Gockenbach M.S. Partial Differential Equations. Analytical and Numerical Methods
Подождите немного. Документ загружается.
598
Appendix
C.
Solutions
to
odd-numbered
exercises
With
/(x)
=
#i,
we
have
On
the
other hand,
so
the
rule
is
exact
in
this case
as
well.
By
symmetry,
the
rule must hold
for
/(x)
=
X2,
which shows
that
the
rule
has
degree
of
precision
at
least
1. To
show
that
the
degree
of
precision
is not
greater
than
1, we
note
that,
for
/(x)
=
xl,
while
3. We
have
On
the
other hand,
the
mapping
from
T
R
to T is
The
Jacobian
is
and its
determinant
is 1.
Therefore, with
we
have
This
last
integral also equals 1/8.
5.
(The reader should consult Exercise 8.4.5 (and
its
solution)
for an
explanation
of
how
an
inhomogeneous Dirichlet condition
is
handled.)
Let G be the
piecewise linear
function
whose nodal value
is
zero everywhere except
at
those boundary nodes lying
on
the
boundary which
are not
free.
At the
nonfree
boundary nodes,
the
value
of
598
Appendix
C.
Solutions
to
odd-numbered exercises
With
f(x)
=
Xl,
we
have
1 1
111-"'1
f =
Xl
dx
2
dx1
TR
0 0
1
1 2 1 1 1
=
(Xl
-
xI)
dX1
= - - - = -.
o 2 3 6
On
the
other hand,
~f
G,
D =
~. ~
=~,
so
the
rule is exact in this case as well. By symmetry,
the
rule
must
hold for
f(x)
=
X2,
which shows
that
the
rule has degree of precision
at
least 1.
To
show
that
the
degree of precision is
not
greater
than
1,
we
note
that,
for
f(x)
=
xi,
while
3.
We have
On
the
other hand,
the
mapping from
TR
to
T is
1
Xl
= 1 +
Zl
+
2"Z2,
X2
=
Z2·
The
Jacobian
is
J=[~
t],
and
its determinant is
1.
Therefore, with
we
have
1 1 1
11l-z1
1
f = 9 = ( 1 +
Zl
+
2"
Z2
)
z~
dZ
2
dz
1.
T
TR
0 0
This last integral also equals
1/8.
5.
(The reader should consult Exercise 8.4.5 (and
its
solution) for an explanation of
how
an
inhomogeneous Dirichlet condition is handled.) Let G
be
the
piecewise linear
function whose nodal value is zero everywhere except
at
those boundary nodes lying
on
the
boundary which are
not
free. At
the
nonfree boundary nodes,
the
value of
Appendix
C.
Solutions
to
odd-numbered
exercises
599
G is the
assigned Dirichlet
data.
To
solve
the
inhomogeneous Dirichlet problem,
we
merely
need
to
modify
the
load vector
by
replacing
by
The
quantity
a(G,
fa) is
nonzero only
if the
free
node
r\f
i
belongs
to a
triangle
that
also contains
a
(nonfree)
boundary node.
The
point here
is
simply
that
this
can
easily
be
determined
from
the
information
in the
data
structure.
As we
loop over
the
triangles,
we can
determine,
for
each, whether
it
contains both
a
free
node
and
a
nonfree
node.
If it
does,
we
modify
the
load vector accordingly.
Section
10.2
1.
Computing
A = LU
requires
n
—
1
steps,
and
step
i
uses
arithmetic operations (this
is
determined
by
simply counting
the
operations
in the
pseudo-code
on
page 473).
The
total
number
of
operations
required
is
This agrees with
the
O(2n
3
/3)
operation count given
in the
text.
The
computation
of
L
-1
b
requires
n
—
1
steps, with
2(n
—
i)
arithmetic operations
nfir
stfvn.
Thfi
tnta.l
is
The final
step
of
back substitution requires
n
steps, with
2(n
—
i) + 1
operations
per
step,
for a
total
of
The
total
for the two
triangular solves
is
2n
2
—
n,
which also agrees with
the
count
given
in the
text.
3.
With
Appendix
C.
Solutions
to
odd-numbered exercises 599
G is
the
assigned Dirichlet data.
To
solve
the
inhomogeneous Dirichlet problem,
we
merely need
to
modify
the
load vector by replacing
by
Fi
= l
JrPi
-
a(G,CPi).
The
quantity a(G,
CPi)
is nonzero only if
the
free node
nfi
belongs
to
a triangle
that
also contains a (nonfree)
boundary
node.
