Stability and Convergence of an Effective Numerical Method for the Time-Space Fractional Fokker-Planck Equation with a Nonlinear Source Term

Abstract

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 𝛼∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order 𝜇∈ (1, 2]. Approximating the CTFD and RSFD using the L1-algorithm and shifted Grünwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

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]: 𝜕𝑢(𝑥,𝑡)𝜕𝑡=𝜕𝜕𝑥𝑉^(𝑥)𝑚𝜂₁+𝐾₁𝜕²𝜕𝑥²𝑢(𝑥,𝑡),(1.1) where 𝑚 is the mass of the diffusing test particle, 𝜂₁ denotes the fraction constant characterising the interaction between the test particle and its embedding, and the force is related to the external potential through 𝐹(𝑥)=𝑑𝑉(𝑥)/𝑑𝑥. 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]: 𝜕𝑢(𝑥,𝑡)𝜕𝑡=−𝑣𝜕𝑢𝜕(𝑥𝑥,𝑡)+𝐾^𝜇𝑐_+𝑎𝐷^𝜇𝑥𝑢(𝑥,𝑡)+𝑐_−𝑥𝐷^𝜇𝑏𝑢(𝑥,𝑡),(1.2) where 𝑣 is the drift of the process, that is, the mean advective velocity; 𝐾_𝜇 is the coefficient of dispersion; _𝑎𝐷^𝜇𝑥 and _𝑥𝐷^𝜇𝑏 are the left and right Riemann-Liouville space fractional derivatives of order 𝜇 given by _𝑎𝐷^𝜇𝑥𝑢(𝑥,𝑡)=1Γ(2−𝜇)𝜕𝜕𝑥²²^𝑥𝑎𝑢(𝜉,𝑡)𝑑𝜉(𝑥−𝜉)^𝜇−1_𝑥,𝐷^𝜇𝑏𝑢(𝑥,𝑡)=1Γ(2−𝜇)𝜕𝜕𝑥²²^𝑏𝑥𝑢(𝜉,𝑡)𝑑𝜉(𝜉−𝑥)^𝜇−1;(1.3)𝑐₊ and 𝑐₋ indicate the relative weight of transition probability; Benson et al. [2, 3] took 𝑐₊=1/2+𝛽/2 and 𝑐₋=1/2−𝛽/2, (−1≤𝛽≤1), which indicate the relative weight forward versus backward transition probability. If 𝑐₊=𝑐₋=−𝑐_𝜇=−1/2cos(𝜋𝜇/2), (1.2) can be rewritten in the following form: 𝜕𝑢(𝑥,𝑡)𝜕𝑡=−𝑣𝜕𝑢𝜕(𝑥𝑥,𝑡)+𝐾^𝜇𝜕^𝜇𝑢(𝑥,𝑡)𝜕|𝑥|^𝜇,(1.4) where 𝜕^𝜇/𝜕|𝑥|^𝜇 is the symmetric space fractional derivative of order 𝜇 (1<𝜇≤2). This is also referred to as the Riesz derivative [8], which contains a left Riemann-Liouville derivative (_𝑎𝐷^𝜇𝑥) and a right Riemann-Liouville derivative (_𝑥𝐷^𝜇𝑏), namely, 𝜕^𝜇𝜕|𝑥|^𝜇𝑢(𝑥,𝑡)=−𝑐_𝜇_𝑎𝐷^𝜇𝑥+_𝑥𝐷^𝜇𝑏𝑢(𝑥,𝑡).(1.5)

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]: 𝜕𝑢(𝑥,𝑡)𝜕𝑡=⁰𝐷^𝑡1−𝛼𝜕𝜕𝑥𝑉^(𝑥)𝑚𝜂_𝛼+𝐾_𝛼𝜕²𝜕𝑥²𝑢(𝑥,𝑡),(1.6) where the Riemann-Liouville operator ₀𝐷^𝑡1−𝛼, (0<𝛼<1) is defined through its operation: ₀𝐷^𝑡1−𝛼𝑢(𝑥,𝑡)=1Γ(𝛼)𝜕𝜕𝑡^𝑡0𝑢(𝑥,𝜂)(𝑡−𝜂)^1−𝛼𝑑𝜂.(1.7)

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 𝑉^(𝑥)=0. 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 𝛼 in the range 0<𝛼≤1 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]: ₀𝐷^𝛼𝑡𝑢(𝑥,𝑡)−𝑢(𝑥,0)𝑡^−𝛼Γ(1−𝛼)=𝜕𝜕𝑥𝑉^(𝑥)𝑚𝜂_𝛼+𝐾_𝛼𝜕²𝜕𝑥²𝑢(𝑥,𝑡).(1.8) By noting that [15] 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=₀𝐷^𝛼𝑡𝑢(𝑥,𝑡)−𝑢(𝑥,0)𝑡^−𝛼Γ(1−𝛼),(1.9) we arrive at 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=𝜕𝜕𝑥𝑉^(𝑥)𝑚𝜂_𝛼+𝐾_𝛼𝜕²𝜕𝑥²𝑢(𝑥,𝑡),(1.10) where 𝜕^𝛼𝑢(𝑥,𝑡)/𝜕𝑡^𝛼 is the Caputo time fractional derivative (CTFD) of order 𝛼 (0<𝛼<1) with starting point at 𝑡=0 defined by [16] 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=1Γ(1−𝛼)^𝑡0𝜕𝑢(𝑥,𝜂)𝜕𝜂(𝑡−𝑑𝜂𝜂)^𝛼.(1.11) The time-space fractional Fokker-Plank equation (TSFFPE), which describes the competition between subdiffusion and Lévy flights, is given by [5] 𝜕𝑢(𝑥,𝑡)𝜕𝑡=⁰𝐷^𝑡1−𝛼𝜕𝜕𝑥𝑉^(𝑥)𝑚𝜂_𝛼+𝐾^𝜇𝛼𝜕^𝜇𝜕|𝑥|^𝜇𝑢(𝑥,𝑡),(1.12) or 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=𝜕𝜕𝑥𝑉^(𝑥)𝑚𝜂_𝛼+𝐾^𝜇𝛼𝜕^𝜇𝜕|𝑥|^𝜇𝑢(𝑥,𝑡),(1.13) where 𝐾^𝜇𝛼 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: 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=−𝜕𝜕𝑥[𝐹(𝑥)𝑢(𝑥,𝑡)]+𝐾^𝜇𝛼𝜕^𝜇𝜕|𝑥|^𝜇𝑢(𝑥,𝑡)−^𝑡0𝑟(𝑡−𝜂)𝑢(𝑥,𝜂)𝑑𝜂,(1.14) where 𝐹(𝑥) is the external force and 𝑟(𝑡) 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 𝑥 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): 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=𝜕𝜕𝑥𝑉^(𝑥)𝑚𝜂_𝛼+𝐾^𝜇𝛼𝜕^𝜇𝜕|𝑥|^𝜇𝑢(𝑥,𝑡)+𝑠(𝑢,𝑥,𝑡)(1.15) subject to the boundary and initial conditions: 𝑢(𝑢𝑎(,𝑥𝑡),0=)=𝑢(𝑢𝑏₀,𝑡)=0,0≤𝑡≤𝑇,(𝑥),𝑎≤𝑥≤𝑏,(1.16) where 𝑝(𝑥)=𝑉^(𝑥)/𝑚𝜂_𝛼 is known as the drift coefficient. The nonlinear source (or absorbent) term 𝑠(𝑢,𝑥,𝑡) is assumed to satisfy the Lipschitz condition: ‖𝑠(𝑢,𝑥,𝑡)−𝑠(𝑣,𝑥,𝑡)‖≤𝐿‖𝑢−𝑣‖.(1.17)

