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)=î‚¸ğœ•ğœ•ğ‘¡ğ‘‰ğœ•ğ‘¥î…ž(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(𝑥,𝑡)𝜕𝑥+𝐾𝜇𝑐+ğ‘Žğ·ğœ‡ğ‘¥ğ‘¢(𝑥,𝑡)+𝑐−𝑥𝐷𝜇𝑏,𝑢(𝑥,𝑡)(1.2) where 𝑣 is the drift of the process, that is, the mean advective velocity; 𝐾𝜇 is the coefficient of dispersion; ğ‘Žğ·ğœ‡ğ‘¥ and 𝑥𝐷𝜇𝑏 are the left and right Riemann-Liouville space fractional derivatives of order 𝜇 given by ğ‘Žğ·ğœ‡ğ‘¥1𝑢(𝑥,𝑡)=𝜕Γ(2−𝜇)2𝜕𝑥2î€œğ‘¥ğ‘Žğ‘¢(𝜉,𝑡)𝑑𝜉(𝑥−𝜉)𝜇−1,𝑥𝐷𝜇𝑏1𝑢(𝑥,𝑡)=𝜕Γ(2−𝜇)2𝜕𝑥2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−=−𝑐𝜇=−1/2cos(𝜋𝜇/2), (1.2) can be rewritten in the following form: 𝜕𝑢(𝑥,𝑡)𝜕𝑡=−𝑣𝜕𝑢(𝑥,𝑡)𝜕𝑥+𝐾𝜇𝜕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 (ğ‘Žğ·ğœ‡ğ‘¥) and a right Riemann-Liouville derivative (𝑥𝐷𝜇𝑏), namely, 𝜕𝜇𝜕|𝑥|𝜇𝑢(𝑥,𝑡)=âˆ’ğ‘ğœ‡î€·ğ‘Žğ·ğœ‡ğ‘¥+𝑥𝐷𝜇𝑏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)=𝜕𝑡0𝐷𝑡1âˆ’ğ›¼î‚¸ğœ•ğ‘‰ğœ•ğ‘¥î…ž(𝑥)𝑚𝜂𝛼+𝐾𝛼𝜕2𝜕𝑥2𝑢(𝑥,𝑡),(1.6) where the Riemann-Liouville operator 0𝐷𝑡1−𝛼, (0<𝛼<1) is defined through its operation: 0𝐷𝑡1−𝛼1𝑢(𝑥,𝑡)=𝜕Γ(𝛼)𝜕𝑡𝑡0𝑢(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 ğ‘‰î…ž(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]: 0𝐷𝛼𝑡𝑢(𝑥,𝑡)−𝑢(𝑥,0)𝑡−𝛼=𝜕Γ(1−𝛼)ğ‘‰ğœ•ğ‘¥î…ž(𝑥)𝑚𝜂𝛼+𝐾𝛼𝜕2𝜕𝑥2𝑢(𝑥,𝑡).(1.8) By noting that [15] 𝜕𝛼𝑢(𝑥,𝑡)𝜕𝑡𝛼=0𝐷𝛼𝑡𝑢(𝑥,𝑡)−𝑢(𝑥,0)𝑡−𝛼,Γ(1−𝛼)(1.9) we arrive at 𝜕𝛼𝑢(𝑥,𝑡)𝜕𝑡𝛼=î‚¸ğœ•ğ‘‰ğœ•ğ‘¥î…ž(𝑥)𝑚𝜂𝛼+𝐾𝛼𝜕2𝜕𝑥2𝑢(𝑥,𝑡),(1.10) where 𝜕𝛼𝑢(𝑥,𝑡)/𝜕𝑡𝛼 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)𝜕𝑡𝛼=1Γ(1−𝛼)𝑡0𝜕𝑢(𝑥,𝜂)𝜕𝜂𝑑𝜂(𝑡−𝜂)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)=𝜕𝑡0𝐷𝑡1âˆ’ğ›¼î‚¸ğœ•ğ‘‰ğœ•ğ‘¥î…ž(𝑥)𝑚𝜂𝛼+𝐾𝜇𝛼𝜕𝜇𝜕|𝑥|𝜇𝑢(𝑥,𝑡),(1.12) or 𝜕𝛼𝑢(𝑥,𝑡)𝜕𝑡𝛼=î‚¸ğœ•ğ‘‰ğœ•ğ‘¥î…ž(𝑥)𝑚𝜂𝛼+𝐾𝜇𝛼𝜕𝜇𝜕|𝑥|𝜇𝑢(𝑥,𝑡),(1.13) where 𝐾𝜇𝛼 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)𝜕𝑡𝛼𝜕=−[]𝜕𝑥𝐹(𝑥)𝑢(𝑥,𝑡)+𝐾𝜇𝛼𝜕𝜇𝜕|𝑥|𝜇𝑢(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)𝜕𝑡𝛼=î‚¸ğœ•ğ‘‰ğœ•ğ‘¥î…ž(𝑥)𝑚𝜂𝛼+𝐾𝜇𝛼𝜕𝜇𝜕|𝑥|𝜇𝑢(𝑥,𝑡)+𝑠(𝑢,𝑥,𝑡)(1.15) subject to the boundary and initial conditions: 𝑢𝑢(ğ‘Ž,𝑡)=𝑢(𝑏,𝑡)=0,0≤𝑡≤𝑇,(𝑥,0)=𝑢0(𝑥),ğ‘Žâ‰¤ğ‘¥â‰¤ğ‘,(1.16) where 𝑝(𝑥)=ğ‘‰î…ž(𝑥)/𝑚𝜂𝛼 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=[ğ‘Ž,𝑏]×[0,𝑇]. In this paper, we suppose that the continuous problem (1.15)-(1.16) has a smooth solution 𝑢(𝑥,𝑡)∈𝐶1+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 𝑥𝑙=ğ‘™â„Ž (𝑙=0,1,…,𝑀) and 𝑡𝑛=𝑛𝜏 (𝑛=0,1,…,𝑁), where ℎ=(ğ‘âˆ’ğ‘Ž)/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,𝑡𝑛+1𝜕𝑡𝛼=𝜏−𝛼Γ(2−𝛼)𝑛𝑗=0𝑏𝑗𝑢𝑥,𝑡𝑛+1−𝑗−𝑢𝑥,𝑡𝑛−𝑗𝜏+𝑂1+𝛼,(2.1) where 𝑏𝑗=(𝑗+1)1−𝛼−𝑗1−𝛼, 𝑗=0,1,2,…,𝑁−1.

