Numerical Solution of Two-Dimensional Time Fractional Mobile/Immobile Equation Using Explicit Group Methods.

In this paper, we shall present the development of two explicit group schemes, namely, fractional explicit group (FEG) and modified fractional explicit group (MFEG) methods for solving the time fractional mobile/immobile equation in two space dimensions. The presented methods are formulated based on two Crank-Nicolson (C-N) finite difference schemes established at two different grid spacings. The stability and convergence of order O ( τ 2 - α + h 2 ) are rigorously proven using Fourier analysis. Several numerical experiments are conducted to verify the efficiency of the proposed methods. Meanwhile, numerical results show that the FEG and MFEG algorithms are able to reduce the computational times and iterations effectively while preserving good accuracy in comparison to the C-N finite difference method.

Introduction

The roots of fractional calculus are nearly as old as those of classical calculus. However, it was only at the turn of the century that interest in fractional calculus exploded. In accordance with this, the first book along with the first conference on fractional calculus and its applications did not appear until 1974. For some researchers, this year is viewed as the beginning of a new era for fractional calculus. Nowdays, a large number of research articles and books are devoted to this topic, and its applications in various areas such as physics, biology, chemistry, finance and economy are witnessing remarkable progress, see [1-11] and the references therein. The popularity of fractional calculus in all branches of science stems from its successful application in modelling physical phenomena, specifically in the form of fractional differential equations. For instance, Zhang et al. [12] introduced an image enhancement approach for single-pixel imaging that is based on fractional order operators. The authors argue that the new fractional approach strikes a good balance between edge detection and noise suppression, making it useful in areas such as feature extraction and object recognition, in addition to military and industrial applications. Aman et al. [13] considered some fractional-order PDEs to study the memory properties pertaining to the behavior of nanofluids, which cannot be addressed by integer-order PDEs. The authors mentioned that their work has useful applications in diverse fields such as physics, engineering and many others. Khan et al. [14] extended the classical Maxwell equation to its counterpart fractional-order Maxwell model via the emerging definition of the Caputo-Fabrizio derivative. The resulting fractional model was then applied to account for the heat transfer due to convection occurring in a generalized Maxwell fluid. Almeida et al. [15] scrutinized two mathematical models namely, the population growth model and the gross domestic product model, by using fractional differential equations described in the frame of Caputo derivative with respect to another function. Alshabanat et al. [16] generalized the concept of the Caputo-Fabrizio fractional derivative and provided some applications to RC-electrical circuits on the basis of fractional differential equations involving the new developed definition. With the help of a fractional-order model, Ali et al. [17] discussed the flow of Jeffrey nanofluid under the effects of thermal radiation and heat/mass transfer. The governing fractional differential equation was described in the Atangana-Baleanu sense. In [18], two epidemic fractional models for describing infection dynamics together with a macroeconomic fractional model were investigated numerically. Uçar et al. [19] yielded approximate numerical solutions for the giving up smoking model expressed in terms of the Atangana-Baleanu fractional differential equation. In another study, Alrabaiah et al. [20] dealt with the phenomenon of tobacco smoking with snuffing class via a system of Caputo-type fractional differential equations. Ndaïrou et al. [21] reported on the mathematical and numerical analysis of a fractional Caputo differential system that approximates the infectious individuals of COVID-19. Some other valuable work on the application of fractional calculus can be found in [22-25].