Let 𝑋 be a Banach space with associated norm ‖𝑢‖. We say that 𝑠∶𝑋→𝑋 is globally Lipschitz continuous if for some 𝐿>0, we have ‖𝑠(𝑢)−𝑠(𝑣)‖≤𝐿‖𝑢−𝑣‖ for all 𝑢,𝑣∈𝑋, and is locally Lipschitz continuous, if the latter holds for ‖𝑢‖,‖𝑣‖≤𝑀 with 𝐿=𝐿(𝑀) for any 𝑀>0 [20].

Let Ω=[𝑎,𝑏]×[0,𝑇]. In this paper, we suppose that the continuous problem (1.15)-(1.16) has a smooth solution 𝑢(𝑥,𝑡)∈𝐶^{1+𝜇,2𝑥,𝑡}(Ω).

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 𝑥_𝑙=𝑙ℎ (𝑙=0,1,…,𝑀) and 𝑡_𝑛=𝑛𝜏 (𝑛=0,1,…,𝑁), where ℎ=(𝑏−𝑎)/𝑀 and 𝜏=𝑇/𝑁 are the spatial and temporal steps, respectively.

Firstly, adopting the L1-algorithm [21], we discretize the Caputo time fractional derivative as 𝜕^𝛼𝑢𝑥,𝑡_𝑛+1𝜕𝑡^𝛼=𝜏^−𝛼Γ(2−𝛼)^𝑛_𝑗=0𝑏_𝑗𝑢𝑥,𝑡_{𝑛+1−𝑗}−𝑢𝑥,𝑡_𝑛−𝑗+𝑂𝜏^1+𝛼,(2.1) where 𝑏_𝑗=(𝑗+1)^1−𝛼−𝑗^1−𝛼, 𝑗=0,1,2,…,𝑁−1.

For the symmetric Riesz space fractional derivative, we use the following shifted Grünwald approximation [22]: 𝜕^𝜇𝑢𝑥_𝑙,𝑡𝜕|𝑥|^𝜇=−ℎ^−𝜇2cos(𝜋𝜇/2)^𝑙+1_𝑖=0𝑤_𝑖𝑢𝑥_{𝑙−𝑖+1},𝑡+^{𝑀−𝑙+1}_𝑖=0𝑤_𝑖𝑢𝑥_{𝑙+𝑖−1},𝑡+𝑂ℎ^𝑞,(2.2) where the coefficients are defined by 𝑤₀=1,𝑤_𝑖=(−1)^𝑖𝜇(𝜇−1)⋯(𝜇−𝑖+1)𝑖!,𝑖=1,2,…,𝑀.(2.3) This formula is not unique because there are many different valid choices for 𝑤_𝑖 that lead to approximations of different orders 𝑞 [23]. The definition (2.2) provides order 𝑞=1.

The first-order spatial derivative can be approximated by the backward difference scheme if 𝑝(𝑥)<0, (otherwise, the forward difference scheme can be used if 𝑝(𝑥)>0): 𝜕𝜕𝑥𝑝𝑥^𝑙𝑢𝑥_𝑙,𝑡=𝑝𝑥^𝑙𝑢𝑥_𝑙,𝑡−𝑝𝑥_𝑙−1𝑢𝑥_𝑙−1,𝑡ℎ+𝑂(ℎ).(2.4)

