Applied Analysis

571 Pages·1986·15.812 MB·English

by  Allan M. Krall (auth.)

Applied Analysis
Mathematics and Its Applications
Managing Editor:
M. HAZE WINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
F. CALOGERO, Unil'ersita deg/i Studi di Roma, Italy
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
G.-c. ROTA, M.I. T., Cambridge Mass., U.S.A.
Allan M. Krall
Department of Mathematics.
Pennsylvania State University, U.S.A.
Applied Analysis
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster / Tokyo
library of Congress Cataloging in Publication Data
Krall, Allan M.
Applied analysis.
(Mathematics and its applications)
Includes index.
I. Mathematical analysis. 2. Numerical analysis. I. Title.
II. Series: Mathematics and its applications (D. Reidel Publishing Company)
QA3OO.K635 1986 515 86-17882
ISBN-13:978-90-277-2342-0 e-ISBN-13:978-94-009-4748-1
DOl: 10.1007/978-94-009-4748-1
Published by D. Reidel Publishing Company
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Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
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In all other countries, sold and distributed
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All Rights Reserved
© 1986 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1st edition 1986
No part of the material protected by this copyright notice may be reproduced or utilized
in any form or by any means, electronic or mechanical, including photocopying,
recording or by any information storage and retrieval system,
without written permission from the copyright owner
EDITOR'S PREFACE
Approach your problems from the right end It isn't that they can't see the solution. It is
and begin with the answers. Then one day, that they can't see the problem.
perhaps you will find the final question.
G.K. Chesterton. The Scandal of Father
'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin',
van Gu!ik.'g The Chinese Maze Murders.
Growing specialization and diversification have brought a host of monographs and textbooks on
increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields
does not grow only by putting forth new branches. It also happens, quite often in fact, that
branches which were thought to be completely disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in various sciences has
changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical
economics; algebraic geometry interacts with physics; the Minkowsky lemma. coding theory and the
structure of water meet one another in packing and covering theory; quantum fields, crystal defects
and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering;
and prediction and electrical engineering can use Stein spaces. And in addition to this there are
such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable
systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the
existing classification schemes. They draw upon widely different sections of mathematics. This pro
gramme, Mathematics and Its Applications, is devoted to new emerging (sub)disciplines and to such
(new) interrelations as exempla gratia:
- a central concept which plays an important role in several different mathematical and/or
scientific specialized areas;
new applications of the results and ideas from one area of scientific endeavour into another;
influences which the results, problems and concepts of one field of enquiry have and have had on
the development of another.
The Mathematics and Its Applications programme tries to make available a careful selection of
books which fit the philosophy outlined above. With such books, which are stimulating rather than
definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better
communication among the practitioners in diversified fields.
Besides the "new" applied mathematics with which the paragraphs above are chiefly concerned,
there still is, of course, the more traditional branch of the field centering around differential equa
tions of one sort or another and which used to be called something like "methods of mathematical
physics" or "boundary value problems". This field has blossomed too, in fact blossomed alarm
ingly, witness for example the success of the Claremont College program, and several similar ones,
the emergence of all kinds of industry-university joint research efforts and the founding of such
entities as the French society for applied mathematics (SMAI) and ECMI (European Consortium of
Mathematics in Industry). Differential equations are still the workhorse of a great deal of physics,
chemistry and engineering mathematics and thanks to great advances in numerics and computers a
large number of problems which were practically untouchable can now be handled adequately.
For this kind of applied work one still needs a thorough understanding of the underlying
mathematics and, of course, with more new applications the background mathematics necessary
also changes and evolves. For example, the Stone-Weierstrass theorem is becoming more and more
important, e.g. in a number of electrical engineering system theory applications. It is this back
ground mathematics that this book aims to provide in a concise and coherent fashion at the upper
undergraduate and lower graduate levels.
The unreasonable effectiveness of mathemat As long as algebra and geometry proceeded
ics in science ... along separate paths, their advance was slow
and thelT applications limited.
Eugene Wigner But when these sciences joined company
they drew from each other fresh vitality and
Well, if you know of a better 'ole, go to it. thencefoIVIard marched on at a rapid pace
towards perfection.
Bruce Bairnsfather
Joseph Louis Lagrange.
What is now proved was once only ima
gined.
Bussum, August 1986
William Blake Michiel Hazewinkel
PREFACE
The purpose of this book is to present to students in
mathematics, engineering and the physical sciences at the upper
undergraduate and beginning graduate levels an introduction to part
of what is fashionably called today applied mathematics. As anyone
who attempts to define the term soon learns, applied mathematics is a
rather large nebulous region sitting between mathematics and various
areas of application. During the past ten years with the advent of
personal computers the region has increased in size greatly, and so
applied mathematics now virtually defies definition.