This article focuses on developing explicit group schemes for the two-dimensional time fractional mobile/immobile equation of the following form:subject to the initial and boundary conditionswhere is the Laplacian operator,f(x, y, t), and g(x, y) are given smooth functions. and is its boundary. () represents the Caputo fractional derivative defined byThe time fractional mobile/immobile Eq. (1) plays a central role in describing various phenomena including heat diffusion and propagation of ocean sounds, among others. Moreover, it has been verified that the long term limit of continuous time random walks can be controlled by the fractional mobile/immobile equation [26], which interprets the probabilistic nature of the latter. In most cases, fractional partial differential equations (PDEs) are difficult to deal with analytically, and therefore, numerical or approximate analytical solutions become indispensable. In the following, we present some recent numerical developments in solving fractional mobile/immobile equations. Liu and Li [27] presented a Crank-Nicolson (C-N) difference scheme for the one-dimensional mobile/immobile equation with time derivative of variable order. Pourbashash et al. [28] proposed a compact difference scheme for solving the one-dimensional time fractional mobile/immobile equation. The stability and convergence of the numerical scheme were proved using Fourier method. Qiu et al. [29] developed a time two-grid method based on a finite difference approach for the nonlinear time fractional mobile/immobile equation in two dimensions. Yang et al. [26] developed the C-N orthogonal spline collocation method for the two-dimensional time fractional mobile/immobile equation. Jiang et al. [30] considered an alternating direction implict (ADI) compact scheme to solve the semilinear time fractional mobile/immobile equation in two dimensions. Yin et al. [31] employed the generalized BDF2- technique in time and finite element in space to deal with the two-dimensional mobile/immobile equation with Riemann-Liouville time derivative. In addition, Chai et al. [32] constructed the high order compact difference schemes for the one-dimensional and two-dimensional time fractional mobile/immobile equations.

Generally speaking, numerical methods for fractional PDEs are abundant, whereas efficient numerical methods leading to rapid convergence, particularly for time fractional mobile/immobile equations, are relatively sparse. This motivates us to develop fast numerical schemes to solve them. This is useful for long time simulations, especially when attempting to solve multi-dimensional fractional problems [33-36]. It is well known that explicit group methods can diminish the computational complexity and reduce the computational time of numerical algorithms effectively [37-44]. However, numerical approximations based on explicit group methods for fractional mobile/immobile equations are still at an early stage of development. The main goal of this paper is to construct two fractional explicit group methods based on C-N difference schemes for solving Eq. (1). The stability in norm and the convergence order , where and h are respectively the temporal and spatial step sizes, will be proved. Numerical experiments to show that the proposed methods are more efficient than the C-N difference method in terms of execution time are also provided.

The outline of this paper is as follows. Section 2 devotes to the derivation of the C-N finite difference scheme. In Sect. 3, the fractional explicit group methods are constructed. The stability and convergence are rigorously proved in Sects. 4 and 5, respectively. In Sect. 6, some numerical simulations are carried out by using the C-N scheme and the two fractional explicit group methods, and some comparisons between three methods are demonstrated. Finally, a brief summary is given in Sect. 7.

The C-N Difference Scheme

In order to discretize the solution domain, we define to be a uniform mesh of domain . Likewise, define to be a uniform mesh of interval [0, T]. , and N are positive integers. Let be the solution value at the grid point . Then, the C-N approximation to Eq. (1) can be expressed as follows:A discrete approximation to the Caputo time fractional derivative is given by the formula [45, 46]whereIn addition, utilizing Taylor expansion, we haveandSubstituting (5-8) into (4), we obtain the following difference equationin which . By disregarding the higher order term in the above equation and replacing with its numerical approximation , the following C-N difference scheme is obtainedwhere and .

Formulation of Group Methods

The Fractional Explicit Group (FEG) Method

Consider the C-N difference scheme (10). Let the mesh points be grouped in blocks of four points as shown in Fig. 1. Then, Eq. (10) is applied to any group of four points so that the following (4 4) system of equations is obtainedwhere The coefficients matrix in 11 can be inverted to get the four-point FEG equationwhereFigure 1 depicts the grouping of mesh points into blocks of four points with mesh size . It is obvious that the execution of Eq. (12) is only applicable for the grouped points. Therefore, the implementation of the FEG method is carried out by applying Eq. (12) to each group in Fig. 1. Before moving to the next time level, we calculate the remaining ungrouped points near the boundaries by utilizing Eq. (10). At each time level, the computation process is carried out in an iterative manner until the final time level N is reached. For convenience, the FEG method is defined in Algorithm 1.