For the symmetric Riesz space fractional derivative, we use the following shifted Grünwald approximation [22]: 𝜕𝜇𝑢𝑥𝑙,𝑡𝜕|𝑥|ğœ‡â„Ž=−−𝜇2cos(𝜋𝜇/2)𝑙+1𝑖=0𝑤𝑖𝑢𝑥𝑙−𝑖+1+,𝑡𝑀−𝑙+1𝑖=0𝑤𝑖𝑢𝑥𝑙+𝑖−1ℎ,𝑡+ğ‘‚ğ‘žî€¸,(2.2) where the coefficients are defined by 𝑤0=1,𝑤𝑖=(−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 𝑤𝑖 that lead to approximations of different orders ğ‘ž [23]. The definition (2.2) provides order ğ‘ž=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): 𝜕𝑝𝑥𝜕𝑥𝑙𝑢𝑥𝑙=𝑝𝑥,𝑡𝑙𝑢𝑥𝑙𝑥,𝑡−𝑝𝑙−1𝑢𝑥𝑙−1,ğ‘¡â„Ž+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,𝑡𝑛+1,𝑥,𝑡𝑛+1𝑢=𝑠𝑥,𝑡𝑛,𝑥,𝑡𝑛+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=0𝑏𝑗𝑢𝑥𝑙,𝑡𝑛+1−𝑗𝑥−𝑢𝑙,𝑡𝑛−𝑗=𝑝𝑥𝑙𝑢𝑥𝑙,𝑡𝑛+1𝑥−𝑝𝑙−1𝑢𝑥𝑙−1,𝑡𝑛+1î€¸â„Žâˆ’ğ¾ğœ‡ğ›¼â„Žâˆ’ğœ‡îƒ¬2cos(𝜋𝜇/2)𝑙+1𝑖=0𝑤𝑖𝑢𝑥𝑙−𝑖+1,𝑡𝑛+1+𝑀−𝑙+1𝑖=0𝑤𝑖𝑢𝑥𝑙+𝑖−1,𝑡𝑛+1𝑢𝑥+𝑠𝑙,𝑡𝑛,𝑥𝑙,𝑡𝑛𝜏+𝑂1+𝛼.+ℎ+𝜏(2.6) After some manipulation, (2.6) can be written in the following form: 𝑢𝑥𝑙,𝑡𝑛+1=𝑏𝑛𝑢𝑥𝑙,𝑡0+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝑢𝑥𝑙,𝑡𝑛−𝑗+𝜇0â„Žî€·ğ‘î€·ğ‘¥ğ‘™î€¸ğ‘¢î€·ğ‘¥ğ‘™,𝑡𝑛+1𝑥−𝑝𝑙−1𝑢𝑥𝑙−1,𝑡𝑛+1−𝜇0𝑟0𝑙+1𝑖=0𝑤𝑖𝑢𝑥𝑙−𝑖+1,𝑡𝑛+1+𝑀−𝑙+1𝑖=0𝑤𝑖𝑢𝑥𝑙+𝑖−1,𝑡𝑛+1+𝜇0𝑠𝑢𝑥𝑙,𝑡𝑛,𝑥𝑙,𝑡𝑛+𝑅𝑙𝑛+1,(2.7) where 𝜇0=𝜏𝛼Γ(2−𝛼)>0, 𝑟0=ğ¾ğœ‡ğ›¼â„Žâˆ’ğœ‡/2cos(𝜋𝜇/2)<0, and ||𝑅𝑙𝑛+1||≤𝐶1𝜏𝛼𝜏1+𝛼.+ℎ+𝜏(2.8)

