On Unconditionally Stable New Modiﬁed Fractional Group Iterative Scheme for the Solution of 2D Time-Fractional Telegraph Model

: In this study, a new modiﬁed group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h -spaced standard and rotated Crank–Nicolson FD approximations. The ﬁndings of new four-point modiﬁed explicit group relaxation method demonstrates the rapid rate of convergence of proposed method as compared to the existing schemes. Numerical tests are performed to test the capability of the group iterative scheme in comparison with the point iterative scheme counter-parts. The stability of the derived modiﬁed group method is proven by the matrix norm algorithm. The obtained results are tabulated and concluded that exact solutions are exactly symmetric with approximate solutions. FEG method. The efﬁcient percentages of


Introduction
Partial differential equations have a wide range of applications in applied sciences, including wave propagation, electric signal propagation, and atomic physics [1][2][3][4][5]. In recent years, several numerical techniques for multidimensional hyperbolic partial differential equations have been developed [6][7][8][9][10][11]. For example, Gao and Chi [12] solved one-spacedimensional linear hyperbolic model based on unconditionally stable difference schemes. Youssef [13] studied the class of fractional functional integro-differential equations of the Caputo-Katugampola type. Many techniques have been used to solve fractional partial differential equations using the Caputo and the Caputo-Fabrizio type operators [14][15][16]. Lai and Liu [17] solved the second order fractional partial differential equation using the Galerkin finite element method and Riesz fractional derivative. Akram et al. [18,19] proposed unconditionally stable methods for the fractional hyperbolic models via B-spline approaches. Several years before this, the unconditionally stable alternating dimension implicit schemes for 2D and three-dimensional hyperbolic equations were derived by Mohanty and Jain [20,21]. Later, Mohanty [22] also proposed new difference approach for the solution of telegraphic equation. Meanwhile, Dehghan and Mehebbi [23,24] suggested numerical approaches for 2D linear hyperbolic equations by applying collocation finite difference (FD) approximations. The other derived numerical schemes for hyperbolic partial differential equations can be seen in [25,26].

Modified Fractional Explicit Group (MFEG) Iterative Scheme
The h-spaced point and group iterative schemes for 2D fractional telegraph equation have been derived by Ali and Ali in [6]. The following standard 2h-spaced iterative scheme can be obtain by utilizing standard 2h-spaced Crank-Nicolson scheme for space-derivatives and Caputo fractional derivatives from Equations (4) and (5) for time-derivatives in Equation (2), we have the following expression, with initial and boundary conditions, The suggested domain is discretized as, υ t (x i , y j , 0) = . From the initial condition, we have υ t (x i , y j , 0 for all i = 2, 4 · · · M x − 2, j = 2, 4 · · · M y − 2 and k = 0, 1, 2, 3 · · · N. When Equation (6) is applied to a group of four points, the result is a 4 × 4 system of equations, as shown below.

Convergence
Let Υ(x i , y j , t k+1/2 ) be the analytical solution of Equation (6) and suppose we represent the truncation error by R k+1/2 at the point (x i , y j , t k+1/2 ) such that Therefore, Equation (11) can be written as, for k = 0, for k > 0, Theorem 2. The scheme defined in Equation (7) is convergent, hence the following estimate is correct.
Proof. We use mathematical induction to prove the theorem by setting (13) for the case when k = 0, where z(α) = c Γ(2−α) . We will prove this result using mathematical induction. Now assume that e s ≤ z(α) R s+1/2 is true for all s = 1, 2, · · · k and we prove it for k + 1.
Assume Equation (14) for the case when k > 0, By collecting the coefficients of p 1 and q and (2) of Lemma (1), we have ω 0 > ω 1 > ω 2 > ω 3 > · · · ω k−1 > ω k . When the number of time levels k is increased, the terms ω k , ω k−1 , ω * k , ω * k−1 approaches to zero. As a results, Since coefficients of p 1 and q in the denominator are larger than the coefficients of p 1 and q in the numerator, therefore Therefore, Multiply and divide the denominator by k 2α , we have the following inequality, Since k → ∞, therefore term (1−α) k α will be zero, hence This completes the proof.

Numerical Problems and Results
Two numerical problems are performed to test the viability of the proposed schemes in solving 2D hyperbolic telegraph differential Equation (2). The numerical tests are run on a PC with a Core 2 Duo 2.8 GHz processor and 2GB of RAM Windows XP SP3 with Cygwin C in Mathematica 11 software. We assume that the step sizes in both x and y directions are the same, i.e., h = ∆x = ∆y in both numerical experiments. Throughout our numerical calculations, we employed the Gauss Seidel method with a relaxation factor of ωe equal to 1. The lin f ty norm was utilized for the convergence criterion, with a tolerance factor of ε = 10 Example 1. Consider the 2D time-fractional telegraph model together where the forcing term is defined by [6],

Example 2.
The 2D telegraph equation of fractional order is given by the following expression [6],
In Tables 1 and 2, it is can be seen that execution time and iteration count decrease as value of α moves towards 2 by providing more accurate results.   The average absolute error and maximum absolute error are calculated by taking the absolute average and maximum absolute value of the column vector of errors, respectively. The maximum absolute error of the exact solution Υ i,j,k and approximate solution υ i,j,k is defined as follows: The temporal convergence order of the proposed method is defined as, γ 1 − order ≈ log 2 e(2τ, h) e(τ, h) , and the spatial convergence order of the proposed method is defined as, Tables 3 and 4 represent the values of maximum absolute error and temporal convergence order of MFEG iterative scheme at various values of α of Example 1 and Example 2. For fixed value of h = π 200 and different values of τ, it is observed that MFEG iterative scheme generates (3 − 2α) temporal convergence order as shown in Tables 3 and 4.  Meanwhile, the Tables 5 and 6 show the values of maximum absolute error and spatial convergence order of MFEG iterative scheme at various values of α of Example 1 and Example 2. For fixed value of τ = 0.001 and different values of h, it shows that MFEG iterative scheme gets second-order spatial accuracy as exposed in Tables 5 and 6.  The ranges of percentages of MFEG method against FSP and FEG methods in terms of execution time and number of iterations of Example 1 at α = 0.55 and Example 2 at α = 0.90 are summarized in Tables 7 and 8, respectively.   Tables 7 and 8 refer to the study of the percentages of MFEG method in term of the execution timings and number of iterations against FSP and FEG methods for solving the 2D fractional telegraph equation. Table 7 Tables 7 and 8.

Conclusions
In this study, the unconditionally stable modified group relaxation method is constructed in the numerical solution of 2D hyperbolic telegraph equation. The MFEG method is derived from the standard Crank-Nicolson FD approximation with 2h grid spacing. It is observed that the suggested algorithm has more efficiency as compared to the other existing methods such as fractional standard point and fractional explicit group methods presented in [6]) in terms of number of iterations and elapsed CPU-timings. Amongst the schemes tested, the MFEG method is proven to require the least computational cost in terms of execution of timings. Moreover, it shown that the derived numerical temporal and spatial convergence order support our theoretical derivations. For a better convergence rate, in future, we will apply splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods for the numerical solution of various types of 2D time-fractional problems.

Funding:
The second author will financially support the paper.
Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: All data generated or analyzed during this study are included in this article.

Acknowledgments:
The author T. Abdeljawad would like to thank Prince Sultan University for the support through TAS research lab.

Conflicts of Interest:
The authors declare that they have no conflict of interest to report regarding the present study.