Gockenbach M.S. Partial Differential Equations. Analytical and Numerical Methods
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Contents
7.2.2
Fourier series solutions
of the
inhomogeneous wave
equation
296
7.2.3 Other boundary conditions
301
7.3
Finite
element methods
for the
wave equation
305
7.3.1
The
wave equation with Dirichlet conditions
....
306
7.3.2
The
wave equation under other boundary conditions
312
7.4
Point sources
and
resonance
318
7.4.1
The
wave equation with
a
point source
318
7.4.2
Another experiment leading
to
resonance
321
7.5
Suggestions
for
further
reading
324
8
Problems
in
multiple spatial dimensions
327
8.1
Physical models
in two or
three
spatial
dimensions
327
8.1.1
The
divergence theorem
328
8.1.2
The
heat
equation
for a
three-dimensional domain
. 330
8.1.3 Boundary conditions
for the
three-dimensional
heat
equation
332
8.1.4
The
heat
equation
in a bar 333
8.1.5
The
heat
equation
in two
dimensions
334
8.1.6
The
wave equation
for a
three-dimensional domain
. 334
8.1.7
The
wave equation
in two
dimensions
334
8.1.8 Equilibrium problems
and
Laplace's equation
....
335
8.1.9 Green's identities
and the
symmetry
of the
Laplacian
336
8.2
Fourier series
on a
rectangular domain
339
8.2.1 Dirichlet boundary conditions
339
8.2.2
Solving
a
boundary value problem
345
8.2.3 Time-dependent problems
346
8.2.4
Other boundary conditions
for the
rectangle
....
348
8.2.5 Neumann boundary conditions
349
8.2.6 Dirichlet
and
Neumann problems
for
Laplace's equa-
tion
352
8.2.7 Fourier series methods
for a
rectangular
box in
three
dimensions
354
8.3
Fourier series
on a
disk
359
8.3.1
The
Laplacian
in
polar coordinates
360
8.3.2 Separation
of
variables
in
polar coordinates
362
8.3.3 Bessel's equation
363
8.3.4 Properties
of the
Bessel
functions
366
8.3.5
The
eigenfunctions
of the
negative Laplacian
on the
disk
368
8.3.6 Solving PDEs
on a
disk
372
8.4
Finite elements
in two
dimensions
377
8.4.1
The
weak
form
of a BVP in
multiple dimensions
. . 377
8.4.2
Galerkin's method
378
8.4.3 Piecewise linear
finite
elements
in two
dimensions
. 379
8.4.4 Finite elements
and
Neumann conditions
388
x Contents
7.2.2 Fourier series solutions of the inhomogeneous wave
equation.
. . . . . . . . . . . . . .
296
7.2.3 Other boundary conditions . . . . . . . . . . 301
7.3 Finite element methods for
the
wave equation . . . . . . 305
7.3.1 The wave equation with Dirichlet conditions 306
7.3.2 The wave equation under other boundary conditions 312
7.4 Point sources and resonance . . . . . . . . . . . . . . 318
7.4.1 The wave equation with a point source . 318
7.4.2 Another experiment leading
to
resonance
321
7.5 Suggestions for further
reading.
. . . . . . . . . . . . 324
8
Problems
in
multiple
spatial
dimensions
327
8.1
Physical models in two or three spatial dimensions. . . . .
..
327
8.1.1 The divergence theorem . . . . . . . . . . . . .
..
328
8.1.2 The heat equation for a three-dimensional domain 330
8.1.3 Boundary conditions for
the
three-dimensional heat
equation.
. . . . . . . . . . . . . . . . 332
8.1.4 The heat equation in a bar
..............
333
8.1.5 The heat equation in two dimensions
.........
334
8.1.6 The wave equation for a three-dimensional
domain.
334
8.1.7 The wave equation in two dimensions
........
334
8.1.8 Equilibrium problems and Laplace's equation . . . . 335
8.1.9 Green's identities and the symmetry of the Laplacian 336
8.2 Fourier series on a rectangular domain
....
339
8.3
8.4
8.2.1 Dirichlet boundary conditions. . . 339
8.2.2 Solving a boundary value problem 345
8.2.3 Time-dependent
problems.
. . . . 346
8.2.4 Other boundary conditions for the rectangle 348
8.2.5 Neumann boundary conditions . . . . . . . . 349
8.2.6 Dirichlet and Neumann problems for Laplace's equa-
tion
...........................
352
8.2.7 Fourier series methods for a rectangular box in three
dimensions . . . . . . . . . . . . . .
Fourier series on a disk . . . . . . . . . . . . . . . . . .
8.3.1 The Laplacian in polar coordinates
.....
8.3.2 Separation of variables in polar coordinates
8.3.3 Bessel's equation
..............
.
8.3.4 Properties of the Bessel functions
....
.
8.3.5 The eigenfunctions of the negative Laplacian on the
disk
..........
.
8.3.6 Solving PDEs on a disk . . . . . . . . . . . . . .
Finite elements in two dimensions . . . . . . . . . . . . . . .
