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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
KIAM HEONG KWA
for all
n
,
n
= 1
,
2
,
···
. So for any
t >
0,
(1.5)

u
(
x,t
)
 ≤
∞
X
n
=1
Me

α
2
nπ
2
t/L
2
=
Me

α
2
π
2
t/L
2
1

e

α
2
π
2
t/L
2
for all
x
∈
[0
,L
]. Since lim
t
→∞
Me

α
2
π
2
t/L
2
1

e

α
2
π
2
t/L
2
= 0, (1.3) follows.
This allows us to solve the following heat conduction problem with
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '11
 Kwa
 Differential Equations, Equations, Partial Differential Equations, Boundary value problem, Boundary conditions

Click to edit the document details