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: 𝑠𝑢𝑥,𝑡_𝑛+1,𝑥,𝑡_𝑛+1=𝑠𝑢𝑥,𝑡_𝑛,𝑥,𝑡_𝑛+𝑂(𝜏).(2.5) 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 𝜏^−𝛼Γ(2−𝛼)^𝑛_𝑗=0=𝑝𝑥𝑏_𝑗𝑢𝑥_𝑙,𝑡_{𝑛+1−𝑗}−𝑢𝑥_𝑙,𝑡_𝑛−𝑗_𝑙𝑢𝑥_𝑙,𝑡_𝑛+1−𝑝𝑥_𝑙−1𝑢𝑥_𝑙−1,𝑡_𝑛+1−𝐾_𝜇𝛼ℎℎ^−𝜇2cos(𝜋𝜇/2)^𝑙+1_𝑖=0𝑤_𝑖𝑢𝑥_{𝑙−𝑖+1},𝑡_𝑛+1+^{𝑀−𝑙+1}_𝑖=0+𝑠𝑢𝑥_𝑙𝑤_𝑖𝑢𝑥_{𝑙+𝑖−1},𝑡_𝑛+1,𝑡_𝑛,𝑥_𝑙,𝑡_𝑛+𝑂𝜏^1+𝛼+ℎ+𝜏.(2.6) After some manipulation, (2.6) can be written in the following form: 𝑢𝑥_𝑙,𝑡_𝑛+1=𝑏_𝑛𝑢𝑥_𝑙,𝑡₀+^𝑛−1_𝑗=0𝑏_𝑗+𝜇₀−𝑏_𝑗+1𝑢𝑥_𝑙,𝑡_𝑛−𝑗ℎ𝑝𝑥^𝑙𝑢𝑥_𝑙−𝜇₀,𝑡_𝑛+1−𝑝𝑥_𝑙−1𝑢𝑥_𝑙−1,𝑡_𝑛+1𝑟₀^𝑙+1_𝑖=0𝑤_𝑖𝑢𝑥_{𝑙−𝑖+1},𝑡_𝑛+1+^{𝑀−𝑙+1}_𝑖=0+𝜇₀𝑤_𝑖𝑢𝑥_{𝑙+𝑖−1},𝑡_𝑛+1𝑠𝑢𝑥_𝑙,𝑡_𝑛,𝑥_𝑙,𝑡_𝑛+𝑅^𝑙𝑛+1,(2.7) where 𝜇₀=𝜏^𝛼Γ(2−𝛼)>0, 𝑟₀=𝐾^𝜇𝛼ℎ^−𝜇/2cos(𝜋𝜇/2)<0, and ||𝑅^𝑙𝑛+1||≤𝐶₁𝜏^𝛼𝜏^1+𝛼+ℎ+𝜏.(2.8)

Let 𝑢^𝑛𝑙 be the numerical approximation of 𝑢(𝑥_𝑙,𝑡_𝑛), and let 𝑠^𝑛𝑙 be the numerical approximation of 𝑠(𝑢(𝑥_𝑙,𝑡_𝑛),𝑥_𝑙,𝑡_𝑛). We obtain the following effective numerical method (ENM) of the TSFFPE-NST (1.15)-(1.16): 𝑢^𝑙𝑛+1=𝑏_𝑛𝑢^0𝑙+^𝑛−1_𝑗=0𝑏_𝑗−𝑏_𝑗+1𝑢^{𝑙𝑛−𝑗}+𝜇⁰ℎ𝑝^𝑙−𝜇₀𝑢^𝑙𝑛+1−𝑝_𝑙−1𝑢^{𝑛+1𝑙−1}𝑟₀^𝑙+1_𝑖=0𝑤_𝑖𝑢^{𝑛+1𝑙−𝑖+1}+^{𝑀−𝑙+1}_𝑖=0𝑤_𝑖𝑢^{𝑛+1𝑙+𝑖−1}+𝜇⁰𝑠^𝑛𝑙(2.9) for 𝑙=1,2,…,𝑀−1, 𝑛=0,1,2,…,𝑁−1. The boundary and initial conditions can be discretised using 𝑢𝑢^𝑛0=𝑢^𝑛𝑀=0,𝑛=0,1,2,…,𝑁,_0𝑙=𝑢₀(𝑙ℎ),𝑙=0,1,2,…,𝑀.(2.10)

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 𝑢^𝑙𝑛+1=𝑏_𝑛𝑢^0𝑙+^𝑛−1_𝑗=0𝑏_𝑗−𝑏_𝑗+1𝑢^{𝑙𝑛−𝑗}+𝜇⁰ℎ𝑝^𝑙−𝜇₀𝑢^𝑙𝑛+1−𝑝_𝑙−1𝑢^{𝑛+1𝑙−1}𝑟₀^𝑙+1_𝑖=0𝑤_𝑖𝑢^{𝑛+1𝑙−𝑖+1}+^{𝑀−𝑙+1}_𝑖=0𝑤_𝑖𝑢^{𝑛+1𝑙+𝑖−1}+𝜇⁰𝑠^𝑙𝑛+1,(2.11) that is, replace 𝑠^𝑛𝑙 in (2.9) with 𝑠^𝑙𝑛+1. This numerical method is stable and convergent when the source term 𝑠(𝑢(𝑥,𝑡),𝑥,𝑡) satisfies the Lipschitz condition (1.17) (see, e.g., [24]).

Lemma 2.2 (see [19]). The coefficients 𝑏_𝑗 satisfy (1)𝑏_𝑗>0 for 𝑗=0,1,2,…,𝑛; (2)1=𝑏₀>𝑏₁>⋯>𝑏_𝑛, 𝑏_𝑛→0 as 𝑛→∞; (3)when 0<𝛼<1, lim_𝑗→∞𝑏^𝑗−1𝑗^𝛼=lim_𝑗→∞𝑗⁻¹1+𝑗⁻¹^1−𝛼−1=11−𝛼.(2.12) Thus, there is a positive constant 𝐶₂ such that 𝑏^𝑗−1≤𝐶₂𝑗^𝛼,𝑗=0,1,2,….(2.13)

Lemma 2.3 (see [25]). The coefficients 𝑤_𝑖 satisfy (1)𝑤₀=0, 𝑤₁=−𝜇<0, and 𝑤_𝑖>0 for 𝑖=2,3,…,𝑀; (2)∑^∞𝑖=0𝑤_𝑖=0, and ∑^𝑛𝑖=0𝑤_𝑖<0 for ∀𝑛∈ℕ.

