Partial Differential Equations through Examples and Exercises

415 Pages·2012·15.285 MB·English

by  Endre Pap, Arpad Takaci, Djurdjica Takaci

Partial Differential Equations through Examples and Exercises
Kluwer Texts in the Mathematical Sciences
VOLUME 18
A Graduate-Level Book Series
http://avaxhome.ws/blogs/ChrisRedfield
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