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Given this Dirichlet problem:

$$\begin{cases} \dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}= -(\cos(x+y)+\cos(x-y)) \\ u(0,y)=\cos(y),\;\; u(\pi,y)=-\cos(y),\;\;u(x,0)=\cos(x),\;\;u\left(x,\frac{\pi}{2}\right)=0, \end{cases}$$ can we apply Fourier's theory or eigenfunctions of Laplacian to express the exact solution?

(I think the answer is no, but I'm not sure why).

Thanks a lot.

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    I think the answer is yes, but I'm not sure why.2012-11-19
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    Any other useful comment, akkk?2012-11-19
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    What happens when you try to use those methods?2012-11-19
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    I asked my question politely.akkkk answer was not. I came here searching ideas and opinions about my problem. Not to start a discussion. Christopher, the question is if they can be used, or if there's any problem doing so. I think it involves the boundary conditions.2012-11-19
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    How about showing some work??2012-12-06

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