Application of fractional differential equation in economic growth model: A systematic review approach

Muhamad Deni Johansyah; Asep K. Supriatna; Endang Rusyaman; Jumadil Saputra; Muhamad Deni Johansyah; Asep K. Supriatna; Endang Rusyaman; Jumadil Saputra

doi:10.3934/math.2021594

AIMS Mathematics

2021, Volume 6, Issue 9: 10266-10280. doi: 10.3934/math.2021594

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Mini review

Application of fractional differential equation in economic growth model: A systematic review approach

1.
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Indonesia
2.
School of Social and Economic Development, Universiti Malaysia Terengganu, Malaysia

Received: 03 June 2021 Accepted: 07 July 2021 Published: 12 July 2021
MSC : 26A33, 34A08, 34A34, 34B10

In this paper we review the applications of fractional differential equation in economic growth models. This includes the theories about linear and nonlinear fractional differential equation, including the Fractional Riccati Differential Equation (FRDE) and its applications in economic growth models with memory effect. The method used in this study is by comparing related literatures and evaluate them comprehensively. The results of this study are the chronological order of the applications of the Fractional Differential Equation (FDE) in economic growth models and the development on theories of the FDE solutions, including the FRDE forms of economic growth models. This study also provides a comparative analysis on solutions of linear and nonlinear FDE, and approximate solution of economic growth models involving memory effects using various methods. The main contribution of this research is the chonological development of the theory to find necessary and sufficient conditions to guarantee the existence and uniqueness of the FDE in economic growth and the methods to obtain the solution. Some remarks on how further researches can be done are also presented as a general conclusion.

Keywords:

Fractional Order Derivative (FDE),
differential equation,
Fractional Riccati Differential Equation (FRDE),
economic growth model,
memory effect modeling

Citation: Muhamad Deni Johansyah, Asep K. Supriatna, Endang Rusyaman, Jumadil Saputra. Application of fractional differential equation in economic growth model: A systematic review approach[J]. AIMS Mathematics, 2021, 6(9): 10266-10280. doi: 10.3934/math.2021594

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Abstract

1. Introduction

Fractional-order differential equations (FDM) have many applications in various fields of engineering and science, such as chemical and physical phenomena ^[2,3,4,5]. For instance, the fractional-order diffusion equation is used to describe anomalous diffusion phenomena in the transport process through disordered and complex systems including fractal media, fractional kinetic equations regarding slow diffusion, and movement of small molecular along the concentration space ^[6].

In this article, the two-dimensional (2-D) time-fractional sub-diffusion equation (FSDE) with the weak singularity at initial time $t = 0$ is considered as follows:

$\begin{equation} \frac{\partial^{\alpha}V}{\partial t^{\alpha}} = \frac{\partial^{2}V}{\partial x^{2}} +\frac{\partial^{2}V}{\partial y^{2}} +G(x, y, t), \end{equation}$

(1.1)

with subject to conditions

$\begin{equation*} \begin{split} V(x, y, 0) = \zeta_0(x, y), \end{split} \end{equation*}$

and

$\begin{equation*} \begin{split} &V(0, y, t) = \zeta_1(x, y, t), \; V(L, y, t) = \zeta_2(x, y, t), \\& V(x, 0, t) = \zeta_3(x, y, t), \; V(x, L, t) = \zeta_4(x, y, t), \\& 0\leq x, y\leq M, \; 0\leq t\leq N, \end{split} \end{equation*}$

where $\zeta_0$ , $\zeta_1$ , $\zeta_2$ , $\zeta_3$ , $\zeta_4$ are known functions and $\alpha \in (0, 1)$ .

The FSDE can be obtained from the anomalous diffusion system by replacing the time derivative with a fractional derivative $\alpha$ where $0 < \alpha < 1$ . The FSDE is an important class of fractional partial differential equations (PDEs), which is mainly used in the modeling of fractional random walk, the phenomenon of wave propagation, diffusion unification, etc. ^[7,8].

The Caputo derivative of order $\alpha$ is

$\begin{equation} _{0}^{C}D_{t}^{\alpha} f(x) = \frac{1}{\Gamma (1-\alpha)}\int_{0}^{t} \frac{f'(x)}{(t-x)^{\alpha}} dx. \end{equation}$

(1.2)

The Caputo derivative approximated using the $L1$ gives the accuracy $2-\alpha$ ^[9,10], but the presence of kernel $(t-x)^{\alpha}$ produces solutions for Eq (1.1) with the weak-singularity at initial time $t = 0$ , which increases the computation cost and give low convergence rate for the approximate methods on uniform meshes ^[11]. Therefore, to increase the convergence rate many researchers solved the FSDE using different high-order numerical methods. Based on the chronology, several numerical methods are proposed for the solution of 2-D time FSDE (1.1), for example, Cui ^[12] proposed high-order alternating direction implicit (ADI) method, and it is unconditionally stable and convergent with convergence order $O(\nu^{\alpha}+h_0^{4})$ . Zhuang and Liu ^[13] proposed an unconditionally stable and convergent implicit difference scheme. Zhang et al. ^[14] used the Crank-Nicolson-type compact ADI scheme and proved the unconditional stability and convergence having convergence order $O(\nu^{min[2-\frac{\alpha}{2}, 2\alpha]}+h_0{1}^{4}+h_0{2}^{4})$ in $H_0^{1}$ norm. Ji and Sun ^[15] used a high-order numerical scheme to solve (1.1), and proved convergence in $L_{1}(L_{\infty})$ -norm and unconditional stability by the energy method. Wang et al. ^[16] solved 2-D FSDE using C-N alternating direction implicit finite difference method (FDM), where the fractional derivative is discretized using Riemann-Liouville fractional definition and to improve its temporal accuracy they used the Richardson extrapolation algorithm. Also, they proved its unique solvability, unconditional stability, and convergence $O(\nu^{2\gamma} + h_0{x}^{4} + h_0{y}^{4})$ of the scheme. Zhai and Feng ^[17] presented three different compact schemes for a 2-D time-fractional diffusion equation. The Caputo fractional definition is used for fractional derivative. All the schemes are fourth-order accurate for space and second-order accurate for the time variable. The stability of all the schemes is analyzed using Fourier analysis, which shows that two schemes are unconditionally stable, and the third one is conditionally stable.

The advantage of high-order schemes is that it produces more accurate results but at the same time increase the execution timings because of the escalated computational complexity of the scheme. Similarly, the advantage of explicit group methods over the standard point methods that the it considered quarter grid points of the solution domain and the points are considered as iterative points in the iterative process which reduces the computational complexity of the proposed method and hence reduce the execution time per iteration. Since, the computational complexity is greatly reduced using the explicit group method the 2-D time-fractional advection-diffusion, hyperbolic telegraph fractional differential, and fractional diffusion equations etc. ^{[18,19,20,21,22]} with second-order accuracy, therefore, we proposed the grouping strategy with uniform grids for the solution of the 2-D time FSDE with fourth-order accuracy. The purpose of this paper is to solve 2-D time FSDE with the fourth-order explicit group method (FEGM).

The paper is organized as follows: In Section 2, the derivation of the group explicit method from the finite difference method is presented. Section 3 discussed the stability of the proposed scheme, and the convergence of the proposed scheme is presented in Section 4. To show the efficiency of the proposed method, some numerical examples with discussion are presented in Section 5, and finally, Section 6 consists of the conclusion.

2. The fourth-order grouping scheme

First, let us define some notations:

$\begin{equation*} \begin{split} &\delta_{x}^2 V_{i, j}^{k} = V_{i+1, j}^{k}- 2 V_{i, j}^{k} + V_{i-1, j}^{k}, \; V_{i, j}^{k + \frac{1}{2} } = \frac{V_{i, j}^{k+1} + V_{i, j}^{k} }{2}, \\ & x_{i} = i h_0, \; y_{j} = j h_0, \; i, j = 0, 1, 2, 3..., M, \\ & t_{k} = k \nu, \; k = 0, 1, 2, 3..., N, \end{split} \end{equation*}$

where $h_0 = \Delta x = \Delta y = \frac{L}{M}$ represents the space step and $\nu = \frac{T}{N}$ time step.

Since, the Taylor series expansion with respect to $x$ is

$\begin{equation} V_{i+1, j}^{k} = V_{i, j}^{k}+ \frac{h_0}{1!}V_{x}|_{i, j}^{k} + \frac{h_0^{2}}{2!}V_{xx}|_{i, j}^{k} + \frac{h_0^{3}}{3!}V|_{i, j}^{k} + ..., \end{equation}$

(2.1)

$\begin{equation} V_{i-1, j}^{k} = V_{i, j}^{k}- \frac{h_0}{1!} V_{x}|_{i, j}^{k} + \frac{h_0^{2}}{2!} V_{xx}|_{i, j}^{k} - \frac{h_0^{3}}{3!} V_{xxx}|_{i, j}^{k} + ... \end{equation}$

(2.2)

By adding Eqs (2.1) and (2.2) and after rearranging, we get

$\begin{equation} \begin{split} \frac{\partial^2 V_{i, j}^{k}}{\partial x^2}& = \frac{V_{i+1, j}^{k}-2u_{i, j}^{k}+V_{i-1, j}^{k}}{h_0^{2}} + \frac{2h_0^{2}}{4!}\frac{\partial^4 V_{i, j}^{k} }{\partial x^4} +...\\& = \frac{V_{i+1, j}^{k}-2V_{i, j}^{k}+V_{i-1, j}^{k}}{h_0^{2}} + O(h_0^{2}). \end{split} \end{equation}$

(2.3)

Therefore, the Taylor series expansions at points $u_{i+1, j}^{k}$ and $u_{i, j+1}^{k}$ are

$\begin{equation} \frac{\delta^{2}_{x}}{h_{0}^2} V_{i, j}^{k} = \frac{\partial^2 V}{\partial x^2}|_{i, j}^{k} - \frac{h_{0}^{2}}{12}\frac{\partial^4 V}{\partial x^4}|_{i, j}^{k} - \frac{h_{0}^{4}}{360}\frac{\partial^6 V}{\partial x^6}|_{i, j}^{k} + O(h_0^{6}), \end{equation}$

(2.4)

$\begin{equation} \frac{\delta^{2}_{y}}{h_{0}^2} V_{i, j}^{k} = \frac{\partial^2 V}{\partial y^2}|_{i, j}^{k} - \frac{h_{0}^{2}}{12}\frac{\partial^4 V}{\partial y^4}|_{i, j}^{k} + \frac{h_{0}^{4}}{360}\frac{\partial^6 V}{\partial y^6}|_{i, j}^{k} + O(h_0^{6}). \end{equation}$

(2.5)

The difference operator $\delta_{x}^2$ , which maintain the three-point stencil is given by ^[]

$\begin{equation} \begin{split} \frac{\delta_{x}^{2}}{h_0^{2}(1+\frac{1}{12}\delta _{x}^{2})} V_{i, j}^{k} = \left.\begin{matrix} \frac{\partial^{2}V}{\partial x^{2}} \end{matrix}\right|_{i, j}^{k}-\frac{h_0^{4}}{240}\left.\begin{matrix} \frac{\partial^{4 }V}{\partial x^{4}} \end{matrix}\right|_{i, j}^{k} + O(h_0^{6}), \end{split} \end{equation}$

(2.6)

and

$\begin{equation} \begin{split} \frac{\delta_{y}^{2}}{h_0^{2}(1+\frac{1}{12}\delta _{y}^{2})} V_{i, j}^{k} = \left.\begin{matrix} \frac{\partial^{2}V}{\partial y^{2}} \end{matrix}\right|_{i, j}^{k}-\frac{h_0^{4}}{240}\left.\begin{matrix} \frac{\partial^{4 }V}{\partial y^{4}} \end{matrix}\right|_{i, j}^{k} + O(h_0^{6}). \end{split} \end{equation}$

