A Semi-Discretization Method Based on Finite Difference and Differential Transform Methods to Solve the Time-Fractional Telegraph Equation

: The telegraph equation is a hyperbolic partial differential equation that has many applications in symmetric and asymmetric problems. In this paper, the solution of the time-fractional telegraph equation is obtained using a hybrid method. The numerical simulation is performed based on a combination of the ﬁnite difference and differential transform methods, such that at ﬁrst, the equation is semi-discretized along the spatial ordinate, and then the resulting system of ordinary differential equations is solved using the fractional differential transform method. This hybrid technique is tested for some prominent linear and nonlinear examples. It is very simple and has a very small computation time; also, the obtained results demonstrate that the exact solutions are exactly symmetric with approximate solutions. The results of our scheme are compared with the two-dimensional differential transform method. The numerical results show that the proposed method is more accurate and effective than the two-dimensional fractional differential transform technique. Also, the implementation process of this method is very simple, so its computer programming is very fast.


Introduction
The differential transform method (DTM) is an iterative method based on Taylor's series.DTM has been used to solve various differential equations.It was first applied to solve electrical circuit problems.After that, it was used to solve ordinary differential equations (ODEs), partial differential equations (PDEs), fuzzy PDEs, fractional-order ODEs and PDEs, systems of ODEs, systems of PDEs, differential-algebraic equations, and eigenvalue problems [1][2][3][4][5][6][7].Additionally, fractional DTM, which is based on a generalized Taylor's series, has been applied to solve various differential, differential-algebraic, and integral equations of fractional order [8][9][10][11][12][13][14].In this paper, we intend to apply a combination of the finite difference (FD) and fractional differential transform (FDT) methods (FD-FDTM) to solve the one-dimensional time-fractional telegraph equation (FTE).
This work aims to obtain an approximate solution to the FTE (1) using a hybrid method.In 2008, a hybrid method based on the combination of DTM and FDM was presented to solve a nonlinear heat conduction differential equation [34].Also, in 2012, some nonlinear PDEs were solved with the hybrid method [35].Arsalan (2020) applied a hybrid scheme to solve the one-dimensional integer-order telegraph equations [36,37].
We organize the rest of the paper as follows.In Section 2, we present the FDTM and related theorems.We propose our hybrid method to solve FTE in Section 3 and prove its convergence in Section 4. Also, in Section 5, we give some examples and solve them with the proposed method, we draw a conclusion in Section 6.

Fractional Differential Transform Method
The fractional differential transform method (or generalized differential transform method) is based on the fractional Taylor's formula.The α-order fractional Taylor expansion of function u(t) about point t = t 0 is defined as [38] where d α dt α is the α-order Caputo fractional derivative and ( The α-order FDT of the function u(t) about t = t 0 is denoted by U α (k) and defined [9].Therefore, at t = 0, we have where Also, the m-approximation fractional differential transform of u(t) is defined as Theorem 1 ([39]).Suppose that F α (k), G α (k), and H α (k) are the differential transformations of the functions f (t), g(t), and h(t), respectively.Then we have Theorem 2 ([39]).Suppose that f (t) = t λ g(t), where λ > −1 and g(t) has the generalized power series expansion g Theorem 3 ([39]).If f (t) = D γ t 0 g(t), m − 1 < γ ≤ m, and the function g(t) satisfies the conditions in theorem (2), then ).

FD-FDTM for Solving the FTE
Consider the FTE (1).If the x-derivative at (x, t) is replaced by ) and x is considered as a constant, Equation (1) can be written as the following ordinary differential equation We subdivide the interval [a, b] into N equal subintervals of step-length h = b − a N .
Thus, the mesh points x i = a + ih, i = 0, 1, . . ., N are obtained.Now, we write Equation ( 6) at the mesh point x i , i = 1, . . ., N − 1, along with time level t.If we discard the local truncation error O(h 2 ) and denote u i (t) as the approximate solution of v i (t) = v(x i , t), we have the following system of ODEs: We solve the system (7) using FDTM.For this purpose, we consider the solution of equation, u i (t), as follows: where the unknown coefficients U α i (k) are the FDT of u i (t) and should be obtained.By choosing a suitable value for α, assuming q α i (k) as the α-fractional differential transform of q i (t), and by using Theorems 2 and 3, the fractional differential transform of Equation ( 7) leads to the following relation: Now, suppose that G α 1 (k) and G α 2 (k) are the fractional differential transforms of the functions g 1 (t) and g 2 (t), respectively.Therefore, by applying the FDTM to conditions (2) and (3), we have the initial conditions and the boundary conditions We rewrite relations ( 9)-( 11) as follows: Also, according to [9], the unknown coefficients , will be available as follows: Therefore, all the unknown coefficients U α i (k), i = 0, 1, 2, . . ., N, ∀k ≥ 0 are calculated according to the recursive formula (12) and relations ( 13)- (15).

