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.

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 [

The Lie symmetry method was firstly advocated by the Norwegian mathematician Sophus Lie [

In this paper, we consider the following two-dimensional time fractional Fokker-Planck equation

The fractional derivatives described here are in the Riemann-Liouville sense of order

The rest of this paper is organized as follows: in Section

In this section, we briefly review the main points about Lie symmetry analysis of FPDEs [

The infinitesimal generator

The explicit form of the

The function

In the present section, the Lie symmetry analysis method has been applied for deriving the infinitesimal generators of the two-dimensional time fractional Fokker-Planck (

By substituting the expressions

where

For

By solving the above characteristic equation, we obtain the solution

Using the symmetry method, we obtain the following infinitesimals:

The characteristic equation for the infinitesimal generator

By solving the above characteristic equation, we obtain the solution

For

In this section, the conservation laws of the two-dimensional time fractional Fokker-Planck equation have been investigated by using a new conservation theorem [

The formal Lagrangian for Eq. (

where

The Euler–Lagrange operator is defined as

For the case of three independent variables

where

Using the Lie symmetries

Based on the fractional generalizations of the Noether operators, the components of conserved vectors can be presented as follows:

where

For

For

For

For

For

For

In this section, based on the power series method [

Substituting (

The power series solution for Eq. (

In this paper, the invariance properties of the two-dimensional time fractional Fokker-Planck equation with the Riemann-Liouville fractional derivative have been investigated in the sense of Lie point symmetries. Then, the power series method has been applied to get an explicit solution for the two-dimensional time fractional Fokker-Planck equation. For obtaining new components of conserved vectors, a new theorem of conservation law has been employed along with the formal Lagrangian, which allows us to construct conservation laws for the two-dimensional time fractional Fokker-Planck equation. Our results show that the extended Lie group analysis approach and the power series method provide powerful mathematical tools to investigate other FDEs in different fields of applied mathematics. In addition, it shows that the proposed analysis is very efficient to construct conservation laws of the two-dimensional time fractional Fokker-Planck equation. Moreover, we can employ symmetry analysis to the time-space fractional Fokker-Planck equation; it will be valuable as future subject works.

No data were used to support this study.

The authors declare that they have no conflicts of interest.