A new formulation of finite difference and finite volume methods for solving a space fractional convection–diffusion model with fewer error estimates

Abstract

Convection and diffusion are two harmonious physical processes that transfer particles and physical quantities. This paper deals with a new aspect of solving the convection–diffusion equation in fractional order using the finite volume method and the finite difference method. In this context, we present an alternative way for estimating the space fractional derivative by utilizing the fractional Grünwald formula. The proposed methods are conditionally stable with second-order accuracy in space and first-order accuracy in time. Many comparisons are performed to display reliability and capability of the proposed methods. Furthermore, several results and conclusions are provided to indicate appropriateness of the finite volume method in solving the space fractional convection–diffusion equation compared with the finite difference method.

1 Introduction

The mathematical model is one of the most important logical aspects used to help us explain and study physical phenomena. Many physical situations include non-Markov processes, earthquakes, and terrestrial surface transport can be modeled through a class of diffusion equations or wave equations. In particular, the physical quantities are transferred inside physical systems by diffusion and the flow of heat and stochastically changing systems are described. In recent years, researchers have focused on finding effective numerical methods to solve such a type of partial differential equations (PDEs). Different PDEs have been studied and resolved including the space–time fractional diffusion–wave equation [1], the space fractional convection–diffusion equation [2], the space fractional Cauchy equation [3], the fractional heat- and wave-like equations [4], the fractional Korteweg–de Vries equation [5], and so forth [6–11]. Different effective methods have been proposed by scholars such as the reproducing kernel method [12–15], the residual power series method [16–18], the homotopy perturbation method [19], the multi-step approach [20], and some others [21–24].

During the past years, certain numerical methods, such as the finite difference method, the finite volume method, and the finite element method, have been developed to solve some PDEs. The important goal of these methods is to convert the equation to a system of linear algebraic equations. For example, in [25], the time-fractional convection–diffusion equations have been reduced to a system of linear algebraic equations using the Gegenbauer polynomials and Lagrange polynomials space and time expansion, respectively. In [26], Saadatmandi et al. studied the sinc-Legendre collocation technique for solving a class of fractional convection–diffusion equations with variable coefficients. In [27], the finite volume element technique was used to solve the sub-diffusion problems in fractional order. In addition, Jia and Wang used the volume penalization approach to improve the finite volume technique for finding a solution for a space fractional diffusion equation [28]. In [29], Mahto et al. applied the continuation property to propose the approximate solution of the sub-diffusion equation via internal control. In [30], Zhao et al. applied the finite element technique for solving the time-fractional diffusion problem, they introduced an unconditionally stable scheme based on the spatial quasi-Wilson nonconforming; while in [31], the authors derived the optimal error estimates by studying the two mixed finite element methods, conforming and nonconforming, to solve the time-fractional diffusion problem. On the other hand as well, Stynes et al. used a different technique to solve the time-fractional diffusion equation and to find elaborate error boundedness based on the regularity of the exact solution [32]. In [33], Sayevand et al. proposed the finite volume element technique for analyzing the behavior of sub-diffusion equations and proved its stability analysis of obtained solution. In [34], the authors proposed the finite difference discretization based on the weighted trapezoidal rule to approximate the fractional Laplacian and proved the convergence in one dimension. More interesting applications can be found in [35–45].

In this paper, we deal with a new aspect of FVM and FDM to obtain approximate solution for the space fractional convection–diffusion equation (SFCDE), in which the space fractional derivative is represented in the Riemann–Liouville sense. More specifically, let us consider the SFCDE of the following form:

$$\begin{aligned} \frac{\partial\psi ( x,t )}{\partial t} +\epsilon \frac{\partial^{\beta} \psi ( x,t )}{\partial x^{\beta }} =\rho \frac{\partial^{2} \psi ( x,t )}{\partial x^{2}}, \end{aligned}$$

(1)

subject to the initial condition

$$\begin{aligned} \psi ( x,0 ) =g ( x ),\quad 0\leq x\leq L, \end{aligned}$$

(2)

where $t\geq0, x\in [ 0,L ]$, a parameter β describes the order of the space fractional derivative, $0<\beta\leq1$, so that the space fractional derivative is described in the Riemann–Liouville sense, $\psi(x,t)$ is concentration, ϵ and ρ are positive parameters, ρ is a diffusion coefficient, ϵ is the velocity of water flow, and L is the length of the channel. For $\beta=1$, the equation, indeed, reduces to the linear convection–diffusion equation. In particular, the exact solution of the linear convection–diffusion equation of the shape

$$\begin{aligned} \frac{\partial\psi ( x,t )}{\partial t} +\epsilon \frac{\partial\psi ( x,t )}{\partial x} =\rho \frac{\partial^{2} \psi ( x,t )}{\partial x^{2}} \end{aligned}$$

subject to the initial condition $\psi ( x,0 ) =-\sin (\pi x)$ is given in the form

$$\begin{aligned} \psi ( x,t ) =- \sin \bigl( \pi ( x-\epsilon t ) \bigr) e^{-\rho\pi^{2} t}. \end{aligned}$$

The structure of this analysis is as follows. In Sect. 2, some preliminary definitions and theorems related to Grünwald–Letnikov fractional derivative are introduced. Section 3 is devoted to constructing the FDM for FIVPs (1) and (2) along with its theoretical analysis. In Sect. 4, the FVM is proposed to solve FIVPs (1) and (2) along with its theoretical analysis. In Sect. 4, numerical examples are given to express validity and reliability of the methods and to make a comprehensive comparison for both FVM and FDM. Finally, some conclusions are given in Sect. 5.

2 Basic definitions

In the literature there are numerous integrals which are widely used in physics, astronomy, and engineering [46, 47]. Consequently, different types of fractional integrals and derivatives have been found very reliable under some conditions. In this section, we introduce some preliminary definitions and theorems related to the Riemann–Liouville fractional derivative and Grünwald–Letnikov fractional derivative.

Definition 1

The Riemann–Liouville integral of fractional order $\alpha>0$, $J_{a}^{\alpha} \psi ( x )$, is defined as

$$\begin{aligned} J_{a}^{\alpha} \psi ( x ) = \frac{1}{\Gamma ( \alpha )} \int_{a}^{x} ( x-t )^{\alpha-1} \psi ( t ) \,dt, \quad\text{provided that } \psi\in L_{1} [a,b]. \end{aligned}$$

For $\alpha=0$, the Riemann–Liouville integral of fractional order $J_{a}^{0} \psi ( x ) =\psi ( x )$ is the identity operator.

