Toward computational algorithm for time-fractional Fokker–Planck models

Asad Freihet,Rokiah Rozita Ahmad,Shatha Hasan,Mohammad Alaroud,Mohammed Al-Smadi,Ummul Khair Salma Din

doi:10.1177/1687814019881039

Abstract

This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.

Highlights

Recent decades have witnessed great attention toward fractional calculus, which can be considered as a generalization of classical integer-order integration and differentiation
We present the Caputo fractional derivative[3] of order b, which is an alternative operator to the Riemann–Liouville fractional operator as follows
To see the effect of the fractional derivative to Fokker–Planck equation, the tabulated and graphical results for the approximate solutions at different values of fractional order b by using the residual power series method (RPSM) and ADM13 with k = 10 are given in Table 4 and Figure 4

Summary

Introduction

Recent decades have witnessed great attention toward fractional calculus, which can be considered as a generalization of classical integer-order integration and differentiation. Keywords Caputo fractional derivatives, Fokker–Planck equation, residual power series, numerical algorithms The concern of this analysis is to consider the numerical approximate solutions of the Fokker–Planck PDE with time-fractional derivative of the following form

Results

Conclusion