The
point here is simply
that
this can
easily be determined from
the
information in
the
data
structure. As
we
loop over
the
triangles,
we
can determine, for each, whether
it
contains
both
a free node
and
a nonfree node.
If
it
does,
we
modify
the
load vector accordingly.
Section 10.2
1. Computing A = L U requires n - 1 steps,
and
step i uses
(n -
i)(l
+ 2(n - i)) = 2(n - i)2 + (n - i)
arithmetic operations (this is determined by simply counting
the
operations in
the
pseudo-code on page 473).
The
total
number of operations required is
n-1
n-1
n-1
i=l
j=l
j=l
2n
3
n
2
n
--
--
-
3
2
6
This agrees with
the
O(2n
3
/3) operation count given in
the
text.
The
computation of L
-lb
requires n - 1 steps, with 2(n - i) arithmetic operations
per step.
The
total
is
n-l n-l
2
~)n
-1)
=
2"Lj
= n
2
-
n.
i=l
j=l
The
final step of back substitution requires n steps, with
2(
n - i) + 1 operations
per
step, for a
total
of
n
n-l
L {2(n - i) +
1}
= 2 L j + n = n
2
•
i=l
j=l
The
total
for
the
two triangular solves is 2n2 - n, which also agrees with
the
count
given in
the
text.
3.
With
1
cp(x)
=
-x·
Ax
- b . x,
2
600
Appendix
C.
Solutions
to
odd-numbered
exercises
we
have
The
fact
that
A is
symmetric allows
the
simplification
which
was
used above.
We
thus
see
that
with
5.
An
inner product
has
three properties:
(a)
(x, x)
>
0 for all x, and (x, x) = 0 if and
only
if x = 0.
(b)
(x, y) = (y, x) for all x, y.
(c)
(ax +
/3y,
z)
=
a(x,
z) +
/?(y,
z) for all x, y, z and all a, ft.
These properties hold
for
The
third property would
be
true
for any
matrix
A, the
second property obviously
requires
that
A be
symmetric,
and the first
property requires
that
A be
positive
definite.
7.
(a)
With
x
(0)
=
(4,2),
the
negative gradient
is
-V0
(x
(0)
)
=
(-2, -1),
and so we
must minimize
The
minimizer
is a =
5/14,
and so
(b)
At
x
(1)
=
(23/7,23/14),
the
negative gradient
is
(-3/14,3/7),
so we
must
minimize
The
minimizer
is a =
5/6,
and so
The
desired solution
is (3,
2).
(c)
With
x
(0)
=
0, the CG
algorithm produces
x
(1)
=
(2.67456,
2.34024),
with
x^
=
(3, 2), the
exact solution.
600
Appendix
C.
Solutions
to
odd-numbered exercises
we
have
1
<fJ(x
+
y)
=
"2
(x +
y).
A(x
+ y) -
h·
(x + y)
1 1
=
"2x
.
Ax
+ y .
Ax
+
"2
Y
.
Ay
- b . x - b . Y
1
=
<fJ(x)
+
(Ax
-
b)
. y +
"2
Y
.
Ay.
The
fact
that
A is symmetric allows
the
simplification
1 1
"2x
.
Ay
+
"2
Y
.
Ax
=
y.
Ax,
which was used above. We
thus
see
that
<fJ(x
+
y)
=
<fJ(x)
+
V'<fJ(x)
. y + 0
(1IyI\2)
,
with
V'<fJ(x)
=
Ax
-
h.
5.
An inner product has three properties:
(a)
(x, x)
~
0 for all x,
and
(x,
x)
= 0 if
and
only if x =
O.
(b)
(x,y)
=
(y,x)
for all
x,y.
(c)
(ax
+ ,8y, z) =
a(x,
z) + ,8(y, z) for all x, y, z
and
all a,,8.
These properties hold for
(X,y)A = X·
Ay.
The
third
property would be
true
for any
matrix
A,
the
second property obviously
requires
that
A be symmetric,
and
the
first property requires
that
A be positive
definite.
7.
(a)
With
xeD)
= (4,2),
the
negative gradient is
-V'<fJ
(X(D))
=
(-2,
-1),
and
so
we
must minimize
The
minimizer is a = 5/14,
and
so
X(l)
=
(23,
23)
.
7 14
(b) At
x(1)
= (23/7,23/14),
the
negative gradient is
(-3/14,3/7),
so
we
must
minimize
,/,
(X(l)
_
r7,/,
(
(1)))
=.E...-
2 _
~
_ 529.
'/'
av,/,
x 196
a
196
a
28
The
minimizer is a = 5/6,
and
so
(2) =
(87
2)
x
28"
The
desired solution is (3,2).