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: 𝑢^𝑙𝑛+1−𝜇⁰ℎ𝑝^𝑙𝑢^𝑙𝑛+1−𝑝_𝑙−1𝑢^{𝑛+1𝑙−1}+𝜇₀𝑟₀^𝑙+1_𝑖=0𝑤_𝑖𝑢^{𝑛+1𝑙−𝑖+1}+^{𝑀−𝑙+1}_𝑖=0𝑤_𝑖𝑢^{𝑛+1𝑙+𝑖−1}=𝑏_𝑛𝑢^0𝑙+^𝑛−1_𝑗=0𝑏_𝑗−𝑏_𝑗+1𝑢^{𝑙𝑛−𝑗}+𝜇₀𝑠^𝑛𝑙.(3.1)

Let 𝑢^𝑛𝑙̃ be the approximate solution of the ENM (3.1), and let 𝑠^𝑛𝑙̃ be the approximation of 𝑠^𝑛𝑙. Setting 𝜌^𝑛𝑙=𝑢^𝑛𝑙−𝑢^𝑛𝑙̃, we obtain the following roundoff error equation: 𝜌^𝑙𝑛+1−𝜇⁰ℎ𝑝^𝑙𝜌^𝑙𝑛+1−𝑝_𝑙−1𝜌^{𝑛+1𝑙−1}+𝜇₀𝑟₀^𝑙+1_𝑖=0𝑤_𝑖𝜌^{𝑛+1𝑙−𝑖+1}+^{𝑀−𝑙+1}_𝑖=0𝑤_𝑖𝜌^{𝑛+1𝑙+𝑖−1}=𝑏_𝑛𝜌^0𝑙+^𝑛−1_𝑗=0𝑏_𝑗−𝑏_𝑗+1𝜌^{𝑙𝑛−𝑗}+𝜇₀𝑠^𝑛𝑙−𝑠^𝑛𝑙̃(3.2) for 𝑙=1,2,…,𝑀−1; 𝑛=0,1,…,𝑁−1.

We suppose that 𝑝(𝑥)≤0 and that 𝑝(𝑥) decreases monotonically on [𝑎,𝑏]. This is based on the fact that physical considerations and stability dictate that 𝑝^(𝑥)<0 [26, 27].

Assuming ‖𝜌^𝑛‖_∞=max_{1≤𝑙≤𝑀−1}|𝜌^𝑛𝑙|, and using mathematical induction, we obtain the following theorem.

Theorem 3.1. Suppose that 𝜌^𝑛𝑙 (𝑙=1,2,…,𝑀−1, 𝑛=1,2,…,𝑁) is the solution of the roundoff error equation (3.2), and the nonlinear source term 𝑠(𝑢(𝑥,𝑡),𝑥,𝑡) satisfies the Lipschitz condition (1.17), then there is a positive constant 𝐶₀, such that ‖𝜌^𝑛‖_∞≤𝐶₀‖‖𝜌⁰‖‖_∞,𝑛=1,2,…,𝑁.(3.3)

Proof. When 𝑛=1, assume that |𝜌^1𝑙₀|=max{|𝜌¹¹|,|𝜌¹²|,…,|𝜌^1𝑀−1|}. Because 𝑝(𝑥)≤0 and decreases monotonically on [𝑎,𝑏], we have 0≤−𝜇⁰ℎ𝑝^𝑙0−𝑝_𝑙0−1|||𝜌^1𝑙₀|||≤−𝜇₀ℎ𝑝^𝑙0|||𝜌_1𝑙0|||+𝜇₀ℎ𝑝^𝑙0−1|||𝜌_1𝑙0−1|||.(3.4) Using the properties of 𝜔_𝑖 in Lemma 2.3, we have 0≤𝜇₀𝑟₀⎡⎢⎢⎣^𝑙0+1_𝑖=0𝑤_𝑖|||𝜌_1𝑙0|||+^{𝑀−𝑙0+1}_𝑖=0𝑤_𝑖≤𝜇₀|||𝜌_1𝑙0|||⎤⎥⎥⎦𝑟₀⎡⎢⎢⎣^𝑙0+1_𝑖=0𝑤_𝑖|||𝜌_{1𝑙0−𝑖+1}|||+^{𝑀−𝑙0+1}_𝑖=0𝑤_𝑖|||𝜌_{1𝑙0+𝑖−1}|||⎤⎥⎥⎦.(3.5) Combining (3.4) with (3.5), using the Lipschitz condition (1.17) and smooth solution condition, we obtain |||𝜌_1𝑙0|||≤|||𝜌_1𝑙0|||−𝜇₀ℎ𝑝^𝑙0|||𝜌_1𝑙0|||+𝜇₀ℎ𝑝^𝑙0−1+𝜇₀|||𝜌_1𝑙0−1|||𝑟₀⎡⎢⎢⎣^𝑙0+1_𝑖=0𝑤_𝑖|||𝜌_{1𝑙0−𝑖+1}|||+^{𝑀−𝑙0+1}_𝑖=0𝑤_𝑖≤|||||𝜌_1𝑙0|||𝜌_{1𝑙0+𝑖−1}|||⎤⎥⎥⎦−𝜇⁰ℎ𝑝^𝑙0𝜌^1𝑙₀+𝜇⁰ℎ𝑝^𝑙0−1𝜌^1𝑙₀₋₁+𝜇₀𝑟₀⎡⎢⎢⎣^𝑙0+1_𝑖=0𝑤_𝑖𝜌^1𝑙_0−𝑖+1+^{𝑀−𝑙0+1}_𝑖=0𝑤_𝑖𝜌^1𝑙_0+𝑖−1=|||𝑏₀⎤⎥⎥⎦|||||≤𝑏₀𝜌^0𝑙₀+𝜇₀𝑠^0𝑙₀−𝑠̃^0𝑙₀||||||𝜌_0𝑙0|||+𝜇₀𝐿|||𝜌^0𝑙₀|||=1+𝜇₀𝐿|||𝜌^0𝑙₀|||.(3.6) Let 𝐶=1+𝜇₀𝐿. Thus, we obtain ‖‖𝜌¹‖‖_∞≤𝐶‖‖𝜌⁰‖‖_∞.(3.7) Now, suppose that ‖‖𝜌^𝑘‖‖_∞≤𝐶‖‖𝜌⁰‖‖_∞,𝑘=2,…,𝑛.(3.8) By assuming |𝜌^𝑙𝑛+1₀|=max{|𝜌^1𝑛+1|,|𝜌^2𝑛+1|,…,|𝜌^{𝑛+1𝑀−1}|}, we have that |||𝜌_𝑙𝑛+10|||≤|||||𝜌_𝑙𝑛+10−𝜇⁰ℎ𝑝^𝑙0𝜌^𝑙𝑛+1₀−𝑝_𝑙0−1𝜌^𝑙𝑛+1₀₋₁+𝜇⁰𝑟₀⎡⎢⎢⎣^𝑙0+1_𝑖=0𝑤_𝑖𝜌^𝑙𝑛+1_0−𝑖+1+^{𝑀−𝑙0+1}_𝑖=0𝑤_𝑖𝜌^𝑙𝑛+1_0+𝑖−1=|||||𝑏_𝑛⎤⎥⎥⎦|||||𝜌^0𝑙₀+^𝑛−1_𝑗=0𝑏_𝑗≤𝑏_𝑛−𝑏_𝑗+1𝜌^{𝑙𝑛−𝑗}₀+𝜇₀𝑠^𝑛𝑙₀−𝑠̃^𝑛𝑙₀||||||||𝜌_0𝑙0|||+^𝑛−1_𝑗=0𝑏_𝑗−𝑏_𝑗+1|||𝜌^{𝑙𝑛−𝑗}₀|||+𝜇₀𝐿|||𝜌^𝑛𝑙₀|||.(3.9) Using (3.7) and (3.8), we have |||𝜌_𝑙𝑛+10|||≤𝑏_𝑛|||𝜌_0𝑙0|||+𝐶^𝑛−1_𝑗=0𝑏_𝑗−𝑏_𝑗+1|||𝜌^0𝑙₀|||+𝐶𝜇₀𝐿|||𝜌^0𝑙₀=𝑏_𝑛||||||𝜌_0𝑙0|||+𝐶𝑏₀−𝑏_𝑛|||𝜌^0𝑙₀|||+𝐶𝜇₀𝐿|||𝜌^0𝑙₀=𝑏_𝑛|||𝜇₀𝐿+𝐶²|||𝜌^0𝑙₀|||.(3.10) Let 𝐶₀=𝑏_𝑛𝜇₀𝐿+𝐶². Hence, we have ‖‖𝜌^𝑛+1‖‖_∞≤𝐶₀‖‖𝜌⁰‖‖_∞.(3.11) 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 𝑠(𝑢(𝑥,𝑡),𝑥,𝑡) satisfies the Lipschitz condition (1.17) and that the drift coefficient 𝑝(𝑥)≤0 decreases monotonically on [𝑎,𝑏], the ENM defined by (2.9)-(2.10) is stable.

