then the critical point is called a center . (See gure 123.1.d for a typical picture.) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 528 III Approximate Analytical Methods/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /./. /./././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /./././././././././././././././././././././. /./././././././././././././././././././. /. d et ermin an ttrace st a b lespiral u nst a b lespiralst a b leno d e u nst a b leno d esaddle s /././. /././././././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././././. /././././././././././././. /././././././. /./././. /./././. /./. /././. /./. /././. /./. /././. /./. /./. /. /./. /./. /. /. /./. /./. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /./. /.Figure 123.2: The dierent types of behavior in the phase plane, as a function of the trace and determinant of the 2 2 matrix. Conjugate complex numbers but not pure imaginary, then the critical point is called a spiral or a focus . (See gure 123.1.e for a typical picture.) In each of the gures, an arrow points in the direction of increasing t. For each case illustrated, there exist systems in which the arrows are pointing in the opposite direction from what we have illustrated. Each solution of equation (123.2) (corresponding to dierent initial conditions) describes a single trajectory. Every trajectory must Go to innity or Approach a limit cycle (see page 78) or Tend to a critical point. If the solution goes to innity, then the solution is said to be unstable , otherwise it is said to be stable . Example 1 Consider the simple linear dierential equation system dx dt=ab cd x: For this equation, the eigenvalues satisfy equation (123.3), which we write in the form 2−T+
=0 , w h e r e Tis the trace of the matrix ( T= a+d) and
is the determinant (
= ad−bc). The eigenvalues, and the qualitative picture of the phase plane, can be deduced from Tand
. Figure 123.2 shows the type of behavior to expect for dierent values ofTand
. The curve gure 123.2 is given by determinant= (trace) 2; only centers can occur along this curve. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 123. Graphical Analysis: The Phase Plane 529 Example 2 Consider the nonlinear autonomous second order ordinary dierential equation d2x dt2+dx dt+!2sinx=0; (123.4) which can be written as the coupled system dx dt=y; dy dt=−y−!2sinx:(123.5) For the equations in equation (123.5) there are innitely many critical points at the locations fx=n,y=0jn=0;1;2;:::g. To analyze the behavior near the point ( k;0) the new variables ey=y,ex=x−kare introduced. In these new variables, the system in equation (123.5) can be approximated by dex dt=ey; dey dt=−ey+(−1)k+1!2ex;(123.6) whenexandeyare both small. From equation (123.3) the characteristic equation for equation (123.6) becomes 2++!2(−1)k=0; with the roots 1=−+p 2+(−1)k+14!2 2; 2=−−p 2+(−1)k+14!2 2: If we now assume that >0a n d2>4!2,t h e n Forkeven,1<0a n d2<0. Hence, the point is a node. Forkodd,1>0a n d2<0. Hence, the point is a saddle point. With this information, we can draw the phase plane for the system in equation (123.5) (see gure 123.3). Because the system in equation (123.4) is dissipative (i.e., the total \energy" decays), all of the dierent possiblesolutions approach one of the nodes in innite time. The trajectories in the phase plane clearly show this. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 530 III Approximate Analytical Methods Figure 123.3: Phase plane for equation (123.4). Notes 1. In the above, we have presumed that the critical points are isolated ; that is, each critical point has a neighborhood around it in which noother critical points are present. 2. If, in equation (123.2), ad−bcwere equal to zero, then second degree (or higher) terms in the Taylor series of fandgwould be required to determine the behavior near that critical point. See Boyce and DiPrima [3, pages 456{486] for details.Ifad−bc6= 0, then the solution curves of the nonlinear system in equation (123.1) will be qualitatively similar to the solution curves of the linear system in equation (123.2), with the single exception thata center for equation (123.2) may be either a center or a spiral for the system in equations (123.1). 3. A second order autonomous ordinary dierential equation can always be written as a rst order system (see page 146). Also, the general equation of rst order M(x;y)dx+N(x;y)dy= 0 may be written as a system in the form of equation (123.1); i.e., dx dt=N(x;y);dy dt=−M(x;y): 4. The point at innity may be analyzed by changing variables by x1=x x2+y2;y 1=−y x2+y2 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 123. Graphical Analysis: The Phase Plane 531 and then analyzing the point (0 ;0) in thex1;y1-plane. This corre- sponds to the substitution z1=1=z,w h e nz=x+iyis treated as a complex variable. 5. Kath [9] describes a method that combines phase plane techniques with matched asymptotic expansions. This method can be used to analyze second order, nonlinear, non-autonomous, singular boundary value problems. 6. Two dierent graphing programs for showing phase planes on a Mac- intosh computer are DEGraph andPhase Portraits . A review of these programs is in Hartz [5]. A program that runs on IBM personal computers (and compatibles) is Phaser ; see Margolis [10] for a review. 7. A large collect of phase portraits may be found in Borrelli et al. [2]. 8. See also Bender and Orszag [1, pages 171{197], Coddington and Levinson [4, Chapter 15, pages 371{388], and Huntley and Johnson [7, Chapter 8, pages 114{133]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Borrelli, R. L., Coleman, C. S., and Boyce, W. E. Dierential Equations Laboratory Workbook . John Wiley & Sons, New York, 1992. [3]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [4]Coddington, E. A., and Levinson, N. Theory of Ordinary Dierential Equations . McGraw{Hill Book Company, New York, 1955. [5]Hartz, D. Degraph and phase portraits. Notices of the American Mathematical Society 36 , 5 (May/June 1989), 559{561. [6]Hubbard, J., and West, B. MacMath: A Dynamical Systems Software Package . Springer{Verlag, New York, 1991. [7]Huntley, I., and Johnson, R. M. Linear and Nonlinear Dierential Equations . Halstead Press, New York, 1983. [8]Jordan, D. W., and Smith, P. Nonlinear Ordinary Dierential Equations , second ed. Clarendon Press, Oxford, England, 1987. [9]Kath, W. L. Slowly varying phase planes and boundary-layer theory. Stud. Appl. Math. 72 (1985), 2221{239. [10]Margolis, M. S. Phaser. Notices of the American Mathematical Society 37, 4 (April 1990), 430{434. [11]Wang, D. Computer algebraic methods for investigating plane dieren- tial systems of center and focus type. In Computers and Mathematics , E. Kaltofen and S. M. Watt, Eds. Springer{Verlag, New York, 1990, pp. 91{998. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 532 III Approximate Analytical Methods 124. Graphical Analysis: The Tangent Field Applicable to First order ordinary dierential equations. Yields A graphical representation of the solutions corresponding to dierent initial conditions. Idea The qualitative features of the solution of a rst order ordinary dier- ential equation may be ascertained from the tangent eld. Procedure Given a rst order ordinary dierential equation in the form dy dx=f(x;y); (124.1) the procedure is to draw small line segments in the ( x;y) plane, such that the line segment that goes through the point ( x0;y0)h a st h es l o p e f(x0;y0). Note that a slope of mcorresponds to an angle of tan−1m.A f t e r ar e g i o no f( x;y) space has been covered with these small line segments, it should be apparent how the solution curves of equation (124.1) behave.An approximate solution may then be drawn by \connecting up" the line segments that originate from a given point. Constructing the tangent eld by hand is often facilitated by the method of isoclines . In this method, a few curves of the form f(x;y)=C,w i t hC being a constant, are constructed. Along each one of these curves, dy=dx is equal to the constant C. Hence, at every point on these curves, the small line segments all have the same slope. Example 1 Suppose we have the nonlinear ordinary dierential equation dy dx=1−xy2: (124.2) It is straightforward to construct the tangent eld, which is shown in gure 124.1. Every solution of equation (124.2) must be tangent to whatever line segments it passes near. For example, if equation (124.2) had the initial conditiony(0) = 1, then the solution can be approximately traced by starting at the point (0 ;1) and drawing a line that remains tangent to the line segments. For this equation and initial condition, ytends to zero as x tends to innity. This behavior can be seen in gure 124.1. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 124. Graphical Analysis: The Tangent Field 533/, /2 /2 x/, /2 /2 y/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. 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/./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /././. /. /./. /. /./. /././. /. /./. /././. /. /././. /./. /././. /././. /././. /././. /././. /././. /./././. /././././. /././. /./././././. /./././. /././././. /./././././. /./././././. /./././././. /././././././. /./././././. /./././././. /./././././. /./././././. /./././././. /././././. /./././././. /./././. /././././. /./././. /././././. /./././. /././. /././././. /./././. /././. /././. /././././. /././. /././. /././. /./././. /././. /././. /././. /././. /././. /././. /./. /././. /././. /././. /././. /. /././. /././. /././. /./. /././. /././. /. /././. /././. /./. /././. /. /././. /./. /././. /. /././. /./. /././. /. /././. /./. /././. /. /././. /./. /././. /. /./. /././. /. /././. /./. /. /././. /./. /. /././. /./. /. /././. /./. /. /././. /./. /. /././. /./. /. /././. /./. /. /././. /./. /. /.Figure 124.1: Tangent eld for equation (124.2). Example 2 Given the dierential equation dy dx=2x+y; (124.3) we nd that the isoclines are the straight lines 2 x+y=C. Figure 124.2 shows the isoclines, with small line segments superposed, as well as three solutions to equation (124.3). The exact solution to equation (124.3) is y=2 ( 1−x)+Ae−x,w h e r eA is an arbitrary constant. The linear behavior for x0 and the exponential behavior for x<0 can be identied in this gure. Notes 1. Consider drawing a small circle Γ in the ( x;y) plane that surrounds the point ( x0;y0). Traversing the circle counter-clockwise, the di- rection eld will change. In every case, the change in angle must be a multiple of 2 : [angle] Γ=2IΓ,w h e r eIΓis an integer called theindex of the vector eld . Suppose the number of times the slope dy=dx changes from +1to−1ismand number of times it changes from−1 to +1isn. Then the index is equal to ( m−n)=2. The index may be positive, negative, or zero. If Γ surrounds no critical points, then the index is zero. If Γ surrounds a saddle point, then the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 534 III Approximate Analytical Methods/, /2 /2 x/, /2 /2 y/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. 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/./././. /././././././././. /././././././././. /././. /././././././. /././././././././././. /./././././././././././. /./././. /././././././. /././././. /./././. /./././././. /./././././././. /././. /././././././././. /././. /./././././././. /././. /./././././././././. /././. /./././././. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././././././. /./././. /./././. /./././. /./. /././. /././. /././. /././. /. /./. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /. /./. /. /. /. /. /. /./. /. /. /. /. /././. /. /./. /. /. /. /./. /. /. /. /. /. /. /./. /. /. /. /. /. /./. /./. /. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /./. /. /. /. /. /. /. /./. /./. /. /. /./. /. /. /.Figure 124.2: Tangent eld for equation (124.3). index is−1. If Γ surrounds a center, spiral, or node, then the index is +1. If Γ surrounds more than one critical point, then the index is the sum of the indices for each critical point.Equation (124.1) sometimes arises from the autonomous system f_x= F(x;y), _y=G(x;y)g,v i a dy dx=G(x;y) F(x;y). In this case, we have IΓ= 1 2I ΓFdG−GdF F2+G2. See Jordan and Smith [3] for details. 2. Mathematica has the packages PlotField andPlotField3D which can plot two- and three-dimensional vector elds. They containfunctions for plotting gradient and Hamiltonian vector elds. 3. Even rough hand construction of the tangent eld can produce useful qualitative information. 4. See also Bender and Orszag [1, pages 148{149] and Boyce and DiPrima [2, pages 34{35]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [3]Jordan, D. W., and Smith, P. Nonlinear Ordinary Dierential Equations , second ed. Clarendon Press, Oxford, England, 1987. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 125. Harmonic Balance 535 125. Harmonic Balance Applicable to Nonlinear ordinary dierential equations with peri- odic solutions. Yields An approximate solution valid over the entire period. There is a speci- ed procedure for increasing the number of terms and, hence, for increasing the accuracy. Idea Harmonic balance is a way of looking for periodic solutions in nonlin- ear systems by trying to t a truncated Fourier series and choosing the frequency, amplitude, and phases so that any error occurs only in the discarded harmonics. Procedure Suppose we have a dierential equation of the form f(x;xt;xtt;t)=0; (125.1) and we wish to nd a periodic solution of period T. We look for an approximation to equation (125.1) in the form of a truncated Fourier series x(t)’y(t): =a0+NX j=1ajcosj!t+bjsinj!t; where!=2=T. The unknowns to be determined are fa0;aj;bjjj= 1;:::;Ngand possibly T. IfTis known, then we require the 2 N+ 1 unknowns to satisfy the 2N+ 1 algebraic equations ZT 0f(y;yt;ytt;t)s i nk!tdt =0; ZT 0f(y;yt;ytt;t)c o sk!tdt =0;(125.2.a-b) fork=0;1;:::;N . If the period Tis unknown, then there are 2 N+ 2 unknowns to be determined. To nd algebraic equations for these unknowns, we require equation (125.2) to hold for k=0;1;:::;N and, say, equation (125.2.a) for k=N+1 . Example 1 Given the equation d2x dt2+x+dx dt2 =s i nt; (125.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 536 III Approximate Analytical Methods whereis a given constant, we search for a 2 periodic solution. If we takeT=2andN= 2, then we are assuming that x(t)’y(t)=a0+a1cost+a2cos 2t++b1sint+b2sin 2t: (125.4) Using equation (125.3) and equation (125.4) in equation (125.2) produces the set of simultaneous algebraic equations /parenleftbig 4b2 2+b2 1+4a2 2+a2 1
+2a0=0; (b1b2+a1a2)=0; /parenleftbig b2 1−a2 1
−6a2=0; 2(a1b1−a2b1)−1=0; 3b2+a1b1=0: These equations have the unique solution fa0=−(2=3+34=3)=2(9)1=3, a1=0 ,a2=1=2(3)1=3,b1=−31=3=2=3,b2=0g. Hence, the approxi- mation (for N= 2) becomes x(t)’−3 21=3 sint+1 2(3)1=3(cos 2t−3)−(3)1=3 6: (125.5) Note that this approximation indicates the qualitatively correct behavior, at least for small values of .W h e nis small, equation (125.3) is a harmonic oscillator being forced near resonance. This would lead to alarge magnitude solution, which is what equation (125.5) indicates. Example 2 Given the equation d2x dt2+x=c(x2+c o st); we choose N= 1 and look for solutions of period T=2. Using the approximation x(t)’y(t)=a0+a1cost+b1sint; we nd that b1=0 ,a1=−1=2a0anda0=c1=3z=2, wherezsatises the cubic equation c4=3z4−2z3+ 2 = 0. Here, the analytical solution for a0is available (implicitly) but is not very informative. However, if we assume thatjcj1, then it can be shown that a0=c1=3 2h 1+c1=3 6+O(c8=3)i . Example 3 The requirements in equation (125.2) are not the only way in which to obtain useful approximations. Consider the Dung equation, ¨ x+x=x3, with _x(0) = 0. If we presume that x=Acos!t,t h e n x−x3=Acos!t 1−3 4A2 −1 4A3cos 3!t: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 125. Harmonic Balance 537 If we disregard the last, higher order term, then we may write x−x3 x/parenleftbig 1−3 4A2
. With this approximation, the original equation becomes ¨x−/parenleftbig 1−3 4A2
x0 . B e c a u s ew eh a v ep r e s u m e dt h a t x=Acos!t,w e can immediately identify the frequency: !21−3 4A2. Hence, to leading order, our approximate solution becomes xAcos/parenleftbig 1−3 8A2
t. Notes 1. This technique is known in the engineering literature as the describing function method . 2. Strictly speaking, this method may also be used to obtain approxi- mations to dierential equations that do not have periodic solutions. 3. This technique applies, in principle, to equations in which there is no small parameter. However, it may prove that the algebraic equations generated by equation (125.2) are not solvable in closed form unless a perturbation expansion is used (as in Example 2). 4. Mees [7] has a very extensive bibliography, separated into categories (applications, theory, background theory, Hopf bifurcation, and har- monic balance). See also MacDonald [6]. 5. When this method is implemented numerically, it is known as the spectral method (see page 851 or see Gottlieb and Orszag [2] for details). 6. See also Ferri [1], Groves [3], Huntley and Johnson [4, Chapter 12, pages 166{168] and Kundert et al. [5]. References [1]F e r r i ,A .A . On the equivalence of the incremental harmonic balance method and the harmonic balance{Newton Raphson method. J. Appl. Mech. 53 (June 1986), 455{457. [2]Gottlieb, D., and Orszag, S. A. Numerical Analysis of Spectral Methods: Theory and Applications . SIAM, Philadelphia, PA, 1977. [3]G r o v e s ,J r . ,F .R . Numerical solution of nonlinear dierential equations using computer algebra. Int. J. Comp. Math. 13 (1983), 301{309. [4]Huntley, I., and Johnson, R. M. Linear and Nonlinear Dierential Equations . Halstead Press, New York, 1983. [5]Kundert, K. S., Sorkin, G. B., and Sangiovanni-Vincentelli, A. Ap- plying harmonic balance to almost-periodic circuits. IEEE Trans. Microwave Theory and Tech. 36 , 2 (February 1988), 366{378. [6]MacDonald, N. Choices in the harmonic balance technique. J. Phys. A: Math. Gen. 26 (1993), 6367{6377. [7]M e e s ,A .I . Describing functions: Ten years on. IMA J. Appl. Mathematics 32(1984), 221{233. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 538 III Approximate Analytical Methods 126. Homogenization Applicable to \Microscopic" dierential equations. Yields \Macroscopic" dierential equations. Idea By averaging microscopic dierential equations, dierential equations for macroscopic quantities may be determined. Procedure In many elds, the (\microscopic") equations of motion contain more information than is needed by a practitioner who is solving a specic problem. For instance, in a fluid flow problem, it may be that only the mass flow is required, rather than a detailed analysis of the flow eld. Consequently, it is of interest to take an \average" of the \microscopic" dierential equations to obtain a set of dierential equations that describethe \macroscopic" quantities of interest. The average taken could be a time average, a space average, an ensemble average, or an average of some other type. In the homogenization method, it is usually assumed that there is a fast time (or a short length) scale, on which the \microscopic" dierential equations vary. The dependence on this fast scale is usually assumed to be either periodic or random. In mechanics problems, the small length scaleis often the length scale of the inclusions or heterogeneities. Often, a formal procedure for analyzing problems via homogenization is by a multiscaling procedure (see page 605). Example 1 As an example of the general procedure, consider the elliptic problem −X i;j@ @xi a ij(x)@u @xj =f(x); (126.1) in some domain Ω. The equation (126.1) probably came from a system of the form −@pi @xi=f(x); pi=a ij(x)@u @xj;(126.2) via Hamilton’s equations. In equations (126.1) and (126.2), it is now assumed that a ij(x) is of the form aij(x=)a n dt h a t a ij(x) is periodic CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 126. Homogenization 539 inx, with the period in the xivariable being Li. A formal two scale procedure can be dened by (see page 605) yi=xi ; u(x)=u0(x;y)+u1(x;y)+2u2(x;y)+


; p(x)=p0(x;y)+p1(x;y)+2p2(x;y)+


; where p=(p1;p2;:::). In this case, we choose to dene the average of some arbitrary function of xandyto be A(x): =1 L1L2


LnZ A(x;y)dy: (126.3) We integrate over yin equation (126.3) to average over the high frequency component of a function that depends on both xandy. For our example, it is straightforward to show that −@p0 i @xi=f(x); (126.4) where p0=( p0 1;p0 2;:::). Now, if an ah ij(x) can be found such that p0 i=ah ij(x)@u0 @xj; (126.5) thenah ij(x) is said to be the homogenized coecient, and equations (126.4) and (126.5) are the homogenized equations. Example 2 For a more detailed example, consider the equation Au:=−X i;j@ @xi aijx @u @xj+a0x  u =f(x); (126.6) whereaij(y)a n da0(y), with y:=x=, are periodic on the unit cube Y. We assume that the solution can be expanded in the form u=u0 x;x  +u1 x;x  +::: =u0(x;y)+u1(x;y)+::::(126.7) Using the chain rule (i.e., @xibecomes@xi+1 @yi), inserting equation (126.7) into equation (126.6) and equating powers of results in A1u0=0; A1u1=A2u0; A1u2=A2u1+A3u0+f;(126.8.a-c) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 540 III Approximate Analytical Methods where A1=−X i;j@ @yi aij(y)@ @yj ; A2=X i;j@ @yi aij(y)@ @xj +@ @xi aij(y)@ @yj ; A3=X i;j@ @xi aij(y)@ @xj +a0(y): If we dene an averaging operator by M[v]=1 jYjZ Yv(y)dy; then it can be shown that the equation A1v=hwill have a unique solution only ifM[h] = 0 (see the section on alternative theorems, page 15). This condition, applied to equation (126.8.b), indicates that u0=u0(x). This fact simplies equation (126.8.b) to A1u1=−X i;j@ @yi aij(y)@ @yj u1=X i;j@aij(y) @yi@u0(x) @xj=A2u0: Using separation of variables on this results in u1(x;y)=X kzk(y)@u0(x) @xk, wherezk(y) is the unique periodic solution of A1zk=−X i;j@ @yi aij(y)@ @yj zk=X i@aik(y) @yi: (126.9) Equation (126.9) is known as the cell problem . To nally obtain a solution, we require from equation (126.8.c) that M[A2u1+A3u0+f] = 0. This results in −X i;jpij(x)@2u0(x) @xi@xj+M[a0]u0(x)=f(x); (126.10) wherepij(x): =M[aij]−M"X kaik@zj @yk# . Notes 1. Homogenization techniques are often used in fluid mechanics (two phase flow in particular), electric eld theory, and solid mechanics. 2. Homogenization is often the method used in ad hoc \mean eld" theories, \eective media" theories, and \averaged equations." CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 126. Homogenization 541 3. Homogenization seems to be related to renormalization group theory. Renormalization group methods study the asymptotic behavior of a system (i.e., the macroscopic behavior) when the scale of observa- tion is much larger than the scale of microscopic description. SeeGoldenfeld et al. [4] or Nunes da Silva [6]. 4. In Persson and Wyller [7], it is shown that, for a sample prob- lem, homogenization is equivalent to Whitham’s averaged Lagrangian method. 5. Averages, denoted by h
i, are generally required to satisfy \Reynold’s rules" hf+gi=hfi+hgi; hhfigi=hfihgi; hci=c; whenfandgare random or periodic functions and cis a constant. It is also often required that @f @t =@hfi @t be satised for functions fthat are \well behaved." References [1]Avellaneda, M. Iterated homogenization: Dierential eective medium theory and applications. Comm. Pure Appl. Math 60 , 5 (September 1987), 527{554. [2]Burgers, J. M. On some problems of homogenization. Quart. Appl. Math. 35, 4 (January 1978), 421{434. [3]Ericksen, J., Kinderlehrer, D., Kohn, R., and Lions, J.-L. ,E d s . Homogenization and Eective Moduli of Materials and Media . Springer{ Verlag, New York, 1986. [4]Goldenfeld, N., Martin, O., and Oono, Y. Asymptotics of partial dierential equations and the renormalization group. In Asymptotics Beyond All Orders ,S e g u r ,H . et al. , Ed. Plenum Publishing Corp., New York, 1991, pp. 375{383. [5]Larsen, E. W. Two types of homogenization. SIAM J. Appl. Math. 36 ,1 (February 1979), 26{33. [6]Nunes da Silva, J. M. Renormalized vibrations of a loaded spring. Am. J. Phys. 62 , 5 (May 1994), 423{426. [7]Persson, L., and Wyller, J. A note on Whithams method and the homogenization procedure. Physica Scripta 38 (1988), 774{776. [8]Sanchez-Palencia, E. Homogenization method for the study of composite media. In Asymptotic Analysis , J. D. Murray, Ed. Springer{Verlag, New York, 1984, pp. 192{214. [9]Weinan, E. Homogenization of linear and nonlinear transport equations. Comm. Pure Appl. Math 65 (1992), 301{326. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 542 III Approximate Analytical Methods 127. Integral Methods Applicable to Linear and nonlinear partial dierential equations. Yields An approximation of the solution. Idea A sequence of physical approximations may lead to an approximate solution. Procedure There are generally three separate steps in using the common integral approximation techniques: A physical boundary (either natural or imposed mathematically) is assumed to be at some nite distance. A weak form of the equations is assumed to hold, up to the boundary described above. The form of the solution is guessed by the method of undetermined coecients. These concepts are made clear in the following example. Example Suppose we want to approximate the solution of the linear parabolic partial dierential equation ut=uxx;forx>0;t> 0; u(0;x)=u0; @u @x(t;0) =f(t);(127.1.a-c) wheref(t) is some prescribed function. Note that the value of u(t;x) (which physically might represent a temperature) is initially u0.F o r t h e rst approximation, we suppose that there is a nite distance (t)t h a t varies with time, beyond which the temperature is still u0. This assumption is contrary to fact; we know that the diusion equation has an innite propagation speed, and the value of uatallpoints is immediately changed from u0. But the change from u0will be exponentially small at large distances, so we assume it is zero for x(t). This adds the boundary conditions u(t;(t)) =u0; ux(t;(t)) = 0:(127.2.a-b) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 127. Integral Methods 543 Equation (127.2.a) states that the temperature at the boundary x=(t) is always equal to u0. Equation (127.2.b) states that there is no heat flux acrossx=(t); if there was such a flux, then the region beyond x=(t) would not maintain the temperature u=u0. The second approximation is to assume that a weak form of the dier- ential equation will hold. To obtain this weak form, we integrate equation (127.1.a) with respect to xfromx=0t ox=(t)t oo b t a i n Z(t) 0utdx=Z(t) 0uxxdx: This expression can be integrated by parts to obtain d dtZ(t) 0udx−u(t;(t))d(t) dt=h ux(t;(t))−ux(t;0)i =−f(t); (127.3) where we have used equation (127.2.b) and equation (127.1.c). If we dene w(t)=Z(t) 0udx; (127.4) then equation (127.3) can be written as the ordinary dierential equation d dt w−u0 =−f(t): (127.5) Note that, from equation (127.4), the average value of u(t;x) in the region 0x(t)i sg i v e nb y w(t)=(t). Now we must determine (t) from equation (127.5). Before we can solve for(t), however, we need to determine w(t). To determine w(t), we presume some form of the general solution for u(t;x). By the use of undetermined coecients, we suppose that u(t;x) has the form u(t;x)=( a(t)+b(t)x+c(t)x2;for 0<x< (t); u0; forx> (t); wherea(t),b(t)a n dc(t) are all unknowns. If this form is to satisfy equation (127.1.c) and equation (127.2), then it must be restricted to be of the form u(t;x)=( u0−f(t) 2(t)[(t)−x]2;for 0x< (t); u0; forx(t):(127.6) Using this form in equation (127.4) results in w(t)=u0(t)−2(t)f(t) 6. Using this value for w(t) in equation (127.5) results ind dth 2(t)f(t) 6i =f(t). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 544 III Approximate Analytical Methods The solution of this ordinary dierential equation is (t)=s 6 f(t)Zt 0f(s)ds: (127.7) Using this form for (t) in equation (127.6) completes the determination of the approximate solution. For comparison purposes, if f(t) is the constant F, then the temperature atx= 0 is given by (using equations (127.7) and (127.6)) u(t;0)’u0−r 3 2tF: (127.8) Conversely, the exact solution of equation (127.1) can be found by the use of Laplace transforms to be u(t;x)=u0−r Zt 0f(t−)pe−x2=4d,a n d so, whenf(t) is the constant F, the exact solution becomes u(t;0) =u0−q 4 tF. The dierence between this exact solution and the approximation in equation (127.8) is about 9%. Notes 1. This method, in fluid mechanics, is known as the K arman{Pohlausen technique. The distance (t) then represents the thickness of a boundary layer. 2. When this technique is used, it is most often with partial dierential equations that have only a single space variable. 3. This technique is often used in free boundary problems (see page 311). 4. See Ames [1, pages 271{278]. References [1]Ames, W. F. Ad hoc exact techniques for nonlinear partial dierential equations. In Nonlinear Partial Dierential Equations in Engineering ,W .F . Ames, Ed. Academic Press, New York, 1967. [2]Goodman, T. R. Application of integral methods to transient nonlinear heat transfer. In Advances in Heat Transfer ,T .F .I r v i n e ,J r .a n dJ .P .H a r t n e t t , Eds. Academic Press, New York, 1964, pp. 51{122. [3]Riley, D. S., and Duck, P. W. Application of the heat-balance integral method to the freezing of a cuboid. Int. J. Heat Mass. Transfer 20 (1977), 294{296. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 128. Interval Analysis 545 128. Interval Analysis Applicable to Ordinary and partial dierential equations. Yields An analytical approximation with an exact bound on the error. Idea Initially, we bound the solution between an upper and lower bound. Then, iterating a contraction mapping, we generate a sequence of approxi- mations in which the upper bound decreases and the lower bound increases. Procedure We use the interval notation [ a;b] to indicate some number between the values ofaandb. We allow the coecients of polynomials to be intervals. For example, the interval polynomial Q(x)=1+[ 2;3]x2+[−1;4]x3; evaluated at the point x=y,m e a n st h a t min 23 −14/parenleftbig 1+y2+y3
Q(y)max 23 −14/parenleftbig 1+y2+y3
: There exists an algebra of interval polynomials. For example /parenleftbig x+[ 2;3]x3
+/parenleftbig [1;2]x+[ 1;4]x3
=[ 2;3]x+[ 3;7]x3; ([1;3] + [−1;2]x)2=[ 1;9] + [−6;12]x+[ 0;4]x2: IfP(x)a n dQ(x) are interval polynomials, then at any point ywe can write P(y)2[PL;PU]a n dQ(y)2[QL;QU]. We say that P(x) containsQ(x) on some interval [ c;d]i fPLQLandQUPUfor ally2[c;d]. This is denoted by Q(x)P(x). To approximate the solution of an ordinary dierential equation, we search for a contraction mapping (see page 58) that has the form Pk+1= F[Pk], whereF[
] is a functional, Pk+1Pk,a n dPktend to the solution of the dierential equation as k!1 . Example Suppose we want to approximate the solution of y0=y2;y (0) = 1; (128.1) for values of xin the interval [0 ;1=4]. Equation (128.1) can be written as the equivalent integral equation y(x)=1+Zx 0y2(z)dz: (128.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 546 III Approximate Analytical Methods It is easy to see that the solution of equation (128.2) must lie in the interval [1;2] whenx2[0;1=4]. This is because y0is always positive, so ycannot be smaller than 1 (which is what y(0) is) and if it is assumed that y(z0)=2f o r somez02(0;1=4), then a contradiction can be reached by using equation (128.2). We now dene the iteration sequence (the contraction mapping) by Pk+1(x)=1+Zx 0P2 k(z)dz; fork=0;1;2;:::, which is just Picard’s integral formula (see page 618). We start the sequence o by P0(x)=[ 1;2] and then calculate P1(x)=1+Zx 0[1;2]2dz; =1+[ 1;4]x; P2(x)=1+Zx 0(1 + [1;4]z)2dz; =1+Zx 0(1 + [2;8]z+[ 1;16]z2)dz; =1+x+[ 1;4]x2+1 3;16 3 x3; P3(x)=1+x+x2+x3+[ 1;2]x4+


; P4(x)=1+x+x2+x3+x4+ 1;7 5 x5+


: It is easy to show that Pk+1(x)Pk(x)a n dt h a tfPk(x)gconverges to the exact solution y(x) of equation (128.1). Note that, from the Pk(x),exact estimates of the solution are available. For example, from P2(x), we nd, 1:141<y(1=8)<1:198. Notes 1. The exact solution to the system in equation (128.1) is y(x)=1=(1− x), which has the Taylor series: y(x)=1+x+x2+x3+x4+


. 2. To avoid dealing with polynomials of large degree, as in the example, we could observe that xn" 0;1 4n−m# xm; forxin the interval [0 ;1 4]. This allows us to replace xnbyxm,w i t h a coarsening of the bounds. 3. The real power of this method is that it can be applied to dierential equations whose coecients are given by intervals. For example, thiswould be the case in a problem in which a parameter appearing in a dierential equation is known only approximately. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 128. Interval Analysis 547 4. The paper by Ames and Nicklas [2] describes the solution of ellip- tic partial dierential equations, using interval analysis to solve the nite dierence equations produced by a numerical approximation. Schwandt [10] addresses the same issue, but with the use of a vectorcomputer. 5. When solving ordinary dierential equations numerically, using inter- val techniques, the error bounds often exhibit spurious exponential growth due to the dierential equation solver used. Numerical meth- ods have been developed that prevent spurious exponential growth ofthe intervals for linear systems, see Gambill and Skeel [5] for details. 6. The techniques presented in this section can be implemented nu- merically. Interval arithmetic packages are available in Algol (seeGuenther and Marquardt [6]), Fortran (see Yohe [12]), and PASCAL (see Rall [9]). 7. There is an interval computations web page at http://cs.utep.edu/ interval-comp/main.html . The book by Eijgenraam [4] contains several worked examples. The journal Interval Computations is a useful reference. References [1]Alefeld, G., and Herzberger, J. Introduction to Interval Computations . Academic Press, New York, 1983. [2]Ames, W. F., and Nicklas, R. C. Accurate elliptic dierential equation solver. In Accurate Scientic Computations ,W .L .M i r a n k e ra n dR .A . Toupin, Eds., no. 235 in Lecture Notes in Computer Science. Springer{Verlag, New York, 1986, pp. 70{85. [3]Corliss, G. F. Survey of interval algorithms for ordinary dierential equations. Appl. Math. and Comp. 31 (1989), 112{120. [4]Eijgenraam, P. The solution of initial value problems using interval arith- metic. Tech. rep., Mathematisch Centrum, Amsterdam, The Netherlands, 1981. [5]Gambill, T. N., and Skeel, R. D. Logarithmic reduction of the wrapping eect with application to ordinary dierential equations. SIAM J. Numer. Anal. 25 , 1 (February 1988), 153{162. [6]Guenther, G., and Marquardt, G. A programming system for interval arithmetic. In Interval Mathematics 1980 , K. Nickel, Ed. Academic Press, New York, 1980, pp. 355{366. [7]Moore, R. E., and Zuhe, S. An interval version of Chebyshev’s method for nonlinear operator equations. Nonlinear Analysis 7 , 1 (1983), 21{34. [8]Oppenheimer, E. P., and Michel, A. N. Application of interval analysis techniques to linear systems: Part III|Initial value problems. IEEE Trans. Circuits and Systems 35 , 10 (October 1988), 1243{1256. [9]Rall, L. B. An introduction to the scientic computing language Pascal{ SC. Comp. & Maths. with Appls. 14 , 1 (1987), 53{69. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 548 III Approximate Analytical Methods [10]Schwandt, H. Newton-like interval methods for large nonlinear systems of equations on vector computers. Comput. Physics Comm. 37 (1985), 223{ 232. [11]Schwandt, H. Interval arithmetic methods for systems of nonlinear equations arising from discretizations of quasilinear elliptic and parabolicpartial dierential equations. Applied Numerical Math. 3 (1987), 257{287. [12]Yohe, J. M. Software for interval arithmetic: A reasonable portable package. ACM Trans. Math. Software 5 , 1 (March 1979), 50{63. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 129. Least Squares Method 549 129. Least Squares Method Applicable to Ordinary and partial dierential equations. Yields An approximation to the solution. Idea A variational principle is created for a given dierential equation, and then an approximation to the solution with some free parameters is pro- posed. By use of the variational principle, the free parameters are deter-mined. Procedure Given the dierential equation N[u]=0; (129.1) foru(x) in some region of space R, with the homogeneous boundary con- ditions B[u]=0; (129.2) on some portion of the boundary of R, we dene the functional J[v(x)] =Z R N[v(x)]2 dx: (129.3) Notice that J[v(x)]0, for all functions v(x). The solution to equations (129.1) and (129.2) clearly satises J[u]=0 because the integrand is identically equal to zero in this case. Hence, the solutions to equations (129.1) and (129.2) represents a minimum of the functionalJ[
]. Now we choose an approximation to u(x) that has several parameters in it, sayu(x)’w(x;), whereis a vector of parameters. This approximation is chosen in such a way that it satises the conditions in equation (129.2). The parameters in w(x;) are determined by minimizing J[w(x;)]; i.e., by solving the simultaneous system of equations @ @kJ[w(x;)] = 0; fork=1;2;:::: (129.4) Example Suppose we wish to approximate the solution of the two point boundary value problem u00+u+x=0; u(0) = 0;u(1) = 0:(129.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 550 III Approximate Analytical Methods (Note that the exact solution of equation (129.5) is y(x)=sinx sin 1−x.) In this case, we may dene J[v(x)] to be J[v(x)] =Z1 0(v00+v+x)2dx: We choose to approximate the solution of equation (129.5) by u(x)’w(x)=1(x−x2)+2(x−x3): This approximation has been chosen in such a way that the boundary conditions for u(x) are satised. Using w(x) in the functional results in J[w(x)] =1 210 7072 1+ 212112+ 22002 2−3851−7842+7 0 : Forming equation (129.4) for k=1;2, we determine that 1and2must satisfy the simultaneous algebraic equations @J[w(x)] @1=1 210[14141+ 21212−385] = 0; @J[w(x)] @2=1 210[21212+ 44002−784] = 0: These equations have the solution: f1=4448 246137’0:0181;2=413 2437’ 0:1694g. The function w(x), with these values, becomes our approximation. The greatest dierence between the exact solution and the approximatesolution, in the range 0 <x< 1, is atx’0:5215, where the dierence is approximately 0.0016. Notes 1. Note that for the functional in equation (129.3) there may exist, in general, several dierent functions fvk(x)gthat satisfy J[vk(x)] = 0. 2. This method is similar to the Rayleigh{Ritz method (see page 638) in that an approximation is utilized in a variational equation. 3. This method is an example of a weighted residual method (see page 786). 4. This technique is often implemented numerically. 5. See Collatz [2, pages 184 and 220{221]. References [1]Chang, C. L., and Gunzburger, M. D. A subdomain-Galerkin/least squares method for rst-order elliptic systems in the plane. SIAM J. Numer. Anal. 27 , 5 (1990), 1197{1211. [2]Collatz, L. The Numerical Treatment of Dierential Equations . Springer{ Verlag, New York, 1966. [3]Hanke, M. On a least-squares collocation method for linear dierential- algebraic equations. Numer. Math. 54 (1988), 79{90. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 130. Lyapunov Functions 551 130. Lyapunov Functions Applicable to Ordinary and partial dierential equations. Yields Bounds on the solution in phase space. Idea Even without solving a given dierential equation, sometimes we can restrict the solution to be in a certain portion of phase space. Procedure Given a dierential equation, nd a non-negative functional of the solution, which has a non-positive derivative. Then the solution of thedierential equation will remain in a region described by the functional and the initial conditions. Most often, the functional will involve the dependent variable and some of its derivatives. Example 1 Suppose we wish to bound the solution of a damped harmonic oscillator xtt+xt+!2x=0; x(0) =A; xt(0) =B;(130.1) with>0. In this case, we dene the Lyapunov functional to be L[x(t);xt(t);xtt(t);t]=!2x2(t)+x2 t(t): BecauseL[
] is a sum of squares, it cannot be negative. Dierentiating L[
] with respect to tproduces Lt[x;xt;xtt;t]=(!2x2+x2 t)t =2!2xxt+2xtxtt =−2x2 t;(130.2.a-c) where we have used the original dierential equation equation (130.1) to replace the xttterm in equation (130.2.b). Because is positive, Ltis non-positive. Therefore, L[
] is a non-increasing function of t. Hence, !2x2(t)+x2 t(t)=L[x(t);xt(t);xtt(t);t] L[x(0);xt(0);xtt(0);0] !2x2(0) +x2 t(0) !2A2+B2 a prescribed constant :(130.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 552 III Approximate Analytical Methods Therefore, we have found an upper bound for !2x2(t)+x2 t(t) without solving the original equation. Example 2 Suppose we have the wave equation on a nite domain (0 xL) utt=c2uxx; ux(0;t)=ux(L;t)=0;u(x;0) =g(x);(130.4) wherecis a given constant and g(x) is given. In this case, we choose the Lyapunov functional to be V(t)=1 2ZL 0[u2 t+c2u2 x]dx: (130.5) BecauseV(t) is the integral of a non-negative quantity, V(t)i sa l s on o n - negative. Dierentiating V(t) with respect to tproduces Vt=ZL 0[ututt+c2uxuxt]dx =ZL 0[ut(c2uxx)+c2uxuxt]dx =c2ZL 0[utuxx+uxuxt]dx:(130.6.a-c) Integration of the second term in equation (130.6.c) by parts yields Vt=c2ZL 0[utuxx−uxxut]dx+c2uxutx=L x=0 =c2[ux(L;t)ut(L;t)−ux(0;t)ut(0;t)]; or, using the initial conditions in equation (130.4), Vt=0: We conclude that V(t)=V(0), for all values of t. This statement is essen- tially an \energy" statement: The energy (described by equation (130.5)) carried by a wave (described by equation (130.4)) remains constant. Notes 1. Lyapunov functionals are often devised from physical considerations. The Lyapunov functionals in both of these examples represent the \energy" of the system in a mathematical way. 2. Finding Lyapunov functionals is, in general, a dicult task. It is often made easier by considering conservation laws: energy, momentum, etc. The \energy" in example one is notheld constant because of the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 130. Lyapunov Functions 553 dissipation due to the term. If=0 ,t h e nLt= 0 and so equation (130.3) becomes !2x2(t)+x2 t(t)=!2A2+B2: In this case, the energy is constant. 3. There is a constructive method, due to Zubov [10], for obtaining Lyapunov functionals for systems of ordinary dierential equations.The procedure requires the solution of a partial dierential equation, which is derived from the given system of ordinary dierential equa- tions. See Hahn [3, pages 78{82] or Willems [9, pages 42{43] fordetails. Hahn [3] gives an example: A Lyapunov function for the systemf_x=−x+2x 2y,_y=−ygisL=−1+e x p −y2 2−x2 2(1−xy) . 4. A dierent constructive method is described in O guzt¨oreli et al. [7]. A detailed algorithm is given for systems of ordinary dierential equations of the form: f_x=f(t;x;y ), _y=g(t;x;y )g. The Lyapunov function for a modication of the Mathieu dierential equation, ¨ x= (+2cos 2t)x+x2, is derived for the region/parenleftbig x2+y2<2
,w h e r e is a suciently small number. 5. Consider the nonlinear system x0=f(x), with f(0)=0and the Jacobian matrix J(x)=@f @x. If a constant, symmetric, positive denite matrix Pcan be found such that PJ(x)+JT(x)Pis neg- ative denite, then V=xTPxis a Lyapunov function (with V0= xTR JT(zx)P+PJ(zx) dz/bracerightbig x).IfPis chosen to be the identity matrix, then V=xTxwill be a Lyapunov function if all of the eigenvalues of the matrix J(x)+JT(x) are negative. This is known as Krasovskii’s theorem. 6. Burton [2] describes how Lyapunov functions may be constructed for delay dierential equations. 7. \Lyapunov" is sometimes written \Liapunov." 8. See Boyce and DiPrima [1, pages 502{512] and Simmons [8, pages 316{322]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Burton, T. A. Perturbation and delays in dierential equations. SIAM J. Appl. Math. 29 , 3 (November 1975), 422{438. [3]Hahn, W. Theory and Application of Liapunov’s Direct Method .P r e n t i c e { Hall, Inc., Englewood Clis, NJ, 1963. [4]Jordan, D. W., and Smith, P. Nonlinear Ordinary Dierential Equations , second ed. Clarendon Press, Oxford, England, 1987. [5]Kalman, R. E., and Bertram, J. E. Control system analysis and design via the ‘second method’ of Lyapunov, I: Continuous-time systems. J. Basic Engrg. Trans. ASME 82 , 2 (1960), 371{393. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 554 III Approximate Analytical Methods [6]Lasalle, J., and Lefschetz, S. Stability by Liapunov’s Direct Method with Applications . Academic Press, New York, 1961. [7]Oguztoreli, M. N., Lakshmikantham, V., and Leela, S. An algorithm for the construction of Liapunov functions. Nonlinear Analysis 5 , 11 (1981), 1195{1212. [8]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. [9]W i l l e m s ,J .L . Stability Theory of Dynamical Systems . John Wiley & Sons, New York, 1970. [10]Zubov, V. I. Methods of A. M. Lyapunov and their Application .P . Noordho, The Netherlands, 1964. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 131. Equivalent Linearization and Nonlinearization 555 131. Equivalent Linearization and Nonlinearization Applicable to Nonlinear ordinary dierential equations. This technique is most frequently used for ordinary dierential equations with periodic solutions. Yields An approximate periodic solution. Idea We model the given equation by a linear or nonlinear equation for which the exact solution can be found. Procedure Suppose we want to approximate the solution to the nonlinear ordinary dierential equation D[x(t);t]=0; (131.1) whereD[
] is a dierential operator. We represent the initial conditions and boundary conditions for x(t)a sB[x(t)] = 0, and assume that x(t)i s periodic on some interval, say for tfrom 0 toT. We do not need to know Ta priori . We model equation (131.1) by choosing a D[
] that has properties that are \similar" to the properties of D[
]. This can be done by any technique. To allow some generality, we assume that D[
] depends on a set of parameters =(1;2;:::;n). Now we look for a solution y(t;) of D[y(t;);t;]=0;B [y(t;)] = 0 (131.2) that is periodic on the interval [0 ;T]. We will approximate the solution to equation (131.1), x(t), by the solution to equation (131.2), y(t;). For this to be a good approximation, the error made must be small. We dene theerror made in using y(t;)f o rx(t)t ob e E(t;): =D[y(t;);t]: The claim is that x(t)’y(t;) if the \total error" is \small" in some sense. The \total error" could be measured as 1 TZT 0jE(t;)j2dt \mean square error," CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 556 III Approximate Analytical Methods or 1 TZT 0jE(t;)jdt \mean modulus," or maxjE(t;)j \extremum." The \total error" is minimized by choosing the . This is accomplished by dierentiating the total error with respect to iand setting the resulting expression to zero (for i=1;2;:::;n ). Solving these simultaneous algebraic equations yields the desired values of the i. Example 1 Suppose we wish to approximate the periodic solution of the nonlinear ordinary dierential equation D[x(t);t]=x00+ax+bx3+cx5=0; x(0) =A; x0(0) = 0:(131.3) Herefa;b;c;Agare all known, xed constants. We choose to approximate the solution of equation (131.3) by the solution of the linear ordinary dierential equation D[y(t);t;!]=y00+!2y=0; y(0) =A; y0(0) = 0;(131.4) for some (unknown) value of !. In this example, the vector of unknown parametersis the single variable !. The solution to equation (131.4) is y(t)=Acos!t: (131.5) The error in using equation (131.5) for the solution of equation (131.3) is E(t;!)=D[y(t);t]; =y00+ay+by3+cy5; =(a−!2)c o s!t+bcos3!t+ccos5!t: We choose, in this example, to minimize the mean square error. Hence, we dene the total error, E(!), by E(!)=1 TZT 0jE(t;!)j2dt =1 TZT 0[(a−!2)c o s!t+bcos3!t+ccos5!t]2dt:(131.6) Now, what is T? For equation (131.3), we do not know the true period of the solution. But, we are using the solution of equation (131.4) to CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 131. Equivalent Linearization and Nonlinearization 557 approximate the solution of equation (131.3). And, for equation (131.4), the solution has period T=2=!(see equation (131.5)). Hence, to evaluate equation (131.6), we use T=2=!to obtain E(!)= 128!4−(160c+ 192b+ 256a)!2+6 3c2+ (140b+ 160a)c +8 0b2+ 192ab+ 128a2 =256:(131.7) Now, the goal is to minimize the total error. If equation (131.7) is dif- ferentiated with respect to !, and the resulting equation is solved for !, then !2=a+3 4bA2+5 8cA4or!=0: (131.8) Therefore, an approximation to the solution of equation (131.3) is found by using equation (131.8) in equation (131.5): x(t)’Acos" tr a+3 4bA2+5 8cA4# : Example 2 Suppose we wish to approximate the periodic solution of the undamped Dung equation D[x(t);t]=x00+ax+bx3=Bcos!t; (131.9) wherefa;b;B;!gare all known constants. We choose to model the equa- tion (131.9) by the nonlinear equation D[y(t);t]=y00+ay+by3=γcn(t;k); (131.10) where cn(t;k) is the Jacobian elliptic cosine function with modulus k.T h e aandbin equation (131.10) are the same as the aandbin equation (131.9). The three remaining parameters in equation (131.10) that are under ourcontrol arefγ;;kg. The solution to equation (131.10) is known to be (see the look-up solution technique, on page 179) y(t)=cn(t;k); (131.11) where,γ,,a n dkare related by b 3+(a−2)=γ; k2=b2 22: (131.12) These equations determine andk(in principle) in terms of γand.F o r the period of the forcing function in equation (131.10) to match the period of the forcing function in equation (131.9) (which is 2 =!), we also require =2K(k)! ; (131.13) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 558 III Approximate Analytical Methods whereK(k) is the complete elliptic integral of the rst kind with modulus k. We will use equation (131.13) to determine . This leaves us with one adjustable parameter, γ, with which to eect the minimization of the total error. N o ww ec a l c u l a t e E(t;γ)=D[y(t)] =Bcos!t−γcn(t;k): (131.14) If we choose to use the mean square error, with T=2=!, we nd that the total error is minimized for γ=BK (k) 2 E(k)−k02K(k)sechK(k0) 2K(k) ; (131.15) whereE(k) is the complete elliptic integral of the second kind and k0,g i v e n byk02=1−k2, is the complementary modulus. Using equations (131.13), (131.14), and (131.15) in equation (131.11) results in the nal approximation to the steady-state periodic solution of equation (131.9). Notes 1. Note that in example 1 the eective frequency of the approximate solution depends on the initial conditions. This is generally expected in nonlinear problems. 2. For example 2, the approximate solution y(t) correctly tracks the frequency change of the solution when the magnitude of the forcing function is changed. More details on this example may be found in Iwan and Patula [4]. 3. This technique also works well for stochastic equations. In this ap- plication, the denition of the total error should include expectationstaken over all of the random variables. This is sometimes called \statistical linearization." See Beaman [1] for details. 4. This technique extends naturally to systems of equations. In this case, there will be an error associated with each equation fE i(t;)g, and we can dene the total error by E(t;)=P ijEi(t;)j2. 5. This technique can also be used for problems that do not have periodic solutions. The technique often used in this case is to minimize the integral ofjEj2from 0 to1. 6. Dierential operators representing dierential equations may also be linearized directly, without minimizing some error functional. We have the denition: The operator A[
] is linearizable at u0if there exists a bounded linear operator L[
] such that A[u]−A[u0]=L[h]+r,w i t h lim h!0jjrjj jjhjj=0 ,w h e nh=u−u0. See Stakgold [8, pages 578{581] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 131. Equivalent Linearization and Nonlinearization 559 7. See also Hagedorn [3, pages 14{16] and McLachlan [7, Chapter 6, pages 103{112]. References [1]Beaman, J. J. Accuracy of statistical linearization. In New Approaches to Nonlinear Problems in Dynamics , P. J. Holmes, Ed. SIAM, Philadelphia, PA, 1980, pp. 195{207. [2]Caughey, T. Equivalent linearization techniques. J. Acoust. Soc. of America 35, 11 (November 1963), 1706{1711. [3]Hagedorn, P. Non-linear Oscillations . Clarendon Press, Oxford, England, 1982. [4]Iwan, W. D., and Patula, E. J. The merit of dierent error minimization criteria in approximate analysis. J. Appl. Mech. (March 1972), 257{262. [5]Iwan, W. D., and Yang, I.-M. Application of statistical linearization techniques to nonlinear multidegree - of - freedom systems. J. Appl. Mech. (June 1972), 545{550. [6]Kroger, H. Linearization of nonlinear dierential equations by means of Cauchy’s integral. J. Math. Physics 26 , 5 (May 1985), 929{940. [7]McLachlan, N. W. Ordinary Non-Linear Dierential Equations in Engi- neering and Physcial Sciences . Oxford University Press, New York, 1950. [8]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 560 III Approximate Analytical Methods 132. Maximum Principles Applicable to Linear ordinary dierential equations and linear partial dierential equations. Yields Upper or lower bounds on the solution. Idea By the use of a maximum theorem, we can nd bounds on certain types of equations. Procedure There are many theorems applicable to specialized equations and bound- ary conditions, which lead to bounds on the solutions. Maximum principles exist for all types of partial dierential equations (hyperbolic, elliptic, and parabolic) as well as for ordinary dierential equations. We choose toillustrate two theorems. Example 1 A theorem from calculus is Theorem A continuous real-valued function on a bounded closed interval attains its maximum and minimum on the interval. We will use this theorem to bound the solution to an ordinary dierential equation. Consider the equation exy00+x(1−x)y0=( 1+x2)y; (132.1) wherejy(a)jMandjy(b)jM. We claim that, for all xin the nite interval [a;b],y(x) is bounded in magnitude by M. Suppose that y(x) exceededMin some region within the interval [ a;b]. Then there would be maximum value of yon the interval; say it occurs at the pointx=c. Becauseyis a maximum at x=c, we require y0(c)=0 andy00(c)0. But this, with equation (132.1), implies that ecy00(c)= (1 +c2)y(c). This cannot be correct; the right side is positive, but the left side cannot be. Hence, ydoes not exceed Min the interval. It can similarly be shown that ycannot be less than −M. Hencejy(x)jMforxin the interval. Example 2 Ames [1, page 181] has the theorem: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 132. Maximum Principles 561 Letu(x) be a solution of the ordinary dierential equation L[u]=u00+H(x;u;u0)=0; fora<x<b; B1[u]=−u0(a)c o s+u(a)s i n=γ1; B2[u]=−u0(b)c o s+u(b)s i n=γ2; (132.2) where 0=2, 0=2,andare not both zero, H,Hu,Hu0are all continuous, and Hu0. Ifz1andz2satisfy L[z1]0; fora<x<b; B1[z1]γ1; B2[z1]γ2;(132.3) and L[z2]0; fora<x<b; B1[z2]γ1; B2[z2]γ2;(132.4) then we can conclude z2(x)u(x)z1(x); (132.5) fora<x<b . Hence, the solutions to equations (132.3) and (132.4) form bounds on the solution of equation (132.2). As an illustration of this theorem, suppose we want to approximate the solution of the ordinary dierential equation u00−u3=0; for 0<x< 1; u(0) = 0; u(1) = 1: This is in the form of equation (132.2) with a=0 ,b=1 ,===2, γ1=0 ,γ2=1 . W en o t et h a t z1(x)=xsatises equation (132.3) because z00 1−z3 1=−x30; z1(0) = 0; z1(1) = 1: We now search for a z2(x) of the form x. Usingz2(x)=xin equation (132.4) yields (−1)x−2−x30; B1[z2]=0i f>0; B2[z2]=1:(132.6.a-c) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 562 III Approximate Analytical Methods Because of equation (132.6.b), we restrict our search to >0. With this assumption, x2(+1)1f o rxbetween 0 and 1. Hence, equation (132.6.a) will be satised if (−1)1: (132.7) We choose=( 1+p 5)=2, so that equation (132.7) is satised. Hence, we can conclude, from equation (132.5) x(1+p 5)=2u(x)x; for 0<x< 1: (132.8) Notes 1. Any value of larger than (1 +p 5)=2 would also have yielded a bound foru(x) in equation (132.8). The best bound corresponds to the minimal value of , which was the one used. 2. Some of the \classical" maximum principles are (see Protter and Weinberger [5] or Sperb [7, pages 12{21]) Ifu(x) is non-constant and satises u00+b(x)u00i na ni n t e r v a l , andb(x) is bounded, then u(x) attains its maximum on the boundaries of the interval. Ifu(x) is non-constant and satises u00+b(x)u0+h(x)>0i n an interval, and b(x)a n dh(x) are bounded, and h0, then a non-negative minimum of u(x) can occur only on the boundaries of the interval. If the elliptic operator L[
] has bounded coecients and u(x) satises the inequality L[u]=X i;jaij(x)@2u @xi@xj+X ibi(x)@u @xi0 in some bounded domain D,t h e nu(x) cannot assume its maxi- mum at an interior point of Dunlessu(x) is identically constant. IfL[
] is a uniformly elliptic operator with bounded coecients andu(x;t) satises the inequality L[u]−@u @t=X i;jaij(x)@2u @xi@xj+X ibi(x)@u @xi−@u @t0 inD(0;T), whereDis a bounded domain and T<1,t h e n u(x) can attain its maximum only for t= 0 or on@D. 3. A theorem in Durstine and Shaer [3], applicable to ordinary dier- ential equations and partial dierential equations, states LetL[
]a n dB[
] be linear dierential operators such that the equation L[u]+(x)=0; in a domain D; Bj[u]=j; forj=1;2;:::;q on@D; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 132. Maximum Principles 563 has a unique solution u(x), and the Green’s function does not change sign in D.I fw1(x)a n dw2(x)s a t i s f y L[wk]+(x)=k(x);inD; Bj[wk]=j; forj=1;2;:::;q on@D; and 1=2is continuous, 1does not change sign in D, either 1>M2 1morM2 1m> 1, then w1+w1−w2 M−1<u(x)<w 1+w1−w2 m−1: 4. A theorem in Hille [4, pages 87{88], applicable to rst order ordinary dierential equations, states LetF(x;y)a n dG(x;y) be continuous in a region D(which contains the initial data) and suppose that F(x;y)<G(x;y) everywhere in D. Lety(x)a n dz(x) be the solutions of y0=F(x;y);y (x0)=y0; z0=G(x;y);z (x0)=y0: Then, in the region where y(x)a n dz(x) are dened and continuous z(x)<y(x);forx<x 0; y(x)<z(x);forx0<x: 5. A theorem in Ding [2] states Consider the equation ¨ x+g(x)=p(t)w i t h p(t) is continuous and 2 periodic, g(x) is continuously dierentiable and satises limjxj!1g(x) x=1. Ifp(t)i sa ne v e nf u n c t i o n ,o ri f p(t)i so d da n d g(x)i s an even function, then all solutions of this equation arebounded. 6. Other standard boundedness results include (a)Theorem Ifp(x) is continuous, of period L, not identically zero, and satisesR L 0jp(x)jdx4=andRL 0p(x)dx0, then all solutions of u00+p(x)u= 0 are bounded as x!1 . (b)Theorem If all solutions of y0=A(t)yare bounded (where lim t!1Rttr (A)dt >−1) and ifR1jB(t)jdt <1, then all solutions of y0=(A(t)+B(t))yare bounded. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 564 III Approximate Analytical Methods (c)Theorem If all solutions of y0=A(t)yare bounded (where A is a periodic matrix) and ifR1jB(t)jdt<1, then all solutions ofy0=(A(t)+B(t))yare bounded. (d)Theorem If all solutions of y0=Ayare bounded as t!1 (whereAis a constant matrix) and ifR1jB(t)jdt <1,t h e n all solutions of y0=(A+B(t))yare bounded. (e)Theorem If all solutions to y00+f(x)y= are bounded and ifR1jg(x)jdx<1, then all solutions of y00+(f(x)+g(x))y=0 are bounded. (f)Theorem (Comparison of approximate solutions) Consider x0= f(t;x)w h e r efis continuous with Lipschitz constant k. Letu1 andu2be approximate solutions with ju0 1(t)−f(t;u1(t))j1;ju0 2(t)−f(t;u2(t))j2 except where the derivatives are discontinuous. Then, if ju1(t0)− u2(t0)j, it follows that ju1(t)−u2(t)jekjt−t0j+1+2 kh ekjt−t0j−1i : (g)Theorem Lety(x) be any solution of y00−f(x)y=0w i t hf(x) positive and continuous in (0 ;1)a n dxf(x)2L(0;1). Then Cexp −Zx 0[f(z)+1 ]dz [y(x)]2+[y0(x)]2 CexpZx 0[f(z)+1 ]dz ; whereC=[y(0)]2+[y0(0)]2. References [1]Ames, W. F. Ad hoc exact techniques for nonlinear partial dierential equations. In Nonlinear Partial Dierential Equations in Engineering ,W .F . Ames, Ed. Academic Press, New York, 1967. [2]Ding, T. Boundedness of solutions of Dung’s equation. Tech. Rep. 58, University of Minnesota, Minneapolis, Minnesota, 1984. IMA Preprint Series. [3]Durstine, R. M., and Shaffer, D. H. Determination of upper and lower bounds for solutions to linear dierential equations. Quart. Appl. Math 16 ,3 (1958), 315{317. [4]Hille, E. Lectures on Ordinary Dierential Equations . Addison{Wesley Publishing Co., Reading, MA, 1969. [5]Protter, M. H., and Weinberger, H. F. Maximum Principles in Dierential Equations . Springer{Verlag, New York, 1984. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 132. Maximum Principles 565 [6]Sewell, M. J. Maximum and Minimum Principles . Cambridge University Press, New York, 1987. [7]Sperb, R. Maximum Principles and Their Applications . Academic Press, New York, 1981. [8]Varma, A., and Strieder, W. Approximate solutions of non-linear boundary-value problems. IMA J. Appl. Mathematics 34 (1985), 165{171. [9]Walter, W. Dierential and Integral Inequalities . Springer{Verlag, New York, 1970. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 566 III Approximate Analytical Methods 133. McGarvey Iteration Technique Applicable to First order ordinary dierential equations. Yields A sequence of approximations to the solution. Idea The method consists of generating a sequence of functions by a recur- rence relation. The initial function used is arbitrary. Procedure Given the rst order ordinary dierential equation dy dx=f(x;y); (133.1) we chooseT0(x;y)=T0(y) to be an arbitrary function of y. Then we dene the sequence of functions fTn(x;y)gby the recurrence relation Tn(x;y)=−Z f(x;y)@ @yTn−1(x;y) dx: (133.2) If we form Sn(x;y)=nX k=0Tk(x;y), thenSn(x;t) = constant is an approx- imate implicit solution to equation (133.1). As nincreases,Sn(x;y) will converge to the true solution of equation (133.1) if lim n!1@ @yTn(x;y) @ @ySn(x;y)=0: Example Suppose we wish to approximate the solution of the nonlinear ordinary dierential equation dy dx=x+1 y; y(0) = 4: In this case, equation (133.2) becomes Tn(x;y)=−Zx x+1 y@ @yTn−1(x;y)dx: (133.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 133. McGarvey Iteration Technique 567 We chooseT0(y)=y(recall that T0is only a function of y). From equation (133.3), we can calculate T1(x;y)=−1 2x2−x y; T2(x;y)=−1 2x2 y3−1 3x3 y2; T3(x;y)=−1 2x3 y5−13 24x4 y4−2 15x5 y3: Note that we have not used any constants of integration in evaluating the fTng. This part of the analysis is independent of whether or not we choose such constants. We can now calculate S3(x;y)a s S3(x;y) =3X k=0Tk(x;y) =y+ −1 2x2−x y + −1 2x2 y3−1 3x3 y2 + −1 2x3 y5−13 24x4 y4−2 15x5 y3 ; =120y6−60x2y5−120xy4−40x3y3−4x2(4x3+ 15)y2−65x4y−60x3 120y5: Now, for the rst time, we use the initial condition: y(0) = 4. The implicit approximation to the solution of equation (133.1) is then given by S3(x;y)=S3(x0;y0)=S3(0;4); or 120y6−60x2y5−120xy4−40x3y3−4x2(4x3+ 15)y2−65x4y−60x3 120y5=4: (133.4) For any value of x, equation (133.4) is a polynomial in y. Thus, for any x, we can solve for y. For this example, it turns out that the dierence between the implicit solution given by equation (133.4) and the numerical solution is less than 2% for 0 x20. Notes 1. The above example is from McGarvey [1]. 2. This approximation technique may converge in cases where Picard approximations (see page 618) diverge. 3. For certain classes of equations, error estimates can be obtained for this technique. Reference [1]McGarvey, J. F. Approximating the general solution of a dierential equation. SIAM Review 24 , 3 (July 1982), 333{337. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 568 III Approximate Analytical Methods 134. Moment Equations: Closure Applicable to A stochastic dierential equation or a Fokker{ Planck equation (which is a second order parabolic partial dierential equation). Yields A system of ordinary dierential equations from which dierent mo- ments may be determined. Idea Interpreting the solution of the Fokker{Planck equation as a probability density, ordinary dierential equations may sometimes be found for the moments of the random process. Procedure The solution of a Fokker{Planck equation is the probability density P(x;t) of a random process (see page 303). For an N-dimensional random process x=(x1;x2;:::;xN), the Fokker{Planck equation has the form @P @t=−NX i=1@ @xi(ciP)+NX i;j=1@2 @xi@xj(aijP); (134.1) where the coecients fcigandfaijgare, in general, functions of tandx. All of the coecients are determined by the stochastic dierential equationthat created equation (134.1). The expectation of a function of x,s a yf(x), is dened to be the integral off(x)t i m e sP(x;t), integrated over all values of x.T h a ti s , E[f(x(t))] =Z f(x)P(x;t)dx: Note that this expectation is a function of t. If equation (134.1) is multi- plied byf(x) and integrated over all values of x, there results d dtE[f(x(t))] =−NX i=1Z f(x)@ @xi(ciP)dx+NX i;j=1Z f(x)@2 @xi@xj(aijP)dx: (134.2) Often, we may be able to integrate the right-hand side of equation (134.2) by parts to obtain an ordinary dierential equation for E [ f(x)]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 134. Moment Equations: Closure 569 Example The system of stochastic dierential equations dx dt+x=z; x (0) = 0; dz dt+2z=N(t);z(0) = 1; (134.3.a-d) whereN(t) is \white Gaussian noise" corresponds to the Fokker{Planck equation and initial condition @P @t=@ @x[(x−z)P]+2@ @z[zP]+@2 @z2[P]; P(0;x;z)=(x)(z−1); (134.4) for the probability density P(t;x;z ) (see page 303). Suppose we desire the expected value of x(t): E[x(t)] =Z1 −1Z1 −1xP(t;x;z )dxdz: Multiplying equation (134.4) by xand integrating from −1 to1with respect to both xandzproduces d dtE[x(t)] =−E[x(t)] + E [z(t)]; (134.5) where we have made the physically reasonable assumptions that jxjP(t;x;z ) !0a sjxj!1 ,a n db o t hjzjP(t;x;z )!0a n djzjPz(t;x;z )!0a s jzj!1 . These assumptions were required to carry out the integrations by parts in the right-hand side of equation (134.2). Note that equation (134.5) involves the expected value of z.T oo b t a i n an equation for E [ z], equation (134.4) can be multiplied by zand then integrated to obtain d dtE[z(t)] =−2E [z(t)]: (134.6) From equation (134.3.b) and equation (134.3.d), the initial conditions for equation (134.5) and equation (134.6) are E[x(0)] = 0; E[z(0)] = 1: (134.7.a-b) Alternatively, these initial conditions can be obtained directly from the initial conditions in equation (134.4) by taking expectations. If equation (134.6) is solved with equation (134.7.b), then equation (134.5) can be solved with equation (134.7.a) to determine the expectation of bothx(t)a n dz(t) E[z(t)] =e−2t; E[x(t)] =1 3/parenleftbig e−t−e−2t
: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 570 III Approximate Analytical Methods If the second order moments (i.e., fE x2(t) ;E[x(t)z(t)];E z2(t) g) are desired, the equations comparable to equation (134.5) and equation (134.6) are d dt2 4E x2 E[xz] E z23 5=2 4−120 0−31 00 43 52 4E x2 E[xz] E z23 5+2 40 023 5; 2 4E x 2 E[xz] E z23 5 t=0=2 40 0 13 5;(134.8) where we have dropped the explicit dependence on tfor clarity. These equations were obtained by multiplying equation (134.4) by each of x2,xz, andz2, and then integrating with respect to xandz. Notes 1. Another procedure for determining ordinary dierential equations for the moments is described on page 572. 2. It is not always the case that the system of ordinary dierential equa- tions for the moments will close (i.e., there will be mequations for themunknowns). For example, the stochastic dierential equation d2x dt2+dx dt+x+x3=N(t); whereN(t) is white noise, corresponds to the Fokker{Planck equation @P @t=−_x@P @x+@ @_x[( _x+x+x3)P]+@2P @_x2: In this case, the equations for the rst moments become d dtE[x]= E[_x]; d dtE[_x]=−E[_x]+E[x]−E x3 : Therefore, knowledge of E [ x] requires knowledge of E x3 . In this example, the system of ordinary dierential equations that determine E x3 involves the quantity E x5 , etc. However, if is small, then perturbation techniques may be used to approximately solve the moment equations. 3. For systems that do not close, two \closing" approximations that are commonly used are (see Boyce [4]): Gaussian closure (also called \cumulant discard") Correlation discard CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 134. Moment Equations: Closure 571 In the Gaussian closure technique, a high odd cumulant of the probability density is set to zero. This procedure yields an equation for E xk in terms offE xj j0<j<kg. Correlation discard is generally used for equations that have \colored noise"forcing terms. In this approximation technique, some high power of the dependent variable in the stochastic dierential equation and the \colored noise" is assumed to be uncorrelated. Crandall [5] contains a review of non-Gaussian closure tech- niques. See also Ibrahim et al. [6]. 4. For determining the moments of random functions dened by partial dierential equations, see, for instance, Wan’s paper [7]. References [1]Assaf, S. A., and Zirkle, L. D. Approximate analysis of non-linear stochastic systems. Int. J. Control 23 , 4 (1976), 477{492. [2]Bobrik, R. V. Hierarchies of moment equations for the solution of the Schrodinger equation with random potential and their closure. Teoretich- eskaya i Matematicheskaya Fizika 68 , 2 (August 1986), 301{311. [3]Bover, D. C. C. Moment equation methods for nonlinear stochastic systems. J .M a t h .A n a l .A p p l .6 5 (1978), 306{320. [4]Boyce, W. E. Random eigenvalue problems. In Probabilistic Methods in Applied Mathematics , A. T. Bharucha-Reid, Ed. Academic Press, New York, 1968, pp. 1{73. [5]Crandall, S. H. Non-Gaussian closure techniques for stationary random vibration. Int. J. Non-Linear Mechanics 20 , 1 (1985), 1{8. [6]Ibrahim, R. A., Soundararajan, A., and Heo, H. Stochastic response of nonlinear dynamic systems based on a non-Gaussian closure. J. Appl. Mech. 52(December 1985), 965{970. [7]Wan, F. Y. M. Linear partial dierential equations with random forcing. Stud. Appl. Math. 51 , 2 (June 1972), 163{178. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 572 III Approximate Analytical Methods 135. Moment Equations: It^o Calculus Applicable to A set of stochastic dierential equations. Yields A system of ordinary dierential equations from which dierent mo- ments may be determined. Idea Using It^ o calculus, a set of ordinary dierential equations may be de- termined that will describe the moments of a random process. Procedure In the It^ o calculus, there are two dierent types of dierential elements. There aredtterms, which are small; and there are dterms ( Brownian motion terms), which are random. Brownian motion is the integral of white noise ;t h a ti s ,(t)=Rt 0n(s)ds,w h e nn(s) is white noise. We assume the standard scaling: E (d)2 =dt,w h e r eE [
]i st h e expectation operator (taken over the random variables in the system). TheBrownian motion terms also have mean zero: E [ d]=0 . Suppose that x 1(t)a n dx2(t) are random processes described by the two stochastic dierential equations dx1 dt=a1(t)+b1(t)n(t); dx2 dt=a2(t)+b2(t)n(t);(135.1) or dx1=a1(t)dt+b1(t)d; dx2=a2(t)dt+b2(t)d: It^o’s lemma states that d(x1x2)=x1dx2+x2dx1+b1b2dt: (135.2) This relation is dierent from the result in the classical calculus by the inclusion of the last term. This relationship may be used to determinemoment equations for a random process. Example Given the stochastic dierential equation d2y dt2−n(t)y=0; y(0) = 1;y0(0) = 0; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 135. Moment Equations: It^ o Calculus 573 wheren(t) is white noise, we can dene z=dy dtand so obtain the coupled system of stochastic dierential equations dy=zdt; y (0) = 1; dz=yd; z (0) = 0:(135.3) Using It^ o’s lemma repeatedly on equation (135.3), we can derive the following relations d/parenleftbig yL
=LyL−1zdt; d/parenleftbig zK
=K(K−1) 2y2zK−2dt+KzK−1yd:(135.4) If we dene the N+ 1 dierent Nth order moments by GM N(t)=E yN−M(t)zM(t) ;M =0;1;:::;N; then, from equation (135.4), we obtain the set of coupled ordinary dier- ential equations dGM N dt=(N−M)GM+1 N +M(M−1) 2GM−2 N for GM N(t)/bracerightbig . For example, if we choose N= 2, then we obtain the system d dt2 4G0 2 G1 2 G2 23 5=2 4020 001 1003 52 4G0 2 G1 2 G2 23 5; (135.5) with the initial conditions2 4G0 2 G12 G2 23 5 t=0=2 41 003 5: The eigenvalues of the matrix in equation (135.5) are the three cube roots of two. Hence, each of fG 0 2;G12;G22ggrows exponentially in time. Notes 1. Note that the Fokker{Planck equation corresponding to equation (135.3) is 1 2y2Pzz−zPy=Pt; P(y;z;0) =(y−1)(z); whereP(y;z;t ) represents the joint probability density of yandzat timet. 2. The coupled ordinary dierential equations that are derived for the moments in this section are identical to the equations obtained by the method described on page 568. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 574 III Approximate Analytical Methods References [1]Kulkarny, V. A., and White, B. S. Focussing of waves in turbulent inhomogeneous media. Phys. Fluids 251 , 10 (1982), 1770{1784. [2]Schuss, Z. Theory and Applications of Stochastic Dierential Equations . John Wiley & Sons, New York, 1980. [3]Spigler, R. Monte Carlo-type simulation for solving stochastic ordinary dierential equations. Math. and Computers in Simulation 29 (1987), 243{ 251. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 136. Monge’s Method 575 136. Monge’s Method Applicable to Some nonlinear second order partial dierential equations with two independent variables. Yields An exact solution. Idea Application of some algebraic identities and then the use of equation splitting permits some nonlinear partial dierential equations to be solved. Procedure Monge’s method works for some dierential equations of the form R@2z @x2+S@2z @x@y+T@2z @y2=V; or Rr+Ss+Tt=V; (136.1) forz=z(x;y), where, as usual, r=zxx,s=zxy,t=zyy,p=zx,q=zy, andfR;S;T;Vgmay be functions of fp;q;x;y;zg. First, note that we can write dp=pxdx+pydy=rdx+sdy; dq=qxdx+qydy=sdx+tdy:(136.2.a-b) Solving equation (136.2.a) for rand equation (136.2.b) for tand then using these values in equation (136.1), we obtain [Rdpdy +Tdqdx−Vd yd x ]−s R(dy)2−Sdydx +T(dx)2 =0: (136.3) By use of equation splitting (see page 520), we look for the simultaneous solutions to Rdpdy +Tdqdx−Vd yd x =0; R(dy)2−Sdydx +T(dx)2=0:(136.4.a-b) Any solution of equation (136.4) is also a solution of equation (136.3). Such a solution is called an intermediate integral and will depend on an arbitrary constant or function. If we can nd two such integrals, say f(x;y;z;p;q )=A; g (x;y;z;p;q )=B; whereAandBare arbitrary constants, then we may be able to solve for fp=p(x;y;z ),q=q(x;y;z )g. If we could, then we might be able to integrate the Pfaan dierential equation (see page 384) dz=pdx+qdy to determine z=z(x;y). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 576 III Approximate Analytical Methods Example Suppose we have the partial dierential equation y2@2z @x2−2y@2z @x@y+@2z @y2=@z @y+6y; (136.5) which can be written as: y2r−2ys+t=p+6y. Therefore, we have fR=y2,S=−2y,T=1 ,V=p+6yg. The two equations in equation (136.4) then become y2dpdy +dqdx−(p+6y)dydx =0; (ydy+dx)2=0:(136.6.a-b) Equation (136.6.b) can be integrated to obtain 2x+y2=A; (136.7) whereAis an arbitrary constant. Dividing equation (136.6.a) by dx(or, equivalently, by ( −ydy) from equation (136.6.b)), we obtain −ydp+dq−(p+6y)dy=0; which can be integrated to yield −py+q−3y2=(A)=(2x+y2)o r y@z @x−@z @y+3y2=(2x+y2); (136.8) whereis an arbitrary function. Equation (136.8) is an intermediate integral and the only one that equation (136.6) has (due to the double root appearing in equation (136.6.b)). Because we do not have two intermediate integrals, we can not proceed with the derivation in the procedure. However, we can solve equation (136.8) directly to obtain a solution of equation (136.5). Because equation(136.8) is quasilinear, the method of characteristics (see page 432) may be used. The subsidiary equations are dx y=dy −1=dz −3y2+(2x+y2): (136.9) From the rst equality in equation (136.9), we recover the integral in equation (136.7). Using equation (136.7) in the second equality in equation (136.9) yields dy −1=dz −3y2+(A); with the solution z−y3+y(2x+y2)=B,w h e r eBis another arbitrary constant. Hence, a general integral of equation (136.5) is /parenleftbig z−y3+y(2x+y2);2x+y2
=0: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 136. Monge’s Method 577 This leads to a general solution of equation (136.5) z=y3−y(2x+y2)+ (2z+y2); (136.10) whereand are arbitrary functions of their arguments. Notes 1. Because equation splitting was used in going from equation (136.3) to equation (136.4), the solution obtained in equation (136.10) is notthe most general solution. 2. See Ames [1, pages 60{65], Forsyth [2, Volume 6, pages 202{208], Piaggio [3, pages 181{187], and Sneddon [4, pages 131{135]. References [1]Ames, W. F. Ad hoc exact techniques for nonlinear partial dierential equations. In Nonlinear Partial Dierential Equations in Engineering ,W .F . Ames, Ed. Academic Press, New York, 1967. [2]Forsyth, A. R. Theory of Dierential Equations . Dover Publications, Inc., New York, 1959. [3]Piaggio, H. T. H. An Elementary Treatise on Dierential Equations and Their Applications . G. Bell & Sons, Ltd, London, England, 1926. [4]Sneddon, I. N. Elements of Partial Dierential Equations . McGraw{Hill Book Company, New York, 1957. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 578 III Approximate Analytical Methods 137. Newton’s Method Applicable to Ordinary and partial dierential equations. Yields A sequence of approximations to the solution. Idea When a Newton iteration is applied to a nonlinear dierential equation, each step of the iteration requires that a linear dierential equation besolved. Procedure We illustrate the general procedure on an ordinary dierential equation. Suppose we wish to approximate the solution to the rst order ordinarydierential equation G(y 0;y;x)=0; y(0) =y0;(137.1) fory(x)w h e nG(y0;y;x) is a nonlinear function. If an approximate solution of equation (137.1), say yk(x), is known, thenG(y0;y;x) could be expanded about yk(x)t oo b t a i n G(y0;y;x)’G(y0 k;yk;x)+Gy(y0 k;yk;x)(y−yk)+Gy0(y0 k;yk;x)(y0−y0 k) (137.2) to leading order. For the solution to equation (137.1), G(y0;y;x) = 0, and so equation (137.2) becomes (y0−y0 k)Gy0(y0 k;yk;x)+(y−yk)Gy(y0 k;yk;x)’−G(y0 k;yk;x): Therefore, if the linear ordinary dierential equation e0 kGy0+ekGy=−G; ek(0) = 0;(137.3) is solved for the \correction term" ek(x), then dening yk+1(x)=yk(x)+ek(x) should yield a better approximation, yk+1(x), toy(x). Equation (137.3) can be solved exactly by the use of integrating factors (see page 356). However, it needs to be solved only approximately becausethe higher order approximations (i.e., y k+2,yk+3;:::) will correct errors made in solving equation (137.3). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 137. Newton’s Method 579 Special Case In the special case that the original equation is linear in y0and hence of the form G(y0;y;x)=y0−f(x;y)=0; then the denition of yk+1may be succinctly represented as y0 k+1−fy(x;yk(x))yk+1=f(x;yk(x))−fy(x;yk(x))yk(x); yk+1(0) =y0:(137.4) Example Suppose we are looking for an approximation, near x= 0, of the solution to the nonlinear ordinary dierential equation y0+y3=0; y(0) = 1; which has the known exact solution y(x)=( 1+2x)−1=2 =1−x+3 2x2−5 2x3+35 8x4−63 8x5+