Let 𝑢𝑛𝑙 be the numerical approximation of 𝑢(𝑥𝑙,𝑡𝑛), and let 𝑠𝑛𝑙 be the numerical approximation of 𝑠(𝑢(𝑥𝑙,𝑡𝑛),𝑥𝑙,𝑡𝑛). We obtain the following effective numerical method (ENM) of the TSFFPE-NST (1.15)-(1.16): 𝑢𝑙𝑛+1=𝑏𝑛𝑢0𝑙+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝑢𝑙𝑛−𝑗+𝜇0â„Žî€·ğ‘ğ‘™ğ‘¢ğ‘™ğ‘›+1−𝑝𝑙−1𝑢𝑛+1𝑙−1−𝜇0𝑟0𝑙+1𝑖=0𝑤𝑖𝑢𝑛+1𝑙−𝑖+1+𝑀−𝑙+1𝑖=0𝑤𝑖𝑢𝑛+1𝑙+𝑖−1+𝜇0𝑠𝑛𝑙(2.9) for 𝑙=1,2,…,𝑀−1, 𝑛=0,1,2,…,𝑁−1. The boundary and initial conditions can be discretised using 𝑢𝑛0=𝑢𝑛𝑀𝑢=0,𝑛=0,1,2,…,𝑁,0𝑙=𝑢0(ğ‘™â„Ž),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 𝑢𝑙𝑛+1=𝑏𝑛𝑢0𝑙+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝑢𝑙𝑛−𝑗+𝜇0â„Žî€·ğ‘ğ‘™ğ‘¢ğ‘™ğ‘›+1−𝑝𝑙−1𝑢𝑛+1𝑙−1−𝜇0𝑟0𝑙+1𝑖=0𝑤𝑖𝑢𝑛+1𝑙−𝑖+1+𝑀−𝑙+1𝑖=0𝑤𝑖𝑢𝑛+1𝑙+𝑖−1+𝜇0𝑠𝑙𝑛+1,(2.11) that is, replace 𝑠𝑛𝑙 in (2.9) with 𝑠𝑙𝑛+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 𝑏𝑗 satisfy (1)𝑏𝑗>0 for 𝑗=0,1,2,…,𝑛; (2)1=𝑏0>𝑏1>⋯>𝑏𝑛, 𝑏𝑛→0 as ğ‘›â†’âˆž; (3)when 0<𝛼<1, limğ‘—â†’âˆžğ‘ğ‘—âˆ’1𝑗𝛼=limğ‘—â†’âˆžğ‘—âˆ’11+𝑗−11−𝛼=1−1.1−𝛼(2.12) Thus, there is a positive constant 𝐶2 such that 𝑏𝑗−1≤𝐶2𝑗𝛼,𝑗=0,1,2,….(2.13)

