Information Technology Reference
In-Depth Information
(l) Prove that
8
<
Z
1
0k
¤
l;
1=2 k
D
l>0;
1k
D
l
D
0:
cos.kx/ cos.lx/ dx
D
:
0
(m) Let f be a smooth function defined on .0; 1/. We want to compute the Fourier
cosine series of this function; that is, we want to determine constants c
0
;c
1
;:::
such that
X
f.x/
D
c
0
C
c
k
cos.kx/:
(8.107)
k
D
1
Show that if (8.107) holds, then
c
0
D
Z
1
0
f.x/
dx
and
D
2
Z
1
0
c
k
f.x/cos.kx/
dx
for k
D
1;2;::::
(n) Let f be defined as in question (g) and compute the solution of (
8.99
)-(
8.101
)
by hand.
(o) Use the solution formula derived in (n) to show that
u
.x; t /
D
Z
1
0
lim
t
!1
f.y/dy
for all x
2
Œ0; 1;
cf. your numerical experiments in (g).
8.3.2
Variable Coefficients
In this project we will consider a problem with a variable thermal conductivity k.
This means that k
D
k.x/ is a function of the spatial position x.Forthesakeof
simplicity, we will assume that k is smooth and that there exist positive numbers m
and M such that
0<m
k.x/
M
for x
2
Œ0; 1:
In this case, the model for the heat conduction takes the form (see Chap. 7)
u
t
D
.k
u
x
/
x
for x
2
.0; 1/; t > 0;
(8.108)
u
.0; t /
D
u
.1; t /
D
0
for t>0;
(8.109)
u
.x; 0/
D
f.x/
for x
2
.0; 1/;
(8.110)
