Numerical Solutions of Fractional Fokker-Planck Equations Using Iterative Laplace Transform Method

Limei Yan

doi:10.1155/2013/465160

Abstract

A relatively new iterative Laplace transform method, which combines two methods; the iterative method and the Laplace transform method, is applied to obtain the numerical solutions of fractional Fokker-Planck equations. The method gives numerical solutions in the form of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and straightforward when applied to space-time fractional Fokker-Planck equations. The method provides a promising tool for solving space-time fractional partial differential equations.

Highlights

Fractional calculus has attracted much attention for its potential applications in various scientific fields such as fluid mechanics, biology, viscoelasticity, engineering, and other areas of science [1,2,3,4]
A great deal of effort has been spent on constructing of the numerical solutions and many effective methods have been developed such as fractional wavelet method [5,6,7,8], fractional differential transform method [9], fractional operational matrix method [10, 11], fractional improved homotopy perturbation method [12, 13], fractional variational iteration method [14, 15], and fractional Laplace Adomian decomposition method [16, 17]
Jafari et al firstly applied Laplace transform in the iterative method and proposed a new direct method called iterative Laplace transform method [22] to search for numerical solutions of a system of fractional partial differential equations

Summary

Introduction

Fractional calculus has attracted much attention for its potential applications in various scientific fields such as fluid mechanics, biology, viscoelasticity, engineering, and other areas of science [1,2,3,4]. Jafari et al firstly applied Laplace transform in the iterative method and proposed a new direct method called iterative Laplace transform method [22] to search for numerical solutions of a system of fractional partial differential equations. The method is based on Laplace transform, iterative method, Caputo fractional derivative, and symbolic computation By using this method, Jafari and Seifi successfully obtained the numerical solutions of two systems of space-time fractional differential equations. We will use the iterative Laplace transform method to solve space-time fractional FokkerPlanck equations. Fokker-Planck equation has been applied in various natural science fields such as quantum optics, solid-state physics, chemical physics, theoretical biology, and circuit theory.

Preliminaries

The Iterative Laplace Transform Method

Numerical Solutions of Fractional Fokker-Planck Equations

Conclusions