Definition 2

Let n be the smallest integer that exceeds α, then the Riemann–Liouville fractional derivative of order $\alpha> 0$ is defined as

$$\begin{aligned} \mathcal{D}_{a}^{\alpha} \psi ( x ) &= \mathcal{D}^{n} J_{a}^{ ( n-\alpha )} \psi ( x ) \\ &= \frac{1}{\Gamma ( n-\alpha )} \biggl( \frac{d^{n}}{d x^{n}} \biggr) \biggl[ \int_{a}^{x} \frac{\psi ( t )}{ ( x-t )^{\alpha+1-n}} \,dt \biggr]. \end{aligned}$$

(3)

For $\alpha\mathbb{\in N,}$ the Riemann–Liouville fractional derivative of order α is given as $\mathcal{D}_{a}^{\alpha} \psi ( x ) = \frac{d^{\alpha} \psi ( x )}{d x^{\alpha}}$.

Definition 3

Let $\psi\in C^{ \lceil\alpha \rceil} [ a,b ], \alpha> 0$. Then the Grünwald–Letnikov fractional derivative of the function ψ of order α is given by

$$\begin{aligned} \tilde{\mathcal{D}} {}_{a}^{\alpha} \psi ( x ) = \lim _{h\rightarrow0} \frac{1}{h^{\alpha}} \sum_{k=0}^{ [ \frac {x-a}{h} ]} ( -1 )^{k} \binom{\alpha}{k} \psi ( x-kh ),\quad a< x\leq b, \end{aligned}$$

(4)

where $h= \frac{x-a}{N}$.

The following theorem shows the relation between this definition and the Riemann–Liouville fractional derivatives.

Theorem 1

Let $\alpha>0, n= \lceil\alpha \rceil$ and $\psi\in C^{n} [ a,b ]$. Then we have

$$\begin{aligned} \tilde{\mathcal{D}} {}_{a}^{\alpha} \psi ( x ) = \mathcal{D}_{a}^{\alpha} \psi ( x ),\quad a< x\leq b. \end{aligned}$$

Theorem 2

Let $\alpha> 0$ and $\psi\in C [ a,b ]$. Then we have

$$\begin{aligned} J {}_{a}^{\alpha} \psi ( x ) = \lim_{h\longrightarrow0} h^{\alpha} \sum_{k=0}^{ [ \frac{x-a}{h} ]} ( -1 )^{k} \binom{-\alpha}{ k} \psi ( x-kh ),\quad h= \frac{x-a}{N}, a< x\leq b, \end{aligned}$$

(5)

where $( -1 )^{k} \binom{-\alpha}{k} = \frac{\alpha ( \alpha-1 ) ( \alpha-2 ) \dots ( \alpha-k+1 )}{k!} = \frac{\Gamma ( \alpha+k )}{\Gamma ( \alpha ) \Gamma ( k+1 )}$ and $\Gamma(x)$ is the gamma function given by $\Gamma ( x ) = \int_{0}^{\infty} s^{x-1} e^{-s} \,ds$. If the weights are given by $\upsilon_{0}^{\alpha} =1, \upsilon_{1}^{\alpha} =\alpha$ and $\upsilon_{k}^{\alpha} = ( 1- \frac{ ( 1-\alpha )}{k} ) \upsilon_{k-1}^{\alpha},k=2,3,\dots$, then Eq. (5) may be rewritten as

$$\begin{aligned} J {}_{a}^{\alpha} \psi ( x )=\lim_{ h\rightarrow0} h^{\alpha} \sum_{k=0}^{ [ \frac{x-a}{h} ]} \upsilon_{k}^{\alpha} \psi ( x-kh ),\quad h= \frac{x-a}{N}. \end{aligned}$$

(6)

The above formula is used to approximate the fractional integral $J_{a}^{\alpha} \psi ( x )$.

Lemma 1

Let $0<\alpha<1$. Then we have

• 1. $v_{0}^{\alpha} =1$ and $v_{k}^{\alpha} >0$ for $k=1,2,\dots$.

• 2. $v_{k}^{\alpha} - v_{k+1}^{\alpha} >0$ for $=0,1,\dots$.

• 3. $\lim_{k\rightarrow\infty} v_{k}^{\alpha} =0$.

Proof

1. By induction and the recursive definition, we indeed have $v_{0}^{\alpha} =1$, $v_{1}^{\alpha} =\alpha>0$, and

$$\begin{aligned} v_{k}^{\alpha} = \biggl( 1- \frac{ ( 1-\alpha )}{k} \biggr) v_{k-1}^{\alpha},\quad k=2,3,\dots. \end{aligned}$$

(7)

Since $0<\alpha<1$, we have $0< \frac{1-\alpha}{k} < \frac{1}{k} <1$ for $k\geq2$. Therefore, the coefficient $( 1- \frac{ ( 1-\alpha )}{k} )$ in Eq. (7) strictly lies between zero and one.

2. For $k\geq2$, we have

$$\begin{aligned} v_{k}^{\alpha} - v_{k+1}^{\alpha} &= \biggl( 1- \frac{ ( 1-\alpha )}{k} \biggr) v_{k-1}^{\alpha} - \biggl( 1- \frac{ ( 1-\alpha )}{k+1} \biggr) v_{k}^{\alpha} \\ &= \biggl( 1- \frac{ ( 1-\alpha )}{k} \biggr) v_{k-1}^{\alpha} - \biggl( 1- \frac{ ( 1-\alpha )}{k+1} \biggr) \biggl( 1- \frac{ ( 1-\alpha )}{k} \biggr) v_{k-1}^{\alpha} \\ &=\biggl[1- \biggl( 1- \frac{ ( 1-\alpha )}{k+1} \biggr) \biggr] \biggl( 1- \frac{ ( 1-\alpha )}{k} \biggr) v_{k-1}^{\alpha} \\ &= \frac{ ( 1-\alpha )}{k+1} \biggl( 1- \frac{ ( 1-\alpha )}{k} \biggr) v_{k-1}^{\alpha}. \end{aligned}$$

The result is, therefore, positive. Part 3 is straightforward. Hence, we omit the details. This completes the proof of the lemma. □

3 Finite difference scheme formulation

In this section, we propose a new finite difference scheme formulation (FDSF) for solving the SFCDE with constant coefficients

$$\begin{aligned} \frac{\partial\psi(x,t)}{\partial t} +\epsilon\frac{\partial^{\beta} \psi(x,t)}{\partial x^{\beta}} = \rho\frac{\partial^{2} \psi ( x,t )}{\partial x^{2}} \end{aligned}$$