Lemma 2.3 (see [25]). The coefficients 𝑤𝑖 satisfy (1)𝑤0=0, 𝑤1=−𝜇<0, and 𝑤𝑖>0 for 𝑖=2,3,…,𝑀; (2)âˆ‘âˆžğ‘–=0𝑤𝑖=0, and ∑𝑛𝑖=0𝑤𝑖<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: 𝑢𝑙𝑛+1−𝜇0â„Žî€·ğ‘ğ‘™ğ‘¢ğ‘™ğ‘›+1−𝑝𝑙−1𝑢𝑛+1𝑙−1+𝜇0𝑟0𝑙+1𝑖=0𝑤𝑖𝑢𝑛+1𝑙−𝑖+1+𝑀−𝑙+1𝑖=0𝑤𝑖𝑢𝑛+1𝑙+𝑖−1=𝑏𝑛𝑢0𝑙+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝑢𝑙𝑛−𝑗+𝜇0𝑠𝑛𝑙.(3.1)

Let ̃𝑢𝑛𝑙 be the approximate solution of the ENM (3.1), and let ̃𝑠𝑛𝑙 be the approximation of 𝑠𝑛𝑙. Setting 𝜌𝑛𝑙=𝑢𝑛𝑙−̃𝑢𝑛𝑙, we obtain the following roundoff error equation: 𝜌𝑙𝑛+1−𝜇0â„Žî€·ğ‘ğ‘™ğœŒğ‘™ğ‘›+1−𝑝𝑙−1𝜌𝑛+1𝑙−1+𝜇0𝑟0𝑙+1𝑖=0𝑤𝑖𝜌𝑛+1𝑙−𝑖+1+𝑀−𝑙+1𝑖=0𝑤𝑖𝜌𝑛+1𝑙+𝑖−1=𝑏𝑛𝜌0𝑙+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝜌𝑙𝑛−𝑗+𝜇0𝑠𝑛𝑙−̃𝑠𝑛𝑙(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 [ğ‘Ž,𝑏]. This is based on the fact that physical considerations and stability dictate that ğ‘î…ž(𝑥)<0 [26, 27].

Assuming â€–ğœŒğ‘›â€–âˆž=max1≤𝑙≤𝑀−1|𝜌𝑛𝑙|, and using mathematical induction, we obtain the following theorem.

Theorem 3.1. Suppose that 𝜌𝑛𝑙 (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 â€–ğœŒğ‘›â€–âˆžâ‰¤ğ¶0‖‖𝜌0‖‖∞,𝑛=1,2,…,𝑁.(3.3)

