Partial Differential Equations. Theory And Technique

PARTIAL DIFFERENTIAL EQUATIONS Theory and Technique GEORGE F. CARRIER Harvard University Cambridge, Massachusetts CARL E. PEARSON University of Washington Seattle, Washington ACADEMIC PRESS New York San Francisco London A Subsidiary of Harcourt Brace Jovanovich, Publishers 1976 COPYRIGHT © 1976, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 Library of Congress Cataloging in Publication Data Carrier, George F Partial differential equations. Bibliography: p. Includes index. 1. Differential equations, Partial-Numerical solutions. I. Pearson, Carl E.Joint author. II. Title. QA374.C36 515'.353 75-13107 ISBN 0 - 1 2 - 1 6 0 4 5 0 - 0 AMS (MOS) 1970 Subject Classifications: 35-01, 35-02 PRINTED IN THE UNITED STATES OF AMERICA PREFACE This book reflects the authors' experience in teaching partial differential equations, over several years, and at several institutions. The viewpoint is that of the user of mathematics; the emphasis is on the development of perspective and on the acquisition of practical technique. Illustrative examples chosen from a number of fields serve to motivate the discussion and to suggest directions for generalization. We have provided a large number of exercises (some with answers) in order to consolidate and extend the text material. The reader is assumed to have some familiarity with ordinary differential equations of the kind provided by the references listed in the Introduction. Some background in the physical sciences is also assumed, although we have tried to choose examples that are common to a number of fields and which in any event are intuitively straightforward. Although the attitudes and approaches in this book are solely the responsibility of the authors, we are indebted to a number of our colleagues for useful suggestions and ideas. A note of particular appreciation is due to Carolyn Smith, who patiently and meticulously prepared the successive versions of the manuscript, and to Graham Carey, who critically proofread most of the final text. xi INTRODUCTION We collect here some formal definitions and notational conventions. Also, we analyze a preliminary example of a partial differential equation in order to point up some of the differences between ordinary and partial differential equations. The systematic discussion of partial differential equations begins in Chapter 1. We start with the classical second-order equations of diffusion, wave motion, and potential theory and examine the features of each. We then use the ideas of characteristics and canonical forms to show that any second-order linear equation must be one of these three kinds. First-order linear and quasi-linear equations are considered next, and the first half of the book ends with a generalization of previous results to the case of a larger number of dependent or independent variables, and to sets of equations. Included in the second half of the book are separate chapters on Green's functions, eigenvalue problems, and a more extensive survey of the theory of cha