Table Of Content

Partial Differential Equations through Examples and Exercises Kluwer Texts in the Mathematical Sciences VOLUME 18 A Graduate-Level Book Series Partial Differential Equations through Examples and Exercises by Endre Pap Arpad Takaci and Djurdjica Takaci Institute ofM athematics, University ofNovi Sad, Novi Sad, Yugoslavia .. SPRINGER -SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-6349-4 ISBN 978-94-011-5574-8 (eBook) DOI 10.1007/978-94-011-5574-8 Printed an acid-free paper AII Rights Reserved © 1997 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997 Softcover reprint ofthe hardcover Ist edition 1997 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic Of mechanical, incIuding photocopying, recording or by any information storage and retrieval system, without written permis sion from the copyright owner. Contents Preface IX List of symbols Xl 1 Introduction 1 1.1 Basic Notions ......... . 1 1.1.1 Preliminaries ..... . 1 1.1.2 Examples and Exercises 3 1.2 The Cauchy-Kowalevskaya Theorem 12 1.2.1 Preliminaries ....... . 12 1.2.2 Examples and Exercises .. 13 1.3 Equations of Mathematical Physics 15 2 First Order PDEs 17 2.1 Quasi-linear PDEs ...... . 17 2.1.1 Preliminaries ..... . 17 2.1.2 Examples and Exercises 18 2.2 Pfaff's Equations ....... . 32 2.2.1 Preliminaries ..... . 32 2.2.2 Examples and Exercises 33 2.3 Nonlinear First Order PDEs .. 35 2.3.1 Preliminaries ..... . 35 2.3.2 Examples and Exercises 38 3 Classification of the Second Order PDEs 49 3.1 Two Independent Variables .. 49 3.1.1 Preliminaries ..... . 49 3.1.2 Examples and Exercises 53 3.2 n Independent Variables ... . 64 3.2.1 Preliminaries ..... . 64 3.2.2 Examples and Exercises 66 v ci CONTENTS 3.3 Wave, Potential and Heat Equation . . . . . . . . . . . . . . . . . .. 69 4 Hyperbolic Equations 71 4.1 Cauchy Problem for the One-dimensional Wave Equation 71 4.1.1 Preliminaries · . · . · . .. 71 4.1.2 Examples and Exercises · . · . . . .. 72 4.2 Cauchy Problem for the n-dimensional Wave Equation. 80 4.2.1 Preliminaries · .. · . 80 4.2.2 Examples and Exercises · . . . 82 4.3 The Fourier Method of Separation Variables 89 4.3.1 Preliminaries · .. 89 4.3.2 Examples and Exercises 93 4.4 The Sturm-Liouville Problem · 106 4.4.1 Preliminaries · . · . · 106 4.4.2 Examples and Exercises · 109 4.5 Miscellaneous Problems. · 129 4.6 The Vibrating String · 141 5 Elliptic Equations 143 5.1 Dirichlet Problem . · . · . · . · 143 5.1.1 Preliminaries · . · .. · 143 5.1.2 Examples and Exercises · 144 5.2 The Maximum Principle · . · 163 5.2.1 Preliminaries · . · .. · 163 5.2.2 Examples and Exercises · 163 5.3 The Green Function · . · . · 167 5.3.1 Preliminaries · . · . · 167 5.3.2 Examples and Exercises · 168 5.4 The Harmonic Functions · . · 173 5.4.1 Examples and Exercises · 173 5.5 Gravitational Potential · . · .. · 182 6 Parabolic Equations 183 6.1 Cauchy Problem · ... · . · 183 6.1.1 Preliminaries · . · 183 6.1.2 Examples and Exercise · 184 6.2 Mixed Type Problem . · 193 6.2.1 Preliminaries · . · 193 6.2.2 Examples and Exercises · 194 6.3 Heat conduction. · ... · .. .223 CONTENTS vii 7 Numerical Methods 227 7.0.1 Preliminaries ..... . · 227 7.0.2 Examples and Exercises · 230 8 Lebesgue's Integral, Fourier Transform 249 8.1 Lebesgue's Integral and the L2( Q) Space · 249 8.1.1 Preliminaries ..... . .249 8.1.2 Examples and Exercises · 252 8.2 Delta Nets ........... . · 256 8.2.1 Preliminaries ..... . · 256 8.2.2 Examples and Exercises · 257 8.3 The Surface Integrals . . . . . . · 260 8.3.1 Preliminaries ..... . · 260 8.3.2 Examples and Exercises · 261 8.4 The Fourier Transform .... . · 267 8.4.1 Preliminaries ..... . · 267 8.4.2 Examples and Exercises · 269 9 Generalized Derivative and Sobolev Spaces 279 9.1 Generalized Derivative .... . · 279 9.1.1 Preliminaries ..... . · 279 9.1.2 Examples and Exercises · 279 9.2 Sobolev Spaces ........ . · 285 9.2.1 Preliminaries ..... . .285 9.2.2 Examples and Exercises · 286 10 Some Elements from Functional Analysis 303 10.1 Hilbert Space ......... . · 303 10.1.1 Preliminaries ..... . · 303 10.1.2 Examples and Exercises · 305 10.2 The Fredholm Alternatives .. . · 313 10.2.1 Preliminaries ..... . · 313 10.2.2 Examples and Exercises · 314 10.3 Normed Vector Spaces .... . · 321 10.3.1 Preliminaries ..... . · 321 10.3.2 Examples and Exercises · 323 11 Functional Analysis Methods in PDEs 329 11.1 Generalized Dirichlet Problem . · 329 11.1.1 Preliminaries ...... . · 329 11.1. 2 Examples and Exercises . · 330 11.2 The Generalized Mixed Problems · 355 11.2.1 Examples and Exercises . · 355 viii CONTENTS 11.3 Numerical Solutions ...... . · 366 11.3.1 Preliminaries ..... . · 366 11.3.2 Examples and Exercises · 367 11.4 Miscellaneous . . . . . . . . . . · 368 11.4.1 Preliminaries ..... . · 368 11.4.2 Examples and Exercises · 369 12 Distributions in the theory of PDEs 373 12.1 Basic Properties ........ . · 373 12.1.1 Preliminaries ..... . .373 12.1.2 Examples and Exercises · 376 12.2 Fundamental Solutions . . . . . · 390 12.2.1 Preliminaries ..... . .390 12.2.2 Examples and Exercises .390 Bibliography 397 Index 401 Preface The book Partial Differential Equations through Examples and Exercises has evolved from the lectures and exercises that the authors have given for more than fifteen years, mostly for mathematics, computer science, physics and chemistry students. By our best knowledge, the book is a first attempt to present the rather complex subject of partial differential equations (PDEs for short) through active reader-participation. Thus this book is a combination of theory and examples. In the theory of PDEs, on one hand, one has an interplay of several mathematical disciplines, including the theories of analytical functions, harmonic analysis, ODEs, topology and last, but not least, functional analysis, while on the other hand there are various methods, tools and approaches. In view of that, the exposition of new notions and methods in our book is "step by step". A minimal amount of expository theory is included at the beginning of each section Preliminaries with maximum emphasis placed on well selected examples and exercises capturing the essence of the material. Actually, we have divided the problems into two classes termed Examples and Exercises (often containing proofs of the statements from Preliminaries). The examples contain complete solutions, and also serve as a model for solving similar problems, given in the exercises. The readers are left to find the solution in the exercises; the answers, and occasionally, some hints, are still given. The book is implicitly divided in two parts, classical and abstract. In the first (classical) part, the necessary prerequisites are a standard undergraduate course on ODEs, on Riemann's multiple and surface integrals and, of course, on Fourier series. For the second (abstract) part, it would be desirable that the reader is familiar with the elements of Lebesgue integrals and functional analysis (in particular, Hilbert spaces and operator theory). We tried to make the book as self-contained as possible. For that reason, we also included in the Preliminaries and Examples some of the mentioned mathematical tools (see, e.g., elementary proofs of the Closed Graph Theorem, Adjoint Theorem and Uniform Boundedness Theorem in Chapter 10). Many different tools are presented for solving important problems with the basic three partial differential equations: the wave equation, Laplace equation, heat equation and their generalizations. We also give the usual three types of problems with PDEs: initial value problems, boundary value problems and mixed type (eigenvalue) problems. For the solutions of the stated problems, we discuss the three important questions: existence, uniqueness, stability (continuous dependence of solutions upon data). We investigate also three important questions for the solutions of PDEs mostly for applications: construction, regularity and approximation. We present, among other tools, the three principal methods for solving the stated ix x PREFACE problems: Fourier method, Green's function and the energy (variational) method. One of the very useful constructive techniques the Fourier method of separation of variables, is applied first in Chapter 4 for hyperbolic equations with respect to the classical Fourier series, where the eigenfunctions are the sine and cosine functions. In the next step, we generalize this method through the Sturm-Liouville problem also with respect to other systems of orthogonal functions, e.g., Legendre polynomials and Bessel functions. The Fourier method is applied also in Chapters 5 and 6 to elliptic and parabolic equations, respectivily. This theoretical background for these methods is obtained in Chapters 10 and 11 in the language of functional analysis through special spaces as, e.g., Sobolev spaces, with generalized eigenvalues and eigenfunctions. The Fourier analysis is completed in Chapter 8 by the Fourier transform. Most of the book is devoted to second or higher order PDEs. However, for completeness, Chapter 2 treats first order PDEs. In the last Chapter we present a part of the distribution theory, which also covers the theory of Dirac's delta distribution ("delta function"). The majority of the problems are of mathematical character, though we often give physical interpretations (see sections at the ends of Chapter 1, 3, 4, 5 and 6). The numerical approximations and computation of the stated problems are presented in Chapter 7, with an abstract theoretical background in Chapter 11. The book is prepared for undergraduate and graduate students in mathematics, physics, technology, economics and everybody with an interest in partial differential equations for modeling complex systems. We have used Mathematica and Scientific Work Place 2.5. for some calculations and drawings. We are grateful to Prof. Olga Hadzic for her numerous remarks and advice on the text, and to Prof. Darko Kapor on his useful suggestions on the physical aspects of PDEs. Dr Dusanka Perisic made some contributions to Subsections 3.2 and 10.2 and has prepared the Figures 4.1-4.4. It is our pleasure to thank the Institute of Mathematics in Novi Sad for working conditions and financial support. We would like to thank Kluwer Academic Publishers, specially to Dr Paul Roos and Ms Angelique Hempel for their encouragement and patience. Novi Sad, April 1997 ENDRE PAP ARPAD TAKACI DJURDJICA T AKACI