: For this problem we recognize that f(x;y)=−y3and so equation (137.4) becomes y0 k+1+3y2 kyk+1=2y3 k; yk+1(0) = 1:(137.5) If we start with y0=y(0) = 1, then, from equation (137.5) y0 1+3y1=2; y1(0) = 1; with the solution y1(x)=1 3/parenleftbig 2+e−3x
=1−x+3 2x2−3 2x3+


: If we use the approximation y1’1−x, then the equation for y2(from equation (137.5), with k=1 )i s y0 2+3 ( 1−x)2y2=2 ( 1−x)3; y2(0) = 0; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 580 III Approximate Analytical Methods with the solution y2(x)=−2e−x3+3x2−3xZx 0ex3−3x2+3x(x−1)3dx+1 =1−x+3 2x2−5 2x3+35 8x4−261 40x5+


: We see then that y1(x) has the rst 3 terms correct, whereas y2(x) (which used only the rst order information in y1(x)) has the rst 5 terms correct. Notes 1. For symbolic manipulation of the formulae appearing above, Geddes [4] discusses the number of correct terms at each step. 2. Most often, this iterative method will be implemented numerically and not performed analytically. This is because, by hand, it is ofteneasier to nd a Taylor series solution directly (see page 632) than to use Newton iterates. Rice and Boisvert [7, pages 101{111] have a numerical example of using Newton’s method to solve an ellipticequation. 3. Error estimates for Newton’s method (applied to rst order equa- tions) can be found in Mikhlin and Smolitskiy [6, pages 12{16]. 4. When Newton’s method is numerically applied to nonlinear boundary value problems, the method is often called quasilinearization . This is the same algorithm that is obtained when multiple shooting is used (see page 706), and the number of rays becomes very large. See Bellman and Kalaba [3] or Stoer and Bulirsch [8, pages 498{502] fordetails. 5. Geddes [4] showed that the number of correct coecients in a power series solution obtained by this method, when applied to an explicit rst order nonlinear ordinary dierential equation, more than doubles at each step. 6. See also Ascher al.[1, pages 52{55]. References [1]Ascher, U. M., Mattheij, R. M. M., and Russel, R. D. Numerical Solution of Boundary Value Problems for Ordinary Dierential Equations . Prentice{Hall, Inc., Englewood Clis, NJ, 1988. [2]B a r k a t o u ,M .A . Rational Newton algorithm for computing formal solution of linear dierential equations. No. 358 in Lecture Notes in Computer Science. Springer{Verlag, New York, 1989, pp. 183{195. [3]Bellman, R. E., and Kalaba, R. Quasilinearization and Nonlinear Boundary-Value Problems . American Elsevier Publishing Company, New York, 1965. [4]Geddes, K. O. Convergence behavior of the Newton iteration for rst order dierential equations. In Symbolic and Algebraic Computation ,E .W .N g , Ed., EUROSAM 79. Springer{Verlag, New York, 1979, pp. 189{199. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 137. Newton’s Method 581 [5]Guenther, R. B., and Lee, J. W. Convergence of the Newton{Raphson method for boundary value problems of ordinary dierential equations. In Computation Solutions of Nonlinear Systems of Equations . Amer. Math. Soc., Providence, RI, 1990, pp. 257{264. [6]Mikhlin, S. G., and Smolitskiy, K. L. Approximate Methods for Solutions of Dierential and Integral Equations . American Elsevier Publishing Company, New York, 1967. [7]Rice, J. R., and Boisvert, R. F. Solving Elliptic Problems Using ELLPACK . Springer{Verlag, New York, 1985. [8]Stoer, J., and Bulirsch, R. Introduction to Numerical Analysis . Springer{ Verlag, New York, 1976. Translated by R. Bartels, W. Gautschi and C.Witzgall. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 582 III Approximate Analytical Methods 138. Pad e Approximants Applicable to Any type of function (whether or not it comes from a dierential equation). Yields An approximation formula generally valid over an interval, and, often, information about whether singularities exist. Idea A Taylor series can be manipulated to produce information about the existence of singularities. Procedure When a power series representation of a function diverges, it indicates the inability of the power series to approximate the function in a certain region. A theorem of complex analysis states that if a Taylor series ofa function diverges, then that function has singularities in the complex plane. A Pad e approximant is a ratio of polynomials that contains the same information that a truncated power series does. Because the polynomial in the denominator may have roots in the region of interest, the Pad e approximant may accurately indicate the presence of singularities. Suppose we have found the kth order Taylor series solution to a dier- ential equation (see page 632) y k(x)=a0+a1x+a2x2+


+akxk: (138.1) The (N;M )P a d  e approximant, PN M(x), is a ratio of polynomials, with the polynomial in the numerator having degree Nand the polynomial in the denominator having degree M: PN M(x)=B0+B1x+


+BNxN A0+A1x+


+AMxM; (138.2) withN+M+1=k. Without loss of generality, we take A0=1 . T h e remainingN+M+ 1 coecientsfA1,A2,:::,AN,B0,B1,:::,BMgare chosen so that the rst N+M+ 1 terms in the Taylor series expansion of PN M(x) match the rst N+M+ 1 terms of the Taylor series in equation (138.1). Usually we consider only the convergence of the Pad e sequencefPJ 0(x), PJ+1 1(x),PJ+2 2(x),:::ghavingN=M+JandJheld constant while M!1 . The special sequence with J= 0 is called the diagonal sequence . CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 138. Pad e Approximants 583 Example 1 Suppose we wish to approximate the solution of the ordinary dierential equation y0=y2;y (0) = 1: (138.3) Because equation (138.3) is separable (see page 401), the solution to equa- tion (138.3) can be found to be y(x)=1=(1−x). If we tried to nd the Taylor series of y(x) directly from equation (138.3), we would obtain y(x)=1+x+x2+x3+x4+


: (138.4) This geometric series is convergent, of course, only for jxj<1. The solution has a singularity at x= 1, but this fact is not readily apparent from the expansion in equation (138.4). The diagonal sequence of Pad e approximants corresponding to equation (138.4) is P1 1(x)=1 1−x; P2 2(x)=1 1−x; P3 3(x)=1 1−x: Therefore, the diagonal sequence of Pad e approximants recovers the exact solution to the dierential equation from only a few terms in the Taylor series. Of course, this is an exceptional example. Example 2 Suppose we wish to approximate the solution of the ordinary dierential equation y0=1+y2;y (0) = 0: (138.5) Because equation (138.5) is separable, the solution to equation (138.5) can be found to be y(x)=t a nx. If we tried to nd the Taylor series of y(x) directly from equation (138.5), we would nd y(x)=x+x3 3+2x5 15+17x7 315+


: (138.6) Note that the exact solution has singularities at x=(2n+1)=2, whereas the Taylor series approximation does not appear to show this behavior. Using equation (138.6), we can compute the rst few elements of the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 584 III Approximate Analytical Methods diagonal sequence P2 2(x)=3x 3−x2; P3 3(x)=x(x2−15) 3(2x2−5); P4 4(x)=5x(21−2x2) x4−45x2+ 105: Note that these Pad e approximants have singularities where the denomi- nator vanishes: ForP2 2(x), these singularities are at x’1:7. ForP3 3(x), these singularities are at x’1:58. ForP4 4(x), these singularities are at x’1:5712, andx’6:5. We observe that these Pad e approximants are attempting to recover the singularities of the exact solution at x==2a n dx=3=2. Because the Pad e approximants have these singularities, they produce an accurate numerical approximation of the exact solution over a wide range of values. Notes 1. Pad e approximants are not always better than a Taylor series rep- resentation. In fact, it may happen that the Pad e approximants diverge while the Taylor series converges. However, it often happensthatP N M(x) converges (as N;M!1 ) to the true solution of the dierential equation, even when the Taylor series solution diverges! 2. Pad e approximants are also called rational function approximations . 3. Prendergast [7] proposes a technique to nd the Pad e approximants for the solution of a nonlinear dierential equation without rst nd-ing the Taylor series. Martin and Zamudio{Cristi [6] address the same issue but for a smaller class of equations. 4. In Bender and Orszag [1, pages 383{410] is a discussion of computa- tional techniques for computing Pad e approximants numerically. 5. A two-point Pad e approximant is one that utilizes Taylor series infor- mation about two dierent points. Often these points are chosen to be zero and innity. For two-point Pad e approximants the coecients in equation (138.2) are chosen so that both Taylor series will bematched. See Bender and Orszag [1] or Magnus [5] for details. 6. Many symbolic computer languages have a function that nds Pad e approximants analytically when a Taylor series is input. See Czaporand Geddes [3]. 7. The Bulirsch{Stoer method is a numerical method for solving rst or- der ordinary dierential equations using Pad e approximants, Richard- son extrapolation, and the modied midpoint rule. See Press et al. [8, pages 563{568] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 138. Pad e Approximants 585 References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Cuyt, A. Pade Approximants for Operators: Theory and Applications . No. 1065. Springer{Verlag, New York, 1984. [3]Czapor, S. R., and Geddes, K. O. A comparison of algorithms for the symbolic computation of Pade approximants. In EUROSAM ’84 ,i nJ .F i t c h , Ed. Springer{Verlag, New York, 1984, pp. 248{259. [4]Geddes, K. O. Symbolic computation of Pade approximants. ACM Trans. Math. Software 5 , 2 (June 1979), 218{233. [5]Magnus, A. On the structure of the two-point Pade table. In Analytic The- ory of Continued Fractions ,W .B .J o n e s ,W .J .T h r o n ,a n dH .W a a d e l a n d , Eds., no. 932 in Lecture Notes in Mathematics. Springer{Verlag, New York,1982, pp. 176{193. [6]Martin, P., and Zamudio-Cristi, J. Fractional approximation for rst- order dierential equations with polynomial coecients|application to zzzref31refzzz. J. Math. Physics 23 , 12 (Dec 1982), 2276{2280. [7]Prendergast, K. H. Rational approximation for non-linear ordinary dierential equations. In The Riemann Problem, Complete Integrability and Arithmetic Applications , D. Chudnovsky and G. Chudnovsky, Eds., no. 925 in Lecture Notes in Mathematics. Springer{Verlag, New York, 1982. [8]Press, W. H., Flannery, B. P., Teukolsky, S., and Vetterling, W. T. Numerical Recipes . Cambridge University Press, New York, 1986. [9]Reusch, M. F., Ratzan, L., Pomphrey, N., , and Park, W. Diagonal Pade approximations for initial value problems. SIAM J. Sci. Stat. Comput. 9, 5 (September 1988), 829{838. [10]Williamson, R. A. Pade approximations in the numerical solution of hy- perbolic dierential equations. In Pade Approximation and Its Applications, Bad Honnef 1983 , H. Werner and H. J. Bunger, Eds., no. 1071 in Lecture Notes in Mathematics. Springer{Verlag, New York, 1984, pp. 252{264. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 586 III Approximate Analytical Methods 139. Perturbation Method: Method of Averaging Applicable to Nonlinear dierential equations that have a periodic solution and a small parameter. Yields An approximation to the solution, valid over an entire period. Idea Write the solution of a given dierential equation as a function with slowly varying parts. Then, average those slowly varying parts over a complete cycle. Procedure We illustrate the method on a perturbed harmonic oscillator. Suppose we have the equation d2y dt2+y+f y;dy dt =0: (139.1) Note that, when = 0, equation (139.1) is a harmonic oscillator. The solution to equation (139.1), when = 0, is therefore: y(t)=Acos(t+ ) (whereAand  are constants). If is very small, we might expect a similar \looking" solution, so we assume that the solution to equation (139.1) isgiven by y(t)=acos(t+); (139.2) wherea(t)a n d(t) are \slowly varying" (another expression often used is \nearly constant"). Dierentiating equation (139.2) with respect to t yields dy dt=−asin(t+)+da dtcos(t+)−ad dtcos(t+): (139.3) Ifaandare \slowly varying," then da=dt andd=dt will be \small" compared to a. Hence, we set the derivative of y(t)t ob e dy dt=−asin(t+): (139.4) Comparing equation (139.3) to equation (139.4), it is clear that we have made the assumption da dtcos(t+)−ad dtcos(t+)=0: (139.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 139. Perturbation Method: Method of Averaging 587 This gives one equation relating the two unknowns, a(t)a n d(t). Dier- entiating equation (139.4) and using equation (139.4) and equation (139.1) results in the expression −da dtsin(t+)−ad dtcos(t+)=f(acos(t+);−asin(t+)): (139.6) The two equations in equation (139.5) and equation (139.6) can be solved to yield the relations da dt=f(acos(t+);−asin(t+)) sin(t+); d dt= af(acos(t+);−asin(t+)) cos(t+): (139.7) The equations in equation (139.2) and equation (139.7) are still exact. The change of variables from fy;y0gtofa;g(by use of equation (139.2) and equation (139.5)) has been carried out without any approximation being made. The assumptions made have been motivated by the smallness of , but the system is still exact. Now we use the \slowly varying" feature of aandto make the required approximation. If aandare \slowly varying," then the values of da=dt and d=dt should not change much over a single period of the solution. Hence, if we replace the right-hand sides of equation (139.7) by their averages over one period, then the solutions for a(t)a n d(t) should not be changed very much. Therefore, we approximate the solution of equation (139.7) by the solution of da dt=F(a);d dt= aG(a); (139.8) where F(a)=1 2Z2 0f(acos(t+);−asin(t+)) sin(t+)d; G(a)=1 2Z2 0f(acos(t+);−asin(t+)) cos(t+)d:(139.9) The prescription is to evaluate equation (139.9) and then to solve equation (139.8). Knowing a(t)a n d(t), we can evaluate equation (139.2) and so recover an approximation to y(t). Example 1 For the Van de Pol oscillator d2y dt2+y+(y2−1)dy dt=0; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 588 III Approximate Analytical Methods we identify f(y;y0)=(y2−1)y0. Evaluating equation (139.9) with this f results inF(a)=a 2−a3 8andG(a) = 0. This, in turn, allows us to solve equation (139.8). We nd a2(t)=4 1+ 4 a2 0−1 e−t; (t)=0; wherea0=a(0);0=(0). Note that as t!1 the approximation in equation (139.2) tends to a sinusoidally varying function of magnitude two. Example 2 For Dung’s equation d2y dt2+y+y3=0; we identify f(y;y0)=y3. Evaluating equation (139.9) with this fresults inF(a)=0a n dG(a)=3 4a3. This, in turn, allows us to solve equation (139.8). We nd a(t)=a0; (t)=0+3 8a2 0t: Notes 1. This method is also called the method of Krylov{Bogoliubov{Mitropolski. 2. The solution of _ x=f(t;x; ) (whenfhas period 2 int)c a nb e approximated by averaging. The solution by kth order averaging is always valid with error O(k) on time intervals of length O(1=). Accuracy is improved in two cases (see Murdock and Wang [5]): If the average of fvanishes (i.e.,1 2R2 0f(t;x; )dt= 0), then thekth order averaging approximation is valid with error O(k−1) for intervals of length O(1=2). If the solutions approach a hyperbolic sink (exponentially at- tracting rest point), then the kth order averaging approxmima- tion has error O(ek) for all future time. This is known as the Sanchez{Palencia theorem. 3. There are many ways in which averaging techniques can be applied to dierential equations; we have illustrated only one technique. An- other useful technique is the method of averaged Lagrangians (see Whitham [9]). This technique is applied by nding the Lagrangiancorresponding to a given dierential equation (see page 61), assuming an expansion of the Lagrangian that contains slowly varying functions and a small parameter, and, at each order of the small parameter,solving the dierential equation corresponding to that term of the Lagrangian. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 139. Perturbation Method: Method of Averaging 589 4. Macsyma [4] has a package ( avgpode ) that implements the method of averaging for ordinary dierential equations. 5. See also Kevorkian and Cole [3, pages 279{287], Nayfeh [6, Chapter 5, pages 159{227], and Rand and Armbruster [7, Chapter 5, pages 107{131]. References [1]Golec, J., and Ladde, G. Euler-type approximation for systems of stochastic dierential equations. J. Appl. Math. Simulation 2 , 4 (1989), 239{ 249. [2]Gromyak, M. I. Justication of a scheme for averaging of hyperbolic systems with fast and slow variables. A mixed problem. Ukrain. Mat. Zh. 38 , 5 (1986), 575{582. [3]Kevorkian, J., and Cole, J. D. Perturbation Methods in Applied Mathematics . Springer{Verlag, New York, 1981. [4]Macsyma .Mathematics Reference Manual . Macsyma, Arlington, MA, 1993. [5]Murdock, J., and Wang, L.-C. Validity of the multiple scale method for very long intervals. Z angew Math Phys 47 (1996), 760{789. [6]Nayfeh, A. H. Perturbation Methods . John Wiley & Sons, New York, 1973. [7]Rand, R. H., and Armbruster, D. Perturbation Methods, Bifurcation Theory and Computer Algebra . No. 65 in Applied Mathematical Sciences. Springer{Verlag, New York, 1987. [8]Sanders, J. A., and Verhulst, F. Averaging Methods in Nonlinear Dynamical Systems . Springer{Verlag, New York, 1985. [9]Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 590 III Approximate Analytical Methods 140. Perturbation Method: Boundary Layer Method Applicable to Dierential equations with a small parameter present for which regular perturbation series are inadequate. Yields This singular perturbation technique yields an expansion of the solution in terms of the small parameter. Idea If a regular perturbation series cannot match all the boundary condi- tions in a dierential equation, there may be one or more regions where the solution is rapidly varying. Procedure Given a dierential equation with a small parameter , attempt to nd a solution in the form of a regular perturbation series (see page 610). Callthis the \outer" solution. If the \outer" solution cannot match all of the initial conditions or boundary conditions, then attempt to place \boundary layers" (regions of rapid variation) near one or more of the boundaries. Inside of each boundary layer, the solution will vary smoothly (in a stretched variable) from the value of a \outer" solution to the value on the boundary. If multiple \outer" solutions exist, then there may be internal boundary layers (called shocks ). These internal boundary layers will change the solution smoothly from one \outer" solution to another. Example Consider the constant coecient ordinary dierential equation d2y dx2+dy dx+y=0; y(0) = 4;y(1) = 5;(140.1) whereis a number much smaller than one. Initially, we look for an \outer" solution in the form of a regular perturbation series (see page 610) y=youter =y0+y1+2y2+


: (140.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 140. Perturbation Method: Boundary Layer Method 591 Using equation (140.2) in equation (140.1) and setting the coecients of dierent powers of to zero produces the sequence of equations dy0 dx+y0=0; dy1 dx+y1=−d2y0 dx2; ...(140.3.a-b) with the boundary conditions y0(0) = 4;y 0(1) = 5; yi(0) = 0;yi(1) = 0;fori=1;2;3;::::(140.4.a-b) The most general solution of equation (140.3.a) is y0(x)=Ce−x(140.5) for some constant C. This solution cannot satisfy both of the boundary conditions in equation (140.4.a); so, we suspect the existence of a boundary layer. First, we search for a boundary layer near x= 0. If it is not possible to place one there, then we would attempt to place one near the other boundary, at x= 1. Because a change of order one is expected to take place in a thin xregion, we scale xso that the width of the thin region becomes of order one (in the new variable ex) ex=x : (140.6) (In other problems, the scaling may be dierent; it may be that ex=x=, whereis an integer or a fraction.) Using the new independent variable as dened by equation (140.6), the equation (140.1) may be written as d2y dex2+dy dex+y=0: (140.7) The solution of this equation is called the \inner" solution. If we search for a regular perturbation series solution to equation (140.7), in the formof equation (140.2), then the sequence of equations begins d 2y0 dex2+dy0 dex=0; d2y1 dex2+dy1 dex=−y0; ...(140.8.a-b) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 592 III Approximate Analytical Methods Using the general solution to equation (140.8.a), we have yinner(ex)=y0(ex)+O() =D+Ee−ex+O();(140.9) whereDandEare constants. Because we have assumed that the boundary layer is at x=0 ,t h e \inner" solution in equation (140.9) must satisfy the boundary condition atx= 0 (i.e.,yinner(0) = 4). The \outer" solution does not extend to x=0 (because the boundary layer is present) but does extend to x= 1. Hence, the solution in equation (140.5) must satisfy the boundary condition at x=1 ;t h a ti s , youter(1) = 5. Evaluating equation (140.5) and equation (140.9) at their respective boundaries results in youter(x)=5e1−x+O(); yinner(ex)=( 4−E)+Ee−ex+O():(140.10) To determine the constant E, we need a \matching principle." The \match- ing principle" is needed to ensure continuity of the solution as it changesfromy inner toyouter. Because the transition occurs for xjust larger than zero, we require lim x!0+yinner(x) = lim x!0+youter(x); which we dene to be ymatch . Writingyinner in terms ofexand assuming thatis arbitrarily small, this statement can be written as lim ex!1yinner(ex) = lim x!0+youter(x): (140.11) Sometimes this is called an intermediate expansion because the matching occurs on an intermediate scale. Using the solutions from equation (140.10) in equation (140.11), we determine that E=4−5e. Finally, we need to combine yinner andyouter together to obtain a uniformly valid approximation, yuniform , over the entire interval: x2[0;1]. The uniform approximation is dened to be the sum of yinner plusyouter, minus the overlap value. That is, yuniform =yinner+youter−ymatch =h 5e+( 4−5e)e−exi + 5e1−x −[5e]+O() =( 4−5e)e−ex+5e1−x+O() =( 4−5e)e−x=+5e1−x+O():(140.12) Figure 140.1 has graphs of the exact solution of equation (140.1) and the approximate solution given by equation (140.12) for =0:1. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 140. Perturbation Method: Boundary Layer Method 593/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. 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/./././. /././././. /./././. /././. /././././. /./././. /././. /././././. /./././. /././. /././././. /././. /./././. /././. /././././. /././. /./././. /././. /././. /././././. /././. /././. /./.Figure 140.1: A comparison of the exact solution of equation (140.1) with the approximation in equation (140.12) for =0:1. Notes 1. The exact solution to equation (140.1) is given by y(x)=1 er2−er1[(4er2−5)er1x+( 5−4er1)er2x]; (140.13) wherer1=/parenleftbig −1−p1−4
=2andr2=/parenleftbig −1+p1−4
=2.F o r small values of ,r1−1=andr2−1. Using these approxima- tions in equation (140.13) and expanding everything to leading order, results in equation (140.12). 2. If the example were carried to second order in , then we would have found youter = 5e1−x + 5(1−x)e1−x +O(2); yinner =h 5e+( 4−5e)e−exi +h 5e 1−e−ex −5eex+( 4−5e)exe−exi +O(2); yuniform =h 5e1−x+( 4−5e)(1 +x)e−x=i +h 5e1−x(1−x)−e1−x=i +O(2): 3. In the example, we could have expected trouble initially. The original equation is of second order and so needs two boundary conditions.But the rst order term in the regular perturbation series, equation (140.3.a), is a dierential equation of rst order, so it would be unlikely to match the two boundary conditions. 4. If it were not possible to match the \inner" and \outer" solutions in equation (140.11), then we would have tried to put a boundary layer CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 594 III Approximate Analytical Methods atx= 1. To do this, we scale xso that it has a large variation near x=1 ,s a ybx=( 1−x)=. Using this new distance scale, the leading order terms in the \outer" and \inner" solutions would have the form of equation (140.5) and equation (140.9). Now, however, the outersolution would extend to x=0( s ot h a t y outer =4e−x), whereas the inner solution would extend to x=1( s ot h a t yinner =( 5−E)+Ee−bx). At this point, we nd that we cannot perform the necessary matching. We have lim x!1−youter(x)=4e−1, but lim x!1−yinner(x) = lim bx!1yinner(bx)=8 >< >:5; ifE=0; 1;ifE> 0; −1;ifE< 0: We conclude that there is no boundary layer near x= 1, at least with the scalingbx=( 1−x)=. 5. Sometimes a boundary layer can appear in the middle of the region of interest. As an example of a \shock" or an \interior transition layer," consider the problem y00+xy0= 0 with the boundary values y(−1) = 1 and y(1) = 2. The solution to this problem is y(x;)= 1 2 3+erf(x=2p) erf(1=2p) . Note the following limits, which indicate the non-uniformity of convergence: lim x!0+lim !0+y(x;)=2; lim x!0−lim !0+y(x;)=1; lim x!0y(x;)=3 2: 6. Kevorkian and Cole [2, pages 20{50 and 370{387] analyze the system: y00−xy0+y=0; y(−1) = 1;y (1) = 2 and show that it has boundary layers at both x=−1a n dx=1 . 7. For certain forms of simple equations, it is possible to predict the ex- istence of boundary layers and other phenomena for generic boundary conditions. Table 140.1 shows the behavior that can be expected fromthe equation y 00−p(x)y0−q(x)y=g(x);axb y(a)=; y (b)=;(140.14) whenis small and positive. For each case, there are simple examples that exhibit the predicted behavior. For example: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 140. Perturbation Method: Boundary Layer Method 595 Conditions on p(x) Type of solution p(x)6=0o naxb: (a)p(x)<0 Boundary layer at x=a (b)p(x)>0 Boundary layer at x=b p(x)=0 : (c)q(x)>0 Boundary layers at x=aandx=b (d)q(x)<0 Rapidly oscillating solution (e)q(x) changes sign Classical turning point p06=q,p(0) = 0 only at x=0 : (f)p0(0)<0 No boundary layers, interior layer at x=0 (g)p0(0)>0 Boundary layers at x=aandx=b, no interior layer at x=0 Table 140.1: Possible behaviors for equation (140.14). Equation Boundary conditions Solution y00+y0=0y(−1) = 0y(1) = 1y(x)=e(1−x)=−e2= 1−e2= y00−y0=0y(−1) = 0y(1) = 1y(x)=e(x−1)=−e−2= 1−e−2= y00−y=0y(−1) = 0y(1) = 1y(x)=ep(x−1)−ep(x+3) 1−e4p y00+y=0y(−1) = 0y(1) = 1y(x)=sin((x+1)p) sin(2p) For non-generic boundary conditions, other solutions are possible. For example, equation (140.1) ts case (a) in table 140.1, which predicts the existence of a boundary layer near x= 0. However, if the boundary conditions for equation (140.1) had been y(0) =y(1) = 4, then the solution would have been y(x) = 4, with no boundary layers present. 8. A classic example showing the dependence of the solution on the boundary conditions is in Kevorkian and Cole [2, Section 2.5]. This nonlinear equation, y00+yy0−y=0; y(0) =A; y (0) =B;(140.15) has the solution behaviors shown in gure 140.2. 9. There are many matching principles that can be used to determine the unknown constants in the \inner" and \outer" solutions. One that is used in Van Dyke [9, page 64] is Then-term expansion of the inner solution (written in the outer variables) to m-terms is equal to the the m-term expansion of the outer solution (written in the inner variables) to n-terms. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 596 III Approximate Analytical Methods/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./. /./././././././././././././././././././, /1A /1 Bsh oc k lay erso ccur in t hi s regionr igh tb o u n d ay lay erso ccur in t hi s region left b ou n d ay lay erso ccur in t hi s regioncor n er an d trans it ion lay erso ccur in t hi s region /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. Figure 140.2: Dierent possible solutions to equation (140.15) for varying boundary conditions. 10. Sometimes there can be multiple boundary layers at a single bound- ary. That is, there are several layers of boundary layers (each with a dierent scaling) before the \outer" solution is matched to the value at the boundary. 11. There exist special numerical procedures that can be used for equa- tions that have boundary layers. See, for instance, Miranker [5, Chapter 5, pages 88{108]. 12. Lo [4] presents a technique for calculating many terms in an asymp- totic expansion. The computer language Macsyma is used to performthe asymptotic matching at each stage. 13. This method is sometimes called the method of matched asymptotic expansions . 14. See also Bender and Orszag [1, Chapter 9, pages 417{483] Nayfeh [6, Chapter 4, pages 110{158], and Van Dyke [9, Chapter 5, pages 77{98]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Kevorkian, J., and Cole, J. D. Perturbation Methods in Applied Mathematics . Springer{Verlag, New York, 1981. [3]Lagerstrom, P. A. Matched Asymptotic Expansions . Springer{Verlag, New York, 1988. [4]Lo, L. L. Asymptotic matching by the symbolic manipulator MACSYMA. J. Comput. Physics 61 (1985), 38{50. [5]Miranker, W. L. Numerical Methods for Sti Equations .D . R e i d e l Publishing Co., Boston, MA, 1981. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 140. Perturbation Method: Boundary Layer Method 597 [6]Nayfeh, A. H. Perturbation Methods . John Wiley & Sons, New York, 1973. [7]Pearson, C. E. On a dierential equation of boundary layer type. J. Math. and Physics 47 (1968), 134{154. [8]Roberts, S. M. Further examples of the boundary value technique in singular perturbation problems. J. Math. Anal. Appl. 133 (1988), 411{436. [9]Van Dyke, M. Perturbation Methods in Fluid Mechanics . The Parabolic Press, Stanford, CA, 1975. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 598 III Approximate Analytical Methods 141. Perturbation Method: Functional Iteration Applicable to Dierential equations with a \small" term and ho- mogeneous initial conditions or boundary conditions. Without the \small" term, the dierential equation must be a linear and have a known Green’s function. Yields A sequence of approximations. Idea If the given equation is only a \small" perturbation from a linear equa- tion (with a known Green’s function), then we may obtain an equivalent integral equation. This integral equation may be expanded methodically. Diagrams are often used to keep track of the terms. Procedure We will illustrate the general technique on a specic class of partial dierential equations. Suppose we have the dierential equation @ @t=H(t;x;@x)+V(x;@x)+A(x); (0;x)=0; (t;0) =(t;1) = 0;(141.1) for the unknown (t;x), whereHandVare functionals. Let us presume that, in some sense, jjVjjjjHjj. If the solution G(t;x;y)o f @G @t=H(t;x;@x)G+(x−y); G(0;x;y)=0;G (t;0;y)=G(t;1;y)=0;(141.2) is known, then the solution to equation (141.1) can be written as the equivalent integral equation (t;x)=Z1 0G(t;x;y)[A(x)+V(x;@x)(t;x)]x=ydy =0(t;x)+Z1 0G(t;x;y)V(y;@y)(t;y);d y;(141.3.a-b) where0(t;x): =R1 0G(t;x;y)A(y)dy. This is because G(t;x;y) is a Green’s function (see page 318) and superposition can be used (note that the bound- ary conditions in equation (141.1) and equation (141.2) are homogeneous). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 141. Perturbation Method: Functional Iteration 599 If(t;y), as determined by the right-hand side of equation (141.3.b), is utilized in the integral in equation (141.3.b), then we obtain (t;x)=0(t;x)+Z1 0G(t;x;y)0(t;y)dy +Z1 0dyZ1 0du[G(t;x;y)V(y;@y)] [G(t;y;u)V(u;@u)](t;u):(141.4) If(t;u), as determined by the right-hand side of equation (141.3.b), is utilized in the double integral in equation (141.4) and the process repeated, then we nd (t;x)=0(t;x)+Z1 0G(t;x;y)0(t;y)dy +Z1 0dyZ1 0du[G(t;x;y)V(y;@y)] [G(t;y;u)V(u;@u)]0(t;u) +Z1 0dyZ1 0duZ1 0dv[G(t;x;y)V(y;@y)] [G(t;y;u)V(u;@u)] [G(t;u;v)V(v;@v)]0(t;v)+


: (141.5) Hence, we have produced a \natural" expansion of the solution to equation (141.1). Because writing the integrals in equation (141.5) becomes tedious, diagrams are often utilized. In a fairly obvious notation, we may write equation (141.5) as (t;x)=0(t;x)+F1+F2+F3+


; (141.6) where each Fiis represented by a diagram in gure 141.1. The diagrams used in this method are never anything more than a shorthand nota- tion for mathematical expressions. For each specic problem in which diagrammatic techniques are used, the diagrams must be appropriatelydened. In this example, a node on a diagram corresponds to the operation [G(t;;−)V(−;@ −)], and each line indicates an integral. Example 1 We now show how functional iteration method can be used to approxi- mate the solution of an ordinary dierential equation, with a small param-eter present. Given the dierential equation with boundary conditions for (x) d 2 dx2=[1−]; (0) =(1) = 0;(141.7) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 600 III Approximate Analytical Methods/#1E /= /#1E/0 /+/#0F/#0F yx /./. /././. /././. /././. /././. /././. /. /././. /././. /././. /././. /././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./+/#0F/#0F /#0F uyx /./. /././. /././. /././. /././. /././. /. /././. /././. /././. /././. /././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /././. /././. /././. /././. /././. /. /././. /././. /././. /././. /././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /+/#0F/#0F /#0F /#0F vuyx /./. /././. /././. /././. /././. /././. /. /././. /././. /././. /././. /././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /././. /././. /././. /././. /././. /././. /. /././. /././. /././. /././. /././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /././. /././. /././. /././. /././. /././. /. /././. /././. /././. /././. /././. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /+ /#01/#01/#01 Figure 141.1: Diagrammatic representation of the solution in equation (141.6). we rst note that the exact solution is given by (x)=1−cospx+cosp−1 sinpsinpx; =x2−x 2 −2x4−2x3+x 24 +O(3):(141.8.a-b) The Green’s function that we need, G(x;y), will satisfy the equation d2G dx2=(x−y); G(0) =G(1) = 0; and is given by (see the example for the Green’s function method, on page 321) G(x;y)=( x(y−1) for 0xy; y(x−1) fory<x1: The dierential equation (141.7) can then be written as an integral equa- tion, using this Green’s function, as (x)=Z1 0G(x;y)[ 1−(y)]dy =0(x)−Z1 0G(x;y)(y)dy;(141.9.a-b) where 0(x): =Z1 0G(x;y)dy =Z1 xx(y−1)dy+Zx 0y(x−1)dy =x2−x 2: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 141. Perturbation Method: Functional Iteration 601 If the value of (x) (as dened by the right-hand side of equation (141.9.b)) is inserted for the function (y) in equation (141.9.b), the natural expansion arises (x)=0(x)−2Z1 0G(x;y)0(y)dy +3Z1 0G(x;y)Z1 0G(y;z)0(z)dzdy−O(4);(141.10) which can be represented by (x)=0(x)+F1+F2+F3+


; where thefFigare given in gure 141.1. In this example, a node on a diagram corresponds to multiplying by G(;) (for some specic and ) and each line segment indicates an integration. It is easy to evaluate the rst few diagrams, that is, to evaluate the rst few terms in equation (141.10). The approximation obtained from equation (141.10) is identical to the expansion in equation (141.8.b). Example 2 The Green’s function is needed so that the solution of the original dierential equation may be written in terms of an integral (as in equation (141.3.b) or equation (141.9.b)). For a rst order equation, though, anintegral representation can be found immediately. In this example, we analyze a nonlinear rst order dierential equation to indicate more fully how the diagrams may be used. Consider the nonlinear ordinary dierential equation dz dt=f(t)+g(t)z2; z(0) = 0 in which the nonlinear term (i.e., the g(t) function) is \small." This equation may be integrated directly to obtain z(t)=Zt 0f()d+Zt 0g()z2()d: (141.11) If the value of z(t) from the left-hand side of equation (141.11) is used in the right-hand side, then z(t)=Zt 0f()d+Zt 0g()Z 0f(1)d12 d +2Zt 0g()Z 0f(1)d1Z 0g(2)z2(2)d2 d +Zt 0g()Z 0g(2)z2(2)d22 d:(141.12) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 602 III Approximate Analytical Methods/#0F/, /! /8/#3C/: /#02/#0F/#0F /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /, /! H /#28 t /, /#1C /#29/#02 /, /! Rf /#28 /#1C /#29 d/#1C/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /, /! Rg /#28 /#1C /#29 d/#1C Figure 141.2: Rules for creating and interpreting diagrams. A \natural" perturbation expansion would be to keep the rst two terms in the right-hand side of equation (141.12) and to assume that the last two terms are \small." If jz(t)j1, then this may well be the case because the last two terms involve jzj2whereas the rst two terms involve jzj. The functional iteration technique can be used to derive equation (141.12) and the higher order extensions from diagrams. We need two sets of rules: One set of rules describes how the diagrams may be computed; the otherset of rules describes how the diagrams are to be turned into mathematical expressions. If we use the rules in gure 141.2 (where H(
) denotes the Heaviside function), then the rst two steps in the diagrammatic solutiontoz(t) (from equation (141.11)) are given by the diagrams in gure 141.3. Note that the third and fourth diagrams in gure 141.3 represent the same mathematical expression because they are topologically equivalent.The purpose of the Heaviside function is to restrict the range of integration. By careful inspection, the mathematical expressions associated with the last set of diagrams will be seen to be identical to equation (141.12). Notes 1. In the physics literature, the Green’s function is sometimes called thepropagator . This is usually written in terms of a path integral , G=R eiS=h,w h e r eSis the action, dened to be the integral of the Lagrangian. The diagrams produced in this context are sometimes called Feynman diagrams . 2. When nonlinear equations are approximated by this technique, as in example 2, keeping track of the terms in the expansion that are of the some order is greatly facilitated by some shorthand notation. The diagrams presented above perform such a task. 3. In more complicated problems, the diagrams will have several dier- ent types of line segments and several dierent types of nodes. 4. Often an \algebra of diagrams" is created, so that diagrams can be added, subtracted and multiplied without recourse to the mathemat-ical expression that each diagram represents. This would require amplication of the rules that were used in example 2. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 141. Perturbation Method: Functional Iteration 603z /#28 t /#29 /=/#0F t/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /#28/1/#29/=/#02 t/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /+ /#0F/#0F t /#1C/1/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /#28/2/#29/=/#02 t/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /+ /#02/#02 t /#1C/1/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /+/#02 /#0F/#0Ft /#1C/1 /#1C/2/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./#28/3/#29/+ /#02/#0F /#0F t /#1C/1/#1C/2 /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /+ /#0F/#0F /#0F/#0F t /#1C/1/#1C/2 /#1C/2/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./#28/4/#29/#28/5/#29 Figure 141.3: Two steps in the diagrammatic expansion of equation (141.11). 5. Presented in this section has been just one type of functional iteration; there are many others. For example, Picard iteration (see page 618) is a functional iteration method. Another method is a decomposition method frequently used by Adomian [2]. 6. This technique is particularly important in problems in which there is no \small" parameter. In these cases, the formally correct dia- grammatic expansion may be algebraically approximated by exactly summing certain classes of diagrams. See Mattuck [6] for details. References [1]Abrikosov, A. A., Gorkov, L. P., and Dzyaloshinski, I. E. Methods of Quantum Field Theory in Statistical Physics . Dover Publications, Inc., New York, 1963. [2]Adomian, G. Stochastic Systems . Academic Press, New York, 1983. [3]Drouffe, J.-M., and Saclay, C. Computer algebra as a research tool in physics. In EUROCAL ’85 , B. Buchberger and B. F. Caviness, Eds. Springer{ Verlag, New York, 1985, pp. 58{67. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 604 III Approximate Analytical Methods [4]Fishman, L., and McCoy, J. J. Factorization and path integration of the Helmholtz equation: Numerical algorithms. J. Acoust. Soc. Am. 81 ,5( M a y 1987), 1355{1376. [5]Houard, J. C., and Irac-Astaud, M. A new approach to perturbation theory: Star diagrams. J. Math. Physics 24 , 8 (Aug 1983), 1997{2005. [6]Mattuck, R. D. A Guide to Feynman Diagrams in the Many-Body Problem . Academic Press, New York, 1976. [7]Pascual, P., and Tarrach, R. QCD: Renormalization for the Practitioner . Springer{Verlag, New York, 1984. [8]Schulman, L. S. Techniques and Applications of Path Integration .J o h n Wiley & Sons, New York, 1981. [9]Srinivasan, S. K., and Vasudevan, R. Introduction to Random Dierential Equations and Their Applications . American Elsevier Publishing Company, New York, 1971. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 142. Perturbation Method: Multiple Scales 605 142. Perturbation Method: Multiple Scales Applicable to Nonlinear dierential equations that have a small parameter present. Yields An approximation to the solution. Idea This is a singular perturbation technique, applicable to problems for which regular perturbation techniques fail. The assumption in this tech- nique is that the solution depends on more than one \length" (or \time") scale. Procedure We presume that the solution depends on two (or more) dierent length (or time) scales. By trying dierent possibilities, we determine what theselength scales are. These dierent length scales are treated as dependent variables when transforming the given ordinary dierential equation into a partial dierential equation, but then the length scales are treated as independent variables when solving the equations. The dependent variable is then expanded in a regular perturbation series (see page 610), where each functions in the series depends on all of the dierent length scales. The dierent orders of are collected, and the sequential set of partial dierential equations is solved. As these equation are solved, the requirement is that each successive term must vanish no slower (as tends to zero) than the previous term. Example Suppose we have the ordinary dierential equation y00+y0=2; y(0) = 0;y(1) = 1;(142.1) fory(x;). We immediately recognize that equation (142.1) is likely to be a singular perturbation problem. This is because, when we set equal to zero, the equation becomes a rst order dierential equation, and it is very unlikely that the solution of this equation (which depends on a singleconstant) will match both boundary conditions. We rst need to determine what the proper length scales are for this problem. We guess that, for this problem, the proper length scales areu:=xandv:=x=. If we had guessed incorrectly, then we would not be able to carry out all of the calculations. First, equation (142.1) must be CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 606 III Approximate Analytical Methods written in terms of these new variables. Writingd dxas d dx=du dx@ @u+dv dx@ @v =@ @u+1 @ @v ; the equation (142.1) becomes @ @u+1 @ @v2 y+@ @u+1 @ @v y=2: (142.2) We now propose the expansion of y(x;) as a regular perturbation series in the dependent variables uandv y(x;)=y0(u;v)+y1(u;v)+2y2(u;v)+


: (142.3) Using equation (142.3) in equation (142.2) and equating the dierent pow- ers ofresults in an innite sequence of equations, of which the rst three are O(−1):@2y0 @v2+@y0 @v=0; O(0):@2y1 @v2+@y1 @v=2−2@2y0 @u@v−@y0 @u; O(1):@2y2 @v2+@y2 @v=−2@2y1 @u@v−@y1 @u−@2y0 @u2:(142.4.a-c) The rst partial dierential equation can be solved to determine y0(u;v)=A(u)+B(u)e−v; (142.5) whereA(u)a n dB(u) are arbitrary functions of u. The second equation then becomes @2y1 @v2+@y1 @v=2−A0(u)+B0(u)e−v; (142.6) which has the solution y1(u;v)=[ 2−A0(u)]v+vB0(u)e−v+D(u)+E(u)e−v; (142.7) whereD(u)a n dE(u) are arbitrary functions. Now, we use our solvability condition, which states that the higher order terms will vanish no slowerthan the lower order terms. For y 1(u;v) (as given in equation (142.7)) to vanish no slower than y0(u;v) (as given in equation (142.5)), we require that 2−A0(u)=0a n dB0(u) = 0. Otherwise, for x6=0a n d1, the terms iny1would be larger than the terms in y0(because, in this case, v1). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 142. Perturbation Method: Multiple Scales 607/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. 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Figure 142.1: A comparison of the exact solution to equation (142.1) (given by equation (142.10)) and the approximate solution in equation (142.9), when=0:5. Using these two constraints, we determine that A(u)=2u+A0and B(u)=B0,w h e r eA0andB0are constants. Hence, the rst order solution becomes y0(u;v)=( 2u+A0)+B0e−v: (142.8) Going back to the original variable (i.e., x), the leading term in the solution fory(x;) is (from equation (142.3) and equation (142.8)) y(x;)y0(x)=( 2x+A0)+B0e−x=: This expression can be matched to both of the boundary conditions in equation (142.1) to determine that y(x;)2x− 1−e−x= : (142.9) The exact solution to equation (142.1) is given by y(x;)=2x−1−e−x= 1−e−1=: (142.10) Hence, we see that the approximate analysis has correctly obtained the rst term in the expansion as tends to zero. Figure 142.1 has a comparison of equations (142.9) and (142.10) when =0:5. Notes 1. It was not really necessary to solve equation (142.6) for y1to obtain the constraints on A(u)a n dB(u). By analysis of the equation fory1, with an eye toward obtaining solutions that do not grow withv, the same conditions could have been obtained. This is an important procedure in more complicated problems for which explicitsolutions are not easy to nd. See the section on alternative theorems, beginning on page 15. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 608 III Approximate Analytical Methods 2. Any problem that can be solved by matched asymptotic expansions can also be solved by multiple scales, although the procedure may require more work. 3. Rubenfeld [8] gives an account of why the method of multiple scales sometimes gives incorrect results. 4. Fateman [3] describes a Macsyma program that will automatically utilize the method of multiple scales to approximate the solution of dierential equations. 5. The method of multiple scales is often called two timing . 6. A Macsyma package to perform these computations is described in Len [5]. 7. The choice of length scales depends on the particular problem. For some problems, three (or more) length scales may be appropriate. Each length scale may have a complicated dependence on the param- eter. 8. The method of multiple scales does not result in an answer that is valid over an indenitely long range. If, for instance, the twoscales arexandx, then the solution is valid, generally, for x= O( −1). The solution of _ x=f(t;x; ) (whenfhas period 2 int) can be approximated by the use of multiple scales. The kth order approximation using the two time scales tand=tare valid with errorO(k) for intervals of length O(1=). It is often believed that adding a third scale =2twill result in solution valid for O(1=2); this is incorrect (see Murdock and Wang [6]). 9. See also Bender and Orszag [1, Chapter 11, pages 544{568], Kevorkian and Cole [4, pages 115{151], and Nayfeh [7, Chapter 6, pages 228{ 307]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Brackbill, J. U., and Cohen, B. I. Multiple Time Scales . Academic Press, New York, 1985. [3]Fateman, R. J. An approach to automatic asymptotic expansions. In Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Compu- tation (SYMSAC ’76) , R. D. Jenks, Ed. ACM, New York, 1976, pp. 365{371. [4]Kevorkian, J., and Cole, J. D. Perturbation Methods in Applied Mathematics . Springer{Verlag, New York, 1981. [5]Len, J. L. Analysis of ODEs using the multiple scales perturbation method. Macsyma Newsletter (April 1989), 4{10. [6]Murdock, J., and Wang, L.-C. Validity of the multiple scale method for very long intervals. Z angew Math Phys 47 (1996), 760{789. [7]Nayfeh, A. H. Perturbation Methods . John Wiley & Sons, New York, 1973. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 142. Perturbation Method: Multiple Scales 609 [8]Rubenfeld, L. A. On a derivative-expansion technique and some comments on multiple scaling in the asymptotic approximation of solutions of certain dierential equations. SIAM Review 20 , 1 (January 1978), 79{105. [9]Sanders, J. A., and Verhulst, F. Averaging Methods in Nonlinear Dynamical Systems . Springer{Verlag, New York, 1985. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 610 III Approximate Analytical Methods 143. Perturbation Method: Regular Perturbation Applicable to Dierential equations with a small parameter. Yields A series of terms of decreasing magnitude that approximate the solution of the original dierential equation. Idea When an equation is changed by only a small amount, the solution will often only change by a small amount. Procedure Expand the dependent variables in a power series depending on the small parameter in the problem. Substitute this series into the originalequation(s), the boundary condition(s), and the initial condition(s). Ex- pand everything in a Taylor series, equate the terms corresponding to dif- ferent powers of the small parameter, and solve the equations sequentially. Example Suppose we have the equation y00+y0+y=0; y(0) = 1;y0(0) = 0;(143.1) whereis a number whose magnitude is much smaller than 1. We suppose that the solution to equation (143.1), y(x;) ,c a nb ee x p a n d e di nap o w e r series inas follows y(x;)=y0(x)+y1(x)+2y2(x)+


: (143.2) Then, using equation (143.2) in equation (143.1), we obtain (y00 0+y00 1+


)+(y0 0+y0 1+


)+(y0+y1+


)=0; y0(0) +y1(0) +2y2(0) +


=1; y0 0(0) +y0 1(0) +2y0 2(0) +


=0:(143.3) Equating powers of in equation (143.3) to zero produces the sequence of equations O(0): y00 0+y0=0; y0(0) = 1; y0 0(0) = 0;(143.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 143. Perturbation Method: Regular Perturbation 611/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. 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/./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././././. /././././. /././. /./././. /././. /./. /././. /. /././. /./. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. 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Figure 143.1: Comparison of the exact solution and the two term approximation to equation (143.1), when =0:25. and O(1): y00 1+y1=−y0 0; y1(0) = 1; y0 1(0) = 0:(143.5) The solution to equation (143.4) is y0(x)=c o sx: (143.6) Using equation (143.6) in equation (143.5), we must now solve the equation y00 1+y1=s i nx; y1(0) = 1; y0 1(0) = 0:(143.7) The solution to equation (143.7) is y1(x)=1 2(sinx−xcosx): (143.8) Therefore, the solution for y(x;) is approximately (using equations (143.6) and (143.8) in equation (143.2)) y(x;)=c o sx+ 2(sinx−xcosx)+O(2): (143.9) We could continue this process indenitely and calculate as many terms as were needed to obtain a desired accuracy. Figure 143.1 is a comparisonof the rst two terms of equation (143.9), when =0:25, with the exact solution. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 612 III Approximate Analytical Methods Notes 1. The exact solution to equation (143.1) is given by y(x;)=p 4−2e−x=2sin xr 1−2 4! +e−x=2cos xr 1−2 4! ; which can be expanded for small to yield y(x;)=c o sx+ 2(sinx−xcosx)+O(2): 2. This method will notwork on all equations that have a small param- eter. As a simple example, consider y00+y=0;y (0) = 1;y(1) = 2: (143.10) In this example, the rst order equation (corresponding to equation (143.4)) is y0=0;y(0) = 1;y(1) = 2: Clearly, this equation has no solution. Hence, the expansion in (143.3), must not be adequate to represent the solution of equation(143.10). 3. In deriving equations (143.4) and equation (143.5) from equation (143.3), it was implicitly assumed that each of jy 1(x)j,jy0 1(x)j,a n d jy00 1(x)jareO(1). Observe that this will notbe the case when x= O(1=) (see equation (143.8)). Hence, we conclude that only when x1=can equation (143.9) be a good approximation to the solution of equation (143.1). If an approximation to the solution is desired over a larger range of xvalues, then the method of multiple scales might be used (see page 605). Secular terms is the name given to terms that become large and prevent a perturbation expansion from being valid. 4. If the solution to a dierential equation is not analytic at =0 ,t h e n the solution can notbe expanded in the form of equation (143.2). Often, the best procedure is to utilize an expansion of the form y(x;)=y0(x)+1()y1(x)+2()y2(x)+:::; and then determine the scaling functions figas thefyigare deter- mined. It is frequently the case that the figare given by terms of the formfnlogmg. Terms with m6= 0 are sometimes called switchback terms (see Lagerstrom and Reinelt [4] or Van Dyke [7, pages 9{20 and 200{202]). 5. The functional iteration method (see page 598) produces the same terms that would be obtained by a regular perturbation expansion.The benet of the diagrammatic method is that it allows easier manipulation of the terms. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 143. Perturbation Method: Regular Perturbation 613 6. See Bender and Orszag [1, pages 319{335], Farlow [2, Lesson 46, pages 370{378], Kevorkian and Cole [3, pages 17{20], and Lin and Segel [5, pages 45{55 and 225{241]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [3]Kevorkian, J., and Cole, J. D. Perturbation Methods in Applied Mathematics . Springer{Verlag, New York, 1981. [4]Lagerstrom, P. A., and Reinelt, D. A. Note on logarithmic switchback terms in regular and singular perturbation expansion. SIAM J. Appl. Math. 44, 3 (June 1984), 451{462. [5]Lin, C. C., and Segel, L. A. Mathematics Applied to Deterministic Problems in the Natural Sciences . The MacMillan Company, New York, 1974. [6]Nayfeh, A. H. Perturbation Methods . John Wiley & Sons, New York, 1973. [7]Van Dyke, M. Perturbation Methods in Fluid Mechanics . The Parabolic Press, Stanford, CA, 1975. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 614 III Approximate Analytical Methods 144. Perturbation Method: Strained Coordinates Applicable to Dierential equations that have a small parameter present. Yields An approximation to the solution, valid on a long time scale. Idea A regular perturbation expansion may give the correct answer but at the wrong location. By scaling the dependent variable and one or more of the independent variables by the small parameter, the solution may beapproximated at the correct location. Procedure If the regular perturbation solution to a dierential equation has secular terms but the original equation has bounded solutions, then the regular perturbation approximation is not valid for large values of the independent variables. One way to obtain a solution that is valid for longer scales is by \straining the coordinates"; that is, expanding the dependent variable and one or more of the independent variables in terms of the small parameter. To completely specify the arbitrary functions and constants that arise, use the maxim: \Higher order approximation shall be no more singular than the rst." Example Suppose we wish to approximate the solution to the nonlinear dieren- tial equation d2y dt2+!2y=y3; y(0) = 1;y0(0) = 0:(144.1) This equation can be integrated once by rst multiplying by y0.T h e r e - sulting rst order dierential equation can be integrated in terms of elliptic functions. The explicit solution indicates that the solution is periodic. If a regular perturbation technique is attempted, then the resulting equations can be solved in the usual manner (see page 610) to determine that y(t;)=c o s!t+3 8t !sin!t−1 32!2 cos 3!t−cos!t +O(2): Note that the second term in this solution becomes unbounded as tin- creases. Hence, secular terms are present. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 144. Perturbation Method: Strained Coordinates 615 In the method of straining, both the dependent variable and the inde- pendent variable are expanded in terms of . For this example, we presume the expansion has the form t=t(;)=+t1()+O(2); y=y(;)=y0()+y1()+O(2):(144.2.a-b) Noting that the derivative with respect to tcan be replaced with a deriva- tive with respect to by d dt=( 1−t0 1+


)d d; (where a prime (0) denotes dierentiation with respect to ), we nd that equation (144.1) can be turned into a sequence of equations, with each equation involving the next yk() term. The rst two equations are d2y0 d2+!2y0=0; d2y1 d2+!2y1=y3 0+2t0 1d2y0 d2+t00 1dt0 d:(144.3) The boundary conditions are similarly expanded. We nd y0(0) = 1;dy0 d(0) = 0; y1(0) +t1(0)dy0 d(0) = 0; dy1 d(0)−t0 1(0)dy0 d(0) +t1(0) +dy0 d(0) = 0:(144.4) Now we proceed to solve the equations sequentially, just as in the regular perturbation method. The rst equation in (144.3) with the rst pair of boundary conditions in equation (144.4) yields y0()=c o s!: (144.5) Using this value for y0(), the next equation (144.3) (which is for y1()) becomes d2y1 d2+!2y1=1 4cos 3!+3 4−2!2t0 1 cos!−!t00 1sin!: (144.6) To prevent y1() from having any secular terms, this equation cannot be forced at resonance. This means that the right-hand side of equation (144.6) cannot have any terms that are solutions of the homogeneous CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 616 III Approximate Analytical Methods equation. To keep the right-hand side of equation (144.6) from having any cos!or sin!terms, we choose 3 4−2!2t0 1 =0 o rt1=3 8!2: (144.7) If we now solve equation (144.6), there will be no secular terms. Utilizing equation (144.7) in equation (144.2.a) results in t=+3 8!2+


; or = 1−3 8!2 t+


: Using this last expression for in equation (144.5) results in our nal form of the rst order approximation y0(t)=c o s !−3 8! t : Notes 1. Another common application of this method is to dierential equa- tions whose solutions are well behaved, but approximations by a regular perturbation scheme produce singular terms. For example, the dierential equation (x+u)du dx+u=0;u (1) = 1; (144.8) has a solution that is well behaved at x= 0, but the regular pertur- bation series u(x;)=u0(x)+u1(x)+:::yieldsu0=x−1,u1= 1 2/parenleftbig x−1−x−3
, and higher order terms that are even more singular at x= 0. Applying strained coordinate techniques results in the exact solution of equation (144.8): u(x)=/parenleftbig −x+p x2+2+2
=. This solution shows that u(x) cannot be expanded in a power series in  nearx=0 . 2. The paper by Roberts and Shipman [7] concerns itself with equations of the form [f(x)+y]dy dx+q(x)y=r(x) on the interval 0 <x< 1, withy(1) =c, when the method of straining does notwork. 3. This technique is a useful tool in many areas, including the theory of boundary layers and the structure and propagation of shock waves. 4. This technique is also known as the Lighthill method, the Lindstedt method, and the Poincar e{Lighthill method. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 144. Perturbation Method: Strained Coordinates 617 5. The computer language Macsyma [5] has a package ( lindst )f o r automatically implementing this technique, see Len [3] for details. 6. See also Goldstein and Braun [2, pages 306{311], Nayfeh [6, Chap- ter 3, pages 56{109], and Van Dyke [8, Chapter 6, pages 99{120]. References [1]Comstock, C. The Poincare{Lighthill perturbation technique and its generalizations. SIAM Review 14 , 3 (July 1972), 433{446. [2]Goldstein, M. E., and Braun, W. H. Advanced Methods for the Solution of Dierential Equations . U.S. Government Printing Oce, Washington, D.C., 1973. NASA SP-316. [3]Len, J. L. Perturbation solution of ODEs in MACSYMA: Lindstedt’s method. MACSYMA Newsletter 5 , 2 (April 8), 6{9. [4]Lighthill, M. J. A technique for rendering approximate solutions to physical problems uniformly valid. Z. Flugwiss 9 (1961), 267{275. [5]Macsyma .Mathematics Reference Manual . Macsyma, Arlington, MA, 1993. [6]Nayfeh, A. H. Perturbation Methods . John Wiley & Sons, New York, 1973. [7]Roberts, S. M., and Shipman, J. S. An iteration perturbation technique. J. Comput. Physics 16 (1974), 285{297. [8]Van Dyke, M. Perturbation Methods in Fluid Mechanics . The Parabolic Press, Stanford, CA, 1975. [9]Whitham, G. The flow pattern of a supersonic projectile. Comm. Pure Appl. Math 5 (1952), 301{348. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 618 III Approximate Analytical Methods 145. Picard Iteration Applicable to Dierential equations, a single equation, or a sys- tem. Yields A sequence of approximations to the solution. Idea We can write an ordinary dierential equation as a xed point formula. If we have a starting guess, we can iterate the equation to nd an approx-imate solution to the original equation. Procedure Suppose we have the rst order ordinary dierential equation dy dx=f(y;x); with the initial condition y(x0)=y0. This equation can be written as the integral equation y(x)=y0+Zx x0f(y(z);z)dz: (145.1) Note that equation (145.1) already incorporates the initial conditions. If we had a guess of y(x), sayy1(x), then we might be able to improve our guess by forming y2(x) as follows y2(x)=y0+Zx x0f(y1(z);z)dz: Then, knowing y2(x), we could form y3(x) by the same technique. We can continue this process indenitely, each time using the formula yn+1(x)=y0+Zx x0f(yn(z);z)dz: (145.2) What we take for y1(x) is arbitrary; it is often easiest to take y1(x)=y0. Example Suppose we have the following ordinary dierential equation dy dx=x2+y2; withy(0) = 1. In this case, the iteration formula, equation (145.2), becomes yn+1(x)=1+Zx 0[z2+yn(z)2]dz: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 145. Picard Iteration 619 If we takey1(x) = 1, then we nd y2(x)=1+x+1 3x3; y3(x)=1+x+x2+2 3x3+