(2.7)

The fractional derivative is approximated using the Central difference formula as ^[24]

$\begin{equation*} \begin{split} \frac{\partial^\alpha }{\partial t^\alpha} V(x, y, t)|_{i, j}^{k+\frac{1}{2}} = &\frac{1}{\Gamma (1-\alpha)}\int_{0}^{t_{k+\frac{1}{2}}}V_t(x, y, \varepsilon) (t_{k+\frac{1}{2}}-\varepsilon)^{-\alpha}\partial\varepsilon \\ = &\frac{1}{\Gamma (1-\alpha)}\left [ \int_{0}^{t_{k+\frac{1}{2}}}V_t(x, y, \varepsilon) \left((k+\frac{1}{2}) \nu-\varepsilon\right)^{-\alpha}\partial\varepsilon \right.\\& \left. + \int_{t_k}^{t_{t_{k+\frac{1}{2}}}}\left (\frac{V_{i, j}^{k+1}-V_{i, j}^k}{\nu} + O (\nu)\right ) \left((k+\frac{1}{2})\nu-\varepsilon\right)^{-\alpha}\partial\varepsilon \right ] \\ = &\frac{1}{\Gamma (1-\alpha)} \sum\limits_{s_0 = 1}^{k}\left [ \int_{(s_0-1)\nu}^{s\nu} \frac{V_{i, j}^{s_0} -V_{i, j}^{s_0-1}}{\nu} + (\varepsilon -t_{s_0-\frac{1}{2}})V_{tt}(x_i, y_j, c_{s_0}) \right.\\& \times \left. ((k+\frac{1}{2}\nu-\varepsilon)^{-\alpha}\partial\varepsilon )\right ] + \int_{k\nu}^{(k+\frac{1}{2})\nu}\left ( \frac{V_{i, j}^{k+1}-V_{i, j}^k}{\nu} + O (\nu)\right ) \left((k+\frac{1}{2})\nu-\varepsilon\right)^{-\alpha}\partial\varepsilon \\ = &\frac{1}{\Gamma (1-\alpha)} \sum\limits_{s_0 = 1}^{k} \frac{V_{i, j}^{s_0} -V_{i, j}^{s_0-1}}{\nu} \int_{(s_0-1)\nu}^{s\nu} \left((k+\frac{1}{2})\nu - \varepsilon \right)^{-\alpha}\partial \varepsilon \\& + \frac{1}{\Gamma (1-\alpha)} \sum\limits_{s_0 = 1}^{k} \int_{(s-1)\nu}^{s\nu} (\varepsilon -t_{s_0 -\frac{1}{2}})V_{tt}(x_i, y_j, c_{s_0}) \left ((k+\frac{1}{2})\nu-\varepsilon\right)^{-\alpha}\partial\varepsilon \\& +\frac{1}{\Gamma(1-\alpha)} \left [ \frac{V_{i, j}^{k+1}-V_{i, j}^{k}}{\nu} +O(\nu)\right ] \int_{k\nu}^{(k+\frac{1}{2})\nu} \left[ \left((k+\frac{1}{2})\nu-\varepsilon \right)^{-\alpha}-\varepsilon \right]\partial \varepsilon \\ = &\frac{1}{\nu^\alpha (1-\alpha)\Gamma (1-\alpha) } \sum\limits_{s_0 = 1}^{k} \left [V_{i, j}^{s_0} -V_{i, j}^{s_0-1} \right ] \left [ (k-s_0+\frac{3}{2})^{1-\alpha} -(k-s_0+\frac{1}{2})^{1-\alpha} \right ] \\& + \frac{1}{\nu^\alpha (1-\alpha) \Gamma (1-\alpha) } (V_{i, j}^{k+1}-V_{i, j}^{k})\frac{1}{2^{1-\alpha}} \\& + \frac{1}{\Gamma (1-\alpha)} \sum\limits_{s_0 = 1}^{k} \int_{(s-1)\nu}^{s\nu} (\varepsilon -t_{s_0 -\frac{1}{2}})V_{tt}(x_i, y_j, c_{s_0}) \left((k+\frac{1}{2})\nu-\varepsilon \right)^{-\alpha}\partial\varepsilon \\& + \frac{1}{\Gamma (1-\alpha)(1-\alpha)2^{1-\alpha}}O(\nu)^{2-\alpha}. \end{split} \end{equation*}$

Therefore, after some simplifications, the Crank-Nicolson (C-N) Caputo fractional derivative

$\begin{equation} \begin{array}{c} \frac{\partial^{\alpha} }{\partial t^{\alpha}}\left.\begin{matrix} V(x, y, t) \end{matrix}\right|_{i, j}^{k+\frac{1}{2}} = a_{1}\left.\begin{matrix} V \end{matrix}\right|_{i, j}^{k}+\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s}) \left.\begin{matrix} V \end{matrix}\right|_{i, j}^{s} -a_{k} \left.\begin{matrix} V \end{matrix}\right|_{i, j}^{0}+ \sigma \frac{V_{i, j}^{k+1}+V_{i, j}^{k}}{2^{1-\alpha}}+O(\nu^{2-\alpha}),\\ \sigma = \frac{1}{\nu^{\alpha}\Gamma(2-\alpha)}, \; a_{s} = \sigma\left( (s+\frac{1}{2})^{1-\alpha}-(s-\frac{1}{2})^{1-\alpha} \right), \; s = 0, 1, 2, ..., k. \end{array}\end{equation}$

(2.8)

Now using Eqs (2.6)–(2.8) and C-N or standard point (SP) scheme at $V(x_{i}, y_{j}, t_{k+\frac{1}{2}})$ , the standard fourth-order finite difference scheme for Eq (1.1) is as follows:

$\begin{equation} \begin{split} &a_{1}V_{i, j}^{k}+\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s})V_{i, j}^{s}-a_{k}V_{i, j}^{0}+\sigma\frac{V_{i, j}^{k+1}+V_{i, j}^{k}}{2^{1-\alpha}}\\ = &\left( 1+\frac{1}{12}\delta_{x}^{2} \right)^{-1} \times \frac{\delta_{x}^{2}}{h_0^{2}}V_{i, j}^{k+\frac{1}{2}} +\left( 1+\frac{1}{12}\delta_{y}^{2} \right)^{-1} \frac{\delta_{y}^{2}}{h_0^{2}}V_{i, j}^{k+\frac{1}{2}} + f_{i, j}^{k+\frac{1}{2}} + O(\nu^{2-\alpha}+h_0^{4}). \end{split} \end{equation}$

(2.9)

Substituting the values of $\delta_{x}^{2}$ , $\delta_{y}^{2}$ and $V_{i, j}^{k+\frac{1}{2}}$ into Eq (2.9), and after rearranging we get the standard point SP compact scheme:

$\begin{equation} \begin{split} \lambda_{1}V_{i, j}^{k+1} = & \lambda_{2}(V_{i+1, j}^{k+1}+ V_{i-1, j}^{k+1}+ V_{i, j+1}^{k+1}+V_{i, j-1}^{k+1})+\lambda_{3}(V_{i+1, j+1}^{k+1}+ V_{i-1, j+1}^{k+1} \\& +V_{i+1, j-1}^{k+1}+V_{i-1, j-1}^{k+1})+\lambda_{4}V_{i, j}^{k}+\lambda_{5}(V_{i+1, j}^{k}+ V_{i-1, j}^{k}+ V_{i, j+1}^{k} \\ & +V_{i, j-1}^{k})+\lambda_{6}(V_{i+1, j+1}^{k}+ V_{i-1, j+1}^{k}+ V_{i+1, j-1}^{k}+V_{i-1, j-1}^{k}) \\& +\frac{25}{18}h_{0}^{2}f_{i, j}^{k+\frac{1}{2}}+\frac{5}{36}h_{0}^{2}(f_{i+1, j}^{k+\frac{1}{t}}+f_{i-1, j}^{k+\frac{1}{t}}+f_{i, j+1}^{k+\frac{1}{t}}+f_{i, j-1}^{k+\frac{1}{t}}) \\& +\frac{h_{0}^{2}}{72}(f_{i+1, j+1}^{k+\frac{1}{2}}+f_{i-1, j+1}^{k+\frac{1}{2}}+f_{i+1, j-1}^{k+\frac{1}{2}}+f_{i-1, j-1}^{k+\frac{1}{2}}) \\& -\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s}) \left(\frac{25}{18}h_0^{2}V_{i, j}^{s}+\frac{5}{36}h_0^{2}(V_{i+1, j}^{s}+V_{i-1, j}^{s}+V_{i, j+1}^{s} \right. \\& \left. +V_{i, j-1}^{s})+\frac{h_0^{2}}{72}(V_{i+1, j+1}^{s}+V_{i-1, j+1}^{s}+V_{i+1, j-1}^{s}+V_{i-1, j-1}^{s})\right) + O(\nu^{2-\alpha}+h_0^{4}), \end{split} \end{equation}$

(2.10)

where

$\begin{equation*} \begin{split} &g_0 = \frac{\sigma}{2^{1-\alpha}}, \; g_1 = a_{1}-g_0, \; \lambda_{1} = \frac{1}{72}(240+100h_0^{2}g_0), \; \lambda_{2} = \frac{1}{72}(48-10h_0^{2}g_0), \; \\& \lambda_{3} = \frac{1}{72}(12-h_0^{2}g_0), \; \lambda_{4} = -\frac{1}{72}(240+100h_0^{2}g_1), \; \lambda_{5} = \frac{1}{72}(48-10h_0^{2}g_1), \; \\& \lambda_{6} = \frac{1}{72}(12-h_0^{2}g_1). \end{split} \end{equation*}$

Now, using Eq (2.10) will give the following system for the group of four points:

$\begin{equation} \begin{bmatrix} \lambda_{1} & -\lambda_{2} & -\lambda_{3} & -\lambda_{2}\\ -\lambda_{2}&\lambda_{1} & -\lambda_{2} & -\lambda_{3} \\ -\lambda_{3}& -\lambda_{2} & \lambda_{1} & -\lambda_{2} \\ -\lambda_{2}& -\lambda_{3} & -\lambda_{2} & \lambda_{1} \end{bmatrix} \begin{bmatrix} V_{i, j}^{k+1}\\ V_{i+1, j}^{k+1}\\ V_{i+1, j+1}^{k+1}\\ V_{i, j+1}^{k+1} \end{bmatrix} = \begin{bmatrix} rhs_{i, j}\\ rhs_{i+1, j}\\ rhs_{i+1, j+1}\\ rhs_{i, j+1} \end{bmatrix}, \end{equation}$

(2.11)

where

$\begin{equation*} \begin{split} rhs_{i, j} = & \lambda_{2}(V_{i-1, j}^{k+1}+V_{i, j-1}^{k+1})+\lambda_{3} (V_{i-1, j+1}^{k+1}+V_{i+1, j-1}^{k+1}+V_{i-1, j-1}^{k+1})+\lambda_{4}V_{i, j}^{k}\\& +\lambda_{5}(V_{i+1, j}^{k}+V_{i-1, j}^{k}+V_{i, j+1}^{k}+V_{i, j-1}^{k})+\lambda_{6}(V_{i+1, j+1}^{k}+V_{i-1, j+1}^{k} \\& +V_{i+1, j-1}^{k}+V_{i-1, j-1}^{k})+\frac{25}{18}h_0^{2}f_{i, j}^{k+\frac{1}{2}}+\frac{5}{36}h_0^{2}(f_{i+1, j}^{k+\frac{1}{2}}+f_{i-1, j}^{k+\frac{1}{2}}+f_{i, j+1}^{k+\frac{1}{2}} \\& +f_{i, j-1}^{k+\frac{1}{2}})+ \frac{h_0^{2}}{72}(f_{i+1, j+1}^{k+\frac{1}{2}}+f_{i-1, j+1}^{k+\frac{1}{2}}+f_{i+1, j-1}^{k+\frac{1}{2}}+f_{i-1, j-1}^{k+\frac{1}{2}}) - F_{i, j}, \end{split} \end{equation*}$

