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 . Orsingher and Beghin  have presented the Fourier transform of the fundamental solutions to time-fractional telegraph equations of order 2α. In , the time-fractional advection dispersion equations have been presented. In , Liu has studied fractional difference approximations for time-fractional telegraph equation. Modanli and Akgül  have worked the second-order partial differential equations by two accurate methods. Finally, Modanli and Akgul  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, r1(x), r2(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
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
Then we have
Next we should determine the matrices αn+1 and β n+1 above. Using the Dirichlet boundary conditition
Thus, we get
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.
The difference scheme (5) is stable.
From the method , we should prove that ρ(αn) < 1, 1 ≤ n ≤ M.
If ρ(αn) < 1, let us calculate ρ(αn+1).
We know that αni = ρ(αn) and 0 ≤ ρ(αn) < 1 for 2 ≤ i ≤ N + 1. Then, we can obtain that ρ(αn+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 (α1) = 0, ρ(α2) = 0.0482, ρ(α3) = 0.0484, ρ(α4) = 0.0485, ρ(α5) = 0.0482, ρ(α6) = 0.0487, ρ(α7) = 0.0487, ρ(α8) = 0.0475, ρ(α9) = 0.0487 and ρ(α10) = 0.0486.
These final results prove the stability estimation of the Theorem 1.
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) = (t3 + 1)sinx. 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:
|The difference scheme (8)||In method |
|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|
|N = 225,M = 15||1.1||3.4496 × 10−4|
|N = 400,M = 20||1.1||2.0762 × 10−4|
We have compared Crank-Nicholson finite difference scheme method by the theta method  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 .
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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