Remark 3.3. If 𝑝(𝑥)>0 and decreases monotonically on [𝑎,𝑏], 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 𝑝(𝑥) 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 (𝜕/𝜕𝑥)[𝑝(𝑥)𝑢(𝑥,𝑡)]=(𝑑𝑝/𝑑𝑥)𝑢(𝑥,𝑡)+𝑝(𝑥)(𝜕𝑢(𝑥,𝑡)/𝜕𝑥), which enables us to group (𝑑𝑝/𝑑𝑥)𝑢(𝑥,𝑡) together with the nonlinear source term 𝑠(𝑢,𝑥,𝑡) to obtain a new nonlinear source term 𝑠^∗(𝑢,𝑥,𝑡)=𝑠(𝑢,𝑥,𝑡)+(𝑑𝑝/𝑑𝑥)𝑢(𝑥,𝑡). This way we can weaken the assumption on 𝑝(𝑥) 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 1−𝜇₀𝐿>0, which is independent of the spatial step. In fact, when the time step is small, the condition 1−𝜇₀𝐿>0 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 𝑢(𝑥_𝑙,𝑡_𝑛) be the exact solution of the TSFFPE-NST (1.15)-(1.16) at mesh point (𝑥_𝑙,𝑡_𝑛), and let 𝑢^𝑛𝑙 be the numerical solution of the TSFFPE-NST (1.15)-(1.16) computed using the ENM (2.9)-(2.10). Define 𝜂^𝑛𝑙=𝑢(𝑥_𝑙,𝑡_𝑛)−𝑢^𝑛𝑙 and 𝐘^𝑛=(𝜂^𝑛1,𝜂^𝑛1,…,𝜂^𝑛𝑀−1)^𝑇. Subtracting (2.9) from (2.7) leads to 𝜂^𝑙𝑛+1−𝜇⁰ℎ𝑝^𝑙𝜂^𝑙𝑛+1−𝑝_𝑙−1𝜂^{𝑛+1𝑙−1}+𝜇₀𝑟₀^𝑙+1_𝑖=0𝑤_𝑖𝜂^{𝑛+1𝑙−𝑖+1}+^{𝑀−𝑙+1}_𝑖=0𝑤_𝑖𝜂^{𝑛+1𝑙+𝑖−1}=𝑏_𝑛𝜂^0𝑙+^𝑛−1_𝑗=0𝑏_𝑗−𝑏_𝑗+1𝜂^{𝑙𝑛−𝑗}+𝜇₀𝑠𝑢𝑥_𝑙,𝑡_𝑛,𝑥_𝑙,𝑡_𝑛−𝑠^𝑛𝑙+𝑅^𝑙𝑛+1,(4.1) where 𝑙=1,2,…,𝑀−1; 𝑛=0,1,…,𝑁−1.

Assuming that ‖𝐘^𝑛‖_∞=max_{1≤𝑙≤𝑀−1}|𝜂^𝑛𝑙| and using mathematical induction, we obtain the following theorem.

Theorem 4.1. Assuming the nonlinear source term 𝑠(𝑢(𝑥,𝑡),𝑥,𝑡) satisfies the Lipschitz condition (1.17), and the drift coefficient 𝑝(𝑥)≤0 decreases monotonically on [𝑎,𝑏], the ENM defined by (2.9)-(2.10) is convergent, and there exists a positive constant 𝐶^∗, such that ‖𝐘^𝑛‖_∞≤𝐶^∗𝜏^1+𝛼+ℎ+𝜏,𝑛=1,2,…,𝑁.(4.2)

