16.
u
t
=
ku
xx
u
x
(0
, t
) = 0
u
x
(
L, t
) = 0
u
(
x,
0) =
f
(
x
)
u
(
x, t
) =
∞
n
=0
u
n
cos
n π x
L
e
−
k
(
n π
L
)
2
t
u
(
x,
0) =
∞
n
=0
u
n
cos
n π x
L
≡
f
(
x
)
u
n
are the coeﬃcients of expanding
f
(
x
) in terms of Fourier cosine series.
u
0
=
1
L
L
0
f
(
x
)
dx
u
n
=
2
L
L
0
f
(
x
) cos
nπ
L
xdx
b.
The equilibrium is when
u
xx
= 0 subject to the same boundary conditions. The solution
is then obtained by integration with respect to
x
u
x
=
K
and
K
= 0 because of the boundary conditions.
Now integrate again to get
u
=
C
. This means that the temperature is constant. There is no other condition to fix this
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 Fall '08
 BELL,D
 Fourier Series, Periodic function, Leonhard Euler

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