; y4(x)=1+x+x2+4 3x3+5 6x4+


; y5(x)=1+x+x2+4 3x3+7 6x4+16 15x8+


:(145.3) The Taylor series solution of this problem (see page 632) begins y(x)=1+x+x2+4 3x3+7 6x4+6 5x5+


: Hence, each successive approximation in equation (145.3) appears to have one more correct term. Notes 1. The successive approximations found by this method are not guaran- teed to converge. 2. This method can also be used on systems of rst order ordinary dierential equations. For example, the scheme corresponding to thesystem dy dt=f(y;z;t );y (0) =y0; dz dt=g(y;z;t );z (0) =z0; is yn+1(t)=y0+Zt 0f(yn(t);zn(t);t)dt; zn+1(t)=z0+Zt 0g(yn(t);zn(t);t)dt: 3. Picard iteration can be applied to ordinary dierential equations of nth order without rst writing the equation as a rst order system. For example, the second order ordinary dierential equation y00=f(t;y(t);y0(t)); y(a)=A; y (b)=B; has the convenient iteration scheme yn+1(x)=A+(x−a)y0 n(a)+Zx a(x−t)f(t;yn(t);y0 n(t))dt wherey0(x)=A+(x−a)(B−A)=(b−a). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 620 III Approximate Analytical Methods 4. It is also possible to approximate partial dierential equations by this technique. For example, the elliptic equation r2u=f x;y;u;@u @x;@u @y has the natural iteration formula r2un=f x;y;un−1;@un−1 @x;@un−1 @y . Iyanaga and Kawada [2, page 998] have technical conditions for when this scheme will converge to the solution of the original equation. Rice and Boisvert [5, pages 79{82] illustrate this technique with the use of ELLPACK. 5. See also Boyce and DiPrima [1, pages 97{103] and Simmons [6, pages 418{422]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Iyanaga, S., and Kawada, Y. Encyclopedic Dictionary of Mathematics . MIT Press, Cambridge, MA, 1980. [3]Lal, M., and Moffatt, D. Picard’s successive approximation for non-linear two-point boundary value problems. J. Comput. Appl. Math. 8 , 4 (1982), 233{236. [4]Ozis, T. The extension of Picards’s successive approximation for constructing two-side bounds for the solutions of diferential equations. J. Comput. Appl. Math. 39 (1992), 7{14. [5]Rice, J. R., and Boisvert, R. F. Solving Elliptic Problems Using ELLPACK . Springer{Verlag, New York, 1985. [6]Simmons, G. F. Dierential Equations with Applications and Historical Notes . McGraw{Hill Book Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 146. Reversion Method 621 146. Reversion Method Applicable to Forced nonlinear ordinary dierential equations. Yields A local approximation. Idea To derive the method, we assume a certain parameter is small and develop a perturbation expansion in that parameter. In practice, we usethe formulae obtained by this method when the parameter is equal to 1. Procedure Suppose that the general nonlinear dierential equation whose solution we wish to approximate near the initial value is given by D1y+D2y2+


+D5y5+


=k(x); (146.1) where thefDigrepresent dierential operators. We seek y=y(x), where kis a constant and (x) is a known forcing function. For this method to work, we require that D16=0 . We assume that y(x) is analytic and kis suciently small so that the solution to equation (146.1) can be expanded in a power series in k.T h a t is, we take y(x)=a1(x)k+a2(x)k2+a3(x)k3+


: (146.2) Using equation (146.2) in equation (146.1) and equating powers of kresults in an innite sequence of equations for the fai(x)g. This sequence of equations begins D1a1=(x); D1a2=−D2a2 1; D1a3=−[2D2a1a2+D3a3 1]; D1a4=−[D2(a2 2+2a1a3)+3D3a2 1a2+D4a4 1]:(146.3.a-d) The reversion method is to assume the solution to equation (146.1) can be represented in the form of equation (146.2) when k= 1 and the coecients are given by equation (146.3). Example Suppose we have the following nonlinear ordinary dierential equation dv dx+v2=x; v (0) =v0; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 622 III Approximate Analytical Methods and we seek an approximation near x= 0. Changing variables to y=v−v0 changes the equation into dy dx+2v0y+y2=x−v2 0;y (0) = 0; (146.4) which simplies the initial condition. Comparing equation (146.4) to equa- tion (146.1), we make the identications D1=d dx+2v0; D 2=; k=1; (x)=x−v2 0: From equation (146.3.a), we obtain the following equation for a1:D1a1= (x), or d dx+2v0 a1=x−v2 0: (146.5) Becausev(0) = 0, we will take a1(0) =a2(0) =


= 0. The solution to equation (146.5) with a1(0) = 0 is a1=x 2v0+e−2v0x−1 4v2 02+v2 0e−2v0x−1 2v0; which was obtained by using a Laplace transform (see page 350). The function a2can be determined from equation (146.3.b) d dx+2v0 a2=a2 1=x 2v0+e−2v0x−1 4v2 02+v2 0e−2v0x−1 2v02 ; witha2(0) = 0. This can also be solved by using Laplace transforms. Proceeding in this way, many terms in the series equation (146.2) can beevaluated. Notes 1. The above example is from Pipes and Harvill [1, pages 653{665]. 2. The extension of equation (146.3) can be found in Orstrand [2], which lists formulae for the rst 13 terms. References [1]Pipes, L. A., and Harvill, L. R. Applied Mathematics for Engineers and Physicists . McGraw{Hill Book Company, New York, 1970. [2]V a nO r s t r a n d ,C .E . Philosophical Magazine 19 (1910), 366. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 147. Singular Solutions 623 147. Singular Solutions Applicable to Nonlinear ordinary dierential equations. Yields A singular solution. Idea Singular solutions may exist where the implicit function theorem does not hold in dierential algebraic equations. Procedure The algebraic ordinary dierential equation F(x;y;y0;:::;y(n)) = 0 (147.1) can often be explicitely solved for the y(n)term to determine that y(n)=G1(x;y;:::;y(n−1)) y(n)=G2(x;y;:::;y(n−1)) ... y(n)=Gi(x;y;:::;y(n−1)):(147.2) By the implicit function theorem, if@F @y(n)(x;y;y0;:::;y(n))6= 0, then the solutions in equation (147.2) are the only solutions possible. However, at those points where@F @y(n)(x;y;y0;:::;y(n)) = 0, there exists the possibility of singular solutions. If they(n)term is algebraically eliminated from the two equations F(x;y;y0;:::;y(n))=0; @F @y(n)(x;y;y0;:::;y(n))=0; then an equation of the form H(x;y;y0;:::;y(n−1)) = 0 (147.3) results. This is called the p-discriminant equation . Its solution(s) describe thesingular loci . After equation (147.3) is solved to determine possible singular solutions, it must be veried that they are, in fact, actual solutions to the original equation (147.1). Typically, the solution to equation (147.3), being adierential equation of ( n−1)-st order, will involve only n−1 arbitrary constants. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 624 III Approximate Analytical Methods Example Given the nonlinear rst order ordinary dierential equation F(x;y;y0)=xy02−3yy0+9x2=0; (147.4) it is straightforward to compute @F @y0=2xy0−3y=0: (147.5) Eliminating the y0term between equation (147.4) and equation (147.5) results in y=2x3=2: (147.6) In this case, both of the solutions in equation (147.6) satisfy equation (147.4). Note that the singular solutions in equation (147.6) do not depend on any constants, even though equation (147.4) was a rst order dierential equation. Notes 1. The general nth order ordinary dierential equation, linear in the nth derivative term, U(x;y;y0;:::;y(n−1))y(n)+V(x;y;y0;:::;y(n−1))=0; has the singular solution y=z(x)i fz(x) satises both of U(x;z;z0;:::;z(n−1))=0; V(x;z;z0;:::;z(n−1))=0: 2. Another way to determine singular solutions of the dierential equa- tionf(x;y;y0) = 0 is to obtain the general solution (x;y;C )=0 (whereCis an arbitrary constant) and then formally eliminate C between the two equations (x;y;C )=0; @ @x(x;y;C )=0: The resulting equation, which only involves xandy, is called the c-discriminant equation . For example, the dierential equation y02+4−4y= 0 has the general solutiony(x)=1+(x−C)2; hence(x;y;C )=y−1−(x+C)2. Forming the c-discriminant results in the singular solution y=1 . 3. In general (see Piaggio [7, pages 65{79 and 192{201]), the p-discriminant equation will contain the envelope of the solutions, the cusp-locus and the tac-locus squared. The c-discriminant equation will contain the envelope of the solutions, the cusp-locus cubed and the node-locussquared. Of these, only the envelope is a solution to the original dierential equation. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 147. Singular Solutions 625 4. Some envelope solutions of dierential equations may be found by use of Lie groups; see Bluman [1]. 5. For polynomial functions, the algebraic elimination in the computa- tion of thec-discriminant (or the p-discriminant) can be done by the use of resultants (see page 50). 6. See also El’sgol’ts [2, pages 81{88], Goldstein and Braun [3, pages 18{24], Ince [4, pages 83{91], and Murphy [6, pages 74{80]. References [1]Bluman, G. Invariant solution for ordinary dierential equations. SIAM J. Appl. Math. 50 , 6 (December 1990), 1706{1715. [2]El’sgol’ts, L. E. Dierential Equations and the Calculus of Variations . MIR Publishers, Moscow, USSR, 1970. [3]Goldstein, M. E., and Braun, W. H. Advanced Methods for the Solution of Dierential Equations . U.S. Government Printing Oce, Washington, D.C., 1973. NASA SP-316. [4]I n c e ,E .L . Ordinary Dierential Equations . Dover Publications, Inc., New York, 1964. [5]Kaplan, W. Ordinary Dierential Equations . Addison{Wesley Publishing Co., Reading, MA, 1958. [6]Murphy, G. M. Ordinary Dierential Equations and Their Solution .D .V a n Nostrand Company, Inc., New York, 1960. [7]Piaggio, H. T. H. An Elementary Treatise on Dierential Equations and Their Applications . G. Bell & Sons, Ltd, London, England, 1926. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 626 III Approximate Analytical Methods 148. Soliton-Type Solutions Applicable to Partial dierential equations with wave-like solu- tions, often partial dierential equations with only two independent vari- ables. Yields Knowledge of whether solitons can be present. Idea See if there is a solitary wave solution to the partial dierential equation. This indicates the possibility that the equation has solitons for solutions. Procedure A solitary wave is a localized, traveling wave; many nonlinear partial dierential equations have solutions of this type. A soliton is a solitary wave that exhibits particle-like behavior. The particle-like properties in-clude stability, localizability, and nite energy. A soliton is best described, however, in terms of its interaction with other solitary waves. We say that an equation possesses solitons when two or more colliding solitary wavesdo not break up and disperse but, instead, become more solitary waves. In this technique, we change variables in such a way as to make such a solitary wave more apparent. If the original partial dierential equation were in the independent variables xandt, we search for a solution of the formu(x−ct). Herecrepresents the wave speed; if c>0(c<0), then u(x−ct) represents a wave traveling to the right (left). Note that many partial dierential equations have solitary waves as solutions; most of thesepartial dierential equations do notexhibit soliton behavior. Example One representation of the Korteweg{de Vries (KdV) equation is given by ut+uux+uxxx=0: (148.1) We change the independent variables from fx;tgtof;gvia (see page 168)f=t,=x−ct:g. This change of variable turns equation (148.1) into u−cu+uu+u=0: (148.2) If we now presume that equation (148.1) admits a wave-like solution, we can then take u(;)=v()=v(x−ct). By assuming this functional form foru(;), equation (148.2) becomes cv+vv+v=0: (148.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 148. Soliton-Type Solutions 627 Equation (148.3) is an autonomous ordinary dierential equation. Hence, the order can be reduced by 1 (see page 230). In fact, for the equation (148.3), the exact solution can be obtained. Equation (148.3) can be integrated with respect to to obtain −cv+1 2v2+v=A; whereAis an arbitrary constant. This last equation, when multiplied by v, can be integrated again to obtain −1 2cv2+1 6v3+(v)2=Av+B; (148.4) whereBis another arbitrary constant. Equation (148.4) can be solved algebraically for vand then this rst order ordinary dierential equation can be integrated in terms of elliptic functions (see Abramowitz and Stegun [2]). Hence, we have shown that the KdV equation has solitary waves as solution. For a soliton type solution to exist for equation (148.1), it must be determined that a solution of equation (148.4) exists that is localized(i.e., diers appreciably from zero only in a bounded region). Finally, to actually show that the KdV has solitons, the interaction of these solitary waves must be investigated. From a much deeper analysis (see, for example,Whitham [9, Chapter 17, pages 577{620]) it is possible to show that the Korteweg{de Vries equation possesses solitons as solutions. In fact, the KdV equation can have, as its solutions, any number of solitons. Notes 1. The technique that we have presented is no more than using similarity variables (see page 497) to obtain a solution of a specic form. Of course, the boundary conditions must admit a traveling wave solution, as well as the equations. 2. The wave speed ( cin the Example) often must be determined as part of the solution. In the above example, it would be determined by the boundary conditions (as would AandB). Typically, in nonlinear problems, the velocity is amplitude dependent. 3. See Ablowitz and Segur [1, Chapter 17, pages 587{607]. References [1]Ablowitz, M. J., and Segur, H. Solitons and the Inverse Scattering Transform . SIAM, Philadelphia, PA, 1981. [2]Abramowitz, M., and Stegun, I. A. Handbook of Mathematical Functions . National Bureau of Standards, Washington, D.C., 1964. [3]Calogero, F., and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations . North{Holland Publishing Co., New York, 1982. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 628 III Approximate Analytical Methods [4]Dodd, R. K., Eilbeck, J. C., and Morris, H. C. Solitons and Nonlinear Wave Equations . Academic Press, New York, 1982. [5]Drazin, P. G., and Johnson, R. S. Solitons: An Introduction . Cambridge University Press, New York, 1989. [6]Eckhaus, W., and Harten, A. V. The Inverse Scattering Transformation and the Theory of Solitons . North{Holland Publishing Co., New York, 1981. [7]Lamb, G. L. Elements of Soliton Theory . John Wiley & Sons, New York, 1980. [8]Newell, A. C. Solitons in Mathematics and Physics . SIAM, Philadelphia, PA, 1985. [9]Whitham, G. B. Linear and Nonlinear Waves . Interscience Publishers, Inc., New York, 1974. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 149. Stochastic Limit Theorems 629 149. Stochastic Limit Theorems Applicable to Linear dierential equations that contain a small parameter and a random forcing term of a certain form. Yields A Fokker{Planck equation. Idea Some equations do not have a \white noise" forcing term and so a Fokker{Planck equation cannot be directly constructed (see page 303).However, it is often true that random forcing terms behave like \white noise" in some asymptotic limit. Hence, in this limit, a Fokker{Planck equation can be constructed. Procedure IfF(x;t;) is a \suciently random" mean zero function then, as  tends to zero, the form 1 F x;t;t 2 (149.1) behaves, in a certain sense, like a \white noise" term (see Papanicolaou and Kohler [4]). Using the \white noise" equivalent of equation (149.1), a Fokker{Planck equation can be obtained in the variables fx;tg. Hence, the prescription is to change a given equation so it has a term in the form of equation (149.1) and then obtain and analyze the corresponding Fokker{Planck equation. Example Using the geometric optics approximation to the wave equation, the scaled position and velocity of a ray in a weakly random medium satisfy dx dt=v; dv dt=1 F x;t 2 ; after a ray has traveled a long distance in the random medium. Here F(
) is a random function with mean zero (it represents the wave speed perturbation at any point). Assuming a \mixing condition" on F, which is a statement about how random F(
) is, the theorem in Papanicolaou and Kohler [4] can be used in the limit of going to zero. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 630 III Approximate Analytical Methods Using this theorem, it can be shown that the probability density of the solution to equation (149.1) converges weakly to the solution of the following Fokker{Planck equation γ@2P @v2−@P @x=@P @t; where the number γis dened by γ2=−R1 0E[F(0;y)(F(0;0)]dy,a n d E[
] is the expectation operator. The details of the derivation are beyond the scope of this book. More details may be found in Kulkarny and White [3]. Notes 1. There are many dierent limit theorems that yield a \white noise" limit. For example, Keston and Papanicolaou’s paper [1] is concernedwith random dierential equations of the form dx dt=1 2v; dv dt=1 F(x;v): 2. The theorems in Keston and Papanicolaou [1] and in Papanicolaou and Kohler [4] have many technical requirements that must be satis- ed. The \mixing condition" requirement has been veried for only a few physical process. 3. For some limit theorems, the Fokker{Planck formalism can be elimi- nated completely. For example, in Khas’minskii [2]m it is shown that the solution to the problem dx dt=F(x;t;!; );x (0) =x0; in an interval of order O(1=), can be uniformly approximated by the solution to the problemdx dt=F(x),x(0) =x0,w h e r e F(x) := lim T!11 TZT 0E[F(x;t;!; )]dt; if the stochastic process F(x;t;!; ) satises the law of large numbers for xedx. 4. Pardoux [5] nds a white noise limit of a partial dierential equation. References [1]Keston, H., and Papanicolaou, G. A limit theorem for stochastic acceleration. Comm. Math. Physics 78 (1980), 19{63. [2]Khas’minskii, R. Z. A limit theorem for the solutions of dierential equations with random right-hand sides. Theory Prob. Appl. 11 , 3 (1966), 390{405. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 149. Stochastic Limit Theorems 631 [3]Kulkarny, V. A., and White, B. S. Focussing of waves in turbulent inhomogeneous media. Phys. Fluids 251 , 10 (1982), 1770{1784. [4]Papanicolaou, G., and Kohler, W. Asymptotic theory of mixing stochastic ordinary dierential equations. Comm. Pure Appl. Math 27 (1974), 641{668. [5]Pardoux, E. Asymptotic analysis of a semi-linear PDE with wide-band noise disturbances. In Stochastic Space{Time Models and Limit Theorems , L. Arnold and P. Kotelenz, Eds. D. Reidel Publishing Co., Boston, MA, 1985, pp. 227{242. [6]Van Den Broeck, C. Stochastic limit theorems: Some examples from nonequilibrium physics. In Stochastic Space{Time Models and Limit The- orems , L. Arnold and P. Kotelenz, Eds. D. Reidel Publishing Co., Boston, MA, 1985, pp. 179{189. [7]White, B., and Franklin, J. A limit theorem for stochastic two-point boundary value problems of ordinary dierential equations. Comm. Pure Appl. Math 32 (1979), 253{276. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 632 III Approximate Analytical Methods 150. Taylor Series Solutions Applicable to Initial value problems, both ordinary dierential equations and partial dierential equations. Yields An approximation to the solution near a point. Idea For an initial value problem, a Taylor series expansion can give an approximate solution. Procedure We will illustrate the general procedure on a rst order linear ordinary dierential equation. Suppose we have the dierential equation y0(x)=F(x;y); (150.1) (where0indicates dierentiation with respect to x) with the initial condi- tiony(a)=y0,w h e r eF(x;y) is a known function. Evaluating equation (150.1) at x=a, we can determine y0(a)=F(a;y0). Dierentiating equation (150.1) with respect to x, and using the chain rule, results in y00(x)=Fx(x;y)+Fy(x;y)yx: (150.2) Now equation (150.2) can be evaluated at x=ato explicitly determine y00(a)=Fx(a;y(a)) +Fy(a;y(a))yx(a) =Fx(a;y0)+Fy(a;y0)F(a;y0); w h e r ew eh a v eu s e d y0(a)=F(a;y0). We can continue this process of dierentiating equation (150.1) and evaluating the result to determine the nth derivative of y(x)a tt h ep o i n t x=a. The result will involve only the partial derivatives of F(x;y)a n d the numerical values aandy0. Knowing these values allows us to construct the Taylor series expansion of y(x)a b o u tx=aby use of y(x)=y(a)+y0(a) 1!(x−a)1+y00(a) 2!(x−a)2+y000(a) 3!(x−a)3+


: (150.3) Example Suppose we wish to approximate the solution of the nonlinear initial value problem y0=x2−y2; y(0) = 1:(150.4.a-b) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 150. Taylor Series Solutions 633 From equation (150.4), it is straightforward to compute y00=2x−2yy0; y000=2−2(y0)2−2yy00; y0000=−6y0y00−2yy000; ...(150.5) Using equation (150.4.b), we evaluate equation (150.4.a) and then equation (150.5) sequentially, at x= 0, to determine y0(0) =−1; y00(0) = 2; y000(0) =−4; y0000(0) = 20; ...(150.6) Using the values from equation (150.6) in equation (150.3), with a=0 ,t h e solution of equation (150.4) for y(x)n e a rx=0i sg i v e nb y =1−x+2 2!x2−4 3!x3+20 4!x4+


=1−x+x2−2 3x3+5 6x4+


: Notes 1. This method may be applied to higher order equations and systems of equations. 2. The method of series solution (see page 403), when used at an ordi- nary point, also yields a Taylor series solution. 3. The Taylor series worked out by this method can be used to compute Pade approximates to the solution. These Pad e approximates may give information about singularities of the exact solution (see page582). Fern andez et al. [3] have developed a dierent technique for determining the location of singular points by postulating a form of the singularity. 4. A direct representation of the Taylor series may be obtained by implicit dierentiation. We nd that the solution to the dierentialequationy 0=f(t;y), withy(0) = 0, has the Lie series representation y(t)=1X n=1tn n!@ @t+f(t;z)@ @zn z z=0: (150.7) See Igumnov [6] for a computationally ecient way to determine y(t) from equation (150.7) when f(t;y) has a known Taylor series. Finizio and Ladas [4, pages 293{298] also describe a numerical scheme. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 634 III Approximate Analytical Methods 5. The numerical technique of analytical continuation (see page 698) combines Taylor series at several dierent points to approximate the solution of a dierential equation in a large region. 6. Taylor’s theorem has been generalized in a way in which the general term is a fractional derivative (see Osler [9] for details). 7. Corliss and Chang [2] describe a Fortran program for solving ordinary dierential equations by the use of Taylor series. 8. Macsyma [8] has a package ( taylorode) which computes Taylor series solutions of ordinary dierential equations. References [1]Corliss, G., and Lowery, D. Choosing a stepsize for Taylor series methods for solving ODE’s. J. Comput. Appl. Math. 3 , 4 (1977), 251{256. [2]Corliss, G. F., and Chang, Y. F. Solving ordinary dierential equations using Taylor series. ACM Trans. Math. Software 8 (1982), 114{144. [3]Fernandez, F. M., Arteca, G. A., and Castro, E. A. Singular points from Taylor series. J. Math. Physics 28 , 2 (Feb 1987), 323{329. [4]Finizio, N., and Ladas, G. Ordinary Dierential Equations with Modern Applications . Wadsworth Publishing Company, Belmont, CA, 1982. [5]Hunter, C., and Guerrieri, B. Deducing the properties of singularities of functions from their Taylor series coecients. SIAM J. Appl. Math. 39 , 2 (October 1980), 248{263. [6]Igumnov, V. P. Representation of solutions of dierential equations by modied Lie series. Dierential Equations 20 (1984), 683{688. [7]Kochavi, E., and Segev, R. Numerical solution of eld problems by nonconforming Taylor discretization. Appl. Math. Modeling 15 (March 1991), 152{157. [8]Macsyma .Mathematics Reference Manual . Macsyma, Arlington, MA, 1993. [9]Osler, T. J. Taylor’s series generalized for fractional derivatives and applications. SIAM Review 2 , 1 (February 1971), 37|48. [10]Razzaghi, M., and Razzaghi, M. Solution of linear two-point boundary value problems via Taylor series. J. Franklin Inst. 326 , 4 (1989), 511{521. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 151. Variational Method: Eigenvalue Approximation 635 151. Variational Method: Eigenvalue Approximation Applicable to Dierential equations with eigenvalues to be deter- mined. Yields Estimates for the eigenvalues. Idea If we guess approximate eigenfunctions, then we will obtain approxi- mations to the eigenvalues. The \better" we guess the eigenfunctions, the better the estimates of the eigenvalues will be. Procedure Although the procedure is quite general, we will discuss it in the spe- cic context of a Sturm{Liouville equation. Suppose we have the Sturm{ Liouville equation on the interval [ a;b] L[y]=d dx p(x)dy dx −s(x)y=−r(x)y; (151.1) withp(x)>0,s(x)0, andy(a)=y(b) = 0. If we expand y(x)a s y(x)=1X n=1cnn(x); (151.2) where thefn(x)gare an arbitrary set of complete functions that vanish atx=aandx=b,a n dt h efcngare constants, then the fcngmust satisfy 1X n=1(Amn−Rmn)cn=0; (151.3) form=1;2;:::,w h e r e Amn=Zb a[p(x)0 m(x)0 n(x)+s(x)m(x)n(x)]dx; Rmn=Zb ar(x)m(x)n(x)dx:(151.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 636 III Approximate Analytical Methods Equation (151.3) is obtained by substituting equation (151.2) into equation (151.1), multiplying the result by m(x), integrating with respect to xfrom atob, and using integration by parts. If the fn(x)gare the eigenfunctions of theL[y] operator in equation (151.1), then the matrices AandRare diagonal matrices and the eigenvalues figare easily obtained. If, instead of equation (151.2), we use the nite sum y(x)=NX n=1cn n(x); (151.5) where thef n(x)gare chosen to satisfy the boundary conditions, then equation (151.3) becomes NX n=1/parenleftbig Amn−Rmn
cn=0; (151.6) form=1;2;:::;N . In this equation, AandRare given by equation (151.4) withk(x) replaced by k(x). For equation (151.6) to have a non-trivial solution, must satisfy jA− Rj=0; whereAis the matrix formed out of the AmnandRis the matrix formed out of theRmn.I f t h ef k(x)gthat we have have chosen are \close" to the actual eigenfunctions of equation (151.1), then the fkgobtained from equation (151) will be \close" to the eigenvalues fkgof equation (151.1). It is always true that the smallest from equation (151) is larger than the smallest of equation (151.1). Example Suppose an approximation to the smallest eigenvalues of the Sturm{ Liouville system y00=−y; y(−1) =y(1) = 0(151.7) is desired. Equation (151.7) has the same form as equation (151.1), with p(x)=1 ,s(x)=0 ,r(x)=1 ,a=−1, andb=1 . W eg u e s st h a t y(x)c a n be well approximated by y(x)=c1(1−x2); which is equation (151.5) with N=1a n d 1(x)=( 1−x2). Using equation (151.4), we calculate A11=Z1 −1(−2x)(−2x)dx=8 3; R11=Z1 −1(1−x2)(1−x2)dx=16 15:(151.8) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 151. Variational Method: Eigenvalue Approximation 637 Using equation (151.8) in equation (151.6) yields the eigenvalue equation for,8 3−16 15= 0, and therefore, =2:5. For this example, it turns out that the smallest eigenvalue is exactly =2=4’2:467, which corresponds to the eigenfunction (x)=c o s (x=2). Notes 1. For the Sturm{Liouville equation (151.1), it can be shown that =(−pyyx)b a+Rb a/parenleftbig p(y0)2+sy2
dx Rb ary2dx: This is known as the Rayleigh quotient . This can be used to estimate the lowest eigenvalue because 1min u(x)" (−puux)b a+Rb a p(u0)2+su2/bracerightbig dx Rb aru2dx# ; where1represents the smallest eigenvalue, and the minimization is taken over all continuous functions that satisfy the boundary condi- tions associated with equation (151.1) (but not necessarily the dier- ential equation itself). See Haberman [2, pages 172{176 and 224{226]for details. 2. There are similar relations for the eigenvalues of partial dierential equations, which are also called the Rayleigh quotient. (See Butkov [1] for details.) For example, for the Helmholtz equation in a bounded region,r 2u+u= 0 there is the relation (see Haberman [2]) =−H uru
nds+RR Rjruj2dxdy RR Ru2dxdy: 3. This section’s example is from Butkov [1, pages 573{586]. 4. See also Zauderer [4, pages 450{483]. References [1]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [2]Haberman, R. Elementary Applied Partial Dierential Equations .P r e n t i c e { Hall, Inc., Englewood Clis, NJ, 1968. [3]Weinberger, H. F. Variational Methods for Eigenvalue Approximation . SIAM, Philadelphia, PA, 1974. [4]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 638 III Approximate Analytical Methods 152. Variational Method: Rayleigh{Ritz Applicable to Dierential equations that come from a variational principle. Yields An approximation valid over an interval. Idea The variational expression from which a dierential equation is derived can be used to approximate the solution. Procedure Most equations of mathematical physics and engineering arise from a variational principle (see page 418). For example, the rst variation of J[u]=ZZ D/parenleftbig u2 x+u2 y+2uf
dxdy (152.1) (also known as the Euler{Lagrange equation associated with equation (152.1)) is given by J=uxx+uyy−f=0: Hence, the solution to uxx+uyy=f; in the region D; u=g; on the boundary of D; is given by that function u(x;y)t h a te q u a l s gon the boundary and mini- mizes equation (152.1). The Rayleigh{Ritz method is to determine the functional that a dif- ferential equation comes from and then to nd an approximate minimum. This is done by choosing a sequence of functions f1;2;:::;ngand then forming uN(x;y)=a11(x;y)+a22(x;y)+


+ann(x;y); (152.2) where thefaigare unknown. Of course, the fkgmust be chosen in such a way that the boundary conditions are satised. Now, the faigare chosen in such a way that the functional will be minimized. Specically, using equation (152.2) in equation (152.1) (or the appropriate variational princi- pal), thefaigare chosen by solving the simultaneous system of equations given by @ @aiJ[uN]=0; fori=1;:::;N: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 152. Variational Method: Rayleigh{Ritz 639 This will often be a simultaneous system of polynomial equations. If thefigin equation (152.2) are chosen \well," then uNwill tend to uasn!1 . Example 1 Suppose we wish to approximate the solution to the following Poisson equation in the unit square uxx+uyy=s i nx; for 0<x< 1;0<y< 1; u=0; onx=0;x=1;y=0;y=1: (152.3.a-b) The above equation comes from the variational principle J=0 ,w h e r e J[u]=Z1 0Z1 0/parenleftbig u2 x+u2 y+2usinx
dxdy: (152.4) We choose to approximate u(x;y) by a linear combination of 1(x;y)=x(1−x)y(1−y); 2(x;y)=x2(1−x)y(1−y); 3(x;y)=x(1−x)y2(1−y): Note that each of the figvanish on the boundary of the square, and so u3will also (as equation (152.3.b) requires). Using equation (152.2) (with N= 3) in equation (152.4) results in the minimization of the functionh 243a2 3+/parenleftbig 353a2+7 03a1+ 2100
+2 43a2 2 +/parenleftbig 703a1+ 2100
a2+7 03a2 1+ 4200a1i =31503:(152.5) Dierentiating equation (152.5) with respect to each of a1,a2,a n da3 results in the linear system of equations 2 41403703703 703483353 7033534833 52 4a1 a2 a33 5=2 4−4200 −2100 −21003 5; with the solution fa1=−30 3,a2=0 ,a3=0g. Using these values in equation (152.2) yields an approximation to the solution of equation (152.3). Note that the exact solution to the problem in equation (152.3) can be found by nite Fourier transforms (see page 344) to be u(x;y)=sinx 2sinh[sinhy+ sinh((1−y))−sinh]: (152.6) Figure 152.1 has a comparison of the exact solution in (152.6) and the approximate solution found above. This gure compares the values of u(0:1;y)a n du3(0:1;y)a syvaries from 0 to 1. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 640 III Approximate Analytical Methods/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. 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/./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././././. /././././. /./././././. /././. /././. /./././. /././. /./. /././. /././. /././. /. /././. /./. /././. /. /./. /././. /. /./. /. /././. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /. /./. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /. Figure 152.1: A comparison of the exact solution in equation (152.6) and the approximate solution in equation (152.2), when x=0:1. Example 2 A variation of this method, due to Kantorovich, is to choose the fkg to depend only on yand to allow the fakgto depend on x. For example, to approximate the solution of the Poisson equation uxx+uyy=−2; for 0<x< 1;0<y< 1; u=0; onx=0;x=1;y=−1;y=1;(152.7.a-b) which corresponds to the rst variation of J[u]=Z1 0Z1 −1/parenleftbig u2 x+u2 y−4u
dxdy; (152.8) we choose u(x;y)v(x;y)=f(x)(y2−1): (152.9) wheref(x) is unknown. Using equation (152.9) in equation (152.8) results in J[v]=Z1 016 15f02+8 3f2+16 3 dx; (152.10) which must now be minimized. The rst variation of equation (152.10) yields the following dierential equation for f(x) f00−5 2f=5 2: (152.11) The function f(x) must satisfy f(0) =f(1) = 0 for equation (152.7.b) to be satised. Solving equation (152.11) with these boundary conditions results in f(x)=−1+c o s hx+1−cosh sinh sinhx; (152.12) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 152. Variational Method: Rayleigh{Ritz 641 where=p 10=2. Combining equation (152.12) with equation (152.9) results in the nal approximation to equation (152.7). Notes 1. The Rayleigh{Ritz method also works for ordinary dierential equa- tions. For example, the variational principle corresponding to J[u]=R1 0[(y0)2+y2]dxisJ=y00+y=0 . 2. This method is an example of a weighted residual method (see page 786). 3. This technique is often implemented numerically. 4. Example 2 is from Casti and Kalaba [2, pages 68{69]. 5. See also Butkov [1, pages 573{586], Farlow [3, Lesson 45, pages 362{ 369], Kantorovich and Krylov [4, Chapter 4, pages 241{357], Mikhlinand Smolitskiy [5, Chapter 3, pages 147{269], Stakgold [6, pages 539{ 544], and Zauderer [7, pages 470{483]. References [1]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [2]Casti, J., and Kalaba, R. Imbedding Methods in Applied Mathematics . Addison{Wesley Publishing Co., Reading, MA, 1973. [3]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [4]Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis . Interscience Publishers, Inc., New York, 1958. [5]Mikhlin, S. G., and Smolitskiy, K. L. Approximate Methods for Solutions of Dierential and Integral Equations . American Elsevier Publishing Company, New York, 1967. [6]Stakgold, I. Green’s Functions and Boundary Value Problems . John Wiley & Sons, New York, 1979. [7]Zauderer, E. Partial Dierential Equations of Applied Mathematics .J o h n Wiley & Sons, New York, 1983. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 642 III Approximate Analytical Methods 153. WKB Method Applicable to Linear dierential equations. Yields A global approximation. Idea The solution of an ordinary dierential equation near an irregular singu- lar point is often in the form of an exponential. Conversely, an exponential will often be a good approximation to an ordinary dierential equation(even one without an irregular singular point.) Procedure If a given ordinary dierential equation does not have a small parameter in it, multiply the highest order derivative term by a \small" parameter 2. This turns the equation into a singularly perturbed dierential equation. Later, we will set equal to 1, and recover the original equation. Given a singularly perturbed linear ordinary dierential equation (of any order) L[y] = 0, look for a solution of the form y(x)exp" 1 1X n=0nSn(x)# ; (153.1) where we consider =() to be a small number. The technique is to use the approximation in (153.1) in the original equation and then apply dominant balance (see page 517) to determine a dierential equation for S0(x). Solve this equation for S0(x). Then, using this solution for S0(x), apply dominate balance again to determine the next largest term. This will be a dierential equation for the unknown S1(x). Solve this equation, and then iterate this procedure to determine several of thefSi(x)g. In order for the WKB approximation to be valid on an interval, we require that nSn+11a s!0a n dt h a t Sn+1(x)=Sn(x) be a bounded function of xon the given interval (for n=1;2;:::). If these do not hold, the expansion procedure is not valid. Note that if we have = 1, the constraints on fSigbecome constraints on the interval where the approximation is valid. Special Case For the singularly perturbed linear second order ordinary dierential equation 2y00=Q(x)y; (153.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 153. WKB Method 643 withQ(x)6= 0, we use equation (153.1) in equation (153.2) to determine 2 2(S0 0)2+22 S0 0S0 1+2 S00 0+


=Q(x); (153.3) where the exponential term common to both sides has been factored out. The largest terms in equation (153.3) are ( S0 0)22=2andQ(x). Because Q(x) is presumed to be of order one, we must have =and (S0 0)2=Q(x), or S0(x)=Zxp Q(t)dt: (153.4) Using=and equation (153.4) in equation (153.3) and applying domi- nant balance again, yields a rst order dierential equation for S1(x) 2S0 0S0 1+S00 0=0; which can be integrated directly to yield S1(x)=−1 4logQ(x): (153.5) Using equation (153.4) and equation (153.5) in equation (153.1), we deter- mine the leading order approximation to the solution of equation (153.1) to be y(x)C1[Q(x)]−1=4exp1 Zxp Q(t)dt +C2[Q(x)]−1=4exp −1 Zxp Q(t)dt ; (153.6) for some constants C1andC2. If a higher order approximation was desired, it is easy to derive that S2(x)=ZxQ00 8Q3=2−5(Q0)2 32Q5=2 dt; S3(x)=Q00 16Q2+5(Q0)2 64Q3; because all of the equations for the higher order fSi(x)gare of rst order. Mari ca n dT o m i  c [10] show that equation (153.6) is the correct asymp- totic result ifR1pQdt =1andR1Q02Q−5=2dt<1. Example Given the Airy equation y00=xy; (153.7) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 644 III Approximate Analytical Methods we introduce a small parameter 2and write equation (153.7) as 2y00=xy. This is now an equation of the same form as equation (153.2), with Q(x)= x. Hence, the approximation in equation (153.6) (with = 1) yields y(x)C1x−1=4exp2 3x3=2 +C2x−1=4exp −2 3x3=2 : (153.8) If we had included the S2(x) term, the approximation would be y(x)C1x−1=4exp2 3x3=2 1+5 48x−3=2 +C2x−1=4exp −2 3x3=2 1−5 48x−3=2 : (153.9) In both equations (153.8) and (153.9), the approximations are valid only asx!1 . Notes 1. WKB stands for Wentzel, Kramers, and Brillouin. This method is also sometimes called the WKBJ method or the Jereys method. 2. The eigenvalue problem z00+2V(x)z=0w i t hz(0) =z(l)=0 can be analyzed by the WKB method. Using equation (153.6), the approximate solution is z(x)=A(x)s i n −−1Rxp V(t)dt+(x) . The eigenvaluesfigare determined by where the oscillatory function vanishes. To leading order, as n!1 , the eigenvalues satisfy n= n=L ,w h e r eL=Rl 0p V(t)dt. A correction to this formula is in Lindblom and Robiscoe [7]. 3. Ludwig [8] illustrates how the WKB method may be applied to partial dierential equations. 4. The WKB approximation results in an asymptotic series. Hence, as more terms are taken in equation (153.1), the result may diverge. 5. WKB is a singular perturbation technique and boundary layer theory (see page 590) may be derived from it. 6. The approximation y(x)’exph S0(x) i is often called the geometrical optics approximation . The approximation y(x)’exph S0(x) +S1(x)i is often called the physical optics approximation . 7. For the linear ODE of degree n,dny dxn=Q(x)y, the physical optics approximation is y(x)’exph S0(x) +S1(x)i with=1=nand S0=!Zx [Q(x)]1=ndt; S 1=1−n 2nlogQ(x); where!is any of the nth roots of unity (i.e., !n=1 ) . CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 153. WKB Method 645 8. In regions where Q(x) does not vanish, the classical WKB solutions of equation (153.2) in equation (153.6) are valid. Points where Q(x) is equal to zero are called turning points ortransition points ;t h e solutions in (153.6) are not valid at these points. However, the Langer connection formula shows how the solution on each side of a turning point may be connected. Consider equation (153.2) when Q(x) has a single, simple zero at x= 0 and is monotonically increasing everywhere. We presume the boundary condition y(1) = 0, to avoid the exponentially growing solution in equation (153.6) when x!1 . Consider a region that contains the turning point x= 0. Dividing this region into three smaller regions (with the turning point in the center region), asymp-totic approximation may be obtained in each region. (Use WKB in the two outer regions, linearize Q(x) in the center region, and write the answer in terms of Airy functions). By appropriate matching (see page 590), the arbitrary constants in these three solutions can be related. Hence, a uniformly valid approximation is given by: y unif(x)=CS1=6 0Q(x)−1=4Ai"3 2S0(x)2=3# ; whereS0(x)=Rx 0p Q(t)dtandCis an arbitrary constant. Many extensions to this simple formula have been found. The ordi- nary dierential equations considered can be of higher order, therecan be multiple turning points, and the turning point need not be simple. Wazwaz [14] considers a singular perturbation problem for a second order ordinary dierential equation with two interior pointsof second order. 9. Note that WKB approximations to the two linearly independent solutions to y 00+a(x)y0+b(x)y= 0 have the form y1(x)c1exp −Zxb(t) a(t)dt ; y2(x)c2 a(x)expZxb(t) a(t)dt−1 Zx a(t)dt ; as!0+. See Bender and Orszag [1, Example 4 in Section 10.1]. 10. Fedoryuk [3] considers the equation y00+f(x;y)=0 . 11. See Bender and Orszag [1, Chapter 10, pages 484{543]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. [2]Farrell, P. A. Sucient conditions for the uniform convergence of a dierence scheme for a singularly perturbed turning point problem. SIAM J. Numer. Anal. 25 , 3 (June 1988), 618{643. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 646 III Approximate Analytical Methods [3]Fedoryuk, M. V. The WKB-method for a non-linear equation of the second order. U.S.S.R. Comput. Maths. Math. Phys. 26 , 1 (1986), 121{128. [4]Giler, S. Generalised WKBJ formulae. J. Phys. A: Math. Gen. 21 (1988), 909{930. [5]Kesarwani, R. N., and Varshni, Y. P. Five-term WKBJ approximation. J. Math. Physics 21 (1980), 90{92. [6]Langer, R. The asymptotic solutions of certain linear dierential equations of the second order. Trans. Amer. Math. Soc. 36 (1934), 90{106. [7]Lindblom, L., and Robiscoe, R. T. Improving the accuracy of WKB eigenvalues. J. Math. Physics 32 , 5 (May 1991), 1254{1258. [8]Ludwig, D. Persistence of dynamical systems under random perturbations. SIAM Review 17 , 4 (October 1975), 605{640. [9]Lynn, R., and Keller, J. B. Uniform asymptotic solutions of second-order linear ordinary dierential equations with turning point. Comm. Pure Appl. Math 23 (1970), 379{408. [10]Maric, V., and Tomic, M. On Liouville{Green (WKB) approximation for second order linear dierential equations. Dierential Integral Equations 1 , 3 (1988), 299{304. [11]McHugh, J. An historical survey of ordinary linear dierential equations with a large parameter and turning points. Arch. Hist. Exact. Sci. 7 (1971), 277{324. [12]Sawi, M. E. On the WKBJ approximation. J. Math. Physics 28 ,3( M a r c h 1987), 556{558. [13]Taylor, J. G. Improved error bounds for the Liouville{Green (or WKB) approximation. J. Math. Anal. Appl. 85 (1982), 79{89. [14]Wazwaz, A.-M. Two turning points of second order. SIAM J. Appl. Math. 50, 3 (June 1990), 883{892. [15]Willner, B., and Rubenfeld, L. A. Uniform asymptotic solutions for a linear ordinary dierential equation with one zzzref52refzzz-th order turning point: Analytic theory. Comm. Pure Appl. Math 26 (1976), 343{367. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 648 IV.A Numerical Methods: Concepts 154. Introduction to Numerical Methods Numerical analysis is a rapidly growing eld, with new techniques being developed constantly. Presented in the last section of this book are some of the more commonly used methods. This section has been separated into three parts: Introductory material about numerical methods Methods that can be used for ordinary dierential equations and, sometimes, also partial dierential equations (When a method in this part can be used for a partial dierential equation, there is a star (*)alongside the method number.) Methods that can be used only for partial dierential equations For some of the numerical methods presented in this section, a short C or Fortran computer program has been given. None of the codes have been optimized for performance. To economize on space, many of the comments that would normally appear in a well-documented computer code have been removed. When a C or Fortran computer code is given, the output is also indicated. Below are some useful thoughts when solving dierential equations numerically. Use prepared software packages whenever possible. Numerical codes are available for solving nearly any type of ordinary dierential equa- tion (see page 654). When writing a computer program, always test it on problems for which you know the solution, either analytically or from a dierent, reliable computer code. Perform numerical calculations with as many digits of precision as is reasonable for ecient execution. However, it is rarely useful to use less than \double precision." The standard way to determine if a numerical scheme is implemented correctly and the mesh sizes are small enough to justify the a priori error estimates is to reduce the size of the mesh and re-run the calculation. The resulting a posteriori error estimates should agree with the a priori error estimates. When choosing a numerical scheme to approximate the solution to a dierential equation, the roundo error should be balanced with the truncation error of the machine being used. A higher order methodwill not give more accurate answers if the major component of the error is due to roundo. Likewise, performing calculations in \double CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 154. Introduction to Numerical Methods 649 precision" will not give more accurate answers if the major component of the error is due to the discretization scheme. As a rule of thumb, to calculate a rst derivative by forward dier- ences, the roundo error and the truncation error will be approx-imately equal (and so accuracy will be high) if the dierence in values used is the square root of the number of signicant digits. For example, if your computer is working with 20 decimal digits of precision, then an accurate numerical approximation to the derivative ofy(t) will be obtained by [ y(t)−y(t+
t)]=
tfor
t’10 −10. Note that several of the methods described in earlier parts of this book may be readily implemented numerically. For some of those methods, references have been given that refer to numerical implementations.No mention of those methods is made in this section. Listed below are, in the author’s opinion, the most useful methods appearing in this last section. These are the methods that might be tried rst when a numerical approximation is required. In the numerical analysis of dierential equations, there are many important topics that are not addressed in this book. These include 1. Numerical boundary conditions for exterior problems (see Hagstrom and Hariharan [1]) 2. Eciency of dierential equation integration techniques (see Hosea and Shampine [2]) 3. Use of splines (see Sallam and Ameen [3]) Most Useful Methods for ODEs Boundary Value Problems: Box Method (page 701) Boundary Value Problems: Shooting Method(page 706) Continuation Method(page 710) Euler’s Forward Method (page 730) Finite Element Method(page 734) Predictor{Corrector Methods (page 759) Runge{Kutta Methods (page 763) Sti Equations(page 770) Weighted Residual Methods(page 786) Most Useful Methods for PDEs Continuation Method(page 710) Finite Element Method(page 734) Weighted Residual Methods(page 786) Elliptic Equations: Finite Dierences (page 805) Elliptic Equations: Relaxation (page 816) Hyperbolic Equations: Method of Characteristics (page 820) Hyperbolic Equations: Finite Dierences (page 824) Method of Lines (page 831) Parabolic Equations: Implicit Method (page 839) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 650 IV.A Numerical Methods: Concepts Pseudospectral Method (page 851) References [1]Hagstrom, T., and Hariharan, S. I. Accurate boundary conditions for exterior problems in gas dynamics. Math. of Comp. 51 , 184 (October 1988), 581{597. [2]Hosea, M. E., and Shampine, L. F. Eciency comparisons of methods for integrating ODEs. Comp. & Maths. with Appls. 28 , 6 (1994), 45{55. [3]Sallam, S., and Ameen, W. Numerical solution of general nth-order dierential equations via splines. Appl. Num. Math. 6 (1989/1990), 225{238. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 155. Denition of Terms for Numerical Methods 651 155. Denition of Terms for Numerical Methods A-stable A linear multistep method is A-stable if all solutions of the dierence equation generated by the application of this method to the scalar test equation, y0=y, tend to zero as x!1 for all complex with Re<0 and for all xed step sizes hwithh>0. Note that an explicit multistep method cannot be A-stable. Computational molecule A computational molecule is a pictorial rep- resentation of a nite dierence scheme for a partial dierential equationin two independent variables. In such a gure, the circles indicate which points are related by a dierence scheme; the value being determined by the dierence scheme is often shown shaded. For example, the computational molecule for the so-called \ve-point star" approximation to the Laplacian, r 2ui;j’1 4(ui+1;j+ui;j+1+ui−1;j+ui;j−1), is shown in gure 155.1.a. The computational molecule for the following explicit nite dierence ap- proximation to ut=uxx ui+1;j−ui;j
t=ui;j+1−2ui;j+ui;j−1 (
x)2 is shown in gure 155.1.b. Consistency of a nite dierence scheme A method is consistent if the truncation errors tend to zero as the mesh is rened (i.e., as the characteristic scales in the mesh f
x;
t;:::gtend to zero). There are two types of consistency: Conditionally consistent If the truncation errors only tend to zero iff
x;
t;:::gtend to zero in a certain way. For example, it may be required that (
x)2<
t. Unconditionally consistent If the truncation errors tend to zero no matter howf
x;
t;:::g, tend to zero. Conservative scheme A conservative numerical scheme is one in which the \total energy" described by the dierential system is conserved during the integration of the system. Dierence scheme A dierence scheme is an approximation of a deriva- tive term at a point by a collection of values near the point. Centered scheme A centered scheme is symmetric about the point at which the derivative is being approximated. For example, y0(x)’y(x+h)−y(x−h) 2h,w h e nh1. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 652 IV.A Numerical Methods: Concepts/. /. /. /./. /. /./. /././. /././././././././././././././././././././. /././././. /././. /. /./././././././././././././././././././. /././. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /././. /././././././././././././././././././././. /././././. /././. /. /./././././././././././././././././././. /././. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /././. /././././././././././././././././././././. /././. /././. /././. /./././././././././././././././././././. /././. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /. /./. /././. /././././././././././././././././././././. /././. /././. /././. /./././././././././././././././././././. /././. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /././. /././././././././././././././././././././. /././././. /././. /. /./././././././././././././././././././. /././. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./. /./. /. /./. /. /. /./. /./. /. /./. /. /. /. /. /./. /./. /. /./. /. /. /. /. /./. /./. /. /./. /. /. /./. /./. /. /./. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /./././././././././././././././././././././. /./././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /././. /./././. /./././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. t or jx or i /#28 a /#29 /. /. /. /./. /. /./. /././. /././././././././././././././././././././. /././././. /././. /././././././././././././././././././././. /././. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /././. /././././././././././././././././././././. /././././. /././. /././././././././././././././././././././. /././. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /././. /././././././././././././././././././././. /././././. /././. /././././././././././././././././././././. /././. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /././. /././././././././././././././././././././. /././. /././. /././././././././././././././././././././././. /././. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./. /./. /. /./. /. /. /./. /./. /. /./. /. /. /. /. /./. /./. /. /./. /. /. /. /. /./. /./. /. /./. /. /. /./. /./. /. /./. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /./././././././././././././././././././././. /./././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /././. /./././. /./././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. t or jx or i /#28 b /#29 Figure 155.1: Computational molecules for two dierent approximations. One-sided scheme A one-sided scheme uses values only from one side of the point at which a derivative is being approximated. Examples are forward and backward dierence schemes. Forward dierence scheme A forward dierence scheme is a one- sided dierence scheme that uses points \ahead" of the point that is being approximated. For example, y0(x)’y(x+h)−y(x) h, whenh1. Backward dierence scheme A backward dierence scheme is a one-sided dierence scheme that uses points \behind" the point that is being approximated. For example, y0(x)’y(x)−y(x−h) h, whenh1. Explicit method An explicit method is one for which there is an explicit formula, at a point, for the value of the unknown terms appearing in the dierential equation. Grid A grid is a set of points, called mesh points , on which the solution of a dierential equation is approximated. If the points are uniformly spaced, then we have a uniform grid ; otherwise we have a non-uniform grid .S e e page 675. Implicit method An implicit method is one for which there is not an explicit formula, at a point, for the value of the unknown terms appearing in the dierential equation. Generally a nonlinear algebraic equation must be solved to determine the value at a given point. Mesh See Grid. Order of a numerical method One less than the exponent in the error term of a method. See page 670. Step size See page 670. Sti equations Sti equations are dierential equations that are ill posed in a computational sense. There are many dierent denitions of stiness, two common ones are CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 155. Denition of Terms for Numerical Methods 653 A system of dierential equations is said to be sti on the interval [0;T] if there exists a component of a solution of the system that has a variation on [0 ;T] that is large compared with 1 =T. A system is sti if there exists more than one scale, with a great dierence in size, on which the solution evolves. For instance, the system of dierential equations y0=Ay(where Ais a constant matrix with eigenvalues i(A)) is sti if max iji(A)jminiji(A)j. Symplectic integration An integration method is said to be symplectic if the state of the (Hamiltonian) system following an integration step couldhave been reached from that before the step by some canonical transfor- mation. The most straightforward way to test if a method is sympletic is to verify the Poisson-bracket relations between the before and afterstates. Given a method that determines u(x), where uandxare both s-dimensional, let Jbe thessJacobian matrix that leads from \before" to \after": J= @(un;xn) @(un−1;xn−1). Now dene the matrix K=0sIs −Is0s .I f JTKJ=K, then the method is sympletic. Truncation error The error when the exact solution is substituted into a nite dierence scheme. See page 670. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 654 IV.A Numerical Methods: Concepts 156. Available Software Applicable to Ordinary and partial dierential equations that are to be approximated numerically. Idea When numerically approximating the solution to a dierential equation, it is best to use commercially available software whenever possible. The routines commonly available for ordinary dierential equations are ade- quate for nearly all types of problems. The routines commonly available forpartial dierential equations are not as well developed. For linear problems with no singularities, however, the available software is very good. There are a multitude of commercially available computer libraries and isolated computer routines available. A taxonomy for dierential equation software has been developed as part of the Guide to Available Mathemat- ical Software (GAMS) project at the National Institute of Standards and Technology (NIST) [5], see table 156.1. GAMS [6] also has a listing of available software. Because good software is readily available, we paraphrase the admoni- tion that Byrne and Hindmarsh [8] give: :::if you are using a 10-line solver for dierential equations :::you should consider using one of the programs referenced in this section. There is now commercially available \software" for dierential equations with no error control, a user-specied stepsize, and no warning messages. We advise against using such programs, even on a small computer. The reasons are straight- forward. For all but trivial problems, such programs cannot be suciently reliable for accurate computational results. When using a prepared software package, it is always useful to test the package on problems similar to the one that you will use the package for. There are many collections of test problems for this purpose, see page 694. Notes 1. Given a new problem to solve numerically, it is often attractive to design new software for this class of problem. However, it is usually more ecient to transform the problem and use well-tested codes. See, for example, Shampine and Zhang [25]. 2. Addison et al. [2] present a decision tree to assist in the process of selecting an appropriate algorithm for the numerical solution of initial value ordinary dierential equations. The decision tree can be usedin an interactive manner. Where possible, the recommended soft- ware routines are in maintained libraries that have been extensively CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 156. Available Software 655 I1 Ordinary dierential equations (ODEs) I1a Initial value problems I1a1 General, nonsti, or mildly sti I1a1a One-step methods (e.g., Runge{Kutta) I1a1b Multistep methods (e.g., Adams predictor-corrector) I1a1c Extrapolation methods (e.g., Bulirsch{Stoer) I1a2 Sti and mixed algebraic-dierential equations I1b Multipoint boundary value problems I1b1 Linear I1b2 Nonlinear I1b3 Eigenvalue (e.g., Sturm{Liouville) I1c Service routines (e.g., interpolation of solutions, error handling, test programs) I2 Partial dierential equations I2a Initial boundary value problems I2a1 Parabolic I2a1a One spatial dimension I2a1b Two or more spatial dimensions I2a2 Hyperbolic I2b Elliptic boundary value problems I2b1 Linear I2b1a Second order I2b1a1 Poisson (Laplace) or Helmholtz equation I2b1a1a Rectangular domain (or topologically rectangular in the coordinate system) I2b1a1b Nonrectangular domain I2b1a2 Other separable problems I2b1a3 Nonseparable problems I2b1c Higher order equations (e.g., biharmonic) I2b2 Nonlinear I2b3 Eigenvalue I2b4 Service routines I2b4a Domain triangulation (search also GAMS class P) I2b4b Solution of discretized elliptic equations Table 156.1: The GAMS taxonomy of dierential equations software tested. Addison et al. [3] contains a decision tree for boundary value problems. 3. Periodically, there are reviews in the literature of software applicable to a specic type of dierential equation. See, for example, Machura and Sweet [17]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 656 IV.A Numerical Methods: Concepts 4. The books by Press et al. [22], contain collections of Fortran, PAS- CAL, and C codes for both ordinary dierential equations and partial dierential equations. 5. Many scientic software routines, including those for dierential equa- tions, may be obtained for free (via electronic mail) from a variety of computer networks. See the article by Dongarra and Grosse [12]. The ACM’s Transactions on Mathematical Software (TOMS) is available at http://gams.nist.gov/toms/Overview.html . Netlib is a collection of mathematical software, papers, and databases. It can be reached at http://www.netlib.org . 6. Numerical methods for rst order PDEs may be found in Pennington and Berzins [20]. 7. Even though it is possible to numerically approximate dierential equations using spreadsheet programs, this is notrecommended; see Enloe [14]. 8. Software for small computers is summarized in Penn [19] and Teles et al. [26], [27]. 9. Software is not listed for all of the GAMS taxonomy classes that have been established. 10. The following computer libraries are referred to in GAMS1. Their inclusion does not constitute an endorsement. Nor does it necessarily imply that unnamed packages are not worth trying. (All of the infor-mation in this note has been obtained from http://gams.nist.gov ). BIHAR A package of Fortran subprograms for the generalized bihar- monic equation in rectangular geometry and polar coordinates subject to rst kind boundary conditions. Distributed by netlib,seehttp://www.netlib.org/bihar . CMLIB The NIST Core Math LIBrary (CMLIB) is a collection of high- quality, easily transportable Fortran subroutine sublibraries solv- ing standard problems in many areas of mathematics and statis- tics (approximately 750 subroutines and functions). It is dis-tributed by the Center for Computing and Applied Mathematics at NIST. The source for CMLIB has come from {BVSUP: see Scott and Watts [24] {CDRIV and SDRIV: see Kahaner et al. [16] {DEPAC: Code developed by Shampine and Watts. {FISHPAK: Code developed by Swarztrauber and Sweet. {SDASSL: see Petzold [21] {VHS3: Code developed by Sweet. 1Identication of commercial products does not imply recommendation or endorse- ment by NIST. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 156. Available Software 657 CRAYFISHPAK A highly vectorized Fortran subroutine library for the solution of separable elliptic partial dierential equations (e.g. Poisson’sequation). Cartesian (2D and 3D), polar, cylindrical, spherical, surface spherical, and spherical cross-section geometries are sup- ported, as well as both centered and staggered nite dierence grids. Distributed by Green Mountain Software, Boulder, CO. DIFFPACK A set of object-oriented libraries for solving partial dierential equations and several Unix utilities for general software man- agement and numerical programming. Aimed at rapid proto- typing of simulators based on PDEs while still oering high eciency. Implemented in C++, the libraries are organized into several layers: Basic Tools, Linear Algebra Tools, Dp Kernel,Dp Utilities, and Dp Applications. Distributed by netlib, see http://www.netlib.org/diffpack . The Dipack home page ishttp://www.oslo.sintef.no/avd/33/3340/diffpack . ELLPACK Solves linear elliptic boundary value problems in general 2D domains and in 3D boxes. Includes a problem-description lan-guage (a Fortran extension) allowing equations, domains, solu- tion methods, and options to be specied at a very high level, but flexible enough to to do special processing (to solve nonlinear problems, for example). Incorporates over 50 problem solving modules for discretization, equation reordering, linear equationsolution, etc. Distributed by Purdue Research Foundation, W. Lafayette, IN. This package is described in the book by Rice and Boisvert [23], see also http;//www.cs,purdue/ellpack . FISHPACK A package of Fortran subprograms for separable elliptic par- tial dierential equations. Distributed by netlib, see http:// www.netlib.org/fishpack . IMSLM The IMSL MATH/LIBRARY is a Fortran subprogram library for solving problems in applied mathematics (approximately 700 subroutines and functions.) Distributed by Visual Numerics of Houston, TX. MANPAK Utility programs for computations with submanifolds of R n implicitly dened by a system of nonlinear equations. Includes subroutines for a wide variety algebraically explicit dierential algebraic equations (DAEs); that is, DAEs in which either the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 658 IV.A Numerical Methods: Concepts algebraic equations and/or variables are explicitly specied. Dis- tributed by netlib, see http://www.netlib.org/contin/manpak . NAG A Fortran subroutine library for solving standard problems in many areas of mathematics, statistics, and optimization (ap- proximately 1000 subroutines.) Distributed by NAG, Downers Grove, IL. NMS A collection of high-quality, portable Fortran subroutines for solving common computational problems in mathematics, engi-neering, and statistics. From the book by Kahaner et al. [16]. ODE A collection of software for solving initial and boundary value problems for ordinary dierential equations. Distributed by netlib, see http://www.netlib.org/ode . ODEPACK and SODEPACK A collection of Fortran solvers for the initial value problem for ordinary dierential equation systems. It currently includes six solvers, suitable for both sti and nonsti systems, andincludes solvers for systems given in linearly implicit form as well as solvers for systems given in explicit form. (Available in single- and double-precision versions.) Distributed by netlib, see http://www.netlib.org/odepack . PDELIB A small collection of Fortran subroutines which solve general systems of nonlinear initial-boundary-value partial dierential equations in one or two space dimensions. Each routine is basedupon the method of lines. PDES Software to solve many types of partial dierential equations collected from a variety of sources. Distributed by netlib, see http://www.netlib.org/pdes . PLTMG and DPLTMG A Fortran package for solving an elliptic partial dierential equa- tion in general regions of the plane. It features adaptive local mesh renement, multigrid iteration, and a pseudo-arclengthcontinuation option for parameter dependencies The package includes an initial mesh generator and several graphics packages. (Available in single- and double-precision versions.) Distributed by netlib, see http://www.netlib.org/pltmg . PORT CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 156. Available Software 659 A Fortran subprogram library for solving a variety of mathe- matical problems. Distributed by Lucent Technologies, Liberty Corner, NJ. SLATEC The SLATEC Common Mathematical Library is a collection of Fortran subprograms for a wide variety of mathematical prob-lems. A primary impetus for the library development was to provide portable, non-proprietary, mathematical software for su- percomputers at a consortium of government-sponsored research laboratories. Distributed by the Energy Science and Technology Software Center, Oak Ridge, TN. TOMS The Collected Algorithms of the ACM, published by the journal ACM Transactions on Mathematical Software. Distributed by netlib, see http://www.netlib.org/toms . References [1]Adams, J. C. Mudpack | Multigrid portable Fortran software for the ecient solution of linear partial dierential equations. Appl. Math. and Comp. 34 , 2 (1989), 113{146. [2]Addison, C. A., Enright, W. H., Gaffney, P. W., Gladwell, I., and Hanson, P. M. A decision tree for the numerical solution of initial value ordinary dierential equations. ACM Trans. Math. Software 17 , 1 (March 1991), 1{10. [3]Addison, C. A., Enright, W. H., Gaffney, P. W., Gladwell, I., and Hanson, P. M. A decision tree for the numerical solution of boundary value ordinary dierential equations, SMU Math Report 89-7, Southern Methodist University, Dallas, TX. [4]Bank, R. E. PLTMG: A Software Package for Solving Elliptic Partial Dierential Equations . SIAM, Philadelphia, PA, 1990. [5]Boisvert, R. F., Howe, S. E., and Kahaner, D. K. GAMS: A framework for the management of scientic software. ACM Trans. Math. Software 11 , 4 (December 1985), 313{355. [6]Boisvert, R. F., Howe, S. E., Kahaner, D. K., and Springmann, J. L. Guide to available mathematical software. Tech. rep., National Institute of Standards and Technology, Gaithersburg, MD, March 1990. Center forComputing and Applied Mathematics NISTIR 90-4237. [7]Boisvert, R. F., and Sweet, R. A. Mathematical software for elliptic boundary value problems. In Sources and Development of Mathematical Software , W. R. Cowell, Ed. Prentice{Hall, Inc., Englewood Clis, NJ, 1984, pp. 200{263. [8]Byrne, G. D., and Hindmarsh, A. C. Sti ODE solvers: A review of current and coming attractions. J. Comput. Physics 70 (1987), 1{62. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 660 IV.A Numerical Methods: Concepts [9]Childs, B., Scott, M., Daniel, J. W., Denman, E., and Nelson, P.,E d s . Codes for Boundary-Value Problems in Ordinary Dierential Equations . Springer{Verlag, New York, 1979. [10]Delves, L. M., McKerrell, A., and Peters, S. A. Performance of GEM2 on the ELLPACK problem population. Internat. J. Numer. Methods Eng. 23 (1986), 229{238. [11]Dew, P. M., and Walsh, J. E. A set of library routines for solving parabolic equations in one space variable. ACM Trans. Math. Software 7 ,3 (Sept 1981), 295{314. [12]Dongarra, J. J., and Grosse, E. Distribution of mathematical software via electronic mail. Comm. of the ACM 30 , 5 (May 1987), 403{407. [13]Dyksen, W. R., and Ribbens, C. J. Interactive ELLPACK: An interactive problem{solving environment for elliptic partial dierential equations. ACM Trans. Math. Software 13 , 2 (June 1987), 113{132. [14]Enloe, C. L. Solving coupled, nonlinear dierential equations with commercial spreadsheets. Computers in Physics (Jan/Feb 1989), 75{76. [15]Gaffney, P. W. A performance evaluation of some FORTRAN subroutines for the solution of sti oscillatory ordinary dierential equations. ACM Trans. Math. Software 10 , 1 (March 1984), 58{72. [16]Kahaner, D., Moler, C., and Nash, S. Numerical Methods and Software . Prentice{Hall, Inc., Englewood Clis, NJ, 1989. [17]Machura, M., and Sweet, R. A. A survey of software for partial dierential equations. ACM Trans. Math. Software 6 , 4 (Dec 1980), 461{488. [18]Melgaard, D. K., and Sincovec, R. F. General software for two- dimensional nonlinear partial dierential equations. ACM Trans. Math. Software 7 , 1 (March 1981), 106{125. [19]Penn, H. L. A review of dierential equations software. Collegiate Microcomputer 6 (1988), 33{42. [20]Pennington, S. V., and Berzins, M. New NAG library software for rst-order partial dierential equations. ACM Trans. Math. Software 20 ,1 (March 1994), 63{99. [21]Petzold, L. R. Dierential/algebraic equations are not ODE’s. SIAM J. Sci. Stat. Comput. 3 , 3 (1982), 367{384. [22]Press, W. H., Flannery, B. P., Teukolsky, S., and Vetterling, W. T. Numerical Recipes . Cambridge University Press, New York, 1986. [23]Rice, J. R., and Boisvert, R. F. Solving Elliptic Problems Using ELLPACK . Springer{Verlag, New York, 1985. [24]Scott, M. R., and Watts, H. A. Computational solutions of linear two- point boundary value problems via orthonormalization. SIAM J. Numer. Anal. 14 (1977), 40{70. [25]Shampine, L. F., and Zhang, W. Ecient integration of ordinary dierential equations by transformations. Comp. & Maths. with Appls. 15 (3 1988), 213{220. [26]Teles, E., Penn, H. L., and Wilkin, J. ODE software for the IBM PC. College Math. J. 21 , 3 (May 1990), 242{245. [27]Teles, E., Penn, H. L., and Wilkin, J. ODE software for the Macintosh. College Math. J. 21 , 4 (September 1990), 330{332. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 157. Finite Dierence Formulas 661 157. Finite Dierence Formulas Applicable to Dierential equations that will be solved by the method of nite dierences. Idea A table of nite dierence formulas for some common grids and common equations can be useful. Procedure Given a dierential equation to be approximated by nite dierences and a grid (see page 675) on which the solution is desired, replace every derivative by a nite dierence approximation to that derivative. Standard nite dierence formulas presume that there is an underlying uniform gridwith a spacing of h. (In two dimensions, the uniform grid spacing is commonly taken to be hin one direction and kin another direction). In the formulas for ordinary dierential equation systems y 0=f(x;y), we use the shorthand notation xn=x0+nh,yn=y(xn),fn=f(xn;yn), andvnyn. In the formulas for partial dierential equation systems L[z]=f(x;y;z) (whereL[ ] is a two-dimensional dierential operator), we use the shorthand notationxn=x0+nh,yn=y0+nk,xn;m=(xn;ym),zn;m=z(xn;ym), fn;m=f(xn;ym;zn;m), and vn;mzn;m. In this section we include tables of formulas for the following cases: One Dimension: Rectilinear Grid Two Dimensions: Rectilinear Grid Two Dimensions: Irregular Grid Two Dimensions: Triangular Grid Numerical Schemes for the ODE: y0=f(x;y) Explicit Numerical Schemes for the PDE: aux+ut=0 Implicit Numerical Schemes for the PDE: aux+ut=S(x;t) Numerical Schemes for the PDE: F(u)x+ut=0 Numerical Schemes for the PDE: ux=utt 157.1 One Dimension: Rectilinear Grid The following is a list of nite dierence formulas of dierent accuracies for a grid with uniform spacing. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 662 IV.A Numerical Methods: Concepts 1. Formulas for the rst derivative: f0(x0)=f1−f0 h+O(h) f0(x0)=f1−f−1 2h+O/parenleftbig h2
f0(x0)=−f2+4f1−3f0 2h+O/parenleftbig h2
f0(x0)=−f2+8f1−8f−1+f−2 12h+O/parenleftbig h4
2. Formulas for the second derivative: f00(x0)=f2−2f1+f0 h2+O(h) f00(x0)=f1−2f0+f−1 h2+O/parenleftbig h2
f00(x0)=−f3+4f2−5f1+2f0 h2+O/parenleftbig h2
f00(x0)=−f2+1 6f1−30f0+1 6f−1−f−2 12h2+O/parenleftbig h4
3. Formulas for the third derivative: f000(x0)=f3−3f2+3f1−f0 h3+O(h) f000(x0)=f2−2f1+2f−1−f−2 2h3+O/parenleftbig h2
4. Formulas for the fourth derivative: f(4)(x0)=f4−4f3+6f2−4f1+f0 h4+O(h) f(4)(x0)=f2−4f1+6f0−4f−1+f−2 h4+O/parenleftbig h2
157.2 Two Dimensions: Rectilinear Grid The following is a list of nite dierence formulas of dierent accuracies for rectangular grids with uniform spacing. Other formulas can be obtainedfrom the last list simply by holding one variable constant. 1. Formulas for rst order partial derivatives: f x(x0;0)=1 2h(f1;0−f−1;0)+O/parenleftbig h2
fx(x0;0)=1 4h(f1;1−f−1;1+f1;−1−f−1;−1)+O/parenleftbig h2
CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 157. Finite Dierence Formulas 663/#0F /#0F/#0F /#0F /#0F /#0F/#0F /#0F /#0F u/2u/4 u/0 u/1 u/3 /#12/1 h /#12/3 h /#12/2 h/#12/4 h Figure 157.1: Spacing on an irregular domain. 2. Formulas for second order partial derivatives: fxx(x0;0)=1 3h2(f1;1−2f0;1+f−1;1+f1;0−2f0;0+f−1;0 +f1;−1−2f0;−1+f−1;−1)+O/parenleftbig h2
fxy(x0;0)=1 4h2(f1;1−f1;−1−f−1;1+f−1;−1)+O/parenleftbig h2
3. Formulas for the Laplacian: r2f(x0;0)=1 h2(f1;0+f0;1+f−1;0+f0;−1−4f0;0)+O/parenleftbig h2
r2f(x0;0)=1 12h2(−60f0;0+ 16(f1;0+f0;1+f−1;0+f0;−1) −(f2;0+f0;2+f−2;0+f0;−2)) +O/parenleftbig h4
157.3 Two Dimensions: Irregular Grid Nonuniform grids may be the only way to numerically solve some prac- tical problems involving partial dierential equations. For example, a non- uniform grid may be required near the boundaries of a domain. Also,adaptive grids and moving grids are sometimes more useful than a xed grid (see page 675). The following nite dierence formulas refer to the parameters dened in gure 157.1. 1. Formulas for rst order partial derivatives: @u @x x0;0=u3−u1 h(1+3)+O(h) @u @y x0;0=u2−u4 h(2+4)+O(h) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 664 IV.A Numerical Methods: Concepts/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./. /./. /././. /./././. /./././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /./././././. /./././././././././././././././. /././././. /./././././././././././././././.x ya b c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igure 157.2: Denition of the coordinate system for a triangular domain. 2. Formulas for second order partial derivatives: @2u @x2 x0;0=2 h2u1−u0 1(1+3)+u3−u0 3(1+3) +O(h) @2u @y2 x0;0=2 h2u2−u0 2(2+4)+u4−u0 4(2+4) +O(h) r2u x0;0=@2u @x2+@2u @y2 x0;0 =2 h2u1 1(1+3)+u2 2(2+4)+u3 3(1+3)+u4 4(2+4) −1 13+1 24 u0 +O(h) 157.4 Two Dimensions: Triangular Grid Sometimes it is easier to perform computations on a uniform triangular grid (see gure 157.2). If we represent the three directions on the triangular grid asfa;b;cg, then we can compute the partial derivatives: @u @a=ux; @u @b=1 2ux+p 3 2uy; @u @c=−1 2ux+p 3 2uy;@2u @a2=uxx; @2u @b2=1 4uxx+p 3 2uxy+3 4uyy; @2u @c2=1 4uxx−p 3 2uxy+3 4uyy: These relations may be inverted to yield CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 157. Finite Dierence Formulas 665 Adams{Bashforth, order 2 vn−vn−1=1 2h[3fn−1−fn−2] Adams{Bashforth, order 4 vn−vn−1=1 24h[55fn−1−59fn−2+3 7fn−3−9fn−4] Adams{Moulton, order 4 vn−vn−1=1 24h[9fn+1 9fn−1−5fn−2+fn−3] backward Euler vn−vn−1=hfn Euler’s method vn−vn−1=hfn−1 explicit leapfrog vn+1−vn−1=hfn implicit leapfrog vn−vn−1=1 2h(fn+fn−1) Simpson’s ruleavn−vn−2=1 3h(fn+4fn−1+fn−2) trapezoidal rulebvn−vn−1=1 2h(fn+fn−1) aAlso known as Milne’s method. bAlso known as Heun’s method and as the Adams{Moulton method of order 2. Table 157.1: Numerical schemes for the ODE: y0=f(x;y) ux=@u @a; uy=1p 3@u @b+@u @c ; uxx=@2u @a2;uyy=1 3 2@2u @b2+2@2u @c2−@2u @a2 ; uxy=1p 3@2u @b2−@2u @c2 ; r2u=uxx+uyy=2 3@2u @a2+@2u @b2+@2u @c2 : See Gerald and Wheatley [6, Section 7.9] for a worked example using triangular coordinates. 157.5 Numerical Schemes for the ODE: y /0=f(x;y) Table 157.1 contains some common dierence formulas for the ordinary dierential equation y0=f(x;y). Of these methods, Euler’s method and the leapfrog method are explicit; all the others are implicit methods. 157.6 Explicit Numerical Schemes for the PDE: aux+ ut=0 Table 157.2 contains named explicit dierence formulas for the partial dierential equation aux+ut= 0. DuChateau and Zachmann [3, page 450] also list the local truncation error for each of these methods. In thislisting,his the uniform xspacing, and kis the uniform tspacing. The approximation to u(x n;tj)=u(x0+nh;t 0+jk) is represented by un;j. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 666 IV.A Numerical Methods: Concepts Forward in time, forward in space (FTFS): aun+1;j−un;j h+un;j+1−un;j k=0 Forward in time, centered in space (FTCS) (unstable): aun+1;j−un−1;j 2h+un;j+1−un;j k=0 Forward in time, backward in space (FTBS): aun;j−un−1;j h+un;j+1−un;j k=0 Lax{Friedrichs method: aun+1;j−un−1;j 2h+un;j+1−1 2(un−1;j−un+1;j) k=0 Lax{Wendro method: un;j+1=un;j−ak 2h(un+1;j−un−1;j) +a2k2 2h2(un−1;j−2un;j+un+1;j) Table 157.2: Explicit numerical schemes for the PDE: aux+ut=0 Backward in time, backward in space (BTBS): aun+1;j+1−un;j+1 h+un+1;j+1−un+1;j k=Sn+1;j+1 Backward in time, centered in space (BTCS): aun+1;j+1−un−1;j+1 2h+un;j+1−un;j k=Sn;j+1 Crank{Nicolson: 1 2 aun+1;j+1−un−1;j+1 2h+aun+1;j−un−1;j 2h +un;j+1−un;j k=Sn;j+1=2 Wendro method: 1 2 aun+1;j+1−un;j+1 h+aun+1;j−un;j h +1 2 un+1;j+1−un+1;j k+un;j+1−un;j k =Sn+1=2;j+1=2 Table 157.3: Implicit numerical schemes for the PDE: aux+ut=S(x;t) 157.7 Implicit Numerical Schemes for the PDE: aux+ ut=S(x;t) Table 157.3 contains named implicit dierence formulas for the partial dierential equation aux+ut=S(x;t). DuChateau and Zachmann [3, page 460] also list the local truncation error for each of these methods. In this listing, his the uniform xspacing, and kis the uniform tspacing. The approximation to u(xn;tj)=u(x0+nh;t 0+jk) is represented by un;j,a n d Sn;jis used to represent S(xn;tj). 157.8 Numerical Schemes for the PDE: F(u)x+ut=0 Table 157.4 contains named dierence formulas for the partial dier- ential equation F(u)x+ut= 0 (see DuChateau and Zachmann [3, page 475] for more details). In this listing, his the uniform xspacing,kis the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 157. Finite Dierence Formulas 667 Centered in time{centered in space (unstable): un;j+1=un;j−1 2s(Fn+1;j−Fn−1;j) Lax{Friedrichs method: un;j+1=1 2(un+1;j+un−1;j)−1 2s(Fn+1;j+Fn−1;j) Lax{Wendro method: un;j+1=un;j−1 2s(Fn+1;j−Fn−1;j) +1 2s2 an+1=2;j(Fn+1;j−Fn;j)−an−1=2;j(Fn;j−Fn−1;j) Richtmeyer method: u n+1=2=1 2(un+1;j+un;j)−1 2(Fn+1;j−Fn;j) un;j+1=un;j−s F n+1=2−F n−1 MacCormack method: u n=un;j−s(Fn+1;j−Fn;j) un;j+1=1 2 un;j+u n−s/parenleftbig F n−F n−1
 FTBS upwind method (use when F0(u)>0): un;j+1=un;j+s(Fn−1;j−Fn;j) FTFS upwind method (use when F0(u)<0): un;j+1=un;j−s(Fn+1;j−Fn;j) Table 157.4: Numerical schemes for the PDE: F(u)x+ut=0 uniformtspacing, and the ratio of these is s=k=h. The approximation to u(xn;tj)=u(x0+nh;t 0+jk) is represented by un;jandFm;n:=F(um;n). A star superscript indicates an intermediate result (and F n:=F(u n)). Finally,an:=F0 n=F0(un). Note that some of the left-hand sides of the last listing can be obtained from this listing by taking F(u)=au. 157.9 Numerical Schemes for the PDE: ux=utt Table 157.5 contains named dierence formulas for the partial dier- ential equation ux=utt. Lapidus and Pinder [9] discuss each of these methods in some detail. In this listing, his the uniform xspacing,kis the uniformtspacing, and is dened to be =h=k2. The approximation to u(xn;tj)=u(x0+nh;t 0+jk) is represented by un;j. Notes 1. Fornberg [4] has a simple recursive technique for determining nite dierence formula of high order. 2. For problems with periodic boundary conditions, it is possible to ob- tain nite dierential formulas that are of innite order; see page 851. 3. All of the discretization methods used should be of comparable order. That is, if one term in an equation has a discretization error of O(h2), then there is no reason for another term to have a discretization error ofO(h4). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 668 IV.A Numerical Methods: Concepts Classic explicit approximation: un+1;j=( 1−2)un;j+(un;j+1+un;j−1) DuFort{Frankel explicit approximation: (1 + 2)un+1;j=2(un;j+1+un;j−1)+( 1−2)un−1;j Richardson explicit approximation: un+1;j−un−1;j−2(un;j+1+un;j−1)+4un;j=0 Backward implicit approximation: (1 + 2)un+1;j−(un+1;j+1+un+1;j−1)=un;j Crank{Nicolson implicit approximation: 2(+1 )un+1;j−(un+1;j+1+un+1;j−1)=2 ( 1−)un;j +(un;j+1+un;j−1) Variable weighted implicit approximation (with 0 1): (1 + 2)un+1;j=(1−)(un;j+1+un;j−1) +(un+1;j+1+un+1;j−1)+[ 1−2(1−)]un;j Table 157.5: Numerical schemes for the PDE: ux=utt 4. Note that nonuniform grids may give rise to a number of consis- tency/stability phenomena that have no counterpart on uniform grids. 5. Macsyma [10] has a package ( fdifpde) which derives nite dierence approximations for partial dierential equations. 6. See also Abramowitz and Stegun [1, pages 883{885] and Lapidus and Pinder [9, section 4.3, pages 153{162]. References [1]Abramowitz, M., and Stegun, I. A. Handbook of Mathematical Functions . National Bureau of Standards, Washington, D.C., 1964. [2]Altas, I., and Stephenson, J. W. Finite dierence schemes on irregular meshes. Eighteenth Manitoba Conference on Numerical Mathematics and Computing Winnipeg, Canada 69 (1989), 21{32. [3]DuChateau, P., and Zachmann, D. Applied Partial Dierential Equa- tions . Harper & Row Publishers, New York, 1989. [4]Fornberg, B. Generation of nite dierence formulas on arbitrarily spaced grids. Math. of Comp. 51 , 184 (October 1988), 699{706. [5]Ganzha, V. G., Mazurik, S. I., and Shapeev, V. P. Symbolic manipulations on a computer and their application to generation andinvestigation of dierence schemes. In EUROCAL ’85 ,B .B u c h b e r g e ra n d B. F. Caviness, Eds. Springer{Verlag, New York, 1985, pp. 335{347. [6]Gerald, C. F., and Wheatley, P. O. Applied Numerical Analysis . Addison{Wesley Publishing Co., Reading, MA, 1984. [7]Heinrich, B. Finite Dierence Methods on Irregular Networks . Birkhauser, Basel, Switzerland, 1987. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 157. Finite Dierence Formulas 669 [8]Keller, H. B., and Pereyra, V. Symbolic generation of nite dierence formulas. Math. of Comp. 32 (1978), 955{971. [9]Lapidus, L., and Pinder, G. F. Numerical Solution of Partial Dierential Equations in Science and Engineering . John Wiley & Sons, New York, 1982. [10]Macsyma .Mathematics Reference Manual . Macsyma, Arlington, MA, 1993. [11]Voss, D. A fth-order exponentially tted formula. SIAM J. Math. Anal. 25, 3 (June 1988), 670{678. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 670 IV.A Numerical Methods: Concepts 158. Finite Dierence Methodology Applicable to Dierential equations. Yields A nite dierence scheme that can be used to numerically approximate a given dierential equation. Procedure For the rst order ordinary dierential equation y0=f(x;y), consider the general multistep (or k-step) method N[vn;vn+1;:::;vn+k]: =kX j=0jvn+j−hkX j=0jf(xn+j;vn+j)=0; (158.1) where06=0 ,n=k;k+1;:::andvnis an approximation to y(xn) (where xn=nhandhis a small number called the step size ). We presume the constantsfigandfigare known. If06= 0, then the scheme is an implicit dierence method. If 0=0 , then the scheme is an explicit dierence method. For explicit methods, equation (158.1) can be solved for vnin terms of the other quantities in equation (158.1). The exact solution to the equation y0=f(x;y) will not, in general, satisfyN[yn;yn+1;:::;yn+k]=0( h e r e , yn=y(xn)). Ifh1, then a Taylor series can be employed to show that yn+j=yn+jhy0 n+(jh)2 2y00 n+