(8)

subject to the initial condition

$$\begin{aligned} \psi ( x,0 ) =g ( x ),\quad a\leq x\leq b, \end{aligned}$$

(9)

in which $t\geq0, 0<\beta\leq1, g(x)$ is an analytical smooth function of a spatial x, $\psi ( x,t )$ is an unknown analytical function, ϵ and ρ are positive parameters, and β is a parameter that describes the order of the space fractional, where the space fractional derivative is described in the Riemann–Liouville sense. If $\beta=1$, then the equation becomes a linear convection–diffusion equation. Now, by utilizing Definition 2 and Riemann–Liouville fractional derivative, we have

$$\begin{aligned} \frac{\partial\psi(x,t)}{\partial t} +\epsilon\frac{\partial }{\partial x} J_{a}^{1-\beta} \psi ( x,t ) =\rho\frac{\partial^{2} \psi ( x,t )}{\partial x^{2}}, \end{aligned}$$

(10)

where $J_{a}^{1-\beta}$ is the Riemann–Liouville integral with respect to x. If $\alpha=1-\beta$, then we have $0\leq\alpha<1$. Define $t_{n} =n\tau, n=0,1,\dots, M$; $x_{i} =a+ih, i=0,1,\dots,N$, where τ is the time step and h is the spatial step. Then, we will be using the standard Grünwald formula to approximate the α order fractional Riemann–Liouville integral and the central difference formula to approximate the first and the second derivative as follows:

$$\begin{aligned} &J_{a}^{\alpha} \psi ( x,t ) = h^{\alpha} \sum _{j=0}^{N} v_{j}^{\alpha} \psi(x-jh,t) + o(1), \end{aligned}$$

(11)

$$\begin{aligned} & \frac{\partial\psi(x,t)}{\partial x} \bigg\vert _{x= x_{i}} = \frac{\psi ( x_{i+1},t ) -\psi( x_{i-1},t)}{2 h} + \mathcal{O}\bigl( h^{2} \bigr), \end{aligned}$$

(12)

$$\begin{aligned} & \frac{\partial^{2} \psi(x,t)}{\partial x^{2}} \bigg\vert _{x= x_{i}} = \frac{\psi ( x_{i-1},t ) -2\psi ( x_{i},t ) +\psi( x_{i+1},t)}{h^{2}} + \mathcal{O} \bigl( h^{2} \bigr). \end{aligned}$$

(13)

The starting point for applying the finite difference method by evaluating Eq. (10) at $x= x_{i}$ and using the above equations is that

$$\begin{aligned} \frac{d\psi( x_{i},t)}{dt} ={}&{ -} \frac{\epsilon}{2 h} \Biggl[ h^{\alpha} \sum _{j=0}^{i+1} v_{j}^{\alpha} \psi ( x_{i-j+1},t ) - h^{\alpha} \sum_{j=0}^{i-1} v_{j}^{\alpha} \psi ( x_{i-j-1},t ) \Biggr] \\ &{}+\rho \biggl[ \frac{\psi ( x_{i-1},t ) -2\psi ( x_{i},t ) +\psi ( x_{i+1},t )}{h^{2}} \biggr]. \end{aligned}$$

(14)

Hence, using the standard backward difference to approximate the temporal derivative in Eq. (14) reveals

$$\begin{aligned} \frac{d\psi( x_{i},t)}{dt} \bigg\vert _{t= t_{n+1}} = \frac {\psi ( x_{i}, t_{n+1} ) -\psi( x_{i}, t_{n} )}{\tau} + \mathcal{O}( \tau). \end{aligned}$$

(15)

Letting $\psi_{i}^{n} \approx\psi( x_{i}, t_{n} )$ denotes that the numerical solution implies

$$\begin{aligned} &\frac{\psi_{i}^{n+1} - \psi_{i}^{n}}{\tau} \\ &\quad=- \frac{\epsilon}{2 h} \Biggl[ h^{\alpha} \sum_{j=0}^{i+1} v_{j}^{\alpha} \psi_{i-j+1}^{n+1} - h^{\alpha} \sum_{j=0}^{i-1} v_{j}^{\alpha} \psi_{i-j-1}^{n+1} \Biggr] +\rho \biggl[ \frac{\psi_{i-1}^{n+1} -2 \psi_{i}^{n+1} + \psi_{i+1}^{n+1}}{h^{2}} \biggr]. \end{aligned}$$

(16)

By collecting like terms, Eq. (16) can be rewritten as

$$\begin{aligned} \frac{\psi_{i}^{n+1} - \psi_{i}^{n}}{\tau} + \sum_{j=0}^{N} b_{ij} \psi_{j}^{n+1} =0, \end{aligned}$$

(17)

where

$$\begin{aligned} b_{ij} = \textstyle\begin{cases} \frac{\epsilon h^{\alpha-1} [ v_{i-j+1}^{\alpha} - v_{i-j-1}^{\alpha } ]}{2},& j< i-1,\\ \frac{\epsilon h^{\alpha-1} [ v_{2}^{\alpha} - v_{0}^{\alpha} ]}{2} - \frac{\rho}{h^{2}},& j=i-1, \\ \frac{\epsilon h^{\alpha-1} v_{1}^{\alpha}}{2} + \frac{2\rho }{h^{2}},& j=i, \\ \frac{\epsilon h^{\alpha-1} v_{0}^{\alpha}}{2} - \frac{\rho }{h^{2}},& j=i+1,\\ 0,& j>i+1. \end{cases}\displaystyle \end{aligned}$$

(18)

Denoting the numerical solution vector by $\psi^{n} =[ \psi_{0}^{n}, \psi_{1}^{n},\dots, \psi_{N}^{n} ]$, the vector equation becomes

$$\begin{aligned} ( I+\tau A ) \psi^{n+1} = \psi^{n}, \end{aligned}$$

(19)

where A is a matrix with elements $a_{ij} = b_{ij}$. Hence, the above given equation can be formatted as

$$\begin{aligned} \psi^{n+1} =M \psi^{n}, \end{aligned}$$

(20)

where $M= (I+\tau A)^{-1}$ is the iteration matrix.

The next theorems show that the finite difference scheme FDS expressed in Eq. (20) is conditionally stable and consistent with first-order accurate in time and second-order accurate in space.