(c)
With
xeD)
= 0,
the
CG algorithm produces
X(l)
==
(2.67456,2.34024),
with
X(2)
= (3,2),
the
exact solution.
Appendix
C.
Solutions
to
odd-numbered
exercises
601
9.
If y = x -
x
(0)
,
then
x = y +
x
(0)
and so
We
can
therefore replace
b by b
—
Ax^
0)
,
apply
the CG
algorithm
to
estimate
y,
and
add
x^
to get the
estimate
of x.
Section
10.3
1.
Let the
vector-valued
function
F be
defined
by
F(x)
=
(/(x),0).
Then
so,
by the
divergence theorem,
If
0
G
C'o
0
(fi),
then
the
boundary integral vanishes,
and we
obtain
The
derivation
for
d/dx-2
is
exactly analogous.
3.
A
direct computation shows
that
while
Thus
the
L
2
error
is
independent
of m,
n,
while
the
H
l
error becomes arbitrarily
large
as m, n
—>
oo.
Section
10.4
1.
(a) Let u, v
€
R
n
contain
the
nodal values
of
piecewise linear
functions
Uh,Vh,
respectively,
so
that
It
follows
that
Appendix
C.
Solutions
to
odd-numbered exercises
601
9.
If
y = x -
x(O),
then
x = y +
x(O)
and
so
Ax
= b
=>
Ay
+ Ax(O) = b
=>
Ay
= b - Ax(O).
We can therefore replace b by b - Ax(O), apply
the
CG algorithm
to
estimate
y,
and
add
x(O)
to
get
the
estimate of
x.
Section 10.3
1. Let
the
vector-valued function F
be
defined by
F(x)
=
(f(x),
0).
Then
'\7. (¢F) =
cf/fVl.
F + '\7¢. F = ¢
!f
+
!¢
f,
vXl
VXl
so, by
the
divergence theorem,
r
¢~f
+ r
~¢
f=
r
'\7'(¢F)=l
¢F.n=l
¢fnl.
in
VXl
in
VXl
in
an an
If
¢ E C8"([2),
then
the
boundary integral vanishes,
and
we
obtain
r
¢~
- _ r
o¢
f
in
OXI
-
in
OXI
•
The
derivation for 0/
OX2
is exactly analogous.
3.
A direct computation shows
that
Ilf -
911L2
=
~
for all integers
m,n,
while
IIf
-
911Hl
=
~Jl
+ m
2
Jr2
+ n
2
Jr2
for all
m,n.
Thus
the
L2
error is independent of
m,
n,
while
the
HI
error becomes arbitrarily
large as m,
n
-+
00.
Section 10.4
1. (a) Let
u,
vERn
contain
the
nodal values of piecewise linear functions
'Uh,
Vh,
respectively,
so
that
n n
'Uh = L
'Ui¢i,
Vh
= L
Vi¢i.
i=1
i=1
It
follows
that
n n
i=1
j=1
=u·Mv.
602
Appendix
C.
Solutions
to
odd-numbered
exercises
Figure
C.47.
The
coarsest
mesh
from Exercise
10-4-3.
(b)
As
shown
in the
text,
if u and v are
generalized eigenvectors
for the
problem
Ku
=
AMu, then
L
T
u
and
L
T
v
are
orthogonal
in the
Euclidean norm.
But
then
Thus
UH
and
VH
are
orthogonal
in the L
2
norm.
3.
(a)
Using
the
same method
as in
Example 10.5,
we find
that
the
smallest eigen-
value
on the
pentagon
is
approximately 18.9. (The estimates obtained
on five
successively
finer
meshes
were
21.2794,
19.5574, 19.0818, 18.9601, 18.9294.
The
coarsest mesh
is
shown
in
Figure C.47.)
(b)
It
would
be
reasonable
to
hypothesize
that
where
Ai
=
18.2
is the
smallest eigenvalue
on the
disk
of
area
1.
(The value
of
Ai
was
found
near
the end of
Section 8.3.)
(c)
Repeating
the
calculation again
for a
regular decagon
of
area
1, we find
that
the
smallest
eigevalue
is
approximately 18.3. (The values obtained
on
four
successively
finer
meshes were 19.0328, 18.5055, 18.3268, 18.2755.)
602
Appendix
C.
Solutions
to
odd-numbered exercises
(b) As shown
in
the
text,
if u
and
v are generalized eigenvectors for
the
problem
Ku
= AMu,
then
LT
u
and
LTv
are
orthogonal in
the
Euclidean norm.
But
then
o = L T U • LTV = U .
