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