Information Technology Reference
In-Depth Information
(f) Show that
Z
1
u
.x; t /
dx
D
Z
1
0
f.x/
dx
for t
0:
0
(g) If heat can neither leave nor enter a body ˝ (
D
.0; 1/) then the temperature
throughout ˝ will approach a constant temperature C as time increases. Thus,
our conjecture is that
u
.x; t /
!
C as t
!1
(8.104)
for all x
2
.0; 1/. Furthermore, it seems reasonable that C equals the average of
the initial temperature distribution f .
Let
f.x/
D
3:14
C
cos.2x/
for x
2
.0; 1/
and define
C
D
Z
1
0
f.x/
dx
:
Design a numerical experiment suitable for testing the hypothesis (8.104).
Perform several experiments and comment on your results.
We will now modify the method of separation of variables, presented in Sect.
8.2
,
to handle Neumann boundary conditions. It turns out that this can be accomplished
in a rather straight forward manner.
(h) Apply the ansatz
u
.x; t /
D
X.x/T .t/
to (
8.99
)-(
8.100
) and show that this leads to the two eigenvalue problems
T
0
.t /
D
T .t /;
(8.105)
X
00
.x/
D
X.x/;
X
0
.0/
D
0 and X
0
.1/
D
0;
(8.106)
where is a constant.
(i) Show that
T.t/
D
ce
t
solves (
8.105
) for any constant c.
(j) Show that
X.x/
D
cos.kx/
satisfies (
8.106
)fork
D
0; 1; 2; : : :
(k) Use the results obtained in (i) and (j) and with the super-positioning principle
to conclude that any convergent series of the form
X
c
k
e
k
2
2
t
cos.kx/
u
.x; t /
D
c
0
C
k
D
1
defines a formal solution of (
8.99
)and(
8.100
).