8.4.1 The weak form of a BVP in multiple dimensions
8.4.2 Galerkin's method . . . . . . . . . . . . . . . . .
8.4.3 Piecewise linear finite elements in two dimensions
8.4.4 Finite elements and Neumann conditions
.....
354
359
360
362
363
366
368
372
377
377
378
379
388
Contents
xi
8.4.5 Inhomogeneous boundary conditions
389
8.5
Suggestions
for
further
reading
392
9
More
about
Fourier
series
393
9.1 The
complex Fourier series
394
9.1.1 Complex inner products
395
9.1.2 Orthogonality
of the
complex exponentials
396
9.1.3 Representing
functions
with complex Fourier series
. 397
9.1.4
The
complex Fourier series
of a
real-valued
function
398
9.2
Fourier series
and the FFT 401
9.2.1 Using
the
trapezoidal rule
to
estimate
Fourier
coef-
ficients
. 402
9.2.2
The
discrete Fourier transform
404
9.2.3
A
note about using packaged
FFT
routines
409
9.2.4
Fast
transforms
and
other boundary conditions;
the
discrete sine transform
410
9.2.5
Computing
the DST
using
the FFT 411
9.3
Relationship
of
sine
and
cosine series
to the
full
Fourier series
. 415
9.4
Pointwise convergence
of
Fourier series
419
9.4.1 Modes
of
convergence
for
sequences
of
functions
. . 419
9.4.2
Pointwise convergence
of the
complex Fourier series
422
9.5
Uniform
convergence
of
Fourier series
436
9.5.1
Rate
of
decay
of
Fourier
coefficients
436
9.5.2
Uniform
convergence
439
9.5.3
A
note
about
Gibbs's phenomenon
443
9.6
Mean-square convergence
of
Fourier series
444
9.6.1
The
space
L
2
(-l,l)
445
9.6.2
Mean-square convergence
of
Fourier series
448
9.6.3 Cauchy sequences
and
completeness
450
9.7
A
note about general eigenvalue problems
455
9.8
Suggestions
for
further
reading
459
10
More
about
finite
element
methods
461
10.1 Implementation
of
finite
element methods
461
10.1.1
Describing
a
triangulation
462
10.1.2
Computing
the
stiffness
matrix
465
10.1.3
Computing
the
load vector
467
10.1.4
Quadrature
467
10.2
Solving sparse linear systems
473
10.2.1 Gaussian elimination
for
dense systems
473
10.2.2
Direct solution
of
banded systems
475
10.2.3
Direct
solution
of
general
sparse
systems
477
10.2.4
Iterative solution
of
sparse linear systems
478
10.2.5
The
conjugate gradient algorithm
482
10.2.6 Convergence
of the CG
algorithm
485
10.2.7 Preconditioned
CG 486
Contents
9
10
8.5
8.4.5 Inhomogeneous boundary conditions
Suggestions for further reading . . . . . . . . . .
More
about
Fourier
series
9.1
The complex Fourier series
...............
.
9.1.1 Complex inner products
..........
.
9.1.2 Orthogonality of the complex exponentials
9.1.3 Representing functions with complex Fourier series.
9.1.4 The complex Fourier series of a real-valued function
9.2 Fourier series and the
FFT
....................
.
9.2.1 Using the trapezoidal rule
to
estimate Fourier coef-
ficients
..
'
.................
.
9.2.2 The discrete Fourier transform
...........
.
9.2.3 A note about using packaged
FFT
routines
....
.
9.2.4 Fast transforms and other boundary conditions; the
discrete sine
transform.
. . . . . . . . . . . . . . .
9.2.5 Computing
the
DST using the
FFT
.......
.
9.3 Relationship of sine and cosine series
to
the full Fourier series
9.4 Pointwise convergence of Fourier series . . . . . . . . . . . . .
9.4.1 Modes of convergence for sequences of functions .
9.4.2 Pointwise convergence of
the
complex Fourier series
9.5 Uniform convergence of Fourier series
.....
.
9.5.1 Rate of decay of Fourier coefficients
9.5.2 Uniform convergence
........
.
9.5.3 A note about Gibbs's phenomenon .
9.6 Mean-square convergence of Fourier series
..
.
9.6.1 The space
L2(
-£,
£)
........
.
9.6.2 Mean-square convergen·ce of Fourier series .
9.6.3 Cauchy sequences and completeness
9.7 A note about general eigenvalue problems
9.8 Suggestions for further
reading.
. . . . . .
More
about
finite
element
methods
10.1 Implementation of finite element methods
10.1.1 Describing a triangulation
...
10.1.2 Computing
the
stiffness matrix
10.1.3 Computing
the
load vector
10.1.4 Quadrature
..........
.
10.2 Solving sparse linear systems
.......
.
10.2.1 Gaussian elimination for dense systems
10.2.2 Direct solution of banded systems . . .
10.2.3 Direct solution of general sparse systems
10.2.4 Iterative solution of sparse linear systems
10.2.5 The conjugate gradient algorithm
10.2.6 Convergence of
the
CG algorithm
10.2.7 Preconditioned CG
........
.
