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Solved Problems

that satisfies the inhomogeneous boundary conditions.

q(x)
00
µ = − , µ (0) = µ (h) = 0
p
p
p
κ
00
µ = 0, µ (0) = a, µ (h) = b
h
h
h
We find the particular solution µ with the method of Green functions.
p
00
G = δ(x − ξ), G(0|ξ) = G(h|ξ) = 0.
We find homogeneous solutions which respectively satisfy the left and right homo-
geneous boundary conditions.

y = x, y = h − x
2
1
Then we compute the Wronskian of these solutions and write down the Green
function.

x h − x

W = = −h
1 −1

1
G(x|ξ) = − x (h − x )
<
>
h
The homogeneous solution that satisfies the inhomogeneous boundary conditions
is
b − a
µ = a + x
h
h
Now we have the equilibrium temperature.

b − a Z h 1 q(ξ)
µ = a + x + − x (h − x ) − dξ
>
<
h 0 h κ
b − a h − x Z x x Z h
µ = a + x + ξq(ξ) dξ + (h − ξ)q(ξ) dξ
h hκ 0 hκ x

Let v denote the deviation from the equilibrium temperature.

u = µ + v

v satisfies a heat equation with homogeneous boundary conditions and no source
term.
v = κv , v(0, t) = v(h, t) = 0, v(x, 0) = f(x) − µ(x)
xx
t
We solve the problem for v with separation of variables.

v = X(x)T(t)
0
00
XT = κX T
T 0 X 00
= = −λ
κT X

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