International Journal of Mathematics and Mathematical Sciences

Volume 2016, Article ID 1414595, 7 pages

http://dx.doi.org/10.1155/2016/1414595

## Double Laplace Transform Method for Solving Space and Time Fractional Telegraph Equations

^{1}Department of Mathematics, Datta Meghe Institute of Engineering Technology and Research, Wardha, India^{2}Department of Mathematics, Government Science College, Gadchiroli, India

Received 17 June 2016; Revised 23 September 2016; Accepted 5 October 2016

Academic Editor: Irena Lasiecka

Copyright © 2016 Ranjit R. Dhunde and G. L. Waghmare. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Double Laplace transform method is applied to find exact solutions of linear/nonlinear space-time fractional telegraph equations in terms of Mittag-Leffler functions subject to initial and boundary conditions. Furthermore, we give illustrative examples to demonstrate the efficiency of the method.

#### 1. Introduction

The telegraph equation developed by Oliver Heaviside in 1880 is widely used in Science and Engineering. Its applications arise in signal analysis for transmission and propagation of electrical signals and also modelling reaction diffusion.

In recent years, great interest has been developed in fractional differential equation because of its frequent appearance in fluid mechanics, mathematical biology, electrochemistry, and physics. A space-time fractional telegraph equation is obtained from the classical telegraph equation by replacing the time and space derivative terms by fractional derivatives.

There are various methods developed to solve fractional telegraph equations. Orsingher and Xuelei [1] and Orsingher and Beghin [2] considered the space and time fractional telegraph equations and obtained the Fourier transform of their fundamental solution. Momani [3] and Garg and Sharma [4] used Adomian decomposition method developed by Adomian in [5] for solving homogeneous and nonhomogeneous space-time fractional telegraph equation. Chen et al. [6] implemented separation of variables method for deriving the analytical solutions of the nonhomogeneous time fractional telegraph equation under Dirichlet, Neumann, and Robin boundary conditions. Huang [7] considered the combine Fourier-Laplace transform to solve time fractional telegraph equation.

Variational iteration method is proposed by He [8] and used by Sevimlican [9] for solving space and time fractional telegraph equations. Yildirim in [10] used a homotopy perturbation method and Das et al. in [11] used a homotopy analysis method to obtain approximate analytical solution of fractional telegraph equation. Garg et al. [12] and Galue [13] used generalised differential transform method to derive the solution of space-time fractional telegraph equations in terms of Mittag-Leffler functions. Srivastava et al. [14] applied reduced differential transform method to solve Caputo time fractional order hyperbolic telegraph equation.

In recent years, significant attention has been given by many authors towards the study of fractional telegraph equations by using single Laplace transform combined with variational iteration method, homotopy analysis method, and homotopy perturbation method. Khan et al. [15], Kumar et al. [16], and Prakash [17] applied a combination of single Laplace transform and homotopy perturbation method to obtain analytic and approximate solutions of the space-time fractional telegraph equations. Alawad et al. [18] used a combination of single Laplace transform and variational iteration method for finding exact solutions of space-time fractional telegraph equations in terms of Mittag-Leffler functions. Kumar in [19] coupled single Laplace transform and homotopy analysis method for the solution of space-fractional telegraph equation.

To our knowledge, solving fractional partial differential equations using the double Laplace transform is still seen in very little proportionate or no work is available in the literature. So, the main objective of this paper is to find the exact solutions of homogeneous and nonhomogeneous space-time fractional telegraph equations in terms of Mittag-Leffler functions subject to initial and boundary conditions, by means of double Laplace transform.

#### 2. A Brief Introduction of Double Laplace Transform and Caputo Fractional Derivative

Let be a function of two variables and defined in the positive quadrant of the -plane. The double Laplace transform of the function as given by Sneddon [20] is defined by whenever that integral exists. Here and are complex numbers.

From this definition we deduce The inverse double Laplace transform is defined as in [21, 22] by the complex double integral formula:where must be an analytic function for all and in the region defined by the inequalities Re and Re , where and are real constants to be chosen suitably.

The double Laplace transform formulas for the partial derivatives of an arbitrary integer order as in [23] are

*Definition 1. *The Caputo fractional derivative of function is defined in [18] asThe double Laplace transform formulas for the partial fractional Caputo derivatives as in [23] are

*Definition 2. *The Mittag-Leffler function is defined by The single Laplace transform of the function takes the form

#### 3. Double Laplace Transform Method

Consider the following general multiterms fractional telegraph equation as in [18]:with initial conditions,and boundary conditions,Here are constants and is given function.