Convergence of FD-FDTM for FTE
Here, we discuss the convergence of FD-FDTM for solving the FTE (1).First, we present the following lemma [40].
For 0 < δ j < 1, we can write Thus, for n ≥ m ≥ k 0 , we have If we let δ = max{δ k 0 , δ k 0 +1 , . . . ,δ m , δ m+1 , . . . ,δ n } , the following relation is obtained: Theorem 4. Suppose that v(x i , t) is the exact solution of the FTE at point (x i , t), u i (t) is the exact solution of Equation (7), and U m i (t) = ∑ m k=0 ϕ k (t), is the m-approximation of u i (t) as the approximate solution of FTE at point (x i , t).Also, suppose for some k 0 ∈ N 0 and for every n ≥ m ≥ k 0 , ∃0 < δ i < 1, such that ϕ i+1 ≤ δ i+1 ϕ i , where ϕ i = max t |ϕ i (t)|.Then the solution U m i (t) converges to the exact solution, v i (t), as m → ∞.Furthermore, for some a < ξ < b the maximum absolute error of the m-series, U m i (t), as an approximation of the FTE's exact solution satisfies the following relation: where δ = max{δ k 0 , δ k 0 +1 , . . ., δ n }.
Proof.We can write where v(x i , t) and u i (t) are the solutions of Equations ( 6) and (7), respectively.Also, Equation ( 6) has been obtained by replacing the second x-derivative of v(x, t) with the central FD formula in Equation (1).Therefore, for some a < ξ < b we can write From relation (16), for n ≥ m ≥ k 0 , we have and since 0 ≤ δ < 1, then 1 − δ n−m < 1, so we have If n approaches ∞, then s n → u i (t) and we have in the other words, By replacing relations (18) and (19) in relation (17), the theorem is proved.

Numerical Examples
In this section, we give some examples to show the efficiency and convenience of the mentioned method.The examples include linear and non-linear FTEs.We present the results of FD-FDTM for solving the examples and calculate the maximum absolute error (MAE) for different values of N using the following formula: Also, we compare the results of FD-FDTM with two-dimensional FDTM (2D-FDTM).Moreover, we obtain the rate of convergence (ROC) of FD-FDTM with the following formula: ).
We show the results of our method for solving Example 1 in Tables 1 and 2. Table 1 contains the maximum absolute error of the obtained solution using the FD-FDTM for γ = 1, m = 3, and different values of N at t = 0.001.Also, we compared the results of FD-FDTM with twodimensional FDTM.Table 1 shows the our method is more accurate than the two-dimensional DTM.Also, we can see that as N increases, the error decreases and the numerical ROC confirms the theoretical ROC.Table 2 compares the approximate solution of FD-FDTM and 2D-FDTM at t = 0.01, for γ = 0.75, m = 10, and N = 10. Figure 1 shows the comparison between the exact solution for γ = 1 and the results of FD-FDTM for γ = 0.5, 0.7, 0.8, 0.9.Example 2. In this example, we consider a non-homogeneous FTE with the following conditions: The exact solution of (23) for γ = 1 with conditions (24) and ( 25) is v(x, t) = e −2t sinh(x).
To compare the exact and numerical solution of Example 2 obtained using FD-FDTM and 2D-FDTM, we compute the MAE of the obtained solution at t = 0.001, and present them in Table 3.Also, we show the results of the FD-FDTM and 2D-FDTM at t = 0.001, for γ = 0.6, m = 15, and N = 10 in Table 4. Figure 2 shows the comparison between the exact solution for γ = 1 and the numerical solution for γ = 0.5, 0.7, 0.8, 0.9.
The Red line is exact solution (γ=1)  Example 3. In this example, we solve the following nonlinear FTE [23]: with the following conditions The exact solution of Equation ( 26) for γ = 1 with conditions ( 27) and (28) is v(x, t) = e x−2t .

Conclusions
In this work, a hybrid method has been used to solve the linear and non-linear timefractional telegraph equation approximately.The central difference method has been applied to discretize the spatial derivative and the fractional differential transform method has been used to solve the obtained system of fractional ordinary equations.A convergence analysis of the mentioned method has been conducted.The numerical results show that the proposed hybrid method is more accurate and effective than two-dimensional FDTM.Also, the implementation process of this method is very simple, so its computer programming is very fast.

Table 1 .
The MAE for Example 1 at t = 0.001, for γ = 1, m = 3, and different values of N.

Table 3 .
The MAE for Example 2 at t = 0.001, for γ = 1, m = 3, and different values of N.

Table 5 .
The MAE for Example 3 at t = 0.01, for γ = 1, m = 3, and different values of N.