Traveling Wave Analysis ofPartial Differential

Equations

Traveling Wave Analysis ofPartial Differential

EquationsNumerical and Analytical Methods

with MATLABr and Maple™

Graham W. GriffithsCity University, London, UK

William E. SchiesserLehigh University, Bethlehem, PA, USA

AMSTERDAM • BOSTON • HEIDELBERG • LONDONNEW YORK • OXFORD • PARIS • SAN DIEGO

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Dedication

To our teachers, with respect and appreciation.

Table of Contents

Preface xi

1. Introduction to Traveling Wave Analysis 1

Traveling Wave Solutions 1

Residual Function Solutions 4

References 6

2. Linear Advection Equation 7

Smooth Solutions 7

Solutions with Sharp Gradients or Discontinuities 20

Appendix 37

References 44

3. Linear Diffusion Equation 47

Reference 55

4. A Linear Convection Diffusion Reaction Equation 57

Reference 65

5. Diffusion Equation with Nonlinear Source Terms 67

Appendix 1 102

Appendix 2 106

References 109

Traveling Wave Analysis of Partial Differential Equations. DOI: 10.1016/B978-0-12-384652-5.xxxxx-xCopyright © 2011 by Elsevier Inc. All rights reserved.

vii

viii Table of Contents

6. Burgers–Huxley Equation 111

Appendix 118

References 121

7. Burgers–Fisher Equation 123

Appendix 128

Reference 133

8. Fisher–Kolmogorov Equation 135

Appendix 141

References 146

9. Fitzhugh–Nagumo Equation 147

Appendix 164

References 171

10. Kolmogorov–Petrovskii–Piskunov Equation 173

Appendix 179

References 183

11. Kuramoto–Sivashinsky Equation 185

Appendix 192

References 195

12. Kawahara Equation 197

Appendix 1 217

Appendix 2 234

References 237

13. Regularized Long Wave Equation 239

Appendix 254

References 260

Table of Contents ix

14. Extended Bernoulli Equation 261

Appendix 271

References 273

15. Hyperbolic Liouville Equation 275

Appendix 284

References 292

16. Sine-Gordon Equation 293

Appendix 301

References 307

17. Mth-Order Klein–Gordon Equation 309

Appendix 336

References 338

18. Boussinesq Equation 339

Appendix 370

References 374

19. Modified Wave Equation 377

Appendix 387

Reference 389

Appendix: Analytical Solution Methods for TravelingWave Problems 391

A.1 Introduction 391

A.2 Tanh Method 391

A.3 Exp Method 411

A.4 Riccati Equation Method 418

A.5 Direct Integration 429

A.6 Factorization 430

x Table of Contents

A.7 Additional Solutions by Addition of Arbitrary Constants 435

A.8 Other Methods 435

A.9 Maple Built-In Procedure TWSolutions 436

References 437

Index 441

Preface

Partial differential equations (PDEs) have been developed and used in science and engineer-ing for more than 200 years, yet they remain a very active area of research because of boththeir role in mathematics and their application to virtually all areas of science and engineer-ing. This research has been spurred by the relatively recent development of computer solutionmethods for PDEs. These have extended PDE applications such that we can now quantifybroad areas of physical, chemical, and biological phenomena. The current development of PDEsolution methods is an active area of research that has benefited greatly from advances in com-puter hardware and software, and the growing interest in addressing PDE models of increasingcomplexity.

A large class of models now being actively studied are of a type and complexity such thattheir solutions are usually beyond traditional mathematical analysis. Consequently, numeri-cal methods have to be employed. These numerical methods, some of which are still beingdeveloped, require testing and validation. This is often achieved by studying PDEs that haveknown exact analytical solutions. The development of analytical solutions is also an active areaof research, with many advances being reported recently, particularly for systems describedby nonlinear PDEs. Thus, the development of analytical solutions directly supports the devel-opment of numerical methods by providing a spectrum of test problems that can be used toevaluate numerical methods.

This book surveys some of these new developments in analytical and numerical meth-ods and is aimed at senior undergraduates, postgraduates, and professionals in the fields ofengineering, mathematics, and the sciences. It relates these new developments through theexposition of a series of traveling wave solutions to complex PDE problems. The PDEs that havebeen selected are largely named in the sense that they are generally closely linked to their orig-inal contributors. These names usually reflect the fact that the PDEs are widely recognized andare of fundamental importance to the understanding of many application areas. Each chapterfollows the general format:. The PDE and its associated auxiliary conditions (initial conditions (ICs) and boundary

conditions (BCs)) are stated.. A series of routines is discussed with detailed explanations of the code and how it relates tothe PDE. They are written in Matlab but have been specifically programmed so that theycan be easily converted to equivalent routines in other languages. The routines have thefollowing common features:– The numerical procedure is the method of lines (MOL) in which the boundary value

(spatial) partial derivatives are replaced with algebraic approximations, in the presentcase finite differences (FDs), although other approximations such as finite elements(FEs), finite volumes (FVs), and spectral methods (SMs) could be used. The FDapproximations are implemented in a series of library routines; the details of how theseroutines were developed are given as an introduction to facilitate the development ofnew routines that may be required for particular PDE applications.

