The Analytical Solution of Telegraph Equation of Space-Fractional Order Derivative by the Aboodh Transform Method

Mohamed Elarbi Benattia ,Kacem Belghaba

doi:10.28924/2291-8639-18-2020-243

Abstract

In this article, an analytical solution based on the series expansion method is proposed to solve the telegraph equation of space - fractional order (TESFO), namely the Aboodh transformation method (ATM) subjected to the appropriate initial condition. Using ATM, it is possible to find exact solution or a closed approximate solution of a differential equation. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.

Highlights

In the last few decades, fractional calculus found many applications in various fields of physical sciences such as viscoelasticity, diffusion, control, relaxation processes and so on [1]
An analytical solution based on the series expansion method is proposed to solve the telegraph equation of space - fractional order (TESFO), namely the Aboodh transformation method (ATM) subjected to the appropriate initial condition
Sometimes they can be better modeled by hyperbolic equations such as the telegraph equation, which have parabolic asymptotic

Summary

Introduction

In the last few decades, fractional calculus found many applications in various fields of physical sciences such as viscoelasticity, diffusion, control, relaxation processes and so on [1]. Aboodh Transform; fractional differential equation; Caputo fractional derivative; telegraph equation. The main objective of this paper is to introduce a new analytical and approximate solution of spatial fractional telegraphic equations using the Aboodh transformation method(ATM), where in [5] authors proposed a Sumudu transformation method (STM) which is used to solve this equation. An Aboodh transform is defined for functions of exponential order. The Aboodh transform denoted by the operator A(:) is defined by the integral equation:. The Aboodh transform A [Dxαf (x)] of the fractional derivative using the Caputo idea of the function is given by:. The Aboodh transform is linear, i.e., if α and β are any constants and f (t) and g(t) are functions defined over the set F above, .

Procedure Solution Using ATM for Solving Linear TESFO

Conclusion