$\begin{equation*} \begin{split} rhs_{i+1, j} = & \lambda_{2}(V_{i+2, j}^{k+1}+V_{i+1, j-1}^{k+1})+\lambda_{3} (V_{i+2, j+1}^{k+1}+V_{i+2, j-1}^{k+1}+V_{i, j-1}^{k+1})+\lambda_{4} V_{i+1, j}^{k}\\& +\lambda_{5}(V_{i+2, j}^{k}+V_{i, j}^{k}+V_{i+1, j+1}^{k}+V_{i+1, j-1}^{k})+\lambda_{6}(V_{i+2, j+1}^{k}+V_{i, j+1}^{k} \\& +V_{i+2, j-1}^{k}+V_{i, j-1}^{k})+\frac{25}{18}h_0^{2}f_{i+1, j}^{k+\frac{1}{2}}+\frac{5}{36}h_0^{2}(f_{i+2, j}^{k+\frac{1}{2}}+f_{i, j}^{k+\frac{1}{2}}+f_{i+1, j+1}^{k+\frac{1}{2}} \\& +f_{i+1, j-1}^{k+\frac{1}{2}})+ \frac{h_0^{2}}{72}(f_{i+2, j+1}^{k+\frac{1}{2}}+f_{i, j+1}^{k+\frac{1}{2}}+f_{i+2, j-1}^{k+\frac{1}{2}}+f_{i, j-1}^{k+\frac{1}{2}})- F_{i+1, j}, \end{split} \end{equation*}$

$\begin{equation*} \begin{split} rhs_{i+1, j+1} = & \lambda_{2}(V_{i+2, j+1}^{k+1}+V_{i+1, j+2}^{k+1})+\lambda_{3} (V_{i+2, j+2}^{k+1}+V_{i, j+2}^{k+1}+V_{i+2, j}^{k+1})+\lambda_{4} V_{i+1, j+1}^{k}\\& +\lambda_{5}(V_{i+2, j+1}^{k}+V_{i, j+1}^{k}+V_{i+1, j+2}^{k}+V_{i+1, j}^{k})+\lambda_{6}(V_{i+2, j+2}^{k}+V_{i, j+2}^{k} \\& +V_{i+2, j}^{k}+V_{i, j}^{k})+\frac{25}{18}h_0^{2}f_{i+1, j+1}^{k+\frac{1}{2}}+\frac{5}{36}h_0^{2}(f_{i+2, j+1}^{k+\frac{1}{2}}+f_{i, j+1}^{k+\frac{1}{2}}+f_{i+1, j+2}^{k+\frac{1}{2}} \\& +f_{i+1, j}^{k+\frac{1}{2}})+ \frac{h_0^{2}}{72}(f_{i+2, j+2}^{k+\frac{1}{2}}+f_{i, j+2}^{k+\frac{1}{2}}+f_{i+2, j}^{k+\frac{1}{2}}+f_{i, j}^{k+\frac{1}{2}})- F_{i+1, j+1}, \end{split} \end{equation*}$

$\begin{equation*} \begin{split} rhs_{i, j+1} = & \lambda_{2}(V_{i-1, j+1}^{k+1}+V_{i, j+2}^{k+1})+\lambda_{3} (V_{i+1, j+2}^{k+1}+V_{i-1, j+2}^{k+1}+V_{i-1, j}^{k+1})+\lambda_{4}V_{i, j+1}^{k}\\& +\lambda_{5}(V_{i+1, j+1}^{k}+V_{i-1, j+1}^{k}+V_{i, j+2}^{k}+V_{i, j}^{k})+\lambda_{6}(V_{i+1, j+2}^{k}+V_{i-1, j+2}^{k} \\& +V_{i+1, j}^{k}+V_{i-1, j}^{k})+\frac{25}{18}h_0^{2}f_{i, j+1}^{k+\frac{1}{2}}+\frac{5}{36}h_0^{2}(f_{i+1, j+1}^{k+\frac{1}{2}}+f_{i-1, j+1}^{k+\frac{1}{2}}+f_{i, j+2}^{k+\frac{1}{2}} \\& +f_{i, j}^{k+\frac{1}{2}})+ \frac{h_0^{2}}{72}(f_{i+1, j+2}^{k+\frac{1}{2}}+f_{i-1, j+2}^{k+\frac{1}{2}}+f_{i+1, j}^{k+\frac{1}{2}}+f_{i-1, j}^{k+\frac{1}{2}})- F_{i, j+1}, \end{split} \end{equation*}$

and

$\begin{equation*} \begin{split} F_{i, j} = &\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s}) \left(\frac{25}{18}h_0^{2}V_{i, j}^{s}+\frac{5}{36}h_0^{2}(V_{i+1, j}^{s}+V_{i-1, j}^{s}+V_{i, j+1}^{s} \right. \\& \left. +V_{i, j-1}^{s})+\frac{h_0^{2}}{72}(V_{i+1, j+1}^{s}+V_{i-1, j+1}^{s}+V_{i+1, j-1}^{s}+V_{i-1, j-1}^{s})\right). \end{split} \end{equation*}$

Similarly, the inverted matrix equation (2.11) will give explicit group equation

$\begin{equation} \begin{bmatrix} V_{i, j}^{k+1}\\ V_{i+1, j}^{k+1}\\ V_{i+1, j+1}^{k+1}\\ V_{i, j+1}^{k+1} \end{bmatrix} = \frac{1}{d}\begin{bmatrix} \phi_{1} & \phi_{2} & \phi_{3} & \phi_{2} \\ \phi_{2} & \phi_{1} & \phi_{2} & \phi_{3} \\ \phi_{3} & \phi_{2} & \phi_{1} & \phi_{2} \\ \phi_{2} & \phi_{3} & \phi_{2} & \phi_{1} \end{bmatrix} \begin{bmatrix} rhs_{i, j}\\ rhs_{i+1, j}\\ rhs_{i+1, j+1}\\ rhs_{i, j+1} \end{bmatrix}, \end{equation}$

(2.12)

where

$\begin{equation*} \begin{split} &\phi_{1} = \lambda_{1}^{3}-2\lambda_{1}\lambda_{2}^{2}-2\lambda_{2}^{2}\lambda_{3}-\lambda_{1}\lambda_{3}^{2}, \; \phi_{2} = \lambda_{1}^{2}\lambda_{2}+2\lambda_{1}\lambda_{2}\lambda_{3}+\lambda_{2}\lambda_{3}^{2}, \\& \phi_{3} = 2\lambda_{1}\lambda_{2}^{2}+\lambda_{1}^{2}\lambda_{3}+2\lambda_{2}^{2}\lambda_{3}-\lambda_{3}^{3}, \; d = (-4\lambda_{2}^{2}+(\lambda_{1}-\lambda_{3})^{2})(\lambda_{1}+\lambda_{3})^{2}. \end{split} \end{equation*}$

In the proposed method, firstly, group of four points are computed for the different iterations using Eq (2.12) till the required convergence is attained. After the required convergence, the SP compact scheme Eq (2.10) is used directly once for computing of reaming points. and show the grid points on the x-y plane for FDM and FEGM at various time levels when $m = 9$ respectively.

Figure 1. Four points in computation of grouping method.

DownLoad: Full-Size Img PowerPoint

Figure 2. Different points in the FEGM at time levels

$k + 1, k, k-1, ..., 1$ with mesh size

$m = 9$ .

DownLoad: Full-Size Img PowerPoint

3. Stability of the FEGM

In this section, the stability of the proposed method is discussed.

The Eq (2.12) can also be written as

$\begin{equation} \begin{split} & AV^{1} = BV^{0}+f^{\frac{1}{2}}, \; k = 0, \\& AV^{k+1} = BV^{k}-h_0^{2}\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s})CV^{s}+h_0^{2}f^{k+\frac{1}{2}}, \; k > 0, \end{split} \end{equation}$

(3.1)

where

$\begin{equation*} \begin{split} &A = \begin{bmatrix} R_{1} & R_{3} & & & 0\\ R_{2}& R_{1} & R_{3} & & \\ & R_{2} & R_{1} & & \\ & & & \ddots & R_{3}\\ 0 & & &R_{2} & R_{1} \end{bmatrix}, \; B = \begin{bmatrix} P_{1} & P_{3} & & & 0\\ P_{2}& P_{1} & P_{3} & & \\ & P_{2} & P_{1} & & \\ & & & \ddots & P_{3}\\ 0 & & &P_{2} & P_{1} \end{bmatrix}, \end{split} \end{equation*}$

$\begin{equation*} \begin{split} &R_{1} = \begin{bmatrix} G_{1} & G_{3} & & & \\ G_{2}& G_{1} & G_{3} & & \\ & G_{2} & G_{1} & & \\ & & & \ddots & G_{3}\\ & & &G_{2} & G_{1} \end{bmatrix}, \; C = \begin{bmatrix} Q_{1} & Q_{3} & & & 0\\ Q_{2}& Q_{1} & Q_{3} & & \\ & Q_{2} & Q_{1} & & \\ & & & \ddots & Q_{3}\\ 0 & & &Q_{2} & Q_{1} \end{bmatrix}, \; f = \begin{bmatrix} K_1\\ K_1\\ \vdots \\ K_1\\ K_1 \end{bmatrix}, \end{split} \end{equation*}$

$\begin{equation*} R_{2} = \begin{bmatrix} G_{6} & G_{4} & & & \\ G_{8}& G_{6} & G_{4} & & \\ & G_{8} & G_{6} & & \\ & & & \ddots & G_{4}\\ & & &G_{8} & G_{6} \end{bmatrix}, \; R_{3} = \begin{bmatrix} G_{7} & G_{9} & & & \\ G_{5}& G_{7} & G_{9} & & \\ & G_{5} & G_{7} & & \\ & & & \ddots & G_{9}\\ & & &G_{5} & G_{7} \end{bmatrix}, \end{equation*}$

$\begin{equation*} \begin{split} P_{1} = \begin{bmatrix} H_{1} & H_{3} & & & \\ H_{2}& H_{1} & H_{3} & & \\ & H_{2} & H_{1} & & \\ & & & \ddots & H_{3}\\ & & &H_{2} & H_{1} \end{bmatrix}, \; P_{2} = \begin{bmatrix} H_{6} & H_{4} & & & \\ H_{8}& H_{6} & H_{4} & & \\ & H_{8} & H_{6} & & \\ & & & \ddots & H_{4}\\ & & &H_{8} & H_{6} \end{bmatrix}, \end{split} \end{equation*}$

$\begin{equation*} P_{3} = \begin{bmatrix} H_{7} & H_{9} & & & \\ H_{5}& H_{7} & H_{9} & & \\ & H_{5} & H_{7} & & \\ & & & \ddots & H_{9}\\ & & &H_{5} & H_{7} \end{bmatrix}, \; Q_{1} = \begin{bmatrix} L_{1} & L_{3} & & & \\ L_{2}& L_{1} & L_{3} & & \\ & L_{2} & L_{1} & & \\ & & & \ddots & L_{3}\\ & & &L_{2} & L_{1} \end{bmatrix}, \end{equation*}$