LL
TV
= U . M v =
(Uh'
Vh).
Thus
Uh
and
Vh
are
orthogonal in
the
L2
norm.
3. (a) Using
the
same
method
as
in
Example
10.5, we find
that
the
smallest eigen-
value on
the
pentagon
is approximately 18.9.
(The
estimates
obtained
on five
successively finer meshes were 21.2794, 19.5574, 19.0818, 18.9601, 18.9294.
The
coarsest mesh is shown in Figure C.47.)
(b)
It
would
be
reasonable
to
hypothesize
that
A(n)
-t
Al
I ,
where
Al
~
18.2 is
the
smallest eigenvalue on
the
disk
of
area
1.
(The
value
of
Al
was found
near
the
end
of
Section 8.3.)
(c)
Repeating
the
calculation again for a regular decagon
of
area
1, we find
that
the
smallest eigevalue is approximately 18.3.
(The
values
obtained
on four
successively finer meshes were 19.0328, 18.5055, 18.3268, 18.2755.)
0.8.------..----.......-----..-----,
0.6
0.4
0.2
x
N
0
-0.2
-0.4
-0.6
_0.8l...-----L------'-----"-----~
-0.5
0 0.5
x
1
Figure
C.4T. The coarsest mesh from Exercise 10.4.3.
Bibliography
[1]
Stuart
S.
Antman.
The
equations
for
large vibrations
of
strings. American
Mathematical
Monthly,
87:359-370,
1980.
[2]
Kendall
E.
Atkinson.
An
Introduction
to
Numerical Analysis. John Wiley
&
Sons,
New
York,
2nd
edition, 1989.
[3]
Eric
B.
Becker, Graham
F.
Carey,
and J.
Tinsley Oden. Finite Elements.
Prentice-Hall, Englewood
Cliffs,
1981.
[4]
T. A.
Bick. Elementary Boundary
Value
Problems. Marcel Dekker,
New
York,
1993.
[5]
William
E.
Boyce
and
Richard
C.
DiPrima. Elementary
Differential
Equations
and
Boundary
Value
Problems. John Wiley
&
Sons,
New
York,
7th
edition,
2000.
[6]
Susanne
C.
Brenner
and L.
Ridgway
Scott.
The
Mathematical
Theory
of
Finite
Element
Methods.
Springer-Verlag,
New
York, 1994.
[7]
W. L.
Briggs
and V. E.
Henson.
The
DFT:
An
Owner's Manual
for the
Discrete
Fourier
Transform.
Society
for
Industrial
and
Applied Mathematics,
Philadel-
phia, 1995.
[8]
E. O.
Brigham.
The
Fast
Fourier
Transform
and Its
Applications. Prentice-
Hall, Englewood
Cliffs,
1988.
[9]
Michael
A.
Celia
and
William
G.
Gray. Numerical
Methods
for
Differential
Equations: Fundamental
Concepts
for
Scientific
and
Engineering Applications.
Prentice-Hall, Englewood
Cliffs,
1992.
[10]
Ruel
V.
Churchill
and
James
Ward Brown. Fourier Series
and
Boundary
Value
Problems. McGraw-Hill,
New
York,
5th
edition, 1993.
[11] Philippe
G.
Ciarlet.
The
Finite Element
Method
for
Elliptic Problems. North
Holland, Amsterdam, 1978.
[12]
J. W.
Cooley
and J. W.
Tukey.
An
algorithm
for the
machine computation
of
complex Fourier series. Mathematics
of
Computation,
19:297-301,
1965.
603
Bibliography
[1]
Stuart
S.
Antman. The equations for large vibrations of strings. American
Mathematical Monthly,
87:359-370, 1980.
[2]
Kendall E. Atkinson.
An
Introduction to Numerical Analysis. John Wiley &
Sons,
New
York, 2nd edition, 1989.
[3]
Eric B. Becker, Graham F. Carey, and J. Tinsley Oden. Finite Elements.
Prentice-Hall, Englewood Cliffs, 1981.
[4]
T.
A.
Bick. Elementary Boundary Value Problems. Marcel Dekker, New York,
1993.
[5]
William E. Boyce
and
Richard C. DiPrima. Elementary Differential Equations
and Boundary Value Problems.
John Wiley & Sons, New York, 7th edition,
2000.
[6]
Susanne
C.
Brenner and
L.
Ridgway Scott. The Mathematical Theory
of
Finite
Element Methods.
Springer-Verlag, New York, 1994.
[7]
W.
L.
Briggs and
V.
E. Henson. The DFT:
An
Owner's Manual for the Discrete
Fourier Transform.
