Efficient Numerical Solution of Multidimensional Fokker-Planck Equations Associated With Chaotic and Nonlinear Random Vibrations

Abstract

In the presented paper the response of nonlinear random oscillators is calculated by the numerical solution of the associated Fokker-Planck-Equation. This approach can be used to approximate the invariant probability density of chaotic nonlinear oscillators or to calculate the response of a nonlinear oscillator, driven by additive white noise. A weighted residual approach is proposed to solve the Fokker-Planck-Equations numerically. By this method the partial differential equation is transformed to a coupled set of linear ordinary differential equations. These equations can be derived easily, because the integrals associated to the Galerkin procedure can be solved analytically. Considering stationary densities the solution is given by a generalized Eigenproblem. Utilizing sparse matrix techniques the solution can be calculated very efficiently.