$\begin{equation*} Q_{2} = \begin{bmatrix} L_{6} & L_{4} & & & \\ L_{8}& L_{6} & L_{4} & & \\ & L_{8} & L_{6} & & \\ & & & \ddots & L_{4}\\ & & &L_{8} & L_{6} \end{bmatrix}, \; Q_{3} = \begin{bmatrix} L_{7} & L_{9} & & & \\ L_{5}& L_{7} & L_{9} & & \\ & L_{5} & L_{7} & & \\ & & & \ddots & L_{9}\\ & & &L_{5} & L_{7} \end{bmatrix}, \; K_1 = \begin{bmatrix} W_1\\ W_1\\ \vdots \\ W_1\\ W_1 \end{bmatrix}, \end{equation*}$

$\begin{equation*} G_{1} = \begin{bmatrix} \lambda_{1} & -\lambda_{2} & -\lambda_{3} & -\lambda_{2}\\ -\lambda_{2}&\lambda_{1} & -\lambda_{2} & -\lambda_{3} \\ -\lambda_{3}& -\lambda_{2} & \lambda_{1} & -\lambda_{2} \\ -\lambda_{2}& -\lambda_{3} & -\lambda_{2} & \lambda_{1} \end{bmatrix}, \; G_{2} = \begin{bmatrix} 0 & 0 & -\lambda_{3} & -\lambda_{2}\\ 0&0 & -\lambda_{2} & -\lambda_{3} \\ 0& 0 &0 & 0 \\ 0& 0 &0 & 0 \end{bmatrix}, \; G_{3} = \begin{bmatrix} 0 & 0 & 0& 0\\ 0&0 & 0 & 0 \\ -\lambda_{3}& -\lambda_{2} &0 & 0 \\ -\lambda_{2}& -\lambda_{3} &0 & 0 \end{bmatrix}, \end{equation*}$

$\begin{equation*} G_{4} = \begin{bmatrix} 0 & 0 & 0& 0\\ 0&0 & 0 & 0 \\ 0& 0 &0 & 0 \\ 0& -\lambda_{3} &0 & 0 \end{bmatrix}, \; G_{5} = \begin{bmatrix} 0 & 0 & 0& 0\\ 0&0 & 0 & -\lambda_{3} \\ 0& 0 &0 & 0 \\ 0& 0 &0 & 0 \end{bmatrix}, \; G_{6} = \begin{bmatrix} 0 & -\lambda_{2}& -\lambda_{3}& 0\\ 0&0 & 0 & 0 \\ 0& 0 &0 & 0 \\ 0& -\lambda_{3} &-\lambda_{2} & 0 \end{bmatrix}, \end{equation*}$

$\begin{equation*} G_{7} = \begin{bmatrix} 0 & 0& 0& 0\\ -\lambda_{2}&0 & 0 & -\lambda_{3}\\ -\lambda_{3}& 0 &0 & -\lambda_{2} \\ 0&0 &0 & 0 \end{bmatrix}, \; G_{8} = \begin{bmatrix} 0 & 0& -\lambda_{3}& 0\\ 0&0 & 0 & 0\\ 0& 0 &0 & 0 \\ 0&0 &0 & 0 \end{bmatrix}, \; G_{9} = \begin{bmatrix} 0 & 0& 0& 0\\ 0&0 & 0 & 0\\ -\lambda_{3}& 0 &0 & 0 \\ 0&0 &0 & 0 \end{bmatrix}, \end{equation*}$

$\begin{equation*} H_{1} = \begin{bmatrix} \lambda_{4} & \lambda_{5} & \lambda_{6} & \lambda_{5}\\ \lambda_{5}&\lambda_{4} & \lambda_{5} & \lambda_{6} \\ \lambda_{6}& \lambda_{5} & \lambda_{4} & \lambda_{5} \\ \lambda_{5}& \lambda_{6} & \lambda_{5} & \lambda_{4} \end{bmatrix}, \; H_{2} = \begin{bmatrix} 0 & 0 & \lambda_{6} & \lambda_{5}\\ 0&0 & \lambda_{5} & \lambda_{6} \\ 0& 0 &0 & 0 \\ 0& 0 &0 & 0 \end{bmatrix}, \; H_{3} = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0&0 & 0 & 0\\ \lambda_{6}& \lambda_{5} &0 & 0 \\ \lambda_{5}& \lambda_{6} &0 & 0 \end{bmatrix}, \end{equation*}$

$\begin{equation*} \begin{split} & H_{4} = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0&0 & 0 & 0\\ 0& 0 &0 & 0 \\ 0& \lambda_{6} &0 & 0 \end{bmatrix}, \; H_{5} = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0&0 & 0 & \lambda_{6}\\ 0& 0 &0 & 0 \\ 0& 0 &0 & 0 \end{bmatrix}, \; H_{6} = \begin{bmatrix} 0 & \lambda_{5} & \lambda_{6} & 0\\ 0&0 & 0 & 0\\ 0& 0 &0 & 0 \\ 0& \lambda_{6} &\lambda_{5} & 0 \end{bmatrix}, \\ \end{split} \end{equation*}$

$\begin{equation*} H_{7} = \begin{bmatrix} 0 & 0 & 0& 0\\ \lambda_{5} &0 & 0 & \lambda_{6} \\ \lambda_{6} & 0 &0 & \lambda_{5} \\ 0& 0&0 & 0 \end{bmatrix}, \; H_{8} = \begin{bmatrix} 0 & 0 & \lambda_{6}& 0\\ 0 &0 & 0 & 0 \\ 0 & 0 &0 & 0 \\ 0& 0&0 & 0 \end{bmatrix}, \; H_{9} = \begin{bmatrix} 0 & 0 & 0& 0\\ 0 &0 & 0 & 0 \\ \lambda_{6} & 0 &0 & 0 \\ 0& 0&0 & 0 \end{bmatrix}, \end{equation*}$

$\begin{equation*} L_{1} = \frac{1}{18}\begin{bmatrix} 25 & \frac{5}{2} & \frac{1}{4} & \frac{5}{4}\\ \frac{5}{2} & 25 & \frac{5}{2}& \frac{1}{4}\\ \frac{1}{4}& \frac{5}{2} & 25 & \frac{5}{2}\\ \frac{5}{2}&\frac{1}{4} & \frac{5}{2} & 25 \end{bmatrix}, \; L_{2} = \frac{1}{18}\begin{bmatrix} 0 & 0 & \frac{1}{4} & \frac{5}{4}\\ 0 & 0 & \frac{5}{2}& \frac{1}{4}\\ 0& 0 & 0 & 0\\ 0&0 & 0 &0 \end{bmatrix}, \; L_{3} = \frac{1}{18}\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0& 0\\ \frac{1}{4}& \frac{5}{4} & 0 & 0\\ \frac{5}{4}&\frac{1}{4} & 0 &0 \end{bmatrix}, \end{equation*}$

$\begin{equation*} L_{4} = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0& 0\\ 0& 0 & 0 & 0\\ 0&\frac{1}{72} & 0 &0 \end{bmatrix}, \; L_{5} = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0& \frac{1}{72}\\ 0& 0 & 0 & 0\\ 0& 0& 0 &0 \end{bmatrix}, \; L_{6} = \frac{1}{18}\begin{bmatrix} 0 & \frac{5}{2} & \frac{1}{4} & 0\\ 0 & 0 & 0& 0\\ 0& 0 & 0 & 0\\ 0& \frac{1}{4}& \frac{5}{2}&0 \end{bmatrix}, \end{equation*}$

$\begin{equation*} \begin{split} & L_{7} = \frac{1}{18}\begin{bmatrix} 0 & 0 & 0 & 0\\ \frac{5}{2} & 0 & 0&\frac{1}{4}\\ \frac{1}{4}& 0 & 0 & \frac{5}{2}\\ 0& 0& 0&0 \end{bmatrix}, \; L_{8} = \begin{bmatrix} 0 & 0 & \frac{1}{72} & 0\\ 0 & 0 & 0&0\\ 0& 0 & 0 &0\\ 0& 0& 0&0 \end{bmatrix}, \; L_{9} = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0&0\\ \frac{1}{72}& 0 & 0 &0\\ 0& 0& 0&0 \end{bmatrix}, \; W_1 = \begin{bmatrix} f_{i, j}\\ f_{i+1, j}\\ f_{i+1, j+1}\\ f_{i, j+1} \end{bmatrix}. \end{split} \end{equation*}$

It can observe that Eq (3.1) form the particular structure as

$\begin{equation*} [A_{(N-2)^2 \times (N-2)^2} ]V^{k+1} = [B_{(N-2)^2 \times (N-2)^2}]V^{k}-h_0^{2}\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s})[C_{(N-2)^2 \times (N-2)^2}]V^{s}+h_0^{2}f^{k+\frac{1}{2}}. \end{equation*}$

Proposition 3.1. The proposed scheme Eq (2.12) is unconditionally stable.

Proof. Let $V_{i, j}^{k}$ represents approximate and $v_{i, j}^{k}$ represents exact solutions for the time FSDE respectively, then the error is defined as $\epsilon_{i, j}^{k} = v_{i, j}^{k}-V_{i, j}^{k}$ . So, from Eq (3.1),

$\begin{equation} \begin{split} & AE^{1} = BE^{0}, \; k = 0, \\& AE^{k+1} = BE^{k}-h_0^{2}\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a{k-s})CE^{s}, \; k > 0, \end{split} \end{equation}$

(3.2)

where

$\begin{equation*} \begin{split} &E^{k+1} = \begin{bmatrix} E_{1}^{k+1}\\ E_{2}^{k+1}\\ \vdots \\ E_{m-2}^{k+1}\\ E_{m-1}^{k+1}\\ \end{bmatrix}, \; E_{i}^{k+1} = \begin{bmatrix} \epsilon_{1}^{k+1}\\ \epsilon_{2}^{k+1}\\ \vdots \\ \epsilon_{m-2}^{k+1}\\ \epsilon_{m-1}^{k+1}\\ \end{bmatrix}, \; \epsilon_{i}^{k+1} = \begin{bmatrix} \epsilon_{i, j}^{k+1}\\ \epsilon_{i+1, j}^{k+1}\\ \epsilon_{i+1, j+1}^{k+1}\\ \epsilon_{i, j+1}^{k+1}\\ \end{bmatrix}, \; i, j = 1, 2, ..., m-1. \end{split} \end{equation*}$

From Eq (3.1) we know

$\begin{equation} \; \; \; \; \; \; A = G_{1}I+(G_{2}+G_{3})Q+G_{6}I+ (G_{4}+G_{8})Q+G_{7}I+(G_{5}+G_{9})Q, \end{equation}$

(3.3)

$\begin{equation} \; \; \; \; \; \; B = H_{1}I+(H_{2}+H_{3})Q+H_{6}I+ (H_{4}+H_{8})Q+H_{7}I+(H_{5}+H_{9})Q, \end{equation}$

(3.4)

$\begin{equation} C = L_{1}I+(L_{2}+L_{3})Q+L_{6}I+ (L_{4}+L_{8})Q+L_{7}I+(L_{5}+L_{9})Q, \end{equation}$

(3.5)

where $I$ and $Q$ are two matrices, $I$ represents identity matrix and $Q$ represents unity values having each diagonal forthwith above and below the main diagonal, and elsewhere zero.