Applying the double Laplace transform on both sides of (9), we getwhere .

Further, applying single Laplace transform to initial (10) and boundary conditions (11), we getBy substituting (13) in (12) and simplifying, we obtainApplying inverse double Laplace transform to (14), we obtain the solution of (9) in the formHere we assume that the inverse double Laplace transform of each term in the right side of (15) exists.

#### 4. Illustrative Examples

In this section, we demonstrate the applicability of the previous method by giving examples.

*Example 1. *By substituting , , , , , and in (9), subject to the initial and boundary conditions,a homogeneous space-fractional telegraph equation.

Taking single Laplace transform to initial (17) and boundary conditions (18), we getSubstituting above in (15), we get solution of (16):Simplifying, we obtainwhich agrees with the solution already obtained in [18].

If we take then we get the exact solution of standard telegraph equation:

*Example 2. *By substituting , , ,, , and in (9), subject to the initial and boundary conditions,a space-fractional nonhomogeneous telegraph equation.

Taking single Laplace transform to initial (24) and boundary conditions (25), we getTaking double Laplace transform of , we haveSubstituting above in (15), we get solution of (23):Rearranging, we haveSimplifying, we obtainwhich agrees with the solution already obtained in [18] for .

For , then .

*Example 3. *By substituting , , , , , and in (9), subject to the initial and boundary conditions,a time fractional telegraph equation in [24].

Taking single Laplace transform to initial (32) and boundary conditions (33), we getTaking double Laplace transform of , we haveSubstituting above in (15), we get solution of (31):Simplifying, we obtain

*Example 4. *By substituting , , , , and in (9), subject to the initial and boundary conditions,a homogeneous time fractional telegraph equation in [25].

Taking single Laplace transform to initial (39) and boundary conditions (40), we getSubstituting above in (15), we get solution of (38):Simplifying, we obtainwhich agrees with the solution already obtained in [25].

*Example 5. *By substituting , , and in (9), subject to the initial and boundary conditions,a nonhomogeneous time fractional telegraph equation in [25].

Taking single Laplace transform to initial (45) and boundary conditions (46), we getTaking double Laplace transform of , we haveSubstituting above in (15), we get solution of (44):Simplifying, we obtainwhich agrees with the solution already obtained in [25].

*Example 6. *Consider the following space-fractional-order nonlinear telegraph equation:under the initial conditions,and boundary conditions,Applying the double Laplace transform on both sides of (51), we getFurther, applying single Laplace transform to initial (52) and boundary conditions (53), we getBy substituting (55) in (54) and simplifying, we obtainApplying inverse double Laplace transform of (57), we getNow we apply the Iterative method as in [26];Substituting (59) in (58), we getThe nonlinear term is decomposed asSubstituting (61) in (60), we getThen we define the recurrence relations asand so on.

Therefore, we obtain the solution of (51) as follows:This is the required exact solution of (51).

#### 5. Conclusion

We have applied double Laplace transform to obtain exact solutions of linear/nonlinear space-time fractional telegraph equations. All of the examples considered show that double Laplace transform method is capable of reducing the volume of computational work as compared to other methods. It may be concluded that DLT technique solves the problems without using Adomian polynomials, Lagrange multiplier value, He’s polynomials, and small parameters.

#### Competing Interests

The authors declare no competing interests regarding the publication of this paper.