Society for Industrial and Applied Mathematics, Philadel-
phia,1995.
[8]
E.
O.
Brigham. The Fast Fourier Transform and Its Applications. Prentice-
Hall, Englewood Cliffs, 1988.
[9]
Michael
A.
Celia and William G. Gray. Numerical Methods for Differential
Equations: Fundamental Concepts for Scientific and Engineering Applications.
Prentice-Hall, Englewood Cliffs, 1992.
[10]
Ruel
V.
Churchill
and
James Ward Brown. Fourier Series and Boundary Value
Problems.
McGraw-Hill, New York, 5th edition, 1993.
[11]
Philippe
G.
Ciarlet. The Finite Element Method for Elliptic Problems. North
Holland, Amsterdam, 1978.
[12]
J. W. Cooley
and
J. W. Tukey. An algorithm for
the
machine computation of
complex Fourier series.
Mathematics
of
Computation, 19:297-301,1965.
603
[13]
James
W.
Demmel.
Applied
Numerical Linear
Algebra.
Society
for
Industrial
and
Applied Mathematics, Philadelphia, 1997.
[14]
G. B.
Folland. Introduction
to
Partial
Differential
Equations. Princeton Uni-
versity
Press,
Princeton,
2nd
edition,
1995.
[15]
Gerald
B.
Folland. Fourier Analysis
and Its
Applications. Brooks/Cole, Pacific
Grove, 1992.
[16]
George
E.
Forsythe, Michael
A.
Malcolm,
and
Cleve
B.
Moler.
Computer Meth-
ods
for
Mathematical Computations. Prentice-Hall, Englewood
Cliffs,
1977.
[17]
Philip
Gillett.
Calculus
and
Analytic Geometry.
D. C.
Heath, Lexington, 1984.
[18]
Jack
L.
Goldberg
and
Merle
Potter.
Differential
Equations:
A
Systems
Ap-
proach.
Prentice-Hall, Upper Saddle River, 1997.
[19]
Gene
H.
Golub
and
Charles
F. Van
Loan. Matrix Computations. Johns Hopkins
University Press, Baltimore,
3rd
edition, 1996.
[20]
Michael Greenberg.
Advanced
Engineering Mathematics. Prentice-Hall, Upper
Saddle River,
2nd
edition, 1998.
[21]
M. E.
Gurtin.
An
Introduction
to
Continuum Mechanics. Academic
Press,
New
York, 1981.
[22]
Richard Haberman. Elementary
Applied
Partial
Differential
Equations: With
Fourier
Series
and
Boundary
Value
Problems. Prentice-Hall, Upper Saddle
River, 1998.
[23]
William
W.
Hager.
Applied
Numerical Linear
Algebra.
Prentice-Hall, Engle-
wood
Cliffs,
1988.
[24]
J.
Ray
Hanna
and
John
H.
Rowland. Fourier Series,
Transforms,
and
Bound-
ary
Value
Problems. John Wiley
&
Sons,
New
York,
2nd
edition, 1990.
[25]
M. T.
Heideman,
D. H.
Johnson,
and C. S.
Burrus.
Gauss
and the
history
of
the
fast
Fourier
transform. Archive
for
History
of
Exact Science,
34:265-277,
1985.
[26]
Morris
Hirsch
and
Stephen
Smale.
Differential
Equations, Dynamical
Systems,
and
Linear
Algebra.
Academic
Press,
New
York, 1974.
[27]
Claes
Johnson. Numerical Solution
of
Partial
Differential
Equations
by the
Finite Element
Method.
Cambridge University
Press,
Cambridge, 1987.
[28]
David
W.
Kammler.
A
First Course
in
Fourier Analysis. Prentice-Hall, Upper
Saddle River, 2000.
[29]
Wilfred
Kaplan.
Advanced
Calculus.
Addison-Wesley,
Reading,
4th
edition,
1994.
604
Bibliography
604
Bibliography
[13]
James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial
and
Applied Mathematics, Philadelphia, 1997.
[14]
G. B. Folland. Introduction to Partial Differential Equations. Princeton Uni-
versity Press, Princeton, 2nd edition, 1995.
[15]
Gerald B. Folland. Fourier Analysis and Its Applications. Brooks/Cole, Pacific
Grove, 1992.
[16]
George E. Forsythe, Michael A. Malcolm,
and
Cleve B. Moler. Computer Meth-
ods
for Mathematical Computations. Prentice-Hall, Englewood Cliffs, 1977.
[17]
Philip Gillett. Calculus and Analytic Geometry. D. C. Heath, Lexington, 1984.