Proof. Assume |𝑅^𝑘0_𝑙0|=max_{1≤𝑙≤𝑀−1,1≤𝑛≤𝑁}|𝑅^𝑛𝑙|. Following a similar argument to that presented above for the stability analysis of the ENM (2.9)-(2.10), when 𝑛=1, assuming that |𝜂^1𝑙₀|=max{|𝜂¹¹|,|𝜂¹²|,…,|𝜂^1𝑀−1|}, we have |||𝜂_1𝑙0|||≤|||𝑏₀𝜂^0𝑙₀+𝜇₀𝑠𝑢𝑥^𝑙₀,𝑡₀,𝑥_𝑙0,𝑡₀−𝑠^0𝑙₀+𝑅^1𝑙₀|||.(4.3) Utilising 𝐘⁰=0, the Lipschitz condition (1.17), and smooth solution condition, we obtain |||𝜂_1𝑙0|||≤𝑏₀|||𝜂_0𝑙0|||+𝜇₀𝐿|||𝜂^0𝑙₀|||+|||𝑅_𝑘0𝑙0|||=|||𝑅_𝑘0𝑙0|||.(4.4) Thus, ‖‖𝐘¹‖‖_∞≤|||𝑅^𝑘0_𝑙0|||.(4.5) Now, suppose that ‖‖𝐘^𝑘‖‖_∞≤𝑏^−1𝑘−1|||𝑅_𝑘0𝑙0|||,𝑘=1,2,…,𝑛.(4.6) Using Lemma 2.2, 𝑏_𝑘>𝑏_𝑘+1, we have ‖‖𝐘^𝑘‖‖_∞≤𝑏^𝑛−1|||𝑅_𝑘0𝑙0|||.(4.7) Similarly, assuming |𝜂^𝑙𝑛+1₀|=max{|𝜂^1𝑛+1|,|𝜂^2𝑛+1|,…,|𝜂^{𝑛+1𝑀−1}|}, we have |||𝜂_𝑙𝑛+10|||≤|||||𝑏_𝑛𝜂^0𝑙₀+^𝑛−1_𝑗=0𝑏_𝑗−𝑏_𝑗+1𝜂^{𝑙𝑛−𝑗}₀+𝜇₀𝑠𝑢𝑥^𝑙₀,𝑡_𝑛,𝑥_𝑙0,𝑡_𝑛−𝑠^𝑛𝑙₀+𝑅^𝑙𝑛+1₀|||||.(4.8) Utilising 𝐘⁰=0, the Lipschitz condition (1.17), and smooth solution condition, we obtain |||𝜂_𝑙𝑛+10|||≤𝑏_𝑛−1𝑏₀−𝑏_𝑛|||𝑅^𝑘0_𝑙0|||+𝜇₀𝐿𝑏^𝑛−1=𝑏_𝑛−1|||𝑅_𝑘0𝑙0|||+|||𝑅_𝑘0𝑙0|||𝑏₀−𝑏_𝑛+𝜇₀𝐿+𝑏_𝑛|||𝑅^𝑘0_𝑙0=𝑏_𝑛−1|||𝑏₀+𝜇₀𝐿|||𝑅^𝑘0_𝑙0|||=𝐶𝑏_𝑛−1|||𝑅_𝑘0𝑙0|||.(4.9) Hence, ‖‖𝐘^𝑛+1‖‖_∞≤𝐶𝑏^𝑛−1|||𝑅_𝑘0𝑙0|||.(4.10) Finally, utilising (2.8) and Lemma 2.2, 𝑏^𝑛−1≤𝐶₂𝑛^𝛼, we obtain the result on the convergence of the ENM (2.9)-(2.10), namely, ‖𝐘^𝑛‖_∞≤≤𝐶𝐶𝐶_1∗𝐶₂𝑛^𝛼𝜏^𝛼𝜏^1+𝛼+ℎ+𝜏𝜏^1+𝛼+ℎ+𝜏(4.11) for 𝑛=1,2,…,𝑁.

Remark 4.2. 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 convergent when 1−𝜇₀𝐿>0, which is independent of the spatial step. In fact, when the time step is small, the condition 1−𝜇₀𝐿>0 is generally satisfied.

5. Numerical Results

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.

Example 5.1. Consider the following TSFFPE: 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=−𝜐𝜕𝜕𝑥+𝐾^𝜇𝛼𝜕^𝜇𝑢(𝑥,0𝑢)(𝑎=,𝑡𝐾)=_𝜇𝛼𝑢(𝑏,𝑡)=𝜕0|𝑥,|^𝜇0𝑢≤(𝑥𝑡,≤𝑡)𝑇+,𝑓(𝑥,𝑡),(𝑥−𝑎)²(𝑏−𝑥)²,𝑎≤𝑥≤𝑏,(5.1) where 𝑓(𝑥,𝑡)=(1+𝛼)𝜐Γ(1+𝛼)𝑡(𝑥−𝑎)²(𝑏−𝑥)²+𝐾^𝜇𝛼𝐾^𝜇𝛼+𝜐𝑡^1+𝛼+2𝜐2𝐾co_𝜇𝛼s(𝜋𝜇/2)[𝑔(𝑥−𝑎)+𝑔(𝑏−𝑥)]𝑔(𝑥)=4!+𝜐𝑡^1+𝛼(𝑥−𝑎)(𝑏−𝑥)(𝑎+𝑏−2𝑥),Γ(5−𝜇)𝑥^4−𝜇−2(𝑏−𝑎)3!Γ(4−𝜇)𝑥^3−𝜇+(𝑏−𝑎)²2Γ(3−𝜇)𝑥^2−𝜇.(5.2) The exact solution of the TSFFPE (5.1) is found to be 𝑢(𝑥,𝑡)=𝐾^𝜇𝛼+𝜐𝑡^1+𝛼(𝑥−𝑎)²(𝑏−𝑥)²,(5.3) which can be verified by direct fractional differentiation of the given solution, and substituting into the fractional differential equation.
In this example, we take 𝑎=0, 𝑏=1, 𝐾^𝜇𝛼=25, 𝜐=1, 𝛼=0.8, and 𝜇=1.9. From Figure 1, it can be seen that the numerical solution using the ENM is in good agreement with the exact solution at different times 𝑇, with ℎ=1/40 and 𝜏=1/40. The maximum errors of the ENM at time 𝑇=1.0 are presented in Table 1. It can be seen that the ENM is stable and convergent for solving the TSFFPE (5.1). The errors, as our theory indicated, satisfy the relationship error≤(𝜏^1+𝛼+ℎ+𝜏).

Table 1 

Maximum error behavior versus grid size reduction for Example 5.1 at 𝑇=1.0.

Figure 1 

Comparison of the numerical solution with the exact solution for Example 5.1 at 𝑇=1.0,3.0,5.0. 𝑇 increases in the direction of the arrow.