#### References

- E. Orsingher and Z. Xuelei, “The space-fractional telegraph equation and the related fractional telegraph process,”
*Chinese Annals Mathematics B*, vol. 24, no. 1, pp. 45–56, 2003. View at Google Scholar - E. Orsingher and L. Beghin, “Time-fractional telegraph equations and telegraph processes with Brownian time,”
*Probability Theory and Related Fields*, vol. 128, no. 1, pp. 141–160, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Momani, “Analytic and approximate solutions of the space- and time-fractional telegraph equations,”
*Applied Mathematics and Computation*, vol. 170, no. 2, pp. 1126–1134, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Garg and A. Sharma, “Solution of space-time fractional telegraph equation by Adomian decomposition method,”
*Journal of Inequalities and Special Functions*, vol. 2, no. 1, pp. 1–7, 2011. View at Google Scholar · View at MathSciNet - G. Adomian,
*Solving Frontier Problems of Physics: The Decomposition Method*, vol. 60, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - J. Chen, F. Liu, and V. Anh, “Analytical solution for the time-fractional telegraph equation by the method of separating variables,”
*Journal of Mathematical Analysis and Applications*, vol. 338, no. 2, pp. 1364–1377, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Huang, “Analytical solution for the time-fractional telegraph equation,”
*Journal of Applied Mathematics*, vol. 2009, Article ID 890158, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus - J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,”
*International Journal of Non-Linear Mechanics*, vol. 34, no. 4, pp. 699–708, 1999. View at Publisher · View at Google Scholar · View at Scopus - A. Sevimlican, “An approximation to solution of space and time fractional telegraph equations by He's variational iteration method,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 290631, 10 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Yildirim, “He's homotopy perturbation method for solving the space- and time-fractional telegraph equations,”
*International Journal of Computer Mathematics*, vol. 87, no. 13, pp. 2998–3006, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Das, K. Vishal, P. K. Gupta, and A. Yildirim, “An approximate analytical solution of time-fractional telegraph equation,”
*Applied Mathematics and Computation*, vol. 217, no. 18, pp. 7405–7411, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Garg, P. Manohar, and S. L. Kalla, “Generalized differential transform method to space-time fractional telegraph equation,”
*International Journal of Differential Equations*, vol. 2011, Article ID 548982, 9 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - L. Galue, “Solution of some fractional order telegraph equations,”
*Revista Colombiana de Mathematicas*, vol. 48, no. 2, pp. 247–267, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - V. K. Srivastava, M. K. Awasthi, and M. Tamsir, “RDTM solution of Caputo time fractional-order hyperbolic telegraph equation,”
*AIP Advances*, vol. 3, no. 3, 2013. View at Publisher · View at Google Scholar · View at Scopus - Y. Khan, J. Diblik, N. Faraz, and Z. Smarda, “An efficient new perturbative Laplace method for space-time fractional telegraph equations,”
*Advances in Difference Equations*, vol. 2012, article 204, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. Kumar, J. Singh, and S. Kumar, “Analytic and approximate solutions of space-time fractional telegraph equations via laplace transform,”
*Walailak Journal of Science and Technology*, vol. 11, no. 8, pp. 711–728, 2014. View at Google Scholar · View at Scopus - A. Prakash, “Analytical method for space-fractional telegraph equation by homotopy perturbation transform method,”
*Nonlinear Engineering*, vol. 5, no. 2, pp. 123–128, 2016. View at Publisher · View at Google Scholar - F. A. Alawad, E. A. Yousif, and A. I. Arbab, “A new technique of Laplace variational iteration method for solving space-time fractional telegraph equations,”
*International Journal of Differential Equations*, vol. 2013, Article ID 256593, 10 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus - S. Kumar, “A new analytical modelling for fractional telegraph equation via Laplace transform,”
*Applied Mathematical Modelling*, vol. 38, no. 13, pp. 3154–3163, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - I. N. Sneddon,
*The Use of Integral Transforms*, Tata Mcgraw Hill, 1974. - L. Debnath and D. Bhatta,
*Integral Transforms and Their Applications*, CRC Press, Taylor & Francis Group, Boca Raton, Fla, USA, 3rd edition, 2015. View at MathSciNet - L. Debnath, “The double Laplace transforms and their properties with applications to functional, integral and partial differential equations,”
*International Journal of Applied and Computational Mathematics*, vol. 2, no. 2, pp. 223–241, 2016. View at Publisher · View at Google Scholar · View at MathSciNet - A. M. Anwar, F. Jarad, D. Baleanu, and F. Ayaz, “Fractional Caputo heat equation within the double Laplace transform,”
*Romanian Journal of Physics*, vol. 58, no. 1-2, pp. 15–22, 2013. View at Google Scholar · View at MathSciNet - S. Sarwar and M. M. Rashidi, “Approximate solution of two-term fractional-order diffusion, wave-diffusion, and telegraph models arising in mathematical physics using optimal homotopy asymptotic method,”
*Waves in Random and Complex Media*, vol. 26, no. 3, pp. 365–382, 2016. View at Publisher · View at Google Scholar · View at MathSciNet - R. Joice Nirmala and K. Balachandran, “Analysis of solutions of time fractional telegraph equation,”
*Journal of the Korean Society for Industrial and Applied Mathematics*, vol. 18, no. 3, pp. 209–224, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - R. R. Dhunde and G. L. Waghmare, “Double Laplace transform combined with Iterative method for solving non-linear telegraph equation,”
*Journal of the Indian Mathematical Society*, vol. 83, no. 3-4, pp. 221–230, 2016. View at Google Scholar