XI
389
392
393
394
395
396
397
398
401
402
404
409
410
411
415
419
419
422
436
436
439
443
444
445
448
450
455
459
461
461
462
465
467
467
473
473
475
477
478
482
485
486
xii
Contents
10.3
An
outline
of the
convergence theory
for finite
element methods
488
10.3.1 The Sobolev space 489
10.3.2 Bes t approximation
in the
energy norm
491
10.3.3
Approximation
by
piecewise polynomials
491
10.3.4
Elliptic regularity
and L
2
estimates
492
10.4
Finite
element methods
for
eigenvalue problems
494
10.5 Suggestions
for
further
reading
499
A
Proof
of
Theorem 3.47
501
B
Shifting
the
data
in two
dimensions
505
B.O.I
Inhomogeneous Dirichlet conditions
on a
rectangle
. 505
B.0.2
Inhomogeneous Neumann conditions
on a
rectangle
508
C
Solutions
to
odd-numbered exercises
515
Bibliography
603
Index
607
H1(
xii Contents
10.3 An outline of the convergence theory for finite element methods 488
10.3.1 The Sobolev space
HJ
(0) . . . . . . . . . 489
10.3.2 Best approximation in
the
energy norm .
491
10.3.3 Approximation by piecewise polynomials
491
10.3.4 Elliptic regularity and
L2
estimates 492
10.4 Finite element methods for eigenvalue problems 494
10.5 Suggestions for further reading . . . . . . . . . . 499
A
Proof
of
Theorem
3.47
501
B
Shifting
the
data
in
two
dimensions
505
B.0.1 Inhomogeneous Dirichlet conditions on a rectangle 505
B.0.2 Inhomogeneous Neumann conditions on a rectangle 508
C
Solutions
to
odd-numbered
exercises
515
Bibliography
603
Index
607
Foreword
Newton
and
other seventeenth-century scientists introduced
differential
equa-
tions
to
describe
the
fundamental laws
of
physics.
The
intellectual
and
technological
implications
of
this innovation
are
enormous:
the
differential
equation description
of
physics
is the
foundation upon which
all
modern
quantitative
science
and
engineer-
ing
rests.
Differential
equations describe relations between physical quantities—
forces,
masses, positions, etc.—and their
rates
of
change with respect
to
space
and
time, i.e. their derivatives.
To
calculate
the
implications
of
these relations requires
the
"solution"
or
integration
of the
differential
equation: since derivatives approxi-
mate
the
ratios
of
small changes, solution
of
differential
equations amounts
to the
summing
together
of
many small changes—arbitrarily many,
in
principle—to com-
pute
the
effect
of a
large change. Until roughly
the
middle
of the
twentieth century,
this meant
the
derivation
of
formulas expressing solutions
of
differential
equations
as
algebraic combinations
of the
elementary functions
of
calculus—polynomials,
trigonometric functions,
the
logarithm
and the
exponential,
and
certain more
ex-
otic functions. Such formulas
are
feasible only
for a
limited selection
of
problems,
and
even then
often
involve
infinite
sums ("series") which
can
only
be
evaluated
approximately,
and
whose evaluation
was
(until
not so
long ago) quite laborious.
The
digital revolution
has
changed
all
that.
Fast
computation, with inexpen-
sive
machines performing many millions
of
arithmetic operations
per
second, makes
numerical
methods
for the
solution
of
differential
equations practical: these meth-
ods
give
useful
approximate solutions
to
differential
equations
by
literally adding
up
the
effects
of
many microscopic changes
to
approximate
the
effect
of a
large change.
Numerical
methods have been used
to
solve certain relatively "small"
differential
equations since
the
eighteenth century—notably those
that
occur
in the
computa-
tion
of
planetary orbits
and
other astronomical calculations. Systematic
use of the
computer
has
vastly expanded
the
range
of
differential
equation models
that
can be
coaxed into making predictions
of
scientific
and
engineering significance. Analytical
techniques developed over
the
last
three
centuries
to
solve special problem classes
have also
attained
new
significance when viewed
as
numerical methods. Optimal
design
of
ships
and
aircraft, prediction
of
underground
fluid
migration, synthesis
of
new
drugs,
and
numerous other critical tasks
all
rely—sometimes tacitly—upon
the
modeling
of
complex phenomena
by
differential
equations
and the
computational
approximation
of
their solutions.
Just
as
significantly,
mathematical advances
un-
derlying
computational techniques have fundamentally changed
the
ways
in
which
scientists
and
engineers think
about
differential
equations
and
their implications.
xiii
Foreword
Newton and other seventeenth-century scientists introduced differential equa-
tions
to
describe
the
fundamental laws of physics. The intellectual and technological
implications of this innovation are enormous: the differential equation description of
physics
is
the foundation upon which all modern quantitative science and engineer-
ing rests. Differential equations describe relations between physical
quantities-
forces, masses, positions,
etc.-and
their rates of change with respect to space and
time, i.e. their derivatives. To calculate the implications of these relations requires
the "solution" or integration of
the
differential equation: since derivatives approxi-
mate
the ratios of small changes, solution of differential equations amounts
to
the
summing together of many small
changes-arbitrarily
many, in
principle-to
com-
pute the effect of a large change. Until roughly the middle of the twentieth century,
this meant the derivation of formulas expressing solutions of differential equations
as algebraic combinations of the elementary functions
of
calculus-polynomials,
trigonometric functions, the logarithm
and
the exponential, and certain more ex-
otic functions. Such formulas are feasible only for a limited selection of problems,
and even then often involve infinite sums ("series") which can only be evaluated
approximately, and whose evaluation was (until not
so
long ago) quite laborious.
