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In-Depth Information
Exercise 8.3.
(a) Compute the Fourier coefficients
f
c
k
g
of the function
f.x/
D
x
2
x
3
for x
2
.0; 1/:
(b) Use Fourier's method to compute a formal solution of the following problem
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
(8.89)
u
.0; t /
D
u
.1; t /
D
0
for t>0;
(8.90)
u
.x; 0/
D
x
2
x
3
for x
2
.0; 1/:
(8.91)
(c) Derive an explicit scheme for (
8.89
)-(
8.91
).
(d) Write a computer program that implements the scheme in (c).
(e) Use the program from (d) to graph an approximation of
u
.x; 2/
for x
2
Œ0; 1:
(8.92)
(f) Consider the formal solution that you derived in (b). Write a computer program
that uses the first 50 terms of this sine series to compute an approximation of
u
.x; 2/ for x
2
Œ0; 1.
(g) Use the computer code that you developed in (f) to graph a second approxima-
tion of (8.92).
˘
Exercise 8.4.
(a) Compute the Fourier sine series of the function
f.x/
D
e
x
for x
2
.0; 1/:
(b) Write a function
26
that, for a given positive integer N and x
2
.0; 1/, computes
the sum of the first N terms of the Fourier sine series in (a).
(c) Write a computer program that calls the function in (b) and plots the partial sum
of the Fourier series. The code should take N as an input parameter. Make plots
for N
D
5; 10; 50.
˘
Exercise 8.5.
Let k and l be positive integers. Apply the identity (8.71) to prove
that
Z
1
sin.kx/ sin.lx/
dx
D
0k
¤
l;
1=2 k
D
l:
0
˘
Exercise 8.6.
So far we have only considered the method of separation of variables
for the diffusion equation on the spatial interval Œ0; 1. What about other intervals,
say, Œ0; L,whereL is a positive constant? Can we generalize the theory developed
above to handle such cases? We will now investigate this question.
26
Computer code!
