Abstract
In the present article, fractional-order telegraph equations are solved by using the Laplace-Adomian decomposition method. The Caputo operator is used to define the fractional derivative. Series form solutions are obtained for fractional-order telegraph equations by using the proposed method. Some numerical examples are presented to understand the procedure of the Laplace-Adomian decomposition method. As the Laplace-Adomian decomposition procedure has shown the least volume of calculations and high rate of convergence compared to other analytical techniques, the Laplace-Adomian decomposition method is considered to be one of the best analytical techniques for solving fractional-order, non-linear partial differential equations—particularly the fractional-order telegraph equation.
Highlights
Fractional calculus has significant roles in different areas of applied mathematics, due to its various applications in the modeling of different physical phenomena in science and engineering.The most general concept of derivatives, D α ( f ( x )), where α is a non-integer, has improved the early development of ordinary derivatives
The results revealed the highest agreement with the exact solutions for the problems
It is concluded that Laplace-Adomian decomposition method (LADM) is the best tool for the solution of fractional partial differential equations (FPDEs), as compared to Adomian decomposition method (ADM) and DTM in the literature
Summary
Fractional calculus has significant roles in different areas of applied mathematics, due to its various applications in the modeling of different physical phenomena in science and engineering. Different numerical and analytical techniques have been used to solve fractional-order telegraph equations, such as the Homotopy perturbation method (HPM) and Laplace transform (LT) [25]. The q-Homotopy analysis transform method (q-HATM) is used for the numerical solution fractional-order telegraph equation. The numerical results provided by q-HAM shows convergence toward the exact solution of the problem [26], and the modified adomian decomposition method (MADM) [27] and reduced differential transform method (RDTM) are used to solve second-order hyperbolic telegraph equations and the fractional-order hyperbolic telegraph equation, respectively. Generalized finite difference/spectral Galerkin methods have been discussed for the numerical solution of fractional telegraph equations.
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