The
digital revolution has changed all
that.
Fast computation, with inexpen-
sive machines performing many millions of arithmetic operations per second, makes
numerical methods for the solution of differential equations practical: these meth-
ods give useful approximate solutions
to
differential equations by literally adding up
the
effects of many microscopic changes to approximate the effect of a large change.
Numerical methods have been used
to
solve certain relatively "small" differential
equations since the eighteenth
century-notably
those
that
occur in the computa-
tion of planetary orbits
and
other astronomical calculations. Systematic use of the
computer has vastly expanded the range of differential equation models
that
can be
coaxed into making predictions of scientific and engineering significance. Analytical
techniques developed over the last three centuries to solve special problem classes
have also attained new significance when viewed as numerical methods. Optimal
design of ships
and
aircraft, prediction of underground fluid migration, synthesis of
new drugs,
and
numerous other critical tasks all
rely-sometimes
tacitly-upon
the
modeling of complex phenomena by differential equations
and
the
computational
approximation of their solutions.
Just
as significantly, mathematical advances un-
derlying computational techniques have fundamentally changed the ways in which
scientists and engineers think about differential equations
and
their implications.
XIII
xiv
Foreword
Until
recently this
sea
change
in
theory
and
practice
has
enjoyed
little
reflec-
tion
in the
teaching
of
differential
equations
in
undergraduate classes
at
universities.
While mention
of
computer techniques began showing
up in
textbooks published
or
revised
in the
1970s,
the
view
of the
subject propounded
by
most textbooks would
have seemed conventional
in the
1920s.
The
book
you
hold
in
your hands, along
with
a few
others published
in
recent years, notably
Gil
Strang's
Introduction
to
Applied
Mathematics, represents
a new
approach
to
differential
equations
at the
undergraduate level.
It
presents computation
as an
integral part
of the
study
of
differential
equations.
It is not so
much
that
computational exercises must
be
part
of
the
syllabus—this
text
can be
used entirely without
any
student involvement
in
computation
at
all, though
a
class taught
that
way
would
miss
a
great deal
of the
possible
impact.
Rather,
the
concepts underlying
the
analysis
and
implementation
of
numerical methods assume
an
importance equal
to
that
of
solutions
in
terms
of
series
and
elementary
functions.
In
fact,
many
of
these concepts
are
equally
effec-
tive
in
explaining
the
workings
of the
series expansion methods
as
well.
This book
devotes considerable
effort
to
these "classical" methods, side
by
side with mod-
ern
numerical approaches (particularly
the
finite
element method).
The
"classical"
series
expansions provide both
a
means
to
understand
the
essential nature
of the
physical phenomena modeled
by the
equations,
and
effective
numerical methods
for
those special problems
to
which
they apply.
Perhaps surprisingly, some
of the
most important concepts
in the
modern
viewpoint
on
differential
equations
are
algebraic:
the
ideas
of
vector, vector space,
and
other components
of
linear
algebra
are
central, even
in the
development
of
more conventional
parts
of the
subject such
as
series solutions.
The
present book
uses
linear algebra
as a
unifying
principle
in
both theory
and
computation, just
as
working
scientists, engineers,
and
mathematicians
do.
This
book, along with
a
number
of
others
like
it
published
in
recent years, dif-
fers
from
earlier undergraduate textbooks
on
differential
equations
in yet
another
respect. Especially
in the
middle years
of the
last century, mathematical instruc-
tion
in
American universities tended
to
relegate
the
physical context
for
differential
equations
and
other topics
to the
background.
The
"big three"
differential
equa-
tions
of
science
and
engineering—the Laplace, wave,
and
heat
equations,
to
which
the
bulk
of
this book
is
devoted—have appeared
in
many texts with
at
most
a
cur-
sory
nod to
their physical origins
and
meaning
in
applications.
In
part,
this trend
reflected
the
development
of the
theory
of
differential
equations
as a
self-contained
arena
of
mathematical research. This development
has
been extremely
fruitful,
and
indeed
is the
source
of
many
of the new
ideas which underlie
the
effectiveness
of
modern numerical methods.
However,
it has
also
led to
generations
of
textbooks
which
present
differential
equations
as a
self-contained subject,
at
most distantly
related
to the
other intellectual disciplines
in
which
differential
equations play
a
crucial role.
The
present
text,
in
contrast, includes physically
and
mathematically
substantial
derivations
of
each
differential
equation,
often
in
several contexts, along
with
examples
and
homework problems
which
illustrate
how
differential
equations
really arise
in
science
and
engineering.
