^{1}

^{1, 2}

^{3}

^{1}

^{2}

^{3}

In this paper, the exact solutions of space-time fractional telegraph equations are given in terms of Mittage-Leffler functions via a combination of Laplace transform and variational iteration method. New techniques are used to overcome the difficulties arising in identifying the general Lagrange multiplier. As a special case, the obtained solutions reduce to the solutions of standard telegraph equations of the integer orders.

Fractional differential equations are widely used in many branches of sciences. Many phenomena in engineering, physics, chemistry and other sciences can be described successfully using fractional calculus. Nonlinear oscillation of earthquake, acoustics, electromagnetism, electrochemistry, diffusion processes and signal processing can be modeled by fractional equations [

The fractional telegraph equations have been investigated by many authors in recent years. Garg et al. [

Recently, a method that combined the Laplace transform and variational iteration method (LVIM) has been introduced by many authors in solving various types of problems. Abassy et al. [

In this paper, the authors extend Laplace variational iteration method (LVIM) and apply it to space-time one-dimensional fractional telegraph equations in a half-space domain (signaling problem). This approach enables us to overcome the difficulties that arise in finding the general Lagrange multiplier.

In Section

The Caputo fractional derivative of order

From Definition

The Laplace transform of fractional order derivative, is defined by [

The Mittag-Leffler function with two parameters is defined by [

It follows Definition

The generalized Mittag-Leffler function is defined by [

He [

Consider the following general multiterms fractional telegraph equation:

The new approach of the Laplace variational iteration technique is based on the following steps.

Removing the fractional derivative of order

Differentiating the results obtained in Step

By applying Laplace transform with respect to

Consider the following space-fractional homogenous telegraph equation:

Applying the Laplace transform with respect to

The surface plot of

Consider the following space-time fractional nonhomogenous telegraph equation:

Taking Laplace transform with respect to

The solution surface of this example is graphically presented in Figure

The surface plot of

Consider the following space-time fractional nonhomogenous telegraph equation:

Taking Laplace transform with respect to

The solution surface of this example is graphically presented in Figure

The surface plot of

In this paper, a combined form of Laplace transform and variational iteration method is presented to handle space-time fractional telegraph equations in a half-space domain. The space and time derivatives are considered in the Caputo sense. Certain techniques are used to overcome the complexity of identifying the general Lagrange multiplier. The solutions are obtained in series form that rapidly converges in a closed exact formula with simply computable terms. The calculations are simple and straightforward. The method was tested on three examples on different situations. The technique is powerful, reliable, and efficient. This technique can be extended to solve various linear and nonlinear fractional problems in applied science.