PARTIAL DIFFERENTIAL EQUATIONS - Sharif

1 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 ofRozousse, Bulgaria WeWorld Scientific N E W 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.

2 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 Introductionwith 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, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

3 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.

4 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 Lu, Scientific Editor of WSPC, for the continuous support, advice and active interest in the development of the sec- ond edition.

5 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.

6 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.

7 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.

8 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.

9 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.

10 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, Grammatikopoulos, Grossinho, E. Ifantis, M. Kon, G. Ladas, N. Popivanov, P. Popivanov, Sficas and P. Siafarikas who reviewed the book and offered helpful comments and valuable suggestions for its improve- ment.