Fig. 1

Grouping of mesh points for the FEG method with

The Modified Fractional Explicit Group (MFEG) Method

Another approximation scheme for Eq. (1) is obtained by Taylor’s expansion and considering points at mesh size of 2h. The C-N difference scheme with 2h spacing is as followsOn simplification with being the numerical approximation of , the following equation is obtainedwhere and .

Now applying the C-N difference formula with 2h spacing (14) to any group of four points will lead to the following () systemwhere ,The above system can be rewritten in explicit form which results in the four-point MFEG equationwhereIn the MFEG method, the mesh points of the discretized solution domain are arranged into several groups. Each group comprise four points of type (shown in Fig. 2). The MFEG Eq. (16) is applied to generate iterations on each group in Fig. 2. Prior going to the next time level, the remaining points of the mesh ( and ) can be obtained directly once utilizing the following sequence: The MFEG method is illustrated in Algorithm 2. From the described solution procedure, it may be observed that the MFEG method involves only a quarter of the mesh points in the iterative process at each time level. Due to this reduction of points that take part in the iterative process, the MFEG method is expected to save more CPU time in solving Eq. (1).

Fig. 2

Grouping of mesh points for the MFEG method with

The skewed C-N difference formula obtained by rotating the axis 45 clockwise is used for points. With and , the skewed C-N difference scheme for Eq. (1) is given by

The C-N difference formula (10) is utilized for points.

Stability Analysis

This section is devoted to the stability of the C-N difference schemes (10) and (14), which are the basis of the FEG and MFEG methods, respectively. The following lemma is introduced for the convenience of our analysis

Lemma 1

The coefficients in Eq. (14) satisfy

.

.

Stability of the h-Spaced C-N Numerical Scheme

Let be the approximate solution of (10), and defineThen by setting (18) into (10), we getThe Fourier series for iswhere and the Fourier coefficients is given byIntroducing the following normApplying the Parseval’s equalitywe getWith and , and based on the above analysis, we assume that the solutions of Eq. (19) is of the following form

Lemma 2

Suppose is defined by (20). If , then we have

Proof

Setting into Eq. (19), we obtainUpon simplification, we can getwhereWe start with in Eq. (27). Since , thenNow, assume that , . We need to prove this for . Utilizing Eq. (27), we getUsing induction hypothesis and lemma 1, we obtainConsider the following two cases:

Case 1. If , then we haveCase 2. , then we haveHere,The proof is completed by mathematical induction.

Theorem 1

The C-N difference scheme (10) is stable if .

Utilizing lemma 2 and Parseval’s equality, we obtainThus, the difference scheme (10) is stable.

Stability of the 2h-Spaced C-N Numerical Scheme

Let be the approximate solution of (14), and defineSubstituting (25) into (14), we obtainThe Fourier series for iswhere and the Fourier coefficients are given byIntroducing the following normApplying the Parseval’s equalitywe getBased on the above analysis, we assume that the solution of Eq. (26) is of the following form

Lemma 3

Suppose is defined by (27). If , then we have

Substituting into Eq. (26), we getAfter simplification, we can getwhereLetting in Eq. (31) and using that , we obtainNow, suppose that , . We must prove it holds for . Using Eq. (31), we getBy the induction hypothesis and lemma 1, we getConsider the following two cases:

Case 1. If , then we haveCase 2. , then we haveHere,This completes the proof.

Theorem 2

The C-N difference scheme (14) is stable if .

Utilizing lemma 3 and Parseval’s equality, we obtainwhich means that the difference scheme (14) is stable.