With
the
exception
of a
part
of the
chapter
on
ordinary
differential
equations
which
begins
the
book, this
text
concerns itself exclusively with linear problems—
XIV
Foreword
Until recently this sea change in theory
and
practice has enjoyed little reflec-
tion in the
teaching of differential equations in undergraduate classes
at
universities.
While mention of computer techniques began showing up in textbooks published or
revised in
the
1970s, the view of
the
subject propounded by most textbooks would
have seemed conventional in
the
1920s. The book you hold in your hands, along
with a
few
others published in recent years, notably Gil Strang's Introduction to
Applied Mathematics,
represents a new approach
to
differential equations
at
the
undergraduate level.
It
presents computation as
an
integral
part
of
the
study of
differential equations.
It
is
not so much
that
computational exercises must be
part
of the
syllabus-this
text
can be used entirely without any student involvement in
computation
at
all, though a class
taught
that
way would miss a great deal of the
possible impact. Rather, the concepts underlying the analysis and implementation
of numerical methods assume
an
importance equal
to
that
of solutions in terms of
series and elementary functions.
In
fact, many of these concepts are equally effec-
tive in explaining the workings of
the
series expansion methods as well. This book
devotes considerable effort to these "classical" methods, side by side with mod-
ern numerical approaches (particularly the finite element method).
The
"classical"
series expansions provide
both
a means
to
understand the essential
nature
of
the
physical phenomena modeled by
the
equations, and effective numerical methods for
those special problems to which they apply.
Perhaps surprisingly, some of
the
most important concepts in
the
modern
viewpoint on differential equations are
algebraic: the ideas of vector, vector space,
and other components of
linear algebra are central, even in
the
development of
more conventional
parts
of the subject such as series solutions. The present book
uses linear algebra as a unifying principle in
both
theory and computation,
just
as
working scientists, engineers, and mathematicians do.
This book, along with a number of others like
it
published in recent years, dif-
fers from earlier undergraduate textbooks on differential equations in yet another
respect. Especially in the middle years of
the
last century, mathematical instruc-
tion in American universities tended
to
relegate the physical context for differential
equations and other topics
to
the background.
The
"big three" differential equa-
tions of science and
engineering-the
Laplace, wave, and heat equations, to which
the
bulk of this book
is
devoted-have
appeared in many texts with
at
most a cur-
sory nod
to
their physical origins
and
meaning in applications. In
part,
this
trend
reflected the development of the theory of differential equations as a self-contained
arena
of mathematical research. This development has been extremely fruitful, and
indeed
is
the
source of many of
the
new ideas which underlie the effectiveness of
modern numerical methods. However,
it
has also led to generations of textbooks
which present differential equations as a self-contained subject,
at
most distantly
related
to
the other intellectual disciplines in which differential equations
playa
crucial role. The present text, in contrast, includes physically and mathematically
substantial derivations of each differential equation, often in several contexts, along
with examples and homework problems which illustrate how differential equations
really arise in science and engineering.
With
the
exception of a
part
of
the
chapter on ordinary differential equations
which begins
the
book, this
text
concerns itself exclusively with linear
problems-
Foreword
xv
that
is,
problems
for
which sums
and
scalar
multiples
of
solutions
are
also
solutions,
if
not of the
same problem then
of a
closely related problem.
It
might
be
argued
that
such
a
conventional choice
is odd in a
book which otherwise breaks with recent
tradition
in
several important respects. Many
if not
most
of the
important
problems
which
concern contemporary scientists
and
engineers
are
nonlinear.
For
example,
al-
most
all
problems involving
fluid flow and
chemical reaction
are
nonlinear,
and
these
phenomena
and
their
simulation
and
analysis
are of
central concern
to
many mod-
ern
engineering disciplines. While series solutions
are
restricted
to
linear problems,
numerical
methods
are in
principle just
as
applicable
to
nonlinear
as to
linear prob-
lems
(though many technical challenges arise
in
making them work
well).
However,
it is
also true
that
not
only
are the
classic linear
differential
equations—Laplace,
wave,
and
heat—models
for
generic classes
of
physical phenomena (potentials, wave
motion,
and
diffusion)
to
which many nonlinear processes belong,
but
also their
so-
lutions
are the
building blocks
of
methods
for
solving complex, nonlinear problems.
Therefore
the
choice
to
concentrate
on
these linear problems seems very
natural
for
a first
course
in
differential
equations.
The
computational viewpoint
in the
theory
and
application
of
differential
equations
is
very important
to a
large segment
of
SIAM membership,
and
indeed
SIAM
members have been
in the
forefront
of its
development.
It is
therefore
gratify-
ing
that
SIAM should play
a
leading role
in
bringing this important
part
of
modern
applied mathematics into
the
classroom
by
making
the
present volume available.
William
W.
Symes
Rice
University
Foreword
xv
that
is, problems for which sums and scalar multiples of solutions are also solutions,
if
not
of
the
same problem then of a closely related problem.