At most universities, however, there is a course which covers
the material to be presented here. It is usually given the title of
methods of mathematical physics, boundary value problems of
mathematical physics, engineering mathematics, or some perturbation
of some of these words. The course may be taught by a physicist, a
mathematician or an engineer with slightly different emphasis. In
all cases, the material of the course consists of some portion of
mathematics having application to the physical sciences: the basic
facts concerning ordinary and partial differential equations and the
background necessary for proper understanding. These are the
subjects of this book.
We would like to mildly say that our point of view will be that
of the mathematician, and not the physicist or engineer. That is to
say, our interest is primarily, but not exclusively, with the
mathematics. Applications as such occupy a secondary role as
motivation and as a check to see if what we do is reasonable. (The
term is deliberately vaguely used.) We shall use theorems to state
our results and proofs to illustrate the techniques involved, as a
mathematician would. Finally we remark that those who wish to see a
rigorous development of the applications should consult courses in
those disciplines. Such courses are much more difficult than the
mere mathematics would indicate, and it is this in part which makes
physics and engineering fields distinct from mathematics.
The first portion of the book is concerned with certain
inequalities, linear spaces and linear operators. The background
necessary for what is to follow. Next using the contraction mapping
theorem, existence and uniqueness of solutions of systems of ordinary
differential equations are proved, with special emphasis on equations
of second order.
vi
PREFACE vii
Next follows the Stone-Weierstrass theorem, which is of
fundamental importance, but is not fully appreciated by most appliers
of mathematics. Then comes an introduction to Hilbert space, the
primary setting for most of the remainder of the book. We return to
differential equations to study regular and singular Sturm-Liouville
problems, with examples given by the classical orthogonal
polynomials, Fourier series and Fourier integrals.
The second portion of the book uses a distributional setting to
discuss the classical partial differential equations of second order
occurring in physics and engineering: The Laplace equation, the heat
equation and the wave equation. Special emphasis is placed on series
and integral solutions as well as the Green's functions for problems
with Dirichlet, Neumann and mixed boundary conditions. The results
from earlier chapters are used extensively throughout these
discussions.
Finally in two appendices we present von Neumann's derivation of
the spectral resolution of an unbounded, self-adjoint operator in a
Hilbert space, then a derivation of the heat, wave and Laplace
equations and their accompanying boundary conditions.
The book is, therefore, something akin to a ladder. The
foundation lies in linear spaces and the theory of linear operators.
At the top sit problems involving partial differential equations.
A word is certainly in order concerning the origin of the book.
While the author was a graduate student at the University of
Virginia, he had the exceptional privelege of attending a class in
ordinary and partial differential equations given by Marvin Rosenblum.
Professor Rosenblum's class was later used most successfully as a
model for a similar course taught by the author at The Pennsylvania
State University, and the notes resulting from the author's attempts
at Penn State are the basis for the book. The author would like to
express his grateful appreciation for the inspiration given by
Professor Rosenblum some years ago. Without his inspiration the book
would not have been writeen.
A number of people have helped over the years with the
development of the book. Many thanks to R.C. Brown and H.L. Krall,
who helped with earlier versions of the book, and Brenda Yucas, who
typed it. The author is more recently indebted to Lance Littlejohn
who has assisted greatly with proofreading the current version. A
special thank you must go to Sarah LeBlanc who has done such a
beautiful job in preparing the current manuscript for publication.
My thanks too to Dr. D.J. Larner and D. Reidel Publishing
Company for agreeing to publish the book, thus making it available
for use to a far larger audience than would have otherwise been
possible.