Convergence Analysis

In this section, the convergence of the proposed methods is investigated. By subtracting Eq. (10) from Eq. (9) and defining , the following error equation is obtainedFrom Eq. (9), there is a positive constant such thatSimilar to the stability analysis, and can be expanded in Fourier series aswhere the Fourier coefficients and areBy making use of the Parseval’s equality and norm, we obtainNext, we suppose the solutions of Eq. (32) are as followsUsing the assumptions in (36) and considering (32), we obtainAfter simplification, we havewhere and are as defined in the previous section.

Due to the convergence of the series in the right hand side of Eq. (35), there is a positive constant such that

Lemma 4

Suppose and satisfy Eq. (38). If , then we have

Proof

Noticing that . Then, for , we haveNow, assume that , . We need to prove this for . From Eq. (38), we haveUtilizing induction hypothesis, lemma 1 and Eq. (39), we getIf , then we haveIf , then we haveHere,This completes the proof.

Theorem 3

The C-N difference scheme (10) is convergent and the order of convergence is .

Using lemma 4 and Parseval’s equality, we obtainNoticing that and letting . From Eq. (33), we havewhich completes the proof.

Theorem 4

The C-N difference scheme (14) is convergent and the order of convergence is .

The proof is similar to Theorem 3.

Numerical Experiments

In this part, three test problems with known exact solutions are simulated on Windows 10 (64 bit) Intel (R) Core (TM) i7-8550U CPU 2.00 GHz, 8GB of RAM utilizing MATLAB software. The numerical results are obtained at three different final times, namely, , and with the spatial domain being restricted to . The proposed methods are combined with the Gauss-Seidel iterative scheme with error tolerance of to solve the test problems of the form in Eq. (1). Throughout the discussion, we let u, , and denote the exact, C-N, FEG and MFEG solutions, respectively. In addition, the computational orders of the presented methods are calculated using the formula [47]

Test Problem 6.1

In the first problem, consider the source term as followswhere the exact solution is given by

Tables 1, 2 and 3 show the numerical results of the elapsed time (in seconds), number of iterations (Ite) and maximum absolute error () for the C-N, FEG and MFEG schemes with , , =0.1, 0.3, 0.7 for different sizes of space steps, respectively. Figure 3 presents a comparison of the exact solution and other numerical solutions obtained at , , and . In addition, the graphical error representation of the C-N scheme for , , and is depicted in Fig. 4 while its second order convergence can be observed from the results in Table 4. From the data in these tables and figures, the convergence of the presented methods is confirmed. Comparing the resolution of the tested methods, it may be observed that the C-N and FEG schemes have almost the same degree of accuracy while the magnitude of absolute errors for the MFEG method is slightly larger than the said schemes. This is because the iterative process for the C-N and FEG schemes is implemented on a mesh size of h while the iterative process of the MFEG method is carried out on a mesh size of 2h; hence, increasing the error term by 4. Figure 5 compares the computing times and iteration numbers of the tested methods and shows that the FEG and MFEG methods are more efficient than the C-N difference scheme in terms of execution time and number of iterations.

Table 1

Comparison between C-N, FEG and MFEG methods for Test Problem 6.1 with , and

h^-1	C-N			FEG			MFEG
	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)
10	0.4175	23	2.2757E-02	0.3356	15	2.2754E-02	0.0945	6	8.6466E-02
18	2.3110	52	6.9678E-03	1.0515	31	6.9591E-03	0.2451	12	2.7489E-02
26	5.8888	92	3.3755E-03	4.1425	53	3.3677E-03	0.4943	19	1.3548E-02
34	18.2440	140	1.9430E-03	10.1393	80	1.9427E-03	1.1403	28	7.9289E-03
42	39.2082	196	1.2113E-03	20.9560	111	1.2394E-03	2.6987	37	5.1846E-03