We commence to prove the strictly diagonally dominant property for the iteration matrix M.

Theorem 3

Let $\epsilon>0,\rho>0, \textit{ and } 0\leq\alpha <1$ satisfy $\rho> \frac{\epsilon h^{\alpha+1}}{2}$, where h is the spatial step. Then the coefficients $a_{ij}$ satisfy the inequality

$$\begin{aligned} \vert a_{ii} \vert > \mathop{\sum_{j=0}}_{ j\neq i}^{N} \vert a_{ij} \vert ,\quad i=1,2,\dots,N. \end{aligned}$$

Proof

Let $i=1,2,\dots,N$ be given. Then consider the sum

$$\begin{aligned} \mathop{\sum_{j=0}}_{j\neq i}^{N} \vert a_{ij} \vert & = \sum_{j=0}^{i-2} \vert a_{ij} \vert + \vert a_{i,i-1} \vert + \vert a_{i,i+1} \vert + \sum_{j=i+2}^{N} \vert a_{ij} \vert \\ &= \sum_{j=0}^{i-2} \biggl\vert \frac{\epsilon h^{\alpha-1} [ v_{i-j+1}^{\alpha} - v_{i-j-1}^{\alpha} ]}{2} \biggr\vert + \biggl\vert \frac{\epsilon h^{\alpha-1} [ v_{2}^{\alpha} - v_{0}^{\alpha} ]}{2} - \frac{\rho}{h^{2}} \biggr\vert + \biggl\vert \frac{\epsilon h^{\alpha-1} v_{0}^{\alpha}}{2} - \frac{\rho}{h^{2}} \biggr\vert . \end{aligned}$$

Therefore, the previous lemma and the hypotheses $\rho> \frac {\epsilon h^{\alpha+1}}{2}$ guarantee that each term is negative. So, we have

$$\begin{aligned} \mathop{\sum_{j=0}}_{j\neq i}^{N} \vert a_{ij} \vert = \sum_{j=0}^{i-2} \frac{\epsilon h^{\alpha-1} [ v_{i-j-1}^{\alpha} - v_{i-j+1}^{\alpha} ]}{2} + \biggl( \frac{\rho}{h^{2}} - \frac{\epsilon h^{\alpha-1} [ v_{2}^{\alpha} - v_{0}^{\alpha} ]}{2} \biggr) + \biggl( \frac{\rho}{h^{2}} - \frac {\epsilon h^{\alpha-1} v_{0}^{\alpha}}{2} \biggr). \end{aligned}$$

Replacing the finite sum with the infinite sum yields

$$\begin{aligned} \mathop{\sum_{j=0}}_{ j\neq i}^{N} \vert a_{ij} \vert < \sum_{j=-\infty}^{i-2} \frac {\epsilon h^{\alpha-1} [ v_{i-j-1}^{\alpha} - v_{i-j+1}^{\alpha} ]}{2} + \biggl( \frac{\rho}{h^{2}} - \frac{\epsilon h^{\alpha-1} [ v_{2}^{\alpha} - v_{0}^{\alpha} ]}{2} \biggr) + \biggl( \frac{\rho}{h^{2}} - \frac {\epsilon h^{\alpha-1} v_{0}^{\alpha}}{2} \biggr). \end{aligned}$$

Therefore, the telescoping sum has the form $( v_{1}^{\alpha} - v_{3}^{\alpha} ) + ( v_{2}^{\alpha} - v_{4}^{\alpha} ) + ( v_{3}^{\alpha} - v_{5}^{\alpha} ) + ( v_{4}^{\alpha} - v_{6}^{\alpha} )$. Now, b y Lemma 1, this equals $v_{1}^{\alpha} + v_{2}^{\alpha}$. Hence, we have obtained

$$\begin{aligned} \mathop{\sum_{j=0}}_{ j\neq i}^{N} \vert a_{ij} \vert &< \frac{\epsilon h^{\alpha-1} [ v_{1}^{\alpha} + v_{2}^{\alpha} ]}{2} + \biggl( \frac{\rho}{h^{2}} - \frac{\epsilon h^{\alpha-1} [ v_{2}^{\alpha} - v_{0}^{\alpha} ]}{2} \biggr) + \biggl( \frac{\rho}{h^{2}} - \frac{\epsilon h^{\alpha-1} v_{0}^{\alpha}}{2} \biggr) \\ &= \frac{\epsilon h^{\alpha-1} v_{1}^{\alpha}}{2} + \frac{2\rho }{h^{2}} = \vert a_{ii} \vert . \end{aligned}$$

This completes the proof of the theorem. □

Corollary 1

The iteration matrix M in Scheme (20) is convergent, so the scheme itself is conditionally stable.

Proof

By the previous theorem, we infer that A is strictly diagonally dominant with positive diagonal elements, and so is $I+\tau A$. Therefore, the iteration matrix $M= (I+\tau A)^{-1}$ exists, and its spectral radius satisfies

$$\begin{aligned} \rho ( M ) = \rho ( I+\tau A )^{-1} = \bigl( 1+\tau \rho ( A ) \bigr)^{-1} < 1. \end{aligned}$$

The proof is therefore completed. □

Theorem 4

Numerical Scheme (17) is consistent with second-order accuracy in space and first-order one in time.

Proof

By using Eqs. (11), (12), (13), and (15), we write Eq. (10) at ($x_{i}, t_{n+1}$) as follows:

$$\begin{aligned} &\frac{\psi_{i}^{n+1} - \psi_{i}^{n}}{\tau} +\mathcal{O}( \tau) \\ &\quad=- \frac{\epsilon}{2 h} \Biggl[ h^{\alpha} \sum_{j=0}^{i+1} v_{j}^{\alpha} \psi_{i-j+1}^{n+1} +o ( 1 ) - h^{\alpha} \sum_{j=0}^{i-1} v_{j}^{\alpha} \psi_{i-j-1}^{n+1} +o ( 1 ) \Biggr] \\ &\qquad{}+ \mathcal{O} \bigl( h^{2} \bigr) +\rho \biggl[ \frac{\psi_{i-1}^{n+1} -2 \psi_{i}^{n+1} + \psi_{i+1}^{n+1}}{h^{2}} \biggr] +\mathcal{O} \bigl( h^{2} \bigr). \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \frac{\partial\psi( x_{i},t)}{\partial t} \bigg\vert _{t= t_{n+1}} =-\epsilon \frac{\partial}{\partial x} \bigl[ J_{a}^{1-\beta} \psi ( x, t_{n+1} ) \bigr] \bigg\vert _{x= x_{i}} +\rho \frac{\partial^{2} \psi ( x, t_{n+1} )}{\partial x^{2}} \bigg\vert _{x= x_{i}}, \end{aligned}$$

which, indeed, leads to Eq. (10) at ($x_{i}, t_{n+1}$). The proof of the theorem is complete. □

Corollary 2

Numerical Scheme (17) is consistent and stable.

