10.6: OTHER HEAT CONDUCTION PROBLEMS
KIAM HEONG KWA
1.
Nonhomogeneous Boundary Conditions
Recall that the solution to the heat conduction problem
α
2
u
xx
=
u
t
,
0
< x < L, t >
0
,
(1.1a)
with the initial condition
u
(
x,
0) =
f
(
x
)
,
0
≤
x
≤
L,
(1.1b)
where
f
(
x
) and
f
0
(
x
) are piecewise continuous, and the homogeneous
boundary conditions
u
(0
,t
) = 0
, u
(
L,t
) = 0
, t >
0
,
(1.1c)
is given by
u
(
x,t
) =
∞
X
n
=1
c
n
e

α
2
n
2
π
2
t/L
2
sin
nπx
L
,
(1.2a)
where
c
n
=
2
L
Z
L
0
f
(
x
) sin
nπx
L
dx.
(1.2b)
Since
f
(
x
) is piecewise continuous in [0
,L
], it is bounded. Hence so are
c
n
,
n
= 1
,
2
,
···
, uniformly bounded in view of (1.2b). It follows that
(1.3)
lim
t
→∞
u
(
x,t
) = 0
,
0
≤
x
≤
L,
because of the negative exponential factor in every term of the series
(1.2a). Explicitly, let

f
(
x
)
 ≤
M
2
for all
x
∈
[0
,L
]. Then
(1.4)

c
n
 ≤
2
L
Z
L
0
M
2
±
±
±
sin
nπx
L
±
±
±
dx
=
M
Date
: February 19, 2011.
1