Analytical Solution of Space-Time Fractional Fokker-Planck Equation by Homotopy Perturbation Sumudu Transform Method

Ravi Shanker Dubey,Badr Saad T Alkahtani,Abdon Atangana

doi:10.1155/2015/780929

Abstract

An efficient approach based on homotopy perturbation method by using Sumudu transform is proposed to solve some linear and nonlinear space-time fractional Fokker-Planck equations (FPEs) in closed form. The space and time fractional derivatives are considered in Caputo sense. The homotopy perturbation Sumudu transform method (HPSTM) is a combined form of Sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. Some examples show that the HPSTM is an effective tool for solving many space time fractional partial differential equations.

Highlights

Fokker-Planck equation (FPE) was introduced by Adriaan Fokker and Max Planck to describe the time evolution of the probability density function of position and velocity of a particle, which is one of the classical widely used equations of statistical physics [1]
FPE arises in a number of different fields in natural sciences; Brownian motion [2] and the diffusion model of chemical reactions [3] are largely employed, in various generalized forms, in physics, chemistry, engineering, and biology [1]
The FPE arises in kinetic theory [4] where it describes the evolution of the one-particle distribution function of a dilute gas with long-range collisions, such as a Coulomb gas

Summary

Introduction

Fokker-Planck equation (FPE) was introduced by Adriaan Fokker and Max Planck to describe the time evolution of the probability density function of position and velocity of a particle, which is one of the classical widely used equations of statistical physics [1]. The FPE arises in kinetic theory [4] where it describes the evolution of the one-particle distribution function of a dilute gas with long-range collisions, such as a Coulomb gas Some applications of this type of equations can be worked out in the works of He and Wu [5], Jumarie [6], Kamitani and Matsuba [7], Xu et al [8], and Zak [9]. The Fokker-Planck equation with fractional space derivative is a particular case of anomalous diffusion and Levy flights (see[16,17,18,19]) This equation is called nonlinear FPE with space-time fractional derivatives [20]:. In the present paper we obtain closed form solutions of a linear-nonlinear time fractional FPE using homotopy perturbation Sumudu transform method (HPSTM); see [25]

Some Mathematical Preliminaries and Definitions

Solution of Fokker-Planck Fractional Differential Equation

Numerical Examples

Conclusion