By the fundamental theorem of numerical methods for differential equations, the given numerical scheme is convergent.

4 Finite volume scheme formulation

In this section, we propose a novel finite volume scheme formulation (FVSF) for solving the SFCDE with constant coefficients

$$\begin{aligned} \frac{\partial\psi( x, t )}{\partial t} + \epsilon \frac{\partial^{\beta} \psi( x, t )}{\partial x^{\beta}} = \rho \frac{\partial^{2} \psi ( x, t )}{\partial x^{2}}, \end{aligned}$$

(21)

subject to the initial condition

$$\begin{aligned} \psi ( x,0 ) = g ( x ),\quad a \leq x \leq b, \end{aligned}$$

(22)

where $t \geq0, 0< \beta\leq1, g ( x )$ is an analytical smooth function of spatial x, $\psi ( x, t )$ is an unknown analytical function, ϵ and ρ are positive parameters, and β is a parameter that describes the order of the space fractional, where the space fractional derivative is described in the Riemann–Liouville sense. If $\beta=1$, it becomes a linear convection–diffusion equation.

Now, to establish the finite volume scheme, we need to partition the finite domain $\Omega=[ a, b ]$ to $N +1$ uniformly spaced nodes $x_{i} = a + ih, i =0,1,\dots, N$, where the spatial step is $h = \frac{b - a}{N}$. Thus, by utilizing the Riemann–Liouville fractional derivative, we have

$$\begin{aligned} \frac{\partial\psi( x, t )}{\partial t} + \epsilon\frac{\partial }{\partial x} J_{a}^{1- \beta} \psi ( x, t ) = \rho \frac{\partial^{2} \psi ( x, t )}{\partial x^{2}},\quad 0< \beta \leq1, \end{aligned}$$

(23)

where $J_{a}^{1- \beta}$ is the Riemann–Liouville integral with respect to x. Setting $\alpha=1- \beta$ gives $0\leq\alpha<1$. Integrating Eq. (23) over the ith control volume [$x_{i - \frac{1}{2}}, x_{i + \frac{1}{2}}$] suggests

$$\begin{aligned} \int_{x_{i - \frac{1}{2}}}^{x_{i + \frac{1}{2}}} \frac{\partial\psi( x, t )}{\partial t} \,dx= \int_{x_{i - \frac{1}{2}}}^{x_{i + \frac{1}{2}}} \frac{\partial}{\partial x} \biggl[- \epsilon J_{a}^{\alpha} \psi ( x, t ) + \rho\frac{\partial u ( x, t )}{\partial x} \biggr] \,dx. \end{aligned}$$

(24)

Dividing each side by h gives the standard finite volume discretization

$$\begin{aligned} \frac{d \overline{\psi}_{i} ( t )}{dt} = \frac{\epsilon}{h} \bigl[ J_{a}^{\alpha} \psi( x_{i - \frac{1}{2}}, t ) - J_{a}^{\alpha} \psi( x_{i + \frac{1}{2}}, t ) \bigr] + \frac{\rho}{h} \biggl[ \frac{\partial\psi ( x_{i + \frac{1}{2}}, t )}{\partial x} - \frac{\partial\psi ( x_{i - \frac{1}{2}}, t )}{\partial x} \biggr], \end{aligned}$$

(25)

where

$$\begin{aligned} \overline{\psi}_{i} ( t ) = \frac{1}{h} \int_{x_{i - \frac{1}{2}}}^{x_{i + \frac{1}{2}}} \psi ( x, t ) \,dx \end{aligned}$$

is the control volume averages of $\psi( x, t )$.

Using the fractionally-shift Grünwald formula to approximate $J_{a}^{\alpha} \psi( x, t )$ yields

$$\begin{aligned} J {}_{a}^{\alpha} \psi ( x, t ) = \lim h^{\alpha}_{ h \rightarrow 0} \sum_{j =0}^{ [ \frac{x - a}{h} + p ]} \frac{\Gamma( \alpha + j )}{\Gamma ( \alpha ) \Gamma( j +1)} \psi \bigl( x - ( j - p ) h, t \bigr), \end{aligned}$$

(26)

in which $h = \frac{x - a}{N}, x \in ( a, b ]$. Hence, the Riemann–Liouville integral includes the order term

$$\begin{aligned} J_{a}^{\alpha} \psi ( x, t ) = h^{\alpha} \sum _{j =0}^{N + p} v_{j}^{\alpha} \psi \bigl( x - ( j - p ) h, t \bigr) + o ( 1 ). \end{aligned}$$

(27)

Therefore, due to certain needs of the fractional shift $p = \frac{1}{2}$, we write

$$\begin{aligned} & J_{a}^{\alpha} \psi( x, t ) \vert_{x = x_{i - \frac{1}{2}}} = J_{a}^{\alpha} \psi ( x_{i - \frac{1}{2}}, t ) + h^{\alpha} \sum_{j =0}^{i} v_{j}^{\alpha} \psi ( x_{i - j}, t ) + o (1), \end{aligned}$$

(28)

$$\begin{aligned} & J_{a}^{\alpha} \psi( x, t ) \vert_{x = x_{i + \frac{1}{2}}} = J_{a}^{\alpha} \psi ( x_{i + \frac{1}{2}}, t ) + h^{\alpha} \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi ( x_{i - j +1}, t ) + o (1), \end{aligned}$$

(29)

where $v_{0}^{\alpha} =1, v_{1}^{\alpha} = \alpha$, and $v_{j}^{\alpha} = ( 1- \frac{ ( 1- \alpha )}{j} ) v_{j -1}^{\alpha}, j =2,3,\dots$. For $\frac{\partial\psi( x, t )}{\partial x}$, use the identity

$$\begin{aligned} \frac{\partial\psi ( x_{i}, t )}{\partial x} = \frac{\psi ( x_{i +1}, t ) - \psi ( x_{i -1}, t )}{2 h} +\mathcal{O} \bigl( h^{2} \bigr) \end{aligned}$$

