Cell Renormalized Fokker–Planck equation method (CR-FPK) for fractional order nonlinear system

Abstract

In this paper, the random response of nonlinear dynamic systems with fractional order term are examined. In this context, an efficient approach called CR-FPK method is proposed to numerically obtain transient stochastic response for those nonlinear systems under white-noise excitation in which damping is of fractional order. This approach is based on the extension of stochastic averaging method and the process of cell renormalization. By first revisit the process of equivalent linearization and stochastic averaging, the Markov property of envelope response is firstly settled. Then the idea of cell renormalization is introduced to reconstruct the derivative moments and the corresponding Fokker–Planck–Kolmogorov (FPK) equation of the envelope process. The transient and stationary probability density function (PDF) of the envelope process is finally obtained by solving the transient FPK equation. The reliability of this method is verified by using the benchmark results of analytical method and Monte Carlo Simulation. The conclusion and the issues to be further studied are presented in the last part.