The power series expansion formalism is used to construct analytical approximations for the propagator of the partial differential equation of a generic type. The present approach is limited to systems with polynomial coefficients. Three typical two-dimensional examples, a Henon-Heiles anharmonic resonating system, a system-bath Hamiltonian, and a Fokker-Planck chaotic model are considered. All results are in excellent agreement with those of an established numerical scheme in the field. It is found that the power series expansion method accurately describes the dynamics of very anharmonic processes in the whole time domain.