Generalized Differential Transform Method to Space‐Time Fractional Telegraph Equation

Mridula Garg,Shyam L Kalla,Pratibha Manohar

doi:10.1155/2011/548982

Abstract

We use generalized differential transform method (GDTM) to derive the solution of space‐time fractional telegraph equation in closed form. The space and time fractional derivatives are considered in Caputo sense and the solution is obtained in terms of Mittag‐Leffler functions.

Highlights

Differential equations of fractional order have been successfully employed for modeling the so called anomalous phenomena during last two decades
There has been an intensive development of the theory of fractional differential equations 1–4
The differential transform method was proposed by Zhou 17 to solve linear and nonlinear initial value problems in electric circuit analysis. This method constructs an analytical solution in the form of a polynomial

Summary

Introduction

Differential equations of fractional order have been successfully employed for modeling the so called anomalous phenomena during last two decades. The differential transform method was proposed by Zhou to solve linear and nonlinear initial value problems in electric circuit analysis This method constructs an analytical solution in the form of a polynomial. The classical telegraph equation is a partial differential equation with constant coefficients given by 21 utt − c2uxx aut bu 0, 1.1 where a, b and c are constants This equation is used in modeling reaction diffusion and signal analysis for propagation of electrical signals in a cable of transmission line 21, 22. Orsingher and Beghin presented that the transition function of a symmetric process with discontinuous trajectories satisfies the space-fractional telegraph equation Several techniques such as transform method, Adomian decomposition method, juxtaposition of transforms, generalized differential transform method, variational iteration method, and homotopy perturbation method have been used to solve space or time fractional telegraph equation. We make an attempt to solve homogeneous and nonhomogeneous space-time fractional telegraph equation by means of generalized differential transform method

Preliminaries

Γ αk 1 Γ βh 1