(30)

to write

$$\begin{aligned} &\frac{\partial\psi ( x_{i - \frac{1}{2}}, t )}{\partial x} \approx\frac{\psi( x_{i + \frac{1}{2}}, t )- \psi( x_{i - \frac {3}{2}}, t )}{2 h}, \end{aligned}$$

(31)

$$\begin{aligned} &\frac{\partial\psi ( x_{i + \frac{1}{2}}, t )}{\partial x} \approx\frac{\psi ( x_{i + \frac{3}{2}}, t ) - \psi ( x_{i - \frac{1}{2}}, t )}{2 h}. \end{aligned}$$

(32)

For Eqs. (31) and (32), use the standard averaging scheme $\psi ( x_{i + \frac{1}{2}}, t ) \approx\frac{[ \psi ( x_{i}, t ) + \psi ( x_{i +1}, t ) ]}{2}$ to construct approximations of first derivative in terms of function values at the nodes $x_{j}$:

$$\begin{aligned} &\frac{\partial\psi ( x_{i - \frac{1}{2}}, t )}{\partial x} \approx\frac{1}{4 h} \bigl[ \psi ( x_{i}, t ) + \psi ( x_{i +1}, t ) - \psi ( x_{i -2}, t ) - \psi ( x_{i -1}, t ) \bigr], \end{aligned}$$

(33)

$$\begin{aligned} &\frac{\partial\psi ( x_{i + \frac{1}{2}}, t )}{\partial x} \approx\frac{1}{4 h} \bigl[ \psi ( x_{i +1}, t ) + \psi ( x_{i +2}, t ) - \psi ( x_{i -1}, t ) - \psi ( x_{i}, t ) \bigr]. \end{aligned}$$

(34)

So, Eq. (25) can be approximated as

$$\begin{aligned} \frac{d \overline{\psi}_{i} ( t )}{dt}={}&\epsilon h^{\alpha-1} \Biggl[ \sum _{j =0}^{i} v_{j}^{\alpha} \psi ( x_{i - j}, t ) - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi ( x_{i - j +1}, t ) \Biggr] \\ &{}+ \frac{\rho}{4 h^{2}} \bigl[ \psi ( x_{i -2}, t ) -2 \psi ( x_{i}, t ) + \psi( x_{i +2}, t ) \bigr]. \end{aligned}$$

(35)

Notice that if $\psi( x, t )$ is a smooth function, then the value of the control volume averages $\overline{\psi}_{i} ( t )$ will agree with the value of $\psi( x, t )$ at the midpoint of the interval [$x_{i - \frac{1}{2}}, x_{i + \frac{1}{2}}$] to $\mathcal{O} ( h^{2} )$. So, we can rewrite Eq. (35) as

$$\begin{aligned} \frac{d\psi ( x_{i}, t )}{dt}={}&\epsilon h^{\alpha-1} \Biggl[ \sum _{j =0}^{i} v_{j}^{\alpha} \psi ( x_{i - j}, t ) - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi ( x_{i - j +1}, t ) \Biggr] \\ &{}+ \frac{\rho}{4 h^{2}} \bigl[ \psi ( x_{i -2}, t ) -2 \psi ( x_{i}, t ) + \psi ( x_{i +2}, t ) \bigr]. \end{aligned}$$

(36)

Now, define a temporal partition $t_{n} = n\tau, n =0,1,\dots$, where τ is the time step, and use the standard backward difference to approximate the temporal derivative in Eq. (36) such that

$$\begin{aligned} \frac{d\psi ( x_{i}, t )}{dt} \bigg\vert _{t = t_{n +1}} = \frac{\psi ( x_{i}, t_{n +1} ) - \psi ( x_{i}, t_{n} )}{\tau} + \mathcal{O} ( \tau ). \end{aligned}$$

Let $\psi_{i}^{n} \approx\psi( x_{i}, t_{n} )$ denote the numerical solution. Then we have

$$\begin{aligned} \frac{\psi_{i}^{n +1} - \psi_{i}^{n}}{\tau} ={}&\epsilon h^{\alpha-1} \Biggl[ \sum _{j =0}^{i} v_{j}^{\alpha} \psi_{i - j}^{n +1} - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi_{i - j +1}^{n +1} \Biggr] \\ &{}+ \frac{\rho}{4 h^{2}} \bigl[ \psi_{i -2}^{n +1} -2 \psi_{i}^{n +1} + \psi_{i +2}^{n +1} \bigr]. \end{aligned}$$

(37)

By collecting like terms, we rewrite Eq. (37) in the form

$$\begin{aligned} \frac{\psi_{i}^{n +1} - \psi_{i}^{n}}{\tau} = \frac{1}{h} \sum_{j =0}^{N} k_{ij} \psi_{j}^{n +1},\quad i =0,1,2,\dots, N, \end{aligned}$$

(38)

where $k_{ij}$ is given by

$$\begin{aligned} k_{ij}= \textstyle\begin{cases} \epsilon h^{\alpha} [ v_{i - j}^{\alpha} - v_{i - j +1}^{\alpha} ],& j < i -2, \\ \epsilon h^{\alpha} [ v_{2}^{\alpha} - v_{3}^{\alpha} ] + \frac{\rho}{4 h},& j = i -2, \\ \epsilon h^{\alpha} [ v_{1}^{\alpha} - v_{2}^{\alpha} ],& j = i -1, \\ \epsilon h^{\alpha} [ v_{0}^{\alpha} - v_{1}^{\alpha} ] - \frac{\rho}{2 h},& j = i, \\ \epsilon h^{\alpha} [ - v_{0}^{\alpha} ],& j = i +1, \\ \frac{\rho}{4 h},& j = i +2, \\ 0,& j > i +2. \end{cases}\displaystyle \end{aligned}$$

The vector of the numerical solution $\psi^{n} =[ \psi_{0}^{n}, \psi_{1}^{n},\dots, \psi_{N}^{n} ]$ can be denoted by

$$\begin{aligned} \biggl( I + \frac{\tau}{h} A \biggr) \psi^{n +1} = \psi^{n}, \end{aligned}$$

(39)

where the matrix A has the elements $a_{ij} = - k_{ij}$.

The next two theorems explain that the finite volume scheme described in Eq. (37) is conditionally stable and consistent with first-order accuracy in time and second-order accuracy in space.

