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Partial Differential Equations: an Introduction with Mathematica And PARTIAL DIFFERENTIAL EQUATIONS (Second Edition) An Introduction with Mathernatica and MAPLE This page intentionally left blank PARTIAL DIFFERENTIAL EQUATIONS (Scond Edition) An Introduction with Mathematica and MAPLE Ioannis P Stavroulakis University of Ioannina, Greece Stepan A Tersian University of Rozousse, Bulgaria WeWorld Scientific NEW JERSEY 6 LONDON * SINGAPORE * BElJlNG SHANGHAI * HONG KONG * TAIPEI CHENNAI Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. PARTIAL DIFFERENTIAL EQUATIONS An Introduction with Mathematica and Maple (Second Edition) Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book or parts thereoj may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-238-815-X Printed in Singapore. To our wives Georgia and Mariam and our children Petros, Maria-Christina and Ioannis and Takuhi and Lusina This page intentionally left blank Preface In this second edition the section “Weak Derivatives and Weak Solutions” was removed to Chapter 5 to be together with advanced concepts such as discontinuous solutions of nonlinear conservation laws. The figures were re- arranged, many points in the text were improved and the errors in the first edit ion were corrected. Many thanks are due to G. Barbatis for his comments. Also many thanks to our graduate students over several semesters who worked through the text and the exercises making useful suggestions. The second author would like to thank National Research Fund in Bulgaria for the support by the Grant MM 904/99. Special thanks are due to Dr J.T. Lu, Scientific Editor of WSPC, for the continuous support, advice and active interest in the development of the sec- ond edition. September, 2003 Ioannis P. Stavroulakis, Stepan A. Tersian vii This page intentionally left blank Preface to the First Edit ion This textbook is a self-contained introduction to Partial Differential Equa- tions (PDEs). It is designed for undergraduate and first year graduate students who are mathematics, physics, engineering or, in general, science majors. The goal is to give an introduction to the basic equations of mathematical physics and the properties of their solutions, based on classical calculus and ordinary differential equations. Advanced concepts such as weak solutions and discontinuous solutions of nonlinear conservation laws are also considered. Although much of the material contained in this book can be found in standard textbooks, the treatment here is reduced to the following features: 0 To consider first and second order linear classical PDEs, as well as to present some ideas for nonlinear equations. 0 To give explicit formulae and derive properties of solutions for problems with homogeneous and inhomogeneous equations; without boundaries and with boundaries. To consider the one dimensional spatial case before going on to two and three dimensional cases. 0 To illustrate the effects for different problems with model examples: To use Mathematics software products as Mathematzca and MAPLE in ScientifiCWorkPlacE in both graphical and computational aspects; To give a number of exercises completing the explanation to some advanced problems. The book consists of eight Chapters, each one divided into several sections. In Chapter I we present the theory of first-order PDEs, linear, quasilinear, nonlinear, the method of characteristics and the Cauchy problem. In Chapter I1 we give the classification of second-order PDEs in two variables based on the method of characteristics. A classification of almost-linear second-order PDEs in n-variables is also given. Chapter I11 is concerned with the one dimensional wave equation on the whole line, half-line and the mixed problem using the reflection method. The inhomogeneous equation as well as weak derivatives ix X Preface to the First Edition and weak solutions of the wave equation are also discussed. In Chapter IV the one dimensional diffusion equation is presented. The Maximum-minimum principle, the Poisson formula with applications and the reflection method are given. Chapter V contains an introduction to the theory of shock waves and conservation laws. Burgers’ equation and Hopf-Cole transformation are discussed. The notion of weak solutions, Riemann problem, discontionuous solutions and Rankine-Hugoniot condition are considered. In Chapter VI the Laplace equation on the plane and space is considered. Maximum principles, the mean value property, Green’s identities and the representation formulae are given. Green’s functions for the half-space and sphere are discussed, as