Traveling Wave Analysis of Partial Differential Equations

Transcription

1 Traveling Wave Analysis of Partial Differential Equations

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3 Traveling Wave Analysis of Partial Differential Equations Numerical and Analytical Methods with MATLAB R and Maple Graham W. Griffiths City University, London, UK William E. Schiesser Lehigh University, Bethlehem, PA, USA AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Academic Press is an imprint of Elsevier

4 Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK Copyright 2012 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our Web site This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). MATLAB R is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book s use or discussion of MATLAB R software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB R software. Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data Application submitted British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: For information on all Academic Press publications visit our Web site at Typeset by: diacritech, India Printed in the United States of America

5 To our teachers, with respect and appreciation. Dedication

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7 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 Linear Diffusion Equation 47 Reference A Linear Convection Diffusion Reaction Equation 57 Reference Diffusion Equation with Nonlinear Source Terms 67 Appendix Appendix References 109 Traveling Wave Analysis of Partial Differential Equations. DOI: /B xxxxx-x Copyright 2011 by Elsevier Inc. All rights reserved. vii

8 viii Table of Contents 6. Burgers Huxley Equation 111 Appendix 118 References Burgers Fisher Equation 123 Appendix 128 Reference Fisher Kolmogorov Equation 135 Appendix 141 References Fitzhugh Nagumo Equation 147 Appendix 164 References Kolmogorov Petrovskii Piskunov Equation 173 Appendix 179 References Kuramoto Sivashinsky Equation 185 Appendix 192 References Kawahara Equation 197 Appendix Appendix References Regularized Long Wave Equation 239 Appendix 254 References 260

9 Table of Contents ix 14. Extended Bernoulli Equation 261 Appendix 271 References Hyperbolic Liouville Equation 275 Appendix 284 References Sine-Gordon Equation 293 Appendix 301 References Mth-Order Klein Gordon Equation 309 Appendix 336 References Boussinesq Equation 339 Appendix 370 References Modified Wave Equation 377 Appendix 387 Reference 389 Appendix: Analytical Solution Methods for Traveling Wave 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

10 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

11 Preface Partial differential equations (PDEs) have been developed and used in science and engineering for more than 200 years, yet they remain a very active area of research because of both their role in mathematics and their application to virtually all areas of science and engineering. This research has been spurred by the relatively recent development of computer solution methods for PDEs. These have extended PDE applications such that we can now quantify broad areas of physical, chemical, and biological phenomena. The current development of PDE solution methods is an active area of research that has benefited greatly from advances in computer hardware and software, and the growing interest in addressing PDE models of increasing complexity. A large class of models now being actively studied are of a type and complexity such that their solutions are usually beyond traditional mathematical analysis. Consequently, numerical methods have to be employed. These numerical methods, some of which are still being developed, require testing and validation. This is often achieved by studying PDEs that have known exact analytical solutions. The development of analytical solutions is also an active area of research, with many advances being reported recently, particularly for systems described by nonlinear PDEs. Thus, the development of analytical solutions directly supports the development of numerical methods by providing a spectrum of test problems that can be used to evaluate numerical methods. This book surveys some of these new developments in analytical and numerical methods and is aimed at senior undergraduates, postgraduates, and professionals in the fields of engineering, mathematics, and the sciences. It relates these new developments through the exposition of a series of traveling wave solutions to complex PDE problems. The PDEs that have been selected are largely named in the sense that they are generally closely linked to their original contributors. These names usually reflect the fact that the PDEs are widely recognized and are of fundamental importance to the understanding of many application areas. Each chapter follows 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 to the PDE. They are written in Matlab but have been specifically programmed so that they can be easily converted to equivalent routines in other languages. The routines have the following 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 present case finite differences (FDs), although other approximations such as finite elements (FEs), finite volumes (FVs), and spectral methods (SMs) could be used. The FD approximations are implemented in a series of library routines; the details of how these routines were developed are given as an introduction to facilitate the development of new routines that may be required for particular PDE applications. Traveling Wave Analysis of Partial Differential Equations. DOI: /B xxxxx-x Copyright 2011 by Elsevier Inc. All rights reserved. xi

