Numerical Green's functions constructed via the eigenfunction expansion method are derived to solve for non-classical (Mindlin) plate problems. The associated eigenvalue problem enables the expression of the Green's function as a series of eigenfunctions which are approximated by a series of polynomials that satisfy the homogeneous boundary conditions to which the plates are subjected. In order to construct the approximate Green's function consisting of polynomials, a computer algebra system (Mathematica) has been extensively used reducing substantially the amount of algebra involved in the calculation. The results are favorably compared with values found in the literature.