Example 5.2. Consider the following TSFFPE-NST: 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=𝐾^𝜇𝛼𝜕^𝜇𝜕|𝑥|^𝜇𝑢(𝑥,𝑡)−𝛾Γ(𝛽)^𝑡0𝑢(−5,𝑡)=𝑢𝑢(𝑥(,50,)𝑡)==𝛿0(,𝑥).0≤(𝑡−𝑡≤𝜉)𝑇,^𝛽−1𝑢(𝑥,𝜉)𝑑𝜉,(5.4)
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 𝛽=0.5, 𝛾=1, and 𝐾^𝜇𝛼=1. Figures 2–4 show the changes in the solution profiles of the TSFFPE-NST (5.4) when 𝛼 and 𝜇 are changed from integer to fraction at different times 𝑇. We see that the solution profile of the fractional order model is characterized by a sharp peak and a heavy tail. The peak height in Figure 2 (𝛼=1.0 and 𝜇=2.0) decreases more rapidly than that in Figure 3 (𝛼=0.8 and 𝜇=1.8). Furthermore, when we choose 𝛼=0.5 and 𝜇=1.5, a more interesting result can be observed; that is, the peak height in Figure 2 decreases more slowly than that shown in Figure 4 at the early time 𝑇=0.1, but this trend reverses for the later times 𝑇=0.5 and 𝑇=1.0. Hence, the TSFFPE-NST (5.4) may be useful to investigate several physical processes in the absence of an external force field by choosing appropriate 𝛼 and 𝜇.

Figure 2 

Numerical solutions for Example 5.2 with 𝛼=1.0 and 𝜇=2.0 at different times 𝑇=0.1,0.5,1.0.

Figure 3 

Numerical solutions for Example 5.2 with 𝛼=0.8 and 𝜇=1.8 at different times 𝑇=0.1,0.5,1.0.

Figure 4 

Numerical solutions for Example 5.2 with 𝛼=0.5 and 𝜇=1.5 at different times 𝑇=0.1,0.5,1.0.

Example 5.3. Consider the following TSFFPE-NST: 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=𝜕𝜕𝑥𝑝(𝑥)+𝐾^𝜇𝛼𝜕^𝜇𝜕|𝑥|^𝜇𝑢(𝑥,𝑡)−Γ𝛾(𝛽)^𝑡0𝑢(−5,𝑡)=𝑢𝑢(𝑥(,50,)𝑡)==𝛿0(,𝑥).0≤𝑡≤𝑇(𝑡,−𝜉)^𝛽−1𝑢(𝑥,𝜉)𝑑𝜉,(5.5)
This example of the TSFFPE-NST incorporates the external force term with 𝑝(𝑥)=−1 and an absorbent term. The formula to approximate the absorbent term is presented in the appendix. Here, we take 𝛽=0.5, 𝛾=1, and 𝐾^𝜇𝛼=1. Figures 5–7 show the changes in the solution profiles of the TSFFPE-NST (5.5) when 𝛼 and 𝜇 are changed from integer order to fractional order at different times 𝑇. Again, we see that the solution profile of the fractional order model is characterized by a sharp peak and a heavy tail. Furthermore, due to the presence of the external force term with 𝑝(𝑥)=−1, the solution profiles are shifted to the right. It is worthwhile to note that the peak of the integer order model in Figure 5 (𝛼=1.0 and 𝜇=2.0) moves to the right as time increases, but the peak of the fractional order model in Figure 6 (𝛼=0.8 and 𝜇=1.8) and Figure 7 (𝛼=0.5 and 𝜇=1.5) does not move.
We also see that the peak heights in Figures 5 and 6 remain almost the same for increasing time. The peak height in Figure 5 decreases more slowly than that shown in Figure 7 at the early time 𝑇=0.1, but this trend reverses for the later times 𝑇=0.5 and 𝑇=1.0. Hence, the TSFFPE-NST (5.5) may be useful to investigate several physical processes within an external force field by choosing appropriate 𝛼 and 𝜇.

Figure 5 

Numerical solutions for Example 5.3 with 𝛼=1.0 and 𝜇=2.0 at different times 𝑇=0.1,0.5,1.0.

Figure 6 

Numerical solutions for Example 5.3 with 𝛼=0.8 and 𝜇=1.8 at different times 𝑇=0.1,0.5,1.0.

Figure 7 

Numerical solutions for Example 5.3 with 𝛼=0.5 and 𝜇=1.5 at different times 𝑇=0.1,0.5,1.0.

Example 5.4. Consider the following TSFFPE-NST: 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=𝜕𝜕𝑥𝑝(𝑥)+𝐾^𝜇𝛼𝜕^𝜇𝑢𝑢((𝑥0,,0𝑡))==𝑢𝑥𝜕₂(5|𝑥,|𝑡^𝜇)=𝑢0(𝑥,,𝑡)0+≤𝑟𝑡𝑢≤(𝑥𝑇,𝑡,)1−𝑢(𝐾𝑥,𝑡),(5−𝑥)²,0≤𝑥≤5.(5.6) In applications to population biology, 𝑢(𝑥,𝑡) is the population density at location 𝑥∈ℛ and time 𝑡>0. The nonlinear source term 𝑠(𝑢(𝑥,𝑡),𝑥,𝑡)=𝑟𝑢(𝑥,𝑡)(1−𝑢(𝑥,𝑡)/𝐾) is Fisher's growth term that models population growth, where 𝑟 is the intrinsic growth rate of a species and 𝐾 is the environmental carrying capacity, representing the maximum sustainable population density [20, 28, 29].
In this example, we take 𝑟=0.2, 𝐾=1. Figure 8 shows the solution behavior when 𝛼=0.8, 𝜇=1.6 at different times 𝑇=0.1,0.5,1.0, while Figure 9 shows the solution behavior with different values of 𝛼 between 0 and 1 and fixed value of 𝜇=1.8 at time 𝑇=1.0. Figure 9 also shows that the system exhibits anomalous diffusion behavior and that the solution continuously depends on the time and space fractional derivatives. Although the source term for Fisher's equation 𝑠(𝑢(𝑥,𝑡),𝑥,𝑡)=𝑟𝑢(𝑥,𝑡)(1−𝑢(𝑥,𝑡)/𝐾) is not globally Lipschitz continuous, the solution of the discrete numerical method still yields bounds on the solution of the continuous problem and the solution of the numerical method (ENM) converges to the unique solution of the continuous problem (5.6) as the time and space steps tend to zero [30].

