1. Introduction

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) [1]: where is the mass of the diffusing test particle, denotes the fraction constant characterising the interaction between the test particle and its embedding, and the force is related to the external potential through . The FPE (1.1) is well studied for a variety of potential types, and the respective results have found wide application. In many studies of diffusion processes where the diffusion takes place in a highly nonhomogeneous medium, the traditional FPE may not be adequate [2, 3]. The nonhomogeneities of the medium may alter the laws of Markov diffusion in a fundamental way. In particular, the corresponding probability density of the concentration field may have a heavier tail than the Gaussian density, and its correlation function may decay to zero at a much slower rate than the usual exponential rate of Markov diffusion, resulting in long-range dependence. This phenomenon is known as anomalous diffusion [4]. Fractional derivatives play a key role in modeling particle transport in anomalous diffusion including the space fractional Fokker-Planck (advection-dispersion) equation describing Lévy flights, the time fractional Fokker-Planck equation depicting traps, and the time-space fractional equation characterizing the competition between Lévy flights and traps [5, 6]. Different assumptions on this probability density function lead to a variety of time-space fractional Fokker-Planck equations (TSFFPEs).

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 [2, 3, 7]: where is the drift of the process, that is, the mean advective velocity; is the coefficient of dispersion; and are the left and right Riemann-Liouville space fractional derivatives of order given by and indicate the relative weight of transition probability; Benson et al. [2, 3] took and , , which indicate the relative weight forward versus backward transition probability. If , (1.2) can be rewritten in the following form: where is the symmetric space fractional derivative of order (). This is also referred to as the Riesz derivative [8], which contains a left Riemann-Liouville derivative () and a right Riemann-Liouville derivative (), namely,

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) [5, 9]: where the Riemann-Liouville operator , is defined through its operation:

Yuste and Acedo [10] proposed an explicit finite difference method and a new von Neumann-type stability analysis for the anomalous subdiffusion equation (1.6) with . However, they did not give a convergence analysis and pointed out the difficulty of this task when implicit methods are considered. Langlands and Henry [11] also investigated this problem and proposed an implicit numerical L1-approximation scheme and discussed the accuracy and stability of this scheme. However, the global accuracy of the implicit numerical scheme has not been derived and it seems that the unconditional stability for all in the range has not been established. Recently, Chen et al. [12] presented a Fourier method for the anomalous subdiffusion equation, and they gave the stability analysis and the global accuracy analysis of the difference approximation scheme. Zhuang et al. [13] also proposed an implicit numerical method and an analytical technique for the anomalous subdiffusion equation. Chen et al. [14] proposed implicit and explicit numerical approximation schemes for the Stokes' first problem for a heated generalized second grade fluid with fractional derivatives. The stability and convergence of the numerical scheme are discussed using a Fourier method. A Richardson extrapolation technique for improving the order of convergence of the implicit scheme is presented. However, effective numerical methods and error analysis for the time-space fractional Fokker-Planck equation with a nonlinear source term are still in their infancy and are open problems.

Equation (1.6) can be written as the following equivalent form [15]: By noting that [15] we arrive at where is the Caputo time fractional derivative (CTFD) of order () with starting point at defined by [16] The time-space fractional Fokker-Plank equation (TSFFPE), which describes the competition between subdiffusion and Lévy flights, is given by [5] or where denotes the anomalous diffusion coefficient.

Schot et al. [17] investigated a fractional diffusion equation that employs time and space fractional derivatives by taking an absorbent (or source) term and an external force into account, which can be described by the following time-space fractional Fokker-Plank equation with an absorbent term and a linear external force: where is the external force and is a time-dependent absorbent term, which may be related to a reaction diffusion process.

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 [2, 3, 18] and the analytical solution provides a general representation in terms of Green's functions. We note that the representation of Green's functions is mostly expressed as convergent expansions in negative and positive power series. These special functions are not suitable for numerical evaluation when is sufficiently small or sufficiently large. Therefore, a new numerical strategy is important for solving these equations. Although numerical methods for the time fractional Fokker-Planck type equation, the space fractional Fokker-Plank type equation, and the time-space fractional Fokker-Planck type equation have been considered [7, 15, 19], numerical methods and stability and convergence analysis for the FFPE are quite limited and difficult. In fact, published papers on the numerical methods for the FFPE are sparse. We are unaware of any other published work on numerical methods for the time-space fractional Fokker-Planck type equation with a nonlinear source term. This motivates us to consider an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term and to investigate its stability and convergence.

In this paper, we consider the following time-space fractional Fokker-Planck equation with a nonlinear source term (TSFFPE-NST): subject to the boundary and initial conditions: where is known as the drift coefficient. The nonlinear source (or absorbent) term is assumed to satisfy the Lipschitz condition:

Let be a Banach space with associated norm . We say that is globally Lipschitz continuous if for some , we have for all , and is locally Lipschitz continuous, if the latter holds for with for any [20].

Let . In this paper, we suppose that the continuous problem (1.15)-(1.16) has a smooth solution .

