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 of Mathematics, 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

List of symbols

1 Introduction 1.1 Basic Notions ......... .

1.1.1 Preliminaries ..... . 1.1.2 Examples and Exercises

1.2 The Cauchy-Kowalevskaya Theorem 1.2.1 Preliminaries ....... . 1.2.2 Examples and Exercises ..

1.3 Equations of Mathematical Physics

2 First Order PDEs 2.1 Quasi-linear PDEs ...... .

2.1.1 Preliminaries ..... . 2.1.2 Examples and Exercises

2.2 Pfaff's Equations ....... . 2.2.1 Preliminaries ..... . 2.2.2 Examples and Exercises

2.3 Nonlinear First Order PDEs .. 2.3.1 Preliminaries ..... . 2.3.2 Examples and Exercises

3 Classification of the Second Order PDEs 3.1 Two Independent Variables ..

3.1.1 Preliminaries ..... . 3.1.2 Examples and Exercises

3.2 n Independent Variables ... . 3.2.1 Preliminaries ..... . 3.2.2 Examples and Exercises

v

IX

Xl

1 1 1 3

12 12 13 15

17 17 17 18 32 32 33 35 35 38

49 49 49 53 64 64 66

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

7 Numerical Methods 7.0.1 Preliminaries ..... . 7.0.2 Examples and Exercises

8 Lebesgue's Integral, Fourier Transform 8.1 Lebesgue's Integral and the L 2 ( Q) Space

8.1.1 Preliminaries ..... . 8.1.2 Examples and Exercises

8.2 Delta Nets ........... . 8.2.1 Preliminaries ..... . 8.2.2 Examples and Exercises

8.3 The Surface Integrals . . . . . . 8.3.1 Preliminaries ..... . 8.3.2 Examples and Exercises

8.4 The Fourier Transform .... . 8.4.1 Preliminaries ..... . 8.4.2 Examples and Exercises

9 Generalized Derivative and Sobolev Spaces 9.1 Generalized Derivative .... .

9.1.1 Preliminaries ..... . 9.1.2 Examples and Exercises

9.2 Sobolev Spaces ........ . 9.2.1 Preliminaries ..... . 9.2.2 Examples and Exercises

10 Some Elements from Functional Analysis 10.1 Hilbert Space ......... .

10.1.1 Preliminaries ..... . 10.1.2 Examples and Exercises

10.2 The Fredholm Alternatives .. . 10.2.1 Preliminaries ..... . 10.2.2 Examples and Exercises

10.3 Normed Vector Spaces .... . 10.3.1 Preliminaries ..... . 10.3.2 Examples and Exercises

11 Functional Analysis Methods in PDEs 11.1 Generalized Dirichlet Problem .

11.1.1 Preliminaries ...... . 11.1. 2 Examples and Exercises .

11.2 The Generalized Mixed Problems 11.2.1 Examples and Exercises .

vii

227 · 227 · 230

249 · 249 .249 · 252 · 256 · 256 · 257 · 260 · 260 · 261 · 267 · 267 · 269

279 · 279 · 279 · 279 · 285 .285 · 286

303 · 303 · 303 · 305 · 313 · 313 · 314 · 321 · 321 · 323

329 · 329 · 329 · 330 · 355 · 355

viii

11.3 Numerical Solutions ...... . 11.3.1 Preliminaries ..... . 11.3.2 Examples and Exercises

11.4 Miscellaneous . . . . . . . . . . 11.4.1 Preliminaries ..... . 11.4.2 Examples and Exercises

12 Distributions in the theory of PDEs 12.1 Basic Properties ........ .

12.1.1 Preliminaries ..... . 12.1.2 Examples and Exercises

12.2 Fundamental Solutions . . . . . 12.2.1 Preliminaries ..... . 12.2.2 Examples and Exercises

Bibliography

Index

CONTENTS

· 366 · 366 · 367 · 368 · 368 · 369

373 · 373 .373 · 376 · 390 .390 .390

397

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

List of Symbols

N Z Z+ R Rn

C iRz ~z

z AC XA n u \ ~

Q 8Q Q Q'ccQ x

Ixl a

z~ 10'1 a! x'" liminf lim sup

set of natural numbers set of integers set of non-negative integers set of real numbers n-dimensional real Euclidean space set of complex numbers

= real part of a complex number z = imaginary part of a complex number z

A imaginary unit complement of the set A characteristic function of the set A intersection of sets union of sets set difference a-algebra of subsets of a set X region of Rn border of the region Q closure of the region Q closure of Q' is a subset of Q (XI, ... ,xn ) ERn

= /x? + ... + x~ = (0'1, ... ,an) multi-index, where ai E Z+, i = 1,2, ... , n

set of multi-indices 0'1 + ... + an a1!" . an! Xfl ... x~n limes inferior limes superior

xi

xii

Dj

D='V

~

(fIg) suppf Ck(Q)

COO(O) C;'(O)

L2 (Q) Ilfll£2(Q) Wk(Q)

Ilfllwk(Q) ok

W (Q)

8 'D(O) 'D'(O)

o OXj

(D1, ... ,Dn )

02 02 --2 + ... + --2 (Laplace operator) ox} oXn the scalar product support of a function or distribution

LIST OF SYMBOLS

space of continuous functions on Q with continuous derivatives of order:::; k space of infinitely differentiable functions over an open set 0 space of infinitely differentiable functions over an open set 0 with compact support space of measurable functions f with fQ If(x)i2 dx < 00

(JQ If(x)12 dx)1/2 Sobolev space of order k

( L fQ IDCXf(xWdx)1/2 Icxl~k

---:=;=:-Sobolev space which is C(f(Q) (closure with respect to II· IIwk(Q)) delta distribution (" delta function") space of test functions over the open set 0 space of distributions over the open set 0