%0 Journal Article
%T Interval Wavelet Numerical Method on Fokker-Planck Equations for Nonlinear Random System
%A Li-wei Liu
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/651357
%X 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. 1. 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. It is named after Adriaan Fokker and Max Planck and is also known as the Kolmogorov forward equation (diffusion), named after Andrey Kolmogorov, who first introduced it in a 1931 paper. When applied to particle position distributions, it is better known as the Smoluchowski equation. The case with zero diffusion is known in statistical mechanics as Liouville equation. In order to describe subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath, Metzler et al. introduced a fractional Fokker-Planck equation (FFPE) which was shown to obey generalized Einstein relations, and its stationary solution is the Boltzmann distribution [1]. 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. In practice, the most difficult problem of using Fokker-Planck equation method is how to solve the equation. For general nonlinear system, it is very difficult to obtain the exact solution. Various numerical methods have been used to solve the Fokker-Planck equation, such as the weighted residual method [2], the finite element method [3], and the path integral method [4] and so on. 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
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