Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space
fractional Fokker-Planck equations with a nonlinear source term (TSFFPENST), which involve the Caputo time fractional derivative (CTFD) of order

The Fokker-Planck equation (FPE) has commonly been used to describe the Brownian motion of particles. Normal diffusion in an external force field is often modeled in terms of the following Fokker-Planck equation (FPE) [

TSFFPE has been successfully used for modeling relevant physical processes. When the fractional differential equation is used to describe the asymptotic behavior of continuous time random walks, its solution corresponds to the Lévy walks, generalizing the Brownian motion to the Lévy motion. The following space fractional Fokker-Planck equation has been considered [

As a model for subdiffusion in the presence of an external field, a time fractional extension of the FPE has been introduced as the time fractional Fokker-Planck equation (TFFPE) [

Yuste and Acedo [

Equation (

Schot et al. [

The fractional Fokker-Planck equations (FFPEs) have been recently treated by many authors and are presented as a useful approach for the description of transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. The analytical solution of FFPE is only possible in simple and special cases [

In this paper, we consider the following time-space fractional Fokker-Planck equation with a nonlinear source term (TSFFPE-NST):

Let

Let

The rest of this paper is organized as follows. In Section

In this section, we present an effective numerical method to simulate the solution behavior of the TSFFPE-NST (

Firstly, adopting the L1-algorithm [

For the symmetric Riesz space fractional derivative, we use the following shifted Grünwald approximation [

The first-order spatial derivative can be approximated by the backward difference scheme if

The nonlinear source term can be discretised either explicitly or implicitly. In this paper, we use an explicit method and evaluate the nonlinear source term at the previous time step:

Thus, using (

Let

If we use the implicit method to approximate the nonlinear source term, the numerical method of the TSFFPE-NST can be written as

The coefficients

when

The coefficients

In this section, we analyze the stability of the ENM (

Let

We suppose that

Assuming

Suppose that

When

Applying Theorem

Assuming that the nonlinear source term

If

In fact, for the case

If we use an implicit method to approximate the nonlinear source term, as shown in Remark

In this section, we analyze the convergence of the ENM (

Assuming that

Assuming the nonlinear source term

Assume

If we use an implicit method to approximate the nonlinear source term, as shown in Remark

In this section, we present four numerical examples of the TSFFPE to demonstrate the accuracy of our theoretical analysis. We also use our solution method to illustrate the changes in solution behavior that arise when the exponent is varied from integer order to fractional order and to identify the differences between solutions with and without the external force term.

Consider the following TSFFPE:

In this example, we take

Maximum error behavior versus grid size reduction for Example

Maximum error | |
---|---|

4.8148E-2 | |

1.0111E-2 | |

2.0587E-3 | |

7.3019E-4 |

Comparison of the numerical solution with the exact solution for Example

Consider the following TSFFPE-NST:

This example is a TSFFPE-NST without the external force term. In fact, it reduces to the fractional diffusion equation with an absorbent term. The formulae to approximate the absorbent term are presented in the appendix. Here, we take

Numerical solutions for Example

Numerical solutions for Example

Numerical solutions for Example

Consider the following TSFFPE-NST:

This example of the TSFFPE-NST incorporates the external force term with

We also see that the peak heights in Figures

Numerical solutions for Example

Numerical solutions for Example

Numerical solutions for Example

Consider the following TSFFPE-NST:

In this example, we take

Numerical solutions for Example

Numerical solutions for Example

In this paper, we have proposed an effective numerical method to solve the TSFFPE-NST and proved that the ENM is stable and convergent provided that the nonlinear source term satisfies the Lipschitz condition, the solution of the continuous problem satisfies the smooth solution condition, and

Let us start from (

Now setting

Applying the Mean Value Theorem (M.V.T) for integration yields

Also, we have

Now, substituting (

This research has been supported by a Ph.D. Fee Waiver Scholarship and a School of Mathematical Sciences Scholarship, QUT. The authors also wish to thank the referees for their constructive comments and suggestions.