Principles of applied mathematics : transformation and approximation

577 Pages·2018·9.97 MB·English

by  Keener

PRINCIPLES OF APPLIED
MATHEMATICS
Transformation and Approximation
James P. Keener
Department of Mathematics
University of Utah
The Advanced Book Program
CRC Press
Taylor &Francis Group
Boca Raton London New York
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First published 1988 by Perseus Books Publishing, L.L.C.
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Library of Congress Cataloging-in-Publication Data
Keener, James P.
Principles of applied mathematics : transformation and
approximation / James P. Keener.
p.(cid:9) cm.
Includes index.
1. Transformations (Mathematics) 2. Asymptotic expansions.
I. Title.
QA601.K4 1987
515.7'23-dc19(cid:9) 87-18630
CIP
ISBN 13: 978-0-201-48363-5 (pbk)
ISBN 13: 978-0-201-15674-4 (hbk)
This book was typeset in Textures using a Macintosh SE computer.
To my three best friends, Kristine,
Samantha, and Justin, who have adjusted
extremely well to having a mathematician
as husband and father.

PREFACE
Applied mathematics should read like good mystery, with an intriguing begin-
ning, a clever but systematic middle, and a satisfying resolution at the end.
Often, however, the resolution of one mystery opens up a whole new problem,
and the process starts all over. For the applied mathematical scientist, there
is the goal to explain or predict the behavior of some physical situation. One
begins by constructing a mathematical model which captures the essential fea-
tures of the problem without masking its content with overwhelming detail.
Then comes the analysis of the model where every possible tool is tried, and
some new tools developed, in order to understand the behavior of the model
as thoroughly as possible. Finally, one must interpret and compare these re-
sults with real world facts. Sometimes this comparison is quite satisfactory,
but most often one discovers that important features of the problem are not
adequately accounted for, and the process begins again.
Although every problem has its own distinctive features, through the years
it has become apparent that there are a group of tools that are essential to
the analysis of problems in many disciplines. This book is about those tools.
But more than being just a grab bag of tools and techniques, the purpose of
this book is to show that there is a systematic, even esthetic, explanation of
how these classical tools work and fit together as a unit.
Much of applied mathematical analysis can be summarized by the obser-
vation that we continually attempt to reduce our problems to ones that we
already know how to solve.
This philosophy is illustrated by an anecdote (apocryphal, I hope) of a
mathematician and an engineer who, as chance would have it, took a gourmet
cooking class together. As this course started with the basics, the first lesson
was on how to boil a pot of water. The instructor presented the students with
two pots of water, one on the counter, one on a cold burner, while another
burner was already quite hot, but empty.
In order to test their cooking aptitudes, the instructor first asked the
engineer to demonstrate to the rest of the class how to boil the pot of water
that was already sitting on the stove. He naturally moved the pot carefully
v
vi(cid:9) PREFACE
from the cold burner to the hot burner, to the appreciation of his classmates.
To be sure the class understood the process, the pot was moved back to its
original spot on the cold burner at which time the mathematician was asked
to demonstrate how to heat the water in the pot sitting on the counter. He
promptly and confidently exchanged the position of the two pots, placing the
pot from the counter onto the cold burner and the pot from the burner onto
the counter. He then stepped back from the stove, expecting an appreciative
response from his mates. Baffled by his actions, the instructor asked him
to explain what he had done, and he replied naturally, that he had simply
reduced his problem to one that everyone already knew how to solve.
We shall also make it our goal to reduce problems to those which we
already know how to solve.
We can illustrate this underlying idea with simple examples. We know
that to solve the algebraic equation 3x = 2, we multiply both sides of the
equation with the inverse of the "operator" 3, namely 1/3, to obtain x = 2/3.
The same is true if we wish to solve the matrix equation Ax = f where
)(cid:9) ( )
A = (1 f=
3 2
Namely, if we know A-1, the inverse of the matrix operator A, we multiply
both sides of the equation by A-1 to obtain x = A-1 f . For this problem,
A-1 = 1 ( 3 —1 and(cid:9) x — (cid:9)5
8(cid:9) 3(cid:9)
If we make it our goal to invert many kinds of linear operators, including
matrix, integral, and differential operators, we will certainly be able to solve
many types of problems. However, there is an approach to calculating the
inverse operator that also gives us geometrical insight. We try to transform the
original problem into a simpler problem which we already know how to solve.