Proof. When 𝑛=1, assume that |𝜌1𝑙0|=max{|𝜌11|,|𝜌12|,…,|𝜌1𝑀−1|}. Because 𝑝(𝑥)≤0 and decreases monotonically on [ğ‘Ž,𝑏], we have 𝜇0≤−0â„Žî€·ğ‘ğ‘™0−𝑝𝑙0−1|||𝜌1𝑙0|||𝜇≤−0â„Žğ‘ğ‘™0|||𝜌1𝑙0|||+𝜇0â„Žğ‘ğ‘™0−1|||𝜌1𝑙0−1|||.(3.4) Using the properties of 𝜔𝑖 in Lemma 2.3, we have 0≤𝜇0𝑟0âŽ¡âŽ¢âŽ¢âŽ£ğ‘™0+1𝑖=0𝑤𝑖|||𝜌1𝑙0|||+𝑀−𝑙0+1𝑖=0𝑤𝑖|||𝜌1𝑙0|||âŽ¤âŽ¥âŽ¥âŽ¦â‰¤ğœ‡0𝑟0âŽ¡âŽ¢âŽ¢âŽ£ğ‘™0+1𝑖=0𝑤𝑖|||𝜌1𝑙0−𝑖+1|||+𝑀−𝑙0+1𝑖=0𝑤𝑖|||𝜌1𝑙0+𝑖−1|||⎤⎥⎥⎦.(3.5) Combining (3.4) with (3.5), using the Lipschitz condition (1.17) and smooth solution condition, we obtain |||𝜌1𝑙0|||≤|||𝜌1𝑙0|||−𝜇0â„Žğ‘ğ‘™0|||𝜌1𝑙0|||+𝜇0â„Žğ‘ğ‘™0−1|||𝜌1𝑙0−1|||+𝜇0𝑟0âŽ¡âŽ¢âŽ¢âŽ£ğ‘™0+1𝑖=0𝑤𝑖|||𝜌1𝑙0−𝑖+1|||+𝑀−𝑙0+1𝑖=0𝑤𝑖|||𝜌1𝑙0+𝑖−1|||⎤⎥⎥⎦≤|||||𝜌1𝑙0−𝜇0â„Žğ‘ğ‘™0𝜌1𝑙0+𝜇0â„Žğ‘ğ‘™0−1𝜌1𝑙0−1+𝜇0𝑟0âŽ¡âŽ¢âŽ¢âŽ£ğ‘™0+1𝑖=0𝑤𝑖𝜌1𝑙0−𝑖+1+𝑀−𝑙0+1𝑖=0𝑤𝑖𝜌1𝑙0+𝑖−1⎤⎥⎥⎦|||||=|||𝑏0𝜌0𝑙0+𝜇0𝑠0𝑙0−̃𝑠0𝑙0|||≤𝑏0|||𝜌0𝑙0|||+𝜇0𝐿|||𝜌0𝑙0|||=1+𝜇0𝐿|||𝜌0𝑙0|||.(3.6) Let 𝐶=1+𝜇0𝐿. Thus, we obtain ‖‖𝜌1â€–â€–âˆžâ€–â€–ğœŒâ‰¤ğ¶0‖‖∞.(3.7) Now, suppose that â€–â€–ğœŒğ‘˜â€–â€–âˆžâ€–â€–ğœŒâ‰¤ğ¶0‖‖∞,𝑘=2,…,𝑛.(3.8) By assuming |𝜌𝑙𝑛+10|=max{|𝜌1𝑛+1|,|𝜌2𝑛+1|,…,|𝜌𝑛+1𝑀−1|}, we have that |||𝜌𝑙𝑛+10|||≤|||||𝜌𝑙𝑛+10−𝜇0â„Žî‚€ğ‘ğ‘™0𝜌𝑙𝑛+10−𝑝𝑙0−1𝜌𝑙𝑛+10−1+𝜇0𝑟0âŽ¡âŽ¢âŽ¢âŽ£ğ‘™0+1𝑖=0𝑤𝑖𝜌𝑙𝑛+10−𝑖+1+𝑀−𝑙0+1𝑖=0𝑤𝑖𝜌𝑙𝑛+10+𝑖−1⎤⎥⎥⎦|||||=|||||𝑏𝑛𝜌0𝑙0+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝜌𝑙𝑛−𝑗0+𝜇0𝑠𝑛𝑙0−̃𝑠𝑛𝑙0|||||≤𝑏𝑛|||𝜌0𝑙0|||+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1|||𝜌𝑙𝑛−𝑗0|||+𝜇0𝐿|||𝜌𝑛𝑙0|||.(3.9) Using (3.7) and (3.8), we have |||𝜌𝑙𝑛+10|||≤𝑏𝑛|||𝜌0𝑙0|||+𝐶𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1|||𝜌0𝑙0|||+𝐶𝜇0𝐿|||𝜌0𝑙0|||=𝑏𝑛|||𝜌0𝑙0|||𝑏+𝐶0−𝑏𝑛|||𝜌0𝑙0|||+𝐶𝜇0𝐿|||𝜌0𝑙0|||=𝑏𝑛𝜇0𝐿+𝐶2|||𝜌0𝑙0|||.(3.10) Let 𝐶0=𝑏𝑛𝜇0𝐿+𝐶2. Hence, we have ‖‖𝜌𝑛+1â€–â€–âˆžâ‰¤ğ¶0‖‖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 [ğ‘Ž,𝑏], the ENM defined by (2.9)-(2.10) is stable.

Remark 3.3. If 𝑝(𝑥)>0 and decreases monotonically on [ğ‘Ž,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(𝑥𝑙,𝑡𝑛) be the exact solution of the TSFFPE-NST (1.15)-(1.16) at mesh point (𝑥𝑙,𝑡𝑛), and let 𝑢𝑛𝑙 be the numerical solution of the TSFFPE-NST (1.15)-(1.16) computed using the ENM (2.9)-(2.10). Define 𝜂𝑛𝑙=𝑢(𝑥𝑙,𝑡𝑛)−𝑢𝑛𝑙 and 𝐘𝑛=(𝜂𝑛1,𝜂𝑛1,…,𝜂𝑛𝑀−1)𝑇. Subtracting (2.9) from (2.7) leads to 𝜂𝑙𝑛+1−𝜇0â„Žî€·ğ‘ğ‘™ğœ‚ğ‘™ğ‘›+1−𝑝𝑙−1𝜂𝑛+1𝑙−1+𝜇0𝑟0𝑙+1𝑖=0𝑤𝑖𝜂𝑛+1𝑙−𝑖+1+𝑀−𝑙+1𝑖=0𝑤𝑖𝜂𝑛+1𝑙+𝑖−1=𝑏𝑛𝜂0𝑙+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝜂𝑙𝑛−𝑗+𝜇0𝑠𝑢𝑥𝑙,𝑡𝑛,𝑥𝑙,𝑡𝑛−𝑠𝑛𝑙+𝑅𝑙𝑛+1,(4.1) where 𝑙=1,2,…,𝑀−1; 𝑛=0,1,…,𝑁−1.