It
might be argued
that
such a conventional choice
is
odd
in a book which otherwise breaks with recent
tradition in several
important
respects. Many
if
not most
of
the
important
problems
which concern contemporary scientists
and
engineers are nonlinear. For example, al-
most all problems involving fluid
flow
and
chemical reaction are nonlinear,
and
these
phenomena
and
their simulation and analysis are of central concern
to
many mod-
ern engineering disciplines. While series solutions are restricted
to
linear problems,
numerical methods are in principle
just
as applicable
to
nonlinear as
to
linear prob-
lems (though many technical challenges arise in making them work well). However,
it
is also
true
that
not
only are
the
classic linear differential
equations-Laplace,
wave,
and
heat-models
for generic classes of physical phenomena (potentials, wave
motion, and diffusion)
to
which many nonlinear processes belong,
but
also their so-
lutions are
the
building blocks of methods for solving complex, nonlinear problems.
Therefore
the
choice
to
concentrate on these linear problems seems very
natural
for
a first course in differential equations.
The
computational viewpoint in the theory
and
application of differential
equations is very
important
to
a large segment of SIAM membership, and indeed
SIAM members have been in
the
forefront of its development.
It
is therefore gratify-
ing
that
SIAM should
playa
leading role in bringing this
important
part
of modern
applied mathematics into
the
classroom by making
the
present volume available.
William W. Symes
Rice University
This page intentionally left blank
This page intentionally left blank
Preface
This introductory
text
on
partial
differential
equations (PDEs)
has
several
features
that
are not
found
in
other texts
at
this level, including:
•
equal emphasis
on
classical
and
modern techniques.
• the
explicit
use of the
language
and
results
of
linear algebra.
•
examples
and
exercises analyzing realistic experiments (with correct physical
parameters
and
units).
• a
recognition
that
mathematical
software
forms
a
part
of the
arsenal
of
both
students
and
professional mathematicians.
In
this
preface,
I
will discuss
these
features
and
offer
suggestions
for
getting
the
most
out of the
text.
Classical
and
modern techniques
Undergraduate courses
on
PDEs tend
to
focus
on
Fourier series methods
and
sep-
aration
of
variables. These techniques
are
still
useful
after
two
centuries because
they
offer
a
great deal
of
insight into those problems
to
which
they apply. How-
ever,
the
subject
of
PDEs
has
driven much
of the
research
in
both pure
and
applied
mathematics
in the
last
century,
and
students ought
to be
exposed
to
some more
modern techniques
as
well.
The
limitation
of the
Fourier series technique
is its
restricted applicability:
it can be
used only
for
equations with constant
coefficients
and
only
on
certain
simple geometries.
To
complement
the
classical topic
of
Fourier series,
I
present
the
finite
element method,
a
modern,
powerful,
and flexible
approach
to
solving PDEs.
Although many introductory
texts
include some discussion
of finite
elements
(or
finite
differences,
a
competing computational methodology),
the
modern approach
tends
to
receive less
attention
and a
subordinate place
in the
exposition.
In
this
text,
I
have
put
equal weight
on
Fourier series
and finite
elements.
Linear
algebra
Both linear
and
nonlinear
differential
equations occur
as
models
of
physical phenom-
ena of
great importance
in
science
and
engineering. However, most introductory
xvii
Preface
This introductory
text
on partial differential equations (PDEs) has several
features
that
are not found in other texts
at
this level, including:
• equal emphasis on classical and modern techniques.
• the explicit use of
the
language and results of linear algebra.
• examples and exercises analyzing realistic experiments (with correct physical
parameters and units).
• a recognition
that
mathematical software forms a
part
of the arsenal of
both
students and professional mathematicians.
In
this preface, I will discuss these features and offer suggestions for getting
the
most out of the text.
Classical and modern techniques
Undergraduate courses on PDEs tend
to
focus on Fourier series methods and sep-
aration of variables. These techniques are still useful after two centuries because
they offer a great deal of insight into those problems
to
which they apply. How-
ever, the subject of PDEs has driven much of the research in
both
pure and applied
mathematics in the last century, and students ought
to
be exposed to some more
modern techniques as well.
The limitation of the Fourier series technique
is
its restricted applicability:
it
can be used only for equations with constant coefficients and only on certain
simple geometries. To complement the classical topic of Fourier series, I present the
finite element method, a modern, powerful, and flexible approach
to
solving PDEs.
Although many introductory texts include some discussion of finite elements (or
finite differences, a competing computational methodology), the modern approach
tends
to
receive less attention and a subordinate place in the exposition. In this
text, I have
put
equal weight on Fourier series and finite elements.
Linear algebra
Both linear and nonlinear differential equations occur as models of physical phenom-
ena of great importance in science
and
engineering. However, most introductory
XVII
xviii
Preface
texts
focus
on
linear equations,
and
mine
is no
exception. There
are
several reasons
why
this should
be so. The
study
of
PDEs
is
difficult,
and it
makes sense
to
begin
with
the
simpler linear equations
before
moving
on to the
more
difficult
nonlinear
equations. Moreover, linear equations
are
much better understood. Finally, much
of
what
is
known about nonlinear
differential
equations depends
on the
analysis
of
linear
differential
equations,
so
this material
is
prerequisite
for
moving
on to
nonlinear equations.