Suppose maximum eigenvalues are represented with $\psi$ , $\chi$ and $\eta$ for the matrices $A$ , $B$ and $C$ respectively, then using Mathematica software, we get

$\begin{equation} \begin{split} &\psi = \frac{9}{8} \left(g_1 h_0^2+4\right), \\& \chi = (\frac{29}{6} + \frac{79}{72}h_0^{2}g_0), \\& \eta = \frac{121}{72}. \end{split} \end{equation}$

(3.6)

From Eq (3.2), when k = 0,

$\begin{equation*} \begin{split} &E^{1} = A^{-1}BE^{0}, \\& \left \| E^{1} \right \|\leq \left \| A^{-1}B \right \|\left \| E^{0} \right \| = \frac{132+121h^{2}g_1}{348+79h^{2}g_0}\left \| E^{0} \right \|. \end{split} \end{equation*}$

But since $a_1 = \sigma ((\frac{3}{2})^{1-\alpha}- (\frac{1}{2})^{1-\alpha}) = g_0(3^{1-\alpha}-1)$ and $g_1 = a_1-g_0 = g_0(3^{1-\alpha}-2)$ . Also we know that $3^{1-\alpha} < 3$ , so,

$\begin{equation*} \begin{split} & 3^{1-\alpha} -2 < 1, \\& g_0(3^{1-\alpha} -2) < g_0, \; \; \; \; \; \because g_0 > 0, \\& g_1 < g_0. \end{split} \end{equation*}$

Hence,

$\begin{equation*} \left \| E^{1} \right \|\leq \left \| E^{0} \right \|, \;\;\;\; \because g_0 > g_1. \end{equation*}$

Suppose

$\begin{equation} \left \| E^{r} \right \|\leq \left \| E^{0} \right \|, \;\; r = 2, 3, ..., k, \end{equation}$

(3.7)

and for $r = k+1$ ,

$\begin{align*} \left \| E^{k+1} \right \|& = \left \| A^{-1}\left ( BE^{k}-h_0^{2}\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s})CE^{s}) \right )\right \|\\& \leq \left \| A^{-1}B \right \|\left \| E^{k} \right \|+h_0^{2}\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s})\left \| A^{-1}C \right \|\left \| E^{s} \right \|\\& \leq \left ( \left \| A^{-1}B \right \|+h_0^{2}\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s})\left \| A^{-1}C \right \|\right )\left \| E^{0} \right \| \;\;\;\;\;\;\text{by using Eq. (3.7)}\\& = \left(\frac{324+81h_0^{2}g_1}{348+h_0^{2}g_0} +\frac{121h_0^{2}(a_{k}-a_{1})}{72(348+79h_0^{2}g_0)} \right)\left \| E^{0} \right \| \\& = \left( \frac{324+81h_0^{2}g_1+1.68h_0^{2}(a_{k}-a_{1})}{348+79h_0^{2}g_0}\right) \left \| E^{0} \right \| \\& = \left ( \frac{132+(72.02)h^{2}(g_1+(a_{k}-a_{1}))}{348+79h_0^{2}g_0} \right )\left \| E^{0} \right \|. \end{align*}$

$\left \| E^{k+1} \right \|\leq\left \| E^{0} \right \|, \;\;\; \;\; \;\; \because (a_{k}-a_{1}) < 0.$

So, by mathematical induction, we prove that FEGM is unconditionally stable.

4. Convergence analysis

Suppose $e_{i, j}^{k+\frac{1}{2}}, \; e_{i+1, j}^{k+\frac{1}{2}}, \; e_{i+1, j+1}^{k+\frac{1}{2}}$ and $e_{i, j+1}^{k+\frac{1}{2}}$ represent different truncation errors, then,

$R^{k+\frac{1}{2}} = \{R_{1, 1}^{k+\frac{1}{2}}, R_{1, 2}^{k+\frac{1}{2}}, ..., R_{1, \frac{M_2 -1}{4}}^{k+\frac{1}{2}}, R_{2, 1}^{k+\frac{1}{2}}, R_{2, 2}^{k+\frac{1}{2}}, ..., R_{\frac{M_1 -1}{4}, \frac{M_2 -1}{4}}^{k+\frac{1}{2}}\},$

where

$R_{i, j}^{k+\frac{1}{2}} = \{e_{i, j}^{k+\frac{1}{2}}, e_{i+1, j}^{k+\frac{1}{2}}, e_{i+1, j+1}^{k+\frac{1}{2}}, e_{i, j+1}^{k+\frac{1}{2}}\}, \; i, j = \{1, 2, ..., \frac{M -1}{4}\},$

so from Eq (2.10) we have

$\begin{equation} \left \| R^{k+\frac{1}{2}} \right \|\leq \varphi_{0} (\nu^{2-\gamma}+h_0^{4} ), \end{equation}$

(4.1)

where $\varphi_{0}$ is a constant.

Proposition 4.1. The FEGS equation (2.12) is unconditionally convergent with the order of convergence $O(\nu^{2-\alpha} + h^{4})$ .

Proof. Since from Eq (4.1),

$\begin{equation} \left \| R^{(k-1)+\frac{1}{2}} \right \| \leq \varphi_{0} (\nu^{2-\alpha}+h_0^{4} ), \end{equation}$

(4.2)

then,

$\begin{align*} &\left \| R^{(k-1)+\frac{1}{2}} \right \|-\left \| R^{k+\frac{1}{2}} \right \| \leq 0, \\& \left \| R^{(k-1)+\frac{1}{2}} \right \|\leq \left \| R^{k+\frac{1}{2}} \right \|. \end{align*}$

(4.3)

Since $E^{0} = 0$ , then from Eq (2.10), we have

$\begin{equation} \begin{split} &AE^{1} = R^{\frac{1}{2}}, \; k = 0, \\& AE^{k+1} = BE^{k}- h_0^{2}\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s})CE^{s}+R^{k+\frac{1}{2}}, \; k > 0. \end{split} \end{equation}$

(4.4)

When $k = 0$ ,

$\begin{equation*} \begin{split} &AE^{1} = R^{\frac{1}{2}}, \\& \left \| E^{1} \right \|\leq \left \| A^{-1} \right \|\left \| R^{\frac{1}{2}} \right \| = \frac{1}{\lambda_{1}+2\lambda_{2}+\lambda_{3}}\left \| R^{\frac{1}{2}} \right \| = \frac{1}{348+79h_0^{2}g_0}\left \| R^{\frac{1}{2}} \right \|, \\& \left \| E^{1} \right \|\leq\mu_{0}\left \| R^{\frac{1}{2}} \right \|, \; \text{where} \; \mu_{0} = \frac{1}{348+79h_0^{2}g_0} \; \; \text{and}\; \; \mu_{0} \in (0, 1) , \\& \left \| E^{1} \right \|\leq \left \| R^{\frac{1}{2}} \right \|. \end{split} \end{equation*}$

Assume that

$\begin{equation} \left \| E^{s} \right \|\leq \left \| R^{(s-1)+\frac{1}{2}} \right \|, \; s = 2, 3, ..., k, \end{equation}$

(4.5)

and now from Eq (4.4),

$\begin{equation} \begin{split} AE^{k+1} = BE^{k}-h_0^{2}\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s})CE^{s}+R^{k+\frac{1}{2}}. \end{split} \end{equation}$

(4.6)

By taking norm function on both sides of Eq (4.6),

$\begin{align*} \left \| E^{k+1} \right \| &\leq \left \| A^{-1}B \right \|\left \| E^{k}\right \|-h_0^{2}\sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s}) \left \| A^{-1}C\right \| \left \| E^{s} \right \| + \left \| A^{-1} \right \| \left \| R^{k+\frac{1}{2}} \right \| \\& \leq \left( \frac{324+81h_0^{2}g_1}{348+79h_0^{2}g_0}+\frac{1.68h_0^{2}(a_{k}-a_{1})}{348+79h_0^{2}g_0}+\frac{1}{348+79h_0^{2}g_0}\right) \left \| R^{k+\frac{1}{2}}\right\| \; \text{(by using Eqs (4.3) and (4.5))}\\& = \left( \frac{325+81h_0^{2}g_1+1.68h_0^{2}(a_{k}-a_{1})}{348+79h_0^{2}g_0}\right)\left \| R^{k+\frac{1}{2}} \right \| \\& = \gamma \left \| R^{k+\frac{1}{2}} \right \|, \end{align*}$

where $\gamma = \frac{133+121h^{2}g_1+1.68h_0^{2}(a_{k}-a_{1})}{348+79h_0^{2}g_0}$ , but since $h \in (0, 1)$ , $g_0 > g_1$ , and $(a_1-a_k) < 0$ , then $\gamma \in (0, 1)$ , therefore,

$\begin{equation*} \begin{split} \left \| E^{k+1} \right \|&\leq \left \| R^{k+\frac{1}{2}} \right \| \leq \varphi_{0} (\nu^{2-\alpha}+h_0^{4}). \end{split} \end{equation*}$

Therefore, we get

$\begin{equation*} \left \| E^{k+1} \right \|\leq \varphi_{0} (\nu^{2-\alpha}+h_0^{4}), \; \forall k = 0, 1, 2..., N-1. \end{equation*}$

Thus the proposed scheme is conditionally stable.

5. Solvability of the proposed scheme

The proposed scheme can be written in matrix form:

$\begin{equation} \begin{split} & \mathcal{G}_1 V^{1} = \mathcal{G}_2 V^0 +\mathcal{G}_3 \Upsilon^{\frac{1}{2}}, \; \; k = 0, \\& \mathcal{G}_1 V^{k+1} = \mathcal{G}_2 V^k +\mathcal{G}_3 \Upsilon^{k+\frac{1}{2}} - \sum\limits_{s = 1}^{k-1}(a_{k-s+1}-a_{k-s}) \mathcal{G}_3 V^{s}, \; \; k \geq 1, \\& V_{i, j}^{0} = b_0(x_i, y_j), \; 1\leq i \leq M, \; 1\leq j \leq M, \\& V_{0, j}^{k} = b_1(0, y_j), \; 1\leq j \leq M, \; 0\leq k \leq N, \\& V_{L, j}^{k} = b_2(L, y_j), \; 1\leq j \leq M, \; 0\leq k \leq N, \\& V_{i, 0}^{k} = b_3(x_i, 0), \; 1\leq i \leq M, \; 0\leq k \leq N, \\& V_{i, L}^{k} = b_4(x_i, L), \; 1\leq i \leq M, \; 0\leq k \leq N, \end{split} \end{equation}$

(5.1)

where

$\begin{equation*} \mathcal{G}_1 = \begin{bmatrix} \lambda_{1} &-\lambda_{2} & & \cdots & -\lambda_{2} &-\lambda_{3} & \cdots & & 0\\ -\lambda_{2} & \lambda_{1} &-\lambda_{2} & & -\lambda_{3} & -\lambda_{2} & -\lambda_{3} & &\vdots \\ &-\lambda_{2} & \lambda_{1} & -\lambda_{2} & \cdots & -\lambda_{3} & -\lambda_{2} & -\lambda_{3} & \\ \vdots &\cdots & -\lambda_{2} & \lambda_{1}&-\lambda_{2} & & -\lambda_{3} & -\lambda_{2} &-\lambda_{3} \\ & & & -\lambda_{2} & \lambda_{1} & -\lambda_{2} & & -\lambda_{3} &-\lambda_{2} \\ \vdots &\cdots & & & -\lambda_{2} & \lambda_{1} & -\lambda_{2} & & \\ & & & & & -\lambda_{2} & \lambda_{1} & -\lambda_{2} & \\ & & & & & & -\lambda_{2} & \lambda_{1} &-\lambda_{2} \\ 0& & \cdots & &\cdots & & & -\lambda_{2} & \lambda_{1} \end{bmatrix}, \end{equation*}$