[18]
Jack
L. Goldberg
and
Merle
Potter.
Differential Equations: A Systems Ap-
proach. Prentice-Hall, Upper Saddle River, 1997.
[19]
Gene H. Golub
and
Charles
F.
Van Loan. Matrix Computations. Johns Hopkins
University Press, Baltimore,
3rd
edition, 1996.
[20]
Michael Greenberg. Advanced Engineering Mathematics. Prentice-Hall, Upper
Saddle River, 2nd edition, 1998.
[21]
M. E. Gurtin.
An
Introduction
to
Continuum Mechanics. Academic Press,
New York, 1981.
[22]
Richard Haberman. Elementary Applied Partial Differential Equations: With
Fourier Series and Boundary Value Problems.
Prentice-Hall, Upper Saddle
River, 1998.
[23]
William W. Hager. Applied Numerical Linear Algebra. Prentice-Hall, Engle-
wood Cliffs, 1988.
[24]
J.
Ray
Hanna
and
John
H. Rowland. Fourier Series, Transforms, and Bound-
ary Value Problems.
John
Wiley & Sons, New York, 2nd edition, 1990.
[25]
M.
T.
Heideman, D. H. Johnson,
and
C. S. Burrus. Gauss
and
the
history
of
the
fast Fourier transform. Archive for History
of
Exact Science, 34:265-277,
1985.
[26]
Morris Hirsch
and
Stephen Smale. Differential Equations, Dynamical Systems,
and Linear Algebra.
Academic Press, New York, 1974.
[27]
Claes Johnson. Numerical Solution
of
Partial Differential Equations
by
the
Finite Element Method.
Cambridge University Press, Cambridge, 1987.
[28]
David W. Kammler. A First Course in Fourier Analysis. Prentice-Hall, Upper
Saddle River, 2000.
[29]
Wilfred Kaplan. Advanced Calculus. Addison-Wesley, Reading,
4th
edition,
1994.
[30]
David Kincaid
and
Ward Cheney. Numerical Analysis. Brooks/Cole, Pacific
Grove,
2nd
edition, 1996.
[31]
Patrick
M.
Knupp
and
Stanly Steinberg.
The
Fundamentals
of
Grid
Genera-
tion.
CRC
Press,
Boca
Raton,
1993.
[32]
Young
W.
Kwon
and
Hyochoong Bang.
The
Finite Element
Method
Using
MATLAB.
CRC
Press, Boca Raton, 1997.
[33]
J. D.
Lambert. Numerical
Methods
for
Ordinary
Differential
Systems. John
Wiley
&
Sons, Chichester, 1991.
[34]
David
C.
Lay. Linear
Algebra
and Its
Applications. Addison Wesley Longman,
Reading,
2nd
(updated) edition, 2000.
[35]
David
R.
Lide,
editor.
CRC
Handbook
of
Chemistry
and
Physics.
CRC
Press,
Boca Raton, 1998.
[36]
C. C. Lin and L. A.
Segel.
Mathematics
Applied
to
Deterministic Problems
in
the
Natural Sciences. Society
for
Industrial
and
Applied Mathematics,
Philadel-
phia, 1988.
[37]
Jerrold
E.
Marsden
and
Anthony
J.
Tromba.
Vector
Calculus.
W. H.
Freeman,
New
York,
4th
edition, 1996.
[38]
R. M. M.
Mattheij
and J.
Molenaar. Ordinary
Differential
Equations
in
Theory
and
Practice. John Wiley
&
Sons Ltd., Chichester, 1996. Reprinted
by the
Society
for
Industrial
and
Applied Mathematics, Philadelphia, 2002.
[39]
Robert
C.
McOwen.
Partial
Differential
Equations:
Methods
and
Applications.
Prentice-Hall, Upper Saddle River, 1996.
[40]
Carl
D.
Meyer. Matrix Analysis
and
Applied
Linear Algebra. Society
for In-
dustrial
and
Applied Mathematics, Philadelphia, 2000.
[41] Jeffrey
Rauch. Partial
Differential
Equations.
Springer-Verlag,
New
York,
1991.
[42]
L. F.
Shampine
and M. W.
Reichelt.
The
MATLAB
ODE
suite.
SIAM
Journal
on
Scientific
Computing,
18:1-22,
1997.
[43]
Lawrence
E.
Shampine. Numerical Solutions
of
Ordinary
Differential
Equa-
tions. Chapman
&
Hall,
New
York, 1994.
[44]
Gilbert Strang. Linear
Algebra
and Its
Applications. Harcourt, Brace,
Jo-
vanovich,
San
Diego,
3rd
edition, 1988.