: Using this expansion, a Taylor series can be taken of N[yn;yn+1;:::;yn+k] to obtain N[yn;yn+1;:::;yn+k]=kX j=0jyn−j−hkX j=0jf(xn−j;yn−j) =hp+1Rn+O(hp+2);(158.2) for some numbers pandRn. Ifp1, then the method is said to be consistent .I f a m e t h o d i s consistent, then pis called the order of the method . We say that \the method ispth order accurate." The term hp+1Rnis called the truncation error . A theorem of numerical analysis states that there exist methods of orderp=2k. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 158. Finite Dierence Methodology 671 The rst and second characteristic polynomials of the method in equa- tion (158.1) are dened as (x)a n d(x), where (x)=kX j=0jxj; (x)=kX j=0jxj: If equation (158.1) is consistent, then it follows that (1) = 0 and 0(1) = (1). Ifp>k + 2, then the method will always be unstable (stability for the discretization of ordinary dierential equations is dened on page 683).Specically, if kis odd, then p=k+ 1 is the largest psuch that there is a stable method. Also, if kis even, then p=k+ 2 is the largest psuch that there is a stable method. If a dierence method is stable and is of pth order accuracy, then jv n−ynj=o(hp) in any nite interval, 0 xL. Many nite dierence formulas are tabulated on page 661. For example, for Euler’s method and the trapezoidal rule, k= 1. For Simpson’s rule, k=2a n dp= 4. To obtain a discretization for a dierential equation, it is possible to obtain a nite dierence formula for every term in thedierential equation and then combine these formulas in the obvious man- ner. (Just replace each term in the dierential equation with its nite dierence approximation.) However, combining formulas in this way for partial dierential equations|without understanding the underlying phys- ics of the problem and the approximations|can quickly produce resultsthat are unrelated to the true problem (see page 27). Example There are many procedures for generating nite dierence formulas for the terms appearing in dierential equations; we illustrate one straightfor- ward method. Suppose we want to nd an approximation to f0(x0), given the valuesf(x0−h)a n df(x0+h). We write f0(x0)=f(x0−h)+f(x0+h)+e(x0;h); (158.3) whereandare constants to be determined, and e(x0;h)r e p r e s e n t st h e error term. Taking a Taylor series of the right-hand side of equation (158.3) (and using f0to represent f(x0),f0 0forf0(x0), etc.), we nd f0 0= f0−hf0 0+h2 2f00 0−h3 6f000 0+O(h4) + f0+hf0 0+h2 2f00 0+h3 6f000 0+O(h4) +e(x0;h): If we choose =−, then this simplies to f0 0= 2hf0 0+h3 3f000 0+O(h4) +e(x0;h): CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 672 IV.A Numerical Methods: Concepts Finally, if we choose =1=2h, then we obtain f0 0=f0 0+h2 6f000+O(h3)+ e(x0;h). Hence,e(x0;h)=O(h2). Putting all of this together, we have the nite dierence approximation f0(x0)=f(x0+h)−f(x0−h) 2h+O(h2): This formula could be used to approximate the ordinary dierential equationy0=y2, on a uniform mesh, by u(x0+h)−u(x0−h) 2h=u2(x0); whereu(x)y(x). Usingx0:=nhandun:=u(nh) in this formula, we ndun+1−un−1 2h=u2 n. This can be manipulated into the explicit formula: un+1=un−1+2hu2 n. Notes 1. Observe that a dierence scheme can be stable and still not be con- sistent. Stability and accuracy are two entirely dierent concerns. 2. The Dahlquist relations are pX j=0jjk=−kpX j=0jjk−1: (158.4) If they hold for k=0;1;:::;p , then we have (compare with equation (158.1)) pX j=0jy(t−jh)=pX j=0jy0(t−jh)+O/parenleftbig hp+1
: 3. Finite dierence schemes can be looked up (see page 661 or Isaacson and Keller [4, Chapter 8, pages 364{43)]) or they can be constructed as needed (see Lapidus and Pinder [8, pages 153{162] or Ganzha et al.[1]). 4. When approximating a dierential equation on a bounded interval, the limith!0,n!1 ,nhxed, is of interest. If the local error of a discretization scheme (as determined by equation (158.2)) is O(hp+1), then the global error (the error at the end of the integration) will be O(hp). 5. Obrechko methods utilize derivatives of yin forming the nite dif- ference scheme. The k-step Obrechko method using the rst m derivatives of ymay be written kX j=0jyn+j=mX i=1hikX j=0ijy(i) n+j: See Lambert [7] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 158. Finite Dierence Methodology 673 6. Often, a dierential equation will have invariants that remain con- stant during the evolution of the dierential equation. For example, in a conservative system the energy should remain constant. A nu- merical scheme should be used that ensures that these invariantsremain constant. See symplectic methods (page 780) and Gear [2]. 7. State-of-the-art software packages for ordinary dierential equations do not use a single discretization scheme with a xed step size. Rather, they vary their order (i.e., they choose from a collection of discretization formulas) and they vary the step size. Ideally, theoptimal step size and order are determined at each step; this is an important aspect of the code’s eciency (see page 770). 8. To determine if a nite dierence scheme for a partial dierential equation is stable, see either the Courant consistency criterion (page 688) or the Von Neumann stability test (page 692). 9. There are other types of nite dierence approximations that are not in the form of equation (158.1). See, for example, the cosine method (see page 716), the predictor{corrector method (see page 759), or themethod of Runge{Kutta (see page 763). 10. There are many useful theorems in numerical analysis concerning methods for specic equations. For example; a method for u t=ux with non-negative coecients cannot have an accuracy of p>1. See Iserles and Strang [5]. 11. Energy propagation under dispersive partial dierential equations travels with the group velocity . Even if an equation is non-dispersive, any nite dierence approximation to it will be dispersive. Hence,study of the group velocity is an important part of the analysis of a nite dierence scheme. See Trefethen [10] for details. References [1]Ganzha, V. G., Mazurik, S. I., and Shapeev, V. P. Symbolic manipulations on a computer and their application to generation andinvestigation of dierence schemes. In EUROCAL ’85 ,B .B u c h b e r g e ra n d B. F. Caviness, Eds. Springer{Verlag, New York, 1985, pp. 335{347. [2]G e a r ,C .W . Maintaining solution invariants in the numerical solution of ODEs. SIAM J. Sci. Stat. Comput. 7 , 3 (July 1986), 734{743. [3]Godunov, S. K., and Ryabenkii, V. S. Dierence Schemes: An Introduction to the Underlying Theory . North{Holland Publishing Co., New York, 1987. Translates by E. M. Gelbard. [4]Isaacson, E., and Keller, H. B. Analysis of Numerical Methods .J o h n Wiley & Sons, New York, 1966. [5]Iserles, A., and Strang, G. The optimal accuracy of dierence schemes. Trans. Amer. Math. Soc. 277 , 2 (June 1983), 779{803. [6]Jackson, K. R. The convergence of integrand{approximation formulas for the numerical solution of IVPs for ODEs. SIAM J. Numer. Anal. 25 ,1 (February 1988), 163{188. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 674 IV.A Numerical Methods: Concepts [7]Lambert, J. D. Computational Methods in Ordinary Dierential Equations . Cambridge University Press, New York, 1973. [8]Lapidus, L., and Pinder, G. F. Numerical Solution of Partial Dierential Equations in Science and Engineering . John Wiley & Sons, New York, 1982. [9]Shampine, L. F. Implementation of implicit formulas for the solution of ODEs. SIAM J. Sci. Stat. Comput. 1 , 1 (March 1980), 103{118. [10]Trefethen, L. N. Group velocity in nite dierence schemes. SIAM Review 24, 2 (April 1982), 113{136. [11]Van Niekerk, F. D. Non-linear one step methods for initial value problems. Comp. & Maths. with Appls. 13 , 4 (1987), 367{371. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 159. Grid Generation 675 159. Grid Generation Applicable to Ordinary and partial dierential equations. Yields A grid on which a dierential equation may be numerically approxi- mated. Procedure When a dierential equation is going to be approximated numerically, the points at which the values of the dependent variable will be determinedmust be specied. This collection of points forms the grid,o rmesh . The most common computational grids are those in rectilinear coor- dinates or polar coordinates (see gure 159.1). These can be used when the domain of a problem naturally ts one of these geometries. For other domains, an appropriate computational grid must be determined. Thereare many ways in which to construct a grid for a specic equation on a specic domain. There are many considerations that go into choosing a grid for a specic problem. The grid should be easy to generate, and the algebraic equations used on the grid (usually nite dierences or nite elements) must be easy to generate. (On page 664 we have indicated how nite dierence approximations may be found on triangular grids.) For nite elementmethods, it is common to use triangulated grids or grids composed of simple objects like triangles and rectangles. See example 3 in the section on nite element methods (on page 739) for an example. Ideally, there should be many grid points where the solution (or its derivatives) are rapidly changing. Some grids naturally lend themselves togrid renement in certain regions; this can be useful in adaptive techniques. Example 1 For domains that can be described by combinations of simple geometric regions, a grid may be easy to nd. See gure 159.2 for a simple compu-tational grid for a domain that can be conveniently decomposed into a rectangle and a semicircle. In this gure we have also illustrated how the grid may be modied if it is found that the solution shows great variationin the upper left region of the domain. Example 2 There are many ways in which a grid may be found for a domain. Figure 159.3, taken from Rice [8], shows six dierent grids for a single irregularlyshaped domain. The rst three grids (A, B, C) show dierent possibilities: Grid A is a simple triangulation of the domain. Grid B is a uniform rectilinear grid on the domain. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 676 IV.A Numerical Methods: Concepts/#0F/#0F/#0F/#0F/#0F/#0F /#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /./. /. /. /. /. /. /. /. /./. /. /. /./. /. /./. /./. /./. /./. /././. /././. /././././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././././. /././. /././. /./././././. /./././././././././. /./././././././././././././././././././././././././././././././././././././././././. /././././././. /./././. /./. /././. /./. /./. /./. /. /./. /. /./. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /./. /. /./. /. /./. /./. /./. /././. /./././. /././././././././././././././././././././././././././././. /./././././. /./././././././././. /././././. /././. /././. /././. /././. /././. /././././. /././././././././././././././././././././././././././././././././././././././. /././. /././. /./. /./. /./. /./. /. /./. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /./. /. /./. /./. /./. /././. /./././././././././././././././././././././././././././././. /././././. /././. /././. /././. /././././. /././. /././././././././././././././././././././././././././././././. /./././. /./. /./. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /./. /./. /./././././././././././././././././././././././. /././. /././. /./././././././././././././. /././././././././././././. /././. /./. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /././././././././././././././. /./././././././././././././. /././. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./#0F /#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F/#0F /#0F /#0F /#0F /#0F/#0F /#0F /#0F /#0F /#0F/#0F/#0F /#0F/#0F/#0F /#0F /#0F /#0F/#0F /#0F /#0F /#0F/#0F /#0F /#0F /#0F/#0F /#0F /#0F /#0FFigure 159.1: Two common computational grids, for rectilinear coordinates and for polar coordinates./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /./. /. /. /. /. /./. /. /./. /. /./. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. 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/. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./#0F /#0F/#0F/#0F/#0F/#0F/#0F /#0F/#0F/#0F/#0F/#0F/#0F /#0F/#0F/#0F/#0F/#0F/#0F /#0F/#0F/#0F/#0F/#0F/#0F /#0F/#0F/#0F/#0F/#0F/#0F /#0F/#0F/#0F/#0F/#0F/#0F /#0F /#0F /#0F /#0F/#0F /#0F/#0F /#0F/#0F /#0F/#0F /#0F/#0F /#0F /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /././. /. /./. /. /./. /. /. /. /. /. /. /. /. /./. /. 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Figure 159.2: A domain, a possible grid on that domain, and a rened grid on that domain. Grid C is a uniform rectilinear mapping, logically mapped to the domain. The second three grids (D, E, F) indicate how the the rst three grids can adapt to some diculties near the right boundary. Notes 1. One of the greatest obstacles in generating numerical solution to fluid dynamics problems is the diculty in geometrically describing complex congurations with computational grids. 2. Conformal mappings (see page 441) are frequently used to construct computational grids. 3. The multigrid method (see page 752) uses a sequence of grids, of vary- ing coarseness, to approximate the solution of a dierential equation. 4. Robert Schneiders maintains a comprehensive web site on mesh gener- ation, see http://www-users.informatik.rwth-aachen.de/~roberts/ meshgeneration.html . This site includes Information on meshing research A directory of people working on mesh generation, Latest news on mesh generation A list of programs (both public domain and commercial, more than 100 are mentioned { for most a URL is listed) Information on conferences and short courses Literature on mesh generation Open positions CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 159. Grid Generation 677 Figure 159.3: Six dierent grids for a domain (from Rice, J. R. Parallel Methods for Partial Dierential Equations. In The Characteristics of Parallel Algorithms , L. H. Jamieson, D. B. Gannon, and R. J. Douglass, Eds. MIT Press, 1987.) Information on related topics (e.g., CFD, scientic computing, and computational geometry) References [1]Abrahamsson, L. Orthogonal grid generation for two dimensional ducts. J. Comput. Appl. Math. 34 (1991), 305{314. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 678 IV.A Numerical Methods: Concepts [2]Atlas, I., Manohar, R., and Stephenson, J. W. Adaptive mesh generation using quadratures. Congr. Numer. 62 (1988), 37{45. [3]Atlas, I., and Stephenson, J. W. A two-dimensional adaptive mesh generation method. J. Comput. Physics 94 (1991), 201{224. [4]Castillo, J. E. ,E d . Mathematical Aspects of Numerical Grid Generation . SIAM, Philadelphia, PA, 1991. [5]Eisman, P. R. Adaptive grid generation. Comp. Meth. Appl. Mech. Eng. 64(1987), 321{376. [6]Mitchell, W. F. A comparison of adaptive renement techniques for elliptic problems. ACM Trans. Math. Software 15 , 4 (Dec 1989), 326{347. [7]Pardhanani, A., and Carey, G. F. Optimization of computational grids. Num. Meth. Part. Di. Eqs. 4 , 2 (1988), 95{117. [8]Rice, J. R. Parallel methods for partial dierential equations. In The Characteristics of Parallel Algorithms , L. H. Jamieson, D. B. Gannon, and R. J. Douglass, Eds. MIT Press, Cambridge, MA, 1987, pp. 209{231. [9]Sparis, P. D. A method for generating boundary-orthogonal curvilinear coordinate systems using the biharmonic equation. J. Comput. Physics 61 (1985), 445{462. [10]Thompson, J. F. Special issue on numerical grid generation. Appl. Math. and Comp. 10{11 (1982). [11]Thompson, J. F., Warsi, Z. U. A., and Mastin, C. W. Numerical Grid Generation Foundations and Applications . North{Holland Publishing Co., New York, 1985. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 160. Richardson Extrapolation 679 160. Richardson Extrapolation Applicable to Approximation techniques for dierential equations. Yields A procedure for increasing the accuracy. Procedure Suppose that a grid with a characteristic spacing his used to nu- merically approximate the solution of a dierential equation. Then the approximation u(x;h) at the point xin the domain will satisfy u(x;h)=y(x)+Rm(x)hm+O(hm+1); (160.1) wherey(x) is the true solution to the dierential equation, mis the order of the method, and the other terms represent the error (see page 670). If the approximation scheme is kept the same, but the characteristic spacing of the grid is changed from htok,t h e n u(x;k)=y(x)+Rm(x)km+O(km+1): (160.2) Equations (160.1) and (160.2) can be combined to yield the approximation v(x;h;k): =kmu(x;h)−hmu(x;k) km−hm=y(x)+O(khm;hkm): Note that v(x;h;k) is one more order accurate than either u(x;h)o r u(x;k). This process may be iterated to increase the accuracy even more. In some cases, the order of the method, and hence min equation (160.1), will be unknown. The Richardson extrapolation method may still be used,by either estimating mnumerically, or by using the Shanks transformation. The Shanks transformation uses three successive terms of the form A n= A1+hnto estimate A1via A1=An+1An−1−A2 n An+1+An−1−2An: This transformation may also be iterated; see Bender and Orszag [1, page 369] for details. Example 1 Given the dierential equation dy dx=y; y (0) = 1; we might choose to approximate the solution by Euler’s method un+1;h=( 1+h)un;h;u 0;h=1; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 680 IV.A Numerical Methods: Concepts whereun;h’y(nh), and the step size satises h1. Observe that our notation explicitly shows the dependence of the approximation on the grid size. Doing a detailed analysis, we can determine that un;h=y(x)−x 2 h+O(h2); (160.3) wherex=nhand hence (here we choose k=h=2) u2n;h=2=y(x)−x 2h 2+O(h2): (160.4) Combining equation (160.3) and equation (160.4) results in wn;h:= 2un;h−u2n;h=2=y(x)+O(h2); which is a numerical approximation that is second order accurate. Because hwas reduced by a factor of 2 in going from equation (160.3) to equation (160.4),nhad to be increased by a factor of 2 to maintain the same physical location,x. Example 2 Suppose we have the dierential equation dy dx=ty t2+1;y(0) = 1: (160.5) The exact solution to equation (160.5) is y(t)=p 1+t2. Hence,y(1) =p 21:41421. Approximating equation (160.5) by use of Euler’s method with a step size of h, we can obtain an approximation to the solution at t=1 ,uhy(1). Ashdecreases, this approximation should becomes better. In table 160.1, we show the values of uhthat are obtained when the h’s are made successively smaller by a factor of 2. Even though the last value is not very close top 2, we can improve the accuracy by using transformations. The rst application of Richardson extrapolation is dened by (because Euler’s method is rst order accurate) uh;R:=2uh−u2h 2−1. The second application of Richardson extrapolation is dened by uh;RR:=4uh;R−u2h;R 4−1. The rst application of the Shanks transformation is dened by uh;S:=u2huh=2−u2 h u2h+uh=2−2uh: The second application then uses the numbers uh;Sin the same formula to obtainuh;SS. As expected, the transformed values are much closer to the true value of y(1). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 160. Richardson Extrapolation 681 huhuh;Ruh;RRuh;Suh;SS 0.200 1.45847 0.100 1.43792 1.41738 1.41198 0.050 1.42646 1.41499 1.41420 1.41376 1.41420 0.025 1.42043 1.41441 1.41421 1.41411 0.012 1.41735 1.41426 1.41421 Table 160.1: Numerical approximations to the solution of equation (160.5) (More accurate results are obtained by applying Richardson extrapolation and the Shanks transformation to this data.) Notes 1. In the example, the quantity R1(x) could be explicitly determined. However, to utilize this method, this value does not have to be known explicitly. 2. To numerically approximate the solution to y0=f(x;y), the modied midpoint method determines y(x+nh), giveny(x), by z0=y(x); z1=z0+hf0(x;z0); zm+1=zm−1+2hf0(x+mh;zm); form=1;2;:::;n−1; y(x+nh)’1 2[zn+zn−1+hf0(x+nh;zn)]; wherehis a small step size. This method is of second order but has an error that only involves even powers ofh. Hence, each Richardson extrapolation of this method increases the order by 2. See Press et al.[8, pages 83{86] for more details. 3. Richardson extrapolation is often referred to as deferred approach to the limit . 4. This method also works for non-uniform grids if every interval is subdivided. 5. Some functions are not well approximated by polynomials but are well approximated by rational functions (see the section on Pad e approximants, page 582). Instead of using a polynomial t for theerror term (as in equation (160.1)), a rational function approximation could be made|this is the basis of the Bulirsch{Stoer method. See Press et al. [8, pages 563{568] for more details. 6. See also Isaacson and Keller [5, pages 372{374]. References [1]Bender, C. M., and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers . McGraw{Hill Book Company, New York, 1978. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 682 IV.A Numerical Methods: Concepts [2]C a s h ,J .R . On the numerical integration of nonlinear two-point boundary value problems using iterated deferred corrections: II. The development and analysis of highly stable deferred correction. SIAM J. Numer. Anal. 25 ,4 (1988), 862{882. [3]Christiansen, E., and Petersen, H. G. Estimation of convergence orders in repeated Richardson extrapolation. BIT 29 (1989), 48{59. [4]Deuflhard, P. Recent progress in extrapolation methods for ordinary dierential equations. SIAM Review 27 , 4 (December 1985), 505{535. [5]Isaacson, E., and Keller, H. B. Analysis of Numerical Methods .J o h n Wiley & Sons, New York, 1966. [6]Lindberg, B. Compact deferred correction formulas. In Numerical Integration of Dierential Equations and Large Linear Systems ,J .H i n z e , Ed. Springer{Verlag, New York, 1982, pp. 220{233. [7]meier, R. F. On Richardson extrapolation for nite dierence methods on regular grids. Numer. Math. 55 (1989), 451{462. [8]Press, W. H., Flannery, B. P., Teukolsky, S., and Vetterling, W. T. Numerical Recipes . Cambridge University Press, New York, 1986. [9]Richardson, L. F. The approximate arithmetical solution by nite dierences of physical problems involving dierential equations. Philos. Trans. Roy. Soc. London Ser. A 210 (1910), 307{357. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 161. Stability: ODE Approximations 683 161. Stability: ODE Approximations Applicable to Ordinary dierential equations. Yields It is straightforward to determine if a nite dierence scheme is stable. Idea If a nite dierence scheme is stable, then a locally good approximation yields a globally good approximation (provided the dierential equation is well posed). Procedure 1 Dierence schemes for ordinary dierential equations may be stable or unstable. The denition closely parallels the denition for the stability andwell-posedness of a dierential equation. A stable dierence scheme is one in which small changes in the initial and boundary data do not change the solution greatly. An unstable dierence scheme is one that shows great sensitivity to the initial and boundary data. To determine if the dierence scheme for an ordinary dierential equa- tion is stable (or zero-stable ), we apply the scheme to the equation y 0=0 (which has only a constant solution) and determine if the nite dierence approximation stays bounded. Suppose we have the following dierencescheme for the rst order equation y 0=f(x;y): pX j=0jvn+j−hpX j=0jf(xn+j;vn+j)=0; (161.1) wherevnis an approximation to y(xn)( a n dxn=nhforn=1;2;:::). Applying the above scheme to the test equation is equivalent to usingf(x;y) = 0 in equation (161.1). This results in pX j=0jvn−j=0: (161.2) The method is said to be stable if all solutions of equation (161.2) are uniformly bounded for all nand all initial data fv0;v1;:::;vp−1g. The dierence equation (161.2) has solutions of the form vn=n. Using vn=nin equation (161.2) results in the characteristic equation for  n()=pX j=0jn−j=0: (161.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 684 IV.A Numerical Methods: Concepts It is easily shown that the method is unstable if any of the roots to equation (161.3) have magnitudes greater than 1, or if there is a multiple root whose magnitude is equal to 1. Procedure 2 Sometimes \stability" is dened in terms of how the approximate so- lution to the equation y0=ybehaves. Using f(y;x)=yand then vn=n, we are led to the stability polynomial . The stability polynomial associated with equation (161.1) is dened to be (r;h)=()−h(), wherehrepresentshand(x)a n d(x) represent the rst and second characteristic polynomials (see page 671). Using the stability polynomial, we have the following denitions (see Lambert [10, pages 409{431]): The method in equation (161.1) is said to be absolutely stable f o rag i v e nhif, for thath, all the roots of (r;h)s a t i s f yjrsj<1 fors=1;2;:::;p ,a n dt ob e absolutely unstable otherwise. An interval (a;b) of the real line is said to be an interval of absolute stability if the method is absolutely stable for all h2(a;b). The method in equation (161.1) is said to be relatively stable for a givenhif, for thath, the roots of (r;h)s a t i s f yjrsj<jr1j fors=2;3;:::;p ,a n dt ob e relatively unstable otherwise. An interval (a;b) of the real line is said to be an interval of relative stability if the method is relatively stable for all h2(a;b). Using these denitions, we dene the method in equation (161.1) to be absolutely/relatively stable in a region Rof the complex plane if, for all h2R, the roots of the stability polynomial (r;h) have the required associated property (dened above). Using the notion of stability in a region, we dene the following types of stability: Am e t h o dh a s A-stability iffhj<(h)<0gR . Am e t h o dh a s A()-stability iffhj−<−arg(h)<gR . Am e t h o dh a s A0-stability iffhj=(h)=0;<(h)<0gR . A picture of the region Ris known as a stability diagram .W h e n approximating a dierential equation on a bounded interval, the limit n!1 ,hxed, is of interest. The stability diagram will indicate allowable values forh. Example 1 Euler’s method for the ordinary dierential equation y0=f(x;y)c o n - sists of the approximation: vn+1−vn=hf(xn;vn). To determine if this method is stable, we apply this method to the equation y0= 0 to determine the dierence scheme vn−vn−1=0: (161.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 161. Stability: ODE Approximations 685/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././. /./././. /././././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./././././././././././././././././././././#3C h /= h/. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /. /./. /. /./. /./. /./. /./. /./. /././. /./././. /./././././././. /./././././././././././././././././././././././././././././././././././././././././././././. /././././. /././././. /././. /././. /././. /././. /././. /././. /././././././././././././. /./././././././././././././././././././././././././././././././././././././././././././././. /./././././. /./././. /././. /./. /./. /./. /./. /./. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /./. /. /. /. /././. /. /././#0F/, /1 /././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././. /#0Fh /= h /#28 /2/+/3 i /#29 /. /. /. /. /. /. /./. /. /. /. /. /. /./. /./././././././././. /././././././././. /./././././././././. /././././././././././. /./././././././././././. /././././././././././././. /./././././././././././././. /././././././././././././. /./././././././././././././. /././././././././././././././. /./././././././././././././. /././././././././././././././. /./././././././././././././././. /././././././././././././././. /./././././././././././././. /././././././././././././././. /./././././././././././././. /././././././././././././. /./././././././././././././. /././././././././././././. /./././././././././././. /././././././././././. /./././././././././. /././././././././. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././. /./././. /././././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./././././././././././././././././././././#3C h /= h/./. /././. /./././././. /././././. /././. /././././././././././././././././././././././././././././././././././././././././././././././. /./././././././. /././. /././. /././. /./. /./. /./. /./. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /./. /. /./. /. /./. /./. /./. /./. /././. /././. /./././. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././. /././. /././././. /./././. /./././././././././././././././././././././././. /././././././././././././././././././././././././. /./././././././././././././././././././././././. /././././././././././././././././././././././././. /./././././././././././././././././././././././. /././././././././././././././././././././././././. /./././././././././././././././././././././././. /./././././././././././././././././././././././. /././././././././././././././././././././././././. /././././././././././././././././././././././././. /./././././././././././././././././././././././. /././././././././././././././././././././././././. /./././././././././././././././././././././././. /././././././././././././././././././././././././. /./././././././././././././././././././././././. /././././././././././././././././././././././././. /././././././././././././././././././././././. /./././././././././././././. /./././././././././. /././././././././. /././././././. /././././././. /./././././. /./././././. /././././. /././././. /././././. /././././. /././././. /././././. /././././. /././././. /./././. /././././. /././././. /././././. /././././. /././././. /././././. /./././././. /./././././. /./././././. /./././././. /././././././. /././././././. /././././././././. /./././././././././. /./././././././././././. /././././././././././././. /./././././././././. /././././././././. /././././././. /././././././. /./././././. /./././././. /././././. /././././. /././././. /././././. /././././. /././././. /././././. /././././. /./././. /././././. /././././. /././././. /././././. /././././. /././././. /./././././. /./././././. /./././././. /./././././. /././././././. /././././././. /././././././././. /./././././././././. /././././././././././.Figure 161.1: Stability diagrams for Euler’s method (left) and Euler’s backward method (right). Region of absolute stability is shown shaded. Usingvn=nin equation (161.4) results in the characteristic equation ()=n−n−1=0; which has the roots =1a n d= 0 (with multiplicity n−1). Because the only root with magnitude 1, = 1, has multiplicity 1, and all the other roots have magnitudes less than 1, Euler’s method is a stable method. Example 2 Applying Euler’s method to the equation y0=f(x;y)=y, we compute vn+1=vn+hfn =vn+hvn =( 1+h)vn: Hence, the region of absolute stability is given by R= hj1+h1/bracerightbig , see gure 161.1.a. Applying Euler’s backwards method to the equation y0=f(x;y)=y, we compute yn+1=yn+hfn+1 =yn+hyn+1 =yn 1−h: Hence, the region of absolute stability is given by R= h1 j1−hj1 , see gure 161.1.b. Stability diagrams can be used to determine allowable step sizes. If we were to integrate the ordinary dierential equation y0=( 2 + 3i)y using Euler’s method, then the maximum allowable (real) step size that CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 686 IV.A Numerical Methods: Concepts will produce an absolutely stable method is h=4 13; see gure 161.1.a. Stability diagrams are also used to qualitatively compare dierent dierence schemes. Notes 1. Observe that a dierence scheme can be stable and still not be con- sistent. Stability and accuracy are two entirely dierent concerns. 2. For a stability analysis of second order ordinary dierential equations, see Gear [6]. 3. Generally, the sequence of methods, fone step methods, iteration methods, implicit methods g, demonstrate progressively better stabil- ity. That is, it is generally true that larger step sizes can be taken for implicit methods than for explicit methods. 4. Karim and Ismail [8] present ve dierent ways in which to determine the stability of a dierence scheme. They all lead to the same con-clusion, but, on certain classes of equations, some methods are easier to apply than others. 5. To determine whether a nite dierence scheme for a partial dif- ferential equation is stable, see either the Courant{Friedrichs{Lewy consistency criterion (page 688) or the Von Neumann stability test (page 692). 6. There are are many useful theorems in numerical analysis concerning the stability of methods for specic equations. For example, an A-stable method cannot have accuracy p>2. See Dahlquist [3]. 7. A consistent method is called stiy stable if (1) for some constant D< 0, all solutions of the dierence equation generated by the application of this method to the scalar test equation, y 0=y,t e n d to zero asn!1 for all complex with Re<D and for all xed step sizeshwithh>0; and (2) there is an open set Swhose closure contains the origin and the method is stable for h2S. Here,h represents the grid spacing. 8. Mathematica has the package OrderStar which displays order stars for both absolute and relative stability. 9. There are many other types of stability that have been dened. A partial ordering of some common types of stability is given by the following list (see Butcher [2]): algebraic stability ) EuclideanAN-stability) strongAN-stability ) weakAN-stability)A-stability References [1]Burrage, K. (k,l)-algebraic stability of Runge{Kutta methods. IMA J. Num. Analysis 8 , 3 (1988), 385{400. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 161. Stability: ODE Approximations 687 [2]Butcher, J. C. Linear and non-linear stability for general linear methods. BIT 27 (1987), 182{189. [3]Dahlquist, G. A special stability problem for linear multistep methods. BIT 3 (1963), 27{43. [4]Dekker, K., and Verwer, J. G. Stability of Runge{Kutta Methods for Sti Nonlinear Systems . North{Holland Publishing Co., New York, 1984. [5]Ganzha, V. G., and Liska, R. Application of the REDUCE computer algebra system to stability analysis of dierence schemes. In Computers and Mathematics , E. Kaltofen and S. M. Watt, Eds. Springer{Verlag, New York, 1990, pp. 119{129. [6]Gear, C. W. The stability of numerical methods for second order ordinary dierential equations. SIAM J. Numer. Anal. 15 , 1 (February 1978), 188{ 197. [7]Iserles, A. Stability and dynamics of numerical methods for nonlinear ordinary dierential equations. IMA J. Num. Analysis 10 (1990), 1{30. [8]Karim, A. I. A., and Ismail, G. A. The stability of multi-step formulae for solving dierential equations. Int. J. Comp. Math. 13 (1983), 53{67. [9]Lambert, J. D. Computational Methods in Ordinary Dierential Equations . Cambridge University Press, New York, 1973. [10]Lambert, J. D. Developments in stability theory for ordinary dierential equations. In The State of the Art in Numerical Analysis , A. Iserles and M. J. D. Powell, Eds. Clarendon Press, Oxford, England, 1987. [11]Wanner, G. Order stars and stability. In The State of the Art in Numerical Analysis , A. Iserles and M. J. D. Powel, Eds. Clarendon Press, Oxford, England, 1987, pp. 451{471. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 688 IV.A Numerical Methods: Concepts 162. Stability: Courant Criterion Applicable to Hyperbolic partial dierential equations. Yields A statement about whether or not a dierence scheme may converge to the exact solution of a hyperbolic equation. Idea The \numerical domain of dependence" for a hyperbolic equation must include the actual domain of dependence in order for the numerical ap- proximation of the solution to converge to the true solution. Procedure A hyperbolic partial dierential equation has characteristics (see page 432). Generally, the dependent variables will satisfy ordinary dierential equations along the characteristics. These characteristics will propagate from the curves along which the initial data are given to every point inthe domain. Given a specic point at which the solution is desired, the characteristics through that point must be determined. If a numerical scheme for a hyperbolic equation attempts to compute a numerical approximation to the solution at a point, then all of the relevant characteristics must be present or the method may not converge to thecorrect solution. Example Suppose we have the wave equation utt=c2uxx; (162.1) foru(x;t), where the constant crepresents the wave speed. The initial conditions for equation (162.1) are assumed to be u(x;0) =f(x); ut(x;0) =g(x): We denevn;j=u(tn;xj), wheretn:=n
tandxj:=j
x. If a second order centered dierence scheme is used, then equation (162.1) might be approximated as un+1;j−2un;j+un−1;j (
t)2=c2un;j+1−2un;j+un;j−1 (
x)2; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 162. Stability: Courant Criterion 689/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./././. /././././././././././././././././././. /././././././././././././././././././././.x or j t or nC/, /: x /, ct /= const an tx /, /#01 x /#01 t t /= const an t /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././. /./././. /./././. /./././. /./././. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././. /./././././././././././././././././././././././././././././././././././././././././././././././. /./././././././././././. /./././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././././././././././././././././././././././. /././././././././././././././././././././././././. /./././././././././././././././././././././././. /././././././././././././././././././././././. /./././././././././././././././././././././. /./././././././././././././././././././././. /././././././././././././././././././././. /./././././././././././././././././././. /././././././././././././././././././. /./././././././././././././././././. /././././././././././././././././. /././././././././././././././././. /./././././././././././././././. /././././././././././././././. /./././././././././././././. /./././././././././././././. /././././././././././././. /./././././././././././. /././././././././././. /./././././././././. /././././././././. /././././././././. /./././././././. /././././././. /./././././. /./././././. /././././. /./././. /././. /./. /. /. Figure 162.1: Characteristics (indicated by dashed lines) that are included in the numerical domain of dependence (shown shaded). which can be manipulated into the explicit formula un+1;j=2" 1− c
t
x2# un;j+ c
t
x2 (un;j+1+un;j−1)−un−1;j: (162.2) Hence, the value of un+1;jdepends onfun;j+kjk=0;1gandun−1;j. Applying equation (162.2) to itself, we see that the value of un+1;jdepends onfun−1;j+kjk=0;1;2g. Applying equation (162.2) again, we see that the value of un+1;jdepends onfun−2;j+kjk=0;1;2;3g. In general, the value of un+1;jwill depend on the points fu0;j+kjk= 0;1;:::;ng. These points along the initial curve (where the initial data are given) describe the numerical domain of dependence . See gure 162.1. The characteristics of equation (162.1) are the two curves (shown dashed in the gures) C−:x−ct=xi; C+:x+ct=xi; wherexiis any point on the initial curve. Hence, the value of u(tn;xj) will depend on the values of u(0;xk)f o rxk=xi−ctandxk=xi+ct. If these values are not included in the numerical domain of dependence, then the numerical approximation will, generally, give the incorrect answer.This is simply because the numerical approximation does not use the data that are important in solving the problem. The two dierent possible scenarios are shown in gures 162.1 and 162.2. In gure 162.1, the characteristics are included in the numerical domain of dependence (i.e.,/parenleftbig
t
x
is less than 1). Because of this, the method may converge to the exact solution. In gure 162.2, the characteristics are not included in the numerical domain of dependence (i.e.,/parenleftbig
t
x
is greater CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 690 IV.A Numerical Methods: Concepts/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./././. /././././././././././././././././././. /././././././././././././././././././././.x or j t or nC/, /: x /, ct /= const an tx /, /#01 x /#01 t t /= const an t /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././. /./././. /./././. /./././. /./././. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././. /./././././././././././././././././././././././././././././././././././././././././././././././. /./././././././././././. /./././././././././././././././././././././././././././././././././././././. /. /./. /. /. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /. /./././. /././. /./././././././././././././././. /././. /././././. /././. /././././././././././. /././././. /././. /./././././././././././././././. /././. /././././. /././. /././././././././././. /././././. /././. /./././././././././././././././. /././. /././././. /././. /././././././././././. /././././. /././. /./././././././././././././././. /././. /././././. /././. /././././././././././././. /./. /./././././. /./././././. /./././././. /./././././. /././././. /././././. /././././. /././././. /././././. /./././. /./././. /./././. /./././. /./././. /././. /././. /././. /././. /././. /./. /./. /./. /./. /./. /. /. /. /. /. Figure 162.2: Characteristics (indicated by dashed lines) that are not included in the numerical domain of dependence (shown shaded). than 1). Because of this, the method cannot , in general, converge to the exact solution of equation (162.1). In summary, for this example, if
xand
tare chosen so that c
t
x>1, then the method cannot converge to the exact solution. c
t
x<1, then the method may converge to the exact solution. Notes 1. This condition is also known as the Courant{Friedrichs{Lewy or CFL condition. The theorem proved by Courant et al. [1] is: There are no explicit, unconditionally stable, consistant nite dierence schemes for hyperbolic systems of partial dierential equations. 2. Of course, more complicated hyperbolic problems will require a more detailed analysis. 3. Another test that can be used to determine the stability of a nite dif- ference scheme for partial dierential equations is the Von Neumannstability test (see page 692). 4. To determine if the dierence scheme for an ordinary dierential equation is stable, see page 670. 5. See also Davis [2, pages 45{47] and Isaacson and Keller [4, page 489] References [1]Courant, R., Friedrichs, K. O., and Lewy, H. Uber dir partiellen dierenzengleichungen der mathematischen physik. Mathematische Annalen 100(1928), 32{74. [2]Davis, J. L. Finite Dierence Methods in Dynamics of Continuous Media . The MacMillan Company, New York, 1986. [3]Gottlieb, D., and Tadmor, E. The CFL condition for spectral approxi- mations to hyperbolic initial-boundary value problems. Math. of Comp. 56 , 194 (April 1991), 565{588. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 162. Stability: Courant Criterion 691 [4]Isaacson, E., and Keller, H. B. Analysis of Numerical Methods .J o h n Wiley & Sons, New York, 1966. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 692 IV.A Numerical Methods: Concepts 163. Stability: Von Neumann Test Applicable to Finite dierence schemes for partial dierential equations. Yields Knowledge of whether the dierence scheme is stable. Procedure The Von Neumann test determines whether the dierence scheme for a partial dierential equation is stable. For dierence schemes with constant coecients, the test consists of examining all exponential solutions todetermine whether they grow exponentially in the time variable even when the initial values are bounded functions of the space variable. If any of them do increase without limit, then the method is unstable . Otherwise, it is stable . This test can also be applied to equations with variable coecients by introducing new, constant coecients equal to the frozen values of the original ones at some specic point of interest. Example If the parabolic equation ut=uxxis discretized via ut’1 kh u(x;t+k)−u(x;t)i ; uxx’1 h2h u(x+h;t)−2u(x;t)+u(x−h;t)i ; andvm;nis used to represent u(mh;nk ), then the recurrence relation um;n+1=um;n+k h2(um+1;n−2um;n+um−1;n) (163.1) is obtained. To investigate all possible bounded exponential type solutions, we choose um;n=eimein: (163.2) Substituting equation (163.2) into equation (163.1) results in the relation ei=1−4k h2sin2 2 ; (163.3) which must be satised for and. It can be shown that the imaginary part ofwill be non-negative (and hence the method is stable) if k h21 2: (163.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 163. Stability: Von Neumann Test 693 Notes 1. A stability test for hyperbolic partial dierential equations is the Courant{Friedrichs{Lewy consistency criterion (see page 688). 2. The Lax{Richtmyer equivalence theorem is the fundamental theorem in the theory of nite dierence schemes for initial value problems: A consistent nite dierence scheme for a partial dierential equation for which the initial value problem is well posed is convergent if and only if it is stable. 3. To determine whether the dierence scheme for an ordinary dieren- tial equation is stable, see page 670. 4. See also Davis [1, pages 47{50], Garabedian [2, page 469 and page 477], Gottlieb and Orszag [3, pages 48{50], Isaacson and Keller [4,pages 523{529], and Lapidus and Pinder [5, pages 170{179]. References [1]Davis, J. L. Finite Dierence Methods in Dynamics of Continuous Media . The MacMillan Company, New York, 1986. [2]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [3]Gottlieb, D., and Orszag, S. A. Numerical Analysis of Spectral Methods: Theory and Applications . SIAM, Philadelphia, PA, 1977. [4]Isaacson, E., and Keller, H. B. Analysis of Numerical Methods .J o h n Wiley & Sons, New York, 1966. [5]Lapidus, L., and Pinder, G. F. Numerical Solution of Partial Dierential Equations in Science and Engineering . John Wiley & Sons, New York, 1982. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 694 IV.A Numerical Methods: Concepts 164. Testing Dierential Equation Routines Applicable to Numerical approximations to dierential equations. Idea Many dierential equations have been used as examples to test dier- ential equation solvers. Procedure As new dierential equation integration techniques are developed, they are compared to existing techniques in terms of accuracy and eciency. Many specic dierential equation have been used as examples to indicate the performance of new algorithms and implementations. Tabulated below are some of those dierential equations. 1. A test case that is often used to test computer codes for boundary value problems is Troesch’s problem (see Roberts and Shipman [5]): d2y dt2−nsinhny=0; y(0) = 0;y(1) = 1: 2. Carroll [1] tests ODE system solvers with the equations (only some are listed below): (a)y0 1=−6y1+5y2+2s i nx,y0 2=9 4y1−95y2, withy1(0) =y2(0) = 0. (b)y0=−y−y2andy(0) = 1 for =f1000;800;0:001;−10g. (c)y0 1=−y2+( 1−y2 1−y2 2),y0 2=y1+( 1−y2 1−y2 2), withy1(0) = 1,y2(0) = 0. (d)y0 1=−y1,y0 2=y2 1−2y2,w i t hy1(0) =y2(0) = 5. 3. Marletta [2] tests Sturm{Liouville problem solvers with the equations: (a)−y00+/parenleftbig2 x2−1 x
y=yforx2(0;1). (b)−y00+/parenleftbig 9e−2x−18e−x
y=yforx2(−1;1). (c)−/parenleftbig (1−x2)y0
0=yforx2(−1;1). (Legendre’s equation) (d)−y00−/parenleftbig 4000e−1:7(x−1:3)−2000e−3:4(x−1:3)−2 x2
y=y forx2(0;1). (Morse potential) (e)−y00+/parenleftbig x2+x4
y=yforx2(−1;1). (f)−(xy0)0−/parenleftbig1 4sec2x
y=yforx2(−=2;=2). (g)−y00−y x=yforx2(0;1). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 164. Testing Dierential Equation Routines 695 (h)− y0 p 1−x20 =p 1−x2yforx2(−1;1). (i)−y00+/parenleftbig −2cos 2x+2sin22x
y=y, withy(=2) =y(−=2) = 0. (Coey{Evans equation) (j)−y00+y=w(x)ywherew(x)=( 0f o rx2 0;1 2 1f o rx21 2;1, y(0) =y(1) = 0. (k)−y00+xy=y 4. Shampine [6] tests sti ODE solvers with the system y0 1=−0:04y1+1 04y2y3; y1(0) = 1 y0 2=0:04y1−104y2y3−3
107y2 2;y2(0) = 0 y0 3=3
107y2 2; y3(0) = 0 5. Rice and Boisvert [4] have established a population of elliptic PDEs for testing purposes. It is divided into two groups, based on thedomain geometry. There are 56 PDE problems dened on rectangular regions, most of which depend on parameters that control features of the problem. The problems themselves have diering Operator type (Poisson, Helmholtz, self-adjoint, constant coecient, general) Boundary conditions (Dirichlet, Neumann, mixed) Solution features (entire, analytic, singular, peak, oscillatory, boundary layer, wave front, singularities, irregular, discontinuities, computationally complex) Some of these problems are: (a)u xx+uyy=1w i t hu= 0 on the unit square ( x=0;1a n d y=0;1). (b)uxx+uyy=6xyex+y(xy+x+y−3) withu= 0 on the unit square (x=0;1a n dy=0;1). (c)uxx+uyy x2+2ux x+uy x2tan3y=−100 withu=0o nx=0:1a n d x=1 ,a n du=0o ny=0:1a n dy=1 . (d) (exyux)x+(e−xyuy)y−u 1+x+y=fwithu= 0 on the unit square (x=0;1a n dy=0;1). For this problem f(x;y)i sc h o s e ns o that the exact solution is u=3exysinxsiny=4. (e)uxx+uyy+3uy 5−y=fwithu=0o nx=0:5a n dy=1. For this problem f(x;y) is chosen so that the exact solution has the formu=( 1−y2)(1−4x2)(5−y)3(a+by). (f)uxx+( 1+y2)uyy−ux−(1 +y2)uy=fwith CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 696 IV.A Numerical Methods: Concepts u+ux=0:27eyonx=0 , u−ux=0o nx=1 , u+uy=0:27exony=0 , u−uy=0:135(log 2−1)(x2−x)2ony=1 . For this problem f(x;y) is chosen so that the exact solution is u=0:135(ex+y+(x2−x)2log(1 +y2). References [1]Carroll, J. A composite integration scheme for the nmerical solution of systems of ordinary dierential equations. J. Comput. Appl. Math. 25 (1989), 1{13. [2]Marletta, M. Certication of algorithm 700. Numerical tests of the SLEIGN software for Sturm{Liouville problems. ACM Trans. Math. Software 17, 4 (December 1991), 481{490. [3]Penn, H. L. A review of dierential equations software. Collegiate Microcomputer 6 (1988), 33{42. [4]Rice, J. R., and Boisvert, R. F. Solving Elliptic Problems Using ELLPACK . Springer{Verlag, New York, 1985. [5]Roberts, S. M., and Shipman, J. S. Two-Point Boundary Value Problems: Shooting Methods . American Elsevier Publishing Company, New York, 1972. [6]Shampine, L. F. Ill-conditioned matrices and the integration of sti ODEs. J. Comput. Appl. Math. 48 (1993), 279{292. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 698 IV.B Numerical Methods for ODEs 165. Analytic Continuation Applicable to Initial value ordinary dierential equations, a single equation, or a system. Yields A numerical approximation in the form of a Taylor series. Idea If the Taylor series of a function is known at a single point, then the Taylor series of that function may be found at another (nearby) point. This process may be repeated until a particular value is reached. Procedure Given a system of initial value ordinary dierential equations, the method is to replace each dependent variable present by a Taylor series centered at a certain origin. The coecients in each Taylor series are re- garded as unknown quantities. The ordinary dierential equations are used to obtain a set of recurrence relations from which the unknown coecientsmay be calculated. Thus, a formal power series solution may be determined to an initial value problem, and the series will be convergent in some region about theorigin. Then, the truncated power series are evaluated at some point within the region of convergence. At this new point, initial values for the system are obtained from the already obtained Taylor series. Using these initialvalues, the recurrence relations then yield a second series solution valid in a region about the new origin. This procedure can be iterated and the solution at a given point may be determined via a sequence of Taylor series. This algorithm is a numerical version of the process of analytic continuation. Example Suppose we have the system of ordinary dierential equations y0=y2+z; y (0) = 1; z0=z2;z (0) = 1: This system can be rewritten as the dierential/algebraic system a=y2;b =a+z; c =z2; y0=b; z0=c;(165.1) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 165. Analytic Continuation699 withb=2a n da=c=y=z=1w h e nt= 0. If we dene the Taylor series coecients fa(j) k;b(j) k;c(j) k;y(j) k;z(j) kgby the expansions a(t)=1X k=0a(j) k(t−tj)k;b(t)=1X k=0b(j) k(t−tj)k; c(t)=1X k=0c(j) k(t−tj)k;y(t)=1X k=0y(j) k(t−tj)k; z(t)=1X k=0z(j) k(t−tj)k;(165.2) then, using equation (165.2) in equation (165.1), the following recurrence relations can be obtained a(j) k=kX n=0y(j) ny(j) k−n;b(j) k=a(j) k+z(j) k; c(j) k=kX n=0z(j) nz(j) k−n;y(j) k=b(j) k=(k+1 ); z(j) k=c(j) k=(k+1 ):(165.3) The initial conditions give the starting values: fj=0 ,t0=0 ,a(0) 0= c(0) 0=y(0) 0=z(0) 0=1 ,b(0) 0=2g. To determine the Taylor series about the pointt0= 0, equation (165.3) is iterated for k=1;2;:::;M . The number of terms in each Taylor series required for a specied numerical accuracyMmay be determined dynamically or xed beforehand (if an appropriate analysis has been done). Then a new point t 1is chosen. A Taylor series for each of a,b,c,y, andzis then found about this new point by taking j= 1 and determining the initial conditions from. a(1) 0=MX k=0a(0) k(t1−t0)k;b(1) 0=MX k=0b(0) k(t1−t0)k; :::: The recurrence relations in equation (165.3) are then iterated again. This process can be repeated indenitely. Notes 1. Holubec and Stauer [5] continue a Frobenius series instead of a Taylor series. This works particularly well on ordinary dierential equations with regular singular points. They also discuss the appro- priate step size to take at each stage in the calculation. 2. A Fortran computer program that generates the recurrence relations and then solves the system is described in Corliss and Chang [4]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 700 IV.B Numerical Methods for ODEs 3. Sometimes several hundred coecients are required to obtain an accurate answer with this method. This is especially true when the expansion point for the Taylor series is near a singularity. 4. Interval bounds (see page 545) for the Taylor series coecients are discussed in Moore [7, Chapter 11]. 5. This technique has been extended to parabolic equations in Chang [1]. References [1]Chang, Y. F. Solution of parabolic partial dierential equations. In Proceedings of the Sixth Manitoba Conference on Numerical Mathematics , B. L. Hartnell and H. C. Williams, Eds. Utilitas Mathematics Publishing,Winnipeg, Canada, 1977, pp. 127{134. [2]Chang, Y. F. Solving sti systems by Taylor series. Appl. Math. and Comp. 31(1989), 251{269. [3]Corliss, G., and Lowery, D. Choosing a stepsize for Taylor series methods for solving ODE’s. J. Comput. Appl. Math. 3 , 4 (1977), 251{256. [4]Corliss, G. F., and Chang, Y. F. Solving ordinary dierential equations using Taylor series. ACM Trans. Math. Software 8 (1982), 114{144. [5]Holubec, A., and Stauffer, A. D. Ecient solution of dierential equations by analytic continuation. J. Phys. A: Math. Gen. 18 (1985), 2141{ 2149. [6]Holubec, A., Stauffer, A. D., Acacia, P., and Stauffer, J. A. Asymptotic shooting method for the solution of dierential equations.J. Phys. A: Math. Gen. 23 (1990), 4081{4095. [7]Moore, R. E. Interval Analysis . Prentice{Hall, Inc., Englewood Clis, NJ, 1966. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 166. Boundary Value Problems: Box Method 701 166. Boundary Value Problems: Box Method Applicable to Boundary value problems for ordinary dierential equations. Yields A numerical approximation of the solution. Idea Using nite dierences, the solution to a boundary value problem is determined (simultaneously) everywhere on the interval of interest. Procedure We will illustrate the procedure on the general second order linear ordinary dierential equation. The same technique can be used, with only slight modications, to systems of higher order ordinary dierential equations, with the boundary data given virtually anywhere in the intervalof interest. Given the second order linear ordinary dierential equation a(x)y 00+b(x)y0+c(x)y=d(x); y(xL)=yL;y(xU)=yU;(166.1.a-b) we introduce the variable z(x)=y0(x) and write equation (166.1) as the system d dxy z =" z d−cy−bz a# : (166.2) Now, we choose a grid, not necessarily uniform, on the interval ( xL;xU), sayxL=x1<x2<