AMK
April 17, 1986
TABLE OF CONTENTS
PREFACE .vi
I. SOME INEQUALITIES .1
1. Young's Inequality .1
2. Holder's Inequality. .4
3. Minkowski's Inequality .6
4. A Relation between Different Norms .7
II. LINEAR SPACES AND LINEAR OPERATORS .15
1. Linear Spaces .15
2. Linear Operators .21
3. Norms and Banach Spaces. .26
4. Operator Convergence .32
III. EXISTENCE AND UNIQUENESS THEOREMS. .39
1. The Contraction Mapping Theorem. .39
2. Existence and Uniqueness of Solutions for
Ordinary Differential Equations. .48
3. First Order Linear Systems .55
4. n-th Order Differential Equations. .60
5. Some Extensions .64
IV. LINEAR ORDINARY DIFFERENTIAL EQUATIONS .68
1. First Order Linear Systems .68
2. Fundamental Matrices .70
3. Nonhomogeneous Syste.s .74
4. n-th Order Equations ....... . .76
5. Nonhomogeneous n-th Order Equations. .79
6. Reduction of Order .82
7. Constant Coefficients. .85
V. SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS .98
1. A Brief Review ... .98
2. The Adjoint Operator .99
3. An Oscillation Theorem .104
4. The Regular Sturm-Liouville Problem. .106
5. The Inverse Problem. Green's Functions .107
VI. THE STONE-WEIERSTRASS THEOREM. .117
1. Preliminary Remarks. .117
2. Algebras and Subalgebras .119
3. The Stone-Weierstrass Theorem. .120
4. Extensions and Special Cases .124
x TABLE OF CONTENTS
VII. HILBERT SPACES .127
1. Hermitian Forms. .127
2. Inner Product Spaces .131
3. Hilbert Spaces ... .134
4. Orthogonal Subs paces ..... .139
5. Continuous Linear Functionals. .145
6. Fourier Expansions ..... . .147
7. Isometric Hilbert Spaces .. . .153
VIII. LINEAR OPERATORS ON A HILBERT SPACE .158
1. Regular Operators on a Hilbert Space .158
2. Bilinear Forms, the Adjoint Operator .161
3. Self-Adjoint Operators .166
4. Projections . . . . . . . .172
5. Some Spectral Theorems . . .176
6. Operator Convergence . . . . . .184
7. The Spectral Resolution of a Self-Adjoint Operator .194
8. The Spectral Resolution of a Normal Operator . .202
9. The Spectral Resolution of a Unitary Operator. .207
IX. COMPACT OPERATORS ON A HILBERT SPACE .217
1. Compact Operators .217
2. Some Special Examples . . . .220
3. The Spectrum of a Compact Self-Adjoint Operator .225
4. The Spectral Resolution of a Compact,
Self-Adjoint Operator . . . . . .228
5. The Regular Sturm-Liouville Problem .232
X. SPECIAL FUNCTIONS .243
1. Orthogonal Polynomials. .243
2. The Legendre Polynomials .248
3. The Laguerre Polynomials .252
4. The Hermite Polynomials .255
5. Bessel Functions .259
XI. THE FOURIER INTEGRAL .268
1. The Lebesgue Integral .268
2. The Fourier Integral in Ll(-«>,oo} .279
3. The Fourier Integral in L2(-oo,oo} .293
XII. THE SINGULAR STURM-LIOUVILLE PROBLEM .308
1. Circles under Bilinear Transformations .310
2. Helly's Convergence Theorems. .314
3. Limit Points and Limit Circles .318
4. The Limit Point Case. ·.325
5. The Limit Circle Case .347
6. Examples ... .352
XIII. AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS .364
1. The Cauchy-Kowaleski Theorem .364
2. First Order Equations .367
3. Second Order Equations .372
4. Green's Formula .379
TABLE OF CONTENTS xi
XIV. DISTRIBUTIONS. .387
1. Test Functions and Distributions .387
2. Limits of Distributions .. .394
3. Fourier Transforms of Distributions .. .403
4. Applications of Distributions to Ordinary
Differential Equations . . . .408
5. Applications of Distributions to Partial
Differential Equations .417
XV. LAPLACE'S EQUATION ........ . .430
1. Introduction, Well Posed Problems .. .430
2. Dirichlet, Neumann, and Mixed Boundary
Value Problems . .. .... . .434
3. The Dirichlet Problem. . ... . .438
4. The Dirichlet Problem on the Unit Circle .449
5. Other Examples .456
XVI. THE HEAT EQUATION .470
1. Introduction, the Cauchy Problem. . . . . .470
2. The Cauchy Problem with Dirichlet Boundary Data .473
3. The Solution to the Nonhomogeneous Cauchy Problem .475
4. Examples. . . . . . .478
5. Homogeneous Problems .484
XVII. THE WAVE EQUATION .490
1. Introduction, the Cauchy Problem . .490
2. Solutions in I, 2 and 3 Dimensions . ..... . .499
3. The Solution to the Nonhomogeneous Cauchy Problem .513
4. Examples ....... . .517
APPENDIX I THE SPECTRAL RESOLUTION OF AN UNBOUNDED SELF-
ADJOINT OPERATOR. . . .532
1. Unbounded Linear Operators .532
2. The Graph of an Operator. .534
3. Symmetric and Self-Adjoint Operators .... .536
4. The Spectral Resolution of an Unbounded Self
Adjoint Operator . . . .539
APPENDIX II THE DERIVATION OF THE HEAT, WAVE AND
LAPACE EQUATIONS .546
1. The Heat Equation .546
2. Boundary Conditions .549
3. The Wave Equation .551
4. Boundary Conditions .553
5. Laplace's Equation .554
INDEX .555