Table 2

Comparison between C-N, FEG and MFEG methods for Test Problem 6.1 with , and

h^-1	C-N			FEG			MFEG
	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)
10	0.3476	22	2.2785E-02	0.2483	15	2.2778E-02	0.0920	6	8.6576E-02
18	1.6289	51	6.9747E-03	1.0755	31	6.9638E-03	0.2450	12	2.7522E-02
26	6.5401	90	3.3776E-03	3.8171	52	3.3693E-03	0.6294	19	1.3561E-02
34	18.1564	138	1.9419E-03	9.9377	79	1.9423E-03	1.2881	27	7.9373E-03
42	39.9924	194	1.2115E-03	21.2502	110	1.2385E-03	2.1786	37	5.1877E-03

Table 3

Comparison between C-N, FEG and MFEG methods for Test Problem 6.1 with , and

h^-1	C-N			FEG			MFEG
	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)
10	0.4374	20	2.2831E-02	0.3357	18	2.2823E-02	0.1252	6	8.6801E-02
18	1.5666	47	6.9786E-03	0.924	28	6.9697E-03	0.2853	11	2.7583E-02
26	4.4705	83	3.3703E-03	2.9016	48	3.3642E-03	0.5462	17	1.3582E-02
34	11.6037	126	1.9280E-03	6.6387	72	1.9324E-03	1.1091	25	7.9436E-03
42	28.1122	177	1.1913E-03	13.7709	101	1.2262E-03	1.7506	34	5.1873E-03

Fig. 3

Comparison of exact and numerical solutions for Test Problem 6.1 with , , , and

Fig. 4

Surface error plot of the C-N method for Test Problem 6.1 with , , and

Table 4

Maximum errors and computational orders of the C-N method for Test problem 6.1 with and

α=0.1			α=0.5			α=0.9
h^-1	E_∞(τ,h)	C-Order	h^-1	E_∞(τ,h)	C-Order	h^-1	E_∞(τ,h)	C-Order
6	3.6027E-03	-	6	3.5845E-03	-	6	3.5656E-03	-
10	1.3960E-03	1.8560	10	1.3888E-03	1.8562	10	1.3818E-03	1.8557
14	7.1168E-04	2.0024	14	7.0804E-04	2.0022	14	7.0537E-04	1.9984
18	4.2523E-04	2.0492	18	4.2318E-04	2.0480	18	4.1537E-04	2.1071

Fig. 5

The graphs of (a) elapsed time and (b) number of iterations of the proposed methods for Test Problem 6.1 with

Test Problem 6.2

In the second problem, we consider the source term given byand the corresponding exact solution

The initial and boundary conditions are extracted from the exact solution. The exact, C-N, FEG and MFEG solutions at , , and are portrayed in Fig. 6, while Fig. 7 demonstrates the surface of absolute errors of the FEG method for and , and . From the figures, it can be observed that the numerical solutions match well with the exact solution. Tables 5, 6 and 7 display the numerical results of the proposed methods with , and =0.3, 0.5, 0.7 for various space steps, respectively. Furthermore, Fig. 8 illustrates the computational times and number of iterations pertained to the three tested methods. The elapsed time and iterations number in Tables 5, 6 and 7 and Fig. 8 indicate that the computational cost of the fractional group methods (FEG and MFEG) is lower than the C-N scheme, which show the effectiveness of the former. Table 8 show the numerical errors and computational orders computed using the FEG method. Just as we expect, the second order convergence can be seen from the data recorded in this table.

Fig. 6

Comparison of exact and numerical solutions for Test Problem 6.2 with , , , and

Fig. 7

Surface error plot of the FEG method for Test Problem 6.2 with , , and

Table 5

Comparison between C-N, FEG and MFEG methods for Test Problem 6.2 with , and

h^-1	C-N			FEG			MFEG
	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)
10	0.8609	40	1.6822E-02	0.3782	24	1.6841E-02	0.1286	9	6.4272E-02
18	3.7968	107	5.0513E-03	2.6769	60	5.1250E-03	0.3748	19	2.0512E-02
26	14.3701	199	2.1594E-03	8.6991	111	2.3224E-03	1.2356	35	9.8791E-03
34	42.6815	312	8.8053E-04	22.9559	175	1.1698E-03	2.0979	54	5.7423E-03
42	57.6235	444	1.2502E-04	43.8745	249	5.0909E-04	4.3400	77	3.7024E-03

