Use the method of separation of variables to show that the 1-D heat conduction problem: find
u
=
u
(
x
,
t
) so that
¶
u
¶
t
(
x
,
t
) -
a
¶
2
u
¶
x
2
(
x
,
t
)
=
0, 0 <
x
<
l
,
t
³
0
u
(0,
t
)
=
0,
t
³
0
u
(
l
,
t
)
=
0,
t
³
0
u
(
x
,0)
=
f
(
x
), 0
£
x
£
l
has the solution
u
(
x
,
t
) =
¥
å
n
= 1
c
n
sin
n
p
x
l
exp
é
ê
ë
-
a
æ
ç
è
n
p
l
ö
÷
ø
2
t
ù
ú
û
where
c
n
=
2
l
ó
õ
l
0
f
(
x
) sin
n
p
x
l
dx
.
File translated from T
E
X by
T
T
H
, version 2.22.
On 13 Jun 1999, 00:56.