<xN=xU. At each one of the grid points, some nite dierence scheme is chosen to approximate the equations in equation (166.2). The scheme used can vary from point to point. For instance, ifEuler’s method is used for every point, then y z k+1=y z k+(xk+1−xk)" z d−cy−bz a# k(166.3) to rst order, where yk=y(xk),zk=z(xk), and similarly for fak;bk;ck;dkg. From equation (166.1.b). the values y1=yLandyN=yUare known. To determine all of the fzkg, and the remaining fykg, all of the relations in equation (166.3) (i.e., for k=1;2;:::;N ) should be combined into one CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 702 IV.B Numerical Methods for ODEs large matrix equation. First, for ease of notation, dene hk=xk+1−xk, ek=dk=ak,fk=ck=akandgk=bk=ak. In these new variables, equation (166.3) may be written as yk+1=yk+hkzk; zk+1=zk+hk(ek−fkyk−gkzk):(166.4) Combining all of the equations in (166.4) results in 2 6666641h 1−100 0 ::: h1f1−1+h1g1 01 0 0 ::: 001 h2−10::: 00 h2f2−1+h2g201 ......3 7777752 6666666664y 1 z1 y2 z2 y3 ... zN3 7777777775=2 66666666640 h 1e1 0 h2e2 0 ... hNeN3 7777777775: To this matrix equation should be added two more rows, one corresponding toy 1=yLand one corresponding to yN=yU. With these two rows, there results an 2 N2Nmatrix equation. This equation can be solved to determine a numerical approximation to the solution at all of the gridpoints. Example The second order linear ordinary dierential equation y00+y=3; y(0) = 3;y 2 =2;(166.5) has the solution y=3−sinx. We use the box method to numerically approximate this solution. Writing equation (166.5) as a system results in d dxy z =z 3−y : (166.6) We choose a uniform grid: xn=(n−1)hforn=1;2;3;4w i t hh==6. Deningyn=y(xn)a n dzn=z(xn), and using Euler’s method, equation (166.6) may be approximated as yn+1=yn+hzn; zn+1=zn+h(3−yn):(166.7) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 166. Boundary Value Problems: Box Method 703 Combining all of the equations in equation (166.7) for n=1;2;3;4 results in 2 66666641h−1 0000 0 h−1 01000 0 00 1 h−10 00 00h−1 010 0 0 0001 h−10 0 000 h−1013 77777752 66666666664y 1 z1 y2 z2 y3 z3 y4 z43 77777777775=2 66666640 3h 0 3h 0 3h3 7777775: Then the following two rows are added, to incorporate the known values of y(0) andy(=2)  10000000 000000102 66666666664y 1 z1 y2 z2 y3 z3 y4 z43 77777777775= 3 2 : The Fortran program in program 166.1 numerically approximates the solution to the above equation. Note that this program uses a linear equation solver LSOLVE , whose source code is not listed. The output of the program is Here is the approximate solution: 3.000 -0.701 2.633 -0.701 2.266 -0.509 2.000 -0.124 Here is the exact solution 3.000 -1.000 2.500 -0.866 2.134 -0.500 2.000 0.000 The values for ynare only accurate to one decimal place in this example. Putting more points in the interval would decrease the error, as would using a higher order method in place of Euler’s method. Notes 1. In our example, if the two rows corresponding to the boundary terms were added to the matrix equation at the correct locations, the re- sulting matrix would be banded. 2. This technique is recommended for sti boundary value problems because many points can be added where the solution undergoes large changes and dierent discretization schemes may be used in dierent regions. 3. For nonlinear equations or nonlinear boundary conditions, this method can be used iteratively by linearizing the nonlinear terms at each step. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 704 IV.B Numerical Methods for ODEs DIMENSION ARRAY(8,18),SOLN(8),RHS(8),NROW(100) PI=3.1415926NPOINT=8H=PI/2.* 2./FLOAT(NPOINT-2)DO 10 J=1,NPOINTDO 10 K=1,NPOINT 10 ARRAY(J,K)=0.0 C Create the matrix ARRAY(1,1)=1.0 RHS(1 )=3.0 ARRAY(NPOINT,NPOINT-1)=1.0 RHS(NPOINT )=2.0 J=1 20 J=J+1 IF( J .GE. NPOINT ) GOTO 30 C Here is the Y-equation ARRAY(J,J-1)=1ARRAY(J,J )=HARRAY(J,J+1)=-1 RHS(J )=0 J=J+1 C Here is the Z-equation ARRAY(J,J-2)=H ARRAY(J,J-1)=-1 ARRAY(J,J+1)=1 RHS(J )=3.0*H GOTO 20 C Solve the matrix system30 CALL LSOLVE(NPOINT,ARRAY,SOLN,RHS,NROW,IFSING,NPOINT) WRITE(6,5) (SOLN(J),J=1,NPOINT) 5 FORMAT(’ Here is the approximate solution:’,/,8(1x,F8.3) ) C Compute the exact solution for comparison J=1DO 40 JJ=1,NPOINT/2SOLN(J )=3.0-SIN( H*FLOAT(JJ-1) )SOLN(J+1)= -COS( H*FLOAT(JJ-1) ) 40 J=J+2 WRITE(6,15) (SOLN(J),J=1,NPOINT) 15 FORMAT(’ Here is the exact solution:’,/,8(1x,F8.3) ) END Program 166.1: Fortran program for box method. 4. Other techniques for solving boundary value problems include collo- cation (see page 514), shooting (see page 706), and invariant imbed- ding (see page 747). 5. Ascher et al. [1], Daniel [2], and Mattheij [5] all have discussions of dierent techniques that can be applied to boundary value problems. 6. See also Isaacson and Keller [4, pages 427{432] and Roberts and Shipman [6, Chapter 8, pages 201{231]. References [1]Ascher, U. M., Mattheij, R. M. M., and Russel, R. D. Numerical CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 166. Boundary Value Problems: Box Method 705 Solution of Boundary Value Problems for Ordinary Dierential Equations . Prentice{Hall, Inc., Englewood Clis, NJ, 1988. [2]Daniel, J. W. A road map of methods for approximating solutions of two-point boundary-value problems. In Codes for Boundary-Value Problems in Ordinary Dierential Equations , B. Childs, M. Scott, J. W. Daniel, E. Denman, and P. Nelson, Eds. Springer{Verlag, New York, 1979, pp. 1{ 18. [3]Gregory, J., and Zeman, M. Spline matrices and their applications to some higher order methods for boundary value problems. SIAM J. Numer. Anal. 25 , 2 (April 1988), 399{410. [4]Isaacson, E., and Keller, H. B. Analysis of Numerical Methods .J o h n Wiley & Sons, New York, 1966. [5]Mattheij, R. M. M. Decoupling and stability of algorithms for boundary value problems. SIAM Review 27 , 1 (March 1985), 1{44. [6]Roberts, S. M., and Shipman, J. S. Two-Point Boundary Value Problems: Shooting Methods . American Elsevier Publishing Company, New York, 1972. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 706 IV.B Numerical Methods for ODEs 167. Boundary Value Problems: Shooting Method Applicable to Nonlinear boundary value problems for ordinary dierential equations. Yields A numerical approximation to the solution. Idea Using Newton’s method, the correct initial conditions for a boundary value problem can be determined. Knowing the initial conditions, thedierential equations can be numerically integrated in a straightforward manner. Procedure The general procedure can be illustrated by studying a second order or- dinary dierential equation. Suppose we wish to numerically approximatethe solution y(x) of the equation L[y 00;y0;y;x]=0; y(0) = 0;y(1) =A;(167.1) whereAis a given constant. The dierential equation L[
]=0m a yo rm a y not be a linear dierential equation. If z(x;) is dened to be the solution of L[z00;z0;z;x]=0; z(0;)=0;z0(0;)=;(167.2) theny(x) will be equal to z(x;) for one or more values of . The parameter in equation (167.2) must be determined so that z(1;)=A: Because equation (167.2) is an initial value problem, it is straightforward to integrate it numerically from x=0t ox= 1. See, for instance, Euler’s method (page 730). To use the shooting method, we integrateequation (167.2) numerically for some arbitrary initial guess for ,s a y 0. Ifz(1;0)=A,t h e ny(x)=z(x;0) and we are done. Ifz(1;0)6=A, then a new value of must be chosen, say 1.E q u a t i o n (167.2) is then integrated for this new value of . The process of choosing new values for is repeated until the value of z(1;) is suciently close CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 167. Boundary Value Problems: Shooting Method707 toA. If the new ’s are chosen well, then z(1;) will converge to Aand a numerical approximation to equation (167.1) will have been obtained. One way to choose the sequence of ’s is by Newton’s method n+1=n−z(1;n)−A @ @z(1;) =n: (167.3) A numerical way to implement equation (167.3) might be n+1=n−z(1;n)−A [z(1;n+)−z(1;n)]=; whereis a small number. Example Suppose we have the nonlinear second order ordinary dierential equa- tion y00+2 (y0)2=0; y(0) = 1;y(1) =1 2:(167.4) Because equation (167.4) has no explicit dependence on y, the \dependent variable missing" method (see page 260) can be used to solve this equation exactly. By this technique, the solution of equation (167.4) is found to be y(x)=1+1 2log 1+1−e ex : Hence,y0(0) = (1−e)=2e’−0:31606. By use of the shooting method, a computer program should \discover" thaty0(0)’−0:31607 . The Fortran program in program 167.1 utilizes nite dierences to determine y0(0) for equation (167.4). The equation in equation (167.4) is turned into the two rst order ordinary dierential equations dy dx=z; dz dx=−2y2; and then integrated by the use of Euler’s method (see page 730). An initial guess of y0(0) = 0 is used in the program. The successive approximations of y0(0) appear below: Iteration number 0 value of Y’(0)= 0. Iteration number 1 value of Y’(0)= -0.50000050 Iteration number 2 value of Y’(0)= -0.49857452Iteration number 3 value of Y’(0)= -0.49102421 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 708 IV.B Numerical Methods for ODEs Y0=1.D0 Y1=0.5D0YP0=0.D0 C Perform a Newton iteration 9 times DO 10 NEWT=1,10WRITE(6,5) NEWT-1,YP0 5 FORMAT(’ Iteration number’,I4,’ value of Y’’(0)=’,F13.8)10 YP0=FNEWTON(Y0,Y1,YP0) END C This function performs one Newton step FUNCTION FNEWTON(Y0,Y1,YP0) EPS=0.000001D0 YP01=YP0YP02=YP0+EPSZ1=YAT1(Y0,YP01)Z2=YAT1(Y0,YP02)FNEWTON=YP0-(Z1-Y1)*EPS/(Z2-Z1)RETURN END C This function determines Y(1); when Y(0) and Y’(0) are given FUNCTION YAT1(Y0,YP0)N=20000DX=1.D0/DFLOAT(N)Y=Y0YP=YP0 C This is the actual integration loop DO 10 J=1,N Y=Y +D X*Y P 10 YP= YP + DX * ( -2.D0*YP**2 ) YAT1=YRETURNEND Program 167.1: Fortran program for shooting method Iteration number 4 value of Y’(0)= -0.46366318 Iteration number 5 value of Y’(0)= -0.40465858Iteration number 6 value of Y’(0)= -0.34199798Iteration number 7 value of Y’(0)= -0.31799014Iteration number 8 value of Y’(0)= -0.31608113Iteration number 9 value of Y’(0)= -0.31607109 Note that the computer program required a large number of steps in the interval [0 ;1] in order to achieve the accuracy shown (partly because we used Euler’s method, which is of low order). Notes 1. If this method is applied to a linear equation, the value of y0(0) will converge to the correct value in a single step. 2. It is also possible to simultaneous integrate along several rays at once. This is called the method of multiple shooting . See Diekho et al. [1] or Stoer and Bulirsch [6] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 167. Boundary Value Problems: Shooting Method709 References [1]Diekhoff, H.-J., Lory, P., Oberle, H. J., Pesch, H.-J., Rentrop, P., and Seydel, R. Comparing routines for the numerical solution of initial value problems in ordinary dierential equations in multiple shooting. Numer. Math. 27 (1977), 449{469. [2]Keller, H. B., and Nelson, Jr., P. Hypercube implementations of parallel shooting. Appl. Math. and Comp. 31 (1989), 574{603. [3]Lemmert, R. The shooting method for some nonlinear Sturm{Liouville boundary value problems. Z. Angew. Math. Phys. 40 , 5 (1989), 769{773. [4]Marzulli, P., and Gheri, G. Estimation of the global discretization error in shooting methods for linear boundary value problems. J. Comput. Appl. Math. 28 (1989), 309{314. [5]Roberts, S. M., and Shipman, J. S. On the closed form solution of Troesch’s problem. J. Comput. Physics 21 (1976), 291{304. [6]Stoer, J., and Bulirsch, R. Introduction to Numerical Analysis . Springer{ Verlag, New York, 1976. Translated by R. Bartels, W. Gautschi and C. Witzgall. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 710 IV.B Numerical Methods for ODEs 168. Continuation Method Applicable to Any type of equation: algebraic or dierential, a single equation, or a system. Yields A numerical approximation to the solution. Idea We embed a given problem into a problem with a continuation pa- rameterin it. For one value of (say= 1), we obtain the original equations; whereas for a dierent value of (say=0 )w eh a v ea n \easier" problem. We solve the simpler problem numerically and then slowly vary the continuation parameter from 0 to 1, obtaining a solution at each intermediate value. Procedure After setting up the problem as described above, we dene a metric that tells how well a function satises the problem when the continuation parameter is between 0 and 1. First, we numerically solve the easier problem (at = 0). Then, the continuation parameter is increased by a small amount, and a solution is found by using Newton’s method (thisis accomplished by making the metric as small as possible). We increase  some more and repeat this step until we have arrived at =1 . Example Suppose we wish to solve the following boundary value problem for y=y(x), yxx+ey=0;y (0) = 1;y(=2) = 0: (168.1) We embed equation (168.1) into the problem for v=v(x;), vxx+( 1−)v+ev=0;v (0;)=1;v(=2;)=0: (168.2) Note that when =1 ,w eh a v e v(x;1 ) =y(x) and that, when =0 , the problem for v(x; 0) becomes v(x;0 )xx+v(x;0 )=0;v (0; 0) = 1;v(=2; 0) = 1; with the solution v(x;0 )=c o sx. The technique is to solve (168.2) numerically on a grid of values from 0t o=2. We start with =0a n dv(x;0 ) =c o sxand then increase by a small amount and allow v(x;) to change accordingly. We choose to solve equation (168.1) at the N+1 grid points:fxn=hnj n=0;1;2;:::;Ng,w h e r eh==2N, and we dene v nto be the numerical CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 168. Continuation Method711 approximation to v(x;)a tt h enth gridpoint. We take v 0=1a n dv N=0 so that the boundary conditions in equation (168.2) are always satised. Now, we must dene the metric. We choose  n=v n+1−2v n+v n−1 h2+( 1−)v n+ev n: (168.3) We choose this metric because, when  nis close to zero, equation (168.2) will be approximately satised. This metric was obtained by simply apply-ing a centered second order dierence formula to equation (168.2). The procedure is now as follows (with  0=0 ,k=0 ) : 1. Increase by a small amount (i.e.,k+1=k+). 2. Find thefv ngthat make kn’0. This is best accomplished by Newton’s method. That is, we keep iterating 2 6664vk 2 vk 3... vk N−13 7775 m+1=2 6664vk 2 vk 3... vk N−13 7775 m−J−12 6664k 2 k 3... k N−13 7775 m; whereJis the Jacobian matrix dened by J=@(k 2;k 3;:::;k N−1) @(vk 2;vk 3;:::;vk N−1), until the \dierence" between2 6664vk 2 vk 3 ... vk N−13 7775 m+1and2 6664vk 2 vk 3 ... vk N−13 7775 mis smaller than some predened constant (based on the machine’s numerical capabilities). (a) Note that the Jacobian and the f ngall depend on the values offvkngm. (b) At each stage, when is increased, the values of fvkng0will be given by the last values of vk−1n/bracerightbig . (c) Ifis small enough, then Newton’s method should converge. 3. Ifk6= 1, go back to the rst step. 4. Ifk= 1, then we have found a numerical approximation to the solution of equation (168.1). Notes 1. There are computer codes available that can perform all of the above steps. The only input needed for them is the denition of the f ng. For example, Rheinboldt [3] has the Fortran listing for a continuation package. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 712 IV.B Numerical Methods for ODEs 2. Continuation methods can be used to track dierent solution branches of a problem with bifurcations. If the Jacobian ever becomes singular (i.e., detJ= 0), a bifurcation point is likely. The null space of the Jacobian will indicate which directions are possible for the dierentsolution branches. 3. It is not uncommon in practice to nd that the iteration in equation (168.3) will not converge unless isvery small (at least initially). The better continuation programs available will automatically deter- mine, making it as small as is needed but also increasing it when possible to speed up the calculation. 4. The method of invariant embedding (see page 747) is a specic type of continuation method. 5. Continuation methods are also known as homotopy methods , References [1]Allgower, R. L., and Georg, E. L. Numerical Continuation Methods . Springer{Verlag, New York, 1990. [2]Bolstad, J. H., and Keller, H. B. A multigrid continuation method for elliptic problems with folds. SIAM J. Sci. Stat. Comput. 7 , 4 (October 1986), 1081{1104. [3]Rheinboldt, W. C. Numerical Analysis of Parametrized Nonlinear Equa- tions , vol. 7. John Wiley & Sons, New York, 1986. [4]Rheinboldt, W. C., and Burkardt, J. V. A locally parametrized continuation process. ACM Trans. Math. Software 9 , 2 (June 1983), 215{ 235. [5]Watson, L. T. Numerical linear algebra aspects of globally convergent homotopy methods. SIAM Review 28 , 4 (December 1986), 529{545. [6]Watson, L. T., Billups, S. C., and Morgan, A. P. Algorithm 652: HOMPACK: A suite of codes for globally convergent homotopy algorithms. ACM Trans. Math. Software 13 , 3 (Sept 1987), 281{310. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 169. Continued Fractions 713 169. Continued Fractions Applicable to Linear second order ordinary dierential equations. Yields A solution in terms of a continued fraction. Idea By nding a simple recurrence pattern, we can express the logarithmic derivative of the solution to an ordinary dierential equation in terms of a continued fraction. Procedure Suppose we have a linear second order ordinary dierential equation in the form y=Q0(x)y0+P1(x)y00: (169.1) If equation (169.1) is dierentiated with respect to x, then we obtain y0=Q1(x)y00+P2(x)y000; (169.2) where Q1=Q0+P0 1 1−Q0 0;P 2=P1 1−Q0 0: (169.3) If equation (169.2) is dierentiated with respect to x, then we obtain y00= Q2(x)y000+P3(x)y0000,w h e r eQ2=Q1+P0 2 1−Q0 1,P3=P2 1−Q0 1. This process can be repeated indenitely to obtain y(n)=Qn(x)y(n+1)+Pn+1(x)y(n+2); (169.4) withQn=Qn−1+P0 n 1−Q0 n−1,Pn+1=Pn 1−Q0 n−1. Now, dividing equation (169.1) by y0produces y y0=Q0+P1y00 y0 =Q0+P1 y0=y00 =Q0+P1 Q1+P2y000 y00 =Q0+P1 Q1+P2 Q2+P3y0000 y000;(169.5) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 714 IV.B Numerical Methods for ODEs where we have used equation (169.3) for the third equality and equation (169.4) (with n= 3) for the fourth equality. We can extend the continued fraction in equation (169.5) indenitely. If it terminates, then it represents the reciprocal of the logarithmic derivative of the solution to equation (169.1). If it does not terminate, then it will converge if the following three conditions are satised: 1.Pn!P,Qn!Qasn!1 . 2. The rootsf1;2gof2=Q+Pare of unequal modulus. 3. Ifj2j<j1j, then lim n!1jy(n)j1=n<( j2j−1ifj2j6=0; 1 ifj2j=0: Example Suppose we wish to nd a continued fraction expansion for the recipro- cal of the logarithmic derivative of the equation xy00−xy0−y=0: (169.6) Comparing equation (169.6) with equation (169.1), we identify Q0(x)= −x,P1(x)=x. Using these values in equation (169.4), it is easy to show thatQn=1−x=(n+1 )a n dPn=x=n. Using these values, the partial sums for the continued fraction can be evaluated as F o r1t e r m :−x2+2 x: For 2 terms: −x3+5x x2+3: For 3 terms: −x4+9x2+8 x3+7x; For 4 terms: −x5+1 4x3+3 3x x4+1 2x2+1 5:(169.7) The information in equation (169.7) can be used to approximately evaluatey=y0. Notes 1. This technique has rarely been extended, with any generality, to any types of dierential equations other than linear second order ordinary dierential equations. There has been a generalization to \matrix continued fractions" in Risken [8, Chapter 9]. In Bellman and Wing[2, page 19], continued fractions are used to represent the solution to a Riccati equation. 2. By taking partial sums of the continued fraction in equation (169.5), successively better approximations may be found. Rarely, though, can convergence be checked. See Field’s paper [5]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 169. Continued Fractions 715 3. Continued fractions have been used recently to obtain high accuracy approximations to eigenvalues and functions of mathematical physics; see Barnett [1] or Gerck and d’Oliveira [6]. References [1]Barnett, A. R. High-precision evaluation of the regular and irregular Colounb wavefunctions. J. Comput. Appl. Math. 8 , 1 (1982), 29{33. [2]Bellman, R., and Wing, G. M. An Introduction to Invariant Imbedding . SIAM, Philadelphia, PA, 1992. [3]Ditto, W. L., and Pickett, T. J. Nonperturbative solutions of nonlinear dierential equations using continued fractions. J. Math. Physics 29 , 8 (1988), 1761{1770. [4]Ditto, W. L., and Pickett, T. J. Exact solution of nonlinear dierential equations using continued fractions. Nuovo Cimento B 105 , 4 (1990), 429{ 435. [5]Field, D. A. Estimates of the speed of convergence of continued fraction expansion of functions. Math. of Comp. 13 , 138 (April 1977), 495{502. [6]Gerck, E., and d’Oliveira, A. B. Continued fraction calculation of the eigenvalues of tridiagonal matrices arising from the Schroedinger equation. J. Comput. Appl. Math. 6 , 1 (1980), 81{82. [7]Lentz, W. J. Continued fraction calculation of spherical Bessel functions. Computers in Physics (Jul/Aug 1990), 403{407. [8]Risken, H. The Fokker{Planck Equation . Springer{Verlag, New York, 1984. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 716 IV.B Numerical Methods for ODEs 170. Cosine Method Applicable to Second order linear autonomous equations of a special form. Yields A nite dierence scheme from which a numerical approximation to the solution may be obtained. Idea An exact representation of the solution is found. This exact represen- tation is discretized to obtain an approximate numerical scheme. Procedure Suppose the following second order linear autonomous equation u00+Au=0; u(0) = u0; u0(0) = v0(170.1) is given for u(t), whereAis a positive denite symmetric matrix. The solution to equation (170.1) has the exact representation u(t+k)+u(t−k)=2c o s kA1=2 u(t); wherekrepresents a time step. Note that the cosine of a matrix is another matrix (see Moler and Van Loan [4] for how the exponential of a matrix may be computed). The approximation scheme for (170.1) is based on the use of a rational function,R(
)=P(
)=Q(
), to approximate the cosine term: cos kA1=2 ’R kA1=2 =Q−1 kA1=2 P kA1=2 : Once a rational function has been chosen (i.e., PandQhave been picked), we dene the approximation to u(tj)t ob e wj(wheretj=jk). The recurrence relation for wjis then given by Q kA1=2 (wj+1+wj−1)=2P kA1=2 wj or wj+1=2Q−1 kA1=2 P kA1=2 wj−wj−1: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 170. Cosine Method717 Using Taylor series (see page 632), the rst two values of wcan be found to start the iteration w0=u(0) = u0; w1=u(k)=u(0) +ku0(0) +k2 2!u00(0) +k3 3!u000(0) +::: =u(0) +ku0(0)−k2 2!Au(0)−k3 3!Au0(0) +::: =u0+kv0−k2 2!Au0−k3 3!Av0+k4 4!A2u0+:::;(170.2) where the dierential equation itself has been used to compute the higher order derivatives of u. The number of terms kept in this series should correspond to the accuracy of the rational approximation used for the cosine function. Example Suppose we have u00+ 21 12 u=0; u(0) = 1 −1 ; u0(0) = 2p 3 2p 3 :(170.3) HereA=[21 12] is symmetric and positive denite (its eigenvalues are f1;3g). The exact solution of the system in equation (170.3) can be found by converting it into the following rst order system u v0 =0I −A0u v ; u(0) v(0) =u0 v0 =2 6641 −1 2p 3 2p 33 775; whereIis the 22 identity matrix and v=u0. The solution of this new system (see page 421) is u(t) v(t) =2 664cost+2s i n (p 3t) −cost+2s i n (p 3t) −sint+2p 3c o s (p 3t) sint+2p 3c o s (p 3t)3 775: To use the cosine method, we need to approximate the cosine function. The (2,2) Pad e approximant (see page 582) to the cosine function is cos(z)’12−5z2 12 +z2; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 718 IV.B Numerical Methods for ODEs so that P kA1=2 =1 2I−5k2A; Q kA1=2 =1 2I+k2A: From this we obtain our discretization scheme wj+1=−wj−1+2 ( 1 2I+k2A)−1(12I−5k2A)wj (170.4) =−wj−1+−15k4−96k2+ 144−72k2 −72k2−15k4−96k2+ 144 wj; where=2=3(k2+4 ) (k2+ 12). The Fortran program in program 170.1 implements the above scheme withk=0:25. To evaluate w1, we utilized the rst ve terms in equation (170.2). We chose to compare the output from the numerical approximation scheme to the exact solution when tis a multiple of 5. Even for tas large as 30, the results are accurate to two decimal places. At time 5.00 W(J) = 1.6667 1.0993 EXACT= 1.6680 1.1007 At time 10.00 W(J) = -2.8364 -1.1584 EXACT= -2.8373 -1.1592 At time 15.00 W(J) = 0.7422 2.2617 EXACT= 0.7403 2.2596 At time 20.00 W(J) = 0.2361 -0.5798 EXACT= 0.2413 -0.5749 At time 25.00 W(J) = -0.2625 -2.2450 EXACT= -0.2680 -2.2504 At time 30.00 W(J) = 2.1371 1.8281 EXACT= 2.1386 1.8301 Notes 1. This method has been extended to apply to non-homogeneous prob- lems, equations with time-dependent coecients, and second order hyperbolic equations. 2. Since the iterates in equation (170.4) do not depend linearly on the step sizek, the cosine method is not a multi-step method as dened on page 670. References [1]Bales, L. A., and Douglas, V. A. Cosine methods for nonlinear second- order hyperbolic equations. Math. of Comp. 52 , 186 (1989), 299{319. [2]Bales, L. A., Douglas, V. A., and Serbin, S. M. Cosine methods for second-order hyperbolic equations with time-dependent coecients. Math. of Comp. 45 (July 1985), 65{89. [3]Coleman, J. P. Numerical methods for y"=f(x,y) via rational approxima- tions for the cosine. I M AJ .N u m .A n a l y s i s9 (1989), 145{165. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 170. Cosine Method719 IMPLICIT DOUBLE PRECISION (A-H,O-Z) REAL*8 W(0:1000,2),MAT(2,2),KK=0.25D0TIME=KSQRT3=DSQRT(3.D0) C Set up the initial conditions W(0,1)= 1.D0 W(0,2)=-1.D0 W(1,1)= 1 + K*2*SQRT3 - K**2/2.D0 - K**3*SQRT3 + K**4/24.D0W(1,2)=-1 + K*2*SQRT3 + K**2/2.D0 - K**3*SQRT3 - K**4/24.D0 C Set up the matrix for the recursion ALPHA = 2.D0/( 3.D0*(K**2+4)*(K**2+12) )MAT(1,1)= ALPHA * ( - 15*K**4 - 96*K**2 + 144)MAT(1,2)= ALPHA * ( - 72*K**2 )MAT(2,1)= MAT(1,2) MAT(2,2)= MAT(1,1) C Loop in time DO 10 J=2,120TIME=TIME+KW(J,1)= -W(J-2,1) + MAT(1,1)*W(J-1,1) + MAT(1,2)*W(J-1,2)W(J,2)= -W(J-2,2) + MAT(2,1)*W(J-1,1) + MAT(2,2)*W(J-1,2) C Compute the exact solution also IF( MOD(J,20) .NE. 0 ) GOTO 10 EXACT1= DCOS(TIME) + 2*DSIN(SQRT3*TIME) EXACT2= - DCOS(TIME) + 2*DSIN(SQRT3*TIME)WRITE(6,5) TIME,W(J,1),W(J,2),EXACT1,EXACT2 5 FORMAT(’At time’,F7.2,3x,’W(J) =’,2F9.4,/,18X,’EXACT=’,2F9.4)10 CONTINUE END Program 170.1: Fortran program for cosine method [4]Moler, C., and Van Loan, C. Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 20 , 4 (October 1978), 801{836. [5]Serbin, S. M. Some cosine schemes for second-order systems of ODE’s with time-varying coecients. SIAM J. Sci. Stat. Comput. 6 , 1 (1985), 61{68. [6]Serbin, S. M., and Fisher, A. L. A post-processor for the cosine method. SIAM J. Sci. Stat. Comput. 9 , 1 (January 1988), 14{23. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 720 IV.B Numerical Methods for ODEs 171. Dierential Algebraic Equations Applicable to Dierential algebraic equations, which are dieren- tial equations of the form F(x;y;y0)=0: (171.1) Often, F(
) is nonlinear in the y0term, or F(
) contains a collection of dif- ferential and algebraic equations. A special subcase of dierential algebraic equations is standard ordinary dierential equations, in the common form y0=f(x;y). Yields A numerical approximation to the solution. Idea Dierential algebraic equations are more dicult to solve than standard ordinary dierential equations. These equations are invariably solved ex- clusively by numerical means. One common numerical technique is to use the backward Euler method. That is, equation (171.1) is approximated by F xn+1;yn+1;yn+1−yn xn+1−xn =0; and then the resulting system of nonlinear equations is solved for y1,t h e n y2,e t c . Many special purpose codes have been written for these systems; see the references. There are, however, a few analytic solution techniques for dierential algebraic equations, as the examples show. Example 1 Algebraic dierential equations arise, for instance, in the analysis of mechanical systems. Each component in a mechanical system will have equations of motion, as well as physical constraints (depending on how thegiven component is attached to other components in the system). It is these physical constraints that become algebraic constraints. For example, consider a pendulum consisting of a point mass m, under the influence of gravity g, suspended by a massless rod of length lfrom an CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 171. Dierential Algebraic Equations 721 attachment point taken to be x=0 ,y= 0. The equations of motion are x0=vx; y0=vy; mv0 x=−x; mv0 y=−y−g; x2+y2=l2:(171.2) Here,(t) is the rod tension, and vx(t)a n dvy(t) are thexandyvelocities. Example 2 The dierential equation y=f(y0)=(y0)5+(y0)3+y0+5; (171.3) fory(x), is an example of a dierential algebraic equation. It is impossible for equation (171.3) to be analytically written in the form y0=g(x;y). However, it is possible to solve dierential equations of the form y= f(y0) parametrically. The solution may be written as y=f(t);x =Z t−1f0(t)dt+C; whereCis an arbitrary constant. Hence, equation (171.3) has the solution x=5 4t4+3 2t2+l o gt+C; y=t5+t3+t+5: Example 3 If a dierential algebraic equation is of the form x=f(y0), then the solution may be written parametrically as x=f(t);y =Z tf0(t)dt+C; whereCis an arbitrary constant. Thus, the equation x=(y0)3−y0−1 has the parametric solution x=t3−t−1; y=3 4t4−1 2t2+C: Example 4 If a dierential algebraic equation is of the form f(y0)=0a n dt h e r e exists at least one real root of f(k)=0 ,t h e n y=kx+Cis a solution (where Cis an arbitrary constant). Thus, the equation ( y0)5−6(y0)2−8=0h a s the solution y=2x+C. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 722 IV.B Numerical Methods for ODEs Notes 1. Ifyis a solution to an algebraic dierential equation, then yis called dierentially algebraic .I fuandvare dierentially algebraic functions, then so are u+v,uv,u=v,uv,u−1,du=dt ,a n dRt 0u(s)ds. Hence, all of the elementary functions (e.g., the rational functions,e x,t a n−1, Bessel functions) are dierentially algebraic. Note that the Gamma function (Γ( x)=R1 0tx−1e−tdt)i snota dierentially algebraic function. The Shannon{Pour-El{Lipshitz{Rubel theoremroughly states that the outputs of general purpose analog computers are dierentially algebraic functions. See Rubel [18]. 2. Dierential algebraic equations of the form u 0=f(u;v;t); 0=g(u;v;t); are said to be in semi-explicit form . 3. A class of algebraic dierential equations that are often studied are systems of the form Ey0=Ay+g(t); y(0) = y0;(171.4) whereAandEare given matrices. In the cases of interest, AorE(or both) are singular, but A−Eis not identically zero. For example, the system y0 2=y1+g(x); 0=y2+h(x); is an algebraic dierential equation in the form of equation (171.4). 4. Consider equation (171.4) when sE−Ais a regular matrix pencil (i.e., det(sE−A) is not identically zero). (If sE−Ais not a regular matrix pencil then equation (171.4) is not well posed.) In this case, non-singular matrices PandQcan be found (see Gantmacher [6]) so that, with y=Qz= z1z2Tandh(t)=Pg(t)= h1h2T, equation (171.4) takes the form z0 1+Cz1=h1(t); Nz0 2+z2=h2(t); whereNis a nilpotent matrix of degree n(i.e.,Nn=0a n dNn−16= 0). This is known as Kronecker canonical form. The degree ndenes theindex of the problem in equation (171.4). The index is equal to the size of the largest Jordan block for the eigenvalue zero (i.e., =0 )o f the matrixE−A. If the index is zero, then Eis non-singular and the system is easily solved numerically. Systems with an index greater CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 171. Dierential Algebraic Equations 723 than 1 are algebraically incomplete, which means that the existence and the uniqueness of the solutions are not guaranteed. For example, the equations in equation (171.2) are of index 3.As another example, the dierential algebraic equations (see Roche [15]) y0=f(y;z) 0=g(y;z) are of index 1 if ( @g=@z )−1exists and is bounded in the neighborhood of the exact solution. 5. In Gear and Petzold [8] is the following algorithm in which the in- dex of the problem in equation (171.4) can be reduced to zero by successive dierentiations: (a) IfEis non-singular, go to step (f). (b) Find non-singular matrices PandQsuch thatPEQ =E110T, withE11having full rank. (c) Make the variable substitution y=Qzand multiply the equa- tions from the left by Pgiving E11 0 z0=F11 F21 z+h1(t) h2(t) : (d) Dierentiate the lower part of the system to arrive at the new problem E11 F21 z0= F11 0 z+ h1(t) −h0 2(t) : (e) If the \ E" matrix for the new problem is singular, consider the new problem as the original problem and go to step (b). (f) Done. Note that the index of the original problem is equal to the number of times the above loop must be executed. 6. To indicate how much dierent the solution to algebraic dierential equations can be from standard ordinary dierential equations, con- sider the following amazing theorem in Rubel [16]: Given any continuous function on (−1;1)a n da n y positive continuous function (t)o n(−1;1), there exists aC1solution of the algebraic dierential equation 3y04y00y00002−4y04y0002y0000+6y03y002y000y0000 +2 4y02y004y0000−12y03y00y0003−29y02y003y0002+1 2y007=0; withjy(t)−(t)j<(t) for allt2(−1;1). Hence, anycontinuous function is a \valid" numerical approximation to a solution of the above equation! CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 724 IV.B Numerical Methods for ODEs 7. A Fortran program for approximating the solution to dierential algebraic equations of index 1, 2, and 3 is described in Hairer et al.[9]. This program is freely available via electronic mail. 8. See also Rheinboldt [14, Chapter 10, pages 183{202]. References [1]Ascher, U. M., and Spiter, R. J. Collocation software for boundary value dierential-algebraic equations. SIAM J. Sci. Comput. 15 , 4 (July 1994), 938{952. [2]Brenan, K. E., Campbell, S. L., and Petzold, L. R. Numerical Solution of Initial-Value Problems in Dierential-Algebraic Equations . No. 14 in Classics in Applied Mathematics. SIAM, Philadelphia, PA, 1995. [3]Brenan, K. E., and Petzold, L. R. The numerical solution of higher index dierential/algebraic equations by implicit methods. SIAM J. Numer. Anal. 26 , 4 (August 1989), 976{996. [4]Buck, R. C. The solutions to a smooth PDE can be dense in C[I]. J. Dierential Equations 41 (1981), 239{244. [5]Burrage, K., and Petzold, L. On order reduction for Runge{Kutta methods applied to dierential/algebraic systems and to sti systems of ODES. SIAM J. Numer. Anal. 27 , 2 (April 1990), 447{456. [6]Gantmacher, F. R. The Theory of Matrices , vol. I and II. Chelsea Publishing Company, New York, 1959. [7]Gear, C. W. Dierential algebraic equations, indices, and integral algebraic equations. SIAM J. Numer. Anal. 27 , 6 (December 1990), 1527{1534. [8]Gear, C. W., and Petzold, L. R. ODE methods for the solution of dierential/algebraic systems. SIAM J. Numer. Anal. 21 , 4 (August 1984), 716{728. [9]Hairer, E., Lubich, C., and Roche, M. The Numerical Solution of Dierential-Algebraic Systems by Runge{Kutta Methods . Springer{Verlag, New York, 1989. [10]Hairer, E., and Wanner, G. Solving Ordinary Dierential Equations, Volume II: Sti and Dierential-Algebraic Problems . Springer{Verlag, New York, 1991. [11]Hanke, M. On a least-squares collocation method for linear dierential- algebraic equations. Numer. Math. 54 (1988), 79{90. [12]Leimkuhler, B., Petzold, L. R., and Gear, C. W. Approximation methods for the consistent initialization of dierential algebraic equations.SIAM J. Numer. Anal. 28 , 1 (February 1991), 205{226. [13]Petzold, L., and Lotstedt, P. Numerical solution of nonlinear dierential equations with algebraic constraints. II. Practical implications. SIAM J. Sci. Stat. Comput. 7 , 3 (1986), 720{733. [14]Rheinboldt, W. C. Numerical Analysis of Parameterized Nonlinear Equations . Wiley Interscience, New York, 1986. [15]Roche, M. Rosenbrock methods for dierential algebraic equations. Numer. Math. 52 (1988), 45{6. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 171. Dierential Algebraic Equations 725 [16]Rubel, L. A. A universal dierential equation. B u l l .A m e r .M a t h .S o c .4 (May 1981), 345{349. [17]Rubel, L. A. Solutions of algebraic dierential equations. J. Dierential Equations 49 (1983), 441{452. [18]Rubel, L. A. Some mathematical limitations of the general-purpose analog computer. Advances in Appl. Math. 9 (1988), 22{34. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 726 IV.B Numerical Methods for ODEs 172. Eigenvalue/Eigenfunction Problems Applicable to Sturm{Liouville problems. Yields A numerical method for determining the eigenvalues and eigenfunctions of a regular Sturm{Liouville problem. Idea The Sturm{Liouville operator can be well approximated numerically by a simple discretization. This leads to a set of simultaneous equations, which can be represented as a matrix eigenvalue problem. The eigenval- ues and eigenvectors of this matrix will approximate the eigenvalues and eigenfunctions of the Sturm{Liouville problem. Procedure Suppose we wish to approximate numerically the eigenvalues and eigen- functions of the Sturm{Liouville system (see page 103) (p(x)y0)0+q(x)y=y; y(0) = 0;y(1) = 0;(172.1) forx2[0;1]. We will illustrate how the method of nite dierences can be used to approximate the eigenvalues and eigenvectors. Equation (172.1)can be approximated by D −/parenleftbig pn+1=2D+un
+qnun=hun; u0=0;uN=0;(172.2) whereh=1=N,un’y(nh),n=1;2;:::;N−1, and a function with a subscript of ncorresponds to an evaluation at x=hn. Also, the forward and backward dierencing operators are dened by D−fn:= (fn−fn−1)=h andD+fn:= (fn+1−fn)=h. It can be shown that (see Isaacson and Keller [9, pages 434{436] or Keller [10, Chapter 3, pages 39{48]) j−hjCh2; whereCis some (unknown) constant. Therefore, for a suciently small h, the collection of fhgthat satisfy equation (172.2) will closely approx- imate the collection of eigenvalues fg. The system in equation (172.2) is equivalent to the linear system of equations Auh=h2huh; (172.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 172. Eigenvalue/Eigenfunction Problems 727 where uh=u1::: uN−1TandAis the symmetric matrix 2 6666666664f 1p3=200


00 p3=2f2p5=2 0


00 0p5=2f3p7=2 00 ............... 00 p N−5=2fN−2pN−3=2 0 00


0pN−3=2fN−1pN−1=2 00


00 pN−1=2fN3 7777777775; (172.4) wheref m:=h2qm−(pm−1=2+pm+1=2). Hence, the eigenvalues of (172.4), scaled byh2(see equation (172.3)), will approximate the eigenvalues of (172.1). Note that uh, the eigenvector of (172.4) corresponding to h,i s an approximation to the eigenfunction in (172.1). The eigenvalues andeigenvectors of equation (172.4) can be computed by standard numerical techniques. As Nincreases, more eigenvalues and eigenvectors are found and the accuracy of the lower order eigenvalues (and their associated eigen-functions) increases. Example Consider the simple Sturm{Liouville system y00+y=y; y(0) = 0;y(1) = 0:(172.5) For this system, the eigenfunctions and eigenvalues are given by yn(x)=s i nnx; n=1−n22;(172.6) forn=1;2;:::. Hence, the two eigenvalues with the least magnitude are1=1−2’−8:86 and2=1−42’−38:47. To utilize the numerical technique presented above, we compare equation (172.5) with equation (172.1) to determine that p(x)=1a n dq(x)=1 . IfN=3( s ot h a t h=1=3), then the matrix in equation (172.4) is given by 2 4−17=91 0 1−17=91 01−17=93 5: (172.7) The eigenvalues of the matrix in equation (172.7) are approximately -1.9 and-0.49 . When scaled by h2, the estimates of the smallest eigenvalues of equation (172.5) become 1’-4.3 and2’-17.0 . ForN= 10, the estimates are 1’-7.1 and2’-30.7 , whereas for N= 50 the estimates are 1’-8.5 and2’-36.9 .A sNincreases, the estimates become better. If a higher order scheme were used to discretize(172.2), then smaller values of Nwould be required to obtain a given accuracy. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 728 IV.B Numerical Methods for ODEs Notes 1. Of course, Sturm{Liouville systems other than the one in equation (172.1) can be represented by a simple discretization such as in equa- tion (172.2). More complicated boundary conditions may lead to a non-symmetric matrix in equation (172.3). 2. Many other techniques have been used to approximate the eigenvalues and eigenfunctions of dierential systems. These methods include nite elements, Galerkin methods, invariant embedding, Pr¨ ufer sub- stitution, shooting, and variational methods. See page 635 of this book, Cope [7], or Keller [10]. 3. Many methods (such as the one illustrated here) approximate the kth eigenvalue of a regular Sturm{Liouville problem by the kth eigenvalue of a matrix problem of dimension n. Unfortunately, the accuracy deteriorates as kincreases, and there is no approximation at all for k>n . However, it is possible to obtain approximations for all kthat are uniformly accurate in k, see Shampine [17]. 4. For the eigenvalues fkgof the Sturm{Liouville problem −u00+qu= u,u(0) =u() = 0, when qhas mean zero, Marti [13] gives the boundsk−k2P1;mk−m+P2;mk−2mfork23jjqjjm,w h e r e jjqjjmis the norm of qin a Sobolev space and the P’s are homogeneous polynomials of degree at most 3 in jjqjjm. References [1]Andrew, A. L. Correction of nite element eigenvalues for problems with natural or periodic boundary conditions. BIT 28 , 2 (1988), 254{269. [2]Babuska, I., and Osborn, J. E. Estimates of the errors in eigenvalue and eigenvector approximation by Galerkin methods with particular attentionto the case of multiple eigenvalues. SIAM J. Numer. Anal. 24 , 6 (December 1987), 1249{1276. [3]Bailey, P. B., Garbow, B. S., Kaper, H. G., and Zettl, A. Algorithm 700: A FORTRAN software package for Sturm{Liouville problems. ACM Trans. Math. Software 17 , 4 (December 1991), 500{501. [4]Bailey, P. B., Garbow, B. S., Kaper, H. G., and Zettl, A. Eigenvalue and eigenfunction computations for Sturm{Liouville problems. ACM Trans. Math. Software 17 , 4 (December 1991), 491{499. [5]Bellman, R., and Wing, G. M. An Introduction to Invariant Imbedding . SIAM, Philadelphia, PA, 1992. [6]Berghe, G. V., and Meyer, H. D. Accurate computation of higher Sturm{Liouville eigenvalues. Numer. Math. 59 (1991), 243{254. [7]Cope, D. A uniformly convergent series for Sturm{Liouville eigenvalues. Quart. Appl. Math. 42 , 3 (October 1984), 373{380. [8]G a r t l a n d ,J r . ,E .C . Accurate approximation of eigenvalues and zeros of selected eigenfunctions of regular Sturm{Liouville problems. Math. of Comp. 42 , 166 (April 1984), 427{439. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 172. Eigenvalue/Eigenfunction Problems 729 [9]Isaacson, E., and Keller, H. B. Analysis of Numerical Methods .J o h n Wiley & Sons, New York, 1966. [10]Keller, H. B. Numerical Solutions of Two Point Boundary Value Problems . SIAM, Philadelphia, PA, 1976. [11]Leroy, J. P., and Wallace, R. Extension of the renormalized Numerov method for second-order dierential eigenvalue equations. J. Comput. Physics 67 (1986), 239{252. [12]Marletta, M. Certication of algorithm 700. Numerical tests of the SLEIGN software for Sturm{Liouville problems. ACM Trans. Math. Software 17 , 4 (December 1991), 481{490. [13]Marti, J. T. New upper and lower bounds for the eigenvalues of the Sturm{ Liouville problem. Computing 42 (1989), 239{243. [14]Mikhailov, M. D., and Vulchanov, N. L. Computational procedures for Sturm{Liouville problems. J. Comput. Physics 50 (1983), 323{336. [15]Pruess, S. On shooting algorithms for calculating Sturm{Liouville eigen- values. J. Comput. Physics 75 (1988), 493{497. [16]Pryce, J. D. Numerical Solution of Sturm{Liouville Problems .O x f o r d University Press, New York, 1993. [17]Shampine, L. F. Uniformly accurate Sturm{Liouville eigenvalues. Comput- ing 47 , 3{4 (1992), 379{385. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 730 IV.B Numerical Methods for ODEs 173. Euler’s Forward Method Applicable to Initial value systems of rst order ordinary dier- ential equations. Yields A numerical marching scheme that is rst order accurate. Idea A forward dierence approximation to a derivative can be easily manip- ulated into a numerical scheme. The technique in this section is the mostelementary nite dierence approximation|other techniques are found on page 670. Procedure Given the rst order system d dty(t)=f[t;y(t)]; y(t0)=y0;(173.1.a-b) where yandfare vectors, we numerically approximate dy=dtby [y(t+
t)− y(t)]=
t,w h e r e
tis a small step size . This numerical approximation is rst order accurate. Using this approximation, equation (173.1.a) can berewritten as y(t+
t)’y(t)+
tf[t;y(t)]: (173.2) Hence, to integrate equation (173.1), we iterate equation (173.2) and use the initial conditions from equation (173.1.b) for y(t 0)=y0; y(t0+
t)’y(t0)+
tf[t0;y0(t0)]; y(t0+2
t)’y(t0+
t)+
tf[t0+
t;y0(t0+
t)]; y(t0+3
t)’y(t0+2
t)+
tf[t0+2
t;y0(t0+2
t)]; ... Example Suppose we want to approximate the value of y(1) wheny(t) is dened by dy dt=ty t2+1;y (0) = 1: (173.3) Since this equation is separable, the exact solution is known to be y(t)=p 1+t2. We can use this exact solution to compare the accuracy of the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 173. Euler’s Forward Method 731 void main(void) { RungeKutta(); } void EulerForwardMethod(void) { int j, ndiv = 10;double t = 0.0;double tinit = 0.0;double y = 1.0;double tend = 1.0; double deltat, exact; deltat = (tend - tinit) / ((double)ndiv);/* This is the integration loop */for (j = 1; j <= ndiv; j++){ t = t + deltat;y = y + (deltat * Yprime(t, y));exact = sqrt(1 + (t*t)); printf("T= %6.3f Y= %8.5f Exact solution= %8.5f \n", t, y, exact); } }/* This function specifies the differential equation */double Yprime( double t, double y) { return ( (t*y) / (t*t + 1)); } Program 173.1: C program for Euler method. NDIV= 10 TINIT= 0.D0TEND= 1.D0DELTAT=(TEND-TINIT)/DFLOAT(NDIV)T= 0.0Y= 1.0 C This is the integration loop DO 10 J=1,NDIV T= T + DELTATY= Y + DELTAT * YPRIME(T,Y)EXACT=DSQRT(1+T**2)WRITE(6,5) T,Y,EXACT 5 FORMAT(’ T=’, F6.3,’ Y=’, F8.5,’ Exact solution=’,F8.5)10 CONTINUE END C This function specifies the differential equation FUNCTION YPRIME(T,Y)YPRIME= T*Y / (T**2+1)RETURNEND Program 173.2: Fortran program for Euler method. numerical approximation. The C (Fortran) code in program 173.1 (173.2) uses Euler’s forward method to numerically approximate the solution of equation (173.3). The codes use a step size of
t=0:1. The output from the programs is listed below, with the exact solution alongside forcomparison. The error in the calculated value for y(1) is about 1.7%. T= 0.100 Y= 1.00990 Exact solution= 1.00499 T= 0.200 Y= 1.02932 Exact solution= 1.01980T= 0.300 Y= 1.05765 Exact solution= 1.04403 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 732 IV.B Numerical Methods for ODEs T= 0.400 Y= 1.09412 Exact solution= 1.07703 T= 0.500 Y= 1.13789 Exact solution= 1.11803T= 0.600 Y= 1.18809 Exact solution= 1.16619T= 0.700 Y= 1.24390 Exact solution= 1.22066T= 0.800 Y= 1.30458 Exact solution= 1.28062 T= 0.900 Y= 1.36945 Exact solution= 1.34536 T= 1.000 Y= 1.43792 Exact solution= 1.41421 If the number of steps were increased (so the step size decreased), then the accuracy would improve. For example, if (in the above example)
twas reduced to 0.01 (i.e., NDIV=100 ), then the calculated value of y(1) would be 1.41672 . Hence, the error in the calculated value for y(1) would decrease to about 0.17%. Notes 1. This technique is the easiest to use and program of all the numerical methods presented in this book. A major drawback is that the stepsize
tmay have to be very small for accurate numerical values. 2. There is also a method known as Euler’s backward method . For this implicit method, the dierence scheme is given by y(t+
t)’y(t)+
tf[t;y(t+
t)]: (173.4) In general, equation (173.4) will be nonlinear in y(t+
t). Hence, an iterative scheme (e.g., Newton’s method) must be employed to nd y(t+
t)a te a c hs t e p . 3. The stability properties of Euler’s forward and backward methods are completely dierent. Consider applying each method to the scalar dierential equation y 0=−cy,y(0) =y0,w h e r ecis a positive constant. For Euler’s forward method, we have y(t+
t)’y(t)+
ty0(t); =y(t)−
t(cy(t)); =( 1−c
t)y(t); =y0(1−c
t)t=
t:(173.5) Whereas for Euler’s backward method, we nd y(t+
t)’y(t)+
ty0(t+
t); =y(t)−
t(cy(t+
t)); =y(t) 1+c
t; =y0 (1 +c
t)t=
t:(173.6) Note that the approximation in equation (173.5) diverges in an oscil- latory fashion when
t>2=c, whereas the approximation in equation CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 173. Euler’s Forward Method 733/0 /1 /2 /3 t /0 /1y exact so lu t ionforw ard Euler/: /#01 t /= /: /3forw ard Euler/: /#01 t /= /: /1bac kw ard Euler/: /#01 t /= /: /3 /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././././././././././. /././././. /././././. /./././././././. /./. /././././././. /./././././././. /./././. /./././././././. /././././. /./././././. /././././. /./. /././././. /./. /./. /./. /./././././. /./. /././. /././. /./. /././. /./././. /./. /././. /./. /./. /././. /./. /./././. /./. /././. /./. /./. /././. /./. /././. /./. /. /././. /./. /./. /./. /./. /./. /./. /. /./. /./././. /./. /. /./. /./. /. /./././. /. /./. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /./. /./. /. /. /./. /. /. /./. /./. /./. /. /. /./. /. /. /. /././. /. /./. /. /./. /. /./. /. /./. /./. /. /. /./. /./. /. /. /./. /. /./. /. /./. /. /. /./. /./. /. /. /. /. /./. /./. /. /. /./. /. /. /./. /./. /. /. /. /. /./. /. /. /./. /././. /. /. /. /./. /. /. /. /. /./. /. /./. /. /././. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /././. /. /. /././././././././././././././././././. /././././././././././././././././././. /././././././././././././././././././././. /././././././././././././././. /./././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././. /././././././././. /./././././. /./././. /./././././. /././. /./././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./././. /././. /./. /./. /././. /./. /././. /./. /./. /././. /./. /./. /././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././././././././././././././././././././././././././././././././././././././. /./. /././././././././././././. /./. /././. /././././././././././././. /./. /./././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /./. /./././. /././. /./. /././. /./././. /./. /././. /./././. /././. /./. /./././. /././. /././. /././. /./. /././. /. /././. /./. /././. /././. /. /././. /./. /././. /././. /. /././. /./. /././. /. /./. /./. /./. /./. /. /./. /. /././. /./. /. /./. /. /./. /./. /./. /./. /. /./. /./. /./. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /. /./. /. /./. /. /. /. /./. /. /. /./. /. /. /./. /. /. /. /./. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /. /. /./. /. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././././././././././././././././. /./././././././././. /././././././././. /././././././. /./././././. /./././././. /././././././. /././././././././. /./././././././. /././././././. /./././././. /././././. /././././. /./././././. /././. /././././. /./././. /./././. /./././. /./././. /././. /././. /./././. /././. /./././. /././. /./././. /./. /././. /././. /././. /./. /./././. /./. /././. /././. /./. /././. /././. /./. /./. /./././. /./. /./. /././. /./. /./. /./. /././. /./. /./. /././. /./. /./. /./. /././. /./. /./. /././. /./. /. /./. /././. /./. /./. /. /././. /./. /./. /./. /./. /./. /./. /. /././. /./. /./. /. /././. /./. /. /./. /././. /. /./. /. /./. /././. /. /./. /. /././. /./. /. /./. /. /././. /. /./. /. /././. /. /./. /./. /. /././. /. /./. /. /././. /. /. /./. /. /././. /. /./. /. /./. /./. /./. /. /. /./. /./. /./. /. /./. /. /./. /./. /. /./. /. /./. /./. /. /. /./. /./. /./. /. /. /./. /./. /. /./. /. /. /././. /. /. /./. /. /. /././. /. /. /./. /. /./. /./. /. /. /./. /./. /. /./. /. /. /. /././. /. /. /./. /. /./. /. /./. /. /. /././. /. /. /. /./. /. /./. /. /./. /. /. /. /././. /. /. /. /./. /./. /. /. /./. /. /. /./. /./. /. /. /. /././. /. /. /. /. /./. /./. /. /. /. /./. /. /./. /. /./. /. /. /. /./. /./. /. /. /. /. /././. /. /. /. /. /. /././. /. /. /. /. /././. /. /. /. /. /./. /./. /. /. /. /. /./. /./. /. /. /. /. /./. /./. /. /. /. /. /./. /./. /. /. /. /. /. /././. /. /. /. /. /. /././. /. /. /. /. /. /. /././. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /. /./. /. /. /./. /. /. /. /. /./. /./. /. /. /. /. /. /. /././. /. /. /. /. /. /./. /. /. /./. /. /. /./. /. /. /. /. /. /././. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /./. Figure 173.1: Dierent numerical techniques applied to y0=−6y+5e−t. (173.6) is stable for any value of
t. In particular, if c1( s ot h a t the problem is sti; see page 770), then
tmay have to be very small for Euler’s forward method to be stable, whereas a larger value of
t can be used with Euler’s backward method. 4. As an indication of the dierent convergence properties of Euler’s for- ward and backward methods, consider the equation: _ y=−6y+5e−t. Figure 173.1 shows the exact solution ( y=e−t) and approximations obtained by using Euler’s forward method (for
t=0:3a n d
t= 0:1) and Euler’s backward method (for
t=0:3). On this problem, Euler’s backward method is better than Euler’s forward method for a xed step size. 5. As always, ordinary dierential equations of higher order can be written as a system of rst order equations (see page 146). 6. See also Boyce and DiPrima [1, pages 399{406], Gear [2, pages 10{23], and Press et al. [3, pages 574{577]. References [1]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [2]Gear, C. W. Numerical Initial Value Problems in Ordinary Dierential Equations . Prentice{Hall, Inc., Englewood Clis, NJ, 1971. [3]Press, W. H., Flannery, B. P., Teukolsky, S., and Vetterling, W. T. Numerical Recipes . Cambridge University Press, New York, 1986. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 734 IV.B Numerical Methods for ODEs 174. Finite Element Method Applicable to Dierential equations that arise from variational principles. Principally ordinary dierential equations and elliptic partial dierential equations. Yields A numerical scheme for approximating the solution. Procedure The nite element method is one version of the method of weighted residuals (see page 786). The present method is characterized by having\local elements." The nite element method has a specialized vocabulary; several of the terms below are dened in the example. Given a dierential equation that comes from a variational principle and a domain in which the equation is to be solved the steps are as follows: Discretize the domain into simple shapes (these are the \nite ele- ments"). Dene a basis function  k(x) on each of the nite elements. These basis functions should have bounded support. Assemble the stiness matrix and the load matrix. These depend only on the nite elements chosen and not on the dierential equation to be approximated. Write the given dierential equation as a variational principle. Ap- proximate the unknown in the variational principle by a linear com- bination of the functions dened on the nite elements; that is,u(x)’u N(x): =PN k=1ckk(x). In this last expression, the fckg are unknown and must be determined. Construct element stiness matrices and load vectors element by element. Then assemble these together into the global stiness matrix Aand the global load vector f. Relate the minimization in the variational principle to the minimiza- tion of the quadratic functional I[uN]=cTAc−2cTf: (174.1) WhenAis symmetric (as it frequently is), the minimization of equa- tion (174.1) will occur when cis the solution of Ac=f. In general, A will not be banded or tridiagonal, but it will be sparse. If the original dierential equation was nonlinear, then A=A(c)o rf=f(c). There is a large literature on the nite element method. We choose to illustrate the basic ideas on simple examples: The rst two examples are CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 174. Finite Element Method735 constant coecient second order linear ordinary dierential equations, the third example is for Laplace’s equation. These examples show the major steps involved without the details that a sophisticated implementation requires. Example 1 Suppose we have the constant coecient second order linear ordinary dierential equation L[u]: =−d dx p(x)du dx +q(x)u=f(x) (174.2) on the interval 0 x1. For simplicity, we take p(x)a n dq(x)t ob e constants. For this equation, we take the natural boundary conditions u(0) =u(1) = 0: (174.3) If we useI[v] to represent the \energy" of the system, then we may form I[v]=Z1 0 p(v0(x))2+qv2(x)−2f(x)v(x) dx: (174.4) It is straightforward to show that the rst variation of I[v] (see page 418) yields equations (174.2) and (174.3). Hence, I[v] will be minimized when v=u. Now we set up a uniform grid of N+ 2 points on the interval 0 x1 (i.e.,xn=nhwithh=1=(N+1 )f o rn=0;1;:::;N + 1). We dene the interval (xk;xk+1) to be \nite element number k." We choose as basis functions on the nite elements the linear functions k(x) dened by k(x)=8 >>< >>:x−xk−1 h;forxk−1xxk; xk+1−x h;forxkxxk+1; 0; otherwise.(174.5) These are the \hat functions" shown in gure 174.1. Note that 0 k(x)=8 >>< >>:1 h;forxk−1xxk; −1 h;forxkxxk+1; 0;otherwise. Now we approximate the function that minimizes equation (174.4), u(x), by a linear combination of the k(x). We take u(x)’uN(x): =NX k=1ckk(x); (174.6) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 736 IV.B Numerical Methods for ODEs/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /././. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./././././././././././././././././././././. /././././././././././././././././././././.xk /, /1 xk xk /+/1 x /0 /1 /#1Ek/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. Figure 174.1: The \hat functions" in equation (174.5). where the unknowns fckgmust be determined. Once the fckgare known, then the approximation to u(x) at any point can be found from equation (174.6). That is, on nite element k(i.e., forxk<x<xk+1) uN(x)=ckk(x)+ck+1k+1(x); u0 N(x)=−ck+ck+1 h:(174.7) UsinguN(x)f o rv(x) in equation (174.4) results in I[uN]=NX k=0Zxk+1 xk p(u0 N)2+q(uN)2−2fuN dx; =NX k=0 Is k+Im k+Il k ;(174.8) where Is k:=Zxk+1 xkp(u0 N)2dx=ckck+1 Ks kck ck+1 ; Im k:=Zxk+1 xkq(uN)2dx=ckck+1 Km kck ck+1 ; Il k:=Zxk+1 xk2f(x)uN(x)dx by virtue of equation (174.7). Here Ks kis the element stiness matrix ,a n d Km kis the element mass matrix ; they are dened by Ks k=p h1−1 −11 Km k=qh 621 12 Ifpandqwere not taken to be constants, then these element matrices would not be so simple. A numerical integration would have been required to nd the entries in these matrices. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 174. Finite Element Method737 A numerical integration is required to determine Il k. If, on nite element numberk,f(x) is approximated by f(x)’fkk(x)+fk+1k+1(x), then we ndIl k=/parenleftbig fl k
Tck ck+1 ,w h e r et h e element load vector is dened by fl k=h 32fk+fk+1 fk+2fk+1 . The system can now be assembled element by element. That is, we write a single matrix equation representing equation (174.8). For this example, we nd that I[uN]=cT(K+M)c−2fTc; (174.9) where c=c1c2::: cNT,f=h 6[(f0+4f1+f2)(f1+4f2+f3)::: (fN−2+4fN−1+fN)]T,a n dt h e global stiness matrix Kand the global mass matrix Mare dened by K=p h2 66666666642−100


