Page 663 - Advanced_Engineering_Mathematics o'neil

18.2 Dirichlet Problem for a Rectangle 643

y

(0, K) u(x, K) = f(x) (L, K)

u = 0 u = 0
D

x
u = 0 (L, 0)

FIGURE 18.1 D and the boundary data
for this problem.

and

Y − λY = 0;Y(0) = 0.
The problem for X has eigenvalues and eigenfunctions
2
n π 2 nπx
λ n = and X n (x) = sin
L 2 L
for n = 1,2,···.
The problem for Y becomes
2
n π 2

Y − Y = 0;Y(0) = 0
L 2
with solutions that are constant multiples of
nπy

Y n (y) = sinh .
L
For each positive integer n, we have a function
nπx nπy

u n (x, y) = sin sinh
L L
that is harmonic on D and satisfies the zero boundary conditions on the lower side and vertical
sides of D. For the condition on the top side, use a superposition
∞

nπx nπy
u(x, y) = b n sin sinh .
L L
n=1
We need
∞

nπx nπ K
u(x, K) = f (x) = b n sin sinh .
L L
n=1
This is a Fourier sine expansion of f (x) on [0, L] with coefficient b n sinh(nπ K/L). Thus, choose
b n sinh(nπ K/L) to be the nth Fourier sine coefficient of f (x) on [0, L]:
2 L nπξ
b n sinh(nπ K/L) = f (ξ)sin dξ.
L 0 L
Then
2 L nπξ
b n = f (ξ)sin dξ.
L sinh(nπ K/L) 0 L

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October 14, 2010 15:27 THM/NEIL Page-643 27410_18_ch18_p641-666