Assuming that â€–ğ˜ğ‘›â€–âˆž=max1≤𝑙≤𝑀−1|𝜂𝑛𝑙| 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 [ğ‘Ž,𝑏], the ENM defined by (2.9)-(2.10) is convergent, and there exists a positive constant 𝐶∗, such that â€–ğ˜ğ‘›â€–âˆžâ‰¤ğ¶âˆ—î€·ğœ1+𝛼+ℎ+𝜏,𝑛=1,2,…,𝑁.(4.2)

Proof. Assume |𝑅𝑘0𝑙0|=max1≤𝑙≤𝑀−1,1≤𝑛≤𝑁|𝑅𝑛𝑙|. Following a similar argument to that presented above for the stability analysis of the ENM (2.9)-(2.10), when 𝑛=1, assuming that |𝜂1𝑙0|=max{|𝜂11|,|𝜂12|,…,|𝜂1𝑀−1|}, we have |||𝜂1𝑙0|||≤|||𝑏0𝜂0𝑙0+𝜇0𝑠𝑢𝑥𝑙0,𝑡0,𝑥𝑙0,𝑡0−𝑠0𝑙0+𝑅1𝑙0|||.(4.3) Utilising 𝐘0=0, the Lipschitz condition (1.17), and smooth solution condition, we obtain |||𝜂1𝑙0|||≤𝑏0|||𝜂0𝑙0|||+𝜇0𝐿|||𝜂0𝑙0|||+|||𝑅𝑘0𝑙0|||=|||𝑅𝑘0𝑙0|||.(4.4) Thus, ‖‖𝐘1‖‖∞≤|||𝑅𝑘0𝑙0|||.(4.5) Now, suppose that â€–â€–ğ˜ğ‘˜â€–â€–âˆžâ‰¤ğ‘âˆ’1𝑘−1|||𝑅𝑘0𝑙0|||,𝑘=1,2,…,𝑛.(4.6) Using Lemma 2.2, 𝑏𝑘>𝑏𝑘+1, we have â€–â€–ğ˜ğ‘˜â€–â€–âˆžâ‰¤ğ‘ğ‘›âˆ’1|||𝑅𝑘0𝑙0|||.(4.7) Similarly, assuming |𝜂𝑙𝑛+10|=max{|𝜂1𝑛+1|,|𝜂2𝑛+1|,…,|𝜂𝑛+1𝑀−1|}, we have |||𝜂𝑙𝑛+10|||≤|||||𝑏𝑛𝜂0𝑙0+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝜂𝑙𝑛−𝑗0+𝜇0𝑠𝑢𝑥𝑙0,𝑡𝑛,𝑥𝑙0,𝑡𝑛−𝑠𝑛𝑙0+𝑅𝑙𝑛+10|||||.(4.8) Utilising 𝐘0=0, the Lipschitz condition (1.17), and smooth solution condition, we obtain |||𝜂𝑙𝑛+10|||≤𝑏𝑛−1𝑏0−𝑏𝑛|||𝑅𝑘0𝑙0|||+𝜇0𝐿𝑏𝑛−1|||𝑅𝑘0𝑙0|||+|||𝑅𝑘0𝑙0|||=𝑏𝑛−1𝑏0−𝑏𝑛+𝜇0𝐿+𝑏𝑛|||𝑅𝑘0𝑙0|||=𝑏𝑛−1𝑏0+𝜇0𝐿|||𝑅𝑘0𝑙0|||=𝐶𝑏𝑛−1|||𝑅𝑘0𝑙0|||.(4.9) Hence, ‖‖𝐘𝑛+1â€–â€–âˆžâ‰¤ğ¶ğ‘ğ‘›âˆ’1|||𝑅𝑘0𝑙0|||.(4.10) Finally, utilising (2.8) and Lemma 2.2, 𝑏𝑛−1≤𝐶2𝑛𝛼, we obtain the result on the convergence of the ENM (2.9)-(2.10), namely, â€–ğ˜ğ‘›â€–âˆžâ‰¤ğ¶ğ¶1𝐶2𝑛𝛼𝜏𝛼𝜏1+𝛼+ℎ+𝜏≤𝐶∗𝜏1+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)𝜕𝑡𝛼=𝜕−𝜐𝜕𝑥+𝐾𝜇𝛼𝜕𝜇𝜕|𝑥|𝜇𝑢𝑢𝑢(𝑥,𝑡)+𝑓(𝑥,𝑡),(ğ‘Ž,𝑡)=𝑢(𝑏,𝑡)=0,0≤𝑡≤𝑇,(𝑥,0)=𝐾𝜇𝛼(ğ‘¥âˆ’ğ‘Ž)2(𝑏−𝑥)2,ğ‘Žâ‰¤ğ‘¥â‰¤ğ‘,(5.1) where 𝑓(𝑥,𝑡)=(1+𝛼)𝜐Γ(1+𝛼)𝑡(ğ‘¥âˆ’ğ‘Ž)2(𝑏−𝑥)2+𝐾𝜇𝛼𝐾𝜇𝛼+𝜐𝑡1+𝛼[𝑔]𝐾2cos(𝜋𝜇/2)(ğ‘¥âˆ’ğ‘Ž)+𝑔(𝑏−𝑥)+2𝜐𝜇𝛼+𝜐𝑡1+𝛼(ğ‘¥âˆ’ğ‘Ž)(𝑏−𝑥)(ğ‘Ž+𝑏−2𝑥),𝑔(𝑥)=4!Γ𝑥(5−𝜇)4−𝜇−2(ğ‘âˆ’ğ‘Ž)3!Γ𝑥(4−𝜇)3−𝜇+(ğ‘âˆ’ğ‘Ž)22𝑥Γ(3−𝜇)2−𝜇.(5.2) The exact solution of the TSFFPE (5.1) is found to be 𝐾𝑢(𝑥,𝑡)=𝜇𝛼+𝜐𝑡1+𝛼(ğ‘¥âˆ’ğ‘Ž)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 ğ‘Ž=0, 𝑏=1, 𝐾𝜇𝛼=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≤(𝜏1+𝛼+ℎ+𝜏).