Theorem 5

For $i =0,1,\dots, N$, the numerical scheme $\frac{\psi_{i}^{n +1} - \psi_{i}^{n}}{\tau} = \frac{1}{h} \sum_{j =0}^{N} k_{ij} \psi_{j}^{n +1}$ is conditionally stable.

Proof

Substitute $\psi_{i}^{n} = \hat{\psi}^{n} \exp ( iI\xi )$ into the numerical scheme to get

$$\begin{aligned} \frac{\psi_{i}^{n +1} - \psi_{i}^{n}}{\tau} = \frac{1}{h} \sum_{j =0}^{N} k_{ij} \psi_{j}^{n +1},\quad i =0,1,2,\dots, N. \end{aligned}$$

However, we have

$$\begin{aligned} \hat{\psi}^{n +1} \exp ( iI\xi ) - \psi^{n} \exp ( iI\xi ) = r \sum_{j =0}^{N} k_{ij} \hat{ \psi}^{n +1} \exp ( jI\xi ),\qquad r = \frac{\tau}{h}, \hat{ \psi}^{n +1} = \rho ( \xi ) \hat{\psi}^{n}, \end{aligned}$$

where

$$\begin{aligned} \rho ( \xi ) = \frac{1}{ [ 1- r \sum_{j =0}^{N} k_{ij} \exp ( ( j - i ) I\xi ) ]} \end{aligned}$$

satisfies the von Neumann condition when $\vert 1- r \sum_{j =0}^{N} k_{ij} \exp ( ( j - i ) I\xi) \vert \geq1$. Therefore, by using the reverse triangle inequality, we have

$$\begin{aligned} \Biggl\vert 1- r \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr\vert \geq\Biggl| 1- \Biggl| r \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr| \Biggl| . \end{aligned}$$

Hence, the von Neumann condition will be satisfied whenever

$$\begin{aligned} \Biggl\vert 1-\Biggl\vert r \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr\vert \Biggr\vert \geq1. \end{aligned}$$

That is, either

$$\begin{aligned} 1- \Biggl\vert r \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr\vert \geq1, \end{aligned}$$

which is impossible to hold, or

$$\begin{aligned} 1- \Biggl\vert r \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr\vert \leq-1, \end{aligned}$$

which is equivalent to $\vert \sum_{j =0}^{N} k_{ij} \exp ( ( j - i ) I\xi) \vert \geq\frac{2}{r}$.

Thus, the symbol $\rho ( \xi )$ of the numerical scheme satisfies the von Neumann condition if

$$\begin{aligned} \Biggl\vert \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr\vert \geq \frac{2}{r},\quad \forall i =0,1,2,\dots, N, \end{aligned}$$

which means that the numerical scheme is conditionally stable. This completes the proof of the theorem. □

Theorem 6

Numerical Scheme (37) is consistent with second-order accuracy in space and first-order one in time.

Proof

Insert $\psi( x, t )$ into expression (37) to get

$$\begin{aligned} &\frac{\psi ( x_{i}, t_{n +1} ) - \psi( x_{i}, t_{n} )}{\tau}\\ &\quad=\epsilon h^{\alpha-1} \Biggl[ \sum _{j =0}^{i} v_{j}^{\alpha } \psi( x_{i - j}, t_{n +1} ) - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi( x_{i - j +1}, t_{n +1} ) \Biggr] \\ &\qquad{}+\frac{\rho}{4 h^{2}} \bigl[ \psi( x_{i -2}, t_{n +1} ) -2 \psi( x_{i -2}, t_{n +1} ) + \psi( x_{i +2}, t_{n +1} ) \bigr], \\ &T_{i}^{n} = \frac{\psi ( x_{i}, t_{n +1} ) - \psi( x_{i}, t_{n} )}{\tau} - \epsilon h^{\alpha-1} \Biggl[ \sum_{j =0}^{i} v_{j}^{\alpha} \psi( x_{i - j}, t_{n +1} ) - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi( x_{i - j +1}, t_{n +1} ) \Biggr] \\ &\phantom{T_{i}^{n} =}{}- \frac{\rho}{4 h^{2}} \bigl[ \psi( x_{i -2}, t_{n +1} ) -2 \psi( x_{i -2}, t_{n +1} ) + \psi( x_{i +2}, t_{n +1} ) \bigr]. \end{aligned}$$

By the control volume averages, combined with a temporal backward difference for the time derivative,

$$\begin{aligned} \frac{d \overline{\psi}_{i} ( t_{n +1} )}{dt} = \frac{\psi ( x_{i}, t_{n +1} ) - \psi ( x_{i}, t_{n} )}{\tau} +\mathcal{O} \bigl( \tau+ h^{2} \bigr), \end{aligned}$$

we have

$$\begin{aligned} T_{i}^{n} = {}& \frac{d \overline{\psi}_{i} ( t_{n +1} )}{dt} -\mathcal{O} \bigl( \tau+ h^{2} \bigr) - \epsilon h^{\alpha-1} \Biggl[\sum_{j =0}^{i} v_{j}^{\alpha} \psi_{i - j}^{n +1} - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi_{i - j +1}^{n +1} \Biggr] \\ &{}- \frac{v}{4 h^{2}} \bigl[ \psi_{i -2}^{n +1} -2 \psi_{i}^{n +1} + \psi_{i +2}^{n +1} \bigr]. \end{aligned}$$

Substitute (25), (27), and (30) above, it turns to $T_{i}^{n} =\mathcal{O} ( \tau+ h^{2} )$. Hence, $T_{i}^{n} \rightarrow0$ as $\tau \rightarrow0$ and $h \rightarrow0$. So, the given numerical scheme is consistent with second-order accuracy in space and first-order in time. This completes the proof of the theorem. □

Corollary 3

Numerical Scheme (37) is consistent and stable.

By the fundamental theorem of numerical methods for differential equations, the given numerical scheme is convergent.

5 Numerical experiments

In this section, we introduce the numerical problems to verify the correctness and effectiveness of the proposed schemes. Both schemes can efficiently be solved, but the FVSF is more accurate than the FDSF. Two examples are introduced by comparing the numerical solutions with the exact solutions. The computations are performed by Mathematica Software 11.0.

