  We present a comprehensive study for common second order PDE's in two
dimensional disk-like systems and show how their solution can be approximated
by finding the Green function of an effective one dimensional system. After
elaborating on the formalism, we propose to secure an exact solution via a
Fourier expansion of the Green function, which entails to solve an infinitely
countable system of differential equations for the Green-Fourier modes that in
the simplest case yields the source-free Green distribution. We present results
on non separable systems$-$or such whose solution cannot be obtained by the
usual variable separation technique$-$on both annulus and disc geometries, and
show how the resulting one dimensional Fourier modes potentially generate a
near-exact solution. Numerical solutions will be obtained via finite
differentiation using FDM or FEM with the three-point stencil approximation to
derivatives. Comparing to known exact solutions, our results achieve an
estimated numerical relative error below $10^{-6}$.