Because
we
focus
on
linear equations, linear algebra
is
extremely
useful.
In-
deed,
no
discussion
of
Fourier series
or finite
element methods
can be
complete
unless
it
puts
the
results
in the
proper linear algebraic
framework.
For
example,
both methods produce
the
best approximate solution
from
certain
finite-dimensional
subspaces,
and the
projection
theorem
is
therefore
central
to
both
techniques.
Sym-
metry
is
another
key
feature exploited
by
both methods.
While many texts de-emphasize
the
linear algebraic nature
of the
concepts
and
solution techniques,
I
have chosen
to
make
it
explicit. This decision,
I
believe,
leads
to a
more cohesive course
and a
better
preparation
for
future
study. However,
it
presents certain challenges. Linear algebra does
not
seem
to
receive
the
attention
it
deserves
in
many engineering
and
science programs,
and so
many students
will
take
a
course based
on
this
text
without
the
"prerequisites." Therefore,
I
present
a
fairly
complete overview
of the
necessary material
in
Chapter
3,
Essential Linear
Algebra.
Both
faculty
previewing this
text
and
students taking
a
course
from
it
will
soon realize
that
there
is too
much material
in
Chapter
3 to
cover thoroughly
in the
couple
of
weeks
it can
reasonably occupy
in a
semester course. From experience
I
know
that
conscientious students dislike moving
so
quickly through material
that
they cannot master
it.
However,
one of the
keys
to
using this
text
is to
avoid getting
bogged down
in
Chapter
3.
Students should
try to get
from
it the
"big picture"
and two
essential ideas:
• How to
compute
a
best approximation
to a
vector
from
a
subspace, with
and
without
an
orthogonal basis (Section 3.4).
• How to
solve
a
matrix-vector equation when
the
matrix
is
symmetric
and its
eigenvalues
and
eigenvectors
are
known (Section 3.5).
Having
at
least
begun
to
grasp these ideas, students should move
on to
Chapter
4
even
if
some details
are not
clear.
The
concepts
from
linear algebra
will
become
much
clearer
as
they
are
used throughout
the
remainder
of the
text.
1
I
have taught this course several times using this approach, and, although
students
often
find it
frustrating
at the
beginning,
the
results seem
to be
good.
Realistic problems
The
subject
of
PDEs
is
easier
to
grasp
if one
keeps
in
mind certain
standard
phys-
ical experiments modeled
by the
equations under consideration.
I
have used these
1
Also,
Chapter
4 is
much
easier
going
than Chapter
3, a
welcome
contrast!
xviii
Preface
texts focus on linear equations,
and
mine is no exception. There are several reasons
why this should be so. The
study
of PDEs
is
difficult, and
it
makes sense
to
begin
with
the
simpler linear equations before moving on to the more difficult nonlinear
equations. Moreover, linear equations are much
better
understood. Finally, much
of what
is
known about nonlinear differential equations depends on
the
analysis
of linear differential equations,
so
this material is prerequisite for moving on
to
nonlinear equations.
Because
we
focus on linear equations, linear algebra is extremely useful. In-
deed, no discussion of Fourier series
or
finite element methods can be complete
unless
it
puts
the
results in
the
proper linear algebraic framework. For example,
both
methods produce the best approximate solution from certain finite-dimensional
subspaces, and
the
projection theorem
is
therefore central to
both
techniques. Sym-
metry
is
another key feature exploited by
both
methods.
While many texts de-emphasize the linear algebraic nature of the concepts
and solution techniques, I have chosen to make
it
explicit. This decision, I believe,
leads
to
a more cohesive course and a
better
preparation for future study. However,
it
presents certain challenges. Linear algebra does not seem to receive the attention
it
deserves in many engineering and science programs, and
so
many students will
take a course based on this
text
without the "prerequisites." Therefore, I present
a fairly complete overview of the necessary material in Chapter
3,
Essential Linear
Algebra.
Both faculty previewing this
text
and students taking a course from
it
will
soon realize
that
there
is
too much material in Chapter 3 to cover thoroughly in
the
couple of weeks
it
can reasonably occupy in a semester course. From experience I
know
that
conscientious students dislike moving
so
quickly through material
that
they cannot master it. However, one of
the
keys to using this
text
is
to
avoid getting
bogged down in Chapter
3.
Students should
try
to
get from
it
the "big picture"
and two essential ideas:
• How to compute a best approximation
to
a vector from a subspace, with and
without
an
orthogonal basis (Section 3.4) .
• How
to
solve a matrix-vector equation when the matrix
is
symmetric and its
eigenvalues
and
eigenvectors are known (Section 3.5).
Having
at
least begun to grasp these ideas, students should move on
to
Chapter 4
even if some details are not clear.
The
concepts from linear algebra will become
much clearer as they are used throughout the remainder of the text.
1
I have
taught
this course several times using this approach, and, although
students often find
it
frustrating
at
the beginning,
the
results seem to be good.