$\begin{equation*} \mathcal{G}_2 = \begin{bmatrix} \lambda_{4} &\lambda_{5} & & \cdots & \lambda_{5} &\lambda_{6} & \cdots & & 0\\ \lambda_{5} & \lambda_{4} &\lambda_{5} & & \lambda_{6} & \lambda_{5}& \lambda_{6} & &\vdots \\ &\lambda_{5} & \lambda_{4}& \lambda_{5} & \cdots & \lambda_{6} & \lambda_{5} & \lambda_{6} & \\ \vdots &\cdots & \lambda_{5} & \lambda_{4} &\lambda_{5} & & \lambda_{6} & \lambda_{5} &\lambda_{6} \\ & & & \lambda_{5} & \lambda_{4} & \lambda_{5} & & \lambda_{6} &\lambda_{5} \\ \vdots &\cdots & & & \lambda_{5} & \lambda_{4} & \lambda_{5} & & \\ & & & & & \lambda_{5} & \lambda_{4} & \lambda_{5} & \\ & & & & & & \lambda_{5} & \lambda_{4} &\lambda_{5} \\ 0& & \cdots & &\cdots & & & \lambda_{5} & \lambda_{4} \end{bmatrix}, \end{equation*}$

$\begin{equation*} \mathcal{G}_3 = \begin{bmatrix} \rho_1 &\rho_2 & & \cdots & \rho_2 &\rho_3 & \cdots & & 0\\ \rho_2 & \rho_1 &\rho_2 & & \rho_3 & \rho_2& \rho_3 & &\vdots \\ &\rho_2 & \rho_1& \rho_2 & \cdots & \rho_3 & \rho_2 & \rho_3 & \\ \vdots &\cdots & \rho_2 & \rho_1 &\rho_2 & & \rho_3 & \rho_2 &\rho_3 \\ & & & \rho_2 & \rho_1 & \rho_2 & & \rho_3&\rho_2 \\ \vdots &\cdots & & & \rho_2 & \rho_1 & \rho_2 & & \\ & & & & & \rho_2 & \rho_1 & \rho_2 & \\ & & & & & & \rho_2 & \rho_1 &\rho_2 \\ 0& & \cdots & &\cdots & & & \rho_2 & \rho_1 \end{bmatrix}, \end{equation*}$

$\Upsilon^{k} = [\Upsilon_0^{k}, \Upsilon_1^{k}, \Upsilon_2^{k}, ..., \Upsilon_n^{k}]^T, \; \Upsilon^{k+\frac{1}{2}} = f(x_i, y_j, t_{k+\frac{1}{2}}), \; \rho_1 = \frac{25h_{0}^2}{18}, \; \rho_2 = \frac{5h_{0}^2}{36}\; \text{and} \; \rho_3 = \frac{h_{0}^2}{72} .$

Proposition 5.1. The difference equation (2.12) is uniquely solvable.

Proof. Since $\lambda_{1} = \frac{1}{72}(240+100h_0^{2}g_0)$ , $\lambda_{2} = \frac{1}{72}(48-10h_0^{2}g_0)$ , $\lambda_{3} = \frac{1}{72}(12-h_0^{2}g_0)$ , and $h_0, g_0 > 0$ , then,

$|\lambda_{1}| = \frac{10}{3}+\frac{25 h_0^{2}g_0}{18}$

and

$3|\lambda_{2}|+2|\lambda_{3}| \leq \frac{7}{3} + \frac{4 }{9}h_0^{2}g_0 < \frac{10}{3}+\frac{25 }{18}h_0^{2}g_0 = |\lambda_{1}|.$

Hence, $|\lambda_{1}| > 3|\lambda_{2}|+2|\lambda_{1}|$ , which shows that matrix $G_1$ is strictly diagonally dominant and $G_1$ is non-singular. This completes the proof.

6. Numerical experiments and discussion

The proposed method is simulated using the Intel Core i-7, 2.40GHz GHz, 6GB of RAM with Windows 8 using Mathematica software, and the experiments were done using the proposed method with SOR iterative technique as an acceleration factor ( $\omega = 1.8$ ) with different mesh sizes $(n = 10, 14, 18, 22, 30)$ and different time steps. Furthermore, throughout the experiments, the $L_{\infty}$ -norm convergence criteria $\zeta = 10^{-5}$ is used. Also, the $C_{2}$ -order and $C_1$ -order of convergence are used for the computational order of spatial and temporal convergence using ^[25]

$\begin{equation} C_{2}-order = log_{2}\left ( \frac{\left \| L_{\infty}\left ( 16\nu, 2h_0 \right ) \right \|}{\left \| L_{\infty}\left ( \nu, h_0 \right ) \right \|} \right ), \end{equation}$

(6.1)

$\begin{equation} C_{1}-order = log_{2}\left( \frac{\parallel L_{\infty}(2\nu, h_0) \parallel}{\parallel L_{\infty}(\nu, h_0) \parallel}\right), \end{equation}$

(6.2)

where $L_{\infty}$ is the maximum error.

Some examples are presented below to show the efficiency of FEGM.

Problem 1. ^[26]

$\begin{equation*} \frac{\partial^{\alpha}V}{\partial t^{\alpha}} = \frac{\partial^2 V}{\partial x^2} +\frac{\partial^2 V}{\partial y^2} + \left( \frac{2}{\Gamma(3-\alpha)} t^{2-\alpha}+ 2 t^{2} \right) \sin(x)\sin(y), \; \; \; 0 < \text{x, y} < 1, \; \; \; 0 < t \leq 1, \end{equation*}$

with initial and Dirichlet boundary conditions.

The analytic solution for Problem 1 is

$V(x, y, t) = t^{2}\sin(x)\sin(y).$

Problem 2. ^[12]

$\begin{equation*} \frac{\partial^2 V}{\partial t^2} = \frac{\partial^2 V}{\partial x^1} +\frac{\partial^2 V}{\partial y^2 } + \left( \Gamma(2+\alpha)-2t^{1+\alpha} \right) e^{x+y}, \; \; \; 0 < \text{x, y} < 1, \; \; \; 0 < t \leq 1, \end{equation*}$

with initial and Dirichlet boundary conditions.

The analytic solution for Problem 2 is

$V(x, y, t) = e^{x+y}t^{1+\alpha}.$

The number of iterations, error analysis, and execution times are shown for the comparison between FEGM and SP methods from –. The execution times in FEGM are decreased by $(4.7-28.49)\%$ , $(2-23)\%$ , $(9.16-28.13)\%$ , $(8.9-25.39)\%$ and $(6.98-27.79)\%$ compared to SP method in Tables 1–5 respectively. Similarly, in Table 6, the comparison between the proposed method and high-order standard point method^[27] is presented, which shows the proposed method gives better results. Figures 3 and 4 represent the exact and approximate solution for Problem 1, respectively, which depicts the effectiveness of the FEGM. Likewise, In Figure 5, the compression of execution times between the SP and proposed methods are shown, which shows the proposed method is efficient in terms of execution timings. Table 7 shows the computational complexity per iteration, while the computational effort is shown in Tables 8 and 9, which depict that the FEGM requires less number of operations during computations as compared to the standard SP method. Tables 10 and 11 represent the spatial convergence order for Problems 1 and 2, respectively. Similarly, Tables 12 and 13 represent the temporal convergence order for the first and second Problem respectively, which depict the experimental and theoretical convergence orders in agreement.

Table 1. Numerical results for the Problem 1, when

$\alpha = 0.5$ .

$h_0/\nu$	No. of iteration		Execution time		Maximum-error		Average-error
$h_0/\nu$	FEGM	SP	FEGM	SP	FEGM	SP	FEGM	SP
$h_0=\nu=\frac{1}{10}$	43	49	7.39	7.76	1.6404 $\times10^{-4}$	1.653 $\times10^{-4}$	7.9408 $\times10^{-5}$	7.3891 $\times10^{-5}$
$h_0=\nu=\frac{1}{18}$	41	48	66.19	79.82	6.6571 $\times10^{-5}$	6.5544 $\times10^{-5}$	2.8320 $\times10^{-5}$	2.9081 $\times10^{-5}$
$h_0=\nu=\frac{1}{22}$	42	54	148.1	207.1	4.5739 $\times10^{-5}$	4.6717 $\times10^{-5}$	1.9997 $\times10^{-5}$	2.085 $\times10^{-5}$
$h_0=\nu=\frac{1}{30}$	43	56	530.45	658.68	3.0559 $\times10^{-5}$	3.2268 $\times10^{-5}$	1.1887 $\times10^{-5}$	1.2180 $\times10^{-5}$

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Table 2. Numerical results for the Problem 1, when

$\alpha = 0.75$ .

$h_0/\nu$	No. of iteration		Execution time		Maximum-error		Average-error
$h_0/\nu$	FEGM	SP	FEGM	SP	FEGM	SP	FEGM	SP
$h_0=\nu=\frac{1}{10}$	44	50	7.75	7.94	2.1534 $\times10^{-4}$	2.2065 $\times10^{-4}$	1.0469 $\times10^{-4}$	1.0634 $\times10^{-4}$
$h_0=\nu=\frac{1}{14}$	44	43	27.36	27.28	1.2736 $\times10^{-4}$	1.2786 $\times10^{-4}$	5.9175 $\times10^{-5}$	6.0624 $\times10^{-5}$
$h_0=\nu=\frac{1}{18}$	42	48	68.29	80.18	8.0887 $\times10^{-5}$	8.5709 $\times10^{-5}$	3.0710 $\times10^{-5}$	3.9889 $\times10^{-5}$
$h_0=\nu=\frac{1}{22}$	41	55	150.40	197.87	6.1207 $\times10^{-5}$	6.2773 $\times10^{-5}$	2.8554 $\times10^{-5}$	2.8809 $\times10^{-5}$
$h_0=\nu=\frac{1}{30}$	42	56	523.25	648.55	3.7373 $\times10^{-5}$	4.4510 $\times10^{-5}$	1.6460 $\times10^{-5}$	1.7874 $\times10^{-5}$

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Table 3. Numerical results for the Problem 2, where

$\alpha = 0.1$ .

$h_0/\nu$	No. of iteration		Execution time		Maximum-error		Average-error
$h_0/\nu$	FEGM	SP	FEGM	SP	FEGM	SP	FEGM	SP
$h_0=\nu=\frac{1}{10}$	44	50	7.75	7.94	2.1534 $\times10^{-4}$	2.2065 $\times10^{-4}$	1.0469 $\times10^{-4}$	1.0634 $\times10^{-4}$
$h_0=\nu=\frac{1}{14}$	44	43	27.36	27.28	1.2736 $\times10^{-4}$	1.2786 $\times10^{-4}$	5.9175 $\times10^{-5}$	6.0624 $\times10^{-5}$
$h_0=\nu=\frac{1}{18}$	42	48	68.29	80.18	8.0887 $\times10^{-5}$	8.5709 $\times10^{-5}$	3.0710 $\times10^{-5}$	3.9889 $\times10^{-5}$
$h_0=\nu=\frac{1}{22}$	41	55	150.40	197.87	6.1207 $\times10^{-5}$	6.2773 $\times10^{-5}$	2.8554 $\times10^{-5}$	2.8809 $\times10^{-5}$
$h_0=\nu=\frac{1}{30}$	42	56	523.25	648.55	3.7373 $\times10^{-5}$	4.4510 $\times10^{-5}$	1.6460 $\times10^{-5}$	1.7874 $\times10^{-5}$

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Table 4. Numerical results for the Problem 2, where

$\alpha = 0.5$ .