Table 6

Comparison between C-N, FEG and MFEG methods for Test Problem 6.2 with , and

h^-1	C-N			FEG			MFEG
	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)
10	0.6385	40	2.5633E-02	0.3369	24	2.5655E-02	0.1324	9	9.8056E-02
18	3.9289	109	7.7312E-03	2.6144	61	7.8090E-03	0.3197	20	3.1264E-02
26	15.1236	205	3.4143E-03	8.4977	114	3.5836E-03	1.1475	35	1.5038E-02
34	43.1109	324	1.5899E-03	22.6807	180	1.8797E-03	2.6300	55	8.7478E-03
42	57.5005	463	5.2261E-04	45.1226	258	9.5837E-04	4.0916	79	5.6592E-03

Table 7

Comparison between C-N, FEG and MFEG methods for Test Problem 6.2 with , and

h^-1	C-N			FEG			MFEG
	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)
10	0.7414	39	3.8935E-02	0.4136	24	3.8961E-02	0.1449	9	1.4920E-01
18	2.8969	107	1.1739E-02	1.9721	60	1.1820E-02	0.3785	19	4.7492E-02
26	10.8329	202	5.2526E-03	6.222	112	5.4286E-03	0.8366	35	2.2808E-02
34	35.868	322	2.6066E-03	16.4327	177	2.8908E-03	1.6588	54	1.3244E-02
42	69.2403	462	1.1268E-03	39.3033	255	1.5682E-03	3.3176	77	8.5538E-03

Fig. 8

The graphs of (a) elapsed time and (b) number of iterations of the proposed methods for Test Problem 6.2 with

Table 8

Maximum errors and computational orders of the FEG method for Test problem 6.2 with and

α=0.1			α=0.5			α=0.9
h^-1	E_∞(τ,h)	C-Order	h^-1	E_∞(τ,h)	C-Order	h^-1	E_∞(τ,h)	C-Order
6	2.9156E-03	-	6	2.8109E-03	-	6	2.6749E-03	-
10	1.0516E-03	1.9963	10	1.0135E-03	1.997	10	9.6529E-04	1.9953
14	5.3744E-04	1.995	14	5.1799E-04	1.9948	14	4.9175E-04	2.0045
18	3.1750E-04	2.0943	18	3.0390E-04	2.1219	18	2.9426E-04	2.0433

Test Problem 6.3

In this problem, the following source term is consideredalong with the exact solution

Again, the initial and boundary conditions can be drawn from the exact solution. In Fig. 9, we sketch the exact solution together with the numerical solutions obtained at , , and , while in Fig. 10 we draw the surface figure of maximum absolute errors of the MFEG method for , , and . The figures illustrate the accuracy of the proposed methods. In Tables 9, 10 and 11, experimental results of the C-N, FEG and MFEG methods with , and = 0.3, 0.5, 0.7 for various mesh sizes are recorded, respectively. Both of the elapsed time and the number of iterations versus different mesh sizes are depicted in Fig. 11. Similar to the previous tests, it can be seen that the FEG and MFEG methods result in simulations with rapid convergence and fewer iterations in comparison to the C-N difference scheme. Among the presented schemes, the MFEG method is clearly the fastest one since it uses the least amount of time consumption for simulating the test problems. The numerical errors and convergence orders of the MFEG method are reported in Table 12. The results are in good agreement with the theoretical analysis.