For example, if we rewrite the equation Ax = f as T-'AT (T-1 x) = T-1 f
and choose
T = (1 —1 T_1 ( 1 1 \
1(cid:9) 1(cid:9) 2 —1 1t
we find that
( 4 0
T-1AT =
—2)
0
With the change of variables y =(cid:9) g = T-1 f, the new problem looks
like two of the easy algebraic equations we already know how to solve, namely
4y1 = 3/2, —2y2 = 1/2. This process of separating the coupled equations
into uncoupled equations works only if T is carefully chosen, and exactly how
vii
this is done is still a mystery. Suffice it to say, the original problem has been
transformed, by a carefully chosen change of coordinate system, into a problem
we already know how to solve.
This process of changing coordinate systems is very useful in many other
problems. For example, suppose we wish to solve the boundary value problem
u" — 2u = f(x) with u(0) = u(r) = 0. As we do not yet know how to write
down an inverse operator, we look for an alternative approach. The single most
important step is deciding how to represent the solution u(x). For example,
we might try to represent u and f as polynomials or infinite power series in
x, but we quickly learn that this guess does not simplify the solution process
much. Instead, the "natural" choice is to represent u and f as trigonometric
series,
= CEO(cid:9) ECO f
u(x) uk sin kx,(cid:9) f (x) = k sin kx.
k=1(cid:9) k=1
Using this representation (i.e., coordinate system) we find that the original
differential equation reduces to the infinite number of separated algebraic
equations (k2 + 2) uk = - fk. Since these equations are separated (the kth
equation depends only on the unknown uk), we can solve them just as before,
even though there are an infinite number of equations. We have managed to
simplify this problem by transforming into the correctly chosen coordinate
system.
For many of the problems we encounter in the sciences, there is a natural
way to represent the solution that transforms the problem into a substantially
easier one. All of the well-known special functions, including Legendre poly-
nomials, Bessel functions, Fourier series, Fourier integrals, etc., have as their
common motivation that they are natural for certain problems, and perfectly
ridiculous in others. It is important to know when to choose one transform
over another.
Not all problems can be solved exactly, and it is a mistake to always
look for exact solutions. The second basic technique of applied mathematics
is to reduce hard problems to easier problems by ignoring small terms. For
example, to find the roots of the polynomial x2+ x+ .0001 = 0, we notice that
the equation is very close to the equation x2+ x = 0 which has roots x'= —1,
and x = 0, and we suspect that the roots of the original polynomial are not
too much different from these. Finding how changes in parameters affect the
solution is the goal of perturbation theory and asymptotic analysis, and in
this example we have a regular perturbation problem, since the solution is a
regular (i.e., analytic) function of the "parameter" .0001 .
Not all reductions lead to such obvious conclusions. For example, the
polynomial .0001x2 + x + 1 = 0 is close to the polynomial x + 1 = 0, but
the first has two roots while the second has only one. Where did the second
root go? We know, of course, that there is a very large root that "goes to
infinity" as ".0001 goes to zero," and this example shows that our naive idea
viii PREFACE
of setting all small parameters to zero must be done with care. As we see,
not all problems with small parameters are regular, but some have a singular
behavior. We need to know how to distinguish between regular and singular
approximations, and what to do in each case.
This book is written for beginning graduate students in applied mathe-
matics, science, and engineering, and is appropriate as a one-year course in
applied mathematical techniques (although I have never been able to cover
all of this material in one year). We assume that the students have studied
at an introductory undergraduate level material on linear algebra, ordinary
and partial differential equations, and complex variables. The emphasis of the
book is a working, systematic understanding of classical techniques in a mod-
ern context. Along the way, students are exposed to models from a variety of
disciplines. It is hoped that this course will prepare students for further study
of modern techniques and in-depth modeling in their own specific discipline.
