Computational Methods for Stationary and Nonstationary Fokker–Planck–Kolmogorov Equations with Random Coefficients

Abstract

In this paper, methodological approaches for distribution functions of nonlinear stochastic differential equations (SDEs) with uncertain parameters are elaborated. The stationary and nonstationary Fokker–Planck–Kolmogorov (FPK) equations associated with the considered random equations are investigated. A semi-analytical approach based on the extended exponential closure handling the parameters uncertainty effects is presented. A new numerical method coupling the polynomial chaos with the differential quadrature method (DQM) is elaborated for the stationary case. A compact matrix formulation is established allowing considering a large number of generalized polynomial chaos. For nonstationary random FPK equations, the polynomial chaos is coupled with the quadrature discretization. A symmetric discretization of the main Fokker–Planck operator using a tensor product of the Hilbert state space and the Hilbert probabilistic space has resulted. The obtained eigenvalues and random eigenfunctions are used for a spectral decomposition. The time solution is obtained by the spectral expansion as well as by the random Euler stochastic method. The accuracy, effectiveness and advantages of the developed procedure in analyzing the stationary and nonstationary probabilistic solutions of nonlinear stochastic dynamical systems with uncertain parameters excited by a Gaussian white noise are demonstrated.