Fig. 9

Comparison of exact and numerical solutions for Test Problem 6.3 with , , , and

Fig. 10

Surface error plot of the FEG method for Test Problem 6.3 with , , and

Table 9

Comparison between C-N, FEG and MFEG methods for Test Problem 6.3 with , and

h^-1	C-N			FEG			MFEG
	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)
10	0.9338	56	8.6771E-03	0.4457	33	8.6925E-03	0.1402	11	3.3764E-02
18	3.9919	155	2.5894E-03	2.5803	86	2.6599E-03	0.3867	26	1.0715E-02
26	16.5821	295	1.0450E-03	9.1018	162	1.1852E-03	1.1168	49	5.1784E-03
34	32.0802	471	3.6668E-04	25.2729	258	5.5033E-04	2.4359	77	3.0052E-03
42	64.4505	678	3.8897E-04	55.5592	373	2.1681E-04	4.7049	111	1.9285E-03

Table 10

Comparison between C-N, FEG and MFEG methods for Test Problem 6.3 with , and

h^-1	C-N			FEG			MFEG
	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)
10	0.7929	57	1.4381E-02	0.4910	33	1.4393E-02	0.1278	11	5.5926E-02
18	4.2983	159	4.3691E-03	2.5652	88	4.4432E-03	0.3977	27	1.7753E-02
26	16.7838	304	1.8992E-03	9.7074	166	2.0478E-03	1.1253	49	8.5798E-03
34	34.1840	487	8.2457E-04	25.7286	265	1.0484E-03	2.4011	78	4.9996E-03
42	78.4912	704	2.7888E-04	56.3769	384	5.0444E-04	4.4506	113	3.2386E-03

Table 11

Comparison between C-N, FEG and MFEG methods for Test Problem 6.3 with , and

h^-1	C-N			FEG			MFEG
	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)	Time	Ite	E_∞(τ,h)
10	0.7322	56	2.3677E-02	0.4408	33	2.3693E-02	0.1277	11	9.2293E-02
18	4.1288	157	7.2080E-03	2.5156	87	7.2878E-03	0.3377	26	2.9239E-02
26	16.0169	302	3.2214E-03	8.8714	164	3.3752E-03	1.1323	49	1.4102E-02
34	35.1462	487	1.5531E-03	25.0797	264	1.7958E-03	2.3041	77	8.2001E-03
42	77.8330	707	6.4924E-04	55.6289	383	9.5072E-04	4.4194	112	5.3181E-03

Fig. 11

The graphs of (a) elapsed time and (b) number of iterations of the proposed methods for Test Problem 6.3 with

Table 12

Maximum errors and computational orders of the MFEG method for Test problem 6.3 with and

α=0.1			α=0.5			α=0.9
h^-1	E_∞(τ,h)	C-Order	h^-1	E_∞(τ,h)	C-Order	h^-1	E_∞(τ,h)	C-Order
6	3.1728E-03	-	6	3.0484E-03	-	6	2.8848E-03	-
10	1.2360E-03	1.8455	10	1.1921E-03	1.8380	10	1.1312E-03	1.8327
14	6.4392E-04	1.9379	14	6.2047E-04	1.9407	14	5.8594E-04	1.9550
18	3.9126E-04	1.9824	18	3.7721E-04	1.9803	18	3.5494E-04	1.9946

Conclusion

In the current work, we presented the FEG method and the MFEG method, which are derived based on the combination of two C-N difference schemes with grouping strategies on the standard grid to solve the two-dimensional time fractional mobile/immobile equation. The Fourier analysis method is utilized to prove the stability and convergence in norm. A comparison of the FEG and MFEG methods with the C-N difference scheme is given in terms of computational time, iterations’ number and maximum error. The computational orders of the proposed methods are calculated and found to be in good agreement with the theoretical analysis. Numerical experiments showed that the exact solutions are well matched using all the tested methods. Furthermore, experimental results verified that the FEG and MFEG methods outperform the C-N difference scheme in terms of iterations’ number and execution time, showing their efficiency. Overall, the proposed methods have the advantages of being accurate, computationally efficient and applicable to other types of linear and nonlinear multi-dimensional fractional problems. As an outlook for future, the parallel implementation technique is fairly meaningful to enhance the computational efficiency of the developed methods.