Hence the temperature distribution in the bar is given by
2
22
332
1
(21)
81(21)
(,)sinexp
(21)
n
nx
n
uxtat
nll
∞
=
−
−
⎛⎞
=−
⎜⎟
⎝⎠
−
∑
p
a
p
p
Example 8.13.10 F
ind the temperature in a thin metal rod of length l
with both ends insulated and with initial temperature in the rod sin
(p x/
l ).
Solution The initial boundary value problem consists of the following
1. Partial differential equation for conduction of heat
2
2
2
uu
a
tx
∂∂
=
∂∂
(8.402)
2. Boundary conditions: (Insulation at both ends)
0
0;
x
u
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