laplace's equation - disc

leanne486

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Apr 25, 2006
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What’s the best way to solve this problem?

Solve Laplace's Eqn.

(1/р)(∂u/∂Ф) + (1/р)(∂u/∂р) + (∂u/∂р) = 0

in the semidisc {( р , Ф ): 0<= р <1; -pi < Ф <pi}, w/ boundary conds.

U(1, Ф) = cos Ф.



So far I have

U(р , Ф)= A0 + Σ(n=1, ∞) (р^n)(An cos (n Ф) + Bn sin (n Ф))

Therefore:

U(1, Ф)= A0 + Σ(n=1, ∞) (An cos (n Ф) + Bn sin (n Ф)) Ξ cos Ф.

cos Ф = (3cos(Ф) + cos(3 Ф)) / 4.


I think I need to find the fourier coefficients next but am not sure how to. Thanks.
 
From the boundary condition and the trig identity, it appears that only 2 of the Fourier coefficients are not zero.
 
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