Figure 8 

Numerical solutions for Example 5.4 with 𝛼=0.8 and 𝜇=1.6 at different times 𝑇=0.1,0.5,1.0.

Figure 9 

Numerical solutions for Example 5.4 with fixed 𝜇=1.8 at time 𝑇=1.0, and different values of 𝛼=0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0.

6. Conclusions

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 𝑝^(𝑥) can be either >0 or <0. Numerical experiments have been carried out to support the theoretical claims. These numerical methods can also be used to investigate other types of fractional partial differential equations.

Appendix

Formulae for Examples 5.2 and 5.3

Let us start from (3.1), that is, 𝑢^𝑙𝑛+1−𝜇⁰ℎ𝑝^𝑙𝑢^𝑙𝑛+1−𝑝_𝑙−1𝑢^{𝑛+1𝑙−1}+𝜇₀𝑟₀^𝑙+1_𝑖=0𝑤_𝑖𝑢^{𝑛+1𝑙−𝑖+1}+^{𝑀−𝑙+1}_𝑖=0𝑤_𝑖𝑢^{𝑛+1𝑙+𝑖−1}=𝑏_𝑛𝑢^0𝑙+^𝑛−1_𝑗=0𝑏_𝑗−𝑏_𝑗+1𝑢^{𝑙𝑛−𝑗}+𝜇₀𝑠^𝑛𝑙.(A.1)

Now setting 𝑠^𝑛𝑙=−(𝛾/Γ(𝛽))∫^𝑡𝑛₀(𝑡_𝑛−𝜉)^𝛽−1𝑢(𝑥_𝑙,𝜉)𝑑𝜉, then we have 𝑠^𝑛𝑙≈−𝛾Γ(𝛽)^𝑛−1_𝑗=0^𝑡𝑗+1_𝑡𝑗𝑡_𝑛−𝜉^𝛽−1𝑢𝑥_𝑙,𝜉𝑑𝜉.(A.2)

Applying the Mean Value Theorem (M.V.T) for integration yields 𝑠^𝑛𝑙≈−𝛾Γ(𝛽)^𝑛−1_𝑗=0𝑢𝑥_𝑙,𝜉_𝑗^𝑡𝑗+1_𝑡𝑗𝑡_𝑛−𝜉^𝛽−1𝑑𝜉,where𝑡_𝑗<𝜉_𝑗<𝑡_𝑗+1≈−𝛾Γ(𝛽)^𝑛−1_𝑗=0𝑢^𝑗𝑙+𝑢^𝑙𝑗+12𝑡^𝑛−𝑡_𝑗^𝛽𝛽−𝑡^𝑛−𝑡_𝑗+1^𝛽=−𝛾𝜏_𝛽𝛽Γ(𝛽)⋅2⋅𝛽^𝑛−1_𝑗=0𝑢^𝑗𝑙=−𝛾𝜏_𝛽+𝑢^𝑙𝑗+1(𝑛−𝑗)^𝛽−(𝑛−𝑗−1)^𝛽2Γ(𝛽+1)^𝑛−1_𝑗=0𝑢^{𝑙𝑛−𝑗−1}=−𝜇_1𝑛−1+𝑢^{𝑙𝑛−𝑗}(𝑗+1)^𝛽−𝑗^𝛽_𝑗=0𝑞_𝑗𝑢^{𝑙𝑛−𝑗−1}+𝑢^{𝑙𝑛−𝑗},(A.3) where 𝜇₁=𝛾𝜏^𝛽/2Γ(𝛽+1), 𝑞_𝑗=(𝑗+1)^𝛽−𝑗^𝛽, 𝑗=0,1,….

Also, we have ^𝑙+1_𝑖=0𝑤_𝑖𝑢^{𝑛+1𝑙−𝑖+1}+^{𝑀−𝑙+1}_𝑖=0𝑤_𝑖𝑢^{𝑛+1𝑙+𝑖−1}=^𝑀−1_𝑖=0𝜂_𝑙𝑖𝑢^𝑖𝑛+1,(A.4) where 𝜂_𝑙𝑖=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑤^{𝑙−𝑖+1}𝑤,1≤𝑖≤𝑙−2,₀2𝑤₁+𝑤₂,𝑖=𝑙−1,𝑤,𝑖=𝑙,₀𝑤+𝑤₂,𝑖=𝑙+1,_{𝑖−𝑙+1},𝑙+2≤𝑖≤𝑀−1.(A.5)

Now, substituting (A.3) and (A.4) into (A.1), we obtain the numerical scheme for Example 5.2 as 𝑢^𝑙𝑛+1+𝜇₀𝑟_0𝑀−1_𝑖=0𝜂_𝑙𝑖𝑢^𝑖𝑛+1=𝑏_𝑛𝑢^0𝑙+^𝑛−1_𝑗=0𝑏^𝑗−𝑏_𝑗+1𝑢^{𝑙𝑛−𝑗}−𝜇₁𝑞_𝑗𝑢^{𝑙𝑛−𝑗−1}+𝑢^{𝑙𝑛−𝑗},(A.6) and the numerical scheme for Example 5.3 as 1−𝜇⁰𝑝_𝑙ℎ𝑢^𝑙𝑛+1+𝜇⁰𝑝_𝑙−1ℎ𝑢^{𝑛+1𝑙−1}+𝜇₀𝑟_0𝑀−1_𝑖=0𝜂_𝑙𝑖𝑢^𝑖𝑛+1=𝑏_𝑛𝑢^0𝑙+^𝑛−1_𝑗=0𝑏^𝑗−𝑏_𝑗+1𝑢^{𝑙𝑛−𝑗}−𝜇₁𝑞_𝑗𝑢^{𝑙𝑛−𝑗−1}+𝑢^{𝑙𝑛−𝑗}.(A.7)

Acknowledgments

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.