One book cannot do justice to all of applied mathematics, and in an ef-
fort to keep the amount of material at a manageable size, many important
topics were not included. In fact, each of the twelve chapters could easily be
expanded into an entire book. The topics included here have been selected,
not only for their scientific importance, but also because they allow a logical
flow to the development of the ideas and techniques of transform theory and
asymptotic analysis. The theme of transform theory is introduced for matrices
in Chapter one, and revisited for integral equations in Chapter three, for ordi-
nary differential equations in Chapters four and seven, for partial differential
equations in Chapter eight, and for certain nonlinear evolution equations in
Chapter nine.
Once we know how to solve a wide variety of problems via transform
theory, it becomes appropriate to see what harder problems we can reduce to
those we know how to solve. Thus, in Chapters ten, eleven, and twelve we give
a survey of the three basic areas of asymptotic analysis, namely asymptotic
analysis of integrals, regular perturbation theory and singular perturbation
theory.
Here is a summary of the text, chapter by chapter. In Chapter one, we re-
view the basics of spectral theory for matrices with the goal of understanding
not just the mechanics of how to solve matrix equations, but more impor-
tantly, the geometry of the solution process, and the crucial role played by
eigenvalues and eigenvectors in finding useful changes of coordinate systems.
This usefulness extends to pseudo-inverse operators as well as operators in
Hilbert sp,ace, and so is a particularly important piece of background infor-
mation.
In Chapter two, we extend many of the notions of finite dimensional vector
spaces to function spaces. The main goal is to show how to represent objects
in a function space. Thus, Hilbert spaces and representation of functions in a
Hilbert space are studied. In this context we meet classical sets of functions
such as Fourier series and Legendre polynomials, as well as less well-known
ix
sets such as the Walsh functions, Sinc functions, and finite element bases.
In Chapter three, we explore the strong analogy between integral equa-
tions and matrix equations, and examine again the consequences of spectral
theory. This chapter is more abstract than others as it is an introduction to
functional analysis and compact operator theory, given under the guise of
Fredholm integral equations. The added generality is important as a frame-
work for things to come.
In Chapter four, we develop the tools necessary to use .,pectral decompo-
sitions to solve differential equations. In particular, distributions and Green's
functions are used as the means by which the theory of compact operators
can be applied to differential operators. With these tools in place, the com-
pleteness of eigenfunctions of Sturm-Liouville operators follows directly.
Chapter five is devoted to showing how many classical differential equa-
tions can be derived from a variational principle, and how the eigenvalues of
a differential operator vary as the operator is changed.
Chapter six is a pivotal chapter, since all chapters following it require a
substantial understanding of analytic function theory, and the chapters pre-
ceding it require no such knowledge. In particular, knowing how to integrate
and differentiate analytic functions at a reasonably sophisticated level is in-
dispensable to the remaining text. Section 6.3 (applications to Fluid Flow) is
included because it is a classically important and very lovely subject, but it
plays no role in the remaining development of the book, and could be skipped
if time constraints demand.
Chapter seven continues the development of transform theory and we
show that eigenvalues and eigenfunctions are not always sufficient to build
a transform, and that operators having continuous spectrum require a gen-
eralized construction. It is in this context that Fourier, Mellin, Hankel, and
Z transforms, as well as scattering theory for the Schrodinger operator are
studied.
In Chapter eight we show how to solve linear partial differential and differ-
ence equations, with special emphasis on (as you guessed) transform theory. In
this chapter we are able to make specific use of all the techniques introduced
so far and solve some problems with interesting applications.
Although much of transform theory is rather old, it is by no means dead.
In Chapter nine we show how transform theory has recently been used to
solve certain nonlinear evolution equations. We illustrate the inverse scattering
transform on the Korteweg-deVries equation and the Toda lattice.
In Chapter ten we show how asymptotic methods can be used to approx-
imate the horrendous integral expressions that so often result from transform
techniques.
In Chapter eleven we show how perturbation theory and especially the
study of nonlinear eigenvalue problems uses knowledge of the spectrum of a
linear operator in a fundamental way. The nonlinear problems in this chapter
all have the feature that their solutions are close to the solutions of a nearby
linear problem.