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Dirichlet problem for a rectangle

Posted October 24th, 2010 by vams42
in

hello,

am unable to derive formulas for solving dirichlet problems of the type

$ \nabla^2 =0 , 0<x<1, 0<y<1 , u(x,0)=0, u(x,1)=sin \pi x, u(0,y)=0, u(1,y)=0 $

so could anyone help me out deriving this type of formulas

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reference

On November 7th, 2010 Structure says:

see Neumann problem
You get

$$u(x,y)=\frac{\sinh\pi y}{\sinh \pi}\sin\pi x$$

see Neumann problem

$$u(x,y)=\sum_{n=0}^{+\infty}X_n(x)Y_n(y)=\sum_{n=1}^{+\infty}(A_n \cosh (n\pi y)+B_n \sinh(n\pi y))\sin n\pi x$$

From $ u(x,0)=\sum_{n=1}^{+\infty}A_n\sin n\pi x=0 $ you get $ A_n=0 $ and from $ u(x,1)=\sum_{n=1}^{+\infty}B_n \sinh n\pi\sin n\pi x=\sin x $ all $ B_n=0 $ except $ B_1\sinh \pi=1 $

You get

$$u(x,y)=\frac{\sinh\pi y}{\sinh \pi}\sin\pi x$$

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