## Introduction and Preliminaries

Fractional differential equations have many implementations in finance, [1, 2, 3]. These type engineering, physics and seismology 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 2*α.* 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, *r*_{1}(*x*), *r*_{2}(*x*) are smooth function defined with the space [0,*T* ], *f* (*t,x*) is smooth function defined with the space (0,*L*) × (0,*t*) and *u*(*t,x*) is unknown function with the domain [0,*L*] × [0,*T* ]. For the equation (1), the Crank-Nicholson finite difference scheme method is applied. With using this method, obtained numerical results are very good and efficient for given examples.

The Caputo fractional derivative *t,x*) of order *α* with respect to time is defined as:

*α*=

*n ∈ N*defined as:

First-order approach difference method for the computation of the problem (1) has been presented as:

Next section, we shall give Crank-Nicholson difference scheme for fractional order telegraph differential equation.

## 1 Crank-Nicolson Difference Scheme and its Stabilty

Using the formula (3) and definition of Crank-Nicholson first order difference schemes, we can construct the following difference scheme formula for (1) as:

The formula (4) can be rewriten as:

We can write the above system in matrix form as

*A*

_{(N+1)×(N+1)}and

*B*

_{(N+1)×(N+1)}are the matrices of the following form,

Then we have

Next we should determine the matrices *α*_{n+1} and *β n*+1 above. Using the Dirichlet boundary conditition

*u*

_{0}=

*α*

_{1}

*u*

_{1}+

*β*1. From that, we can choose

*α*

_{1}=

*O*

_{(N+1)×(N+1)}and

*β*

_{1}=

*O*

_{N+1)}. Substitute

*u*

_{n}=

*α*

_{n+1}

*u*

_{n+1}+

*β n*+1 and

*u*

_{n−1}=

*α*

_{n}

*u*

_{n}+

*β n*into the equation (7), then

Thus, we get

Thus, we obtain the following equalities

*≤ n ≤ M.*

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.

The difference scheme (5) *is stable.*

From the method [15], we should prove that *ρ*(*α*_{n}) *<* 1, 1 *≤ n ≤ M.*

If *ρ*(*α*_{n}) *<* 1, let us calculate *ρ*(*α*_{n+1}).

We know that *α*_{ni} = *ρ*(*α*_{n}) and 0 *≤ ρ*(*α*_{n}) *<* 1 for 2 *≤ i ≤ N* + 1. Then, we can obtain that *ρ*(*α*_{n+1}) *<* 1. As a result, we obtain the desired result with induction.

*Using Matlab programming for N* = *M* = 10, *α* = 1.5,0 ≤ *t ≤* 1, 0 *≤ x ≤ π and *

ρ (*α*_{1}) = 0, *ρ*(*α*_{2}) = 0.0482, *ρ*(*α*_{3}) = 0.0484, *ρ*(*α*_{4}) = 0.0485, *ρ*(*α*_{5}) = 0.0482, *ρ*(*α*_{6}) = 0.0487, *ρ*(*α*_{7}) = 0.0487, *ρ*(*α*_{8}) = 0.0475, *ρ*(*α*_{9}) = 0.0487 *and ρ*(*α*_{10}) = 0.0486.

These final results prove the stability estimation of the Theorem 1.

*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.

## 2 Numerical implementation

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:

Error Analysis

The difference scheme (8) | In method [13] | ||

α | |||

N = M = 40 | 1.5 | 0.0040 | 0.0242 |

N = M = 80 | 1.5 | 5.4707 × 10^{−4} | 0.0118 |

N = M = 160 | 1.5 | 0.0022 | 0.0058 |

N = 100, M = 10 | 1.1 | 7.0178 × 10^{−4} | |

1.5 | 0.0045 | ||

1.9 | 0.0093 | ||

N = 225,M = 15 | 1.1 | 3.4496 × 10^{−4} | |

1.5 | 0.0040 | ||

1.9 | 0.0083 | ||

N = 400,M = 20 | 1.1 | 2.0762 × 10^{−4} | |

1.5 | 0.0034 | ||

1.9 | 0.0079 |

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].

## 3 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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