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(a) Let l and k be two positive integers. Show that
sin
kx
L
sin
lx
L
dx
D
0k
ยค
l;
L
2
Z
L
k
D
l:
0
(b) Assume that f is a function defined on .0; L/. We want to compute the Fourier
sine coefficients of this function on this interval. That is, we want to compute
constants c
1
;c
2
;::: such that
c
k
sin
kx
L
X
f.x/
D
for x
2
.0; L/:
k
D
1
Derive the formula
Z
L
f.x/sin
kx
L
dx
:
2
L
c
k
D
0
(c) Show by differentiation that the function
u
.x; t /
D
ce
.k=L/
2
t
sin
kx
L
;
where c is an arbitrary constant, satisfies the diffusion equation
u
t
D
u
xx
for x
2
.0; L/; t > 0:
(d) Compute the Fourier sine series of the function
f.x/
D
e
x
for x
2
.0; L/;
and find a formal solution of the problem
u
t
D
u
xx
for x
2
.0; L/; t > 0;
(8.93)
u
.0; t /
D
u
.L; t /
D
0
for t>0;
(8.94)
u
.x; 0/
D
e
x
for x
2
.0; L/:
(8.95)
(e) Write a computer program that graphs an approximation of the formal solution
that you derived in (d). The program should take L, T ,andN as input param-
eters and plot the N partial sum
u
N
.x; T /, x
2
.0; L/, of the Fourier series of
the formal solution, see (8.80).
(f) Derive and implement an explicit finite difference scheme for (
8.93
)-(
8.95
). The
program should take the discretization parameters x and t , and the time T
(the time at which we want to evaluate the solution) as input parameters. The
