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implicit methods must be applied, since they do not demand similar severe restric-
tions on the discretization parameters. We will define and analyze implicit schemes
in later chapters.
8.2.11
Exercises
Exercise 8.1.
Let c be an arbitrary constant and k an arbitrary integer. Define the
function
u
D
u
.x; t / by
u
.x; t /
D
ce
k
2
2
t
sin.kx/
for x
2
Œ0; 1; t > 0:
(a) Compute
u
t
and
u
xx
and verify that
u
t
D
u
xx
:
(b) Verify that
u
.0; t /
D
u
.1; t /
D
0
for t>0:
(c) Show that
u
.x; 0/
D
c sin.kx/
for x
2
.0; 1/:
˘
Exercise 8.2.
(a) Use formulas (8.74)and(8.75) to compute the Fourier sine
series of the function
f.x/
D
1
for x
2
.0; 1/:
(b) Write a computer program that can plot the function given by the N th partial
sum of the Fourier series of f . Make plots for N
D
3; 7; 80; 100.
(c) Find a formal solution of the problem
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
u
.0; t /
D
u
.1; t /
D
0
for t>0;
u
.x; 0/
D
f.x/
for x
2
.0; 1/;
see (8.76)-(
8.79
).
(d) Write a computer program that can graph the function defined by the first N
terms of the series of the formal solution in (c). The program should take N
and the time t at which you want to graph the approximate solution, as input
parameters. Make plots for N
D
3; 7; 80; 100,att
D
0:25 and t
D
2.
(e) Redo (a)-(d) but this time apply the initial condition
f.x/
D
x
for x
2
.0; 1/:
˘
