A new analytical modelling for fractional telegraph equation via Elzaki transform

The main aim of this paper is to propose a new and simple algorithm for space-fractional telegraph equation, namely new fractional homotopy analysis transform method (FHATM). The fractional homotopy analysis transform method is an innovative adjustment in Elzaki transform algorithm (ETA) and makes the calculation much simpler. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.

The diffrernt type solutions of the fractional telegraph equations have been discussed recently by several authors. For example Momani [5] by using decomposition method, Yildirim [6] by homotopy perturbation method, chen [7] et al. by the method of separable variables, Huang [8] by Cauchy problem, Biazar and Eslami [9] by using diffrerntial transform method, Sunil [10] by using Laplace transform. Our concern in this paper is to consider the space-fractional telegraph equations as where a , b and n are given constants, ) , ( t x f is given function. In this paper, the homotopy analysis transform method (HATM) basically illustrates how the Elzaki transform can be used to approximate the solutions of the linear and nonlinear partial differential equation by manipulating the homotopy analysis method. The proposed method is coupling of the homotopy analysis method and Elzaki transform. Homotopy analysis method (HAM) was first proposed and applied by Liao [11,12,13,14] based on homotopy, a fundamental concept in topology and differential geometry.
The main purpose of this article is introduce a new analytical and approximate solution of space-fractional telegraph equation by means of fractional homotopy analysis transform method, which is coupling of homotopy analysis method and Elzaki transform method.

Fractional calculus
We recall some definitions of fractional derivatives and fractional integrals. Let

Basic definition of fractional calculus
This section, we give some basic definitions and properties:

Definition 2.1 We Consider function in the set
A , defined by [15]: For a given function in the set A , the constant M must be finite number, ]. ,

Definition 2.2 The Elzaki transform of
is defined as [16]: D e c e m b e r 3 1 , 2 0 1 5

Definition 2.3 The Elzaki transform
)] ( [ t f E of the Riemann¨CLiouville fractional is defined as [16]: of the caputo fractional is defined as [16]:

Basic idea of newly fractional homotopy analysis transform method (FHATM)
We consider the following fractional partial differential equation as: is the general nonlinear operator for 3 R r  and ) , ( t r g are continuous functions. For simplicity, we ignore all initial and boundary conditions, which can be treated in similar way. Now the methodology consists of applying Elzaki transform first on both sides of equation (7), we get )]. , According to the differentiation property of the Elzaki transform, we have We define the nonlinear operator [32] )), ; , be an embedding parameter and ) ; , ( q t r  is the real function of r , t and q .
We define the zero order deformation equation is an unknown function.Obviously, when Operating the inverse Elzaki transform on both sides of the equation (14), we get Operating the Elzaki transform on both sides in (16) and after using the differentiation property of Elzaki transform for fractional derivative, we get We now define a nonlinear operator as )]. ; , Using above definition, with assumption Now the solution of equation (18)  .
We choose the initial condition     Operating the Elzaki transform on both sides in (22) and after using the differentiation property of Elzaki transform for fractional derivative, we get We now define a nonlinear operator as Using above definition, with assumption Now the solution of Equation (24)  .
2) ( Operating the Elzaki transform on both sides in (29) and after using the differentiation property of Elzaki transform for fractional derivative, we get Using above definition, with assumption