Example 5.2. Consider the following TSFFPE-NST: 𝜕𝛼𝑢(𝑥,𝑡)𝜕𝑡𝛼=𝐾𝜇𝛼𝜕𝜇𝜕|𝑥|𝜇𝛾𝑢(𝑥,𝑡)−Γ(𝛽)𝑡0(𝑡−𝜉)𝛽−1𝑢(𝑥,𝜉)𝑑𝜉,𝑢(−5,𝑡)=𝑢(5,𝑡)=0,0≤𝑡≤𝑇,𝑢(𝑥,0)=𝛿(𝑥).(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 𝐾𝜇𝛼=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.

Example 5.3. Consider the following TSFFPE-NST: 𝜕𝛼𝑢(𝑥,𝑡)𝜕𝑡𝛼=𝜕𝜕𝑥𝑝(𝑥)+𝐾𝜇𝛼𝜕𝜇𝜕|𝑥|𝜇𝛾𝑢(𝑥,𝑡)−Γ(𝛽)𝑡0(𝑡−𝜉)𝛽−1𝑢𝑢(𝑥,𝜉)𝑑𝜉,𝑢(−5,𝑡)=𝑢(5,𝑡)=0,0≤𝑡≤𝑇,(𝑥,0)=𝛿(𝑥).(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 𝐾𝜇𝛼=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.

Example 5.4. Consider the following TSFFPE-NST: 𝜕𝛼𝑢(𝑥,𝑡)𝜕𝑡𝛼=𝜕𝜕𝑥𝑝(𝑥)+𝐾𝜇𝛼𝜕𝜇𝜕|𝑥|𝜇𝑢(𝑥,𝑡)+𝑟𝑢(𝑥,𝑡)1−𝑢(𝑥,𝑡)𝐾,𝑢(0,𝑡)=𝑢(5,𝑡)=0,0≤𝑡≤𝑇,𝑢(𝑥,0)=𝑥2(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].

