In a domain $\Omega=[-1,1]\times[-1,1]$, consider the Poisson equation \begin{align} -\Delta u=\sin(\pi x)\sin(\pi y) \end{align} inside $\Omega$, and $u(x,y)=0$ for $(x,y)$ in the boundary of $\Omega$.
By inspection i know that \begin{align} u(x,y)=\frac{\sin(\pi x)\sin(\pi y)}{-2\pi^2} \end{align} solves the Poisson equation boundary value problem.
I tried to obtain that solution convolving the source term of Poisson equation with the fundamental solution associated, in this part i was not able to compute a analytical expression for this convolution.