12 xii Preface The resulting system of ordinary differential equations (ODEs) in an initial value variable, typically time in an application, is then integrated numerically using an initial value ODE integrator from the Matlab library. The displayed numerical output also includes the analytical solution and the difference between the numerical and analytical solutions. The agreement between the two solutions is displayed numerically and graphically as a way of demonstrating the validity 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 into the PDE and auxiliary conditions.. Additionally, in several chapters, the analytical solution is derived by relatively new techniques such as the tanh-, exp-, Riccati- or factorization-based methods. The derivation is either by direct application of the analytical method or through the use of the computer algebra system (CAS), Maple. Where Maple is used, the associated code is included in the text along with a description of its main functional elements. This code usually demonstrates the use of our new Maple procedures, which implement various analytical methods that are described in the text. Graphical output from these Maple applications is provided, including a 2D animation (to facilitate insight into and understanding of the solution) and a plot in 3D perspective. Maple is also used in other chapters to confirm analytical solutions from the literature. Where appropriate, the code is provided in the mws file 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 traveling wave analysis by which the PDE (in an Eulerian or fixed frame) is converted to an ODE (in a Lagrangian or moving frame). An analytical solution to the ODE is then derived and the inverse 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 also used in some of the chapters to derive an analytical solution to a PDE that is closely related to the original PDE.. The basic approach of traveling wave analysis, whereby a PDE is transformed to an associated ODE, is also reversed in two chapters. These start with ODEs that are then restated as PDEs that are first and second order in the initial value variable. The analytical solution 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 whether it is first or second order in the initial value variable, the order of the boundary value derivatives, the features of nonlinear terms, and the form of the BCs. In this way, the intention of the final summary is to suggest concepts and computational approaches that can be applied in new PDE applications.. Each chapter concludes with a discussion of the numerical solution, particularly how it conforms to the initial statement of the PDE and its auxiliary conditions; also the numerical solution is evaluated with regard to the magnitude of the errors and how these errors might be reduced through additional computation.. In Chapter 2 we discuss the linear advection equation, one of the simplest PDEs, and show that solutions involving steep gradients or discontinuities can be difficult to achieve numerically. We then illustrate how flux limiters can be employed to improve the fidelity of the numerical solution. A short appendix to this chapter is also included, which briefly discusses some of the background to the ideas behind flux limiters.. A general appendix details the tanh-, exp-, Ricatti-, direct integration-, and factorization-based methods. Maple implementation, by way of newly developed procedures, is included for the tanh-, exp- and Ricatti-based analytical methods. As

13 Preface xiii referred to above, these general features are then referenced for specific applications in appendices to individual chapters. In summary the major focus of this book is the numerical MOL solution of PDEs and the testing of numerical methods with analytical solutions, through a series of applications. The origin of the analytical solutions through traveling wave and residual function analysis provides a framework for the development of analytical solutions to nonlinear PDEs that are now widely reported in the literature. Also in selected chapters, procedures based on the tanh, exp, and Ricatti methods that have recently received major attention are used to illustrate the derivation of analytical solutions. References are provided where appropriate to additional information on the techniques and methods deployed. Our intention is to provide a set of software tools that implement numerical and analytical methods that can be applied to a broad spectrum of problems in PDEs. They are based on the concept of a traveling wave and the central feature of these methods is conversion of the system PDEs to ODEs. The discussion is limited to one-dimensional (1D) PDEs and complements our earlier book A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab, Cambridge University Press, Finally all the code discussed in this book, along with a set of the MOL DSS library routines, is available for download from Graham W. Griffiths Nayland, Suffolk, UK William E. Schiesser Bethlehem, PA, USA June 1, 2010