[45]
Gilbert Strang
and
George
J.
Fix.
An
Analysis
of
the
Finite Element
Method.
Wellesley-Cambridge
Press, Wellesley, 1988.
[46]
Walter
A.
Strauss. Partial
Differential
Equations:
An
Introduction. John Wiley
&
Sons,
New
York, 1992.
Bibliography
605
Bibliography
605
[30]
David Kincaid and Ward Cheney. Numerical Analysis. Brooks/Cole, Pacific
Grove, 2nd edition, 1996.
[31]
Patrick
M.
Knupp and Stanly Steinberg. The Fundamentals
of
Grid Genera-
tion.
CRC Press, Boca Raton, 1993.
[32]
Young W. Kwon and Hyochoong Bang. The Finite Element Method Using
MATLAB.
CRC Press, Boca Raton, 1997.
[33]
J.
D.
Lambert. Numerical Methods for Ordinary Differential Systems. John
Wiley
& Sons, Chichester, 1991.
[34]
David
C.
Lay.
Linear Algebra and Its Applications. Addison Wesley Longman,
Reading, 2nd (updated) edition, 2000.
[35]
David
R.
Lide, editor.
CRC
Handbook
of
Chemistry and Physics. CRC Press,
Boca Raton, 1998.
[36]
C. C.
Lin and
L.
A.
Segel. Mathematics Applied
to
Deterministic Problems in
the Natural Sciences.
Society for Industrial and Applied Mathematics, Philadel-
phia, 1988.
[37]
Jerrold E. Marsden and Anthony J. Tromba. Vector Calculus. W.
H.
Freeman,
New
York, 4th edition, 1996.
[38]
R.
M. M.
Mattheij and J. Molenaar. Ordinary Differential Equations in Theory
and Practice.
John Wiley & Sons Ltd., Chichester, 1996. Reprinted by the
Society for Industrial and Applied Mathematics, Philadelphia, 2002.
[39]
Robert C. McOwen. Partial Differential Equations: Methods and Applications.
Prentice-Hall, Upper Saddle River, 1996.
[40]
Carl D. Meyer. Matrix Analysis and Applied Linear Algebra. Society for In-
dustrial and Applied Mathematics, Philadelphia, 2000.
[41]
Jeffrey Rauch. Partial Differential Equations. Springer-Verlag, New York,
1991.
[42]
L. F. Shampine and M. W. Reichelt. The
MATLAB
ODE suite.
SIAM
Journal
on Scientific Computing,
18:1-22, 1997.
[43]
Lawrence E. Shampine. Numerical Solutions
of
Ordinary Differential Equa-
tions.
Chapman & Hall, New York, 1994.
[44]
Gilbert Strang. Linear Algebra and Its Applications. Harcourt, Brace, Jo-
vanovich, San Diego, 3rd edition, 1988.
[45]
Gilbert Strang
and
George J. Fix.
An
Analysis
of
the Finite Element Method.
Wellesley-Cambridge Press, Wellesley, 1988.
[46]
Walter
A.
Strauss. Partial Differential Equations:
An
Introduction.
John
Wiley
& Sons, New York, 1992.
[47]
John
C.
Strikwerda. Finite
Difference
Schemes
and
Partial
Differential
Equa-
tions. Chapman
&
Hall,
New
York, 1989.
[48]
Paul
N.
Swarztrauber.
Fftpack, 1985.
[49]
Lloyd
N.
Trefethen
and
David
Bau.
Numerical Linear Algebra. Society
for
Industrial
and
Applied Mathematics,
Philadelphia,
1997.
[50]
Charles
Van
Loan. Computational Frameworks
for the
Fast Fourier
Transform.
Society
for
Industrial
and
Applied Mathematics,
Philadelphia,
1992.
[51]
J. S.
Walker. Fast Fourier
Transform.
CRC
Press, Boca
Raton,
2nd
edition,
1996.
[52]
H. F.
Weinberger.
A
First Course
in
Partial
Differential
Equations with Com-
plex
Variables
and
Transform
Methods.
Blaisdell,
New
York, 1965.
[53]
Antoni Zygmund. Trigonometric Series. Cambridge University
Press,
Cam-
bridge,
2nd
edition, 1959.
606
Bibliography
606
Bibliography
[47]
John C. Strikwerda. Finite Difference Schemes and Partial Differential Equa-
tions.
Chapman & Hall, New York, 1989.
[48]
Paul
N.
Swarztrauber. Fftpack, 1985.
[49]
Lloyd
N.
