Robust Numerical Solution of the Transient Fokker-Planck Equation for Nonlinear Dynamical Systems

The finite element method is applied to the solution of the transient Fokker-Planck equation for several often cited nonlinear stochastic systems accurately giving, for the first time, the joint probability density function of the response for a given initial distribution. The method accommodates nonlinearity in both stiffness and damping as well as both additive and multiplicative excitation, although only the former is considered herein. Several systems are examined, including linear, Duffing, and Van der Pol oscillators, to illustrate the robustness and accuracy of the finite element method.

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