Abstract
The exact solution of fractional telegraph partial differential equation of nonlocal boundary value problem is obtained. The theorem of stability estimates is presented for this equation. Difference schemes are constructed for both the implicit finite difference scheme and the Dufort–Frankel finite difference scheme (DFFDS). The stability of these difference schemes for this problem are proved. With the help of these methods, numerical solutions of the fractional telegraph differential equation defined by Caputo fractional derivative for fractional orders alpha=0.1, 0.5, 0.9 are calculated. Numerical results are compared with the exact solution and the accuracy and effectiveness of the proposed methods are investigated.
Highlights
Fractional differential equations lead to various significant applications in engineering, finance, physics, biology and seismology [1,2,3,4]
Modanlı Advances in Difference Equations (2018) 2018:333 worked on an expanded mixed finite element method for the two-sided time-dependent fractional diffusion problem with two-sided Riemann–Liouville fractional derivatives
The stability estimates are shown for the abstract form of this differential equation
Summary
Fractional differential equations lead to various significant applications in engineering, finance, physics, biology and seismology [1,2,3,4]. Modanlı Advances in Difference Equations (2018) 2018:333 worked on an expanded mixed finite element method for the two-sided time-dependent fractional diffusion problem with two-sided Riemann–Liouville fractional derivatives. We shall investigate the following fractional telegraph partial differential equations for nonlocal boundary value problem:. A function u(t) is called a solution of the problem (6) if the following conditions are satisfied: (i) u(t) is a twice continuously differentiable on the interval [0, T]. We shall obtain the formula for the solution of the problem (6) For this purpose, we can rewrite the problem (6) in the following the first-order differential equations systems form:. Using the following formula for the fractional derivative of order 0 < α < 1: Dαt u(t) =. Follows from the previous estimate and the triangle inequality, which completes the proof of the theorem
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