Trefethen and David Bau. Numerical Linear Algebra. Society for
Industrial and Applied Mathematics, Philadelphia, 1997.
[50]
Charles Van Loan. Computational Frameworks for the Fast Fourier Transform.
Society for Industrial
and
Applied Mathematics, Philadelphia, 1992.
[51]
J.
S.
Walker. Fast Fourier Transform. CRC Press, Boca Raton, 2nd edition,
1996.
[52]
H. F. Weinberger. A First Course
in
Partial Differential Equations with Com-
plex Variables and Transform Methods.
Blaisdell, New York, 1965.
[53]
Antoni Zygmund. Trigonometric Series. Cambridge University Press, Cam-
bridge, 2nd edition, 1959.
Index
aluminum,
15, 26,
163, 217,
276
angle between
two
vectors,
55
area
integral,
329
automatic
step
control,
111
back substitution,
474
backward
Euler
method, 118, 263,
310
banded matrix, 386,
475
bar
aluminum,
217,
276
chromium,
278
circular,
245
copper,
228, 243, 244, 264, 265,
278
elastic,
21, 173
heat
equation and,
333
heat
flow in, 9
heterogeneous,
276
iron,
214, 217, 222, 228, 230, 236,
244, 264, 281,
323
steel,
302
basis,
50
orthogonal,
55, 57
orthonormal,
57, 65
Bessel
function
of
order
n,
366
properties,
366
Bessel's equation
of
order
n,
364
Bessel's inequality,
428
best approximation,
61,
147,156,181,
182,
343
in
the
energy norm,
491
bilinear
form,
180,
270
symmetric,
181
body
force,
23
boundary conditions,
134
Dirichlet,
13, 34,
144, 168, 169,
197, 211, 280, 292, 332, 335,
339
essential, 266, 313,
388
for
a
vibrating string,
30
for
the
diffusion
equation,
20
for
the
hanging bar,
24
for
the
heat
equation,
13
homogeneous,
14, 168
inhomogeneous,
14,164,169,197,
222,
243, 273, 298,
391
inhomogeneous Dirichlet,
321
mixed,
13,150,161,
283, 301, 313,
333, 348,
388
natural, 266, 313,
388
Neumann,
13, 27, 34,
152, 231,
273, 275, 282, 332, 335, 349,
388
periodic,
153,
246
Robin,
143
time dependent,
223
boundary value problem,
14, 144
on
a
rectangular domain,
345
BVP,
14
C[a,_b],
33, 58
C
2
(ft),
336
C
k
[a,b],
33
Co
00
(ft),
489
C£(n),
336,
377
C&(fi),
338
carbon monoxide,
220
Cauchy
sequence,
451
Celsius,
19
chain rule,
360
607
Index
aluminum,
15,26,
163, 217,
276
angle between two vectors,
55
area integral, 329
automatic step control,
111
back substitution, 474
backward Euler method, 118,263,310
banded matrix, 386, 475
bar
aluminum, 217, 276
chromium, 278
circular,
245
copper, 228, 243, 244, 264, 265,
278
elastic, 21, 173
heat equation and, 333
heat
flow
in, 9
heterogeneous, 276
iron, 214, 217, 222, 228, 230, 236,
244, 264, 281, 323
steel, 302
basis, 50
orthogonal, 55, 57
orthonormal, 57, 65
Bessel function
of order
n,
366
properties, 366
Bessel's equation
of order
n,
364
Bessel's inequality, 428
best approximation, 61, 147,156,181,
182, 343
in the energy norm,
491
bilinear form, 180, 270
symmetric,
181
body force,
23
607
boundary conditions, 134
Dirichlet, 13, 34, 144, 168, 169,
197, 211, 280, 292, 332, 335,
339
essential, 266, 313, 388
for a vibrating string,
30
for the diffusion equation,
20
for the hanging bar,
24
for
the
heat equation,
13
homogeneous, 14, 168
inhomogeneous, 14, 164, 169, 197,
222, 243, 273, 298,
391
inhomogeneous Dirichlet,
321
mixed, 13, 150, 161, 283, 301, 313,
333,348,388
natural, 266, 313, 388
Neumann, 13, 27, 34, 152, 231,
273,275,282,332,335,349,
388
periodic, 153,
246
Robin, 143
time dependent, 223
boundary value problem,
14,
144
on a rectangular domain, 345
BVP, 14
C[a,
b],
33,
58
e
2
(n),336
ek[a,
b],
33
e~(n),
489
eben),
336, 377
e1
(n)
, 338
carbon monoxide, 220
Cauchy sequence,
451
Celsius, 19
chain rule, 360