00 −12−10


00 0−12−10 0 ............... 00−12−10 00


0−12−1 00


00−123 7777777775; M=qh 62 666666666441 0 0


00 14 1 0


00 01 4 1 00 ............... 00 1 4 1000


01 4 1 00


00 1 43 7777777775: To minimize the expression in equation (174.9), cshould be chosen (becauseK+Mis a symmetric matrix in this example) to satisfy the matrix equation ( K+M)c=f. This is a tridiagonal system of equations and may be solved by standard numerical linear algebra routines. Example 2 This example shows more of the details for a specic application of the nite element method. Suppose that we wish to approximate the solutionof the ordinary dierential equation u 00−u0=ex/parenleftbig e−xu0
0=0; u(0) = 2;u (4) = 1 +e4;(174.10) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 738 IV.B Numerical Methods for ODEs whose exact solution is u(x)=1 +ex. From page 418, we see that the variational principle associated with equation (174.10) is just J=0 ,w h e r e J[v]=Z4 0e−x(v0)2dx: To use the nite element method on the problem in equation (174.10), we choose to use three elements: the intervals [0 ;1], [1;2], and [2;4]. We choose the polynomial basis functions on element [0 ;1];the basis function is f(x)=+x+γx2; on element [1 ;2];the basis function is g(x)=+x+x2; on element [2 ;4];the basis function is h(x)=+x+x2; (174.11) so that our approximation has the form v=8 >< >:f(x) on the interval [0 ;1] g(x) on the interval [1 ;2] h(x) on the interval [2 ;4] Afterf;;γ;;;;;; gare determined, we will have found an approx- imate solution. The equations needed to satisfy the boundary conditions and for our approximation and its rst derivative to be continuous on the interval [0;1] are boundary conditions: f(0) = 2;h (4) = 1 +e4; continuity conditions: f(1) =g(1);f0(1) =g0(1);(174.12) g(2) =h(2);g0(2) =h0(2): Subject to the above constraints, we want to minimize J[v]. Using our chosen set of nite elements and basis functions, we have J[v]=Z1 0e−x(f0)2dx+Z2 1e−x(g0)2dx+Z4 2e−x(h0)2dx =( 4e−8)γ2+4γ+( 8e2−4e)2+4e2+(e2−e)2 +4e2+( 8e2−4e)2+(e2−e)2+(e−1)2; To minimize this last expression, subject to the constraints in equation (174.12), we use Lagrange multipliers. The expression obtained after La- grange multipliers are introduced is dierentiated with respect to each ofthe variables to obtain a linear system of 15 equations (9 equations for the variables in equation (174.11) and 6 equations for the Lagrange multipliers). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 174. Finite Element Method739/0 /1 /2 /3 /4 x /2 /5 /1/0 /2/0 /5/0log uexact solution/#0Cnite elemen t appro ximation/. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./. /. /. /. /./. /. /././. /./. /./. /./. /. /./. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. Figure 174.2: Exact solution and nite element approximation to (174.10). This system can be solved to determine the basis function on each element: f(x)=−3:4508x2+5:3673x+2; g(x)= 4:1836x2−9:9014x+9:6343; h(x)= 8:8416x2−28:5337x+2 8:2666: Figure 174.2 has a comparison of the exact and approximate solutions. At points midway on the elements, we nd u(0:5) = 2:65;u(1:5) = 5:48;u(3) = 21:09; f(0:5) = 3:82;g(1:5) = 4:20;h(3) = 22:24: A more accurate approximation could have been obtained by increasing the degree of the basis functions or by increasing the number of elements. Example 3 Suppose that we want to approximate the solution to r2u= 0 in the rectangle 0 x2;0y1; u(x;0) =f(x);u (1;y)=j(y); u(x;1) =h(x);u (2;y)=g(y):(174.13) For this problem, we choose we use three nite elements; two of these elements (I and II) are triangles and one (III) is a square (see gure174.3). On the dierent elements, we choose to use the following polynomial functions to represent the solution: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 740 IV.B Numerical Methods for ODEs/0 /1 y/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./0 /2 x /. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./././././././././././././././././././././. /./././././././././././././././././././././#0F /#0F /#0F /#0F /#0F /#0F /#0F/#0F/#0F /#0Ff /#28 x /#29 g /#28 y /#29 h /#28 x /#29j /#28 y /#29P/1 P/2 P/3 P/4 P/5 P/6 P/7P/8P/9 P/1/0I II III/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /.Figure 174.3: Finite elements used in example 3. uI=a11+a12x+a13y+a14x2+a15y2+a16xy; uII=a21+a22x+a23y+a24x2+a25y2+a26xy+a27x3+a28y3; uIII=a31+a32x+a33y: Now we must specify how the parameters in these approximate solutions are to be determined. Using a subscript on f,g,a n dhto denote evaluation at a node on gure 174.3, we choose to approximately satisfy the equation and boundary conditions on the individual elements as follows: On element I: uI P4=g4;u I P5=g5; uI P6=h6;r2uI P5=0: On element II: uII P2=f2;uII P3=f3; uII P4=f4;uII P6=h6; r2uII P6=0: (174.14) On element III: uIII P1=f1;uIII P2=f2; uIII P6=h6;uIII P7=h7: To \connect" the elements, we choose the following conditions: @uI @n P8=@uII @n P8;u I P8=uII P8; @uII @n P10=@uIII @n P10;uII P10=uIII P10; (174.15) @uI @x P6=@uIII @x P6: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 174. Finite Element Method741 wherenstands for the normal. To actually carry out the solution technique, we choose the functions on the boundary to be ff(x)=x2,g(y)=4+y−y2,h(x)=x2,j(y)= y−y2g. For these functions, equation (174.13) has the exact solution u(x;y)=x2+y−y2. Solving the linear equations in equations (174.14) and (174.15), we obtain the approximate solution uI=−2y2+( 1 0−4x)y+2x2+x−6; uII=−8y3+2 3y2+( 8x−23)y+2 4x3−107x2+ 156x−72; uIII=x: Comparing this approximate solution to the exact solution, we determine the maximum errors (and their locations) to be Element Maximum error Location I 1 (1/2;3/2) II16 3p 3(1;1−1p 3) III 1/4 (0;1/2) Notes 1. Nearly every part of the nite element procedure that has been pre- sented in example 1 can be generalized. The basis functions do not have to be piecewise linear but could be piecewise quadratic, cubic, or higher order (they were chosento be quadratic in example 2). For physically two-dimensional structures, the \nite elements" can be triangles, quadrilaterals, or polygons with more sides(they can be tetrahedrons, cubes, or more complicated struc- tures for three-dimensional structures). However, the smooth- ness conditions across the boundaries may be dicult to formu- late. Even in one dimension, the \nite elements" do not have to represent intervals of equal length (as in example 2). 2. The approximation to the solution in equation (174.6) will only be C 0, because the basis functions chosen in equation (174.5) are piecewiselinear. The cubic Hermite approximation results in a C 1approxima- tion by choosing the following two basis functions per nite element: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 742 IV.B Numerical Methods for ODEsxk /, /1 xk xk /+/1 x /0 /1 y/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /. /./. /. /./././././././././././././././././././././. /./././././././././././././././././././././#11k/#18k/. /. /./. /. /./. /. /./. /. /././. /./. /. /././. /./. /././. /././. /. /././. /././. /././././. /././. /./././. /./././././. /./././././. /./././././. /././././././././. /././././././././././. /././././././././././././././. /././././././././././././././. /./././././././././. /././././././././. /./././././. /./././././. /././././. /./././. /././. /././././. /././. /././. /././. /. /././. /./. /././. /. /././. /./. /. /./. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /./. /. /././. /./. /././. /. /././. /././. /././. /././. /././. /././. /././././. /./././. /./././././. /./././././. /./././././././. /./././././././././. /./././././././././././././. /././././././././././././././. /./././././././././././. /././././././././. /././././././. /././././. /./././. /././././. /./././. /././././. /././. /././. /././. /././. /. /././. /./. /././. /. /./. /././. /. /./. /. /./. /. /././. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././. /./. /././. /././././././. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././. /./. /././. /. /./. /./.Figure 174.4: The functions for the cubic Hermite approximation. k(x)=8 >>< >>:1−3/parenleftbig x−xk h
2−2/parenleftbigx−xk h
3;forxk−1xxk; 1−3/parenleftbigx−xk h
2+2/parenleftbigx−xk h
3;forxkxxk+1; 0; otherwise, k(x)=8 >>< >>:(x−x k)/parenleftbig 1+x−xk h
2;forxk−1xxk; (x−xk)/parenleftbig 1−x−xk h
2;forxk−1xxk; 0; otherwise. These basis functions are continuous with their rst derivatives at the nodes (endpoints of the intervals); see gure 174.4. Using these functions, an approximation of the form u(x)’uN(x): =NX k=1dkk(x)+ekk(x) is supposed, where the constants fdk;ekgmust be determined. 3. In higher dimensions, smoother approximations are found analo- gously. Basis functions are chosen that are continuous (with several of their derivatives) at the nodes of the \nite elements." The nodescould be the vertices of a square (or cube), or some of the vertices and some points along the edges on the square (or cube). 4. Both Brebbia [3] and Mackerle and Fredriksson [7] have comprehen- sive listings of available software that numerically approximate the solutions of dierential equations by nite elements. 5. Incidentally, by integrating by parts and using the boundary con- ditions in equation (174.3), it can be shown that equation (174.4) is equivalent to I[v]=(v;L[v])−2(f;v), where (g;h): =R 1 0g(x)h(x)dx. 6. In some nite element programs, the discretization errors are con- trolled by letting the diameter of the largest element happroach zero. This is called the h-version of the nite element method .I nt h e CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 174. Finite Element Method743 p-version of the nite element method , the mesh is xed while the degree of the polynomials on the elements is increased (this is also called the global element method ). In thehp-version , both limits are considered simultaneously. See Babu ska [2] for details. 7. Mackerle [6] contains a very large annotated bibliography. 8. A comprehensive listing of nite element resources is maintained by Roger Young and Ian MacPhedran; see http://www.engr.usask.ca/ ~macphed/finite/fe_resources/fe_resources.html . 9. See also Strang [10, pages 428{445]. References [1]Allen, M. B., and Curran, M. C. Adaptive local grid renement algorithms for nite-element collocations. Num. Methods Part. Di. Eqns. 5(1989), 121{132. [2]Babuska, I. The p and h-p versions of the nite element method. The state of the art. In Finite Elements: Theory and Application ,D .L .D w o y e r , M. Y. Hussaini, and R. G. Voigt, Eds. Springer{Verlag, New York, 1988,pp. 199{239. [3]Brebbia, C. A. Finite Element Systems: A Handbook . Springer{Verlag, New York, 1985. [4]Delves, L. M., and Phillips, C. A fast implementation of the global element method. J. Inst. Maths. Applics 25 (1980), 177{197. [5]Lapidus, L., and Pinder, G. F. Numerical Solution of Partial Dierential Equations in Science and Engineering . John Wiley & Sons, New York, 1982. [6]Mackerle, J. Special volume|nite element methods: A guide to information sources. Finite Elements in Analysis and Design 8 , 1{4 (December 1990). [7]Mackerle, J., and Fredriksson, B. Handbook of Finite Element Software: Supercomputers Mainframes, Minicomputers, Microcomputers . Krieger Pub. Co., Melbourne, FL, 1988. [8]Mitchell, A. R., and Wait, R. The Finite Element Method in Dierential Equations . John Wiley & Sons, New York, 1977. [9]Rockey, K. C., Evans, H. R., Griffiths, D. W., and Nethercot, D. A. The Finite Element Method | A Basic Introduction for Engineers , second ed. Halstead Press, New York, 1983. [10]Strang, G. Introduction to Appled Mathematics . Wellesley{Cambridge Press, Wellesley, MA, 1986. [11]Strang, G., and Fix, G. J. An Analysis of the Finite Element Method . Prentice{Hall, Inc., Englewood Clis, NJ, 1973. [12]Wang, P. S. FINGER: A symbolic system for automatic generation of numerical programs in nite element analysis. J. Symbolic. Comp. 2 (1986), 305{316. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 744 IV.B Numerical Methods for ODEs 175. Hybrid Computer Methods Applicable to Ordinary and partial dierential equations. Yields A numerical approximation to the solution. Idea Sometimes the advantages of both digital and analog computers can be used simultaneously on a single dierential equation. Procedure A hybrid computer is one that combines both digital and analog com- puting devices. Generally, in such a conguration, the analog computer isused to perform tasks that are very time consuming on a digital computer. The analog computer is constructed, generally by the user, out of capac- itors, operational ampliers, resistors, and other electronic components.The numbers in an analog computer are represented by electrical quantities such as voltage and amperage. As an example of use, a partial dierential equation can often be approx- imated by a large number of ordinary dierential equations; for example,the method of lines (see page 831) or the Rayleigh{Ritz method (see page 638). Rather than introduce additional approximations in nding solutions of these ordinary dierential equations, an analog computer may be used. In other problems, the analog computer is used to evaluate integrals as they arise. These integrals are often multi-dimensional and would be computationally intensive on a digital computer. The digital computer is nearly always used to control the solution procedure and to determine the discretization and the overall error. Example The block diagram in gure 175.1 shows how the dierential equation d2x dt2+adx dt+bx2=f(t) might be solved by an analog computer. Each of the blocks in this gure is easily implemented by electronic components. The blocks that perform the multiplications will generally have the numerical values of aand−bspecied by potentiometers. These values may be changed by adjusting the potentiometers by hand. Or, these values could be changed by a digital computer. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 175. Hybrid Computer Methods745f /#28 t /#29 /, Rdt/#02 a/#02 /#28 /, b /#29 /, Rdtsquare /+ xx /2/, bx /2 /, a dxdt d /2xdt /2 /, dxdt/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././. /./././. /./././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././. /./././. /./././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././. /./././. /./././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././. /./././. /./././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././. /./././. /./././. /./././. /././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././. /./././. /./././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /./././. /./././. /./././. /./././. /./././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /./././././././././././././././././././././. /./././././././././././././././././././././. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /./././././././././././././././././././././. /././././././././././././././././././././. /././. /././. /./././././././././././././././././././././. /./././. /./. /./. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /././. /././././././././././././././././././././././././././././. /./.Figure 175.1: A block diagram for the analog solution of the dierential equationd2x dt2+adx dt+bx2=f(t). Notes 1. For an example of a hybrid nonlinear parabolic equation solver, see El-Zorkany and Balasubramanian [3]. 2. Recently, hybrid computers have been introduced that do not require the user to \plug" components together; the specication of the analog part of the machine is performed on the digital part of the machine. References [1]Allison, J. S., and Johnson, H. M. Stability of hybrid simulation of dynamic systems. Mathematics and Computers in Simulation 21 (1979), 289{303. [2]Amyot, J. R., and Camire, G. A. Stability of a class of hybrid computer models of dynamical systems. Mathematics and Computers in Simulation 28 (1986), 57{64. [3]El-Zorkany, H. I., and Balasubramanian, R. Hybrid solution of non-linear P.D.E.’s based on a special F.E. approximation I. One dimen- sional problem. In Advances in Computer Methods For Partial Dierential Equations-II , R. Vichnevetsky, Ed., IMACS (AICA). North{Holland Publish- ing Co., New York, 1977, pp. 227{234. [4]El-Zorkany, H. I., and Balasubramanian, R. Hybrid computer solution of PDE’s using Laplace{modied Galerkin approximation. Mathematics and Computers in Simulation 23 (1981), 304{311. [5]El-Zorkany, H. I., and Balasubramanian, R. Hybrid simulation of multidimensional P.D.E.’s via solution of many one dimensional problems 1. Theoretical basis and hybrid implementation. Mathematics and Computers in Simulation 25 (1983), 70{76. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 746 IV.B Numerical Methods for ODEs [6]Lawson, P. A. Contraction mapping techniques applied to the hybrid computer solution of parabolic PDE’s. Mathematics and Computers in Simulation 23 (1981), 299{303. [7]Neundorf, W. Iterative block methods for the hybrid computer solution of the method of lines. Math. and Computers in Simulation 23 , 2 (1981), 142{148. [8]Roubcek, T. Hybrid solution of weakly formulated boundary-value prob- lems. Mathematics and Computers in Simulation 26 (1984), 11{19. [9]Shearer, J. L., Murphy, A. T., and Richardson, H. H. Introduction to System Dynamics . Addison{Wesley Publishing Co., Reading, MA. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 176. Invariant Imbedding747 176. Invariant Imbedding Applicable to Most often, two point boundary value problems for ordinary dierential equations. Yields A new formulation as an initial value problem. Idea Invariant imbedding is a type of continuation method (see page 710). For the usual problems that are treated, the length of the interval of interest is considered to be the continuation parameter. Hence, the endpoint in a two point boundary value problem is treated as a variable. By dierenti-ating with respect to this variable, an initial value problem can be created. Procedure The general technique involves some subtleties, so we choose to illus- trate the technique on a class of two point boundary value problems. More details can be found in Casti and Kalaba [3]. Suppose we have the system of ordinary dierential equations dx dt=a(t)x(t)+b(t)y(t); dy dt=c(t)x(t)+d(t)y(t)+f(t);(176.1) with 1x(0) +2y(0) = 0; 3x(T)+4y(T)=1;(176.2) on the interval t2[0;T], where thefigare constants and fa;b;c;dgare continuous functions. If we think of the endpoint Tas being a variable, then the solution to equations (176.1) and (176.2) can be written, by use of superposition, as x(t)=x(t;T)=u(t;T)+p(t;T); y(t)=y(t;T)=v(t;T)+q(t;T);(176.3) where the functions fu;v;p;qgare dened by du(t;T) dt=a(t)u+b(t)v;  1u(0;T)+2v(0;T)=0; dv(t;T) dt=c(t)u+d(t)v+f(t); 3u(T;T)+4v(T;T)=0;(176.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 748 IV.B Numerical Methods for ODEs and dp(t;T) dt=a(t)p+b(t)q;  1p(0;T)+2q(0;T)=0; dq(t;T) dt=c(t)p+d(t)q;  3p(T;T)+4q(T;T)=1:(176.5) Using algebraic manipulations, the systems in equations (176.4) and (176.5) can be written as initial value systems by the introduction of four new vari-ables. Dene the functions fr;s;m;ngto be the solutions to the following nonlinear ordinary dierential equations: r 0(t)=b(t)s(t)+[a(t)−3b(t)s(t)−4d(t)s(t)]r(t)−[3a(t)+4c(t)]r2(t); s0(t)=c(t)r(t)+[d(t)−3a(t)r(t)−4c(t)r(t)]s(t)−[3b(t)+4d(t)]s2(t); m0(t)=a(t)m(t)+b(t)n(t)−n [3a(t)+4c(t)]m(t) +[3b(t)+4d(t)]n(t)+f(t)o r(t); n0(t)=c(t)m(t)+d(t)n(t)+f(t)−n [3a(t)+4c(t)]m(t) +[3b(t)+4d(t)]n(t)+f(t)o s(t);(176.6) where0denotes dierentiation of a function with respect to its single argument (i.e., the variable t). The initial values for fr;s;m;ngare given by 1r(0) +2s(0) = 0;m (0) = 0; 3r(0) +4s(0) = 1;n(0) = 0:(176.7) Note that we must have 14−236=0i fr(0) ands(0) are to be determined from equation (176.7). Using fr;s;m;ng, the equations for fp;q;u;vgcan now be written as dp(t;T) dT=−n r(T)[3a(T)+4c(T)] +s(T)[3b(T)+4d(T)]o p(t;T); dq(t;T) dT=−n r(T)[3a(T)+4c(T)] +s(T)[3b(T)+4d(T)]o q(t;T); du(t;T) dT=−n m(T)[3a(T)+4c(T)] +n(T)[3b(T)+4d(T)] +f(T)o p(t;T); dv(t;T) dT=−n m(T)[3a(T)+4c(T)] +n(T)[3b(T)+4d(T)] +f(T)o q(t;T):(176.8) The initial conditions for fp;q;u;vgmay be written as p(t;t)=r(t);q (t;t)=s(t); u(t;t)=m(t);v(t;t)=n(t):(176.9) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 176. Invariant Imbedding749 Suppose that the solution of the original system, equations (176.1) and (176.2), is desired at the set of abscissas ft1;t2;t3;:::;tNg,w h e r etN=T, andTis the interval length of interest. The numerical technique is to numerically integrate the equations in equation (176.6) for fr;s;m;ng, from t=0t ot=T. Hence, the values of fr;s;m;ngwill be known at the points ft1;t2;t3;:::;tNg. Now, xt=t1in equations (176.8) and (176.9). Integrate the resulting equations (with respect to T) fromT=t1toT=T. This will yield fp(t1;T),q(t1;T),u(t1;T),v(t1;T)g. If these values are used in equa- tion (176.3), then x(t1;T);y(t1;T) will be determined. Of course, this is the same as x(t1);y(t1). Hence,xandyhave been determined at the rst point of interest, t1. To obtainxandyatt=t2, evaluate equations (176.8) and (176.9) at t=t2and integrate the resulting equations with respect to T(fromt2to T). Repeat this for each of t=t3;t=t4;:::;t =tN. Example Suppose we want to turn the boundary value problem dx dt=1 0y; x (0) = 0; dy dt=1 0x; y (10) = 1; into an initial value problem. Using the above notation, we nd that 1= 4=1 ,2=3=0 ,a(t)=d(t)=f(t)=0 ,b(t)=c(t) = 10, and T= 10. The system in equation (176.6) becomes r0= 10(s−r2); s0= 10(s−1); m0= 10(n−mr); n0=1 0m(1−s);(176.10) with the initial conditions: r(0) = 0,s(0) = 1,m(0) = 0,n(0) = 0. It is easy to see that n(t)=0 ,m(t)=0 ,s(t) = 1, although these equations could have been integrated if this had not been observed. The system in (176.8) becomes dp(t;T) dT=−10r(T)p(t;T); dq(t;T) dT=−10r(T)q(t;T); du(t;T) dT=−10m(T)p(t;T); dv(t;T) dT=−10m(T)q(t;T);(176.11) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 750 IV.B Numerical Methods for ODEs with the initial conditions: p(t;t)=r(t),q(t;t)=s(t),u(t;t)=m(t), v(t;t)=n(t). From the above observation, we conclude that q(t;t)=1 , u(t;T)=0 ,v(t;T) = 0. Using equation (176.3), we nd: x(t)=x(t;10) = p(t;10) andy(t)=y(t;10) =q(t;10). Let us suppose that we want to know the values of xandyfort=2;4;6;8;10. The procedure to follow is 1. Integrate r(t) fromt=0u pt ot= 10 using equation (176.10). Hence, r(2),r(4),r(6),r(8),r(10) will all be known. 2. Setp(2;2) =r(2) andq(2;2) = 1. Integrate equation (176.11) for p(t;T)a n dq(t;T) fromT=2t oT= 10. Then,fp(2;10);q(2;10)g will be known and hence, fx(2);y(2)gwill be known. 3. Setp(4;4) =r(4) andq(4;4) = 1. Integrate equation (176.11) for p(t;T)a n dq(t;T) fromT=4t oT= 10. Then,fp(4;10);q(4;10)g will be known and hence, fx(4);y(4)gwill be known. 4. Repeat steps (2) and (3) for t=6 ,t=8 ,a n dt= 10. Notes 1. The paper by Scott [9] lists several dierent ways in which boundary value problems may be converted into stable initial value problems. 2. Imbedding methods can be used for more than just boundary value problems. This technique can also be applied to nonlinear vari-ational problems, unconstrained nonlinear control processes, con- strained control processes, and Fredholm integral equations. Imbed- ding methods have also been used in hyperbolic and parabolic partialdierential equations. 3. Wasserstrom [11] discusses how imbedding methods can be analyzed as continuation methods (see page 710). 4. Other names for the invariant imbedding approach are \eld method," \factorization method," \method of sweeps," \compound matrix method,"and \Riccati transformation." In this last method, matrix Riccati equations (see page 395) are developed. See Ascher et al. [1] for details. References [1]Ascher, U. M., Mattheij, R. M. M., and Russel, R. D. Numerical Solution of Boundary Value Problems for Ordinary Dierential Equations . Prentice{Hall, Inc., Englewood Clis, NJ, 1988. [2]Bellman, R., and Wing, G. M. An Introduction to Invariant Imbedding . SIAM, Philadelphia, PA, 1992. [3]Casti, J., and Kalaba, R. Imbedding Methods in Applied Mathematics . Addison{Wesley Publishing Co., Reading, MA, 1973. [4]Dieci, L., Osborne, M. R., and Russell, R. D. A Riccati transformation method for solving linear BVPs. I: Theoretical aspects. SIAM J. Numer. Anal. 25 , 5 (October 1988), 1055{1073. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 176. Invariant Imbedding751 [5]Lee, E. S. Quasilinearization and Invariant Imbedding . Academic Press, New York, 1968. [6]Meyer, G. H. Initial Value Methods for Boundary Value Problems: Theory and Application of Invariant Imbedding . Academic Press, New York, 1973. [7]Meyer, G. H. Invariant imbedding for xed and free two point boundary value problems. In Numerical Solutions of Boundary Value Problems for Ordinary Dierential Equations ,A .K .A z i z ,E d .A c a d e m i cP r e s s ,N e wY o r k , 1975, pp. 249{275. [8]Ng, B. S., and Reid, W. H. A numerical method for linear two-point boundary-value problems using compound matrices. J. Comput. Physics 33, 1 (Oct 1979), 70{85. [9]Scott, M. R. On the conversion of boundary{value problems into stable initial{value problems via several invariant imbedding algorithms. In Numerical Solutions of Boundary Value Problems for Ordinary Dierential Equations , A. K. Aziz, Ed. Academic Press, New York, 1975, pp. 89{146. [10]Scott, M. R., and Vandevender, W. H. A comparison of several invariant imbedding algorithms for the solution of two-point boundary-value problems. Appl. Math. and Comp. 1 (1975), 187{218. [11]Wasserstrom, E. Numerical solutions by the continuation method. SIAM Review 15 , 1 (January 1973), 89{119. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 752 IV.B Numerical Methods for ODEs 177. Multigrid Methods Applicable to Ordinary and partial dierential equations. Yields A numerical approximation technique. Idea After dierential equations are discretized for the purpose of approxi- mating the solution numerically, some linear algebraic operations must be performed. Frequently, a system of linear equations may need to be solved(e.g., see pages 701, 805, and 816). If the system of linear equations is large (e.g., when a ne discretization grid is used), then iterative methods are often used to solve the linear equations. Multigrid methods are iterative methods for solving systems of linear equations arising from dierential equations. Generally, dierent grids areused, with only a few iterations per grid. The last approximation on one grid becomes the rst approximation on the next grid. Procedure We sketch the approximation process using the following ordinary dif- ferential equation as motivation: u00(x)−u(x)=−f(x) u(0) = 0;u(1) = 0:(177.1) Consider approximating the solution of equation (177.1) on a uniform grid with a spacing of h(e.g.,xj=jhandvju(xj)). Call this grid Ωh. Using (vj−1−2vj+vj+1)=h2as an approximation to u00(xj), equation (177.1) can be written as 1 h22 6666642+h2−10 0 −12 +h2−1 ......... −12 +h2−1 00 −12 +h23 7777752 6664v 1 v2 ... vN−13 7775=2 6664f1 f2 ... fN−13 7775 (177.2) or simply as Ahvh=fh. (Here a superscript indicates the spacing on the underlying grid.) The solution of the linear system in equation (177.2) can be approxi- mated by any of the standard iteration methods, such as Jacobi’s method or the Gauss{Seidel method (see Golub and Van Loan [5]). Typically, theseiterative methods begin to stall (i.e., the convergence rate decreases) when smooth error modes are present. Because a smooth mode on a ne grid CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 177. Multigrid Methods 753 looks less smooth on a coarser grid, it is advisable to move to a coarser grid. Iterating on this coarser grid will more eectively reduce the error term. The values on this coarse grid are then fed back to the ne grid. To illustrate the process, let I2h hbe the linear operator that performs restriction and maps a vector from Ωhto Ω2h. (For instance, every other value in the vector could be chosen.) Similarly, let Ih 2hbe the linear operator that performs interpolation and maps a vector from Ω2hto Ωh. Let us use the term \Relax on" to mean \iterate some number of times using a standard technique such as Gauss{Seidel." Here, then, is how a multigrid method might be implemented: Relax onAhuh=fhwith an input initial guess vh. Find the residual: rh:=Ahuh−fh. Move to a coarser grid: f2h:=I2h hrh. Relax onA2hu2h=f2hwith the initial guess v2h=0. Find the residual: r2h:=A2hu2h−f2h. Move to a coarser grid: f4h:=I4h 2hr2h. Relax onA4hu4h=f4hwith the initial guess v4h=0. Find the residual: r4h:=A4hu4h−f4h. Move to a coarser grid: f8h:=I8h 4hr4h. ... SolveA2khu2kh=f2khforu2kh(which we call v2kh). ... Revise approximate solution on Ω4h:v4h v4h+I4h 8hv8h. Relax onA4hu4h=f4hwith the initial guess v4h. Revise approximate solution on Ω2h:v2h v2h+I2h 4hv4h. Relax onA2hu2h=f2hwith the initial guess v2h. Revise approximate solution on Ωh:vh vh+Ih 2hv2h. Relax onAhuh=fhwith the initial guess vh. The overall eect is that an approximate solution to the system on the h-grid is input at the top, and a rened approximation to this solution is output at the bottom. Note 1. The multigrid method is applicable to linear algebraic equations. Its importance for dierential equations comes about because dierentialequations can be approximated by solving linear algebraic equations. References [1]Adams, J. Recent enhancements in MUDPACK A multigrid software package for elliptical partial dierential equations. Appl. Math. and Comp. 43 (May 1991), 79{94. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 754 IV.B Numerical Methods for ODEs [2]Biggs, W. L. A Multigrid Tutorial . SIAM, Philadelphia, PA, 1988. [3]Brandt, A., McCormick, S., and Ruge, J. Multigrid method for dierential eigenproblems. SIAM J. Sci. Stat. Comput. 4 , 2 (1983), 244{260. [4]Goldstein, C. I. Multigrid analysis of nite element methods with numerical integration. Math. of Comp. 56 , 194 (April 1991), 409{436. [5]Golub, G. H., and Van Loan, C. F. Matrix Computations ,s e c o n de d .T h e Johns Hopkins University Press, Baltimore, MD, 1989. [6]Hackbusch, W., and Trottenberg, U. Multigrid Methods . Springer{ Verlag, New York, 1982. [7]Jespersen, D. Multigrid Methods for Partial Dierential Equations ,v o l .2 4 ofStudies in Mathematics . Mathematical Association of America, Providence, RI, 1984. [8]McCormick, S. Multigrid Methods: Theory, Applications, and Supercom- puting . Marcel Dekker, New York, 1988. [9]Wesseling, P. An Introduction to Multigrid Methods . John Wiley & Sons, New York, 1992. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 178. Parallel Computer Methods 755 178. Parallel Computer Methods Applicable to All types of dierential equations. Yields Numerical approximations to the solutions. Idea Parallel computers may be used to quickly obtain numerical approxi- mations to dierential equations. Procedure The physical basis for most dierential equations is a local and asyn- chronous model. Hence, it should be possible to numerically approximate a partial dierential equation by processors that are loosely coupled. There are three major ways in which software for dierential equations can exploit parallelism: in coding a method so that it can be performed simultaneously on several processors, in splitting variables (in a multi- variable system) between processors, and in using parallelism in perform- ing the needed algebraic computations (i.e., solving algebraic systems of equations). We illustrate one parallel technique; it uses the rst of thesemethods. Example This example for a two processor MIMD machine is from Iserles and Nrsett [8]. The Butcher-array is a convenient way in which to represent all of the information in a Runge{Kutta method for the equation y0=f(x;y), y(x0)=y0(see page 763). The Butcher array for a four-stage, fourth order Runge{Kutta method is 1/21/2 000 2/3 02/300 1/2−5/25/21/20 1/3−5/34/302/3 −13/2−13/2 Because of the specic sparsity structure of this Butcher array, we can eciently implement this technique on two processors. Given the valuey n, to nd the approximation at the next time step, yn+1, the steps are as follows: Use an iteration technique (perhaps Newton{Raphson) to solve the equations {1=f/parenleftbig tn+1 2h;yn+1 2h1
for1on processor 1, CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 756 IV.B Numerical Methods for ODEs {2=f/parenleftbig tn+2 3h;yn+2 3h2
for2on processor 2. Copy the value of 1to processor 2, and copy the value of 2to processor 1. Use an iteration technique (perhaps Newton{Raphson) to solve the equations {3=f/parenleftbig tn+1 2h;yn+h/parenleftbig −5 21+5 22+1 23

for3on proces- sor 1, {4=f/parenleftbig tn+1 3h;yn+h/parenleftbig −5 31+4 32+2 34

for4on proces- sor 2. Copy4to processor 1 and then form the estimate at the next time value: yn+1=yn+h/parenleftbig3 2(2+4)−1−3
. Notes 1. Many parallel computers can quickly perform matrix operations, such as solving a system of linear equations. Hence, these machines may be used to quickly approximate the solutions to dierential equations by using methods (such as nite dierences and nite elements) that produce large systems of linear algebraic equations.When solving dierential equations numerically, it is not uncommon to have large computational needs. For example, a 50 5050 grid with 5 degrees of freedom per grid point, such as might be obtainedfrom Euler’s equation in fluid dynamics, will lead to matrices of size N= 625;000 and a bandwidth m25000. Even though sparse matrix techniques may be used, the complexity of the problem is very high. However, Rice [21] makes the point that linear algebra approaches are only tangentially relevant to solving partial dierentialequations and are, in fact, often misleading. Numerical analysis of dierential equations begins with the equation itself, not with a discretized version of the equation. 2. Domain decompostion (see page 800) subdivides a large domain (on which an elliptic partial dierential equation is dened) into many smaller domains. A separate processor can then be used on each smaller domain; see Quarteroni [19]. 3. All types of processors have been used to numerically approximate the solutions to dierential equations. Hypercubes have been used by many, including Lustman et al. [12], Mu and Rice [15], and Murthy [16]. The use of neural networks for solving dierential equations is considered in Dissanayake and Phan-Thien [5], Lee and Kang [10], and Meade and Fernandez [13]. By a simple replication of hardware, many Monte-Carlo simula- tions can be performed simultaneously (see pages 810 and 844). This is particularly useful for SIMD machines. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 178. Parallel Computer Methods 757 Lattice gas methods (which use cellular automata) are a method of parallel computation; see page 828. Use of cellular automata to numerically approximate the solution of dierential equations has also been considered in Boghosian and Levermore [3]. It is also possible to build a specialized VLSI circuit to inte- grate a specic set of dierential equations. A special purpose computer for high-speed, high-precision orbital mechanics com- putations has been built; see Applegate et al. [1]. This was used to demonstrate that the orbit of Pluto was chaotic; see Sussmanand Wisdom [22]. It is also possible to construct systolic arrays that solve a class of equations very quickly; see Megson and Evans [14]. 4. The technique described in Garbey and Levine [7] numerically ap- proximates hyperbolic equations by using both characteristics andcellular automata. 5. In the Kolmogorov theory of turbulence, computer memory usage scales asR 9=4and computational work, including time integration, scales asR3,w h e r eRis the Reynolds number. For engineering applications, Reynolds numbers of 103are typical. For geophysical flows, Reynolds numbers of 108are not unusual. See Jackson et al. [9] for details. 6. Lustman et al. [12] considers a parallel machine in which every pro- cessor computes the same problem but with a dierent step size. Extrapolation methods (see page 679) are then employed. References [1]Applegate, J. H., Douglas, M. R., Gursel, Y., Hunterm, P., Seitz, C. L., and Sussman, G. J. A digital orrery. IEEE Trans. Computers C-34 , 9 (September 1985), 822{831. [2]Bellen, A., Vermiglio, R., and Zennaro, M. Parallel ODE-solvers with stepsize control. J. Comput. Appl. Math. 31 (1990), 277{293. [3]Boghosian, B. M., and Levermore, C. D. A cellular automaton for Burgers’ equation. Complex Systems 1 (1987), 17{29. [4]Burrage, K. Parallel and Sequential Methods for Ordinary Dierential Equations . Numerical Mathematics and Science Publication Series. Claren- don Press, Oxford, England, 1995. [5]Dissanayake, M. W. M. G., and Phan-Thien, N. Neural-network-based approximations for solving partial dierential equations. Comm. Num. Meth. Eng. 10 , 3 (March 1994), 195{201. [6]Evans, D. J., and Sanugi, B. B. A parallel Runge{Kutta integration method. Parallel Comput. 11 , 2 (1989), 245{251. [7]Garbey, M., and Levine, D. Massively parallel computation of conserva- tion laws. Parallel Computing 16 (1990), 293{304. [8]Iserles, A., and Nrsett, S. P. On the theory of parallel Runge{Kutta methods. IMA J. Num. Analysis 10 (1990), 463{488. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 758 IV.B Numerical Methods for ODEs [9]Jackson, E., She, Z.-S., and Orszag, S. A. A case study in parallel computing: I. Homogeneous turbulence on a hypercube. J. Scientic Comput. 6 , 1 (March 1991), 27{45. [10]Lee, H., and Kang, I. S. Neural algorithm for solving dierential equations. J. Comput. Physics 91 (1990), 110{131. [11]Lin, A. Parallel algorithms for boundary value problems. J. Parallel & Distrib. Comput. 11 (1991), 284{290. [12]Lustman, L., Neta, B., and Katti, C. P. Solution of ordinary dierential initial value problems on an INTEL hypercube. Comp. & Maths. with Appls. 23, 10 (1992), 65{72. [13]Meade, Jr., A. J., and Fernandez, A. A. Solution of nonlinear ordinary dierential equations by feedforward neural networks. Math. Comput. Modelling 20 , 9 (1994), 19{44. [14]Megson, G. M., and Evans, D. J. Systolic arrays for group explicit methods for solving rst order hyperbolic equations. Parallel Computing 16 (1990), 191{205. [15]Mu, M., and Rice, J. R. A grid-based subtree-subcube assignment strategy for solving partial dierential equations on hypercubes. SIAM J. Sci. Comput. 13 , 3 (May 1992). [16]Murthy, C. S. R. Solving hyperbolic PDE’s on hypercubes. Comp. & Maths. with Appls. 21 , 21 (1991), 79{82. [17]Murthy, C. S. R., and Rajaraman, V. A multiprocessor architecture for solving nonlinear partial dierential equations. Math. and Computers in Simulation 30 (1988), 453{464. [18]Oretga, J. M., and Voigt, R. G. Solution of partial dierential equations on vector and parallel computers. SIAM Review 27 (1985), 149{240. [19]Quarteroni, A. Domain decomposition and parallel processing for the numerical solution of partial dierential equations. Surv. Math. Ind. 1 (1991), 75{118. [20]Ribbens, C. J., Watson, L. T., and Desa, C. Trowards parallel mathematical software for elliptic partial dierential equations. ACM Trans. Math. Software 19 , 4 (December 1993), 457{473. [21]Rice, J. R. Parallel methods for partial dierential equations. In The Characteristics of Parallel Algorithms , L. H. Jamieson, D. B. Gannon, and R. J. Douglass, Eds. MIT Press, Cambridge, MA, 1987, pp. 209{231. [22]Sussman, G. J., and Wisdom, J. Numerical evidence that the motion of Pluto is chaotic. Science (22 July 1988), 433{437. [23]Tam, H. W. \one-stage parallel methods for the numerical solution of ordinary dierential equations" and \two-stage parallel methods for the numerical solution of ordinary dierential equations". SIAM J. Sci. Stat. Comput. 13 , 5 (September 1992), 1039{1061 and 1062{1084. [24]van der Houwen, P. J., and Sommeijer, B. P. Iterated Runge{ Kutta methods on parallel computers. SIAM J. Sci. Stat. Comput. 12 , 5 (September 1991), 1000{1028. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 179. Predictor{Corrector Methods 759 179. Predictor{Corrector Methods Applicable to Ordinary dierential equations of the form y0= f(x;y). Yields A sequence of numerical approximations. Idea To integrate an ordinary dierential equation from a point xnto a new pointxn+1=xn+h, a single formula may be used to predict yn+1. Alternatively, the value of yn+1could be predicted by one formula, and then that value could be rened by an iterative formula (the \corrector"). Procedure For the rst order ordinary dierential equation y0=f(x;y), suppose that the values of xandyare known at the sequence of m+1 p o i n t s fxn−m;:::;xn−1;xng. Then the values of y0are known at those same points (becausey0is determined from xandyviay0=f(x;y)). An interpolatory polynomial of degree mcan be tted to m+ 1 values of xandy0. This polynomial can be used to predict the value of y0in the interval ( xn;xn+1). This, in turn, can be used to predict the value of yn+1by a numerical approximation of the relation yn+1=yn+Zxn+1 xny0(x)dx: (179.1) Such a formula is called an \predictor." A modication of this step can be repeated. The values of xandy0are now known at the m+1poi n t sfxn−m+1;:::;xn;xn+1g. A polynomial can be t through these points, and then the quantity in equation (179.1) can be recomputed. This formula, which furnishes a new estimate of yn+1,i s called a \corrector." The corrector may be used repeatedly. Example One set of predictor{corrector equations is the Adams{Bashforth pre- dictor formula yn+1=yn+h 24/parenleftbig 55y0 n−59y0 n−1+3 7y0 n−2−9y0 n−3
; (179.2) and the Adams{Moulton corrector formula yn+1=yn+h 24/parenleftbig 9y0 n+1+1 9y0 n−5y0 n−1+y0 n−2
; (179.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 760 IV.B Numerical Methods for ODEs wherehis the dierence between adjacent xpoints (The xpoints are assumed to be equally spaced). These equations are fourth order accurate. Example The Fortran program in program 179.1 uses the method in equation (179.2) and (179.3) to approximate the solution to the dierential equation dy dx=1−x+y x;y (1) = 0: (179.4) Because the solution of equation (179.4) is given by y(x)=x(logx−x+ 1) (determined by integrating factors), it is easy to see that the valuesproduced Step number= 4 X= 1.60 Y= -0.2080 Step number= 5 X= 1.80 Y= -0.3820Step number= 6 X= 2.00 Y= -0.6137 Step number= 7 X= 2.20 Y= -0.9054 Step number= 8 X= 2.40 Y= -1.2588Step number= 9 X= 2.60 Y= -1.6756Step number= 10 X= 2.80 Y= -2.1570Step number= 11 X= 3.00 Y= -2.7041Step number= 12 X= 3.20 Y= -3.3179Step number= 13 X= 3.40 Y= -3.9991Step number= 14 X= 3.60 Y= -4.7486 all are correct to the number of decimal places given. Note that the program required that the values of ybe given for x=hj wherej=1;2;3. These \starting" values were obtained by using a Runge{ Kutta method that was fourth order accurate (these calculations are not shown). Notes 1. The corrector formula could be iterated as many times as is necessary to ensure convergence. This is called correcting to convergence .I n general, however, if more than two iterations are required, then the step sizehis probably too large. 2. Given the equation y0=f(x;y), letPindicate an application of a predictor,Ca single application of a corrector, and let Eindicate an evaluation of the function fin terms of known values of its arguments. Correcting to convergence can then be represented by P(EC)1.S e e Lambert [7] for an analysis of P(EC)mandP(EC)mE,w h e r emis a x e dn u m b e r . 3. Note that the predictor{corrector method is a nite dierence scheme that is not a linear multistep method as dened on page 670. 4. To obtain the starting values so that the predictor{corrector pair can be used, Runge{Kutta methods can be used rst. This was done in the example above. When this is done, the Runge{Kutta methodused should be at least as accurate as the predictor{corrector formula used. See Gear [4] for details. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 179. Predictor{Corrector Methods 761 REAL*4 X(100),Y(100),YP(100) C Define the initial values (found by Runge-Kutta) H=0.2X(1)= 1.0Y(1)= 0.0 YP(1)= F(X(1),Y(1)) X(2)= X(1) + H Y(2)=-0.02121 YP(2)= F(X(2),Y(2)) X(3)= X(2) + HY(3)=-0.08894 YP(3)= F(X(3),Y(3)) X(4)= X(3) + HY(4)=-0.20799 YP(4)= F(X(4),Y(4)) C Here is the integration loop DO 10 N=4,14NP1=N+1X(NP1)= X(N) + HY(NP1)= PREDIC(X,Y,YP,N,H)YP(NP1)= F(X(NP1),Y(NP1))Y(NP1)= CORECT(X,Y,YP,N,H)Y(NP1)= CORECT(X,Y,YP,N,H) 10 WRITE(6,5) N,X(N),Y(N) 5 FORMAT(’ Step number=’,I3,’ X=’,F5.2,’ Y=’,F8.4) END C This function has the predictor FUNCTION PREDIC(X,Y,YP,N,H)REAL*4 X(100),Y(100),YP(100)PREDIC=Y(N) + H/24.*(55.*YP(N)-59.*YP(N-1)+37*YP(N-2)-9.*YP(N-3)) RETURN END C This function has the corrector FUNCTION CORECT(X,Y,YP,N,H)REAL*4 X(100),Y(100),YP(100)CORECT=Y(N) + H/24.*(9.*YP(N+1)+19.*YP(N)-5.*YP(N-1)+YP(N-2))RETURNEND C This function has the right hand side of the differential equation FUNCTION F(X,Y)F=1.0-X+Y/XRETURNEND Program 179.1: Fortran program for predictor{corrector method. 5. For the same accuracy, using a predictor{corrector pair to integrate a rst order ordinary dierential equation generally requires fewer eval- uations of the function f(x;y) than a Runge{Kutta method would. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 762 IV.B Numerical Methods for ODEs 6. One set of commonly used predictor{corrector equations is \Milne’s method" yn+1=yn−3+4h 3/parenleftbig 2yn−y0 n−1+2y0 n−2
; yn+1=yn−1+h 3/parenleftbig y0 n+1+4y0 n+y0 n−1
: These equations are also fourth order accurate. Milne’s method is notrecommended because it is subject to an instability problem, in which the errors do nottend to zero as the step size his made smaller. See Gerald and Wheatley [5, pages 314{323] for details. 7. The Adams{Bashforth formulas are a family of linear multistep meth- ods that are often used as predictors for the equation y0=f(x;y). Thek-step xed-stepsize Adams{Bashforth formula yn=yn−1+hkX j=1jf(xn−j;yn−j); is equivalent to yn=yn−1+Rxn xn−1pn(s)ds,w h e r epn(x) is the unique polynomial of degree k−1 that interpolates f(xn−j;yn−j)a txn−j forj=1;:::;k . 8. See also Abramowitz and Stegun [1, formula 25.5.13{25.5.16, pages 896{897], Boyce and DiPrima [2, pages 431{438], and Bronson [3, 232{257]. References [1]Abramowitz, M., and Stegun, I. A. Handbook of Mathematical Functions . National Bureau of Standards, Washington, D.C., 1964. [2]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [3]Bronson, R. Modern Introductory Dierential Equations . Schaum’s Outline Series. McGraw{Hill Book Company, New York, 1973. [4]Gear, C. W. Runge{Kutta starters for multistep methods. ACM Trans. Math. Software 6 , 3 (Sept 1980), 263{279. [5]Gerald, C. F., and Wheatley, P. O. Applied Numerical Analysis . Addison{Wesley Publishing Co., Reading, MA, 1984. [6]Karim, A. I. A., and Ismail, G. A. Nonequidistant modied predictor{ corrector methods for solving systems of dierential equations. Int. J. Comp. Math. 17 (1985), 339{361. [7]Lambert, J. D. Computational Methods in Ordinary Dierential Equations . Cambridge University Press, New York, 1973. [8]Van Der Houwen, P. J., and Sommeijer, B. P. Predictor{corrector methods for periodic second-order initial-value problems. IMA J. Num. Analysis 7 (1987), 407{422. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 180. Runge{Kutta Methods 763 180. Runge{Kutta Methods Applicable to Initial value systems of rst order ordinary dier- ential equations. Yields A numerical approximation to the solution of an initial value system. Idea Given an ordinary dierential equation and an initial value, the value of the dependent variable may be found at the next desired value of theindependent variable by calculating several intermediate values. Procedure Given the rst order ordinary dierential equation y0=f(x;y);y (x0)=y0; (180.1) the value of y(x) at the point x0+hmay be approximated by a weighted average of values of f(x;y) taken at dierent points in the interval x0 xx0+h. The classical Runge{Kutta formula is given by y(x0+h)=y(x0)+h 6(k1+2k2+2k3+k4); (180.2) where k1=f(x0;y0); k2=f(x0+1 2h;y0+1 2k1); k3=f(x0+1 2h;y0+1 2k2); k4=f(x0+h;y0+k3):(180.3) This approximation to y(x0+h) is fourth order accurate. After y(x0+h)h a s been determined, the same formula may be used to determine y(x0+2h). This process may be repeated. The Butcher array is a convenient way in which to represent all of the information in a Runge{Kutta method for the equation y0=f(x;y)w i t h y(x0)=y0. Specically, the s-stage Runge{Kutta scheme (which uses s intermediate values) yn+1=yn+hsX i=1biki; ki:=f0 @xn+cih;yn+hsX j=1aijkj1 A; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 764 IV.B Numerical Methods for ODEs whereh:=xn+1−xn,P ibi=1 ,a n dci=Ps j=1aijfor eachj,i s represented in the tabular form cA bTorc1a11a12


a1s c2a21a22


a2s ............... csas1as2


ass b1b2::: bs Note that an explicit Runge{Kutta scheme has aij=0f o rji (sometimes these zeros are omitted). See Butcher [5, page 163] or Dekker and Verwer [9, Chapter 3] for details. The explicit method in equations (180.2) and (180.3) has the Butcher array (with s=4 ) 0 000 0 1/21/200 0 1/201/200 1 001 0 1/61/31/31/6 Example 1 The C (Fortran) code in program 180.1 (180.2) calculates a numerical approximation to the solution of the equation y0=1−x+y x;y(1) = 0; (180.4) using the method in equations (180.2) and (180.3). It uses a step size h of0.1. The exact solution of equation (180.4), determined by integrating factors, isy(x)=x(logx−x+ 1). Hence, y(2) = 2(log 2−1)’−0:6137. This is the value returned by the programs. Example 2 The derivation of a Runge{Kutta method is instructive because it indi- cates the arbitrary degrees of freedom that exist in Runge{Kutta methods. Given the equation y0=f(t;y) and using yn=y(tn)a n dtn=nhto nd a 2-stage Runge{Kutta scheme, we assume a discrete approximation scheme of the form yn+1=yn+ak1+bk2; k1=hf(tn;yn); k2=hf(tn+h;yn+k1):(180.5) We want to nd fa;b;;gto make the order of this scheme as high as possible. From equation (180.5) we can explicitly write yn+1and then nd a Taylor series expansion: yn+1=yn+ahf(tn;yn)+bhf(tn+h;yn+hf(tn;yn)); =yn+(a+b)hfn+h2(bft+bfyf)n+O/parenleftbig h3
;(180.6) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 180. Runge{Kutta Methods 765 void main(void) { RungeKutta(); } void RungeKutta(void) { int j;double h = 0.1;double x = 1.0;double y = 0.0;for(j=0; j<=9; j++) { y += Runge(x, y, h); x+ =h ;printf("X= %6.2f Y= %7.4f \n", x, y); } }/* This performs one integration step */double Runge(double x, double y, double h) { double fk1, fk2, fk3, fk4; fk1 = F(x , y ); fk2 = F(x + h/2.0, y + h*fk1 / 2.0);fk3 = F(x + h/2.0, y + h*fk2 / 2.0);fk4 = F(x + h , y + h*fk3 );return(h * (fk1 + 2.0*fk2 + 2.0*fk3 + fk4) / 6.0); }/* This function has the right-hand side of the equation */double F(double x, double y) { return(1.0 - x + y/x); } Program 180.1: C program for Runge{Kutta method. H= 0.1 X= 1.0Y= 0.0DO 10 J=1,9Y= Y+RUNGE(X,Y,H)X= X+H 10 WRITE(6,88) X,Y 88 FORMAT(’ X=’,F6.2,’ Y=’,F7.4) END C This function performs one integration step FUNCTION RUNGE(X,Y,H)FK1= F(X, Y)FK2= F(X+H/2.0,Y+H*FK1/2.0)FK3= F(X+H/2.0,Y+H*FK2/2.0)FK4= F(X+H, Y+H*FK3) RUNGE= H*(FK1 + 2.0*FK2 + 2.0*FK3 + FK4)/6.0 RETURNEND C This function has the right hand side of the equation FUNCTION F(X,Y)F= 1.0-X+Y/XRETURNEND Program 180.2: Fortran program for Runge{Kutta method where a subscript of ndenotes evaluation at the point ( tn;yn). From y0=f(t;y) we can directly construct a Taylor expansion in tto nd: yn+1=yn+hfn+h2 2df dt n+O/parenleftbig h3
; =yn+hfn+h2 2(ft+fyf)n+O/parenleftbig h3
;(180.7) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 766 IV.B Numerical Methods for ODEs becausedf dt=ft+fydy dt=ft+fyf. Comparing equations (180.6) and (180.7), we nd the 3 equations a+b=1; b =1 2; b =1 2; (180.8) for the 4 unknowns fa;b;;g. Because these equations are undetermined, there are innitely many second order Runge{Kutta schemes in the form of equation (180.5). Fourth order Runge{Kutta methods result in 11 equations for 13 un- knowns; 2 of the unknowns may be chosen arbitrarily to achieve some goal. For example, a fourth order Runge{Kutta method with a specic sparsitypattern is used in the section on parallel methods (see page 755) to allow a parallel implementation. Example 3 To obtain accurate numerical results when using any method, an esti- mate of the local error must be obtained. This could be done by the stan- dard technique of recomputing the answer with the step size halved; but this requires lots of additional computation. The Runge{Kutta{Fehlberg method is a fth order method that uses 6 functional evaluations and allows an estimate of the error by re-using the same points: k1=hf(xn;yn); k2=hf xn+1 4h;yn+1 4k1 ; k3=hf xn+3 8h;yn+3 32k1+9 32k2 ; k4=hf xn+12 13h;yn+1932 2197k1−7200 2197k2+7296 2197k3 ; k5=hf xn+h;yn+439 216k1−8k2+3680 513k3−845 4104k4 ; k6=hf xn+1 2h;yn−8 27k1+2k2−3544 2565k3+1859 4104k4−11 40k5 ; yn+1=yn+25 216k1+1408 2565k3+2197 4104kk−1 5k5 error1 360k1−128 4275k3−2197 75240k4+1 50k5+2 55k6:(180.9) Notes 1. Iff(x;y) does not depend on y, then the solution of the initial value problem y0=f(x),y(x0)=y0, is just the integral y(x)= y0+Rx x0f(t)dt. The Runge{Kutta method in equation (180.2) then corresponds to the approximation of y(x) by means of Simpson’s rule. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 180. Runge{Kutta Methods 767 2. There are several Runge{Kutta methods for rst order equations. For example, the following scheme for equation (180.1) y(x0+h)=y(x0)+1 2(k1+k2); k1=hf(x0;y0); k2=hf(x0+h;y0+k1);(180.10) is of second order accuracy. A commonly used fourth order accurate method for rst order ordinary dierential equations (dierent from the one in equation (180.3)) is Gill’s method; see Abramowitz and Stegun [1, formula 25.5.12]. 3. There are also implicit Runge{Kutta methods, see Burrage and Butcher [4] or Butcher [5, Chapter 34]. There are also Runge{Kutta methodsfor ordinary dierential equations of orders 2{10. See, for example, Abramowitz and Stegun [1, formulae 25.5.6{25.5.12] or Collatz [8, Section 2.4, pages 61{77]. For example, a Runge{Kutta scheme forthe second order equation y 00=g(x;y;y0);y (x0)=y0;y0(x0)=v0; is given by k1=hg(x0;y0;v0); k2=hg x0+1 2h;y0+1 2hv0+1 8hk1;v0+1 2k1 ; k3=hg x0+1 2h;y0+1 2hv0+1 8hk1;v0+1 2k2 ; k4=hg x0+h;y0+hv0+1 2hk3;v0+k3 ;(180.11) and y(x0+h)=y0+hv0+1 6h(k1+k2+k3); y0(x0+h)=v0+1 6(k1+2k2+2k3+k4): (180.12) This scheme is numerically fourth order accurate. 4. There are also Runge{Kutta methods for systems of rst order ordi- nary dierential equations. For example, the system y0=m(x;y;z );z0=n(x;y;z ) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 768 IV.B Numerical Methods for ODEs of ordinary dierential equations may be numerically approximated by rst calculating k1=hm(x0;y0;z0); l1=hn(x0;y0;z0); k2=hm(x0+h;y0+k1;z0+l1); l2=hn(x0+h;y0+k1;z0+l1);(180.13) and then the updated values are y(x0+h)=y(x0)+1 2(k1+k2); z(x0+h)=z(x0)+1 2(l1+l2):(180.14) This formula is second order accurate. See Dekker and Verwer [9] for details. 5. The Butcher array can represent all multi-linear methods for approx- imating dierential equations. For example The backward Euler method yn+1=yn+hf(tn+h;yn+1)h a s the Butcher array ( s=1 ) 11 1 The trapezoidal rule yn+1=yn+h 2[f(tn+yn)+f(tn+h;yn+1)] has the Butcher array ( s=2 ) 0 00 11/21/2 1/21/2 6. Pseudo Runge{Kutta methods use not only the stages of the current step, but also the stages of the previous step. For example, for theequationy 0=f(x;y) the method has the form: yn+1=yn+sX i=1iKi;n Ki;n=hf0 @xn+mih;yn+sX j=1i;jKj;n−1+i−1X j=1i;jKj;n1 A: See Caira et al. [7] for details. 7. To obtain a Runge{Kutta method with a desired order, a minimum number of stages (i.e., function evaluations) are required. From Butcher [5] we have: d e s i r e d o r d e r :1234567 8 minimal number of stages: 1 2 346791 1 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 180. Runge{Kutta Methods 769 8. Mathematica has the package Butcher that sets up the equations to solve for a Runge{Kutta method, as in (180.8). The method can be choosen to be explicit, implicit, or diagonally implicit. The package can also create Butcher trees. 9. Runge{Kutta methods are always sympletic; see page 780. 10. RKSUITE is a suite of Fortran codes implementing Runge{Kutta methods. See http://www.netlib.org/ode/rksuite/ . 11. The book by Butcher [5] has a very comprehensive account of Runge{ Kutta methods (it includes 96 pages of references!). See also Boyceand DiPrima [3, pages 420{423]. References [1]Abramowitz, M., and Stegun, I. A. Handbook of Mathematical Functions . National Bureau of Standards, Washington, D.C., 1964. [2]Bogacki, P., and Shampine, L. F. Interpolating high-order Runge{Kutta formulas. Comp. & Maths. with Appls. 20 , 3 (1990), 15{24. [3]Boyce, W. E., and DiPrima, R. C. Elementary Dierential Equations and Boundary Value Problems , fourth ed. John Wiley & Sons, New York, 1986. [4]Burrage, K., and Butcher, J. C. Stability criteria for implicit Runge{ Kutta methods. SIAM J. Numer. Anal. 16 , 1 (February 1979), 30{45. [5]Butcher, J. C. The Numerical Analysis of Ordinary Dierential Equations . John Wiley & Sons, New York, 1987. [6]B u t c h e r ,J .C . ,a n dC a s h ,J .R . Towards ecient Runge{Kutta methods for sti systems. SIAM J. Numer. Anal. 27 , 3 (June 1990), 753{761. [7]Caira, R., Costabile, C., and Costabile, F. A class of pseudo Runge{ Kutta methods. BIT 30 (1990), 642{649. [8]Collatz, L. The Numerical Treatment of Dierential Equations . Springer{ Verlag, New York, 1966. [9]Dekker, K., and Verwer, J. G. Stability of Runge{Kutta Methods for Sti Nonlinear Systems . North{Holland Publishing Co., New York, 1984. [10]Evans, D. J., and Sanugi, B. B. A nonlinear Runge{Kutta formula for initial value problems. SIGNUM Newsletter 22 , 3 (July 1987), 27{30. [11]F i ,J .M . Low order practical Runge{Kutta{Nystrom methods. Computing 38(1987), 281{297. [12]Kutta, W. Beitrag zur naherungsweisen integration totaler dierentialgle- ichungen. Zeits. Math. Phys. 46 (1901), 435{453. [13]Runge, C. Ueber die numerische Auflosung von dierentialgleichungen. Math. Ann. 46 (1895), 167{178. [14]Shampine, L. F. Diagnosing stiness for Runge{Kutta methods. SIAM 12 , 2 (March 1991), 260{272. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 770 IV.B Numerical Methods for ODEs 181. Sti Equations Applicable to Sti dierential equations (i.e., equations that evolve on more than one scale). Yields A numerical approximation technique. Idea Since sti equations evolve on dierent scales, the techniques used to numerically approximate the solution should change as the dierent scales become important. This is because the stability aspects of a numerical technique often change as the equation changes (see page 683). Consider, for example, the denition of stiy stable on page 686|as the eigenvaluesof the problem change a method may no longer be stiy stable. Procedure When trying to numerically approximate the solution to a sti dier- ential equation, the step size used in the discretization process should be variable, becoming very small when needed. The discretization formulashould also change in dierent regions to reflect the dierent type of local solution (i.e., exponential growth, exponential decay, algebraic growth, etc.) The step size should be made as small as is needed to obtain a desired accuracy, but it should be increased whenever possible to reduce the total number of computations. The step size should not be allowed to get so large, though, that the discretization technique becomes unstable. A good choice of step size can be determined by monitoring the change in the solution of the dierential equation. For any single step, the change in the function being approximated andall of its derivatives should not become too large. Example Suppose we have the problem d2y dx2+( 1−)dy dx−y=0; y(0) = 2;y0(0) =−1;(181.1) whereis a small positive number. The solution to equation (181.1) is y(x)=ex+e−x; (181.2) which has a steep decrease from x=0t ox’−logand then has a gradual increase; see gure 181.1. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 181. Sti Equations771/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./././. /././././././././././././././././././. /././././././././././././././././././././. /0 /5 /1/0 x /0 /2/4/6/8 /1/0 y/././././././././././././././././././././././. /./././././././././././././././././././././././././././././././././././. /./././././././././././././././././. /./././././. /./././././././././././. /./././././././././. /./././././././././. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /./././././././. /./././././././. /./././././././././././././. /./././././. /./././././. /./././././. /./././. /./././././. /./././. /./././././././././././././././././././././././././././. /./././././././././././. /./././././././. /./././././. /./././././. /./././././. /./././. /././. /././././. /././. /././. /. /././. /././. /././. /./. /././. /././. /. /././. /./. /././. /. /././. /./. /././. /. /./. /./. /. /./. /./. /. /./. /./. /. /./. /. /./. /./. /. /. /././. /. /. /. /./. /./. /. /./. /. /. /. /././. /. /. /. /. /. /. /././. /. /. /. /. /. /. /./. /. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /./. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /.Figure 181.1: The solution to equation (181.1) is y(x)=ex+e−x. When using a simple discretization scheme (e.g., say, Euler’s method), a small step size is required in the region from x=0t ox’− logto resolve the exponential decay. After that region, however, the step size should be increased because the solution is no longer rapidly varying. The Fortran program in program 181.1 implements this numerical idea for=0:01. It uses Euler’s method and a variable step size. The parameter TOLdetermines how much the solution is allowed to change at any step. Note that the change in the solution is dened to also include the changein the value of the derivative. We have chosen TOL=0:01. A few lines of the output of the program are shown below At T= 0.005 DELTAT= 0.0049 Y(T)= 1.9952 Exact value= 1.9952 At T= 0.317 DELTAT= 0.0049 Y(T)= 1.7307 Exact value= 1.7312 At T= 0.327 DELTAT= 0.0098 Y(T)= 1.7237 Exact value= 1.7243 At T= 1.001 DELTAT= 0.0098 Y(T)= 1.3761 Exact value= 1.3776At T= 1.021 DELTAT= 0.0195 Y(T)= 1.3691 Exact value= 1.3707At T= 1.685 DELTAT= 0.0195 Y(T)= 1.2005 Exact value= 1.2025At T= 1.724 DELTAT= 0.0391 Y(T)= 1.1937 Exact value= 1.1958At T= 2.349 DELTAT= 0.0391 Y(T)= 1.1170 Exact value= 1.1193At T= 2.427 DELTAT= 0.0781 Y(T)= 1.1105 Exact value= 1.1129 At T= 2.974 DELTAT= 0.0781 Y(T)= 1.0788 Exact value= 1.0813 At T= 3.130 DELTAT= 0.1563 Y(T)= 1.0728 Exact value= 1.0755At T= 3.599 DELTAT= 0.1563 Y(T)= 1.0613 Exact value= 1.0640At T= 9.849 DELTAT= 0.3125 Y(T)= 1.1034 Exact value= 1.1036At T=10.161 DELTAT= 0.3125 Y(T)= 1.1068 Exact value= 1.1070 During the program execution, the step size, DELTAT , has increased from 0.0049 to 0.3125. Hence, large steps were taken where the solution was not rapidly changing. Notes 1. In the example shown, we can use the same discretization scheme throughout the region of interest|only the step size needs to beadjusted for ecient computation. In other problems, dierent dis- cretization schemes will be needed in dierent regions. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 772 IV.B Numerical Methods for ODEs IMPLICIT DOUBLE PRECISION (A-H,O-Z) TEND=10.0D0EPSLON=0.01D0TOL=0.01D0DELTAT=TENDOLDCHG=1.0D0T=0.0D0 Y=2.0D0 YP=EPSLON-1.D0 C Decrease the size of the time step10 DELTAT=DELTAT/2.D020 IF ( DELTAT .GT. .5D0 ) GOTO 10 CALL STEP(Y,YP,DELTAT,EPSLON,YN,YNP)CHANGE= DSQRT((Y-YN)**2 + (YP-YNP)**2)IF( CHANGE .GT. TOL ) GOTO 10 IF( CHANGE .GT. 2.D0*OLDCHG ) GOTO 10 C Store away the new values T = T + DELTATY=Y NYP= YNPOLDCHG=CHANGEVAL=EXACT(T,EPSLON)WRITE(6,5) T, DELTAT, Y, VAL 5 FORMAT(’ At T=’,F6.3,’ DELTAT=’,F7.4, 1 ’ Y(T)=’,F7.4,’ Exact value=’,F7.4) C Increase the size of the time step DELTAT=2.D0*DELTATIF( T .LT. TEND ) GOTO 20END C This subroutine updates Y and Y’ by Euler’s method SUBROUTINE STEP(Y,YP,DELTAT,EPSLON,YN,YNP) IMPLICIT DOUBLE PRECISION (A-H,O-Z)YN = Y + DELTAT*( YP )YNP= YP + DELTAT*( EPSLON*Y - YP*(1.D0-EPSLON) )RETURNEND C This function computes the exact solution to compare against FUNCTION EXACT(T,EPSLON) IMPLICIT DOUBLE PRECISION (A-H,O-Z) EXACT=DEXP(EPSLON*T)+DEXP(-T)RETURNEND Program 181.1: Fortran program for sti ODEs. 2. If the new independent variable ~ x=xis introduced, then the solution in equation (181.2) may be written as y(x)=e~x+e−~x=.I n this representation of the solution, it is clear that there is a \boundarylayer" near ~ x= 0; see the section on boundary layers (page 590). 3. For an example of how the stability of a method may change as the solution of a dierential equation evolves, see the stability analysisfor Euler’s method on page 732. In the example there, as the value of the positive constant cbecomes smaller, the step size must also CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 181. Sti Equations773 become smaller to ensure stability. 4. Sometimes non-sti methods can solve sti problems, without any special diculty except that they can be computationally expensive. 5. Changing the length of the step size leads to accurate solutions to sti initial value ordinary dierential equations and for partial dierential equations that may be solved by a marching technique. For boundary value ordinary dierential equations or for elliptic partial dierential equations, the analogous technique is to numerically solve the equa- tions on a non-uniform mesh. This mesh should be ne where thesolution is rapidly changing, and coarse elsewhere. 6. It is nottrue that the eigenvalues of the matrix A(t) in the system dy dt=A(t)y (181.3) will determine whether the system is sti or not. For example, the matrix A(t)=−1−9c o s26t+6s i n1 2t 12 cos26t+9 2sin 12t −12 sin26t+9 2sin 12t−1−9s i n26t−6s i n1 2t (181.4) has the constant eigenvalues −1a n d−10, but the solution to equation (181.3) is y=C1e2t cos 6t+2s i n6t 2c o s6t−sin 6t +C2e−13t sin 6t−2c o s6t 2s i n6t+c o s6t ; whereC1andC2are arbitrary constants. Clearly the exponentials e−tande−10tare not present in the solution. Also, the solution may blow up as ttends to innity. Even so, the eigenvalues of the linearized problem are often the most useful piece of information available regarding the conditioning of the system. This exampleis from Dekker and Verwer [3, page 11]. 7. If(t) is dened by =jjyjj 2=yHy, then, using equation (181.3), d dt=yH/parenleftbig A+AH
y.I fmaxrepresents the largest eigenvalue of (A+AH)t h e n(t)0emaxt. Hence, the eigenvalues of ( A+AH) allow bounds to be determined for y(t). For the matrix in equation (181.4), the eigenvalues of ( A+AH) are 4 and−26. 8. An equation is often realized to be sti only after the dierential equation has been numerically integrated. There are tests that canbe performed during the integration procedure to determine whether the equation is sti. See, for example, Gear [5] or Shampine [8]. 9. For a recent review of software for sti equations, see Aiken [1, Chapters 3{4, pages 70{202] or Byrne and Hindmars [2]. 10. See also Ganey [4], Miranker [6], and Petzold [7]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 774 IV.B Numerical Methods for ODEs References [1]Aiken, R. C. Sti Computation . Oxford University Press, New York, 1985. [2]Byrne, G. D., and Hindmarsh, A. C. Sti ODE solvers: A review of current and coming attractions. J. Comput. Physics 70 (1987), 1{62. [3]Dekker, K., and Verwer, J. G. Stability of Runge{Kutta Methods for Sti Nonlinear Systems . North{Holland Publishing Co., New York, 1984. [4]Gaffney, P. W. A performance evaluation of some FORTRAN subroutines for the solution of sti oscillatory ordinary dierential equations. ACM Trans. Math. Software 10 , 1 (March 1984), 58{72. [5]Gear, C. W. Automatic detection and treatment of oscillatory and/or sti ordinary dierential equations. In Numerical Integration of Dierential Equations and Large Linear Systems , J. Hinze, Ed. Springer{Verlag, New York, 1982, pp. 190{206. [6]Miranker, W. L. Numerical Methods for Sti Equations .D . R e i d e l Publishing Co., Boston, MA, 1981. [7]Petzold, L. Automatic selection of methods for solving sti and nonsti systems of ordinary dierential equations. SIAM J. Sci. Stat. Comput. 4 ,1 (March 1983), 136{148. [8]Shampine, L. F. Stiness and nonsti dierential equation solvers, II: Detecting stiness with Runge{Kutta methods. ACM Trans. Math. Software 3, 1 (March 1977), 44{53. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 182. Integrating Stochastic Equations 775 182. Integrating Stochastic Equations Applicable to Stochastic dierential equations. Yields A numerical approximation. Idea The \white Gaussian noise" term in a stochastic dierential equation can be numerically approximated in many dierent ways. Procedure Suppose we have the stochastic dierential equation x0=b(x)+(x)n(t);x (0) =y; (182.1) wheren(t) represents white noise. There exist several numerical approxi- mations for the quantity x(T), whereT=mh,his a (small) time step, and Tis a xed time of order one. Three common numerical approximations of equation (182.1) are ~x(tk+1)=~x(tk)+bkh+kp hk; (182.2) bx(tk+1)=bx(tk)+bkh+kp hk; (182.3) x(tk+1)=x(tk)+ b−1 2@ @x kh+kp hk+1 2 @ @x kh2 k; (182.4) with ~x(0) =bx(0) = x(0) =y,w h e r etk=khand a subscript of kmeans evaluation at the kth point (e.g., bk=b(x(tk))). Thefkgare independent random variables that take on the values +1 and −1 with probability 1/2, while thefkgare independent Gaussian random variables with mean 0 and variance 1. Each of the approximations in equations (182.2){(182.4) have a dierent mean square error for a single step. If E [
] represents the expectation operator, then E (x(h)−~x(h))2 =O(h); E (x(h)−bx(h))2 =O(h2); E (x(h)−x(h))2 =O(h3):(182.5) Hence, equation (182.4) is the most accurate if a sample of x(T) is desired. However, if the mean of a function of x(T) is required, then each of the three approximations in equations (182.2){(182.4) is rst order accu- rate. That is, each of E [ f(~x(T))], E [f(bx(T))], and E [f(x(T))] is equal to CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 776 IV.B Numerical Methods for ODEs E[f(x(T))] +O(h), for general functions f. This next approximation, z(tk+1)=z(tk)+ b−1 2@ @x kh+kp hk+1 2 @ @x kh2 k +1 2b@ @x+1 2@b @x+1 2@ @t+1 42@2 @x2 kh3=2k +1 2b@b @x+1 2@b @t+1 42@2b @x2 kh2; z(0) =y;(182.6) has the better error estimate: E [ f(z(T))] = E [f(x(T))] +O(h2). Note that, in equation (182.6), we have allowed bandto be functions of both tandx. Example Suppose we have the stochastic dierential equation x0=x+n(t);x (0) = 1; (182.7) wheren(t) is white noise, and we want to estimate E x2(1) .T h eF o k k e r { Planck equation corresponding to (182.7) is (see page 303) @P @t=−@ @x(xP)+1 2@2 @x2(P); withP(0;x)=(x−1). By using the method of moments (see page 568), the ordinary dierential equation that describes E x2(t) is given by d dtE x2(t) =2 E x2(t) +1; E x2(0) =1; with the solution E x2(t) =( 3e2t−1)=2. Therefore, E x2(1) =( 3e2− 1)=2’10:58. This is the value that our numerical approximation should produce. To implement the method in equation (182.3), the Fortran program in program 182.1 was constructed. The program takes the results of NTRIAL trials and averages these values together. Note that the program uses a routine called RANDOM , whose source code is not shown, which returns a random value uniformly distributed on the interval from 0 to 1. A similar program was written which implemented the methods in equation (182.2) and equation (182.4). The results are indicated in table182.1. It should be observed that the numerical results are increasingly accurate when the step size his decreased. Notes 1. Gaussian random variables may be generated from uniformly dis- tributed random variables by the classical technique of Box andMuller [1]. This technique has been used in the function ZETA in program 182.1. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 182. Integrating Stochastic Equations 777 NTRIAL h Equation (182.2) Equation (182.3) Equation (182.4) 1000 0.25 8.14 8.40 11.19 1000 0.20 8.61 8.74 11.11 1000 0.10 9.62 9.30 10.59 1000 0.05 10.00 10.16 10.87 5000 0.25 8.14 8.40 11.19 5000 0.20 8.51 8.36 10.60 5000 0.10 9.46 9.35 10.59 5000 0.05 9.97 10.18 10.90 Table 182.1: Numerical comparison of dierent approximation techniques for equation (182.7) C This program is a numerical implementation of equation (3) NTRIAL=1000H=0.05NTIME=20XINIT=1.0 SUMX2=0.0 C Here is the integration loop DO 10 NSTEP=1,NTRIALX=XINITDO 20 K=1,NTIME 20 X=X + X*H + SQRT(H)*ZETA()10 SUMX2=SUMX2 + X**2 AVERAG=SUMX2/FLOAT(NTRIAL) WRITE(6,*) AVERAG END C This function returns a gaussian random variable FUNCTION ZETA()DATA TWOPI/6.2831853/Y1=RANDOM( DSEED )Y2=RANDOM( DSEED )ZETA= SQRT( -2.*ALOG(Y2) ) * COS( TWOPI*Y1 ) RETURN END Program 182.1: Fortran program for stochastic equation integration. 2. Because low numerical accuracy is obtained by this technique, a computer program does not need to work with extended precision arithmetic. 3. Mil shtein [9] and [10] describes equations (182.2){(182.4) and presents a derivation of equation (182.6). He also includes a numerically fastimplementation of equation (182.6) using Runge{Kutta methods. 4. Sun [16] presents a numerical method for approximating the solution to equations of the form −(pu 0)0+(q+r)2u=f,w h e np,qandr are all functions of the independent variable and both andfare random terms. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 778 IV.B Numerical Methods for ODEs Dierential equation Solution du=Audt+dw u(t)=eAtu(0) +Rt 0eA(t−)dw() dx=xdt +xd! x=e(−(1=2)2)t+! dx=1 2xdt+p x2−1d! x=c o s h! dx=−(4ax3−3x2)dt−2xp x−ax2d!x=a=(a+!2) Table 182.2: Test problems for stochatic equation methods 5. Peterson [12] uses the test problems shown in table 182.2 to illustrate a numerical code for integrating stochastic dierential equations. 6. Saito and Mitsui [14] describe 11 dierent numerical schemes for in- tegrating stochastic dierential equations and give stability diagrams based on the test equation dx=xdt +xd! ,x(0) = 1, whose solution isx(t)=e x p/parenleftbig −1 22
t+!(t)/bracerightbig . 7. Hofmann and Mathe [6] study the numerical phenomena when switch- ing from (real) Monte-Carlo simulations to quasi-Monte-Carlo simu-lations (which is what computers carry out). References [1]Box, G. E. P., and Muller, M. E. A note on the generation of random normal deviates. Ann. Math. Statistic 9 (1958), 610{611. [2]Chang, C.-C. Numerical solution of stochastic dierential equations with constant diusion coecients. Math. of Comp. 49 , 180 (October 1987), 523{ 542. [3]Drummond, I. T., Hoch, A., and Morgan, R. R. Numerical integration of stochastic dierential equations with variable diusivity. J. Phys. A: Math. Gen. 19 (1986), 3871{3881. [4]Golec, J., and Ladde, G. Euler-type approximation for systems of stochastic dierential equations. J. Appl. Math. Simulation 2 , 4 (1989), 239{249. [5]Greenside, H. S., and Helfand, E. Numerical integration of stochastic dierential equations - II. The Bell System Technical Journal 60 ,8( O c t o b e r 1981), 1927{1940. [6]Hofmann, N., and Mathe, P. On quasi-Monte Carlo simulation of stochastic dierential equations. Math. of Comp. 66 , 218 (April 1997), 573{ 589. [7]Janssen, R. Dierence methods for stochastic dierential equations with discontinuous coecients. Stochastics 13 (1984), 199{212. [8]Janssen, R. Discretization of the Wiener-process in dierence-methods for stochastic dierential equations. Stochastic Processes and Their Applications 18(1984), 361{369. [9]Milshtein, G. N. Approximate integration of stochastic dierential equations. Theory Prob. Appl. 19 , 4 (1974), 557{562. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 182. Integrating Stochastic Equations 779 [10]Milshtein, G. N. A method of second-order accuracy integration of stochastic dierential equations. Theory Prob. Appl. 23 , 2 (1978), 396{401. [11]N e w t o n ,N .J . Asymptotically ecient Runge{Kutta methods for a class of Ito and Stratonovich equations. SIAM J. Appl. Math. 51 , 2 (April 1991), 542{567. [12]Peterson, W. P. Some experiments on numerical simulations of stochastic dierential equations and a new algorithm. J. Comput. Physics 113 (1994), 75{81. [13]R umelin, W. Numerical treatment of stochastic dierential equations. SIAM J. Numer. Anal. 19 , 3 (June 1982), 604{613. [14]Saito, Y., and Mitsui, T. Stability analysis of numerical schemes for stochastic dierential equations. SIAM J. Numer. Anal. 33 , 6 (December 1996), 254{2267. [15]Spigler, R. Monte Carlo-type simulation for solving stochastic ordinary dierential equations. Math. and Computers in Simulation 29 (1987), 243{ 251. [16]Sun, T.-C. A nite element method for random dierential equations with random coecients. SIAM J. Numer. Anal. 16 , 6 (December 1979), 1019{ 1035. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 780 IV.B Numerical Methods for ODEs 183. Symplectic Integration Applicable to Hamiltonian systems. Yields An appropriate numerical approximation. Idea Hamiltonian systems have invariants that should be maintained during the numerical integration procedure. Procedure Consider an autonomous Hamiltonian system of the form dpi dt=−@H @qi;dqi dt=@H @pi: (183.1) The time evolution of these equations is area preserving or symplectic; equivalently, the flow conserves the two-form dq^dp. A numerical method is called symplectic if, when applied to Hamiltonian problems, it generates numerical solutions that inherit the property of symplecticness. That is, the state of the system following an integration step could have been reachedfrom that before the step by a canonical transformation. There are two main groups of symplectic integrators. The rst group consists of formulae that belong to standard families of numerical methods (e.g., Runge{Kutta methods) and just \happen" to be symplectic. These methods can be applied to general systems of dierential equations. Thesecond group consists of methods derived via generating functions. These methods cannot be applied to general systems of dierential equations, not even small dissipative perturbations of Hamiltonian systems. Procedure 1 The Runge{Kutta method with tableau a11a12


a1s a21a22


a2s ............ as1as2


ass b1b2::: bs (note that the usual fcigdo not appear because the system in equation (183.1) is autonomous) will be symplectic if the coecients satisfy: biaij+bjaji−bibj=0; for 1i;js: (183.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 183. Symplectic Integration 781 Procedure 2 We may choose to integrate the pequations with one Runge{Kutta scheme (using say faij;big), and the qequations with a dierent Runge{ Kutta scheme (using say fAij;Big), with a11a12


a1s a21a22


a2s ............ as1as2


ass b1b2::: bsA11A12


A1s A21A22


A2s ............ As1As2


Ass B1B2::: Bs This scheme will be symplectic if the coecients satisfy biAij+Bjaji−biBj=0; for 1i;js: (183.3) Example 1 A simple example of a rst-order symplectic scheme for H=p2=2+V(q) is (q;p)!(Q;P), where Q=q+(
t)p; P=p−(
t)@V @q(q+(
t)p):(183.4) Example 2 For separable Hamiltonians (i.e., H(p;q)=T(p)+V(q)), Candy and Rozmus [1] list the symplectic integration formulae in table 183.1. These formulae are to be used in the following fashion: Initial conditions: ( p0;q0)a tt=t0, Do fori=1t on; pi=pi−1+biF(qi−1)
t, qi=qi−1+aiP(pi−1)
t, Integrated variables: ( pn;qn)a tt=t0+
t, where F(q)=−rqV(q)a n d P(p)=rpT(p). Example 3 This example illustrates what can happen if a non-symplectic method, such as forward Euler’s method, is used. Consider the Hamiltoninan H= p2=2+ (q) for which the equations of motion aredq dt=panddp dt=−@ @q= F(q). Integrating these equations using forward Euler results in pn+1=pn+hF(qn) qn+1=qn+hpn:(183.5) There are are least three problems with this numerical scheme CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 782 IV.B Numerical Methods for ODEs Order(n)C o e  c i e n t s 1( a1;b1)=( 1;1) 2( a1;a2;b1;b2)=(1/2;1/2;0;1) 3(a1;a2;a3;b1;b2;b3)=(2/3;−2/3;1;7/24;3/4;−1/24) 4 a1=a4=( 2+21=3+2−1=3)=6 a2=a3=( 1−21=3−2−1=3)=6 b1=0;b2=b4=( 2−21=3)−1;b3=( 1−22=3)−1 Table 183.1: Symplectic integration schemes for separable Hamiltonians. 1. The Jacobian, dened by the determinant J=@pn+1 @pn@qn+1 @pn@pn+1 @qn@qn+1 @qn,i st o leading order equal to 1 −h 2F0(qn). A value of J<1( o rJ>1) leads to volume contraction (or expansion), neither of which is a property of a Hamiltonian systems. 2. The equations are not invariant to time reversal. That is, equation (183.5) can be inverted to yield pn=pn+1−hF(qn) qn=qn+1−hpn;(183.6) but this is not (183.5) with hreplaced for−handnandn+1 interchanged. 3. The energy, dened by En=p2 n=2+ (qn), is not independent of n. In fact, En+1=En+h2 2 F2(qn)−p2 nF0(qn) +O(h3): Notes 1. If ( p;q)= (p;q) is a variable transformation, then will be area preserving if and only if the Jacobian determinant is identically unity: @p @p@q @q−@p @q@q @p= 1. This can be written as @(p;q) @(p;q)T J@(p;q) @(p;q)=J whereJ=0I I0 : 2. Symplecticness characterizes Hamiltonian flows; conservation of vol- ume is a much weaker property shared by some non-Hamiltonian systems. Symplectic integrators do not in general conserve the energy(Hamiltonian) of a mechanical system. 3. It is impossible for an algorithm to simultaneously conserve the sym- plectic structure, the momentum map, and the Hamiltonian. Non-symplectic algorithms that conserve both momentum and energy have been studied by Simo and Wong [6]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 183. Symplectic Integration 783 4. The adjoint of a symplectic map, the inverse of a symplectic map, and the composition of two symplectic maps, all are symplectic. 5. A Hamiltonian system of the form f_q=M−1p,_p=−rF(q)g,w i t h Ma symmetric, positive denite matrix can, under the transforma- tionfq7!M1=2q,p7!M−1=2pg, be reduced to an equivalent system withM=I. 6. Zwillinger [8, pages 341{345] describes the exterior calculus in which two-forms are dened. 7. Ben Leimkuhler maintains a web page on symplectic methods; see http://www.math.ukans.edu/~leimkuhl/symplectic.html . References [1]Candy, J., and Rozmus, W. A sympletic integration algorithm for separable Hamiltonian functions. J. Comput. Physics 92 (1991), 230{256. [2]Channell, P. J., and Scovel, C. Symplectic integration of Hamiltonian systems. Nonlinearity 3 (1990), 231{259. [3]Greenspan, D. A counterexample of the usse of energy as a measure of computational accuracy. J. Comput. Physics 91 (1990), 490{494. [4]Qin, M., and Zhu, W. Construction of symplectic schemes for wave equations via hyperbolic functions zzzref37refzzz, zzzref38refzzz and zzzref39refzzz. Computers Math. Applic. 26 , 8 (1993), 1{11. [5]Sanz-Serna, J. M. Sympletic integrators for Hamiltonian problems: an overview. Acta Numerica (1991), 243{286. [6]Simo, J. C., and Wong, K. K. Unconditionally stable algorithms for the orthogonal group that exactly conserve energy and momentum. Internat. J. Numer. Methods Eng. 31 (1989), 19{52. [7]Zhong, G., and Marsden, J. Lie{Poisson Hamilton{Jacobi theory and Lie{Poisson integrators. Phys. Lett. A 133 (1988), 134{139. [8]Zwillinger, D. ,E d . Standard Mathematical Tables and Formulae ,3 0e d . CRC, Boca Raton, FL, 1995. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 784 IV.B Numerical Methods for ODEs 184. Use of Wavelets Applicable to Ordinary and partial dierential equations. Yields A fast numerical scheme. Idea Using a weighted residual method with easily computed basis functions c a nl e a dt oa ne  c i e n tm e t h o d . Procedure Wavelets are one set of functions that can be used with a Galerkin (weighted residual) method; see page 786. Orthogonal wavelets are dened(see Zwillinger [7, pages 663{667]) by specifying a set of parameters fh kg (withhk=0i fk<0o rk>n )t h a ts a t i s f y Normalization:Pn k=0hk=p 2 Orthogonality:P khkhk−2j=20;j Accuracyp:Pn k=0(−1)kkjhk=0f o rj=0;:::;p−1w i t hp>0 Using these parameters, the solution to the equation (x)=p 2nX k=0hk(2x−k); called the scaling function , is guaranteed to exist. For each j0a n df o r k=0;1;:::; 2jsetj;k=2j=2(2jx−k). DeneVjto be the span of fj;kg2j k=0.T h e nVmVm−1