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 ğ‘î…ž(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, 𝑢𝑙𝑛+1−𝜇0â„Žî€·ğ‘ğ‘™ğ‘¢ğ‘™ğ‘›+1−𝑝𝑙−1𝑢𝑛+1𝑙−1+𝜇0𝑟0𝑙+1𝑖=0𝑤𝑖𝑢𝑛+1𝑙−𝑖+1+𝑀−𝑙+1𝑖=0𝑤𝑖𝑢𝑛+1𝑙+𝑖−1=𝑏𝑛𝑢0𝑙+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝑢𝑙𝑛−𝑗+𝜇0𝑠𝑛𝑙.(A.1)

Now setting 𝑠𝑛𝑙∫=−(𝛾/Γ(𝛽))𝑡𝑛0(𝑡𝑛−𝜉)𝛽−1𝑢(𝑥𝑙,𝜉)𝑑𝜉, then we have 𝑠𝑛𝑙𝛾≈−Γ(𝛽)𝑛−1𝑗=0𝑡𝑗+1𝑡𝑗𝑡𝑛−𝜉𝛽−1𝑢𝑥𝑙,𝜉𝑑𝜉.(A.2)

Applying the Mean Value Theorem (M.V.T) for integration yields 𝑠𝑛𝑙𝛾≈−Γ(𝛽)𝑛−1𝑗=0𝑢𝑥𝑙,𝜉𝑗𝑡𝑗+1𝑡𝑗𝑡𝑛−𝜉𝛽−1𝑑𝜉,where𝑡𝑗<𝜉𝑗<𝑡𝑗+1𝛾≈−Γ(𝛽)𝑛−1𝑗=0𝑢𝑗𝑙+𝑢𝑙𝑗+12𝑡𝑛−𝑡𝑗𝛽𝛽−𝑡𝑛−𝑡𝑗+1𝛽𝛽=−𝛾𝜏𝛽Γ(𝛽)⋅2⋅𝛽𝑛−1𝑗=0𝑢𝑗𝑙+𝑢𝑙𝑗+1(𝑛−𝑗)𝛽−(𝑛−𝑗−1)𝛽=−𝛾𝜏𝛽2Γ(𝛽+1)𝑛−1𝑗=0𝑢𝑙𝑛−𝑗−1+𝑢𝑙𝑛−𝑗(𝑗+1)𝛽−𝑗𝛽=−𝜇1𝑛−1𝑗=0ğ‘žğ‘—î‚€ğ‘¢ğ‘™ğ‘›âˆ’ğ‘—âˆ’1+𝑢𝑙𝑛−𝑗,(A.3) where 𝜇1=𝛾𝜏𝛽/2Γ(𝛽+1), ğ‘žğ‘—=(𝑗+1)𝛽−𝑗𝛽, 𝑗=0,1,….

Also, we have 𝑙+1𝑖=0𝑤𝑖𝑢𝑛+1𝑙−𝑖+1+𝑀−𝑙+1𝑖=0𝑤𝑖𝑢𝑛+1𝑙+𝑖−1=𝑀−1𝑖=0𝜂𝑙𝑖𝑢𝑖𝑛+1,(A.4) where 𝜂𝑙𝑖=âŽ§âŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽ©ğ‘¤ğ‘™âˆ’ğ‘–+1𝑤,1≤𝑖≤𝑙−2,0+𝑤2,𝑖=𝑙−1,2𝑤1𝑤,𝑖=𝑙,0+𝑤2𝑤,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 𝑢𝑙𝑛+1+𝜇0𝑟0𝑀−1𝑖=0𝜂𝑙𝑖𝑢𝑖𝑛+1=𝑏𝑛𝑢0𝑙+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝑢𝑙𝑛−𝑗−𝜇1ğ‘žğ‘—î‚€ğ‘¢ğ‘™ğ‘›âˆ’ğ‘—âˆ’1+𝑢𝑙𝑛−𝑗,(A.6) and the numerical scheme for Example 5.3 as 𝜇1−0ğ‘ğ‘™â„Žî‚ğ‘¢ğ‘™ğ‘›+1+𝜇0𝑝𝑙−1â„Žğ‘¢ğ‘›+1𝑙−1+𝜇0𝑟0𝑀−1𝑖=0𝜂𝑙𝑖𝑢𝑖𝑛+1=𝑏𝑛𝑢0𝑙+𝑛−1𝑗=0𝑏𝑗−𝑏𝑗+1𝑢𝑙𝑛−𝑗−𝜇1ğ‘žğ‘—î‚€ğ‘¢ğ‘™ğ‘›âˆ’ğ‘—âˆ’1+𝑢𝑙𝑛−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.