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. In contrast to the usual approach of directly solving the backward Kolmogorov equation, when appropriate boundary conditions are prescribed, the probability density function associated with the first passage problem can be directly obtained. Standard numerical methods are employed, and results are shown to be highly accurate. Several systems are examined, including linear, Duffing, and Van der Pol oscillators.

On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems

Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation

Efficient path integration methods for nonlinear dynamic systems

The method of stochastic averaging, as applied previously to analyze the energy envelope of a nonlinear system under random excitations of independent ideal Gaussian white noises, is generalized to...

Stochastic Averaging of Energy Envelope

A method for the evaluation of the stationary and non-stationary probability density function of non-linear oscillators subjected to random input is presented. The method requires the approximation of the probability density function of the response in terms of C-type Gram-Charlier series expansion. By applying the weighted residual method, the Fokker-Planck equation is reduced to a system of non-linear first order ordinary differential equations, where the unknowns are the coefficients of the series expansion. Furthermore, the relationships between the A-type and C-type Gram-Charlier series coefficient are derived.

Y. Yong

Papers

Exact Stationary-Response Solution for Second Order Nonlinear Systems Under Parametric and External White-Noise Excitations