V1V0. To use the Galerkin method, the dependent variable in the dierential equation is projected into the space of trial functions belonging to Vm. That is, we make the approximation yX kykm;k(x): When the usual inner products are evaluated and orthogonality of the ele- ments is used, linear algebraic equations can be obtained from a dierentialequation. If, at any time, a multiresolution is desired, this can be performed as a postprocessing step or as an adjunct calculation. Notes 1. MathSoft maintains a web site containing wavelet reprints at http:// www.mathsoft.com/wavelets.html . Specic collections of reprints are listed under \Wavelets and Ordinary Dierential Equations" and \Wavelets and Partial Dierential Equations." CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 184. Use of Wavelets 785 2. Jawerth and Sweldens [4] adapt wavelets so they become (bi)orthogonal with respect to the inner product dened by a dierential operator. The stiness matrix in the Galerkin method then becomes diagonal and can be trivially inverted. They also show how to construct anO(N) algorithm for various constant and variable coecient opera- tors. 3. A reason to use wavelet expansions in numerical methods is that in wavelet coordinates dierential operators may be preconditioned by a diagonal matrix. Moreover, a large class of operators, namelyCalder on{Zygmund and pseudo-dierential operators, are sparse in wavelet bases. 4. Wavelets are presently only capable of dealing with the simple bound- ary conditions. This is improving rapidly. 5. The wavelet corresponding to the scaling function (x) is the func- tion (x)=p 2Pn k=0(−1)khn−k(2x−k). Using we dene the functions j;k(x)=2j=2 (2jx−k); these are orthonormal and the entire collectionf j;kg1 j;k=−1forms a basis for L2(R). References [1]Amaratunga, K., Williams, J. R., Qian, S., and Weiss, J. Wavelet{ Galerkin solutions for one-dimensional partial dierential equations. Int. J. Num. Meth. Eng. 27 (1994), 2703{2716. [2]Bacry, E., Mallat, S., and Papanicolaou, G. A wavelet based space{time adaptive numerical method for partial dierential equations. Mathematical Modelling and Numerical Analysis 26 (1992), 793{834. [3]Enquist et al. Fast wavelet based algorithms for linear evolution equations. SIAM J. Sci. Comput. 15 , 4 (July 1994). [4]Jawerth, B., and Sweldens, W. Wavelet multiresolution analyses adapted for the fast solution of boundary value ordinary dierential equations. In Sixth Copper Mountain Conference on Multigrid Methods (1993), N. D. Melson, T. A. Manteuel, and S. F. McCormick, Eds., NASA Conference Publication3224, pp. 259{273. [5]Qian, Z., and Weiss, J. Wavelets and the numerical solution of boundary value problems. Appl. Math. Lett. 6 (1993), 47{52. [6]Xu, J. S., and Shann, W. C. Galerkin{wavelet methods for two-point boundary value problems. Numer. Math. 63 (1992), 123{144. [7]Zwillinger, D. ,E d . Standard Mathematical Tables and Formulae ,3 0e d . CRC, Boca Raton, FL, 1995. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 786 IV.B Numerical Methods for ODEs 185. Weighted Residual Methods Applicable to Ordinary and partial dierential equations. Yields By introducing approximations, this method changes the numerical calculation of An ordinary dierential equation to the numerical calculation of a set of algebraic equations A partial dierential equation to the numerical calculation of a set of ordinary dierential equations Idea We approximate the solution by taking a linear combination of an arbitrarily chosen set of functions. The coecients of the functions, which may be constants or functions themselves, are unknown. We may use any of a number of schemes to nd the numerical values for the unknowncoecients. Procedure We will illustrate the general technique via a specic example. Suppose we have the following partial dierential equation to solve ut−N[u]=0; forx2V; t> 0; u(0;x)=v(x); forx2V; u(t;x)=f(t;x); forx2S; (185.1.a-c) whereN[
] is a dierential operator in xandSis the boundary of V,t h e region in which we seek the solution. We choose a y(t;x) and some set of functions fui(t;x)gwith the prop- erties y(t;x)=f(t;x); forx2S; uj(t;x)=0; forx2S; and then form a trial solution by superposition uT(t;x)=y(t;x)+MX j=1cj(t)uj(t;x): (185.2) Note that the trial solution has been constructed in such a way that it automatically satises equation (185.1.c) but not equations (185.1.a) or CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 185. Weighted Residual Methods787 (185.1.b). If we use the trial solution in the original dierential equation, (185.1.a), then the right-hand side will not be equal to zero but will be equal to some residual REgiven by RE(uT)=(uT)t−N[uT]: (185.3) Instead of this denition of RE, we might equally well have taken the square of equation (185.3). Likewise, the initial condition, equation (185.1.b), willnot be satised, but there will be a residue R Igiven by RI(uT)=v(x)−MX j=1cj(0)uj(0;x): Now, we choose Mweighting functions fwj(x)g. It is the choice of these weighting functions that denes the method. For example, Galerkin: wj=uj; Collocation: wj=(x−xj); least squares: wj=@RE(uT) @cj; subdomain method: wj=( 1;forx2Vj; 0;forx62Vj;(185.4) wherefxjjj=1;2;:::;Mgis a set ofMpoints inVthat must be chosen when collocation is used, and fVjgis a set of disjoint regions whose union is equal toVthat must be chosen when the subdomain method is used. Next, an inner product is dened by (w;z)=Z Vw(x)z(x)dV; (185.5) or something similar. Then, nally, the unknown coecients fcj(t)gwill be determined from the two conditions (wj;RE(uT)) = 0; forj=1;2;:::;M; (wj;RI(uT)) = 0; forj=1;2;:::;M:(185.6.a-b) The condition in equation (185.6.a) generates Msimultaneous ordinary dif- ferential equations for the fcj(t)jj=1;2;:::;Mg, which will generally be nonlinear. The condition in equation (185.6.b) generates Msimultaneous algebraic equations for fcj(0)jj=1;2;:::;Mg, which will generally be nonlinear. The procedure is as follows. We solve equation (185.6.b) for the initial conditions for the fcj(t)g. Using equation (185.6.a), we can then solve the ordinary dierential equations to determine the fcj(t)gfor all values of t. Using these values in equation (185.2), we have found an approximation to equation (185.1). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 788 IV.B Numerical Methods for ODEs Example Suppose we wish to approximate the solution to the equation ut=N[u]=u2+uxx;for 0<x< 1;t > 0 u(0;x)=s i nx=v(x); u(t;0) = 0; u(t;1) = 1: We choose y(t;x)=xanduj(t;x)=s i njx. Our trial solution then becomes the rst Mterms in a Fourier sine series uT(t;x)=x+MX j=1cj(t)s i njx: Approximating u(t;x)b yuT(t;x) the errors in the equation and the initial conditions are RE(uT)=MX j=1c0 j(t)s i njx−2 4x+MX j=1cj(t)s i njx3 52 −MX j=1j22cj(t) sin(jx); RI(uT)=s i nx−MX j=1cj(0) sinjx:(185.7.a-b) These two equations are in xandt. Ideally, we would like to have both expressions in equation (185.7) vanish identically. Because this is not possible (for all xand allt), we choose one of the four methods described in equation (185.4). Using the chosen method, we will obtain ordinary dierential equations for the fcj(t)gand algebraic equations for the fcj(0)g. When these equations are satised, the expressions in equation (185.7) will be \close" to zero. Notes 1. It is also possible to choose the fui(t;x)gto satisfy the dierential equation (185.1) but not the boundary conditions. In this case, the integral in equation (185.5), which denes the inner product, becomes an integral over the boundary. 2. See the separate sections on collocation (page 514), least squares method (page 549), nite element method (page 734), Rayleigh{Ritz method (page 638), and wavelets (page 784). 3. Within the Galerkin framework, it is possible to generate nite ele- ments, nite dierence, and spectral methods. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 185. Weighted Residual Methods789 4. This method can be used to change the calculation of an ordinary dierential equation to the calculation of the solution of algebraic equations. The sequence of steps are the same as for partial dier- ential equations, with the dierence that both sets of equations in(185.6) will be algebraic equations. See the nite element method (page 734) for a worked example involving an ordinary dierential equation. 5. See also Collatz [1, pages 408{418], Kantorovich and Krylov [5, pages 258{283], and Villadsen and Michelsen [6, Chapter 2, pages 67{95]. References [1]Collatz, L. The Numerical Treatment of Dierential Equations . Springer{ Verlag, New York, 1966. [2]F l e t c h e r ,C .A .J . Computational Galerkin Methods . Springer{Verlag, New York, 1984. [3]Gottlieb, D., and Orszag, S. A. Numerical Analysis of Spectral Methods: Theory and Applications . SIAM, Philadelphia, PA, 1977. [4]Haque, M., Baluch, M. H., and Mohsen, M. F. N. Solution of multiple point, nonlinear boundary value problems by method of weighted residuals.Int. J. Comp. Math. 19 (1986), 69{84. [5]Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis . Interscience Publishers, Inc., New York, 1958. [6]Villadsen, J., and Michelsen, M. L. Solution of Dierential Equation Models by Polynomial Approximation . Prentice{Hall, Inc., Englewood Clis, NJ, 1978. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 790 IV.B Numerical Methods for ODEs CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 792 IV.C Numerical Methods for PDEs 186. Boundary Element Method Applicable to Most often linear elliptic partial dierential equa- tions, often Laplace’s equation. Sometimes parabolic, hyperbolic, or non- linear elliptic equations. Yields An integral equation. The solution of the integral equation is used in an integral representation of the solution. Idea The problem of solving a partial dierential equation within ag i v e n domain can be transformed into one solving an equivalent integral equation onthe boundary of the domain. The unknown in the integral equation will be the \charge density" on the boundary of the domain. Procedure Suppose we have Laplace’s equation (general linear elliptic equations have results analogous to those listed below) r2u(x)=0; (186.1) with the Dirichlet or Neumann data u S=f(x)o r@u @n S=g(x); (186.2.a-b) whereSis the boundary of the domain. Dene (x;y)t ob et h ef r e e space Green’s function of equation (186.1). That is, r2 (x;y)=(x−y), where yis an arbitrary point inside the domain. Using Green’s theorem, the solution to equation (186.1) and equation (186.2) can be represented in any of the following forms: u(x)=Z S(z) (x;z)dz; (186.3) u(x)=Z S(z)@ (x;z) @ndz; (186.4) u(x)=Z S (z) (x;z)+(z)@ (x;z) @n dz: (186.5) In these equations, (z)a n d(z) represent surface densities of the \single- layer" potential, (z)a n d(z) represent the surface densities of the \double- layer" potential, zrepresents a point on the boundary, and nrepresents CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 186. Boundary Element Method 793 the outward pointing normal. If (z),(z), or(z)a n d(z) were known, thenu(x) could be computed via one of the above three equations. Note there is not a unique way to represent the solution by equation (186.5); there is a \degree of freedom" in this formulation that may be used forother purposes. It turns out that the single-layer potential is continuous across the boundaryS, whereas the double-layer potential has a jump of (y). This is because, as xtends to the boundary point Pfrom the inside of the domain, u(P)=−1 2(P)+Z S(z)@ (P;z) @ndz: (186.6) Using equation (186.6), a variety of boundary integral equations may be obtained. For example, using equation (186.4) to represent the solution to the Dirichlet problem, if we allow the point xto approach the boundary, we determine from equation (186.6) that f(y)=−1 2(y)+Z S(z)@ (z;y) @ndz: This Fredholm integral equation of the second kind can, in principle, be solved for(y). After(y) is obtained, the value of u(x) may be computed from equation (186.4). If equation (186.3) had been used to represent the solution of the Neumann problem, then, after nding the normal derivative of equation(186.3), the following integral equation for (y) results g(y)=−1 2(y)+Z S(z)@ (z;y) @ndz: After(y) is obtained by solving the above integral equation, the value of u(x) may be computed from equation (186.3). Example Consider Laplace’s equation in the upper half plane, r2u=0f o r−1< x<1and 0<y, with the boundary conditions uy(x;0) = 0−1<x< 0; uy(x;0)−ku(x;0) = 0 0 <x<1; wherekis a constant. The Green’s function, r2 =(x−)(y−), in the upper half plane is (x;y;;)=−1 2logp (x−)2+(y−)2−1 2logp (x−)2+(y+)2; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 794 IV.C Numerical Methods for PDEs so that, on y=0 ,w eh a v e (x;0;;)=−1 2log/parenleftbig (x−)2+2
.N o w equation (186.3) can be simplied to u(x)=−Z S@u(z) @n (x;z)dz. Using the known values of unand in this expression, we nd u(;)=k 2Z1 0u(x;0) log/parenleftbig (x−)2+2
dx: (186.7) If we dene (x)=u(x;0), then evaluation of equation (186.7) at =0 results in ()=k Z1 0(x)l o gjx−jdx: After this integral equation is solved for (x), the solution is given by equation (186.7). Notes 1. Representing the solution in the form of equation (186.5) would be appropriate if the boundary conditions were mixed. 2. This technique has also been applied to the biharmonic equation in several applications. See Ingham and Kelmanson [7] for details. 3. After the boundary integral equation has been formulated, it is often solved numerically. Some numerical techniques for these equations can be found in Banerjee and Buttereld [1]. In practice one nds that the solution to the original elliptic equation could have been de- termined by solving a large sparse matrix system, while the boundaryelement method often requires that a smaller, dense, matrix system be solved to determine the potential. A worked example is shown in Lapidus and Pinder [8, pages 461{481]. 4. The principle advantage of the reformulation in this section is that the dimensionality of the problem is reduced. As in the above exam- ple, a two-dimensional partial dierential equation becomes a one- dimensional integral equation. 5. For problems in innite domains, the behavior at innity is (usually) automatically included in the boundary element formulation. Hence, there is no need for a \remote" boundary simulating an innite distance. See Margulies [9]. 6. The boundary element method has also been applied to parabolic equations; see Duran et al. [5] or Zamani [11]. It has also been applied to some hyperbolic equations; see Brebbia [4, Chapter 12, pages 191{199]. For an application to nonlinear elliptic equations, see Ingham and Kelmanson [7, Chapter 4]. 7. The boundary element method and the nite element method have several features in common. See Brebbia [4, Chapter 9, pages 141{ 158] for a general account of the similarities and dierences. 8. The presentation here has been for the indirect boundary element method. In this formulation, an integral equation for the potential CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 186. Boundary Element Method 795 must be solved and then the solution to the original equation is given by an integral. It is also possible to directly determine an integral equation whose solution also satises the original equation. This is called the direct boundary element method. For example, given Laplace’s equation, r2= 0, if we dene the Green’s function G(x;y) byr2G=(x−y), then by Green’s theorem 1 2(y)=Z (Gr2−r2G)dV =Z G@ @n−@G @n dS: This integral equation can be solved directly for . 9. See Garabedian [6, Section 9.3, pages 334{348]. References [1]Banerjee, R., and Butterfeld, P. K. Boundary Element Methods in Engineering Science . McGraw{Hill Book Company, New York, 1981. [2]Brebbia, C. A. Boundary Element Techniques in Computer Aided Engineering . Martinus Nijho Publishers, Boston, 1984. [3]Brebbia, C. A. ,E d . Topics in Boundary Element Research. Volume 1: Basic Principles and Applications . Springer{Verlag, New York, 1984. [4]Brebbia, C. A. Topics in Boundary Element Research. Volume 2: Time- Dependent and Vibration Problems . Springer{Verlag, New York, 1985. [5]Duran, D., Cross, M., and Lewis, B. A. A preliminary analysis of boundary element methods applied to parabolic partial dierentialequations. In New Developments in Boundary Element Methods ,C .A . Brebbia, Ed. Butterworth{Heinemann, 1981, pp. 179{190. [6]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [7]Ingham, D. B., and Kelmanson, M. A. Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems . Springer{Verlag, New York, 1984. [8]Lapidus, L., and Pinder, G. F. Numerical Solution of Partial Dierential Equations in Science and Engineering . John Wiley & Sons, New York, 1982. [9]Margulies, M. Exact treatment of the exterior problem in the combined FEM{BEM. In New Developments in Boundary Element Methods ,C .A . Brebbia, Ed. Butterworths, London, England, 1980, pp. 43{64. [10]Wardle, L. An introduction to the boundary element method. In Computational Techniques for Dierential Equations ,J .N o y e ,E d .N o r t h { Holland Publishing Co., New York, 1984, pp. 525{549. [11]Zamani, N. Some remarks on the use of the boundary element method in transient heat conduction problems. Mathematics and Computers in Simulation 27 (1985), 61{64. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 796 IV.C Numerical Methods for PDEs 187. Dierential Quadrature Applicable to Nonlinear partial dierential equations, a single equation, or a system. Most often, partial dierential equations in two independent variables. Yields A system of ordinary dierential equations whose solution approximates the solution of the original partial dierential equation(s). Idea All of the derivatives with respect to one or more of the independent variables are replaced by a sum involving the dependent variable. Procedure To illustrate the general technique, we show how it works on a class of partial dierential equations. Suppose we have the partial dierential equation for u(t;x) ut=g(t;x;u;ux;uxx); u(0;x)=h(x);(187.1) ont>0,−1<x<1. Instead of solving equation (187.1) for all values of x, we choose a nite set of xvalues at which the solution will be determined, sayS=fxjjj=1;:::;Ng. We now presume that the rst derivatives with respect to x, at the points inS, can be written as a linear combination of the values inS.T h a ti s , ux(t;xi)’NX j=1aiju(t;xj): (187.2) Viewing equation (187.2) as the linear transformation ux=Au, it seems natural to approximate uxx=Aux=A2u,o r uxx(t;xi)’NX k=1NX j=1aikakju(t;xj): (187.3) Utilizing equations (187.2) and (187.3) in equation (187.1) results in the system of ordinary dierential equations ui t=g0 @t;xi;ui;NX j=1aijuj;NX k=1NX j=1aikakjuj1 A; ui(0) =h(xi); CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 187. Dierential Quadrature 797 fori=1;:::;N ,w h e r eui(t): =u(t;xi). These initial value ordinary dierential equations may be integrated numerically by any scheme. Note that this method is similar to the method of lines (see page 831), except that the aijarenotchosen in such a way that equation (187.2) represents a nite dierence approximation to the derivative. The aijare instead chosen so that equation (187.2) is exact for all polynomials of degree less than or equal to N−1. That is, the aijsatisfy the linear system k(xi)k−1=NX j=1aij(xj)k: (187.4) fork=1;2;:::;N . Example We choose to numerically approximate the solution to the nonlinear partial dierential equation ut=uux; u(0;x)=0:2x2; which has the exact solution u=0:2(x+ut)2,o r u(t;x)=[1−(0:4)tx]−p 1−(0:8)tx (0:4)t2: The program shown in program 187.1 uses twenty xvalues in the interval from 0 to 1. Note that the source code for the linear equation solver (LSOLVE ) is not shown. Some results of the program are shown next: The time is now: 0.5000 Here is the approximate solution at this time value: 0.0005 0.0020 0.0046 0.0083 0.0132 0.0192 0.0264 0.0348 0.0446 0.0556 0.0681 0.0820 0.0974 0.1143 0.1328 0.1530 0.1750 0.1985 0.2241 0.2620 Here is the exact solution at this time value: 0.0005 0.0020 0.0046 0.0083 0.0132 0.0192 0.02640.0349 0.0446 0.0557 0.0682 0.0822 0.0977 0.11470.1334 0.1538 0.1760 0.2000 0.2260 0.2540 The time is now: 0.7500 Here is the approximate solution at this time value: 0.0005 0.0021 0.0047 0.0085 0.0135 0.0198 0.02740.0365 0.0470 0.0591 0.0729 0.0885 0.1060 0.12550.1471 0.1712 0.1977 0.2255 0.2687 0.5368 Here is the exact solution at this time value: 0.0005 0.0021 0.0047 0.0085 0.0135 0.0198 0.02750.0365 0.0471 0.0593 0.0732 0.0889 0.1066 0.1263 0.1484 0.1728 0.2000 0.2301 0.2634 0.3002 Att=0:75, with the last value shown excluded, the relative error in the approximate solution is not more than 4%. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 798 IV.C Numerical Methods for PDEs DIMENSION X(50),U(50),UNEW(50),A(50,50),CORECT(50) DIMENSION SAVE(50,50),COEFF(50,50),RHS(50),NROW(50),SOLN(50) C Set up the parameter values N=20 TIME=0DELTAT=0.05NSTEP=15 C Set up the X points DO 10 J=1,N 10 X(J)=FLOAT(J)/FLOAT(N) C Set up the coefficient matrix DO 20 K=1,NDO 20 J=1,N 20 SAVE(K,J)=X(J)**K C For each I, determine A_[IJ] by solving a system of equations DO 40 I=1,NDO 30 K=1,NRHS(K)=K*X(I)**(K-1)DO 30 J=1,N 30 COEFF(J,K)=SAVE(J,K) CALL LSOLVE(N,COEFF,SOLN,RHS,NROW,IFSING,50) IF( IFSING .NE . 1 ) STOP DO 40 J=1,N 40 A(I,J)=SOLN(J)C Set up the initial conditions DO 50 J=1,N 50 U(J)=U0( X(J) )C This is the loop in time DO 100 LOOPT=1,NSTEPTIME=TIME + DELTAT WRITE(6,5) TIME C Iterate each one of the equations one time step DO 70 J=1,NSUM=0DO 60 K=1,N 60 SUM=SUM + A(J,K)*U(K) 70 UNEW(J)= U(J) + DELTAT * U(J) * SUM DO 80 J=1,N 80 U(J)=UNEW(J)C Write out the approximate answer, and then the exact answer WRITE(6,*) ’ Here is the approximate solution at this time value:’ WRITE(6,15) (U(J), J=1,N) DO 90 J=1,N 90 CORECT(J)=EXACT(TIME, X(J) ) WRITE(6,*) ’ Here is the exact solution at this time value:’ 100 WRITE(6,15) (CORECT(J), J=1,N) 5 FORMAT(’ The time is now:’,F10.4) 15 FORMAT( 30( 1X, 7(F9.4,1X) / ) ) END C This function has the initial conditions FUNCTION U0(X)U0=0.2*X**2 RETURN END C This function has the exact solution FUNCTION EXACT(T,X)TEMP=( 1.0 - (0.4)*T*X ) - SQRT( 1.0 - (0.8)*T*X ) EXACT=TEMP / ( (0.4)*T**2 ) RETURNEND Program 187.1: Fortran program for dierential quadrature. Notes 1. Note that the coecient matrix in equation (187.4) is a Vandermonde matrix. 2. It is not clear that having the xvalues uniformly spaced produces the most accurate results. In Bellman et al. [1] thexvalues are chosen to be the roots of Legendre polynomials. 3. In Bellman et al. [1], a simple error analysis is performed. It is shown, for example, that the error in equation (187.2) is less than KhN−1=(N−1)! if the mesh has a uniform spacing of hand if ju(N)(x)jKin the domain of interest. 4. In Civan and Sliepcevich [3] a weighted sum of terms (similar to the approximation in equation (187.2)) is used to approximate the CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 187. Dierential Quadrature 799 second derivative terms (such as in equation (187.3)). This reduces the computational complexity of the coding. References [1]Bellman, R., Kashef, B. G., and Casti, J. Dierential quadrature: A technique for the rapid solution of nonlinear partial dierential equations. J. Comput. Physics 10 (1972), 40{52. [2]Civan, F., and Sliepcevich, C. M. Solution of the Poisson equation by dierential quadratures. Internat. J. Numer. Methods Eng. 19 (1983), 711{ 724. [3]Civan, F., and Sliepcevich, C. M. Dierential quadrature for multi- dimensional problems. J. Math. Anal. Appl. 101 (1984), 423{443. [4]Civan, F., and Sliepcevich, C. M. On the solution of the Thomas{Fermi equation by dierential quadrature. J. Comput. Physics 56 (1984), 343{348. [5]Naadimuthu, G., Bellman, R., Wang, K. M., and Lee, E. S. Dier- ential quadrature and partial dierential equations: Some numerical results. J .M a t h .A n a l .A p p l .9 8 (1984), 220{235. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 800 IV.C Numerical Methods for PDEs 188. Domain Decomposition Applicable to Elliptic second order partial dierential equations in non-regularly shaped domains. Yields An iterative solution procedure. Idea If the geometric domain in which a partial dierential equation is to be solved can be written as the union of two (or more) regularly shaped domains, then it may be possible to write a recurrence relation for thesolution. Procedure Suppose we wish to numerically approximate the solution to the elliptic equation N[u]=F(x;y;u;ux;uy;uxx;uxy;uyy) = 0 (188.1) in the domain B=B1[B2(see gure 188.1). We presume this is a Dirichlet problem, with the initial data, f(x;y), given on the boundary of B. Dene the part of the boundary of B1(@B1) that is also a boundary of Bto be; the rest of the B1boundary of B1will be denoted by . Likewise, dene the part of the boundary of B2(@B2) that is also a boundary of B to be; the rest of the B2boundary of B2will be denoted by . The solution procedure is to rst solve equation (188.1) only in B1. Then, using this solution, we solve equation (188.1) only in the domain B2. This is used to nd a new solution of equation (188.1) in B1, and then the process is repeated. Initially, the data on the arc are chosen so that the data on @B1 are piecewise continuous. That is, let u1(x;y) be the solution of equation (188.1) with the boundary conditions u1(x;y)=( f(x;y)o n; (x;y)o n; where(x;y) can be chosen in many dierent ways. After u1(x;y) is deter- mined, letv1(x;y) be the solution of equation (188.1) with the boundary conditions v1(x;y)=( f(x;y)o n; u1(x;y)o n: CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 188. Domain Decomposition 801/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /./././././. /././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./././. /./././././././././././././. /././././. /././././././././././././././././././././. /0 /1 /2 u /=/0 x /0 /1/2u /=/0 y u /=/0u /=/0 u /= g /#28 y /#29u /= f /#28 x /#29 Figure 188.1: The domain for equation (188.1). Then an iterative sequence of solutions to equation (188.1) is formed, fuk(x;y),vk(x;y)jk=1 ,2 ,:::gwith uk(x;y)=( f(x;y)o n; vk−1(x;y)o n; vk(x;y)=( f(x;y)o n; uk(x;y)o n: Under fairly general conditions, these functions will converge to the solution of equation (188.1). That is, the limiting uk(x;y) will be the solution to equation (188.1) in the region B1, whereas the limiting vk(x;y) will be the solution to equation (188.1) in the region B2. In Kantorovich and Krylov [6, Chapter 7, pages 616{670], ve assump- tions are given that are required to assure the convergence of the above sequences. They are 1. Equation (188.1), with its boundary conditions, has a unique solution. 2. IfF[u]=F[u] = 0, and u>u on the boundary of the domain, thenu>ueverywhere in the domain. 3. Within the domain, the solution to equation (188.1) is bounded by the values of uon the boundary of the domain. 4. A convergent sequence of uniformly bounded solutions to equation (188.1) converges to a solution to equation (188.1). 5. The boundary data are, at least, piecewise continuous. Generally, non-pathological examples should satisfy these conditions. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 802 IV.C Numerical Methods for PDEs/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /./. /. /. /. /./. /. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /. /./. /. /./. /. /./. /./. /. /./. /./. /. /./. /./. /./. /./. /./. /./. /./. /./. /././. /./. /././. /./. /././. /./././. /././. /././././. /./././././. /././././././././././././././././././././././././././././././././././././././././././././././. /././. /././././././. /././. /. /././. /././././././././././././././././././././././././././././././././././././././././././././././././. /./././././. /././././. /././. /./././. /././. /./. /././. /././. /./. /./. /./. /./. /./. /./. /./. /./. /./. /. /./. /./. /. /./. /./. /. /./. /. /./. /. /./. /. /. /./. /. /./. /. /. /./. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././. /#0B /#0CB/1 /#16/#0C /#16 /#0B B/2 Figure 188.2: The domain for equation (188.2). Example Suppose we want to solve Laplace’s equation in the L-shaped region shown in gure 188.2. For brevity, we dene the following portions of the boundary Γ1=fx=2;0y1g[f 0x2;y=0g[fx=0;0y1g; Γ2=fx=0;0y2g[f 0x1;y=0g[f 0x1;y=2g: Then, the mathematical problem we wish to solve is r2u=0; u=0; on Γ 1; u=0; on Γ 2; u=f(x); onf1x2;y=1g; u=g(y); onfx=1;1y2g:(188.2) For this example, we break up the original domain into two rectangles, one vertical and one horizontal; the overlap region being the unit square. We start with r2u1=0; u1=0; on Γ 1; u1=f(x); onf1x2;y=1g; u1=(x); onf0x1;y=1g:(188.3) Then, our iteration sequence becomes r2vk=0; vk=0; on Γ 2; vk=uk−1(1;y);onfx=1;0y1g; vk=g(y); onfx=1;1y2g;(188.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 188. Domain Decomposition 803 fork=1;2;:::, whereas r2uk=0; uk=0; on Γ 1; uk=f(x); onf1x2;y=1g; uk=vk(x;1); onf0x1;y=1g;(188.5) fork=2;3;:::. In this case, because of the simple geometry, we can analytically write the solution to equation (188.4) and equation (188.5) by the use of Fourier transforms (see page 350). Note rst, if we dene fn(x)=un(x;1) = P1 k=1fnksinkx,t h e nun(x;y)=1X k=1fnk sinh(k=2)sinhkysinkx. Sim- ilarly, if we dene the expansion gn(x)=vn(1;y)=P1 k=1gnksinky, then we obtain the result vn(x;y)=1X k=1gnk sinh(k=2)sinhkxsinky. Using these expansions in equations (188.4) and (188.5), we can readily determine that fnk=Bk+1X s=1Aksgn−1;s; gnk=Ck+1X s=1Aksfn−1;s;(188.6) where Bk=Z2 1f(x) sin(kx= 2)dx; Ck=Z2 1g(y)s i n (ky= 2)dy; Aks=2 1 s2+k2 ssink 2 coshs 2 −kcosk 2 sinhs 2 : In practice, the two recurrence relations in equation (188.6) would be iterated until a stationary value was obtained. Notes 1. This method is usually implemented numerically, with little analysis done on the equations. For the above example, equations (188.3){(188.5) would be approximated numerically by an elliptic equation package. 2. This method also works for coupled systems of elliptic equations. For two unknowns, a guess is made for one of the unknowns, and one of the equations is used to solve for the other unknown. Knowing this CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 804 IV.C Numerical Methods for PDEs second unknown, the rst unknown is approximated numerically by the other equation, and the process is repeated. See Rice and Boisvert [9, pages 121{135] for some examples. 3. The procedure illustrated in this section is called Schwarz’s method , it is only one of several dierent domain decomposition methods (see Glowinski et al. [5]). 4. In Chan et al. [3] it is shown that the convergence rate of the Schwarz alternating procedure, for general second-order elliptic equa- tions, is independent of the aspect ratio for L-shaped, T-shaped, andC-shaped domains. 5. This technique works very well with parallel computers (see page 755), as the numerical problem on each domain can be solved by aseparate processor; see Quarteroni [8]. References [1]Canuto, C., and Funaro, D. The Schwarz algorithm for spectral methods. SIAM J. Numer. Anal. 25 , 1 (February 1988), 24{40. [2]Chan, T. F., Glowinski, R., Periaux, J., and Widlund, O. B. ,E d s . Domain Decomposition Methods . SIAM, Philadelphia, PA, 1989. [3]Chan, T. F., Hou, T. Y., and Lions, P. L. Geometry related convergence results for domain decomposition algorithms. SIAM J. Numer. Anal. 28 ,2 (April 1991), 378{391. [4]Ehrlich, L. W. The numerical Schwarz alternating procedure and SOR. SIAM J. Sci. Stat. Comput. 7 , 3 (July 1986), 989{993. [5]Glowinski, R., Golub, G., Meurant, G., and Periaux, J. ,E d s . First International Symposium on Domain Decomposition Methods for PartialDierential Equations . SIAM, Philadelphia, PA, 1988. [6]Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis . Interscience Publishers, Inc., New York, 1958. [7]Meier, U. Two parallel SOR variants of the Schwarz alternating procedure. Parallel Comp. 3 , 3 (1986), 205{215. [8]Quarteroni, A. Domain decomposition and parallel processing for the numerical solution of partial dierential equations. Surv. Math. Ind. 1 (1991), 75{118. [9]Rice, J. R., and Boisvert, R. F. Solving Elliptic Problems Using ELLPACK . Springer{Verlag, New York, 1985. [10]T a n g ,W .P . Generalized Schwarz splittings. SIAM J. Sci. Stat. Comput. 13, 2 (1992), 573{595. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 189. Elliptic Equations: Finite Dierences 805 189. Elliptic Equations: Finite Dierences Applicable to Elliptic partial dierential equations. Yields A numerical approximation of the solution. Idea By use of nite dierences, a simultaneous system of equations may be determined. The solution of this algebraic system (which is often a linearsystem of equations) yields a numerical approximation to the dierential equation. Procedure The method is simply to use nite dierences everywhere and solve the resulting set of simultaneous equations. Because elliptic equations are boundary value problems, the solution at all points in the domain must bedetermined simultaneously. We choose to illustrate the method on a second order elliptic equation of the form u xx+uyy=f(x;y;u;ux;uy); (189.1) whereandare functions of xandy. We suppose that equation (189.1) applies inside a rectangle with axA,byBand that the boundary conditions for equation (189.1) are u(x;y)=8 >< >:f(y);onx=a; g(y);onx=A; h(x);ony=B;(189.2) @u @y+@u @x+u3=j(x);ony=b; (189.3) whereff;g;h;jgare all known functions. We rst dene a grid that lls the geometric domain (see page 675). For the rectangular geometry given, we choose a rectangular grid with an xspacing of hand ayspacing of k(whereh=(A−a)=(N−1), and k=(B−b)=(M−1)). Here,N(M) is the number of grid points in the x (y) direction (see gure 189.1). Let the numerical approximation to u(x;y) be given by vij(i.e.,vij’u(a+ih;b+jk)). We can then choose virtually any nite dierence approximation to the derivatives appearing in equation (189.1). For instance, one second order approximation to equation (189.1) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 806 IV.C Numerical Methods for PDEs/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././. /./././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /././. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./././. /./././././././././././././. /././././. /././././././././././././././././././././.x/1 /= a xN /= A x y/1 /= b yM /= B y kh Figure 189.1: The numerical grid on which the problem is to be solved. would be ijvi+1;j−2vi;j+vi−1;j h2+ijvi;j+1−2vi;j+vi;j−1 k2 =f a+ih;b+jk;vij;vi+1;j−vi−1;j 2h;vi;j+1−vi;j−1 2k :(189.4) For eachiandj, equation (189.4) represents an algebraic equation among thefvijg. Now the boundary conditions must be incorporated. The boundary conditions in equation (189.2) can be written simply as v0;j=f(b+jk); forj=1;2;:::;M; vN;j=g(b+jk); forj=1;2;:::;M; vi;m=h(a+ih); fori=1;2;:::;N:(189.5) The boundary condition in equation (189.3) can be written as vi;1−vi;0 k+vi+1;0−vi;0 h+(vi;j)3=j(a+ih)f o ri=1;2;:::;N: (189.6) If equation (189.4) is evaluated for j=1;2;:::;M andi=1;2;:::;N , and equation (189.5) and equation (189.6) are included, there results a si- multaneous system of equations for the fvijg. There are as many equations as there are unknowns. This system may then be solved numerically. If the original elliptic equation (189.1) and the boundary conditions are linear in the independent variable, then the resulting system of equations will be linear. For this example, equation (189.6) is not linear (note the(v i;j)3term) because there is a u3term in equation (189.3). The most common type of elliptic systems have linear equations and linear boundary CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 189. Elliptic Equations: Finite Dierences 807/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./././././././././././././././././././././0 /1 /3 /2/3 /1 x /0 /1/3 /2/3 /1 y/#0F /#0F/#0F/#0F/#0F /#0F/#0F/#0F/#0F /#0F/#0F/#0F/#0F /#0F/#0F/#0F /1/1 /3 /2/3/0 /1v/2/3v/2/2/0 /1v/3/3v/3/2/0 /1/1/3 /4/9/0 Figure 189.2: The grid on which equation (189.8) is solved. conditions. For this type of elliptic system, a standard linear equation solver may be used. If the system of linear equations is too large to solve directly, an iterative method may be used (see page 816). Example Suppose we have the linear elliptic equation (x+1 )uxx+(y+1 )2uyy=1+u; (189.7) on 0x1, 0y1w i t h u(0;y)=y; u (1;y)=y2; u(x;0) = 0;u(x;1) = 1:(189.8) If we choose M=N=4( s ot h a t h=k=1=3), then there are 16 points fvijj1i4;1j4gat which to determine an approximation to u(x;y). The pointsfvijji=1o ri=4o rj=1o rj=4gare determined directly by the boundary conditions in equation (189.8). Hence, the onlyunknowns that need to be determined are fv 22;v23;v32;v33g; see gure 189.2. If equation (189.7) is discretized as (ih+1 )vi+1;j−2vi;j+vi−1;j h2 +(jk+1 )2vi;j+1−2vi;j+vi;j−1 k2=1+vij; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 808 IV.C Numerical Methods for PDEs EQUATION. (X+1)*UXX+(Y+1)**2*UYY=1.0 + U BOUNDARY. U=Y ON X=0.0 U=Y**2 ON X=1.0U=0.0 ON Y=0.0U=1.0 ON Y=1.0 GRID. 4 X POINTS 4 Y POINTS DISCRETIZATION. 5 POINT STAR SOLUTION. LINPACK BANDOUTPUT. TABLE(U) PLOT(U) END. Program 189.1: ELLPACK program for an elliptic problem. then the equations for the unknown fvijgmay be written as 2 66457=9−16=9−4=30 −25=92 5=30−4=3 −5=30 7 −16=9 0−5=3−25=993 7752 664v22 v23 v32 v333 775=2 664−5=9 24=9 −22=27 68=273 775: (189.9) The equations in equation (189.9) have the solution v22’0:0131,v23’ 0:3791,v32’−0:0265,v33’0:3419. Notes 1. The computer language ELLPACK (see Rice and Boisvert [4] is a high-level language that allows linear elliptic problems in two orthree dimensions to be entered in an elementary way. The program generates a discretization scheme based on user preference. The geometry in two dimensions can be nearly arbitrary, with holes and other cutouts available. For example, to solve the problem in the example, the entire ELLPACK program is given in program 189.1. The use of ELLPACK for two and three-dimensional problems is highly recommended. There is also a version of ELLPACK available for parallel computation. 2. Picard iteration (see page 618), Newton’s method (see page 578), and Monte-Carlo methods (see page 810) can also be used to numerically approximate the solution to elliptic problems. 3. Boisvert and Sweet [2] have a comprehensive listing of currently available software for solving elliptic problems. 4. See Twizell [5, pages 42{80]. References [1]Birkoff, G., and Lynch, R. Numerical Solution of Elliptic Problems . SIAM, Philadelphia, PA, 1984. [2]Boisvert, R. F., and Sweet, R. A. Mathematical software for elliptic boundary value problems. In Sources and Development of Mathematical CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 189. Elliptic Equations: Finite Dierences 809 Software , W. R. Cowell, Ed. Prentice{Hall, Inc., Englewood Clis, NJ, 1984, pp. 200{263. [3]Dyksen, W. R., and Ribbens, C. J. Interactive ELLPACK: An interactive problem{solving environment for elliptic partial dierential equations. ACM Trans. Math. Software 13 , 2 (June 1987), 113{132. [4]Rice, J. R., and Boisvert, R. F. Solving Elliptic Problems Using ELLPACK . Springer{Verlag, New York, 1985. [5]Twizell, E. H. Computational Methods of Partial Dierential Equations . Ellis Horwood Limited, Chichester, England, 1984. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 810 IV.C Numerical Methods for PDEs 190. Elliptic Equations: Monte-Carlo Method Applicable to Linear elliptic partial dierential equations. Yields A numerical approximation to the solution of a linear elliptic partial dierential equation at a single point. Idea Simulation of the motion of a random particle may be used to approx- imate the solution to linear elliptic equations. Procedure The steps for this method are straightforward. First, we give an overview; then, a more detailed presentation. First, approximate the given elliptic partial dierential equation by a nite dierence method. Rewrite the nite dierence formula as a recursivefunction for the value of the unknown at any given point. Then interpret this recursive formula as a set of transition probabilities that determine the motion of a random particle. Now, write a computer program that will allow many (say K) particles to wander randomly around the domain of interest, based on the transition probabilities found from the dierence formula. Simulate particles one ata time, with every particle starting o at the same point (say the point z). If the boundary data are of the Dirichlet type (i.e., the value of the unknown is prescribed on the boundary), then, when a particle reaches the boundary, stop that particle and store away the value on the boundary. Begin another particle at the point z. If the boundary data are not of the Dirichlet type (i.e., Neumann or mixed boundary conditions) then, when the particles reach the boundary, they will be given a nite probability to leave the boundary, and re-enter the domain of the problem. If the particle leaves the boundary, then continue the iteration process. If it does not leavethe boundary, then the value at the boundary is stored away, and a new particle is started o at the point z. The simulation is nished after all Kparticles have been absorbed into the boundary. If the original elliptic equation was homogeneous, then an approximation to the solution, at the point z, will be given by the average of all the values obtained (recall that when the particles stop at the boundary they obtain a value). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 190. Elliptic Equations: Monte-Carlo Method 811 If the given elliptic equation was not homogeneous, then equation (190.4) shows how to obtain an approximation to the solution. In this latter case, the approximate value of the solution depends on the entire history of the particle. In more detail, we now describe how the technique may be applied to the linear second order elliptical partial dierential equation L[u]=F(x;y); (190.1) with the operator L[
] dened by L[u]=Auxx+2Buxy+Cuyy+Dux+Euy; wherefA;B;C;D;Egare all functions of fx;yg. The operator L[
]m a yb e discretized to yield the approximation L[u]’Ai;jvi+1;j−2vi;j+vi−1;j (
x)2 +2Bi;jvi+1;j+1−vi;j+1−vi+1;j+vi;j (
x)(
y) +Ci;jvi;j+1−2vi;j+vi;j−1 (
y)2 +Di;jvi+1;j−vi;j
x +Ei;jvi;j+1−vi;j
y ;(190.2) wherexi=x0+i(
x),yj=y0+j(
y),vi;j=u(xi;yj), and a subscript ofi;jmeans an evaluation at the point ( xi;yj). If thefΓ
;
gandQi;jare dened by Γi+1;j+1=2Bi;j (
x)(
y) ; Γi+1;j=Ai;j (
x)2−2Bi;j (
x)(
y)+Di;j
x ; Γi;j+1=Ci;j (
y)2−2Bi;j (
x)(
y)+Ei;j
y ; Γi−1;j=Ai;j (
x)2 ; Γi;j−1=Ci;j (
x)2 ; Qi;j=2Ai;j (
x)2−2Bi;j (
x)(
y)+2Ci;j (
y)2+Di;j
x+Ei;j
y ; then, using equation (190.2), equation (190.1) may approximated as Qi;jvi;j=Γi+1;jvi+1;j+Γi+1;j+1vi+1;j+1+Γi;j+1vi;j+1 +Γi−1;jvi−1;j+Γi;j−1vi;j−1−Fi;j; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 812 IV.C Numerical Methods for PDEs or dividing through by Qi;jand dening pi;j=Γi;j=Qi;j, vi;j=pi+1;jvi+1;j+pi+1;j+1vi+1;j+1+pi;j+1vi;j+1 +pi−1;jvi−1;j+pi;j−1vi;j−1−Fi;j Qi;j:(190.3) Because the operator L[
] has been presumed to be elliptic, then
xand
ymay be chosen small enough so that each of the p’s are positive. The p’s also add up to one, and we interpret them as probabilities of taking a step in a specied direction. Specically, equation (190.3) is interpreted as follows: If a particle is at position ( i;j)a ts t e pN, then, With probability pi;j+1, the particle goes to ( i;j+1 )a ts t e p N+1 . With probability pi;j−1, the particle goes to ( i;j−1) at stepN+1 . With probability pi+1;j, the particle goes to ( i+1;j)a ts t e pN+1 . With probability pi−1;j, the particle goes to ( i−1;j)a ts t e pN+1 . With probability pi+1;j+1, the particle goes to ( i+1;j+1 )a ts t e p N+1 . Now, suppose a particle starts at the point P0=zand undergoes a random walk according to the above prescription. After, say, msteps it will hit the boundary. Suppose that the sequence of points that this particlevisits is (P 0;P1;P2;:::;Pm). Then, an unbiased estimator of the value of u(z) for the following elliptic problem L[u]=F(x;y);for all points x;yin the domain R; u=(x;y);for all points x;yon the boundary @R; is given by u(z)’(Pm)−mX j=0F(Pj) Q(Pj): (190.4) In practice, several random paths will be taken, and the average taken to estimateu(z). That is, u(z)’1 KKX k=18 < :(Pk mk)−mkX j=0F(Pk j) Q(Pk j)9 = ;; (190.5) where (Pk 0;Pk 1;:::;Pk mk), represents the path taken by the kth random particle. Example Suppose we wish to numerically approximate the solution to Laplace’s equation in an annulus. We have r2u=0f o ru(r;) with the boundary CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 190. Elliptic Equations: Monte-Carlo Method 813/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././. /./././. /././././. /./././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /././. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /././././././././././././././././././././.x/2 yu /=/6u /=/4/. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /./. /. /./. /./. /./. /././. /././. /././././././. /./././././././././././././././././././././././././././././././././././././././././././. /././. /././././. /././. /././. /. /././././. /././. /././././././././././././. /./././././././././././././././././././././././././././. /././././. /././. /././. /./. /./. /./. /. /./. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /./. /./. /. /./. /. /./. /. /./. /. /. /. /. /. /. /./. /. /./. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /. /./. /. /. /./. /. /. /./. /./. /. /./. /. /./. /./. /./. /. /./. /./. /./. /././. /./. /./. /././. /./. /././. /././. /./././. /./././. /./././. /././././././. /././././././././././. /././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././././././././././././././. /./././././././././././././././././././././. /././././. /./././././././. /././. /././././. /./././././. /././././. /././. /././. /././. /././. /././. /././. /././. /././. /./././././. /././. /./././././././. /././. /./././././././././././././././././. /./././././././././././. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././././. /././././././././. /./././././. /././././. /././. /./././. /././. /././. /./. /././. /./. /././. /./. /./. /./. /./. /. /./. /./. /./. /. /./. /. /./. /. /./. /. /./. /. /. /./. /. /. /./. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /./. /. /. /. /. /. /. /./. /. /./. /. /. /. /. /. /. /./. /. /./. /. /. /. /./. /. /./. /. /. /. /. /././. /. /././#0F z/./././././././././././. /././././././././././././. /././././././././././././. /./././././././././././. /././././././././././././. /././././././././././././. /./././././././././././. /././././././././././././. /././././././././././././. /./././././././././././. /././././././././././././. /././././././././././././. /./././././././././././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./././././././././././././././././././././././././././././././././././././././././././././././././././././././. /. /. /. /./. /. /./. /. /./. /. /. /. /. /. /. /./. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./3 /1/. /././././././././././././././././././././. /./././././././././././././././././././. /././././././././././././././././././././. /./././././././././././././././././././. /././././././././././././././././././. /./././././././././././././././././. /././././././././././././././././. /./././././././././././././././. /././././././././././././././. /./././././././././././././. /././././././././././././. /./././././././././././. /././././././././././. /./././././././. /././. /././. /./././././././. /././././././././././. /./././././././././././. /././././././././././././. /./././././././././././././. /././././././././././././././. /./././././././././././././././. /././././././././././././././././. /./././././././././././././././././. /././././././././././././././././././. /./././././././././././././././././././. /././././././././././././././././././././. /./././././././././././././././././././. /././././././././././././././././././././. /./././././././././. /./././././././. /./././././././. /./././././././. /./././././././. /./././././././. /././././././. /./././././././. /./././././././. /././././././. /./././././././. /./././././././. /./././././././. /./././././././. /./././././././. /./././././././././. /./././././././././. /./././././././. /./././././././. /./././././././. /./././././././. /./././././././. /././././././. /./././././././. /./././././././. /././././././. /./././././././. /./././././././. /./././././././. /./././././././. /./././././././. /././././././././.Figure 190.1: The domain in which Laplace’s equation is solved. conditions u(1;)=4a n d u(3;) = 6. (See gure 190.1.) We will approximate the value of u(z), when z=(r=2;= 0). The exact solution for this problem is u(r) =4+2l og r=log 3, so that u(z) =4+l og2 =log 3’ 5:261. To approximate the solution to this problem numerically, we will follow the steps outlined above. We will use the rectangular variables x andy, rather than the polar coordinate variables rand. Using a standard second order approximation to the Laplacian, we nd r2u’vi+1;j+vi−1;j+vi;j+1+vi;j−1−4vi;j h2=0; (190.6) wherevi;j=u(hi;hj )a n dh1. Equation (190.6) can be manipulated into vi;j=vi+1;j 4+vi−1;j 4+vi;j+1 4+vi;j−1 4: (190.7) We interpret equation (190.7) probabilistically as follows: If a particle is at position ( i;j)a ts t e pN, then, With probability 1/4, the particle goes to ( i;j+1 )a ts t e p N+1 . With probability 1/4, the particle goes to ( i;j−1) at stepN+1 . With probability 1/4, the particle goes to ( i+1;j)a ts t e pN+1 . With probability 1/4, the particle goes to ( i−1;j)a ts t e pN+1 . Program 190.1 has Fortran code that was used to simulate the motion of the particles according to the above probability law. The output of thatprogram is given below for u(r=2;= 0). As more points are taken, the approximation becomes better. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 814 IV.C Numerical Methods for PDEs STEP=0.10 SUM=0.0DO 10 IWALK=1,10000 X=2.0 Y=0.0 20 X=X + SIGN(STEP, RANDOM(DUMMY)-0.5 ) Y=Y + SIGN(STEP, RANDOM(DUMMY)-0.5 )R=SQRT( X**2+Y**2 )IF( R.LT.3 .AND. R.GT.1 ) GOTO 20 C When a particle hits the boundary, sum the value IF( R .LE. 1) SUM=SUM+4 IF( R .GE. 3) SUM=SUM+6 IF( MOD(IWALK,1000) .NE . 0 ) GOTO 10 APPROX=SUM/FLOAT(IWALK)WRITE(6,5) IWALK,APPROX 5 FORMAT(’ Number of particles=’,I5,’ Approximation=’,F7.4)10 CONTINUE END Program 190.1: Fortran program for Monte-Carlo method applied to elliptic equations. Number of particles= 1000 Approximation= 5.3440 Number of particles= 2000 Approximation= 5.3330Number of particles= 3000 Approximation= 5.3200Number of particles= 4000 Approximation= 5.3195 Number of particles= 6000 Approximation= 5.3030 Number of particles= 7000 Approximation= 5.3023Number of particles= 8000 Approximation= 5.2958Number of particles= 9000 Approximation= 5.2944Number of particles=10000 Approximation= 5.2914 Note that the program uses a routine called RANDOM , whose source code is not given, which returns a random value uniformly distributed on theinterval from zero to one. Notes 1. If further accuracy is required, the options are (a) Increase the number of random particles. (b) Make the mesh discretization ner (i.e., reduce h). (c) Do both of the above. If the number of random particles is not very large, then (b) will not help much; and if the mesh is very coarse then, (a) will not help much. Generally, the variance of the answer (a measure of the \scatter") decreases as the number of trials to the minus one half power. 2. Because low numerical accuracy is obtained by this technique, a computer program does not need to work with extended precision arithmetic. 3. Sadeh and Franklin [8] present several worked examples. See also Farlow [5, pages 346{352] and Latt es [6, Chapter 8, pages 158{190]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 190. Elliptic Equations: Monte-Carlo Method 815 References [1]Bhavsar, V. C., and Gujar, U. G. VLSI algorithms for Monte Carlo solutions of partial dierential equations. In Advances in Computer Methods For Partial Dierential Equations , R. Vichnevetsky and R. S. Stepleman, Eds., IMACS. North{Holland Publishing Co., New York, 1984. [2]Bhavsar, V. C., and Isaac, J. R. Design and analysis of parallel Monte Carlo algorithms. SIAM J. Sci. Stat. Comput. 8 , 1 (1987), 573{595. [3]Booth, T. E. Exact Monte Carlo solution of elliptic partial dierential equations. J. Comput. Physics 30 (1981), 396{404. [4]Booth, T. E. Regional Monte Carlo solution of elliptic partial dierential equations. J. Comput. Physics 47 (1982), 281{290. [5]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [6]Lattes, R. Methods of Resolution for Selected Boundary Problems in Mathematical Physics . Gordon and Breach, New York, 1969. [7]Marshall, G. Monte Carlo methods for the solution of nonlinear partial dierential equations. Comput. Physics Comm. 56 (1989), 51{61. [8]Sadeh, E., and Franklin, M. A. Monte Carlo solution of partial dierential equations by special purpose digital computer. IEEE Transactions on Computers C-23 , 4 (April 1974), 389{397. [9]Vrbik, J. Monte Carlo simulation of the general elliptic operator. J. Phys. A: Math. Gen. 20 (1987), 2693{2697. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 814 IV.C Numerical Methods for PDEs STEP=0.10 SUM=0.0DO 10 IWALK=1,10000 X=2.0 Y=0.0 20 X=X + SIGN(STEP, RANDOM(DUMMY)-0.5 ) Y=Y + SIGN(STEP, RANDOM(DUMMY)-0.5 )R=SQRT( X**2+Y**2 )IF( R.LT.3 .AND. R.GT.1 ) GOTO 20 C When a particle hits the boundary, sum the value IF( R .LE. 1) SUM=SUM+4 IF( R .GE. 3) SUM=SUM+6 IF( MOD(IWALK,1000) .NE . 0 ) GOTO 10 APPROX=SUM/FLOAT(IWALK)WRITE(6,5) IWALK,APPROX 5 FORMAT(’ Number of particles=’,I5,’ Approximation=’,F7.4)10 CONTINUE END Program 190.1: Fortran program for Monte-Carlo method applied to elliptic equations. Number of particles= 1000 Approximation= 5.3440 Number of particles= 2000 Approximation= 5.3330Number of particles= 3000 Approximation= 5.3200Number of particles= 4000 Approximation= 5.3195 Number of particles= 6000 Approximation= 5.3030 Number of particles= 7000 Approximation= 5.3023Number of particles= 8000 Approximation= 5.2958Number of particles= 9000 Approximation= 5.2944Number of particles=10000 Approximation= 5.2914 Note that the program uses a routine called RANDOM , whose source code is not given, which returns a random value uniformly distributed on theinterval from zero to one. Notes 1. If further accuracy is required, the options are (a) Increase the number of random particles. (b) Make the mesh discretization ner (i.e., reduce h). (c) Do both of the above. If the number of random particles is not very large, then (b) will not help much; and if the mesh is very coarse then, (a) will not help much. Generally, the variance of the answer (a measure of the \scatter") decreases as the number of trials to the minus one half power. 2. Because low numerical accuracy is obtained by this technique, a computer program does not need to work with extended precision arithmetic. 3. Sadeh and Franklin [8] present several worked examples. See also Farlow [5, pages 346{352] and Latt es [6, Chapter 8, pages 158{190]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 190. Elliptic Equations: Monte-Carlo Method 815 References [1]Bhavsar, V. C., and Gujar, U. G. VLSI algorithms for Monte Carlo solutions of partial dierential equations. In Advances in Computer Methods For Partial Dierential Equations , R. Vichnevetsky and R. S. Stepleman, Eds., IMACS. North{Holland Publishing Co., New York, 1984. [2]Bhavsar, V. C., and Isaac, J. R. Design and analysis of parallel Monte Carlo algorithms. SIAM J. Sci. Stat. Comput. 8 , 1 (1987), 573{595. [3]Booth, T. E. Exact Monte Carlo solution of elliptic partial dierential equations. J. Comput. Physics 30 (1981), 396{404. [4]Booth, T. E. Regional Monte Carlo solution of elliptic partial dierential equations. J. Comput. Physics 47 (1982), 281{290. [5]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [6]Lattes, R. Methods of Resolution for Selected Boundary Problems in Mathematical Physics . Gordon and Breach, New York, 1969. [7]Marshall, G. Monte Carlo methods for the solution of nonlinear partial dierential equations. Comput. Physics Comm. 56 (1989), 51{61. [8]Sadeh, E., and Franklin, M. A. Monte Carlo solution of partial dierential equations by special purpose digital computer. IEEE Transactions on Computers C-23 , 4 (April 1974), 389{397. [9]Vrbik, J. Monte Carlo simulation of the general elliptic operator. J. Phys. A: Math. Gen. 20 (1987), 2693{2697. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 816 IV.C Numerical Methods for PDEs 191. Elliptic Equations: Relaxation Applicable to Elliptic equations, most often Laplace’s equations. Yields A numerical approximation to the solution. Idea The nite dierence scheme for an elliptic equation can be interpreted as a local condition on the value of the solution. This local condition leadsnaturally to an iterative numerical procedure. Procedure Given an elliptic equation, choose a nite dierence formula to approx- imate the equation on a grid in the domain of interest. This formula canbe manipulated into a relation between the value of the unknown at a point and the values of the unknown at neighboring points. Hence, once values have been assigned to every point in the grid, this formula can beused iteratively to update the value at every point. When the values stops changing (to some specied precision), an approximate solution has been found. Example Suppose we want to approximate the solution to Laplace’s equation on a square r2u=0; u(0;y)=0;u(1;y)=0;for 0y1; u(x;0) = 0;u(x;1) = 1;for 0<x< 1:(191.1.a-c) If we choose a grid with a uniform xspacing of
xand a uniform yspacing of
y, then equation (191.1.a) can be discretized as 1 (
x)2(vi+1;j−2vi;j+vi−1;j)+1 (
y)2(vi;j+1−2vi;j+vi;j−1)=0; (191.2) wherevi;j=u(i
x;j
y), fori=1;2;:::; 1=
xandj=1;2;:::; 1=
y. Equation (191.2) can be manipulated to yield vi;j=1 2(1 +2)/parenleftbig 2(vi;j+1+vi;j−1)+vi+1;j+vi−1;j
; (191.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 191. Elliptic Equations: Relaxation 817 REAL*8 V(6,6) C Initializel the grid DO 10 I=2,5DO 10 J=2,5 10 V(I,J)=0.25D0C Here is the boundary data DO 20 K=1,6 V(K,1)=0.0D0 V(K,6)=1.0D0V(1,K)=0.0D0 20 V(6,K)=0.0D0C Perform the iterations EPS=0.0001D0NUM=0 40 NUM=NUM+1 IFLAG=0 DO 30 I=2,5DO 30 J=2,5VNEW= ( V(I+1,J) + V(I-1,J) + V(I,J+1) + V(I,J-1) ) / 4.D0IF( DABS(V(I,J)-VNEW) .GT. EPS ) IFLAG=1 30 V(I,J)=VNEWC Determine if another iteration is required IF( IFLAG .EQ. 1 ) GOTO 40 WRITE(6,5) NUM 5 FORMAT(’ Number of iterations required:’, I5) DO 50 J=1,6 50 WRITE(6,15) (V(I,7-J),I=1,6)15 FORMAT( 7(1X,F9.4) ) END Program 191.1: Fortran program for relaxation method. where=
y=
x. From equation (191.3), we see that vi;jcan be replaced by a weighted average of the values at the neighboring points. Note that this is only true for points interior to the boundary. The numerical technique is this: Initialize the values at all points in the grid (one common choice is to use the averaged value of the independent variable on the boundary); then systematically apply equation (191.3) to all the grid points until the solution converges. In theory, the points to be updated can be chosen in any order. In practice, some choices result in faster convergence. The Fortran code in program 191.1 carries out this prescription for the problem in equation (191.1). In this program, h=0:2,k=0:2, and the number of iterative updates required before the approximation didnot change more than EPS(set to 0.0001) was 16. The output from the computer program is given below Number of iterations required: 16 0. 1.0000 1.0000 1.0000 1.0000 0.0. 0.4545 0.5946 0.5946 0.4545 0. 0. 0.2234 0.3294 0.3294 0.2234 0. 0. 0.1097 0.1703 0.1703 0.1098 0.0. 0.0454 0.0718 0.0719 0.0454 0. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 818 IV.C Numerical Methods for PDEs 0. 0. 0. 0. 0. 0. The symmetry of the solution was to be expected. The exact solution to equation (191.1) can be determined by separation of variables (see page 487). The solution is u(x;y)=4 1X n=1sin [(2n−1)x]sinh [(2n−1)y] sinh [(2n−1)]: As can be veried, the numerical approximation is accurate to two decimal places. Notes 1. The equations in (191.2) can be combined into one large system of linear equations, and then iterative methods can be applied to this system. Each dierent iterative method for a linear system can be interpreted as a relaxation method directly on the grid values. 2. Depending on the equation to which this method is applied and on the ordering in which the updated values are obtained, this technique is called Alternating-direction-implicit (ADI) method Gauss{Seidel or successive iteration scheme Jacobi or simultaneous iteration scheme Liebmann’s method. Successive over-relaxation (SOR) method In the ADI method, the nite dierence approximation to Laplace’s equation may be written r2u’u(2n) i;j−1−2u(2n) i;j+u(2n) i;j+1 (
x)2+u(2n+1) i−1;j−2u(2n+1) i;j +u(2n+1) i+1;j (
y)2=0: The superscripts indicate the iteration number. Hence, the updating is done alternately by rows and columns in the array of values. 3. This method, when applied to the elliptic equation L[u]=0 ,c a n be interpreted as an approximation to the solution of the parabolic equationut=L[u]. By iterating until the solution stops changing, the steady-state solution of the parabolic equation is obtained. This interpretation allows error estimates to be obtained for this method(see Garabedian [4]). 4. See also Farlow [2, pages 304{305], Garabedian [3, pages 485{492], Gerald and Wheatley [4, pages 412{417], Isaacson and Keller [5, pages463{478], and Smith [6, Chapter 5, pages 239{330]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 191. Elliptic Equations: Relaxation 819 References [1]Chan, T. F., and Elman, H. C. Fourier analysis of iterative methods for elliptic problems. SIAM Review 31 , 1 (March 1989), 20{49. [2]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [3]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [4]Gerald, C. F., and Wheatley, P. O. Applied Numerical Analysis . Addison{Wesley Publishing Co., Reading, MA, 1984. [5]Isaacson, E., and Keller, H. B. Analysis of Numerical Methods .J o h n Wiley & Sons, New York, 1966. [6]Smith, R. D. Numerical Solution of Partial Dierential Equations: Finite Dierence Methods , third ed. Clarendon Press, Oxford, England, 1985. [7]Vega-Fernandez, J. M., Duque-Carrillo, J. F., and Pe na-Bernal, J. J. A new way for solving Laplace’s problem (the predictor jump method). J. Math. Physics 26 , 3 (March 1985), 416{419. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 820 IV.C Numerical Methods for PDEs 192. Hyperbolic Equations: Method of Characteristics Applicable to A single hyperbolic equation or a system of hyper- bolic equations. Yields A numerical approximation scheme. Idea The method of characteristics (see page 432) can be used directly to create a numerical scheme to integrate hyperbolic equations. Procedure To simplify the analysis, we will illustrate the method on the second order hyperbolic partial dierential equation auxx+buxy+cuyy+d=0: (192.1) In equation (192.1), the functions fa;b;c;dgare assumed to depend on fx;y,u,ux,uyg. With the usual denitions of p=uxandq=uy,e q u a t i o n (192.1) may be rewritten as the system of equations E1:=apx+bpy+cqy+d=0; E2:=py−qx=0: If we dene E=E1+E2,t h e nEmay be written as E=[apx+(+b)py]+(cqy−qx)+d=0: This, in turn, may be written as E=d ds(p+q)+ d−qd ds =0; (192.2) along the curve dened parametrically by dx ds=a=− ;dy ds=+b=c ; (192.3) if such a curve exists. For consistency in the equations in (192.3), we must chooseto satisfya2−b+c=0 ;t h a ti s , 1;2=bp b2−4ac 2a: (192.4) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 192. Hyperbolic Equations: Method of Characteristics 821 Denef1;2gto be the distinct real roots given in equation (192.4) (if the roots are not distinct and real, then equation (192.1) is not hyperbolic), and denei=−ai. Then equations (192.2) and (192.3) can be written as d ds(p+1q)=− d−qd1 ds on the curve C1; d ds(p+2q)=− d−qd2 ds on the curve C2; (192.5) where the characteristic curves C1andC2are dened by onC1:dx ds=a;dy ds=1+b; onC2:dx ds=a;dy ds=2+b: (192.6.a-b) These two characteristics curves have slopes that vary from point to point and are generally not orthogonal. Knowing fa;b;c;dgallows us to de- terminef1;2gand sof1;2gcan also be determined. Therefore, the characteristics curves can be calculated numerically. Now, ifk1:=p+1qandk2:=p+2qwere known at some common pointR(these values arise naturally from equation (192.5)), then p(R)a n d q(R) can be found by inverting these relations; that is q(R)=k1−k2 1−2; p(R)=1k1−2k2 1−2:(192.7) The numerical procedure is now a straightforward application of the method of characteristics. First, the characteristic curves in equation (192.6) are identied, at some point, by determining iandifrom equa- tion (192.4). Then the equations for k1andk2(from equation (192.5)) are integrated a short distance along the characteristics. From the values of k1andk2, values for pandqmay be determined from equation (192.7). Finally, knowing pandq, the value of u(x;y) can be determined. In more detail, 1. Given values at the points PandQ(see gure 192.1.a), we will determine the values of all the variables at the new point R. 2. Using equation (192.6), determine Rby integrating along character- isticC1fromPand along characteristic C2fromQuntil the curves intersect. 3. Using equation (192.5), integrate k1=p+1qfromPtoRand integratek2=p+2qfromQtoR. Knowingfk1;k2gandf1;2g atRallowsq(R)a n dp(R) to be obtained from equation (192.7). CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 824 IV.C Numerical Methods for PDEs 193. Hyperbolic Equations: Finite Dierences Applicable to Hyperbolic partial dierential equations. Yields A numerical approximation scheme. Idea Finite dierences can be used directly to numerically approximate the solution of a hyperbolic partial dierential equation. Procedure The technique is to replace all of the derivatives appearing in the given hyperbolic partial dierential equation by nite dierence approximations. By rearranging the terms in this new equation, an explicit recurrence formula can generally be obtained. A stability analysis can be performed on this recurrence relation to determine the step sizes that will ensure convergence of the numerical approximation to the true solution. A frequent problem encountered with this method is having enough starting values to begin iterating the recur- rence relation. Starting values can generally be obtained by performingmanipulations of the original equation. Example The hyperbolic equation utt−2uxx=0; (193.1) on the interval 0 <x<L ,f o rt> 0, with the initial and boundary conditions u(0;t)=u(L;t)=0; u(x;0) =f(x); @u @t(x;0) =g(x);(193.2) can be numerically approximated directly by nite dierences. We choose a uniform grid of M+1 points in the xdirection (i.e., xi=ih fori=0;1;2;:::;M withh=L=M ). We choose the step length in the t variable to be kand dene tj=jk. We also choose to use the following centered dierence formulas for uxxandutt utt(xi;tj)=u(xi;tj+1)−2u(xi;tj)+u(xi;tj−1) k2; uxx(xi;tj)=u(xi+1;tj)−2u(xi;tj)+u(xi−1;tj) h2: (193.3) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 193. Hyperbolic Equations: Finite Dierences 825 Each of these formulae is second order accurate. If we dene wi;j= u(xi;tj), then using (193.3) in equation (193.1) results in wi;j+1−2wi;j+wi;j−1 k2−2wi+1;j−2wi;j+wi−1;j h2=0: This last equation can be solved for wi;j+1to dene the recurrence relation wi;j+1=2 ( 1−2)wi;j+2(wi+1;j+wi−1;j)−wi;j−1; (193.4) fori=1;2;:::; (M−1) andj=1;2;:::,w h e r e=k=h . The initial con- ditions and boundary conditions, from equation (193.2), can be represented as w0;j=wM;j=0;j=1;2;:::; wi;0=f(xi);i =1;2;:::;M:(193.5) Now comes the problem of starting the recurrence relation o. Suppose we wish to iterate equation (193.4). The values we rst compute are the fwi;2g, but these require knowledge of fwi;1g, which is not given in equation (193.5). The procedure for obtaining these data is to perform a Taylor series expansion of wi;1. We nd that wi;1=u(xi;t1) =u(xi;k) ’u(xi;0) +k@u @t(xi;0) +k2 2@2u @t2(xi;0) +:::;(193.6) where this last formula is second order accurate if we retain only the terms shown (higher order approximations can also be obtained). Now uttis known in terms of uxxfrom equation (193.1), and u(x;0) is known in terms off(x) from (193.2). Therefore, (193.6) can be simplied to yield wi;1’wi;0+kg(x1)+2k2 2f00(xi): (193.7) Special Case The Fortran program in program 193.1 numerically approximates the solution of the hyperbolic equation uxx−9uxx=0; for 0<x< 1;0<t; u(0;t)=u(1;t)=0; for 0<t; u(x;0) = sinx; for 0x1; ut(x;0) = 0; for 0x1:(193.8) This system has the analytic solution u(x;t)=s i nxcos 3t. The program utilizes M= 10 and the value of kwas chosen to be 0.02. The solution obtained for t= 1 at the points xi=0:1i(fori=0;1;:::; 10) is CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 826 IV.C Numerical Methods for PDEs REAL W(100,100) C Here are the initial values ALPHA=3.FL=1.M=10H=FL/FLOAT(M)FK=0.02 N=1./FK FLAMBD=ALPHA*FK/HCONST=2.*(1.-FLAMBD**2) C Set up the initial/boundary values in the matrix DO 10 J=1,N+1W(1,J)=0. 10 W(M+1,J)=0. DO 20 I=2,M XI=(I-1)*H W(I,1)=F(XI) 20 W(I,2)=W(I,1)+FK*G(XI)+FK**2*FPP(XI)/2.C Here is the recurrence relation DO 30 J=2,NTT=J*FKDO 40 I=2,M 40 W(I,J+1)=CONST*W(I,J)+FLAMBD**2*(W(I+1,J)+W(I-1,J))-W(I,J-1) 30 WRITE(6,5) J,TT,(W(K,J+1), K=1,M+1) 5 FORMAT(’ AT TIME STEP ’,I4,’ (T=’,F7.3,’)’/,4(1X,6(F9.4)/) ) END C These functions compute F(X), F’’(X) and G(X) FUNCTION F(X)F=SIN(3.1415927*X)RETURN END FUNCTION G(X)G=0.RETURNENDFUNCTION FPP(X)FPP=-(3.1415927)**2 * SIN(3.1415927*X)RETURN END Program 193.1: Fortran: nite dierences applied to hyperbolic equa- tions. 0. -0.3082 -0.5862 -0.8069 -0.9485 -0.9973 -0.9485 -0.8069 -0.5862 -0.3082 0. By comparing these values to the exact solution, we observe that the numerical approximation is correct to two decimal places. Notes 1. A stability analysis shows that equation (193.4) is stable if <1. 2. If thek2term in equation (193.7) had been neglected, then the method would have been only a rst order method. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 193. Hyperbolic Equations: Finite Dierences 827 3. See also Burden [1, pages 583{599], Davis [2, pages 42{44], and Garabedian [4, pages 463{475]. References [1]Burden, R. L. Numerical Analysis . PWS Publishers, Boston, MA, 1985. [2]Davis, J. L. Finite Dierence Methods in Dynamics of Continuous Media . The MacMillan Company, New York, 1986. [3]DuChateau, P., and Zachmann, D. Applied Partial Dierential Equations . Harper & Row Publishers, New York, 1989. [4]Garabedian, P. R. Partial Dierential Equations . John Wiley & Sons, New York, 1964. [5]Renaut-Williamson, R. A. Full discretisations of zzzref40refzzz and rational approximations to coshzzzref41refzzz. SIAM J. Numer. Anal. 26 , 2 (April 1989), 338{347. [6]Trefethen, L. N. Instability of dierence models for hyperbolic initial boundary value problems. Comm. Pure Appl. Math 37 , 3 (May 1984), 329{ 367. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 828 IV.C Numerical Methods for PDEs 194. Lattice Gas Dynamics Applicable to Partial dierential equations that physically arise from the motion of \particles." Yields A numerical approximation methodology. Idea Partial dierential equations are usually derived from some microscopic dynamical system. It may be possible to simulate the dynamical system directly without rst formulating dierential equations. Procedure We illustrate the basic ideas behind this method for the case of a fluid. By considering the interacting particles that make up a fluid and using continuum theory, the usual Navier{Stokes equation can be derived (see,e.g., Hasslacher [9]). This equation describes the evolution of the fluid. To numerically approximate the solution to this equation, the equation is discretized, and the resulting algebraic equations are solved on a computer. Because a computer will be used to solve a discrete problem, it may be easier (and faster) to directly simulate the motion of the original, discrete particles. The resulting simulation can mimic all of the eects that fluid systems have. By considering only local interaction laws in the simulation,we are led to use cellular automata to describe the dynamics of the parti- cles. Methods have been found for constructing cellular automata that are microscopically reversible (and thus support a realistic thermodynamics),obey exact conservation laws, and model continuum phenomena. Example We will illustrate one possible set of interaction laws that can be used to simulate gas dynamics; this model goes by the name of HPP. We consider a rectilinear array in which a particle may be present in a cell (indicatedby a dot), or it may be absent (indicated by a blank). At each \time step," the grid is considered in 2 2 blocks. The blocking alternates between even and odd time steps (see gure 194.1). At any time step, a particle in a cellis considered to be moving toward the center of the 2 2 block (see gure 194.1). Hence, a particle in the upper left corner will move to the lower right corner in one time step. On the next time step, because the blocking has changed, this particle will once again be in the upper left of its new block. Hence, it will continue moving on a diagonal path. The particles travel straight, with one exception: When exactly two par- ticles coming together from opposite directions collide, they bounce apartin the other two directions. These interactions are particle-conserving, deterministic, and invertible. In gure 194.2, we have indicated all possi- CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 194. Lattice Gas Dynamics 829ev en t im es t ep s /./././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././././././././././. /./. /././. /./. /./. /./. /./././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /././. /./. /./. /./. /./. /././././././././.od d t im es t ep s /./././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /././././././././. /./. /././. /./. /./. /./. /./././././././././././././././././././././././. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /././. /./. /./. /./. /./. /././././././././. Figure 194.1: The 2 2 blocking of the rectilinear array at dierent time steps./, /!/#0F /#0F/, /!/#0F /#0F/#0F/, /!/#0F /#0F/#0F /#0F/, /!/#0F /#0F/#0F/#0F/#0F /, /!/#0F /#0F/#0F/#0F/#0F /#0F/, /! /#0F /#0F/#0F /#0F Figure 194.2: All possible motions and interactions on the rectilinear grid in one time step (up to rotations). ble interaction possibilities (up to rotations). With just the information presented, it is possible to construct a full-scale simulation of a gas. Notes 1. It should be noted that, for some regimes, a lattice gas may fail to well approximate the Navier{Stokes equation and yet be closer to the actual physics than the Navier{Stokes equation itself. 2. It is possible to amplify the simple example above by having many particles, interaction eects between the dierent particles, exclusion rules, etc. 3. The example above is for a rectilinear grid. The articles by Hasslacher [9] describe the use of hexagonal grids. 4. Papatheodorou and Fokas [12] have shown that \discrete soliton"{ type behavior is possible in cellular automata. 5. Using special purpose hardware, simulation in lattice gas dynamics can be performed very quickly. See Margolus et al. [11]. References [1]Chen, H., Matthaeus, W. H., and Klein, L. W. Theory of multicolor lattice gas: A cellular automaton poisson solver. J. Comput. Physics 88 ,2 (1990), 433{466. [2]Chen, S., Lee, M., Zhao, K. H., and Doolen, G. D. A lattice gas model with temperature. Physica D 37 , 1{3 (1989), 42{59. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 830 IV.C Numerical Methods for PDEs [3]Cottel, G. H., and Mas-Gallic, S. A particle method to solve the Navier{Stokes system. Numer. Math. 57 , 8 (1990), 805{827. [4]Doolen, G., Hasslacher, B., Frisch, U., Orszag, S., and Wolfram, S.,E d s . Lattice Gas Methods for Partial Dierential Equations . Addison{ Wesley Publishing Co., Reading, MA, 1989. [5]Elenin, G. G., and Krylov, V. V. Equilibrium equations for a multi- component nonideal lattice gas on sublattices. Mat. Model. 2 , 1 (1990), 85{104. [6]Enquist, B., and Hou, T. Y. Particle method approximation of oscillatory solutions to hyperbolic dierential equations. SIAM J. Numer. Anal. 26 ,2 (April 1989), 289{319. [7]Frisch, U., d’Humieres, D., Hasslacher, B., Lallemand, P., Pomeau, Y., and Rivet, J. P. Lattice gas hydrodynamics in two and three dimensions. Complex Systems 1 , 4 (1987), 649{707. [8]Frisch, U., Hasslacher, B., and Pomeau, Y. Lattice gas automata for the Navier{Stokes equation. Phys. Rev. Let. 56 (1986), 1505{1508. [9]Hasslacher, B. \Background for lattice gas automata" and \The simple hexagonal model" and \The promise of lattice gas methods". Los Alamos Science (1987), 175{186, 187{200, and 211{217. [10]Lawniczak, A. T., and Kapral, R. ,E d s . Pattern Formation and Lattice Gas Automata , vol. 6 of Fields Institute Communications . Amer. Math. Soc., Providence, RI, 1996. [11]Margolus, N., Toffoli, T., and Vichiniac, G. Cellular-automata supercomputers for fluid-dynamics modeling. PhysRevLet 56 , 16 (21April 1986), 1694{1696. [12]Papatheodorou, T. A., and Fokas, A. S. Evolution theory, periodic particles, and solitons in cellular automata. Stud. Appl. Math. 80 (1989), 165{182. [13]Russo, G. A particle method for collisional kinetic equations. I. Basic theory and one-dimensional results. J. Comput. Physics 87 , 2 (1990), 270{300. [14]Toffoli, T. Cellular automata as an alternative to (rather than an approximation of) dierential equations in modeling physics. Physica D 10(1984), 117{127. [15]Tonegawa, T., Kaburagi, M., and Kanamori, J. Ground state analysis of the lattice gas model with two kinds of particles on the triangular lattice. J. Phys. Soc. Japan 59 , 5 (1990), 1660{1675. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 195. Method of Lines 831 195. Method of Lines Applicable to Elliptic, hyperbolic, and parabolic partial dieren- tial equations. Yields A system of partial dierential equations with one fewer independent variables. Idea The basis of the method is substitution of nite dierences for the derivatives with respect to one independent variable, and retention of the derivatives with respect to the remaining variables. This approach changes a given partial dierential equation into a system of partial dierential equations. Procedure We will illustrate the general method on a second order elliptic partial dierential equation. Suppose the given equation is A@2u @x2+B@2u @x@y+C@2u @y2+D@u @x+E@u @y+Fu=G (195.1) in a domain Ω, where fA;B;C;D;E;F;G gare functions of xandy.B e - cause equation (195.1) is assumed to be elliptic, the necessary data for equation (195.1) are given on the boundary of Ω. If we choose to discretize in the yvariable, then we draw lines parallel to thexaxis, with a constant distance hbetween adjacent lines. (See gure 195.1.) Suppose the lines are specied by y=yk=y0+kh; k =0;1;:::;N: Then, we set y=ykin equation (195.1) and use nite dierences for the derivatives with respect to y. For example, we can use @u @y y=yk’1 h[uk+1(x)−uk(x)]; @2u @x@y y=yk’1 h u0 k+1(x)−u0 k(x) ; @2u @y2 y=yk’1 h2[uk+1(x)−2uk(x)+uk−1(x)];(195.2) whereuk(x) is an approximation to u(x;yk). Using equation (195.2) in equation (195.1) (with y=yk), we obtain a rst order dierential equa- tion involving the unknown functions fuk−1;uk;uk+1g. By taking k= CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 832 IV.C Numerical Methods for PDEs/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././././././. /./. /././././././././././././././././././.x y/0 y/1 y/2 y /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /././././././././././././././././././. /././././././././././././././././. /./. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /. /. /. /./. /. /. /. /. /. /. /. /. /. /.h Figure 195.1: Subdivision of the domain to solve equation (195.1). 0;1;:::;N , we obtain a system of rst order ordinary dierential equations for theN+ 1 unknown functions fu0(x);u1(x);:::;uN(x)g. If equation (195.1) is elliptic and Ω is convex, then the equations will constitute a two point boundary value system. Any standard (numerical) two point ordinary dierential equation system solver can be used to solve this system. Example Suppose we have the following parabolic equation for u(x;t) ut=uxx; u(0;x)=(x); u(t;0) =(t); u(t;1) =(t):(195.3.a-d) We discuss discretizing this equation in both xandt. 1. If we choose to discretize in the xvariable, then we approximate u(t;xn)b yvn(t), wherexn=n=N =n
x. Then we can approxi- mate the derivatives with respect to xin equation (195.3.a) by nite dierences to obtain d dtvn(t)’vn+1(t)−2vn(t)+vn−1(t) (
x)2; (195.4) forn=1;2;:::;N−1. The initial conditions and boundary condi- tions in (195.3) can be written as vn(0) =(n
x); forn=1;2;:::;N−1; v0(t)=(t); vN(t)=(t):(195.5) (See gure 195.2.) If an explicit scheme (say forward Euler’s method) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 195. Method of Lines 833/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /././. /./. /././. /././. /././././././. /./././. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./. /. /./././. /./././././././././././././. /././././. /././././././././././././././././././././.x/0 x/1 x/2 u /= /#11 /#28 x /#29 xN x u /= /#0B /#28 t /#29 t u /= /#0C /#28 t /#29 /#0F /#0F /#0F Figure 195.2: Subdivision of the domain. is chosen to numerically approximate equation (195.4), then the sim- ple formula vn(t+
t)=vn(t)+
t (
x)2[vn+1(t)−2vn(t)+vn−1(t)] (195.6) results. This formula can be iterated with equation (195.5) to nd a numerical approximation to the solution of (195.3). 2. If, instead, we choose to discretize equation (195.3) in the tvariable, then we would approximate u(tk;x)b ywk(x), wheretk=k
t.A p - proximating the tderivatives in equation (195.3) by nite dierences, we obtain wk(x)−wk−1(x)
t=d2 dx2wk(x); (195.7) with the corresponding initial and boundary conditions w0(x)=(x) wm(0) =(m
t); form=0;1;::: wm(1) =(m
t); form=0;1;:::: Note that equation (195.7) is a constant coecient ordinary dieren- tial equation for the dependent variable wk(x). Hence, the explicit solution can be obtained and the dierential system can be replaced by an algebraic system. Notes 1. This method is sometimes called the generalized Kantoravich method . 2. Observe that the recurrence relation in equation (195.6) could have been obtained directly by applying nite dierences to both the x CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 834 IV.C Numerical Methods for PDEs andtderivatives appearing in equation (195.3). This is not a clever use of the method of lines. A better approach would be to use a computer package to solve the initial value system in equations (195.4) and (195.5). This package could use an implicit method forthetderivative, and it could adjust the step size as necessary to reduce the error. References [1]Berzins, M. Global error estimation in the method of lines for parabolic equations. SIAM J. Sci. Stat. Comput. 9 , 4 (July 1988), 687{703. [2]Dew, P. M., and Walsh, J. E. A set of library routines for solving parabolic equations in one space variable. ACM Trans. Math. Software 7 ,3 (Sept 1981), 295{314. [3]Graney, I., and Richardson, A. A. The numerical solution of non-linear partial dierential equations by the method of lines. J. Comput. Appl. Math. 7, 4 (1991), 229{236. [4]Keast, P., and Muir, P. H. EPDCOL: A more ecient PDECOL code. ACM Trans. Math. Software 17 , 2 (June 1991), 153{166. [5]Kreiss, H.-O., and Scherer, G. Method of lines for hyperbolic dierential equations. SIAM J. Numer. Anal. 29 , 3 (June 1992). [6]Melgaard, D. K., and Sincovec, R. F. General software for two- dimensional nonlinear partial dierential equations. ACM Trans. Math. Software 7 , 1 (March 1981), 106{125. [7]Meyer, G. H. The method of lines for Poisson’s equation with nonlinear or free boundary conditions. Numer. Math. 29 (1978), 329{344. [8]Mikhail, M. N. On the validity and stability of the method of lines for the solution of partial dierential equations. Appl. Math. and Comp. 22 (1987), 89{98. [9]Schiesser, W. E. The Numerical Method of Lines . Academic Press, New York, 1991. [10]Walter, W. Dierential and Integral Inequalities . Springer{Verlag, New York, 1970. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 196. Parabolic Equations: Explicit Method 835 196. Parabolic Equations: Explicit Method Applicable to Parabolic partial dierential equations. Yields An explicit numerical scheme. Idea Marching in time is the easiest way to solve a parabolic equation. For this explicit method, the time steps must be small. Procedure Suppose we have the parabolic dierential equation ut=L(u;x;t); u(t0;x)=f(x);(196.1) foru(x;t), whereL(u;x;t) is uniformly elliptic. The easiest way to solve equation (196.1) is by the use of \marching," which is an explicit method. An explicit numerical approximation is determined by taking a forward dierence in the tvariable in equation (196.1) and having no other terms that involve future time values. For example, we can approximate u(x;t) byv(x;t)w h e r ev(x;t) satises v(t+
t;x)=v(t;x)+
tbL(v(t;x);x;t); v(t0;x)=f(x);(196.2) andbL(
) is any reasonable nite dierence approximation to L(v(t;x);x;t) that does not involve v(t+
t;x) (if it did involve this term, then the method would be implicit). The main drawback of this method is that
tmust often be very small for the method to be stable. If j
xjis the smallest discretization step in the evaluation of bL(v(t;x);x;t) then we require
t=O(j
xj2) for equation (196.2) to be a numerically stable technique. More precise restrictions on
tcan be derived from the exact form of L(v(t;x);x;t), and the numerical approximation used for the derivatives. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 836 IV.C Numerical Methods for PDEs Example Suppose we want to numerically approximate the solution to the diu- sion problem ut=uxx; u(t;0) = 0; u(t;1) = 1; u(0;x)=0;(196.3.a-d) fort0w i t h0x1. From the method of Fourier series or separation of variables (see pages 344 and 487), we nd the analytic solution of equation(196.3) to be u(t;x)=x+2 1X n=1(−1)n ne−2n2tsinnx: This exact solution will be used to ascertain the accuracy of the numerical solution. To numerically approximate the solution to (196.3), we use a grid of N points between 0 and 1, fxnjxn=(n−1)
x,n=1;2;:::;Ng,w h e r e
x=1=(N−1). We dene vn(t) to be the approximation of u(t;x)a t thenth grid point: vn(t)’u(t;xn). The initial conditions in equation (196.3.d) can be represented as vm(0) = 0;m =0;1;2;:::;N; whereas the boundary conditions in equations (196.3.b,c) can be repre- sented as v1(t)=0;vN(t)=1: Using a centered second order scheme for the uxxterm and a rst order forward dierence scheme for the utterm, equation (196.3.a) can be dis- cretized as vm(t+
t)=vm(t)+
tvm+1(t)−2vm(t)+vm−1(t) (
x)2 : (196.4) The C (Fortran) code in program 196.1 (196.2) implements the above scheme for N=2 1a n d
t=0:001. We choose to compare the output from the program to the exact solution for t=0:1a n dx=0:5. The exact solution isu(0:1;0:5)’0:2637. Table 196.1 shows the approximate value of u(0:1;0:5), for several dif- ferent choices of Nand
t. From these values, we conclude 1. AsNincreases, the accuracy of the numerical solution increases. 2. As
tdeceases, the accuracy of the numerical solution increases. The dierence equation (196.4) was the example used to demonstrate the Von Neumann stability test (see page 692). It was determined there that the method will be stable if and only if
t=(
x)2is less than 1. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 196. Parabolic Equations: Explicit Method 837 void UPDATE(double *, double, double, int); void main() { double X[1000], V[1000];double DELTAT=0.001, DELTAX, T=0;int J, K, NTIME=100, N=21;DELTAX= (double) 1/(double)(N-1);/* Initialize the grid */ for ( J=1; J<=N; J++ ) { X[J]= (double) (J-1)*DELTAX;V[J]= (double) 0; }V[N]= (double) 1;/* This is the loop for the number of time steps */for ( J=1; J<=NTIME; J++ ) { T += DELTAT; /* Update the grid */ UPDATE(V,DELTAX,DELTAT,N);/* Output the answer */printf("The time is %8.4f\n",T);for ( K=1; K<=N; K++ ) { printf("(%8.4f,%8.4f)\n", X[K],V[K] ); } } }/* This subroutine increments the solution by one time step */ void UPDATE(double *VOLD, double DELTAX, double DELTAT, int N) { double RATIO, VNEW[1000];int J;RATIO=DELTAT/(DELTAX*DELTAX);for ( J=2; J<=N-1; J++ ) { VNEW[J]=VOLD[J] + RATIO*( VOLD[J+1] - 2*VOLD[J] + VOLD[J-1] ); } for ( J=2; J<=N-1; J++ ) { VOLD[J]=VNEW[J]; } } Program 196.1: C: explicit method applied to parabolic equations. N
x
t
t=(
x)2v(0:1;0:5) 50.25 0.05 0.80 0.6400 50.25 0.01 0.16 0.2745 11 0.10 0.005 0.50 0.2628 11 0.10 0.001 0.10 0.2640 21 0.05 0.001 0.40 0.2639 Table 196.1: Approximate value of u(0:1;0:5) for dierent Nand
t.T h e exact value is u(0:1;0:5)’0:2637 Note 1. See also Davis [1, Chapter 4, pages 167{193], Farlow [4, Lesson 38, pages 309{315], Press et al. [6, pages 635{640], Smith [7, Chapters 2 and 3, pages 11{174], and Twizell [8, pages 200{265]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 838 IV.C Numerical Methods for PDEs REAL*8 X(1000),V(1000) DELTAT=0.001D0NTIME=100N=21DELTAX=1.D0/DFLOAT(N-1) C Initialize the grid DO 10 J=1,N X(J)=DFLOAT(J-1)*DELTAX 10 V(J)=0.D0 V(N)=1.D0T=0.D0 C This is the loop for the number of time steps DO 20 J=1,NTIMET=T+DELTAT C Update the grid CALL UPDATE(V,DELTAX,N,DELTAT) C Output the answer20 WRITE(6,5) T, (X(K),V(K),K=1,N)5 FORMAT(’ The time is=’,F8.4,100(/10X,2F8.4) ) END C This subroutine increments the solution by one time step SUBROUTINE UPDATE(VOLD,DELTAX,N,DELTAT)REAL*8 VOLD(1000),VNEW(1000) RATIO=DELTAT/DELTAX**2 DO 100 J=2,N-1 100 VNEW(J)=VOLD(J) + RATIO*( VOLD(J+1) -2.D0 * VOLD(J) + VOLD(J-1) ) DO 200 J=2,N-1 200 VOLD(J)=VNEW(J) RETURNEND Program 196.2: Fortran: explicit method applied to parabolic equations. References [1]Davis, J. L. Finite Dierence Methods in Dynamics of Continuous Media . The MacMillan Company, New York, 1986. [2]DuChateau, P., and Zachmann, D. Applied Partial Dierential Equations . Harper & Row Publishers, New York, 1989. [3]Evans, D. J., and Abdullah, A. R. B. A new explicit method for the solution of zzzref16refzzz. Int. J. Comp. Math. 14 (1983), 325{353. [4]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [5]Gerald, C. F., and Wheatley, P. O. Applied Numerical Analysis . Addison{Wesley Publishing Co., Reading, MA, 1984. [6]Press, W. H., Flannery, B. P., Teukolsky, S., and Vetterling, W. T. Numerical Recipes . Cambridge University Press, New York, 1986. [7]Smith, R. D. Numerical Solution of Partial Dierential Equations: Finite Dierence Methods , third ed. Clarendon Press, Oxford, England, 1985. [8]Twizell, E. H. Computational Methods of Partial Dierential Equations . Ellis Horwood Limited, Chichester, England, 1984. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 197. Parabolic Equations: Implicit Method 839 197. Parabolic Equations: Implicit Method Applicable to Parabolic partial dierential equations. Yields An implicit numerical scheme. Idea An implicit scheme will numerically approximate the solution of a par- abolic equation and allow large time steps to be taken. Procedure Suppose we have the parabolic dierential equation ut=L(u;x;t); u(t0;x)=f(x);(197.1) foru(x;t), whereL(u;x;t) is uniformly elliptic. We desire an implicit dierence scheme that will numerically approximate the solution to equa-tion (197.1). An implicit method is one in which the value of u(t+
t;x) is not determined explicitly by the value of u(t;x) but instead uses both u(t+
t;x)a n du(t;x). For simplicity, we discuss only the case of a single space dimension. The dierence scheme will utilize a uniform grid, with a spacing of
xin the xdirection and a spacing of
tin thetdirection. Dene v n;jto be an approximation to u(tn;xj), wheretn=n
tandxj=j
x. To discretize equation (197.1) in t, we choose to use a forward dierence in thetvariable. That is, ut(tn;xj)=vn+1;j−vn;j
t: Now thexderivatives will be approximated, at any point, by values at time tnand at time tn+1.T h a ti s , ux(tn;xj)=( 1−1)vn+1;j−vn+1;j−1
x+1vn;j−vn;j−1
x; uxx(tn;xj)=( 1−2)vn+1;j+1−2vn+1;j+vn+1;j−1 (
x)2 +2vn;j+1−2vn;j+vn;j−1 (
x)2;(197.2) where1and2are any real numbers between zero and one. For any such values, the scheme in equation (197.2) will be consistent. Note that CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 840 IV.C Numerical Methods for PDEs if1=2= 1, there is only dependence on the values at a previous time step and an explicit method is recovered. If neither 1nor2is equal to one, an implicit dierence scheme results. An implicit scheme often has the advantage that time steps can be taken that are much larger than the time steps that can be taken for an explicit method. More precise restrictions on
tcan be obtained from the form of L(v;x;t) and the values chosen for 1and2in equation (197.2). Example Suppose we want to numerically approximate the solution to the diu- sion problem ut=uxx; u(t;0) = 0; u(t;1) = 1; u(0;x)=0;(197.3.a-d) fort0w i t h0x1. From the method of Fourier series or separation of variables (see pages 344 and 487), we nd the analytic solution to equation (197.3) is u(t;x)=x+2 1X n=1(−1)n ne−2n2tsinnx: (197.4) This exact solution will be used to determine the accuracy of the numerical solution. To numerically approximate the solution to equation (197.3), we use a grid ofNpoints between 0 and 1, fxnjxn=(n−1)
x,n=1;2;:::;Ng, where
x=1=(N−1). The initial conditions in equation (197.3.d) can be represented as v0;j=0;j =0;1;2;:::;N; (197.5) whereas the boundary conditions in equation (197.3.b,c) can be represented as vn;0=0;vn;N=1; forn=1;2;:::: (197.6) We choose to discretize the equation with 1=2=1=2; this produces theCrank{Nicolson scheme . The approximation to equation (197.3.a) is therefore vn+1;j−vn;j
t=1 2vn+1;j+1−2vn+1;j+vn+1;j−1 (
x)2+1 2vn;j+1−2vn;j+vn;j−1 (
x)2; which can be manipulated into −vn+1;j+1+( 2+2)vn+1;j−vn+1;j−1=vn;j+1+( 2−2)vn;j+vn;j−1; (197.7) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 197. Parabolic Equations: Implicit Method 841 N
x
t
t=(
x)2v(0:1;0:5) 50.25 0.01 0.80 0.2526 11 0.10 0.01 1.00 0.2508 11 0.10 0.005 0.50 0.2569 21 0.05 0.01 4.00 0.2507 Table 197.1: Approximate value of u(0:1;0:5) for dierent Nand
t.T h e exact value is u(0:1;0:5)’0:2637 w h e r ew eh a v ed e  n e d =
t=(
x)2. Note that for a given value of n, equation (197.7) is an algebraic equa- tion forvn+1;jand two of its spatial neighbors. Hence, equation (197.7) cannot be used alone to determine vn+1;j. Instead, a system of equations must be solved simultaneously. Utilizing equations (197.5) and (197.6), this system may be written as 2 6666666410 00


