%0 Journal Article
%T Interval Shannon Wavelet Collocation Method for Fractional Fokker-Planck Equation
%A Shu-Li Mei
%A De-Hai Zhu
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/821820
%X Metzler et al. introduced a fractional Fokker-Planck equation (FFPE) describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath. In this paper, we present an interval Shannon wavelet numerical method for the FFPE. In this method, a new concept named ˇ°dynamic interval waveletˇ± is proposed to solve the problem that the numerical solution of the fractional PDE is usually sensitive to boundary conditions. Comparing with the traditional wavelet defined in the interval, the Newton interpolator is employed instead of the Lagrange interpolation operator, so, the extrapolation points in the interval wavelet can be chosen dynamically to restrict the boundary effect without increase of the calculation amount. In order to avoid unlimited increasing of the extrapolation points, both the error tolerance and the condition number are taken as indicators for the dynamic choice of the extrapolation points. Then, combining with the finite difference technology, a new numerical method for the time fractional partial differential equation is constructed. A simple Fokker-Planck equation is taken as an example to illustrate the effectiveness by comparing with the Grunwald-Letnikov central difference approximation (GL-CDA). 1. Introduction Due to the fact that 1/ signal gains the increasing interests in the field of biomedical signal processing and engineering systems [1], the differential equations of fractional order appear more and more frequently in various research areas and engineering applications [2, 3]. As a matter of fact, the applications of fractional differential equations and their corresponding time series have been developed in various fields of sciences and technologies [4, 5] in recent years, ranging from computer science to physics [6, 7]. An effective and easy-to-use method for solving such equations is needed. However, known methods have certain disadvantages. Methods, described in detail in [3] for fractional differential equations of rational order, do not work in the case of arbitrary real order. On the other hand, there is an iteration method described in [8], which allows solution of fractional differential equations of arbitrary real order but it works effectively only for relatively simple equations, in addition to the series method. Up to now, most studies on the numerical methods for the fractional PDEs concentrate on the finite difference methods. Li [9] proposed an analytical method taking the fractal time series as the solution to a differential equation of
%U http://www.hindawi.com/journals/amp/2013/821820/