On Solutions of Fractional order Telegraph Partial Differential Equation by Crank- Nicholson Finite Difference Method

The exact solution is calculated for fractional telegraph partial differential equation depend on initial boundary value problem. Stability estimates are obtained for this equation. Crank-Nicholson difference schemes are constructed for this problem. The stability of difference schemes for this problem is presented. This technique has been applied to deal with fractional telegraph differential equation defined by Caputo fractional derivative for fractional orders α = 1.1, 1.5, 1.9. Numerical results confirm the accuracy and effectiveness of the technique.


Introduction and Preliminaries
Fractional differential equations have many implementations in finance, engineering, physics and seismology [1][2][3]. These type equations are solvable with restpect to variables time and space. Some difference schemes are given for the space-fractional heat equations in [4][5][6][7][18][19][20][21][22]. A new difference scheme for time fractional heat equation based on the Crank-Nicholson method has been presented in [5]. Orsingher and Beghin [14] have presented the Fourier transform of the fundamental solutions to time-fractional telegraph equations of order 2α. In [15], the time-fractional advection dispersion equations have been presented. In [16], Liu has studied fractional difference approximations for time-fractional telegraph equation. Modanli and Akgül [12] have worked the second-order partial differential equations by two accurate methods. Finally, Modanli and Akgul [13] have solved the fractional telegraph differential equations by theta-method. For more details see [23][24][25][26][27].
In this study, the Crank-Nicholson difference schemes method has been applied to fractional derivatives to get numerical results. Now, we examine the following fractional telegraph equations Here, and for α = n ∈ N defined as: Definition 2. First-order approach difference method for the computation of the problem (1) has been presented as: where Next section, we shall give Crank-Nicholson difference scheme for fractional order telegraph differential equation.
Thus, we obtain the following equalities For the stability, implementing the technique of analyzing the eigenvalues of the iteration matrices of the schemes.
Let ρ(A) be the spectral radius of a matrix A, which indicates the maximum of the absolute value of the eigenvalues of the matrix A. We can write the following results. Proof. From the method [15], we should prove that ρ(α n ) < 1, 1 ≤ n ≤ M.
These final results prove the stability estimation of the Theorem 1.
Remark 3. Applying the method in [16,17], we can get the convergence of the method from stability and consistency of the proposed method. Now, we give numerical applications for the fractional telegraph partial differential equation by Crank-Nicholson method.

Numerical implementation
Example . We take into consideration the following fractional telegraph partial differential equation: The exact solution is given as u(t, x) = (t 3 + 1) sin x. We implement difference schemes method to solve the problem. We utilize a procedure of modified Gauss elimination method for difference equation (8). We obtain the maximum norm of the error of the numerical solution by: where u k n = u(t k , x n ) is the approximate solution. The error analysis in Table 1 gives our error analysis for difference schemes method. The difference scheme (8) In method [13]  We have compared Crank-Nicholson finite difference scheme method by the theta method [13] for the variable values N = M = 40, 80, 160. From these comparisons, we see that this method is more effective then the method used in [13].

Conclusion
In this work, stability estimates were presented for fractional telegraph differential equations. Stability inequalities were given for the difference schemes method. We applied the difference schemes-method for investigating fractional telegraph partial differential equations. Approximate solutions were obtained by this method. MATLAB software program was utilized for all results.
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