Realistic problems
The
subject of PDEs
is
easier to grasp
if
one keeps in mind certain
standard
phys-
ical experiments modeled by the equations under consideration. I have used these
1 Also,
Chapter
4 is much easier going
than
Chapter
3,
a welcome contrast!
Preface
xix
models
to
introduce
the
equations
and to aid in
understanding their solutions.
The
models
also show,
of
course,
that
the
subject
of
PDEs
is
worth studying!
To
make
the
applications
as
meaningful
as
possible,
I
have included many
examples
and
exercises posed
in
terms
of
meaningful
experiments with realistic
physical
parameters.
Software
There exists
powerful
mathematical
software
that
can be
used
to
illuminate
the
material
presented
in
this
book. Computer software
is
useful
for at
least
three
reasons:
• It
removes
the
need
to do
tedious computations
that
are
necessary
to
compute
solutions.
Just
as a
calculator eliminates
the
need
to use a
table
and
inter-
polation
to
compute
a
logarithm,
a
computer
algebra
system
can
eliminate
the
need
to
perform
integration
by
parts
several times
in
order
to
evaluate
an
integral. With
the
more mechanical obstacles removed, there
is
more time
to
focus
on
concepts.
•
Problems
that
simply cannot
be
solved
(in a
reasonable time)
by
hand
can of-
ten be
done with
the
assistance
of a
computer. This allows
for
more interesting
assignments.
•
Graphical capabilities allow students
to
visualize
the
results
of
their compu-
tations, improving understanding
and
interpretation.
I
expect students
to use a
software
package such
as
MATLAB, Mathematica,
or
Maple
to
reproduce
the
examples
from
the
text
and to
solve
the
exercises.
I
prefer
not to
introduce
a
particular
software
package
in the
text
itself,
for
at
least
two
reasons.
The
explanation
of the
features
and
usage
of the
software
can
detract
from
the
mathematics. Also,
if the
book
is
based
on a
particular software
package,
then
it can be
difficult
to use
with
a
different
package.
For
these reason,
my
text
does
not
mention
any
software
packages except
in a few
footnotes. However,
since
the use of
software
is, in my
opinion, essential
for a
modern course,
I
have
written tutorials
for
MATLAB,
Mathematica,
and
Maple
that
explain
the
various
capabilities
of
these programs
that
are
relevant
to
this book. These tutorials appear
on
the
accompanying
CD.
Outline
The
core material
in
this
text
is
found
in
Chapters 5-7, which present Fourier series
and
finite
element techniques
for the
three most important
differential
equations
of
mathematical physics: Laplace's equation,
the
heat equation,
and the
wave equa-
tion. Since
the
concepts themselves
are
hard enough, these chapters
are
restricted
to
problems
in a
single
spatial
dimension.
Several
introductory chapters
set the
stage
for
this core. Chapter
1
briefly
defines
the
basic terminology
and
notation
that
will
be
used
in the
text.
Chapter
Preface
XIX
models to introduce the equations and
to
aid in understanding their solutions.
The
models also show, of course,
that
the
subject of
PDEs
is
worth studying!
To make the applications as meaningful as possible, I have included many
examples and exercises posed in terms of meaningful experiments with realistic
physical parameters.
Software
There exists powerful mathematical software
that
can be used
to
illuminate the
material presented in this book. Computer software
is
useful for
at
least three
reasons:
•
It
removes
the
need to do tedious computations
that
are necessary to compute
solutions.
Just
as a calculator eliminates the need
to
use a table and inter-
polation to compute a logarithm, a computer algebra system can eliminate
the need to perform integration by
parts
several times in order
to
evaluate
an
integral.
With
the
more mechanical obstacles removed, there
is
more time to
focus on concepts.
• Problems
that
simply cannot be solved (in a reasonable time) by hand can
of-
ten be done with the assistance of a computer. This allows for more interesting
assignments.
• Graphical capabilities allow students to visualize
the
results of their compu-
tations, improving understanding and interpretation.
I expect students
to
use a software package such as MATLAB, Mathematica, or
Maple
to
reproduce the examples from the
text
and
to
solve the exercises.
I prefer not
to
introduce a particular software package in
the
text
itself, for
at
least two reasons.
The
explanation of the features and usage of the software can
detract from
the
mathematics. Also, if the book is based on a particular software
package,
then
it
can be difficult
to
use with a different package. For these reason, my
text
does not mention any software packages except in a
few
footnotes. However,
since the use of software is, in my opinion, essential for a modern course, I have
written tutorials for MATLAB,
Mathematica, and Maple
that
explain
the
various
capabilities of these programs
that
are relevant to this book. These tutorials appear
on
the
accompanying CD.
Outline
The
core material in this
text
is
found in Chapters 5-7, which present Fourier series
and finite element techniques for the three most important differential equations of
mathematical physics: Laplace's equation, the heat equation,
and
the wave equa-
tion. Since the concepts themselves are
hard
enough, these chapters are restricted
to problems in a single spatial dimension.
Several introductory chapters set the stage for this core. Chapter 1 briefly
defines the basic terminology and notation
that
will be used in
the
text. Chapter