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. – akkkk Nov 19 '12 at 19:37
Any other useful comment, akkk? – Mark_Hoffman Nov 19 '12 at 20:42
What happens when you try to use those methods? – Christopher A. Wong Nov 19 '12 at 22:06
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. – Mark_Hoffman Nov 19 '12 at 22:51
How about showing some work?? – user641 Dec 6 '12 at 21:59