Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

Nisrine Maarouf,Khalid Hilal,Andrea Scapellato

doi:10.1155/2021/2490392

Abstract

The main purpose of this paper is to apply the Lie symmetry analysis method for the two-dimensional time fractional Fokker-Planck (FP) equation in the sense of Riemann–Liouville fractional derivative. The Lie point symmetries are derived to obtain the similarity reductions and explicit solutions of the governing equation. By using the new conservation theorem, the new conserved vectors for the two-dimensional time fractional Fokker-Planck equation have been constructed with a detailed derivation. Finally, we obtain its explicit analytic solutions with the aid of the power series expansion method.

Highlights

Fractional calculus has attracted more attention of many researches in various scientific areas including biology, physics, financial theory, gas dynamics, engineering, fluid mechanics, and other areas of science, see for example [1,2,3,4,5,6,7,8]
The new conservation laws were introduced by Ibragimov [28], based on the notion of Lie symmetry generators without Lagrangian for solving fractional partial differential equations (FPDEs)
The rest of this paper is organized as follows: in Section 2, we review some basic definitions of the Lie Symmetry method for fractional partial differential equations (FPDEs) and its properties

Summary

Introduction

Fractional calculus has attracted more attention of many researches in various scientific areas including biology, physics, financial theory, gas dynamics, engineering, fluid mechanics, and other areas of science, see for example [1,2,3,4,5,6,7,8]. The Lie symmetry method was firstly advocated by the Norwegian mathematician Sophus Lie [21, 22], who has made great achievements in the theories of continuous groups and differential equations. It is an efficient approach and widely employed for solving ordinary differential equations (ODEs), partial differential equations (PDEs), and fractional partial differential equations (FPDEs). This popularity is due to its utility in determining the explicit solutions of both ODEs and PDEs, linearization of some nonlinear equations, reducing the order of independent variables, and so on.

Objectives

Methods

Conclusion