0 −2+2− 0


0 0− 2+2− 0 ............ 00 −2+2− 00


0013 777777752 66666664v n+1;0 vn+1;1 vn+1;2 ... vn+1;N−1 vn+1;N3 77777775= 2 666666640 v n;1+( 2−2)vn;2+vn;3 vn;2+( 2−2)vn;3+vn;4 ... vn;N−2+( 2−2)vn;N−1+vn;N 13 77777775:(197.8) Because this system of linear equations has a banded matrix of width three, the system can be solved very eciently. The Fortran program in program 197.1 implements the above scheme withN=2 1a n d
t=0:01. Note that this program uses a matrix solver, LSOLVE , whose source code is not shown. We choose to compare the output from the program to the exact solution (given in equation (197.4)), for t=0:1a n dx=0:5. The exact solution is u(0:1;0:5)’0:2637. Table 197.1 shows the approximate value of u(0:1;0:5), for several dif- ferent choices of Nand
t. From these values, we conclude 1. AsNincreases, the accuracy of the numerical solution increases. 2. As
tdeceases, the accuracy of the numerical solution increases. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 842 IV.C Numerical Methods for PDEs DIMENSION FMAT(100,100),RHS(100),V(100),X(100),NROW(200) N=21DELTAT=0.01NTIME=5DELTAX=1./DFLOAT(N-1)RHO=DELTAT/DELTAX**2 C Initialize the vector at T=0 DO 10 J=1,N X(J)=DELTAX*(J-1) 10 V(J)=0 T=0.DO 20 JTIME=1,NTIMET=T+DELTAT C Set up the right hand side RHS(1)=0. RHS(N)=1. DO 30 J=2,N-1 30 RHS(J)= RHO*V(J-1)+(2.-2.*RHO)*V(J)+RHO*V(J+1)C Set up the matrix DO 40 J=1,NDO 40 K=1,N 40 FMAT(J,K)=0. FMAT(1,1)=1. FMAT(N,N)=1. DO 50 J=2,N-1FMAT(J,J-1)=-RHOFMAT(J,J )=2.+2.*RHO 50 FMAT(J,J+1)=-RHOC Solve the matrix equation CALL LSOLVE(N,FMAT,V,RHS,NROW,IFSING,100) C Print out the answer 20 WRITE(6,5) T, (X(K),V(K),K=1,N)5 FORMAT(’ Here is the solution at time=’,F8.4,/,90(10X, 2F12.5/)) END Program 197.1: Fortran: implicit method applied to parabolic equations. Notes 1. Observe from table 197.1 that the numerical method used resulted in reasonable approximations when
t=(
x)2was as large as 4. Using the Von Neumann test (see page 692), it can be shown that theCrank{Nicolson scheme is unconditionally stable for any value of
t=(
x) 2. 2. Another way to interpret this solution technique is as a sequence of elliptic problems, with one problem being solved at every time step. For example, given the parabolic system ut=L[u]+f(x;t); onR; t> 0; u=g(x;t); on@R; t> 0; u=u0(x); onR[@R; t =0;(197.9.a-c) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 197. Parabolic Equations: Implicit Method 843 we can take a forward dierence in tto obtainut(t)’u(t)−u(t−
t)
t, which allows equation (197.9) to be rewritten asu(t)−u(t−
t)
t’ L[u(t)]+f(x;t). This is an elliptic equation for u(t) in whichu(x;t−
t) plays the role of a nonhomogeneous forcing term. Hence, the successive time values of u(x;t) may be determined by solving a sequence of elliptic problems. The boundary conditions for each elliptic problem come from equation (197.9.b), whereas the rst value ofu(x;t)i sg i v e nb y u0(x). Rice and Boisvert [5, pages 111{120] present the template of an ELLPACK program that will numericallyapproximate the solution of parabolic equations by sequentially solv- ing elliptic equations. 3. See also Davis [1, Chapter 4, pages 167{193], Farlow [3, Lesson 38, pages 309{315], and Smith [6, Chapters 2 and 3, pages 11{174]. References [1]Davis, J. L. Finite Dierence Methods in Dynamics of Continuous Media . The MacMillan Company, New York, 1986. [2]DuChateau, P., and Zachmann, D. Applied Partial Dierential Equations . Harper & Row Publishers, New York, 1989. [3]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [4]Gerald, C. F., and Wheatley, P. O. Applied Numerical Analysis . Addison{Wesley Publishing Co., Reading, MA, 1984. [5]Rice, J. R., and Boisvert, R. F. Solving Elliptic Problems Using ELLPACK . Springer{Verlag, New York, 1985. [6]Smith, R. D. Numerical Solution of Partial Dierential Equations: Finite Dierence Methods , third ed. Clarendon Press, Oxford, England, 1985. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 844 IV.C Numerical Methods for PDEs 198. Parabolic Equations: Monte-Carlo Method Applicable to Linear parabolic partial dierential equations. Yields A numerical approximation to the solution of a linear parabolic partial dierential equation at a single point. Idea Simulation of the motion of a random particle may be used to approx- imate the solution to linear parabolic equations. Procedure The steps for this method are straightforward. First, we give an overview; then, a more detailed presentation. First, approximate the elliptic part of the given parabolic partial dier- ential equation by a nite dierence method. Rewrite the nite dierence formula as a recursive function for the value of the unknown at any given point. Then interpret this recursive formula as a set of transition prob- abilities that determine the motion of a random particle. By creating a nite dierence scheme for the time derivative in the dierential equation,a natural time scale will be associated with every step of the particle. Now, write a computer program that will allow many (say K) particles to wander randomly around the domain of interest, based on the transitionprobabilities found from the dierence formula. Simulate the particles one at a time, with every particle starting o at the same point (say the point z). If the time step is
t, and the solution is desired at t=T,t h e n the particles will be allowed to wander randomly but for no more than M=T=
tsteps. If the boundary data are of the Dirichlet type (i.e., the value of the unknown is prescribed on the boundary), then, when a particle reaches the boundary, stop that particle and store away the value onthe boundary. Begin another particle at the point z. If the boundary data are not of the Dirichlet type (say Neumann or mixed boundary conditions), then, when the particles reach the boundary, they will be given a nite probability to leave the boundary and re-enter the domain of the problem. If the particle leaves theboundary, continue the iteration process. If it does not leave the boundary, the value at the boundary is stored away, and a new particle is started o at the point z. For parabolic equations there is also the possibility that the particle will not reach the boundary in Msteps. If the particle has not CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 198. Parabolic Equations: Monte-Carlo Method 845 reached the boundary in Msteps, then record the position that is nally reached. Using the initial conditions of the problem, there is a value associated with the point reached. Then, begin a new particle at the point z. If the parabolic equation was homogeneous, a numerical approximation to the solution at the point zwill be given by an average of the Kvalues stored away. If the given equation was not homogeneous, then equation (198.3) is used to obtain an estimate of the solution at the point z. In this case, all points on the path that the particle traversed will be utilized. In more detail, here is how the technique may be applied to the linear parabolic partial dierential equation in the domain R ut=L[u]+F(x;y;t );x ; y2Randt>0; u=(x;y;t );x ; y 2@Randt>0; u(x;y;0) =g(x;y);x ; y 2R; (198.1.a-c) with the operator L[
] dened by L[u]=Auxx+2Buxy+Cuyy+Dux+Euy; wherefA;B;C;D;Egare all functions of fx;y;tg. The operator L[
]m a y be discretized to yield the approximation L[u]’Ai;jvi+1;j;n−2vi;j;n+vi−1;j;n r(
x)2 ) +2Bi;jvi+1;j+1;n−vi;j+1;n−vi+1;j;n+vi;j;n (
x)(
y) +Ci;jvi;j+1;n−2vi;j;n+vi;j−1;n (
y)2 +Di;jvi+1;j;n−vi;j;n
x +Ei;jvi;j+1;n−vi;j;n
y ; wherexi=x0+i(
x),yj=y0+j(
y),tn=n(
t),vi;j;n=u(xi;yj;tn), and a subscript of i;j;n means an evaluation at the point ( xi;yj;tn). If CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 846 IV.C Numerical Methods for PDEs thefΓ
;
;
gandQi;j;nare dened by Γi+1;j+1;n=2Bi;j;n (
x)(
y) ; Γi+1;j;n=Ai;j;n (
x)2−2Bi;j;n (
x)(
y)+Di;j;n
x ; Γi;j+1;n=Ci;j;n (
y)2−2Bi;j;n (
x)(
y)+Ei;j;n
y ; Γi−1;j;n=Ai;j;n (
x)2 ; Γi;j−1;n=Ci;j;n (
x)2 ; Qi;j;n=2Ai;j;n (
x)2−2Bi;j;n (
x)(
y)+2Ci;j;n (
y)2+Di;j;n
x+Ei;j;n
y ; andutis approximated byu(x;y;t+
t)−u(x;y;t)
t, then equation (198.1.a) may be discretized as vi;j;n+1=(
t)h Γi+1;j;nvi+1;j;n+Γi+1;j+1;nvi+1;j+1;n+Γi;j+1;nvi;j+1;n +Γi−1;j;nvi−1;j;n+Γi;j−1;nvi;j−1;ni +[ 1−Qi;j;n(
t)]vi;j;n+(
t)Fi;j;n:(198.2) If we now choose
t=1=Qi;j;n and dene pi;j;n =Γi;j;n=Qi;j;n,t h e n equation (198.2) can be written as vi;j;n+1=pi+1;j;nvi+1;j;n+pi+1;j+1;nvi+1;j+1;n+pi;j+1;nvi;j+1;n +pi−1;j;nvi−1;j;n+pi;j−1;nvi;j−1;n+Fi;j;n Qi;j;n: Note thatp’s add up to 1. We interpret them as probabilities of taking a step in a specied direction. Specically, if a particle is at position ( i;j;n ) at stepn,t h e n With probability pi;j+1;n, the particle goes to ( i;j+1 )a ts t e p n+1 . With probability pi;j−1;n, the particle goes to ( i;j−1) at stepn+1 . With probability pi+1;j;n, the particle goes to ( i+1;j)a ts t e pn+1 . With probability pi−1;j;n, the particle goes to ( i−1;j)a ts t e pn+1 . With probability pi+1;j+1;n, the particle goes to ( i+1;j+1 )a ts t e p n+1 . Now, suppose a particle starts at the point P0=zand undergoes a random walk according to the above prescription. We allow this particle to wander until a time of Thas elapsed. If Qi;j;nis constant, then
tis CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 198. Parabolic Equations: Monte-Carlo Method 847 a constant, and we only need to count the number of steps taken. Either the particle will hit the boundary after, say, Nsteps, or it will not hit the boundary at all in Msteps. Suppose that the sequence of points that this particle visits is ( P0;P1;P2;:::;PN), andN=Mif the boundary has not been reached. Then, an unbiased estimator of the value of u(z)f o rt h e parabolic problem in (198.1) is given by −NX j=0F(Pj) Q(Pj)+( (PN;tN);if the particle reached the boundary ; g(PN); if the particle did not reach the boundary : (198.3) In practice, several random paths will be taken, and the average taken to estimateu(x;y;t ). Example Suppose we wish to numerically approximate the solution to the diu- sion equation in the unit square, at a single point. Suppose we have the partial dierential equation ut=r2u; (198.4) foru(t;x;y ) with the boundary conditions u(t;x;0) =u(t;x;1) = 0; u(t;0;y)=u(t;1;y)=0; u(0;x;y)=1 0:(198.5) The exact solution to equations (198.4) and (198.5) is u(x;y;t )=16 21X n;m=1e−[(2n−1)2+(2m−1)2]t (2m−1)(2n−1)sin [(2m−1)x] sin [(2n−1)y]; (198.6) which was obtained by separation of variables (see page 487). Using equa- tion (198.6) we determine that u(0:6;0:6;0:5)’5:354. We choose the point z=( 0:6;0:6) and try to numerically approximate the solution to equations (198.4) and (198.5) at the point zwhent=0:5. We follow the steps outlined above. Using the standard second order approximation to the Laplacian, (see Abramowitz and Stegun [1, formula 25.3.30]), we nd r2u’ui+1;j;n+ui−1;j;n+ui;j+1;n+ui;j−1;n−4ui;j;n h2=0; whereui;j;n=u(hi;hj;n (
t)) andh1. Using our above approximation to the time derivative, we nd that equation (198.4) may be approximatedas u i;j;n+1−ui;j;n
t=ui+1;j;n+ui−1;j;n+ui;j+1;n+ui;j−1;n−4ui;j;n h2; CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 848 IV.C Numerical Methods for PDEs or (dening γ=
t=h2) ui;j;n+1=γ[ui+1;j;n+ui−1;j;n+ui;j+1;n+ui;j−1;n]+ui;j;n(1−4γ): (198.7) If we choose γ=1=4, then equation (198.7) simplies to ui;j;n+1=ui+1;j 4+ui−1;j 4+ui;j+1 4+ui;j−1 4: (198.8) We interpret equation (198.8) probabilistically as follows: If a particle is at position ( i;j)a ts t e pn,t h e n With probability 1/4, the particle goes to ( i;j+1 )a ts t e p n+1 . With probability 1/4, the particle goes to ( i;j−1) at stepn+1 . With probability 1/4, the particle goes to ( i+1;j)a ts t e pn+1 . With probability 1/4, the particle goes to ( i−1;j)a ts t e pn+1 . The Fortran program in program 198.1 was used to simulate the mo- tion of the particles according to the above probability law. A total ofNSIM random particles were started o. The outcome of that program is given below. As more paths are taken, the approximation becomes better. Obtaining many decimal places of accuracy requires a very large numberof simulations. STEP=0.03000 DT=0.00360 M=138 Average after 10000 particles is: 4.7320Average after 20000 particles is: 4.7845Average after 30000 particles is: 4.7847 Note that the program uses a routine called RANDOM , whose source code is not shown, that returns a random value uniformly distributed on the interval from zero to one. Notes 1. If further accuracy is required, the options are (a) Increase the number of random particles. (b) Make the mesh discretization ner (i.e., decrease h). (c) Do both of the above. If the number of random particles is not very large, then (b) will not help much; and if the mesh is very coarse, then (a) will not help much. Generally, the variance of the answer (a measure of the\scatter") decreases as the number of trials to the minus one half power. 2. Because low numerical accuracy is obtained by this technique, a computer program does not need to work with extended precision arithmetic. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 198. Parabolic Equations: Monte-Carlo Method 849 NSIM=30000 TIME=0.500XHOLD=0.60 YHOLD=0.60 C Specify the step length STEP=0.02 C The step length determines the time step DT=4.*STEP**2 C Determine the number of time steps allowed M=TIME/DTSUM=0. DO 30 IWALK=1,NSIM C Start off a new random walk X=XHOLDY=YHOLDNSTEP=0 10 NSTEP=NSTEP+1C Determine if M steps have been taken yet IF( NSTEP .GT. M ) GOTO 20 C Update the position X=X + SIGN(STEP, RANDOM(DUMMY)-0.5 )Y=Y + SIGN(STEP, RANDOM(DUMMY)-0.5 ) C If the particle escapes the box, start a new particle off IF( X.GT.1 .OR. X.LT.0 ) GOTO 40IF( Y.GT.1 .OR. Y.LT.0 ) GOTO 40 C Otherwise take another step GOTO 10 C Time has run out with the particle still in the grid20 SUM=SUM+1040 IF( MOD(IWALK,10000) .NE. 0 ) GOTO 30 APPROX=SUM/FLOAT(IWALK)WRITE(6,5) IWALK,APPROX 30 APPROX=SUM/FLOAT(NSIM) WRITE(6,5) NSIM,APPROX 5 FORMAT(’ Average after’,I6,’ particles is: ’,F7.4) END Program 198.1: Fortran program for Monte-Carlo method applied to parabolic equations. 3. If the time at which the solution is desired is so large that all of the particles end up at the boundaries, then the quantity really being calculated is the steady-state solution to the parabolic equation. 4. If a parabolic equation is interpreted as a Fokker{Planck equation (see page 303), then It^ o equations can be associated with the parabolic equation. The It^ o equations may be numerically integrated by the technique described on page 775. 5. Another type of Monte-Carlo approach for parabolic equations, using cellular automata, is described in Boghosian and Levermore [3]. 6. Sadeh and Franklin [8] contain several worked examples. See also Farlow [4, pages 346{352]. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 850 IV.C Numerical Methods for PDEs References [1]Abramowitz, M., and Stegun, I. A. Handbook of Mathematical Functions . National Bureau of Standards, Washington, D.C., 1964. [2]Bhavsar, V. C., and Gujar, U. G. VLSI algorithms for Monte Carlo solutions of partial dierential equations. In Advances in Computer Methods For Partial Dierential Equations ,R .V i c h n e v e t s k ya n dR .S .S t e p l e m a n , Eds., IMACS. North{Holland Publishing Co., New York, 1984. [3]Boghosian, B. M., and Levermore, C. D. A cellular automaton for Burgers’ equation. Complex Systems 1 (1987), 17{29. [4]Farlow, S. J. Partial Dierential Equations for Scientists and Engineers . John Wiley & Sons, New York, 1982. [5]Marshall, G. Monte Carlo methods for the solution of nonlinear partial dierential equations. Comput. Physics Comm. 56 (1989), 51{61. [6]Puckett, E. G. Convergence of a random particle method to solutions of the Kolmogorov equation zzzref36refzzz. Math. of Comp. 52 , 186 (April 1989), 615{645. [7]Roberts, S. Convergence of a random walk method for the Burgers equation. Math. of Comp. 52 , 186 (April 1989), 647{673. [8]Sadeh, E., and Franklin, M. A. Monte Carlo solution of partial dier- ential equations by special purpose digital computer. IEEE Transactions on Computers C-23 , 4 (April 1974), 389{397. [9]Sherman, A. S., and Peskin, C. S. A Monte Carlo method for scalar reaction diusion equations. SIAM J. Sci. Stat. Comput. 7 ,4( O c t o b e r 1986), 1360{1372. [10]Sherman, A. S., and Peskin, C. S. Solving the Hodgkin{Huxley equations by a random walk method. SIAM J. Sci. Stat. Comput. 9 , 1 (January 1988), 170{190. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 199. Pseudospectral Method 851 199. Pseudospectral Method Applicable to Most commonly, hyperbolic equations with periodic boundary conditions. Yields A numerical scheme for calculating the spatial derivatives. Idea A numerical nite Fourier transform can be used to obtain dierence schemes that are of innite order. Procedure On a uniformly spaced grid fx1;x2;:::;xNg,w i t hxi+1−xi=h,a numerical approximation to @u=@x at the point xkthat is second order accurate is @u @x x=xk’1 2h(uk+1−uk−1); whereuk=u(xk). A numerical approximation that is fourth order accurate is given by @u @x x=xk’1 3h(uk+1−uk−1)−1 6h(uk+2−uk−2): A numerical approximation that is sixth order accurate is given by @u @x x=xk’1 2h(uk+1−uk−1)−1 3h(uk+2−uk−2)+1 30h(uk+3−uk−3): Methods of arbitrary high order may be constructed. For higher order methods, more points surrounding the point xkwill be utilized. In the limit, the following centered dierence scheme of innite order accuracy isobtained @u @x x=xk=1X j=12(−1)j+1 jh(uk+j−uk−j): (199.1) Eventually, when implementing methods of progressively higher order, the value of u(x)a tap o i n t xk+j,w i t hk+j>N , will be required. If we assume that u(x) is periodic, with period Nh,t h e nu(xi)=u(xi+N). By periodicity, then the value at xj+kis the same as the value at xj+k−N. Hence, methods of arbitrarily high order may be constructed, and only the valuesfu1;u2;:::;uNgwill be utilized. Alternately, for given u(x), a Fourier transform may be taken to deter- mine bu(!)=1p 2Z1 −1u(x)ei!xdx: (199.2) CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 852 IV.C Numerical Methods for PDEs Once determined, bu(!) may be multiplied by −i!, and then an inverse transform taken to yield @u @x=−1p 2Z1 −1i!bu(!)e−i!xd!: (199.3) An informal derivation of this statement is simple; consider dierentiating the formula u(x)=1p 2R1 −1bu(!)e−i!xd!with respect to x. Hence, the rst derivative at every point in a domain may be computed by taking a Fourier transform, multiplying by −i!, and then taking an inverse Fourier transform. By discretizing equations (199.2) and (199.3), the Fourier transforms can be performed by \fast Fourier transforms" (FFTs). The FFT is a fast numerical technique for determining the nite Fourier transform of a function that is dened on a set of equally spacedgrid points. Hence, the derivative at every point in the grid can be computed by taking an FFT, multiplying by the discrete analogue of i!, and then taking an inverse FFT. This approach yields the same numerical scheme given in equation (199.1). Using either technique, a highly accurate nite dierence scheme is generated. This scheme may then be used to numerically approximate the u xterm appearing in a dierential equation. Example Suppose we have the hyperbolic equation for u(x;t) @u @t=@u @x; (199.4) fort0o n0x1 with the periodic boundary conditions u(0;t)=u(1;t); (199.5) and the initial conditions u(x;0) = sin 2x: (199.6) The solution of this system can be determined by the method of char- acteristics (see page 432) to be u(x;t)=s i n 2(x−t). We will compare the solution from our numerical scheme to this exact solution. The pseudospectral method dictates that we take the derivatives of the periodic component ( xin this example) by FFTs. We choose to use a one sided explicit dierence scheme for the time derivative term. Of course, a more accurate derivative expression for the @u=@t term would result in a more accurate numerical approximation (see Gottlieb and Turkel [8]). A Fortran computer program is given in program 199.1 that nds a numerical approximation to the solution of equations (199.4){(199.6). For CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 199. Pseudospectral Method 853 IMPLICIT DOUBLE PRECISION (A-H,O-Z) REAL*8 V(100),X(100),EXACT(100)COMPLEX*16 VV(100),DERIV(100)N=8DELTAT=0.0001D0NTIME=10H=1.D0/DFLOAT(N) PI=3.141592653589D0 W0=1.D0/DFLOAT(N/2-1) C Initialize the vector with the initial conditions DO 10 J=1,NX(J)=DFLOAT(J-1)*H 10 V(J)=DSIN( 2.D0 * PI * X(J) )C Here is the loop in time DO 20 LOOP=1,NTIME TIME=LOOP*DELTAT C take the fourier transform of the V vector DO 30 J=1,N 30 VV(J)=V(J) CALL FFT(N,VV, 1.D0) C multiply by (I W0) NBY2=N/2DO 40 J=1,N 40 DERIV(J)= VV(J) * DCMPLX(0.D0,1.D0) * DFLOAT(-NBY2-1+J) * W0 C Take the inverse Fourier transform CALL FFT(N,DERIV,-1.D0) C Use the derivative values to update the mesh values DO 50 J=1,NV(J)=V(J) + DELTAT*DREAL( DERIV(J) ) 50 EXACT(J)=DSIN( 2.D0*PI*( X(J)-TIME ) ) 20 WRITE(6,5) TIME, (X(K),V(K),EXACT(K),K=1,N) 5 FORMAT(’ Here is the solution at time’,F6.3,/, 1 8(2X,’X=’,F7.4,’ Y(approx)=’,F8.4,’ Y(exact)=’,F8.4/)) END Program 199.1: Fortran program for spectral method. comparison purposes, the exact solution is also printed out. Note that the program calls a subroutine (called FFT(N,V,SIGNI) ), whose source code is not given, to perform the fast Fourier transform. This routine is inputa complex-valued vector Vand returns the same vector, where the values have been modied by V(k)=1 p NNX j=1V(j)e x p 2i(j−1)(k−1)SIGNI N : The last few lines of the program output are shown next: Here is the solution at time 0.001 X= 0. Y(approx)= 0.0010 Y(exact)= -0.0063X= 0.1250 Y(approx)= 0.7078 Y(exact)= 0.7026 X= 0.2500 Y(approx)= 1.0000 Y(exact)= 1.0000 X= 0.3750 Y(approx)= 0.7064 Y(exact)= 0.7115X= 0.5000 Y(approx)= -0.0010 Y(exact)= 0.0063 CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 854 IV.C Numerical Methods for PDEs X= 0.6250 Y(approx)= -0.7078 Y(exact)= -0.7026 X= 0.7500 Y(approx)= -1.0000 Y(exact)= -1.0000X= 0.8750 Y(approx)= -0.7064 Y(exact)= -0.7115 Notes 1. To calculate higher order derivatives, higher powers of ( i!) should be used to multiply ^ u(!). See any book on Fourier transforms (e.g., Butkov [2]). 2. Note that the method, when applied to partial dierential equations, requires that the grid be uniform in every spatial variable in which a FFT is to be taken. 3. This scheme has also been applied to elliptic and parabolic equations, but the results are not much better than using a relatively low order nite dierence scheme. 4. Comparing this method to nite dierences, the pseudospectral method (the nite dierence method) uses a global (local) interpolation of a function, then an approximation of a derivative is made from thisinterpolatory function. 5. Spectral methods are really more general than the limited exposition given here. Theoretically, spectral methods expand the unknownquantities in a series of orthogonal functions; these functions, in turn, result from the solution of a Sturm{Liouville problem. In practice, one considers either a Fourier expansion (as we have done here)| usually for periodic problems|or an expansion in terms of orthogonal polynomials. The Chebyshev polynomials are often used as they areamenable to the fast Fourier transform but also admit more general boundary values than those allowed in Fourier series. The use of Walsh series is discussed in Ohkita and Kobayashi [10]. References [1]Ahner, H. F. Walsh functions and the solution of nonlinear dierential equations. Am. J. Phys. 56 , 7 (July 1988), 628{633. [2]Butkov, E. Mathematical Physics . Addison{Wesley Publishing Co., Reading, MA, 1968. [3]Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. Spectral Methods in Fluid Mechanics . Springer{Verlag, New York, 1987. [4]Cooley, J. W., Lewis, A. W., and Welch, P. D. The fast Fourier transform and its applications. IEEE Transactions on Education E-12 (1969), 27{34. [5]Fornberg, B. High-order nite dierences and the pseudospectral method on staggered grids. SIAM J. Numer. Anal. 12 (August 1990), 904{918. [6]Fornberg, B. A Practical Guide to Pseudospectral Methods . Cambridge Monographs on Applied and Computational Mathematics. Cambridge Uni- versity Press, New York, 1996. [7]Gottlieb, D., and Orszag, S. A. Numerical Analysis of Spectral Methods: Theory and Applications . SIAM, Philadelphia, PA, 1977. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 199. Pseudospectral Method 855 [8]Gottlieb, D., and Turkel, E. On time discretizations for spectral methods. Stud. Appl. Math. 63 (1980), 67{86. [9]Mercier, B. An Introduction to the Numerical Analysis of Spectral Methods . Springer{Verlag, New York, 1989. [10]Ohkita, M., and Kobayashi, Y. Piecewise-linear approximations of solutions of linear dierential equations by Walsh functions. Math. and Computers in Simulation 32 (1990), 297{308. [11]O r s z a g ,S .A . Comparison of pseudospectral and spectral approximation. Stud. Appl. Math. 51 (1979), 253{259. [12]Pickering, M. An Introduction to Fast Fourier Transform Methods for Partial Dierential Equations, with Applications . John Wiley & Sons, New York, 1986. [13]Tadmor, E. Stability analysis of nite-dierence, pseudospectral and Fourier{Galerkin approximations for time-dependent problems. SIAM Review 29 , 4 (1987), 525{555. [14]Tal-Ezer, H. Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26 , 1 (February 1989), 1{11. CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 856 IV.C Numerical Methods for PDEs CD-ROM Handbook of Dierential Equations c/circlecopyrtAcademic Press 1997 Mathematical Nomenclature Applicable to Thesymbolsusedinthisbook. Yields De¯nitions ofallspecialsymbols. Procedure ²Cp[a;b]:Theclassoffunctions thatarecontinuousandhavepcontinuous derivativesontheinterval[a;b]. ²FFourier transform operator. ²H(x):TheHeaviside function orstepfunction; itisde¯ned by H(x)=Zx ¡1±(x)dx=8 >< >:0 ifx<0; 1=2ifx=0; 1 ifx>0: ²=:Theimaginary partofaquantity. ²L:Laplace transform operator. ²O:Wesaythatf(x)=O(g(x))asx!x0ifthereexists apositiveconstan t CandaneighborhoodUofx0suchthatjf(x)j·Cjg(x)jforallxinU. ²o:Wesaythatf(x)=o(g(x))asx!x0if,givenany¹>0,there exists aneighborhoodUofx0suchthatjf(x)j<¹jg(x)jforallxinU. ²p:When z=z(x;y),thenp=zx;when y=y(x),thenp=yx. ²q:When z=z(x;y),thenq=zy. ²r:When z=z(x;y),thenr=zxx. ²s:When z=z(x;y),thens=zxy. ²t:When z=z(x;y),thent=zyy. ²yx(n):Thenthderivativeofywithrespecttox. ²±ij:TheKronec kerdelta, ithasthevalue1ifi=jandthevalue0ifi6=j. ²²:Thisisoften usedtorepresen tasmall numberassumed tobemuchless thanoneinmagnitude. ²±(x):Thedeltafunction; ithastheproperties that±(x)=0forx6=0,butR1 ¡1±(x)dx=1. ²@S:IfSisaregion orvolume, then@Sdenotes itsboundary . ²er:Thespace-time gradien toperator; itisde¯ned byer=[r;@=@t]. ²r2:TheLaplacian; itisde¯ned byr2(Á)=div(grad Á). ²R:Therealnumbers. ²<:Therealpartofaquantity. ²C:Thevector Laplacian; itisde¯ned byCv=grad(div v)¡curlcurlv. ²¤:Thed'Alem bertoperator; itisde¯ned by¤=@2=@t2¡r2. ²´:Asymmetric relation. ²[L;H]:Thecomm utator ofthetwodi®eren tialoperators LandH(see metho d7). ²fu;vg:TheLagrange bracketofthetwoindependen tvariables uandv(see metho d7). ²[f;g]:ThePoisson bracketofthetwofunctions fandg(seemetho d7). ²fy;xg:TheSchwarzian derivativeofywithrespecttox(seemetho d7). 857 ErrorsintheThirdEdition of Handb ookofDi®eren tialEquations byDaniel Zwillinger LAST UPDATED: November22,2000 (1)Section 11,Fixed PointExistence Theorems ,pages 58and59 (a)Thename \Schrauder" should be\Schauder" (b)Thefollowingreference should beadded: J.Schauder,\DerFixpunktsatz inFunktionalraeumen," Studia Math .,2,(1930), 171{180. (Thanks toG.Frieseckeforthese corrections.) (2)Section 27,Canonical Forms,page130,reference number2isnow Bateman, H.Partial Di®erential Equations ofMathematic alPhysics , DoverPublications, NewYork,1944. Whichisincorrect. Thereference should havebeen Bateman, H.Di®erential Equations ,Longmans, Green andCo., NewYork,1926, pages 75-79. (Thanks toAliNejadmala yeriforthiscorrection.) (3)Section 44.1.3, Look-Up Technique ,page189,lastequation beforesec- tion44.2,presen tlyhas y(m)=axy¡m=2 Thisisincorrect, itshould havebeen y(m)=ayx¡m=2 (Thanks toFlavioNocaforthiscorrection.) (4)Section 79,Integrating Functions ,page359,notenumber10,thefol- lowingshould beadded: Thegeneral solution toux=yuyisu=f(x+logy),where fis anarbitrary function. (Thanks toAlain Moussiaux forthisobserv ation.) (5)Section 80,Interchanging Dependen tandIndep enden tVariables , page361,notenumber2,thereference toBender andOrszag should be section 1.5,not1.6. (Thanks toJames Dareforthisobserv ation.) (6)Section 85,Reduction oforder ,page 390,notenumber2,presen tly contains More generally ,iffz1(x);:::;zp(x)garelinearly independen tso- lutions ofequation (85.6), thenthesubstitution y(x)=2 64z1:::zp v z0 1:::z::::::...... z(p) 1:::z(p) p v(p)3 75 reduces equation (85.7) toalinear ordinary di®eren tialequation oforder n¡pforv(x). Thisshould bechanged to More generally ,iffz1(x);:::;zp(x)garelinearly independen tso- lutions ofequation (85.6), thenthesubstitution 1 2 y(x)=2 64z1:::zp z z0 1:::z::::::... z(p) 1:::z(p) p z(p)3 75Á(x) where Á(x)neednotbespeci¯ed, reduces equation (85.6) toa linear ordinary di®eren tialequation oforder n¡pfory(x). Herey(x)canbewritten intheform y(x)=A(x)z(p)+B(x)z(p¡1)+:::;A(x)6=0 anditsderivativeshavetheform y0(x)=A(x)z(p+1)+:::;y00(x)=A(x)z(p+2)+:::; These equations canbeusedtoeliminate fz(p);:::;z(n)gand (85.6) willtaketheform b0y(n¡p)+¢¢¢+bn¡py+V=0 where Vislinear inthefz;z0;:::;z(p¡1)g (Thanks toUnalGoktas forthiscorrection.) (7)Section 93,InverseScattering ,page413,equation (93.1) isnow L[y]=y00+a(x)y0+b(x)=f(x) Whichisincorrect. Thisshould havebeen(note themissing y) L[y]=y00+a(x)y0+b(x)y=f(x) (Thanks toYoungKimforthiscorrection.) (8)Section 106,InverseScattering ,page460,theApplicable tostatemen t should haveattheend havingtheformof(106.2) (Thanks toG.Frieseckeforthisobserv ation.) (9)Section 118,Chaplygin's Metho d,page512,equations (118.5) and(118.6) andthesurrounding textarenow Then de¯ne u1(x)tobethesolution of y0=M(x)y+N(x);y(x0)=y0: (118:5) andde¯ne v1(x)tobethesolution of y0=cM(x)y+bN(x);y(x0)=y0: (118:6) Whichisincorrect. Thisshould havebeen(note thatthede¯nitions have beenswitched): Then de¯ne v1(x)tobethesolution of y0=M(x)y+N(x);y(x0)=y0: (118:5) andde¯ne u1(x)tobethesolution of y0=cM(x)y+bN(x);y(x0)=y0: (118:6) (Thanks toBruno VanderBosscheforthese corrections.) (10)Section 145,Picard Iteration ,page619,notenumberone,thefollowing should beadded: However,thesuccessiv eapproximations areguaran teedtocon- vergetothetruesolution forallxsu±cien tlyclosetozeropro- vided fisacontinuously di®eren tiable function. (Thanks toG.Frieseckeforthisobserv ation.) (11)Section 148,Soliton-T ypeSolutions ,pages 626{627 (a)Inequation (148.3) thetermcv³should be¡cv³. (b)Inequation (148.4) theterm(v³)2should be1 2(v³)2. (c)Anadditional noteshould beadded onpage627tostate 3 With thestandard choice ofA=B=0,thesolution to (148.4) canbesolvedinterms ofelemen taryfunctions: v(x)=3c ¾µ sechµpcx 2¶¶2 (Thanks toG.Frieseckeforthese corrections.) (12)Section 199,Pseudosp ectral Metho d,page851presen tlyhas: @u @x¯¯¯¯ x=xk'1 3h(uk+1¡uk¡1)¡1 6h(uk+2¡uk¡2): and @u @x¯¯¯¯ x=xk'1 2h(uk+1¡uk¡1)¡1 3h(uk+2¡uk¡2)+1 30h(uk+3¡ uk¡3): and @u @x¯¯¯¯ x=xk=1X j=12(¡1)j+1 jh(uk+j¡uk¡j): Whichareallincorrect. They should havebeen: @u @x¯¯¯¯ x=xk'2 3h(uk+1¡uk¡1)¡1 12h(uk+2¡uk¡2): and @u @x¯¯¯¯ x=xk'3 5h(uk+1¡uk¡1)¡3 20h(uk+2¡uk¡2)+1 60h(uk+3¡ uk¡3): and @u @x¯¯¯¯ x=xk=1X j=1(¡1)j+1 jh(uk+j¡uk¡j): (Thanks toDidier Clamond forthese corrections.)