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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 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.
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.
Definition 1
The Caputo fractional derivative D_t^\alpha u\left( {t,x} \right) (t,x) of order α with respect to time is defined as:
\matrix{ {{{{\partial ^\alpha }u(t,x)} \over {\partial {t^\alpha }}} = D_t^\alpha u(t,x)} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {1 \over {\Gamma (n - \alpha )}}\int\limits_0^t {{1 \over {{{(t - p)}^{\alpha - n + 1}}}}{{{\partial ^\alpha }u(p,x)} \over {\partial {p^\alpha }}}dp,(\,n - 1 < \alpha < n)} } \hfill \cr }
and for α = n ∈ N defined as:
D_t^\alpha u(t,x) = {{{\partial ^\alpha }u(t,x)} \over {\partial {t^\alpha }}} = {{{\partial ^n}u(t,x)} \over {\partial {t^n}}}.
Definition 2
First-order approach difference method for the computation of the problem (1) has been presented as:
D_t^\alpha U_n^k \cong g\alpha \tau \sum\limits_{j = 0}^{k - 1} {b_j^{(\alpha )}} (U_n^{k - j} - U_n^{k - j - 1}),
where {g_{\alpha ,\tau }} = {{{\tau ^{2 - \alpha }}} \over {\Gamma \left( {3 - \alpha } \right)}} and b_j^{(\alpha )} = {\left( {j + 1} \right)^{2 - \alpha }} - {j^{2 - \alpha }}.
Next section, we shall give Crank-Nicholson difference scheme for fractional order telegraph differential equation.
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:
\left\{ {\matrix{ {{{u_n^{k + 1} - 2u_n^k + u_n^{k - 1}} \over {{\tau ^2}}} + g\alpha ,\tau \sum\limits_{j = 0}^{k - 1} {b_j^{(\alpha )}\,(u_n^{k - j}} - u_n^{k - j - 1}) + {1 \over 2}(u_n^{k + 1} + u_n^k)} \hfill \cr { - {1 \over {{h^2}}}(\left( {u_{n + 1}^{k + 1} - 2u_n^{k + 1} + u_{n - 1}^{k + 1}} \right) + \left( {u_{n + 1}^k - 2u_n^k + u_{n - 1}^k} \right) = f_n^k,} \hfill \cr {f_n^k = f({t_k},{x_n}),\,1 < \alpha < 2,} \hfill \cr {u_n^0 = {r_1}({x_n}),\,{{u_n^1 - u_n^0} \over \tau } = {r_2}(({x_n}),\,0 \le n \le M,} \hfill \cr {u_0^k = u_M^k = 0,0 \le k \le N.} \hfill \cr } } \right.
Thus, we obtain the following equalities
\matrix{ {{\alpha _{n + 1}} = - {{\left( {B + A{\alpha _n}} \right)}^{ - 1}}A,} \hfill \cr {{\beta _{n + 1}} = {{\left( {B + A{\alpha _n}} \right)}^{ - 1}}\left( {{\varphi _n} - A\beta n} \right),} \hfill \cr }
where 1 ≤ 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.
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.
Remark 2
Using Matlab programming for N = M = 10, α = 1.5,0 ≤ t ≤ 1, 0 ≤ x ≤ π and h = {\pi \over M}, tau = {1 \over N}, we obtain the following spectral radius of a matrix as:
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:
\eqalign{ & \matrix{ {\varepsilon = } & {\max } & {\left| {u(t,x) - u({t_k},{x_n})} \right|} \cr } \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,n = 0,1, \ldots ,M \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,k = 0,1,2, \ldots ,N \cr}
where u_n^k = u\left( {{t_k},{x_n}} \right) is the approximate solution. The error analysis in Table 1 gives our error analysis for difference schemes method.
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.