Partial Differential Equations through Examples and Exercises
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 be
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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 ofMathematics, 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 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
1 1 1 3
12 12 13
15
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
17 17 17 18 32 32 33 35 35 38
3
49
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
49 49 53
64 64 66
CONTENTS
ci
3.3 4
5
6
Wave, Potential and Heat Equation . . . . . . . . . . . . . . . . . .. 69
71
Hyperbolic Equations 4.1 Cauchy Problem for the One-dimensional Wave Equation 4.1.1 Preliminaries · . · . · . .. 4.1.2 Examples and Exercises · . · . . . .. 4.2 Cauchy Problem for the n-dimensional Wave Equation. 4.2.1 Preliminaries · .. · . 4.2.2 Examples and Exercises · . . . 4.3 The Fourier Method of Separation Variables 4.3.1 Preliminaries · .. 4.3.2 Examples and Exercises 4.4 The Sturm-Liouville Problem 4.4.1 Preliminaries ·.·. 4.4.2 Examples and Exercises 4.5 Miscellaneous Problems. 4.6 The Vibrating String
· · · · ·
72 80 80 82 89 89 93 106 106 109 129 141
Elliptic Equations 5.1 Dirichlet Problem . · . · . · . 5.1.1 Preliminaries · . · .. 5.1.2 Examples and Exercises 5.2 The Maximum Principle ·. 5.2.1 Preliminaries · . · .. 5.2.2 Examples and Exercises 5.3 The Green Function ·.·. 5.3.1 Preliminaries ·.·. 5.3.2 Examples and Exercises 5.4 The Harmonic Functions · . 5.4.1 Examples and Exercises 5.5 Gravitational Potential · . · ..
· · · · · · · · · · · ·
143 143 143 144 163 163 163 167 167 168 173 173 182
Parabolic Equations 6.1 Cauchy P