well as Harnack’s inequalities and theorems. In Chapter VII some basic the- orems on Fourier series and orthogonal systems are given. Fourier methods for the wave, diffusion and Laplace equations are also considered. Finally in Chapter VIII two and three dimensional wave and diffusion equations are con- sidered. Kirchoff’s formula and Huygens’ principle as well as Fourier method are presented . Model examples are given illustrated by software products as Muthematicu and MAPLE in ScientifiCWorkPlacE. We also present the programs in Math- ematica for those examples. For further details in Muthemutica the reader is referred to Wolfram [49], Ross [34] and Vvedensky [47]. A special word of gratitude goes to N. Artemiadis, G. Dassios, K. Gopal- samy, M.K. Grammatikopoulos, M.R. Grossinho, E. Ifantis, M. Kon, G. Ladas, N. Popivanov, P. Popivanov, Y.G. Sficas and P. Siafarikas who reviewed the book and offered helpful comments and valuable suggestions for its improve- ment. Many thanks are also due to G. Georgiou, J.R. Graef, G. Karakostas, K. Kyriaki, Th. Kyventidis, A. Raptis, Th. Vidalis for their comments and to T. Kiguradze, G. Kvinikadze, J.H. Shen for their extensive help with the proofreading of the material. The help of S.I. Biltchev, J. Chaparova and M. Karaivanova is gratefully acknowledged. Our deep appreciation to Calouste Gulbenkian Foundation and to the Greek Ministry of National Economy. Special thanks are due to Ms S.H. Gan, Editor of WSPC, for her contin- uous support, advice and active interest in the development of this project. June, 1999 Ioannis P. Stavroulakis, Stepan A. Tersian Contents 1. First-order Partial Differential Equations 1 1.1. Introduction 1 1.2. Linear First-order Equations 4 1.3. The Cauchy Problem for First-order Quasi-linear Equations 11 1.4. General Solutions of Quasi-linear Equations 23 1.5. Fully-nonlinear First-order Equations 28 2. Second-order Partial Differential Equations 39 2.1. Linear Equations 39 2.2. Classification and Canonical Forms of Equations in Two Independent Variables 46 2.3. Classification of Almost-linear Equations in R" 59 3. One Dimensional Wave Equation 67 3.1. The Wave Equation on the Whole Line. D'Alembert Formula 67 3.2. The Wave Equation on the Half-line, Reflection Method 78 3.3. Mixed Problem for the Wave Equation 84 3.4. Inhomogeneous Wave Equation 87 3.5. Conservation of the Energy 92 4. One Dimensional Diffusion Equation 97 4.1. Maximum-minimum Principle for the Diffusion Equation 97 4.2. The Diffusion Equation on the Whole Line 103 4.3. Diffusion on the Half-line 115 4.4. Inhomogeneous Diffusion Equation on the Whole Line 118 5. Weak Solutions, Shock Waves and Conservation Laws 123 5.1. Weak Derivatives and Weak Solutions 123 xi xii Contents 5.2. Conservation Laws 130 5.3. Burgers’ Equation 140 5.4. Weak Solutions. Riemann Problem 153 5.5. Discontinuous Solutions of Conservation Laws. Rankine-Hugoniot Condition 162 6. The Laplace Equation 169 6.1. Harmonic Functions. Maximum-minimum Principle 169 6.2. Green’s Identities 173 6.3. Green’s Functions 182 6.4. Green’s Functions for a Half-space and Sphere 185 6.5. Harnack’s Inequalities and Theorems 193 7. Fourier Series and Fourier Method for PDEs 199 7.1. Fourier Series 199 7.2. Orthonormal Systems. General Fourier Series 217 7.3. Fourier Method for the Diffusion Equation 229 7.4. Fourier Method for the Wave Equation 238 7.5. Fourier Method for the Laplace Equation 243 8. Diffusion and Wave Equations in Higher Dimensions 255 8.1. The Diffusion Equation in Three Dimensional Space 255 8.2. Fourier Method for the Diffusion Equation in Higher Dimensions 262 8.3. Kirchoff’s Formula for the Wave Equation. Huygens’ Principle 269 8.4. Fourier Method for the Wave Equation on the Plane. Nodal Sets 276 References 287 Answers and Hints to Exercises 29 1 Index 301 Chapter 1 First-order Partial Differential Equations 1.1 Introduction Let u = u(q,..., 2,) be a function of n independent variables z1, ...,2,. A Partial Differential Equation (PDE for short) is an equation that contains the independent variables q , ... , Xn, the dependent variable or the unknown function u and its partial derivatives up to some order. It has the form where F is a given function and uXj= au/aXj, uxCixj= a2U/aX;azj, i,j = 1, ...,n are the partial derivatives of u. The order of a PDE is the order of the highest derivative which appears in the equation. A set R in the n-dimensional Euclidean space Rn is called a domain if it is an open and connected set. A region is a set consisting of a domain plus, perhaps, some or all of its boundary points.
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