APPM 4350/5350: Fourier Series and Boundary Value Problems
Homework #2 Friday September 3, 2010
Due: Monday, September 20, 2010
In what follows, and in the future, all arbitrary functions are assumed to be piecewise smooth
unless otherwise speciﬁed.
1. Consider:
∂T
∂t
=
κ
∂
2
T
∂x
2
0
< x < L
where
κ >
0 is constant,
T
(
x
= 0
,t
) =
T
(
x
=
L,t
) = 0
Solve for
T
(
x,t
) with the following initial values:
a)
T
(
x,
0) =
T
0
sin
Nπx
L
,
b)
T
(
x,
0) =
T
0
cos
Nπx
L
where
N
is a nonnegative integer, and
T
0
is a constant.
2. Suppose:
∂T
∂t
=
κ
∂
2
T
∂x
2
0
< x < L
where
κ >
0 is constant,
∂T
∂x
(
x
= 0
,t
) =
T
(
x
=
L,t
) = 0 and
T
(
x,
0) =
f
(
x
). Find
T
(
x,t
)
and the the equilibrium, temperature. Hint: integration or general formulae, which also will be
discussed in class, can be used to show that the eigenfunctions are orthogonal.
3. Consider
∂T
∂t
=
κ
∂
2
T
∂x
2

αT
0
< x < L
where
κ,α >
0 are constant,
∂T
∂x
(
x
= 0
,t
) =
∂T
∂x
(
x
=
L,t
) = 0 and
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 Fall '08
 ABLOWITZ
 Continuous function, Boundary value problem, ∂T ∂2T

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