Interval Wavelet Numerical Method on Fokker-Planck Equations for Nonlinear Random System

Li-Wei Liu

doi:10.1155/2013/651357

Abstract

The Fokker-Planck-Kolmogorov (FPK) equation governs the probability density function (p.d.f.) of the dynamic response of a particular class of linear or nonlinear system to random excitation. An interval wavelet numerical method (IWNM) for nonlinear random systems is proposed using interval Shannon-Gabor wavelet interpolation operator. An FPK equation for nonlinear oscillators and a time fractional Fokker-Planck equation are taken as examples to illustrate its effectiveness and efficiency. Compared with the common wavelet collocation methods, IWNM can decrease the boundary effect greatly. Compared with the finite difference method for the time fractional Fokker-Planck equation, IWNM can improve the calculation precision evidently.

Highlights

The Fokker-Planck equation describes the time evolution of the probability density function of the velocity of a particle, and can be generalized to other observables as well
The Galerkin method for the numerical solution of the stationary Fokker-Planck equation developed by Bhandari and Sherrer is based on taking multiple Hermite polynomial as joint probability density, but the rate of convergence of this method is slow for strong nonlinear system
Based on Galerkin method, a finite element method for FokkerPlanck equation was developed by Langley; this method is more efficient than Galerkin method in computation

Summary

Introduction

The Fokker-Planck equation describes the time evolution of the probability density function of the velocity of a particle, and can be generalized to other observables as well. Many methods for calculating nonlinear random response have been developed by numerous scholars over a long period of time One type of these methods is the diffuseness process theory method, and the primarily one is Fokker-Planck equation method. Compared with Shannon wavelet function, Shannon-Gabor wavelet [13, 14] possesses better compact support property, which has been widely employed in various mechanical analysis fields [15] including solving the FPK equation [16]. The purpose of this paper is to construct an interval ShannonGabor wavelet collocation method on solving Fokker-Planck equation for nonlinear oscillators and a time fractional Fokker-Planck equation describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath [28]

Shannon-Gabor Wavelet Collocation Method

Numerical Experiments

Conclusions