I just want a specific example (preferably in 2-D) of a PDE (preferably steady-state) that I can test which can be solved using one of the Green's functions from HERE. I'm mostly asking as a sanity check, because whenever I try to set up my own problem and test the Green's functions from the table by computing the solution, they don't end up working for me, usually because the RHS of the PDE when plugged in does not allow the integral to converge.
A specific example would look something like: $$\Delta u(x,y) = -2\Big(x(1-x)+y(1-y)\Big); \ \ 0 < x,y < 1 \\ \cases{u(0,y) = 0, \\ u(1,y) = 0, \\ u(x,0) = 0 \\ u(x,1) = 0}$$
Any example will do, I just want one that can use Green's function from the table. Thank you!