$\nu/h_0$	No. of iteration		Execution time		Maximum-error		Average-error
$\nu/h_0$	FEGM	SP	FEGM	SP	FEGM	SP	FEGM	SP
$h_0=\nu=\frac{1}{10}$	46	52	7.65	8.72	2.9442 $\times10^{-4}$	2.8915 $\times10^{-4}$	1.3091 $\times10^{-4}$	1.2849 $\times10^{-4}$
$h_0=\nu=\frac{1}{14}$	48	47	30.93	29.53	1.3893 $\times10^{-4}$	1.4377 $\times10^{-4}$	3.7322 $\times10^{-5}$	3.9455 $\times10^{-5}$
$h_0=\nu=\frac{1}{18}$	48	52	77.2	84.75	8.5499 $\times10^{-5}$	8.5075 $\times10^{-5}$	2.4058 $\times10^{-5}$	2.4340 $\times10^{-5}$
$h_0=\nu=\frac{1}{22}$	47	57	172.53	203.76	5.5120 $\times10^{-5}$	5.5018 $\times10^{-5}$	1.9837 $\times10^{-5}$	1.7813 $\times10^{-5}$
$h_0=\nu=\frac{1}{30}$	47	65	588.74	789.14	2.9259 $\times10^{-5}$	2.9669 $\times10^{-5}$	1.2478 $\times10^{-5}$	1.2549 $\times10^{-5}$

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Table 5. Numerical results for the Problem 2, where

$\alpha = 0.75$ .

$h_0/\nu$	No. of iteration		Execution time		Maximum-error		Average-error
$h_0/\nu$	FEGM	SP	FEGM	SP	FEGM	SP	FEGM	SP
$h_0=\nu=\frac{1}{10}$	47	53	7.92	8.82	2.2868 $\times10^{-4}$	2.2917 $\times10^{-4}$	1.0194 $\times10^{-4}$	1.0189 $\times10^{-4}$
$h_0=\nu=\frac{1}{14}$	48	48	31.23	30.46	2.3640 $\times10^{-4}$	2.3848 $\times10^{-4}$	1.1941 $\times10^{-4}$	1.2003 $\times10^{-4}$
$h_0=\nu=\frac{1}{18}$	49	52	78.34	85.3	2.1361 $\times10^{-4}$	2.1290 $\times10^{-4}$	1.0621 $\times10^{-4}$	1.0863 $\times10^{-4}$
$h_0=\nu=\frac{1}{22}$	48	57	169.15	207.6	1.9686 $\times10^{-4}$	1.8908 $\times10^{-4}$	9.5981 $\times10^{-5}$	9.5644 $\times10^{-5}$
$h_0=\nu=\frac{1}{30}$	48	65	599.97	775.49	1.5244 $\times10^{-4}$	1.4938 $\times10^{-4}$	7.3782 $\times10^{-5}$	7.3922 $\times10^{-5}$

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Table 6. Comparison of the Proposed method with standard point method^[27] for Example 2 when

$\alpha = 0.5$ .

$h_0/\nu$	No. of iteration		Maximum-error		Average-error
$h_0/\nu$	FEGM	^[27]	FEGM	^[27]	FEGM	^[27]
$h_0=\nu=\frac{1}{10}$	43	53	1.6404 $\times10^{-4}$	1.2428 $\times10^{-2}$	7.9408 $\times10^{-5}$	8.8490 $\times10^{-3}$
$h_0=\nu=\frac{1}{18}$	41	52	6.6571 $\times10^{-5}$	7.1213 $\times10^{-3}$	2.8320 $\times10^{-5}$	3.6917 $\times10^{-3}$
$h_0=\nu=\frac{1}{22}$	42	55	4.7739 $\times10^{-5}$	2.6959 $\times10^{-3}$	1.9997 $\times10^{-5}$	1.3580 $\times10^{-3}$
$h_0=\frac{1}{65}, \nu=\frac{1}{35}$	41	58	2.5368 $\times10^{-5}$	2.0605 $\times10^{-3}$	5.0090 $\times10^{-6}$	1.0285 $\times10^{-3}$

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Figure 3. Approximate solution for the Problem 1, where

$h_0 = \nu = \frac{1}{35}$ .

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Figure 4. Exact solution for the Problem 1, where

$h_0 = \nu = \frac{1}{35}$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. Execution time (in sec) for different mesh sizes for the Problems 1 and 2.

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Table 7. The number of computing operations required for the FEGM and SP Technique.

Technique	operations per iteration
Technique	+/-	^*/ $\div$
SP	$(26+8(k-1))m^{2}$	$(8+4(k-1))m^{2}$
FEGM	$(28+8(k-1))(m-1)^{2}$ + $(26+8(k-1))(2m-1)$	$(12+4(k-1))(m-1)^{2}$ + $(8+4(k-1))(2m-1)$

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Table 8. The total computation effort for the Problem 1, where

$\alpha = \frac{1}{2}$ .

$h_0/\nu$	SP method		FEGM
$h_0/\nu$	Number of iteration	Total operations	Number of iteration	Total operations
$h_0=\nu=\frac{1}{10}$	49	695800	43	631498
$h_0=\nu=\frac{1}{18}$	48	3701376	41	3232686
$h_0=\nu=\frac{1}{22}$	54	7474896	42	5924940
$h_0=\nu=\frac{1}{30}$	56	19252800	43	15000378

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Table 9. The total computation effort for the Problem 2, where

$\alpha = \frac{3}{4}$ .

$k/m$	SP method		FEGM
$k/m$	Number of iteration	Total operations	Number of iteration	Total operations
$h_0=\nu=\frac{1}{10}$	53	752600	47	690242
$h_0=\nu=\frac{1}{18}$	52	4009824	49	3863454
$h_0=\nu=\frac{1}{22}$	57	7890168	48	6771360
$h_0=\nu=\frac{1}{30}$	65	22347000	48	16744608

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Table 10. The spatial convergence order for the Problem 1.

$\alpha=0.4$			$\alpha=0.5$
$\nu/h_0$	Maximum error	$C_{2}$ -order	$\nu/h_0$	Maximum error	$C_{2}$ -order
$\nu=h_0=0.5$	1.4394 $\times 10^{-4}$	—	$\nu=h_0=0.5$	3.1088 $\times 10^{-4}$	—
$\nu=0.031$ , $h_0=0.25$	1.0910 $\times 10^{-5}$	3.72	$\nu=0.031$ , $h_0=0.25$	2.2509 $\times 10^{-5}$	3.78
$\nu=h_0=0.25$	2.1929 $\times 10^{-4}$	—	$\nu=h_0=0.25$	3.546 $\times 10^{-4}$	—
$\nu=0.016$ , $h_0=0.12$	9.1371 $\times 10^{-6}$	4.58	$\nu=0.016$ , $h_0=0.12$	2.1414 $\times 10^{-5}$	4.04
$\alpha=0.6$			$\alpha=0.8$
$\nu/h_0$	Maximum error	$C_{2}$ -order	$\nu/h_0$	Maximum error	$C_{2}$ -order
$\nu=h_0=0.5$	5.2435 $\times 10^{-4}$	—	$\nu=h_0=0.5$	9.9004 $\times 10^{-4}$	—
$\nu=0.031$ , $h_0=0.25$	2.7191 $\times 10^{-5}$	4.26	$\nu=0.031$ , $h_0=0.25$	9.9004 $\times 10^{-5}$	4.13
$\nu=h_0=0.25$	5.0421 $\times 10^{-4}$	—	$\nu=h_0=0.25$	7.2673 $\times 10^{-4}$	—
$\nu=0.016$ , $h_0=0.12$	3.1980 $\times 10^{-5}$	3.97	$\nu=0.016$ , $h_0=0.12$	3.7243 $\times 10^{-5}$	4.28

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Table 11. The spatial convergence order for the Problem 2.

$\alpha=0.7$			$\alpha=0.8$
$\nu/h_0$	Maximum error	$C_{2}$ -order	$\nu/h_0$	Maximum error	$C_{2}$ -order
$\nu=h_0=0.5$	1.6072 $\times 10^{-3}$	—	$\nu=h_0=0.5$	3.4202 $\times 10^{-3}$	—
$\nu=0.031$ , $h_0=0.25$	1.0543 $\times 10^{-4}$	3.93	$\nu=0.031$ , $h_0=0.25$	1.7636 $\times 10^{-4}$	4.27
$\nu=h_0=0.25$	1.3545 $\times 10^{-3}$	—	$\nu=h_0=0.25$	1.6955 $\times 10^{-3}$	—
$\nu=0.016$ , $h_0=0.12$	5.9784 $\times 10^{-5}$	4.50	$\nu=0.016$ , $h_0=0.12$	8.2806 $\times 10^{-5}$	4.35
$\alpha=0.3$			$\alpha=0.5$
$\nu/h_0$	Maximum error	$C_{2}$ -order	$\nu/h_0$	Maximum error	$C_{2}$ -order
$\nu=h_0=0.5$	3.832 $\times 10^{-3}$	—	$\nu=h_0=0.5$	8.6771 $\times 10^{-4}$	—
$\nu=0.031$ , $h_0=0.25$	3.2105 $\times 10^{-4}$	3.57	$\nu=0.031$ , $h_0=0.25$	4.8772 $\times 10^{-5}$	4.15
$\nu=h_0=0.25$	4.5916 $\times 10^{-4}$	—	$\nu=h_0=0.25$	4.1734 $\times 10^{-4}$	—
$\nu=0.016$ , $h_0=0.12$	2.5978 $\times 10^{-5}$	4.14	$\nu=0.016$ , $h_0=0.12$	2.3009 $\times 10^{-5}$	4.18

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Table 12. Temporal convergence order for the Problem 1, when

$h_0 = \frac{1}{8}$ .

$\nu$	$\alpha=0.3$		$\alpha=0.8$
$\nu$	$L_{\infty}$	$C_{1}-$ Order	$L_{\infty}$	$C_{1}-$ Order
$\nu=\frac{1}{10}$	$7.8675 \times 10^{-4}$	—	$2.1609\times 10^{-4}$	—
$\nu=\frac{1}{20}$	$3.2208$ $\times 10^{-5}$	1.28	$7.1112$ $\times 10^{-5}$	1.60
$\nu=$ $\frac{1}{40}$	$1.7253$ $\times10^{-5}$	1.29	$2.6203$ $\times 10^{-5}$	1.44
$\nu =\frac{1}{80}$	$6.2057 \times 10^{-6}$	1.72	$8.3392 \times 10^{-6}$	1.65

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Table 13. Temporal convergence order for the Problem 2, when

$h_0 = \frac{1}{8}$ .

$\nu$	$\alpha=0.1$		$\alpha=0.8$
$\nu$	$L_{\infty}$	$C_{1}-$ Order	$L_{\infty}$	$C_{1}-$ Order
$\nu=\frac{1}{10}$	$4.7691\times 10^{-4}$	—	$2.1429\times 10^{-4}$	—
$\nu=\frac{1}{20}$	$1.6086$ $\times 10^{-4}$	1.56	$8.2679$ $\times 10^{-5}$	1.37
$\nu=$ $\frac{1}{40}$	$5.1170$ $\times10^{-5}$	1.65	$3.2827$ $\times 10^{-5}$	1.33
$\nu =\frac{1}{80}$	$1.7848 \times 10^{-5}$	1.51	$1.3010 \times 10^{-5}$	1.51

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7. Conclusions

In this article, the 2-D fractional sub-diffusion equation is solved using the fractional explicit group method with weak singularity at initial time $t = 0$ , where the standard point finite difference scheme is used for the development of the fourth-order grouping scheme. The fractional explicit group method reduces the computational complexity and execution time by comparing it with the standard point fourth-order method without deteriorating the accuracy of the solutions. Furthermore, the unconditional stability and convergence of the proposed scheme are proved using the matrix analysis via mathematical induction, which confirms the feasibility and reliability of the new formulation.

