An efficient new perturbative Laplace method for space-time fractional telegraph equations
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In this paper, we propose a new technique for solving space-time fractional telegraph equations. This method is based on perturbation theory and the Laplace transformation. Fractional Taylor series and fractional initial conditions have been introduced. However, all the previous works avoid the term of fractional initial conditions in the space-time telegraph equations. The results of introducing fractional order initial conditions and the Laplace transform for the studied cases show the high accuracy, simplicity and efficiency of the approach.
KeywordsFractional Order Fractional Derivative Fractional Differential Equation Fractional Brownian Motion Homotopy Analysis Method
Telegraph equations are hyperbolic partial differential equations that are applicable in several fields such as wave propagation , signal analysis , random walk theory , etc. In recent years, there has been a great deal of interest in fractional differential equations [4, 5]. Time-fractional telegraph equations have been studied by Orsingher and Zhao  and Orsingher and Beghin . Telegraph equations apply to high-frequency transmission lines such as telegraph wires and radio frequency conductors. They are also applicable to designing high-voltage transmission lines.
where u can be considered as a function depending on distance (x) and time (t), a and b are constants depending on a given problem and f, φ, ϕ, ψ are known continuous functions. The second equation has been solved by Das et al.  using the homotopy analysis method.
The aim of this paper is to introduce a new method for fractional space-time telegraph equations. This new technique is a combined form of the perturbation method [9, 10, 11, 12, 13, 14] with the Laplace transform. This method is called the perturbation Laplace method (PLM). Moreover, we have introduced fractional order initial conditions for space-time telegraph equations. Point to be noted regarding fractional differential equations is that one should use fractional Taylor series. To make the calculation easy and simple, for the first time, we have used the Laplace transform to solve the systems of equations formed after applying homotopy perturbation instead of applying an inverse operator. Through the Laplace transform of fractional order term, it is easy to judge that one must use fractional order initial conditions. It is easy to judge, by applying the Laplace transformation, that it is essential to use a fractional order initial condition to analyze any physical phenomenon which has been expressed in terms of fractional differential equations. To the best of authors’ knowledge, in the literature on space-fractional telegraph equations , there is no closed form solution for different values of α except for the standard case, i.e., for . The elegance of this article can be attributed to its endeavor of finding the solution in a simple way by considering only the PLM. Two examples which show that only a few iterations are needed to obtain accurate approximate solutions are solved.
2 Fractional calculus theory
We give some basic definitions and properties of the fractional calculus theory proposed by Jumarie  which are used further in this paper.
Proposition 1 (On the decomposition of fractional derivatives)
3 Perturbative Laplace method
Substituting successive iterations in Eq. (7) will give the required result.
4 Space-time fractional telegraph equations
where denotes the Mittag-Leffler function.
In this paper, we have introduced a combination of perturbation and Laplace methods for space-time fractional problem which we called the PLM. We described the method and used it in some fractional telegraph equations in order to show its applicability and validity. We achieved accurate approximations by using only a few numbers of iterations, which reveals efficiency of the new method. The solution very rapidly converges by utilizing the perturbation Laplace method. The PLM is also valid for other fractional differential equations, and this paper can be used as a standard paradigm for other applications.
The second author is supported by Grant P201/11/0768 of the Czech Grant Agency (Prague). The fourth author is supported by Grant FEKT-S-11-2-921 of the Faculty of Electrical Engineering and Communication, Brno University of Technology.
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.