Example 1

Consider the following SFCDE:

$$\begin{aligned} \frac{\partial u ( x,t )}{\partial t} +\epsilon \frac{\partial^{\beta} u ( x,t )}{\partial x^{\beta}} =\rho \frac{\partial^{2} u ( x,t )}{\partial x^{2}} \end{aligned}$$

(40)

subject to the initial condition

$$\begin{aligned} u ( x,0 ) =- \sin ( \pi x ), \end{aligned}$$

(41)

where $x\in [ 1,2.5 ], t\geq0, \epsilon=1, \rho=2$, and $0< \beta\leq1$. For $\beta=1$, the exact solution is $u ( x,t ) =- \sin ( \pi ( x-t ) ) \exp(-2 ( \pi^{2} t ) )$. The proposed schemes are applied to solve the above problem for different values of β such that $\beta\in \{ 0.7, 0.8, 0.9,1 \}$, by fixed $h =0.015625$ and $\tau=0.01$, h and τ are chosen small enough to demonstrate the accuracy of our schemes. In Table 1, we show the numerical results at $t=1$ with varying β such that $\beta\in \{ 0.7, 0.8, 0.9 \}$ and $x\in [ 1, 1.25 ]$ using the FDSF and the FVSF; while the absolute errors between the numerical solution in both schemes and the exact solution are reported in Table 2. When $\beta=1, t=1$, and $x\in [ 1, 1.25 ]$, this table demonstrates that there is an agreement between the approximate solution and the exact solution, which confirms that the proposed schemes are effective. According to our observations, the FVSF is more accurate than the FDSF. The 2D plot of the solutions behavior with varying β such that $\beta\in \{ 0.7, 0.8, 0.9 \}$ is drawn in Fig. 1 at $t=1$ and $x \in [ 1, 2.5 ]$ using the FVSF and the FDSF; while Fig. 2 shows that FVSF provides an excellent agreement with the exact solution rather than the FDSF. Moreover, the 3D plot of the exact solution behavior and the solutions behavior using the FVSF is drawn in Fig. 3 at $\beta=1, t \in [ 0,1 ]$ and $x\in [ 1, 2.5 ]$.

Figure 1

Solutions to Example 1 at time $t=1$ for varying β using FDSF and FVSF

Figure 2

Exact and approximate solutions to Example 1 at time $t=1$ for $\beta=1$ using FDSF and FVSF. Here black and green lines coincide to some extent

Figure 3

Surface plot to Example 1 at $\beta=1$ exact solution (left) and using FVSF (right)

Table 1 Numerical results for Example 1 with varying β at time $t=1$, fixed $h=0.015625$ and $\tau=0.01$ using FDSF and FVSF

Table 2 Exact solution and absolute errors for Example 1 at time $t=1$, $\beta=1$, fixed $h=0.015625$ and $\tau=0.01$ using FDSF and FVSF

Example 2

Consider the following SFCDE:

$$\begin{aligned} \frac{\partial u(x,t)}{\partial t} +\epsilon\frac{\partial^{\beta} u ( x,t )}{\partial x^{\beta}} =\rho\frac{\partial^{2} u ( x,t )}{\partial x^{2}}, \end{aligned}$$

(42)

subject to the initial condition

$$\begin{aligned} u ( x,0 ) = e^{\zeta x}, \end{aligned}$$

(43)

where $x\in [ -2, 4 ], \epsilon=0.1, \rho=0.02, t\geq0, 0<\beta\leq1$, and $\zeta=1.1771243444677$. For $\beta =1$, the exact solution is $u ( x,t ) = e^{\zeta x-mt},m=0.09$. The FDSF and FVSF are tested numerically to solve the above problem for different values of β such that $\beta\in \{ 0.5, 0.7, 0.9, 1 \}$, by fixed $h=0.0625$ and $\tau=0.01$. Here, h and τ are chosen small enough to demonstrate the accuracy of our schemes. The numerical results at $t=1$ with varying β such that $\beta\in \{ 0.5, 0.7, 0.9 \}$ and $x\in [ -2, -1.25 ]$ using the FDSF and the FVSF are reported in Table 3. When $\beta=1, t=1$, and $x\in [ -2, -1.25 ]$, the numerical results are reported in Table 4. These results demonstrate that the FVSF is more significantly accurate than the FDSF, and it can be observed through by the absolute errors between the numerical solution in both schemes and the exact solution. The 2D plot of the solutions behavior with varying β such that $\beta\in \{ 0.5, 0.7, 0.90 \}$ is drawn in Fig. 4 at $t=1$ and $x\in [ -2, 1 ]$ using the FDSF and the FVSF, while Fig. 5 shows that the FVSF provides an excellent agreement with the exact solution rather than the FDSF. Moreover, the 3D plot of the exact solution behavior and the solutions behavior using the FVSF is drawn in Fig. 3 at $\beta=1, t\in [ 0,1 ]$, and $x\in [ -2, 1 ]$. Figure 6 shows the surface plot of the solution behavior at $\beta=1$ by using the FVSF and the FDSF at $t\in [ 0,1 ]$ and $x\in [ -2, 4 ]$.

Figure 4

Solutions to Example 2 at time $t=1$ for varying β using FDSF and FVSF

Figure 5

Exact and approximate solutions using FDSF and FVSF for Example 2 at time $t=1$ and $\beta=1$

Figure 6

Surface plot to Example 2 at $\beta=1$ exact solution (left) and using FVSF (right)

Table 3 Numerical results for Example 2 with varying β at time $t=1$, fixed $h= 0.0625$ and $\tau=0.01$ using FDSF and FVSF

Table 4 Exact solution and absolute errors for Example 2 at time $t=1$, $\beta=1$, fixed $h=0.0625$ and $\tau=0.01$

Table 3 and Table 4 depict that the solution obtained by the FVSF is more accurate and closer to the exact solution compared with the solution obtained by the FDSF in two different experiments. Also, it is easy to increase the nodes in the regular grid to get higher-order accuracy. Thus, it can be concluded that the FVM is effective for solving the SFCDE with a constant coefficient. This implicit scheme can be improved for implementation when tackling differential equations, in specific for problems with or without source terms that are deformed or nonlinear. Also, it can handle general condition without difficulty and can deal with unlimited kinds of the initial condition.

6 Conclusions

The SFCDE is one of the most important equations used to describe physical phenomena. In this paper, the FDM and the FVM have been developed for solving the SFCDE. The space fractional derivative in this equation was considered in the Riemann–Liouville sense. The Grünwald–Letnikov fractional formula has been used for the discretization of space fractional derivative, and in both formulations the implicit scheme has been introduced. It has been shown that both schemes are stable with first-order accuracy in time and second-order accuracy in space. Numerical results have been found using Mathematica software 11.0. Comparing illustrated examples justified the efficiency and accuracy of the proposed methods.