The rest of this paper is organized as follows. In Section 2, the Caputo time fractional derivative (CTFD) and the Riesz space fractional derivative (RSFD) are approximated by the L1-algorithm and the shifted Grünwald method, respectively. An effective numerical method (ENM) for solving the TSFFPE-NST (1.15)-(1.16) is proposed. The stability and convergence of the ENM are discussed in Sections 3 and 4, respectively. In Section 5, numerical experiments are carried out to support the theoretical analysis. Finally, some conclusions are drawn in Section 6.

2. An Effective Numerical Method for the TSFFPE-NST

In this section, we present an effective numerical method to simulate the solution behavior of the TSFFPE-NST (1.15)-(1.16). Let () and (), where and are the spatial and temporal steps, respectively.

Firstly, adopting the L1-algorithm [21], we discretize the Caputo time fractional derivative as where , .

For the symmetric Riesz space fractional derivative, we use the following shifted Grünwald approximation [22]: where the coefficients are defined by This formula is not unique because there are many different valid choices for that lead to approximations of different orders [23]. The definition (2.2) provides order .

The first-order spatial derivative can be approximated by the backward difference scheme if , (otherwise, the forward difference scheme can be used 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: In this way, we avoid solving a nonlinear system at each time step and obtain an unconditionally stable and convergent numerical scheme, as shown in Section 3. However, the shortcoming of the explicit method is that it generates additional temporal error, as shown in (2.5).

Thus, using (2.1)–(2.5), we have After some manipulation, (2.6) can be written in the following form: where , , and

Let be the numerical approximation of , and let be the numerical approximation of . We obtain the following effective numerical method (ENM) of the TSFFPE-NST (1.15)-(1.16): for , . The boundary and initial conditions can be discretised using

Remark 2.1. If we use the implicit method to approximate the nonlinear source term, the numerical method of the TSFFPE-NST can be written as that is, replace in (2.9) with . This numerical method is stable and convergent when the source term satisfies the Lipschitz condition (1.17) (see, e.g., [24]).

Lemma 2.2 (see [19]). The coefficients satisfy (1) for ; (2), as ; (3)when , Thus, there is a positive constant such that

Lemma 2.3 (see [25]). The coefficients satisfy (1), , and for ; (2), and for .

3. Stability of the Effective Numerical Method

In this section, we analyze the stability of the ENM (2.9)-(2.10). Firstly, we rewrite (2.9) in the following form:

Let be the approximate solution of the ENM (3.1), and let be the approximation of . Setting , we obtain the following roundoff error equation: for ; .

We suppose that and that decreases monotonically on . This is based on the fact that physical considerations and stability dictate that [26, 27].

Assuming , and using mathematical induction, we obtain the following theorem.

Theorem 3.1. Suppose that (, ) is the solution of the roundoff error equation (3.2), and the nonlinear source term satisfies the Lipschitz condition (1.17), then there is a positive constant , such that

Proof. When , assume that . Because and decreases monotonically on , we have Using the properties of in Lemma 2.3, we have Combining (3.4) with (3.5), using the Lipschitz condition (1.17) and smooth solution condition, we obtain Let . Thus, we obtain Now, suppose that By assuming , we have that Using (3.7) and (3.8), we have Let . Hence, we have The proof of Theorem 3.1 is completed.

Applying Theorem 3.1, the following theorem of stability is obtained.

Theorem 3.2. Assuming that the nonlinear source term satisfies the Lipschitz condition (1.17) and that the drift coefficient decreases monotonically on , the ENM defined by (2.9)-(2.10) is stable.

Remark 3.3. If and decreases monotonically on , we can use the forward difference method to approximate the first-order spatial derivative and apply a similar analysis of stability.

Remark 3.4. In fact, for the case does not decrease monotonically, we can still obtain a stable numerical scheme by a minor change in our current ENM. We can expand the first term on the RHS of (1.15) as , which enables us to group together with the nonlinear source term to obtain a new nonlinear source term . This way we can weaken the assumption on and the analysis given in this section still can be used.

Remark 3.5. If we use an implicit method to approximate the nonlinear source term, as shown in Remark 2.1, we can prove that the numerical method defined in (2.11) is stable when , which is independent of the spatial step. In fact, when the time step is small, the condition is generally satisfied.

4. Convergence of the Effective Numerical Method

In this section, we analyze the convergence of the ENM (2.9)-(2.10). Let be the exact solution of the TSFFPE-NST (1.15)-(1.16) at mesh point , and let be the numerical solution of the TSFFPE-NST (1.15)-(1.16) computed using the ENM (2.9)-(2.10). Define and . Subtracting (2.9) from (2.7) leads to where ; .

Assuming that and using mathematical induction, we obtain the following theorem.

Theorem 4.1. Assuming the nonlinear source term satisfies the Lipschitz condition (1.17), and the drift coefficient decreases monotonically on , the ENM defined by (2.9)-(2.10) is convergent, and there exists a positive constant , such that

Proof. Assume . Following a similar argument to that presented above for the stability analysis of the ENM (2.9)-(2.10), when , assuming that , we have Utilising , the Lipschitz condition (1.17), and smooth solution condition, we obtain Thus, Now, suppose that Using Lemma 2.2, , we have Similarly, assuming , we have Utilising , the Lipschitz condition (1.17), and smooth solution condition, we obtain Hence, Finally, utilising (2.8) and Lemma 2.2, , we obtain the result on the convergence of the ENM (2.9)-(2.10), namely, for .