Acknowledgments

This work was supported by the Ministry of Higher Education under Fundamental Research Grant Scheme (FRGS/1/2019/STG06/UTM/02/13, FRGS/1/2020/STG05/UTM/02/12).

Conflict of interest

We declare no conflicts of interest in this paper.

References

[1]	R. Herrmann, Fractional calculus: An introduction for physicists, 2 Eds., Singapore: World Scientific Publishing, 2014.
[2]	A. Lateef, C. K. Verma, An analysis of fractional explicit method over black scholes method for pricing option, Global J. Pure Appl. Math., 13 (2017), 5851–5869.
[3]	F. Reza, T. Widodo, The impact of education on economic growth in Indonesia, J. Indones. Econ. Bus., 28 (2013), 23–44.
[4]	G. W. Suter, Review papers are important and worth writing, Environ. Toxicol. Chem., 32(2013), 1929–1930. doi: 10.1002/etc.2316
[5]	V. Tarasova, V. Tarasov, Fractional dynamics of natural growth and memory effect in economics, Eur. Res., 12 (2016), 30–37.
[6]	A. Benlabbes, M. Benbachir, M. Lakrib, Existence solutions of a nonlinear fractional differential equations, J. Adv. Res. Dyn. Control Syst., 6 (2014), 1–12.
[7]	D. Singh, Existence and uniqueness of solution of linear fractional differential equation, Int. J. Recent Res. Aspects, 4 (2017), 6–10.
[8]	Y. Y. Gambo, R. Ameen, F. Jarad, T. Abdeljawad, Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives, Adv. Differ. Equ., 2018 (2018), 134. doi: 10.1186/s13662-018-1594-y
[9]	V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal.-Theor., 69 (2008), 2677–2682. doi: 10.1016/j.na.2007.08.042
[10]	O. S. Odetunde, O. A. Taiwo, A decomposition algorithm for the solution of fractional quadratic Riccati differential equations with Caputo derivatives, Am. J. Comput. Appl. Math., 4 (2014), 83–91.
[11]	W. J. Yuan, A note on the Riccati differential equation, J. Math. Anal. Appl., 277 (2003), 367–374.
[12]	K. Busawon, P. Johnson, Solution of a class of Riccati equations, In: Proceedings of the 8th WSEAS International Conference on Applied Mathematics, 4 (2005), 334–338.
[13]	M. Merdan, On the solutions fractional Riccati differential equation with modified Riemann-Liouville derivative, Int. J. Diff. Equ., 2012 (2012), 346089.
[14]	N. A. Khan, A. Ara, N. A. Khan, Fractional-order Riccati differential equation: analytical approximation and numerical results, Adv. Diff. Equ., 2013 (2013), 185. doi: 10.1186/1687-1847-2013-185
[15]	J. Biazar, M. Didgar, Numerical solution of Riccati equations by the Adomian and asymptotic decomposition methods over extended domains, Int. J. Diff. Equ., 2015 (2015), 580741.
[16]	M. M. Khader, Numerical treatment for solving fractional Riccati differential equation, J. Egypt. Math. Soc., 21 (2013), 32–37. doi: 10.1016/j.joems.2012.09.005
[17]	H. N. A. Ismail, I. K. Youssef, T. M. Rageh, Numerical treatment for solving fractional Riccati differential equation using VIM–Restrictive Padé, J. Sci. Eng. Res., 4 (2017), 276–283.
[18]	M. Hamarsheh, A. I. Ismail, Z. Odibat, An analytic solution for fractional order Riccati equations by using optimal homotopy asymptotic method, Appl. Math. Sci., 10 (2016), 1131–1150.
[19]	T. Harko, F. S. M. Lobo, M. K. Mak, Analytical solutions of the Riccati equation with coefficients satisfying integral or differential conditions with arbitrary functions, Universal J. Appl. Math., 2 (2014), 109–118.
[20]	K. Jaber, S. Al-Tarawneh, Exact solution of Riccati fractional differential equation, Universal J. Appl. Math., 4 (2016), 51–54.
[21]	T. Khaniyev, M. Merdan, On the fractional Riccati differential equation, Int. J. Pure Appl. Math., 107 (2016), 145–160.
[22]	A. A. Bastami, M. R. Belic, N. Z. Petrovic, Special solutions of the Riccati equation with applications to the Gross-Pitaevskii nonlinear PDE, Electron. J. Differ. Equ., 2010 (2010), 1–10.
[23]	G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), 501–544. doi: 10.1016/0022-247X(88)90170-9
[24]	F. Z. Genga, Y. Z. Lin, M. G. Cui, A piecewise variational iteration method for Riccati differential equations, Comput. Math. Appl., 58 (2009), 2518–2522. doi: 10.1016/j.camwa.2009.03.063
[25]	H. Jafari, H. Tajadodi, He's variational iteration method for solving fractional Riccati differential equation, Int. J. Diff. Equ., 2010 (2010), 764738.
[26]	N. Faraz, Y. Khan, H. Jafari, A. Yildirim, M. Madani, Fractional variational iteration method via modified Riemann–Liouville derivative, J. King Saud Univ. Sci., 23 (2011), 413–417. doi: 10.1016/j.jksus.2010.07.025
[27]	S. Bhakelar, V. Daftardar-Gejji, Solving fractional-order logistic equation using a new iterative method, Int. J. Diff. Equ., 2012 (2012), 975829.
[28]	J. S. Duan, R. Rach, D. Baleanu, A. M. Wazwaz, A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Frac. Calc., 3 (2012), 73–99.
[29]	H. Jafari, H. Tajadodi, D. Baleanu, A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials, Fract. Calc. Appl. Anal., 16 (2013), 109–122. doi: 10.2478/s13540-013-0008-9
[30]	K. H. Mohammedali, N. A. Ahmad, F. S. Fadhel, He's variational iteration method for solving Riccati matrix delay differential equations of variable coefficients, In: AIP Conference Proceedings, 1830 (2017), 020029. doi: 10.1063/1.4980892
[31]	A. M. S. Mahdy, G. M. A. Marai, Sumudu decomposition method for solving fractional Riccati equation, JACM, 3 (2018), 42–50.
[32]	V. V. Tarasova, V. E. Tarasov, Elasticity for economic processes with memory: Fractional differential calculus approach, Fract. Diff. Calc., 6 (2016), 219–232.
[33]	V. V. Tarasova, V. E. Tarasov, Fractional dynamics of natural growth and memory effect in economics, Eur. Res., 12 (2016), 30–37.
[34]	I. Tejado, D. Valério, E. Pérez, N. Valério, Fractional calculus in economic growth modelling: the Spanish and Portuguese cases, Int. J. Dynam. Control, 5 (2017), 208–222.
[35]	H. Ming, J. R. Wang, M. Feckan, The application of fractional calculus in Chinese economic growth models, Mathematics, 7 (2019), 665. doi: 10.3390/math7080665
[36]	V. V. Tarasova, V. E. Tarasov, Economic accelerator with memory: Discrete time approach, Probl. Mod. Sci. Educ., 36 (2016), 37–42.
[37]	V. E. Tarasov, V. V. Tarasova, Long and short memory in economics: Fractional-order difference and differentiation, IRA-Int. J. Manage. Soc. Sci., 5 (2016), 327–334.
[38]	J. A. Machado, M. E. Mata, A. M. Lopes, Fractional state space analysis of economic systems, Entropy, 17 (2015), 5402–5421. doi: 10.3390/e17085402
[39]	V. E. Tarasov, Local fractional derivatives of differentiable functions are integer-order derivatives or zero, Int. J. Appl. Comput. Math., 2 (2016), 195–201. doi: 10.1007/s40819-015-0054-6
[40]	V. V. Tarasova, V. E. Tarasov, Economic interpretation of fractional derivatives, Prog. Fract. Differ. Appl., 3 (2017), 1–7. doi: 10.18576/pfda/030101
[41]	V. V. Tarasova, V. E. Tarasov, Logistic map with memory from economic model, Chaos Soliton. Fract., 95 (2017), 84–91. doi: 10.1016/j.chaos.2016.12.012
[42]	V. V. Tarasova, V. E. Tarasov, Concept of dynamic memory in economics, Commun. Nonlinear Sci., 55 (2018), 127–145. doi: 10.1016/j.cnsns.2017.06.032
[43]	D. Luo, J. R. Wang, M. Fečkan, Applying fractional calculus to analyze economic growth modelling, JAMSI, 14 (2018), 25–36.
[44]	R. Pakhira, U. Ghosh, S. Sarkar, Study of memory effects in an inventory model using fractional calculus, Appl. Math. Sci., 12 (2018), 797–824. doi: 10.18576/amis/120414
[45]	V. E. Tarasov, V. V. Tarasova, Criterion of existence of power-law memory for economic processes, Entropy, 20 (2018), 414. doi: 10.3390/e20060414
[46]	R. Pakhira, U. Ghosh, S. Sarkar, Study of memory effect in an inventory model with quadratic type demand rate and salvage value, Appl. Math. Sci., 13 (2019), 209–223. doi: 10.18576/amis/130208
[47]	V. V. Tarasova, V. E. Tarasov, Dynamic Keynesian model of economic growth with memory and lag, Mathematics, 7 (2019), 178. doi: 10.3390/math7020178
[48]	V. E. Tarasov, V. V. Tarasova, Harrod–Domar growth model with memory and distributed lag, Axioms, 8 (2019), 9. doi: 10.3390/axioms8010009
[49]	I. Tejado, E. Pérez, D. Valério, Fractional calculus in economic growth modelling of the group of seven, Fract. Calc. Appl. Anal., 22 (2019), 139–157. doi: 10.1515/fca-2019-0009
[50]	B. Acay, E. Bas, T. Abdeljawad, Fractional economic models based on market equilibrium in the frame of different type kernels, Chaos Soliton. Fract., 130 (2020), 109438. doi: 10.1016/j.chaos.2019.109438
[51]	A. Thiao, N. Sene, Fractional optimal economic control problem described by the generalized fractional order derivative, In: 4 th International Conference on Computational Mathematics and Engineering Sciences, Springer, Cham, 2019, 36–48.
[52]	A. Traore, N. Sene, Model of economic growth in the context of fractional derivative, Alex. Eng. J., 59 (2020), 4843–4850. doi: 10.1016/j.aej.2020.08.047
[53]	Sukono, A. Sambas, S. B. He, H. Liu, S. Vaidyanathan, Y. Y. Hidayat, et al., Dynamical analysis and adaptive fuzzy control for the fractional-order financial risk chaotic system, Adv. Diff. Equ., 2020 (2020), 674.
[54]	S. Vaidyanathan, A. Sambas, S. Kacar, U. Cavusoglu, A new finance chaotic system, its electronic circuit realization, passivity-based synchronization and an application to voice encryption, Nonlinear Eng., 8 (2019), 193–205.
[55]	K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229–248. doi: 10.1006/jmaa.2000.7194

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Application of fractional differential equation in economic growth model: A systematic review approach

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Application of fractional differential equation in economic growth model: A systematic review approach

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