On the Solution of Generalized Space Time Fractional Telegraph Equation

Differential calculus has its great importance in the field of Science and Technology [1].The partial differential equations are one of the most powerful mathematical tools to describe, define, and explain the various phenomena of physics and chemistry and the process to solve them is a new interesting field to study. Telegraph equations are hyperbolic partial differential equations that gained remarkable interest due to their numerous applications in high-frequency transmission lines, propagation of electrical signals, optimization of guided communication system, and the propagation of quantum particles and many other physical and chemical phenomena. They are also applicable to designing high-voltage transmission lines. Recently generalized space time fractional telegraph equations were used to describe number of chemical and physical phenomena with remarkably interesting results. The solutions of generalized space time fractional telegraph equations were obtained by the various numerical methods. Numerous investigations have been done to solve fractional telegraph equations.Themethods, namely, generalized differential transform, Fourier-Laplace transform, separations of variables, perturbation theory, and basic variational iteration, have been used to investigate the solution of homogeneous and nonhomogenous time fractional telegraph equation and the solutions are given in the form ofMittag-Leffler functions [2–6].Nowadays variational iterationmethod is being used to solve many fractional differential equations and the presence of Sumudu transform is making the process rich. In this paper we modified the variational iteration method (VIM) by making use of the Sumudu transform and apply it to generalized space time fractional telegraph equations.This Sumudu variational iteration method enables us to overcome the difficulties that arise in finding the general Lagrange multipliers.


Introduction
Differential calculus has its great importance in the field of Science and Technology [1].The partial differential equations are one of the most powerful mathematical tools to describe, define, and explain the various phenomena of physics and chemistry and the process to solve them is a new interesting field to study.Telegraph equations are hyperbolic partial differential equations that gained remarkable interest due to their numerous applications in high-frequency transmission lines, propagation of electrical signals, optimization of guided communication system, and the propagation of quantum particles and many other physical and chemical phenomena.They are also applicable to designing high-voltage transmission lines.
Recently generalized space time fractional telegraph equations were used to describe number of chemical and physical phenomena with remarkably interesting results.The solutions of generalized space time fractional telegraph equations were obtained by the various numerical methods.Numerous investigations have been done to solve fractional telegraph equations.The methods, namely, generalized differential transform, Fourier-Laplace transform, separations of variables, perturbation theory, and basic variational iteration, have been used to investigate the solution of homogeneous and nonhomogenous time fractional telegraph equation and the solutions are given in the form of Mittag-Leffler functions [2][3][4][5][6].Nowadays variational iteration method is being used to solve many fractional differential equations and the presence of Sumudu transform is making the process rich.
In this paper we modified the variational iteration method (VIM) by making use of the Sumudu transform and apply it to generalized space time fractional telegraph equations.This Sumudu variational iteration method enables us to overcome the difficulties that arise in finding the general Lagrange multipliers.

Variational Iteration Method
The analytical technique used by many researchers to study linear and nonlinear partial differential equations is commonly known as the variational iteration method.This technique of variational iteration was developed by Chinese mathematician He [15].To illustrate the basic idea of this method, we consider the general nonlinear system of the following form: (, ) +  (, ) =  (, ) , where  is the unknown function,  and  are linear and nonlinear operators, respectively, and  is the source term.
The basic character of the method is to construct the following correction functional for (12): where  is called the general Lagrange multiplier [16] and   is the approximate solution of th order.If we have seen the whole process of the Lagrange multipliers in the case of an algebraic equation then solution of the algebraic equation () = 0 can be obtained by an iteration formula: The optimality condition for the extreme  +1 /  = 0 leads to where  is the classical variational operator.
Applying Sumudu transform with respect to  in both sides of (16), Using (8) with  = 0 we get By the formula of ( 14), we get the iteration formula for the above as follows: Putting  = −  , the Lagrange multiplier in (19) becomes Now taking inverse Sumudu transform on both sides of (20), This is the iteration formula for solving generalized space time fractional telegraph equation with the following initial iteration:

Main Results
Theorem 6.Consider the space time fractional homogenous telegraph equation (27)