This article addresses the computation of stationary and transient solutions of the Fokker-Planck equation for nonlinear stochastic processes. We extend a Galerkin-method, which was previously used to compute stationary solutions for nonlinear mechanical systems, by a generalized formulation of the test function space, including the original approach as a limit. The use of weighted test functions improves the performance in the stationary setting and enables the computation of transient solutions. The properties of the resulting linear system of equations are discussed, and results for stationary and transient probability density functions for nonlinear 1D-, 2D- and 4D-systems are presented.