Traveling Wave Analysis of Partial Differential Equations. DOI: 10.1016/B978-0-12-384652-5.xxxxx-xCopyright © 2011 by Elsevier Inc. All rights reserved.

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xii Preface

– The resulting system of ordinary differential equations (ODEs) in an initial valuevariable, typically time in an application, is then integrated numerically using an initialvalue ODE integrator from the Matlab library.

– The displayed numerical output also includes the analytical solution and the differencebetween the numerical and analytical solutions. The agreement between the twosolutions is displayed numerically and graphically as a way of demonstrating thevalidity of the numerical methods.. An analytical solution for the PDE is stated, including a reference to the original source of

the solution, and in some cases, a verification (proof) of the solution by substitution intothe PDE and auxiliary conditions.. Additionally, in several chapters, the analytical solution is derived by relatively newtechniques such as the tanh-, exp-, Riccati- or factorization-based methods. The derivationis either by direct application of the analytical method or through the use of the computeralgebra system (CAS), Maple. Where Maple is used, the associated code is included in thetext along with a description of its main functional elements. This code usuallydemonstrates the use of our new Maple procedures, which implement various analyticalmethods that are described in the text. Graphical output from these Maple applications isprovided, including a 2D animation (to facilitate insight into and understanding of thesolution) and a plot in 3D perspective. Maple is also used in other chapters to confirmanalytical solutions from the literature. Where appropriate, the code is provided in the mwsfile format as well as the mw format so that it will also run in early versions of Maple.. The form of the analytical solution is considered, with particular emphasis on travelingwave analysis by which the PDE (in an Eulerian or fixed frame) is converted to an ODE (in aLagrangian or moving frame). An analytical solution to the ODE is then derived and theinverse coordinate transformation is applied to provide an analytical solution to the PDE.. A second approach to a PDE analytical solution, the method of residual functions, is alsoused in some of the chapters to derive an analytical solution to a PDE that is closely relatedto the original PDE.. The basic approach of traveling wave analysis, whereby a PDE is transformed to anassociated ODE, is also reversed in two chapters. These start with ODEs that are thenrestated as PDEs that are first and second order in the initial value variable. The analyticalsolution to the initial ODE is then provided as an analytical solution to the PDE.. The structure of the PDE is usually revisited briefly with regard to its form, such as whetherit is first or second order in the initial value variable, the order of the boundary valuederivatives, the features of nonlinear terms, and the form of the BCs. In this way, theintention of the final summary is to suggest concepts and computational approaches thatcan be applied in new PDE applications.. Each chapter concludes with a discussion of the numerical solution, particularly how itconforms to the initial statement of the PDE and its auxiliary conditions; also thenumerical solution is evaluated with regard to the magnitude of the errors and how theseerrors might be reduced through additional computation.. In Chapter 2 we discuss the linear advection equation, one of the simplest PDEs, and showthat solutions involving steep gradients or discontinuities can be difficult to achievenumerically. We then illustrate how flux limiters can be employed to improve the fidelity ofthe numerical solution. A short appendix to this chapter is also included, which brieflydiscusses some of the background to the ideas behind flux limiters.. A general appendix details the tanh-, exp-, Ricatti-, direct integration-, andfactorization-based methods. Maple implementation, by way of newly developedprocedures, is included for the tanh-, exp- and Ricatti-based analytical methods. As

Preface xiii

referred to above, these general features are then referenced for specific applications inappendices to individual chapters.

In summary the major focus of this book is the numerical MOL solution of PDEs and thetesting of numerical methods with analytical solutions, through a series of applications. Theorigin of the analytical solutions through traveling wave and residual function analysis providesa framework for the development of analytical solutions to nonlinear PDEs that are now widelyreported in the literature. Also in selected chapters, procedures based on the tanh, exp, andRicatti methods that have recently received major attention are used to illustrate the derivationof analytical solutions. References are provided where appropriate to additional informationon the techniques and methods deployed.

Our intention is to provide a set of software tools that implement numerical and analyticalmethods that can be applied to a broad spectrum of problems in PDEs. They are based on theconcept of a traveling wave and the central feature of these methods is conversion of the systemPDEs to ODEs. The discussion is limited to one-dimensional (1D) PDEs and complements ourearlier book A Compendium of Partial Differential Equation Models: Method of Lines Analysiswith Matlab, Cambridge University Press, 2009.

Finally all the code discussed in this book, along with a set of the MOL DSS library routines,is available for download from www.pdecomp.net.

Graham W. GriffithsNayland, Suffolk, UK

William E. SchiesserBethlehem, PA, USA

June 1, 2010