Equation (8.542) is a half-range Fourier sine series expansion for u
0
in
(0, p) and hence the Fourier constants d
n
are given by
0
0
0
0
0
2
2cos
sin
0if is even
2
(cos1)
4
if is odd.
n
u
nx
dunx dx
n
n
u
n
u
n
n
n
0
−
⎛⎞
==
⎜⎟
⎝⎠
⎧
−
⎪
=−=
⎨
⎪
⎩
∫
p
p
pp
p
p
p
(8.543)
No
w (8.541) reduces to
0
1,3,5
4
(,)sin
ny
n
u
uxynxe
n
−
=
=
∑
p
(8.544)
which is the required solution.
Example 8.13.21 The temperature u (x, y) is maintained at 0°C
along three edges of a square plate of side 100 cm and the four
th edge
is maintained at a constant temperature u
0
until steady-state conditons
pre ...
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