Partial Differential Equations

230 Pages·1971·5.252 MB·English

by  Fritz John (auth.)

Applied
Mathematical
Sciences
EDITORS
Fritz John Joseph P. LaSalle Lawrence Sirovich
Courant Institute 01 Division 01 Division 01
Mathematical Sciences Applied Mathematics Applied Mathematics
New York University Brown University Brown University
New York, N. Y. 10003 Providence, R. I. 02912 Providence, R. I. 02912
EDITORIAL STATEMENT
The mathematization of all sciences, the fading of traditional scientific boun
daries, the impact of computer technology, the growing importance of mathematical
computer modelling and the necessity of scientific planning all create the need both
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The purpose of this series is to provide such books, suitable for the user of
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Verlag and in the same spirit as this series.
MANUSCRIPTS
The Editors welcome all inquiries regarding the submission of manuscripts
for the series. Final preparation of all manuscripts will take place in the editorial
offices of the series in the Division of Applied Mathematics, Brown University, Prov
iden ce, Rhode Island.
Published by SPRINGER-VERLAG NEW YORK INC., 175 Fifth Avenue,
New York, N.Y. 10010.
I
Applied Mathematical Sciences Volume 1
F.John
Partial
Differential
Equations
With 31 Illustrations
Springer-Verlag New York· Heidelberg • Berlin
1971
Fritz John
Courant Institute 01 Mathematical Sciences
New York University, New York, New York
All rights reserved
No part of this book may be translated or reprodueed in any form
without written permission from Springer-Verlag.
© 1971 by Springer-Verlag New York Ine.
Lribrary of Congress Card Number 76-149140
ISBN-13: 978-0-387-90021-6 e-ISBN-13: 978-1-4615-9966-1
DOI: 10.1007/978-1-4615-9966-1
PREFACE
These Notes grew out of a course given by the author in 1952-53. Though
the field of Partial Differential Equations has changed considerably since those
days, particularly under the impact of methods taken from Functional Analysis, the
author feels that the introductory material offered here still is basic for an
understanding of the subject. It supplies the necessary intuitive foundation
which motivates and anticipates abstract formulations of the questions and relates
them to the description of natual phenomena.
In the present edition, only minor corrections have been made in the text.
An Index and up-to-date listing of books recommended for further study have been
added.
Fritz John
New York
November 19, 1970
v
TABLE OF CONTENTS
Introduetion 1
CHAPl'ER I - TEE SINGLE FIRST ORDER EQUATION
1. The linear and quasi-linear equations. 6
2. The general first order equation for a funetion
of two variables. • • • • • • • • • 15
3. The general first order equation for a funetion
of n independent variables. • • • • • 37
CHAPl'ER II - TEE CAUCIIT PROBLEM FOR HIGEER ORDER EQUATIONS
1. Analytie funetions of several real variables •
2. Formulation of the Cauehy problem. The not ion
of eharaeteristies. • • • 54
3. The Cauehy problem for the general non-linear
equation. 71
4. The Cauehy-Kowalewsky theorem. 76
CHAPl'ER 111 - SECOND ORDER EQUATIONS WITH CONSTANT COEFFICIENTS
1. Equations in two independent variables.
Canonieal forms
2. The one-dimensional wave equation. •
3. The wave equation in higher dimensions. Method
of spherieal means. Method of deseent ••••••• 101
4. The inhomogeneous wave equation by Duhamel's
prineiple • • • • • • • • • • • • 110
5. The potential equation in two dimensions • 116
6. The Diriehlet problem. • • •• 127
7. The Green's funetion and the fundamental solution ••••• 145
8. Equations related to the potential equation. • 151
9. Continuation of harmonie functions • •• 167
10. The heat equation. • • • • 170
CHAPl'ER IV - TEE CAUCHY PROBLEM FOR LINEAR HYPERBOLIC EQUATIONS IN GENERAL
1. Riemann's method of integration• ••••••••••••• 186
vii
2. Higher order equations in two independent variables. 196
3. The method of plane waves. • • • 204
LIST OF BOOKS RECOMMENDED FOR FURT HER STUDY 2~
INDEX. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 217
viii
I
Applied Mathematical Sciences Volume 1
INTRODUCTION
A partial differential equation for a function u(x,y, ••• ) with partial
derivatives is arelation of the form
(~ 0,
where F is a given function of the variables x,y, ••• ,u,ux,uy,uxx,.... Onlya
finite number of derivatives shall occur. Needless to say, a function u(x,y, ••• )
is said to be a solution of (1), if in some region of the space of its independent
variables, the function and its derivatives satisfy the equation identically in
x,y, •..•
One may also consider a system of partial differential equations~ in
which case one is concerned with several expressions of the above type containing
one or more unknown functions and their derivatives.
As in the theory of ordinary differential equations a partial differ
ential equation (henceforth abbreviated P.D.E.) is said to be of order n if the
highest order derivatives occurring in F are of the n-th order. One also
classifies the P.D.E. as to the type of function F. In particular, we have the
important linear P.D.E. if F is linear in the unknown function an& its deriva
tives, and the more general quasi-linear P.D.E. if F is linear in at least the
highest order derivatives.
Partial differential equations occur frequently and quite naturally in
the problems of various branches of mathematics, as the following examples show.
Example 1. A necessary and sufficient condition that the expression
(2) M(x,Y)dx + N(x,Y)dY
be a total differential is the condition of integrability
1