On the Solution of High Dimensional Fokker Planck Equations using Orthogonal Polynomial Expansion

Solving the Fokker‐Planck‐Equation for multidimensional nonlinear systems is a great challenge in the field of stochastic dynamics. As for many mechanical systems a general idea about the shape of stationary solutions for the probability density function is known, it seems promising to use an approach that contains this knowledge. This is done using a Galerkin‐method which applies approximate solutions as weighting functions for the expansion of orthogonal polynomials, e.g. generalized Hermite polynomials [1]. As examples, nonlinear oscillators containing cubical restoring (Duffing oscillators) and cubical damping elements are considered. The method is applied to the two‐dimensional problem of a single‐degree‐of‐freedom oscillator and consecutively extended up to dimension ten. Results for probability density functions are presented and compared with results from Monte Carlo simulations. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)