PDE solver ? Which method with FEM on operator matrix?

Hi, I have transformed a recent PDE that was under evaluation here on the forum for a solver, and received some feedback. Some suggest that :

" now you can discretize the operator ussing FEM with Newman boundary condition (no bundary condition at all) to obtain a singular matrix. Then use matlab eigs function to find the eigen vectors asociated to eigen value zero. And you are done, because FEM interpolates the solution, you vector solution (reordered) will be an aproximation of the solution you are looking for :-)"

which is applied on this form

Can that be done in Matlab? If so which steps and solvers should I use?